optimized salt selection for solar thermal latent heat energy storage · 2018-02-26 · materials...
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Optimized Salt Selectionfor
Solar ThermalLatent Heat Energy Storage
Ralf Raud
A thesis submitted in fulfillment of the requirements for a degree ofDoctor of Philosophy
At the Queensland University of TechnologySchool of Chemistry, Physics, and Mechanical Engineering
Science and Engineering Faculty
2017
For all people who live their livesa bit broken inside, hiding flaws from the world.
I see you. I am you. You are beautiful to me.This myth is for you.
i
AbstractAs climate change accelerates, there is increased demand for renewable energysources to be capable of supplying baseload power. To use solar radiationto generate baseload electricity, the concentrated solar thermal power (CSTP)paradigm must undergo severe cost reductions. To this end, every subsystemmust increase in efficiency and decrease in cost. For latent heat thermal energystorage (LHTES) to be competitive, new storage materials must be identifiedfrom the thousands of possible candidates.
In this thesis, salts and their mixtures are investigated as potential LHTESmedia. First, several methods are developed to predict the properties of themixtures based on the properties of the constituent salts. The density and ther-mal conductivity can be predicted by simple mixing models; this is verified viacomparing several literature sources for these measurements. The compositionis predicted via a modified geometric method for both ternary and quaternarymixtures. Finally, the latent heat is estimated via a mixing model, combinedwith estimations for the heat of mixing. Certain salt display markably lowerlatent heats when mixed. This is accounted for by developing new approxima-tions.
Next, the latent heats of three binary and three ternary mixtures are exper-imentally measured. These measurements show that the predictions fall withinthe uncertainty of measurement. In addition, the predicted ternary concen-trations had very similar behaviors to the literature’s suggested compositions.Thus, the prediction of composition is also useful for comparisons.
Because these methods predict the properties to within experimental uncer-tainty, the burden of testing hundreds of mixtures is greatly reduced.
Due to the high temperatures involved, the containment vessel must be con-structed from expensive materials. A known issue is that a plurality of thesystem cost is actually the cost of the containment vessel and heat exchanger.In this thesis an analytic relationship between the mixture properties, heatexchanger geometry, and charging time is developed. Most examinations ofthe performance of the containment vessel rely on numerical or CFD methods,which are precise but time consuming. The analytic solution developed hereis extremely computationally cheap, allowing for fast optimization of the con-tainment vessel design. Together with the cost of the mixture, the least costLHTESS can be determined for each mixture.
Finally, the properties of 563 binary, ternary, and quaternary mixtures areestimated. From this estimation, the minimum containment vessel cost is cal-culated, and the least cost candidates are determined. The senstivity of thesecandidates is elaborated on for a variety of conditions, and the candidates for afive part cascaded LHTESS are enumerated.
The salt mixtures identified in this thesis exceed the cost reduction targetsset for the LHTES paradigm. This will help CSTP compete directly on costwith fossil fuels.
ii
Acknowledgement of Contributions
As required, My Advisor, signed here, signifies that communication has beensighted which verifies that the contributions of co-authors to the works con-tained herein were less than 50%. The work contained herein are original andnovel work completed solely by myself with the following exceptions:
Co-author Contribution Chapter(s)
Rhys Jacob Section 6 of Chapter 2 2
Frank Bruno Advisory role for co-authors, aided inmanuscript editing
2,4
Stuart Bell Aided in acquisition of SEM andEDS, manuscript editing, and pro-vided general supervisory comments
3,6
Kyla Adams Calculated and prepared a singlemixture and collected the associateddata
3
Rodrigo Lima Calculated and prepared a singlemixture and collected the associateddata
3
Michael E. Cholette Created a rigourous proof of method,aided extensively in manuscriptpreparation
4,5
Soheila Riahi Conducted CFD verification of ana-lytic method
4
Wasim Saman advisory role for co-author 4
Geoffrey Will Associate supervisory for all projects,aided in central focusing andmanuscript perparation
2,3,4,5,6
Theodore A. Steinberg Supervisor for all projects, aided incentral focusing and manuscript per-paration
2,3,4,5,6
In addition, the work would not have been possible without the space andequipment provided by the School of Chemistry, Physics and Mechanical Engi-neering at QUT and by the QUT CARF.
iii
Statement of Original Authorship
The work contained in this thesis has not been previously submitted to meetrequirements for an award at this or any other higher education institution. Tothe best of my knowledge and belief, the thesis contains no material previouslypublished or written by another person except where due reference is made.
Singature
Date
iv
QUT Verified Signature
February 2018
Contents
Dedication i
Abstract ii
Acknowledgement of Contributions iii
Statement of Authorship iv
List of Figures vi
List of Tables vii
Nomenclature viii
List of Publications ix
1 Introduction 1
2 Literature Review 6
3 Experimental Verification of Eutectic Composition Prediction 42
4 Design Optimization Method for Latent Heat Thermal EnergyStorage Systems 69
5 Thermal Conductivity Enhancement via Fins or Suspension 100
6 Optimized Salt Selection 109
7 Conclusion 135
8 Bibliography 138
A Appendix 141
v
List of Figures
2.1 A gibbs triangle with the fold and secant of component ‘C’ la-belled.
14
3.1 Diagram of terms used in the compostions calculations method. 50
3.2 Heat flow vs temperature for the three binary mixtures tested. 54
3.3 Heat flow versus temperature for the ternary nitrate mixture. 57
3.4 SEM and EDS of the literature composition of the ternary ni-trate mixture.
57
3.5 SEM and EDS of the predicted composition of the ternary ni-trate mixture.
58
3.6 Heat flow versus temperature for the ternary carbonate mixture. 59
3.7 SEM and EDS of the literature composition of the ternary car-bonate mixture.
60
3.8 SEM and EDS of the predicted composition of the ternary car-bonate mixture.
61
3.9 Heat flow versus temperature for the ternary sodium mixture. 62
3.10 SEM and EDS of the literature composition of the ternarysodium mixture.
63
3.11 SEM and EDS of the predicted composition of the ternarysodium mixture.
64
4.1 Geometry of heat exchanger with variables included. 80
4.2 Discretization of tube into elements with unique time to meltand length.
81
4.3 Temperature differential [between HTF and PCM]. 83
4.4 Optimal cost per kWh as a function of total stored energy fortwo PCM systems in the tube and shell configuration.
89
4.5 Optimal cost per kWh as a function of total stored energy fortwo PCM systems in the finned tube configuration.
90
4.6 Optimal cost per kWh as a function of total stored energy fortwo PCM systems in the finned tube configuration.
91
4.7 Effect of corrosion allowance on optimal cost and the equivalentoptimal cost of containment material.
93
5.1 Total LHTESS cost vs wt% EG for target times and input tem-perature differentials.
104
5.2 Maximum cost per kg of EG for the total system cost to beequal to the cost of 0 %wt EG, as a function of wt% of EG.
104
5.3 System cost of least cost EG PCM vs pure PCM with aluminumfinned tubes.
105
6.1 A schematic outline of the cascaded system. 120
6.2 Cost per MWh for tube and shell geometries [for all predictedmixtures] as a function of melting point.
123
6.3 Cost per MWh for metal finned tube geometries [for all pre-dicted mixtures] as a function of melting point.
126
vi
List of Tables
2.1 Collected results from Brunet at al. via Eq. (3). 12
2.2 Results of method outlined by Sun et al. 14
2.3 Results of method developed by Martynova and Susarev. 16
2.4 Calculations of ternary eutectic temperature. 19
2.5 Calculations of density. 20
2.6 Latent heat as determined by Eq. (15) with ∆Cp,i = 0 and∆Hmix = 0, and by Eq.(17).
23
2.7 Results of Eq (22). 28
2.8 Properties of the NaCl + KCl + BaCl2 eutectic from the litera-ture and predicted via Eqs. (7), (13), and (15).
31
3.1 Single Salt Purity and Source. 46
3.2 A comparison of literature and predicted latent heats for variousbinary eutectics.
53
3.3 Experimental and predicted latent heats for tested salt mixtures. 53
3.4 Predicted and reported compositions for ternary eutectic mix-tures.
55
4.1 Constants used for CFD correlation [and results]. 87
4.2 Constants used for the optimization. 88
4.3 Optimum parameters based on geometry for 1 MWh of storage. 92
5.1 Properties of materials used in optimization. 103
5.2 Properties of the PCM. 103
6.1 Approximations of heats of mixing. 114
6.2 Bulk salt prices, in USD/kg. 117
6.3 Small Selection of Mixture properties. 118
6.4 Constants used for the optimization. 121
6.5 The average least system cost is presented for a variety of condi-tions, based on the mixtures outlined as groups a, b, and c.
124
vii
Nomenclature
common abbreviations
CSTP concentrated solar thermal power
LHTES latent heat thermal energy storage
LHTESS latent heat thermal energy storage system
CFD computational fluid dynamics
PV photovoltaic
PCM phase change material
HTF heat transfer fluid
LCOE levelized cost of electricity
material properties
∆H latent heat of fusion
∆Hmix heat of mixing
ρ density
λ thermal conductivity
T temperature
R universal gas constant
µ chemical activity
Cp specific heat
viii
List of Publications
Journal Publications
• Raud, R. and Jacob, R. and Bruno, F. and Will, G. and Steinberg, T.A “ACritical Review of Eutectic Salt Property Prediction for Latent Heat En-ergy Storage Systems” Renewable and Sustainable Energy Reviews, 2017.
• Raud, R. and Cholette, M. and Riahi, S. and Bruno, F. and Saman, W.and Will, G. and Steinberg, T.A “Design optimization method for tubeand fin latent heat thermal energy storage systems” Energy 2017.
• Raud, R. and Bell, S. and Adams, K. and Lima, R. and Will, G. andSteinberg, T.A “Experimental verification of theoretically estimated com-position and enthalpy of fusion of eutectic salt mixtures” Solar EnergyMaterials and Solar Cells 2018.
• Raud, R. and Bell, S. and Will, G. and Steinberg, T.A “Optimized Salt Se-lection for Solar Thermal Latent Heat Energy Storage” Energy (in prepa-ration).
Refereed Conference Proceedings
• Raud, R. and Cholette, M. and Will, G. and Steinberg, T.A “Analysis ofthe Economic Effect of Nanoparticle Suspension In Phase Change Mate-rials for Latent Heat Thermal Energy Storage” In proceedings of the 10 th
annual Australasian Heat and Mass Transfer Conference, 2016
ix
Chapter 1
Introduction
Background
Concentrated Solar Thermal Power (CSTP) is a process in which mirrors orlenses are used to focus sunlight on a small area, which is quickly heated to avery high temperature. This high temperature energy is then used to drive aheat engine, which in turn generates electricity. CSTP is one of many renew-able energy methods to generate electricity, and is often directly compared toPhotovoltaic (PV) panels. PV uses the photons in solar radiation to generateelectricity directly via the photovoltaic effect. Both of these systems use energyfrom the sun to generate electricity, however, CSTP has distinct advantages.The maximum theoretical obtainable efficiency of CSTP [3] is higher than PV[4] (45% vs 33%). In addition, excess thermal energy generated by the concen-trator can be stored for a targeted $15/kWh, while storing excess electricitycosts at least $100/kWh and loses 18% of the stored energy across the storageand retrieval processes [5]. A recent reliable quote for storing electricity in bat-teries on a grid scale was $250/kWh, which is effectively thrice as expensive asfossil fuels. [6]
The ability to store excess thermal energy efficiently is the primary advantageof CSTP over other renewable energy sources. Several authors have performedeconomic analysis which suggests that storing thermal energy can increase theselling price of generated electricity by 10% [7, 8] as compared to a lack ofstorage. This is due to the ability to generate electricity during peak demand, incontrast to PV which generates the most electricity during peak solar radiation.In this way, PV electricity generation actually presents a large issue for themanaging of power grids, as conventional power sources are throttled downduring peak solar radiation, then quickly turned back on and ramped up toprovide power for peak demand [9]. In some regions where renewable energycan provide a large amount of power to the grid, solar power is disconnectedwhen it provides too much energy during peak solar radiation.
Excess thermal energy can be stored in a several ways; sensible heat, latentheat, or thermochemical. Sensible heat is the storage of heat in the specificheat capacity of a solid or liquid. Latent heat is the storage of heat in the phasechange of a material, typically in the solid-liquid transition. Thermochemicalstorage is the storage of heat in a reversible chemical process. Latent Heat Ther-
1
mal Energy Storage (LHTES) has several advantages over both thermochemicaland sensible heat storage. Because the heat required to change phase is typicallyvery large compared to the heat capacity, latent heat storage is significantly moreenergy dense than sensible heat storage. In addition, more energy is releasedat a single temperature, which allows for the storage system to be optimizedfor the highest efficiency of the electrical generator. Latent heat thermal stor-age can also include sensible storage in either the liquid or solid phases, againincreasing the amount of heat stored. LHTES is less susceptible to chemicaldegradation caused by thermochemical cycling. Finally, LHTES systems can be“cascaded”, that is, several materials with differing melting temperatures canbe selected, and heat can be transfered into and out of the materials in order.A cascaded system further increases the storage efficiency by allowing for theuse of multiple heat engine cycles and can increase the exergy of the system.
Despite the distinct advantages, there are disadvantages to the use of CSTP.The current Levelized Cost Of Electricity (LCOE) for a CSTP plant is about0.22 USD/kWh, or roughly 50% more than the LCOE of PV panels [9, 10]. Toreduce this cost, the U.S Dept. of Energy Sunshot Initiative [11] and the Euro-pean Solar Thermal Electricity Association [12] have both published roadmapsto reduce the cost of CSTP to 0.06 USD/kWh. If this cost target can beachieved, CSTP will be a renewable and green energy source which can com-pete directly on cost with traditional power generation methods such as coal,natural gas, and oil fired power plants.
Motivation
A great deal of research has been done exploring the use of eutectic salts andmetal alloys as storage materials for LHTES. These materials are often referredto as Phase Change Materials (PCM) as they are investigated for their phasechange characteristics. Several reviews have published collections of propertiesof various eutectics [13, 14, 15, 16], but a great many eutectic mixtures arestill completely uncharacterized. In addition, there is some conflicting data;both Kenisarin [13] and Gomez et al.[16] have elaborated on times in which theliterature reports differ from one another. Liu et al.[17] and Jiang et al.[18] haveconducted experiments in which the measured values for the latent heats differsubstantially from the values previously reported in the literature.
The large number of untested mixtures, combined with the uncertainty inavailable data, is a difficult problem to overcome. To best select mixtures fortargeted experimental testing, a solid theoretical basis for estimating their prop-erties is necessary.
Another pressing issue is the cost of the containment material, which Murenet al. [19] have reported as being a large portion (>50%) of the total cost ofthe LHTES. This is a well-documented issue [20, 21, 22]. Thus, optimizingthe design to include the least amount of containment material is necessary todrastically reduce costs. Wen et al. [23] have shown that an increase in thermalconductivity from 1 W/mK to 2 W/mK can reduce the amount of containment
2
material required by nearly 30%. Thus, a more expensive PCM could reducethe total system cost by reducing the cost of the required containment vessel.However, this depends on the balance between the decrease in containment costand increase in PCM cost. This balance requires a systematic optimization toachieve the lowest possible total cost.
Current methods for characterizing the performance of heat exchangers withPCMs rely heavily on iterative numerical methods, with a great number ofstudies using CFD. Each simulation of a charging or discharging cycle takes onthe order of one day of computational time to complete; this makes optimizationunfeasible. To achieve the desired systematic optimization of containment vesseldesign for hundreds of mixtures requires a computationally cheap solution.
Research Questions
In this thesis, the central objective is to determine which PCMs are the mostcost effective for cascaded solar thermal latent heat energy storage. There areissues which must be overcome prior to this, and these issues form the basis forthe research questions:
1. First, due to conflicting data in the literature and a large number ofuntested mixtures, a theoretical basis must be determined for accuratelyassessing the thermal properties of the PCMs.
2. Second, a systematic method to determine the containment vessel costmust be developed, as a large part of the system cost depends on theproperties of the PCM influencing the design of the containment vessel.
3. With these two issues resolved, the combination space of binary, ternary,and quaternary salt mixtures should be catalogued and the total systemcost calculated. From here, the most optimal PCMs can be selected for acascaded system with specific properties.
This thesis answers these three questions in five chapters:
1. In Chapter 2, the literature is reviewed for methods to predict the prop-erties of mixtures of salts. These methods are tuned for accuracy.
2. And then in Chapter 3 experimental testing is undertaken to verify theaccuracy of these methods and the modifications made to them. Question1 is answered via a combination of Chapters 1 and 2.
3. In Chapter 4, the state of the literature on system design is discussed, andan analytic method is presented which quickly and accurately optimizesthe system design parameters for least cost. This optimization can opti-mize either tube and shell or finned tube geometries. Thus, Question 2has a systematic and computationally cheap solution.
3
4. Chapter 5 follows the work of the previous chapter and discusses meth-ods to improve the thermal conductivity of the PCM, which is a majorlimiting factor in storing and releasing the energy. The advantages anddisadvantages of different design geometries and thermal conductivity en-hancement are discussed.
5. Chapter 6 completes the work by generating a list of predicted PCM mix-tures and their properties, and using this to determine the least cost setof mixtures for a cascaded PCM storage solution for a supercritical CO2
Brayton cycle heat engine. This chapter answers the final research ques-tion.
This work presents a step forward in the space of latent heat thermal energystorage systems. First, the uncertainty in salt mixture properties is reduced bydemonstrating that simple models can be used to determine the density, thermalconductivity, concentration, and latent heat of salt mixtures.
These four properties can then be used to minimize the cost of the heatexchanger and containment vessel, subjected to the required performance con-straints. This minimization is accomplished via a novel analytic solution to theperformance characteristics of the heat exchanger.
Finally, the preceeding tools are used to select the most optimal salt mixturesfor the power cycle being investigated by the broader research team.
4
”Dont force me to draw my own conclusions.I do have a very big pencil.”
Terry Prachett[2]
5
Chapter 2
Literature Review
The following paper was originally published in the April 2017 issueof Renewable and Sustainable Energy Reviews. This paper embod-ies the majority of the literature review necessary for this work:in it, the theory behind the prediction of the relevant properties isevaluated and compared to experimental data. Some issues wereidentified that require further research. In particular, prior to thereview conflicting methods for the prediction of composition, melt-ing point, latent heat, and thermal conductivity were identified. Inorder to predict properties, methods must be developed which areaccurate to near experimental uncertainty.
In this review, the methods were compared to experimental data.With some modifications, the prediction for composition, meltingpoint, and latent heat were selected for further testing. The mod-ifications to the prediction methods are presented in the review,beginning the solution for Research Question 1.
6
A Critical Review of Eutectic Salt Property Predictionfor Latent Heat Energy Storage Systems
Ralf Rauda,∗, Rhys Jacobb, Frank Brunob, Geoffrey Willa, Theodore A.Steinberga
aQueensland University of Technology, 2 George Street, Brisbane, QLD 4000, AustraliabUniversity of South Australia,101 Currie St, Adelaide, SA 5001, Australia
Abstract
According to the SunShot initiative, one sixth of the levelized cost of electricity
for Concentrated Solar Thermal Power is thermal energy storage. For this power
generation paradigm to be successful, the cost of every sub-system must be
dramatically reduced. However, the search space for possible storage mediums
is too large for a brute force experimental search to be feasible. Thus, a more
refined approach is necessary. In this paper, eutectic salt combinations are
considered as storage medium. The state of the selection process for these
eutectics is discussed. Various methods to predict the important thermophysical
properties are reported and applied to eutectics whose physical properties are
known. Based on single salt properties, the density of molten salt eutectics can
be predicted, around their melting point, to within 5%. Prediction of the melting
point and composition is accurate to within 7%. However, the estimation of
latent heat for multi-component eutectics is not always accurate, and requires
more work. Finally, the thermal conductivity of multi-component eutectics has
not been well studied; further research is required to corroborate the predictions.
∗Corresponding author
Preprint submitted to Elsevier December 7, 2017
7
1. Introduction
As mitigation of climate change becomes more important, critical exam-
inations of all renewable energy production paradigms must be undertaken.
Amongst the many possible forms of renewable energy production for large elec-
trical grids, Concentrated Solar Thermal Power (CSTP) has the advantage of
cheaply storing solar energy during the day to be converted to electrical energy
during peak load. However, currently, the cost of CSTP is significantly greater
than fossil fuels, making widespread adoption a difficult economic proposition.
The U.S Department of Energy Sunshot Initiative and the European equiva-
lent, ESTELLA, have both investigated the costs associated with CSTP and
have published guidelines for system components such that the system will be
economically competitive with fossil fuels. The economics are based on the
Levelized Cost of Electricity (LCOE), a comparative method which takes into
account capital costs, running costs, and fuel costs over the lifetime of the power
plant. These cost targets take into account certain operating parameters; for
the storage the parameters include a charge and discharge time of less than six
hours and discharge temperatures above 600 °C. In this review, eutectic salts
are considered as phase change materials (PCMs) for latent heat thermal energy
storage systems (LHTESS).
Due to the large number of individual salt species and the even larger number
of mixtures, the combination space for eutectic salts is enormous. This leads to
every temperature range having a number of potential PCM mixtures, which is
an advantage for optimal selection. Kenisarin[1] has published a review with an
exhaustive list of experimentally determined latent heats of fusion of eutectics,
but this review does not exhaust the list of possible combinations. FactsageTM
has published over three hundred binary eutectic phase diagrams; Kenisarin’s
list does not include a significant fraction of these. Indeed, the possible search
8
space for eutectic salts is in the order of hundreds of thousands. This is the
disadvantage of a large search space; to experimentally measure the latent heat
of fusion, or even just the melting point, for these salts would be an undertaking
for several lifetimes. A better method for the selection of salt eutectics must
therefore be developed.
The relatively large heat flux required presents another difficulty in the eval-
uation of eutectic salts as PCMs. Eutectic salts have relatively low thermal
conductivity, which reduces heat transfer. The range of thermal conductivities
is quite large, however, and the effect of the thermal conductivity on the system
cost must be taken into account. A recent review of potential chloride eutectic
PCMs [2] found that this is a necessary step in proper evaluation of PCMs. Some
studies [3, 4] have found that a significant portion of the cost of the LHTESS is
the containment vessel, while other studies have found that increasing the ther-
mal conductivity can lead to dramatic reductions of the system cost [5]. Indeed,
the search for higher thermal conductivity PCMs has driven a great deal of re-
search into composites which can increase the thermal conductivity[6, 7, 8, 9].
Thus, to properly evaluate the PCMs, the thermal conductivity must be known.
Despite several recent reviews collating data on large numbers of salts [1,
2, 10, 11], little data has been collected on the thermal conductivity of molten
salts. This is a conclusion which a number of works cite as causing difficulty
in the accurate assessment of molten salts [12, 13]. Recent works also detail
several issues with experimental determination of the latent heat of salts; Jiang
et al. [14] found salt creep to be an issue which prevented their results to
correlate with other authors. Williams [15] found salt creep and atmospheric
contamination to be a pressing issue which could damage delicate instruments.
Gomez [13] also found atmospheric contamination to be an issue; most salts are
hydroscopic and absorb water even in relatively dry laboratories. This makes
9
accurate property assessment difficult, and may be behind the discrepency in a
number of recorded measurements [1]. Finally, while a large database of eutectic
salt properties exists [16], several studies have not supported its data [13, 17].
With the ever present experimental issues, and to narrow down candidates
for testing, the search space of possible eutectics must be narrowed down. To
accomplish this, the relevant PCM properties must be estimated. First, the
relative composition must be determined as the thermophysical properties are
calculated based on the mass or molar concentrations of individual component
salts. Second, the latent heat of fusion must be accurately estimated. The
primary purpose of a LHTESS is to store energy, thus, the amount of energy
which is able to be stored is the primary driver of the cost of the PCM. Finally,
the density and thermal conductivity of the eutectic informs the design of a
containment vessel; a highly dense and highly conductive PCM is preferable, as
a smaller containment vessel is required to contain the PCM, further reducing
the cost. However, this does not consider the potential corrosive interaction
between the PCM and the containment vessel.
In this paper, several previously presented theories on calculating the concen-
tration, melting point, latent heat, density, and thermal conductivity of molten
eutectic salts are critically evaluated. The results of these theories are compared
to reliable experimental data on binary and ternary eutectics. Gaps in the the-
oretical assessment of the aforementioned properties are discussed; the filling
of these gaps will allow for targeted optimization of the selection of PCMs for
LHTESS.
2. Component Concentrations and Melting Point
The prediction of the melting point and individual component concentration
of a eutectic, based on the component salts, is necessary for the prediction of
other properties as it determines the relative weight of the single salt properties
10
in influencing the properties of the eutectic which is formed. A method to predict
eutectic composition has been suggested by Brunet et al.[18] This method is
based on Raoult's Law:
µi = µ?i +RTln(xiai
). (1)
where µi is the chemical potential, µ?i is the chemical potential of the ideal
component, xi is the molar concentration, and ai is the activity coefficient,
respectively, of component “i.” Brunet assumes that the activity is proportional
to the concentration. The objective is an equilibrium solution, thus, Brunet
assumes µi = 0. With some derivation, Brunet proposes that:
Rln(xi) = −Hi
T+Hi
Ti. (2)
where Ti is the melting temperature of the individual component in Kelvin, T
is the melting temperature of the eutectic, R is the gas constant, and Hi is
the latent heat of the pure substance. Thus, for an n-component mixture, the
following system of equations is developed:
Rln(xi) +Hi
T− Hi
Ti= 0,
n∑
i=1
xi = 1. (3)
which can be solved numerically. Brunet compares results from Eq. (3) to
experimental data. Only results for eutectic salt mixtures are considered here
and are presented in Table 1. The results of Brunet’s method to predict the
composition and temperature do not match the experimental data1, which im-
plies that the theory has a fundemental flaw. This method cannot be used to
11
Table 1: Collected results from Brunet at al. [18] via Eq. (3)
Eutectic Salt Theoretical Concentration Experimental Concentration RMS Error
mol% mol% %
K2SO4 + Li2SO4 24/76 28/72 4
K2SO4 + Li2SO4 + Na2SO4 10/58/32 9/78/13 15.9
LiF + NaF 76/24 50/50 26
LiF+ NaF + KF 61/14/25 47/11/42 12.8
KCl + LiCl 30/70 41/59 11
KBr + LiBr 24/76 40/60 16
Theoretical Temperature Experimental Temperature Error
°C °C %
K2SO4 + Li2SO4 666 535 16.2
K2SO4 + Li2SO4 + Na2SO4 526 512 1.8
LiF + NaF 606 484 16.1
LiF+ NaF + KF 499 454 6.2
KCl + LiCl 467 361 16.7
KBr + LiBr 439 348 14.7
estimate the compostion of new eutectic mixtures.
Sun et al.[19] have used the Wilson Equation to determine the melting point
and composition of eutectics with more than two components. Their method
relies on experimentally verified binary compositions. Wilson[20] expressed the
adjustable components, Λij , in terms of the molar volume of the molecules and
1When predicted temperatures, densities, or thermal conductivites are compared, the fol-lowing equation is used to determine the ”error:”
Error = 100 ·∣∣∣∣Vtheory − Vexperiment
Vexperiment
∣∣∣∣
where V is a value such as temperature or thermal conductivity. In the case of componentconcentrations, as in the following sections, the RMS error is given by:
RMS =
√∑ni (xi,theory − xi,experiment)2
n
12
their interaction energies. Sun et al. solved for these adjustable components
based on experimentally verified binary data, then calculated the activity coef-
ficients for a quaternary eutectic system. These activity coefficients were used
in Eq. (3) to determine the melting point and composition of the eutectic. The
Wilson Equation, for m salts is:
ln(ak) = −ln(m∑
j=1
xjΛkj) + 1−m∑
i=1
( xiΛik∑mj=1 xjΛij
). (4)
In this case, the xi in Eq. (3) is replaced with ai ∗ xi. This method was applied
to the ternary system of Li2CO3 + Na2CO3 + K2CO3, and the results are
shown in Table 2.
Of note is that the binary Li2CO3 + K2CO3 system, used in the calcula-
tion of the ternary carbonate system, has two eutectic points. However, the
composition and temperature of one of those eutectic points, when used to cal-
culate the activity coefficients, did not have a solution. Thus, the 41.6 mol%
Li2CO3 eutectic was used in the calculations presented in Table 2. Again, these
results show little correlation to experimentally verified eutectic compositions
and melting points. This means Sun et al.’s method must be rejected for the
prediction of new eutectic mixtures.
Martynova and Susarev [21] have proposed using a geometric method based
on the triangular Gibbs diagrams to compute the ternary eutectic. This is done
by computing the intersection of the two most stable folds, where the folds join
the binary eutectic point on a side of the Gibbs diagram to the opposite corner.
Fig. 1 is a visual representation of these features. The stability of the fold is
determined by computing the deviation of the corresponding secant, where the
secant is the line joining the other binary eutectic points. First, the derivatives
(Ai,ji ) are calculated:
Ai,ji =2.3
xi,jjln
(Tix
i,ji
T i,jfus
). (5)
13
Table 2: Results of method outlined by Sun et al.[19]
Eutectic Salt Theoretical Concentration Experimental Concentration RMS Error
mol% mol% %
Li2CO3 + Na2CO3 + K2CO3 57.3/28.9/13.9 43.5/31.5/25 10.3
Li2SO4 + Na2SO4 + K2SO4 70.1/28.3/1.6 78/13/9 10.8
LiF + NaF + KF 24.2/13.1/62.7 46.5/11.5/42 17.6
Theoretical Temperature Experimental Temperature Error
K K %
Li2CO3 + Na2CO3 + K2CO3 769 670 14.8
Li2SO4 + Na2SO4 + K2SO4 880 785 12.1
LiF + NaF + KF 736 727 1.2
where xi,ji is the concentration of component i in the binary eutectic of i and j,
and T i,jfus is the melting temperature of said eutectic.
A B
C
Binary Eutectic Point
secant
fold
Figure 1: A gibbs triangle with the fold and secant of component ‘C’ labelled.
The stability (P (i)) of fold i is calculated via:
P (i) = |(Ai,jj −Ai,kk )(Aj,kj +Aj,kk )|. (6)
14
Trunin et al. [22] assert that for systems wherein the minimum P (i) < 0.15 the
following equations are used to determine the component concentrations of the
ternary eutectic:
xx =xx,zx xy,zz
xx,zx xy,zz + xx,zz xy,zy + xx,zz xy,zz
xy =xx,zz xy,zy
xx,zx xy,zz + xx,zz xy,zy + xx,zz xy,zz
xz =xx,zz xy,zz
xx,zx xy,zz + xx,zz xy,zy + xx,zz xy,zz
(7)
where (xyz) is the rotation of (ijk) such that P (x) < P (y) < P (z).
For systems wherein the minimum P (i) ≥ 0.15 a different set of equations
is used:
∆xx,y,zx = (Tx − T y,zfus)
(∆xx,yx xy,zyTx − Ty
+∆xx,zx xy,zzTx − Tz
),
∆xx,yx = |0.5− xx,yx |,
∆xx,zx = |0.5− xx,zx |,
xx = |0.5−∆xx,y,zx |,
xy = (1− xx)xy,zy ,
xz = (1− xx)xy,zz .
(8)
where (xyz) is the rotation of (ijk) such that P (x) < P (y) < P (z). Trunin et al.
further develop this system for several different eutectics. Among their results,
the root mean square error of composition is less than three, and the melting
point is often calculated within 2%. Application of these equations as presented
leads to errors in composition which, while not as egregious as previous methods,
nevertheless can be improved. For example, the ternary carbonate system yields
a composition which has a RMS error of 5.2%, which is an improvement from
15
Table 3: Results of method developed by Martynova and Susarev, Eqs.(7),(8)
Eutectic Experimental Composition Theoretical Composition RMS Error
%mol %mol %
Li2CO3 + Na2CO3 + K2CO3 43.5/31.5/25 40.4/34.9/24.7 2.3
LiF + NaF + KF 46.5/11.5/42 44.5/12.7/42.8 1.4
Li2SO4 + Na2SO4 + K2SO4 78/13/9 78.4/16/5.6 2.6
the 10.3% RMS error of the previous prediction. However, the agreement can
be improved by utilizing Eq. (7) not when P (i) < 0.15, but rather in all cases
where the stability of the three elements is similar; i.e, when |P (x)−P (z)| ≤ 1.5.
In addition, when the element x is the element with the median melting point,
Eq. (8) is modified as such:
xy = |0.5−∆xx,y,zx |,
xx = (1− xx)xx,zx ,
xz = (1− xx)xx,zz .
(9)
Eqs. (7), (8), and (9) have been applied to selected systems. The results of
these computations are presented in Table 3. These results are superior to those
of Brunet et al. and Sun et al. as they predict the composition of the eutectics
to within 5%, and to within 3% in most cases. This is in marked contrast to
the methods presented earlier, which usually contain errors of 12% or more in
the predicted composition.
Susarev and Martynova[21] extended their equations to apply to the pre-
diction of the composition of quaternary eutectics utilizing binary data and
predicted ternary data. Their results had a maximum error in composition of
2% for reciprocal eutectic mixtures. Non-reciprocal eutectic mixtures do not
contain all the possible combinations of species in the definition, which can lead
to errors in prediction as new species can be formed upon melting. The maxi-
16
mum error for non-reciprocal mixtures wherein the ternary data was predicted
based on binary salts was 7%. These results suggest that experimental binary
eutectic data can be utilized to predict composition for reciprocal quaternary
eutectic mixtures, as these errors are similar in size to the errors in the ternary
calculations.
Beilmann et al.[23] performed a thermodynamic investigation of the LiF +
NaF + CaF2 + LaF3 system based on a polynomial estimation of the excess
Gibbs free energy. The excess Gibbs free energy is fitted to experimental data
with a polynomial equation and this equation is used to compute the Gibbs free
energy of higher order systems, which, in turn is used to calculate the phase
diagrams. Beilmann et al.[23] experimentally verified their predicted phase dia-
grams via differential scanning calorimetry. For three ternary systems and one
quaternary system, the experimental solidus and liquidus tempertures varied
by about 5% when compared with the theoretical prediction. This is in very
good agreement with the theory. However, estimating the Gibbs free energy
requires at least one experimental data point for enthalpy of mixing, which
Beilmann et al. were required to choose. Existing literature sometimes reports
extremely variable enthapies of mixing for identical samples. The data obtained
by Macleod et al.[24] is approximately three times the results obtained by Hong
et al.[25] Thus, a careful critic is required of the available enthalpy of mixing
data before any further analysis can be completed. Without this enthalpy of
mixing data, further analysis would be impossible. However, with just the eu-
tectic points of the LiF + CaF2 and NaF + CaF2 systems, Eq. (8) yields a
eutectic composition for LiF + NaF + CaF2 of 53.4/34.2/12.4, which compares
favorably with Beilmann et al.’s prediction of 51.1/36.5/12.4.
Trunin [22] describes the calculation of the melting point for ternary systems
as follows. The calculation is broken up into the following four conditions. To
17
begin, let (αβγ) be the rotation of (ijk) such that Tα > Tβ > Tγ . First, if the
binary eutectic with the lowest melting point (elmp) is not the eutectic formed
by components y and z (ey,z), but does contain α, then the ternary eutectic
temperature is the average of the two individual components with the closest
temperatures. If not, the melting point of the ternary eutectic is determined
via:
Tfusi,j,k = Tφ −
1− xi,j,kφ
1− xφ,θφ(Tθ − Tφ,θfus) (10)
φ and θ are determined via the following three conditionals:
α 6∈ elmp and elmp 6= ey,z
then φ = γ & θ = β
elmp = ey,z and x 6= α
then φ = α & eφ,θ = elmp
elmp = ey,z and x = α
then φ = β & θ = γ
(11)
Despite Trunin et al.’s excellent agreement, Eq. (10) yields poor agreement.
A new equation for the melting point of the ternary eutectic is developed:
T i,j,kfus = Tγ −m‡ · (1− xγ)
m‡ =Tγ − Tα,γfus
xα,γα
xβxβ + xα
+Tγ − T β,γfus
xβ,γβ
xαxβ + xα
(12)
As shown in Table 4 this yields better agreement, however, the absolute errors
are still quite large. Better predictions are necessary.
Numerous programs, such as FactsageTM , Thermo-CalcTM , and OpenCalphadTM
18
Table 4: Calculations of ternary eutectic temperature based on Eqs. (10) and (12).
Eutectic Experimental Temperature Theoretical Temperature via Eq. (10) Error
K K %
Li2CO3 + Na2CO3 + K2CO3 670 1067 37.2
LiF + NaF + KF 727 724 0.4
Li2SO4 + Na2SO4 + K2SO4 785 949 17.3
Experimental Temperature Theoretical Temperature via Eq. (12) Error
K K %
Li2CO3 + Na2CO3 + K2CO3 670 661 1.3
LiF + NaF + KF 727 777 6.9
Li2SO4 + Na2SO4 + K2SO4 785 814 3.7
NaCl + KCl + BaCl2 813 874 7.5
LiF + NaF + CaF2 880 [22] 825 6.3
NaF + KF + SrF2 748 [22] 708 5.3
generate phase diagrams based on several geometric methods or mathematical
formalisms. These programs are widely used in research and industry to predict
thermophysical properties based on the properties of the individual components
and single points of experimental data of mixtures. Beilmann et al.’s inves-
tigation of the quaternary flouride system, discussed earlier, follows the same
formalisms and uses Factsage to complete the calculations. Their accurate re-
sults provide evidence for the efficacy of these systems, however, several recent
studies[13, 17, 14] describe instances where the calculated properties differ from
experimental results.
3. Density
The density of molten salt eutectics is accurately predicted at the melting
point [26, 27, 12, 28]. Artsdalan suggested using a simple mixing model:
ρeutectic =n∑
i
xi ∗ ρi. (13)
where ρi is the density of the component salt extrapolated to the eutectic
temperature based on the linear trend of the single component, and n is the
19
number of components. Table 5 contains results compiled from several sources
[26, 27, 12, 29, 30]. The theoretical density is calculated from data presented by
Janz et al.[29, 30] and Nasch et al. [31] and the data for zirconium tetrafluoride
is taken from Williams et. al. [12]. The average error is less than 3.5% with only
two instances of greater than 6% error, across a wide variety of measurements
and salts. This indicates excellent agreement with the theory, especially when
considering that the linear extrapolations of salt densities often have errors in
excess of 2%.
Table 5: Calculations of density based on Eq.(13) and single salt data.
Eutectic Concentration Temperature Literature Theoretical Error Reference
wt% K g(cm)−3 g(cm)−3 %
LiCl + KCl + CsCl 29/12/59 573.35 2.337 2.553 9.24 [27]
LiCl + CsCl 27/73 613.55 2.51 2.714 8.13 [27]
LiCl + KCl 45/55 653.85 1.694 1.686 0.47 [27]
Li2CO3 + Na2CO3 + K2CO3 32.1/33.4/34.5 670 2.148 2.085 2.93 [28]
LiF + LiCl + LiBr 9.6/22/68.4 773 2.19 2.283 4.25 [28]
Li2SO4 + K2SO4 71.6/28.4 860 2.105 2.106 0.05 [28]
LiF + BeF2 51.7/48.2 733 2.056 1.981 3.6 [12]
LiF + BeF2 + ZrF4 42.4 / 35.7/ 21.9 701 2.295 2.375 3.5 [12]
LiF + KF 30.9/ 69.1 765 2.125 2.097 1.3 [12]
LiF+ NaF + KF 29.2/11.7/59.1 727 2.199 2.133 3 [12]
LiF + RbF 15.8/84.2 743 2.886 3.041 5.3 [12]
NaF + BeF2 54.2/45.8 613 2.144 2.159 0.7 [12]
NaF + ZrF4 25/75 773 3.21 3.308 3 [12]
4. Latent Heat of Fusion
Very little work has been done on the prediction of the latent heat of fusion
based on single salt properties. Kosa et al.[32] published a method which is based
on the assumption that the entropy of a binary eutectic must be equal to the
20
entropy of the individual components. They proposed the following equation:
∆Hfus = Tfus∗(−R ∗ x1 ∗ ln(a1 ∗ x1)−R ∗ x2 ∗ ln(a2 ∗ x2) + ∆Sf,1 + ∆Sf,2) +
Tfus ∗(∫ Tfus
T1
∆Cp,1T
dT +
∫ Tfus
T2
∆Cp,2T
dT
)+ ∆Hmix. (14)
where Tfus is the melting temperature of the eutectic in Kelvin, Ti is the melting
temperature of the ith component, ∆Sf,i =∆Hf,i
Tiis the entropy of fusion for
the ith component, ∆Cp,i is the difference in heat capacity between the solid
and liquid phase for the ith component, ai is the activity coefficient of the ith,
and ∆Hmix is the heat of mixing. Kosa et al. further discussed the practicality
of applying this equation to unstudied binary salt combinations. The heat
of mixing is difficult to determine without studying the latent heat, and the
specific heat of the liquid is difficult to determine for temperatures lower than
the melting temperature of the pure substance. Kosa et al. make the following
simplifying assumptions: ∆Cp,i = 0, the activity coefficient is one, and ∆Hmix
= 0 and then they compare their predictions with the literature for the systems
NaF + Na2SO4 and KF + K2SO4. Their predictions for the above simplifying
conditions are 13% and 0%, respectively, of the literature values. Kosa et al.
further investigate the influence of their simplifying conditions, finding that
there is little effect on the final error with any combination of conditions.
Kosa et al. only examined binary systems, however, so in order to examine
n-component eutectics a generalized Eq.(14) for n-component mixtures is given:
∆Hfus = Tfus ∗n∑
i
(∆Sf,i +
∫ Tfus
Ti
∆CpT
dT −R ∗ xi ∗ ln(ai ∗ xi))
+ ∆Hmix.
(15)
As shown in Table 6, applying Eq.(15) to the system NaCl + Na2SO4 yields
a latent heat of 266 J/g, which compares very favorably with the literature
results. Results that correlate within 9% of literature values for most salts can
21
be calculated by using the assumptions that ai can be determined via Eq. (1),
∆Cp,i = 0, and ∆Hmix = 0. These assumptions are difficult to justify, however,
as they produce extremely inaccurate results for some salts, such as the ternary
system LiF + NaF + KF.
Misra et al.[33] have developed another method to determine the latent heat
of fusion. Their method is based on the assumption that the heat of fusion for
the eutectic is given by the heat of fusion of the individual components, Eq.
(16), plus the heat of mixing of the liquid and solid phases.
The latent heat of the component at the melting point of the eutectic
(∆Hfusi ) is given by:
∆Hfusi = ∆Hi +
∫ Tfus
Ti
∆CpdT. (16)
thus, the latent heat of fusion is given by:
∆Hfus =
n∑
i
∆Hfusi + ∆Hi,j,k
mix . (17)
Misra et al. do not consider the heat of mixing to be able to be approximated
as zero, and thus develop very explicitly the relationship between the heat of
mixing of binary mixtures and the heat of mixing of ternary components:
∆Hi,j,kmix = (1−x3)2(H1,2
mix)
(x1,2
1
x1,22
)+(1−x2)2(H1,3
mix)
(x1,3
1
x1,33
)+(1−x1)2(H2,3
mix)
(x2,3
2
x2,33
).
(18)
where Hi,jmix is the heat of mixing of the binary eutectic, and xi,ji is the con-
centration of component i in the binary eutectic of components i and j. This
explicit relationship allows for the calculation of the heat of fusion to be cal-
culated for ternary mixtures where the heat of mixing is unknown. Systems
where the binary heats of mixing are unknown must use a different method to
determine the latent heat. Misra et al. develop an approximation of the entropy
22
Table 6: Latent heat as determined by Eq. (15) with ∆Cp,i = 0 and ∆Hmix = 0, and byEq.(17)
Eutectic Experimental Eq.(15) Eq.(15) Error Eq.(17) Eq.(17) Error
J(g)−1 J(g)−1 % J(g)−1 %
Li2CO3 + Na2CO3 + K2CO3 276 288 4.3 257 6.9
NaCl + Na2SO4 268 266 0.7
NaCl + KCl + BaCl2 221 230 4.1 286 29.4
LiF+ NaF + KF 402 666 65.7 790 96.5
NaNO3 + NaCl + Na2SO4 177 193 9
NaF + CaF2 + MgF2 512 676 32 574 2 12.1
of mixing based on binary eutectics:
∆Si,jmix = −R(xiln(xi,ji ) + xj ln(xi,jj )) =∆Hi,j
mix
T i,jfus. (19)
The results of Misra et al.’s method as applied to selected salts are also
given in Table 6. This method is not as accurate as Kosa et al. Both methods
fail for the mixture LiF+NaF+KF, and if experimental binary heats of mixing
are used, Eq.(15) yields 441 g−1J and Eq.(17) yields 541 g−1J. Eq. (17) still
has unacceptable error. However, Eq. (15) only overestimates the results by
10% as compared to experimental values[12]. This may be acceptable for some
applications, however, the lack of extensive heat of mixing data makes this of
limited applicability.
Misra et al. go on to describe more detailed methods for computing the heats
of mixing. These methods depend heavily on phase diagrams of constituent par-
tial mixtures, which are difficult to acquire for complicated mixtures. Acquiring
these phase diagrams and heats of mixing is more difficult than experimentally
measuring the latent heat of the desired eutectic.
Thus, no robust solution has been indentified which can be applied to predict
the latent heat of eutectics from single salt data. All of these methods either
23
require extensive additional experimental data, or have unacceptable errors.
5. Thermal Conductivity
The many methods of thermal conductivity measurement have yielded a
very large spread in the reported values for single salt species [34]. This makes
estimating the thermal conducitivity of eutectics more difficult. To establish
a basis for the prediction of eutectics, the state of the literature for thermal
conductivity values of single salts must first be examined.
Nagasaka et al. [34] suggested an equation for the thermal conductivity of
molten NaNO3 based on an extensive critical review of published results, tak-
ing into account the weaknesses of many methods in controlling for convection.
Close to the melting point, this correlation differed from the published data by
up to 10%. Further from the melting point, the drift became more pronounced,
with up to 20% error at larger temperatures. This error was attributed to
weak results from techniques which did not properly take into account convec-
tion or electrical conductance of the molten salt. Nagasaka, Nakazawa, and
Nagashima[35, 36, 37, 38] have also published a critic of the standard methods
for measuring thermal conductivity. They suggested that the forced Rayleigh
scattering method is the preferred method for measuring thermal conductiv-
ity, as this reduces the contribution of convention and radiative heat transfer.
Their results for the measurement of molten chlorides[35], bromides[37], and
iodides[38] are all on the low end of the results reported previously in the liter-
ature. Their results are often as low as 50% of the maximum result reported in
the literature. In addition, they have measured a decline in thermal conductivity
as temperature increases for every molten salt. This observation is counter to
the majority of the experimental literature, but in line with theoretical results
2Misra et al.[33] utilized experimental heats of mixing to predict this.
24
based on first principles[39, 40] This gives little reason to doubt the validity of
their results, despite the large observed differences and the large experimental
uncertainty in their results.
Otsubo et al.[41] published an experimental study on the thermal conductiv-
ity of molten carbonates and their eutectic mixtures. They have also used the
forced Rayleigh scattering method and report results significantly below those
presented elsewhere in the literature. Of note here is that they suggest a thermal
conductivity of 0.567 W(mK)−1 for the ternary eutectic of lithium, sodium and
potassium carbonate. Maru et al. [42] have estimated the thermal conductivity
of this salt as 2.041 W(mK)−1; this not an atypical variance in literature values
for the thermal conductivity of eutectic salts[39, 40]. This large varience makes
assessing the accuracy of predictions difficult, and greatly complicates design
and selection work for molten salt mixtures, as there is little consistency in the
data for single salts. This issue must be resolved before selection of molten salts
can be effectively conducted.
Hossain et al.[39] have developed further a theory for the prediction of the
thermal conductivity of single salts. They base their model on Chandler’s theory
of molten salts as being a collection of hard spheres. Characteristic properties of
the salts can be derived from a single specimen of a species of salts and applied
to the other members of said species via the constant Cλ. The hard sphere
model for thermal conductivity of molten salts is written in a similar fashion
to the relationship of viscosity discussed by Chandler. Essentially, the thermal
conductivity, λ, relates to temperature via the same function as a reference, in
this case liquid argon, with respect to the reduced volume:
λ(t) = Cλ ∗(0.68285− 0.84286x+ 0.66370x2 − 0.21015x3
). (20)
where, Cλ is the characteristic property of the salt, x = (V − Vm)/Vs is the re-
25
duced volume at temperature t, V is the volume at the temperature, Vm is the
volume at the melting temperature, Vs = Naσ32−1/2, Na is avagadro’s number
and σ is the molecular radius. In particular, Hossain et al. predicted Cλ based
on the relationship between Cλ determined from the literature and the molecular
weight of the cation. They claimed to find agreement in predicting the thermal
conductivity of sodium, potassium, and cesium iodides based on the linear rela-
tionship between Cλ and the molecular mass of several chlorides. Their results
are only tabulated in graph form, which makes precise analysis difficult, but
their results appear to suggest error between the theory and experimental data
of about 25%, which is larger than the experimental uncertainty of Nagasaka
et al. In addition, applying this technique to the prediction of bromides fails.
Bromides follow a pattern wherein Cλ increases as molecular mass increases.
Thus, applying Hossain et al.’s technique directly results in an error for CsBr
on the order of 100%, as compared to the experimental results that Hossain et
al. use to justify their theory. For this reason, this theory is not considered
robust for further prediction of salt mixtures.
Gheribi et al.[40] utilize the Boltzmann transport equation and hard sphere’s
theory to derive an expression for the thermal conductivity at the melting point
which depends on the volumetric specific heat, the speed of sound in the molten
salt, and the phonon mean free path. They then assert that the phonon free
path is proportional to the average of the sum of the anionic and cationic radii,
and inversely proportional to the number of atoms per molecule. Their results
suggest a prediction of thermal conductivity given by:
λfus = rKCv,fusUfus3 ∗ n ∗ Vfus
. (21)
where K is a proportionality constant, Cv is the volumetric specific heat at the
melting point, U is the speed of sound at the melting point, V is the molten
26
volume at the melting point, and r is the sum of the average radii. The molten
salt LiCl was used to obtain a proportionality constant of 4.33.
This equation proves to be quite robust, as Gheribi’s predictions for the
iodides LiI, RbI, and CsI fall within experimental error, despite the speed of
sound data being extrapolated from entropy data. For salts with reliable exper-
imental thermal conductivity data, all of Gheribi’s predictions fall within the
experimental uncertainty.
Tufeu et al.[43] measured the thermal conductivity of molten KNO3, NaNO3,
and NaNO2, as well as some mixtures of these salts. They used the coaxial cylin-
der method, however, their results for pure NaNO3 match closely the correlation
suggested by Nagasaka et al.[34], so their results can be considered precise. In
particular, Tufeu et al. recorded the thermal conductivity of HITECTM , a
mixture of all three salts. The following equation has been proposed[44] for
estimating the thermal conductivity of salt mixtures:
λeutectic =n∑
i
xiλi (22)
Table 7 compares the predictions of Eq. (22) utilizing the results of Gheribi
et al. and experimental data obtained by Tufeu et al.[43] and Otsubo et al.[41].
For the nitrate mixtures, the experimental and theoretical results differ by less
than 7%. For the carbonate mixtures, the results of Otsubo et al. differ consid-
erably from the expected results. However, the experimental results of Otsubo
et al. are positively correlated with temperature, which Nagasaka et al.[34] have
described as typical of experiments which do not properly take into account con-
vection. Thus, Table 7 does not provide a definitive indication of the efficacy of
Eq. (22). Multiple temperatures are included as the thermal conductivity varies
with temperature, and comparing the varience between experiments and theory
demonstrates whether the error is constant or potentially related to experimen-
27
Table 7: Results of Eq (22).
Salt Mixture Concentration Temperature Experimental Theoretical Error
Thermal Conductivity Thermal Conductivity
mol% K W(mK)−1 W(mK)−1 %
NaNO3 + KNO3 30/70 543.3 0.433 0.453 4.6
566.7 0.425 0.448 5.4
589.5 0.429 0.444 3.5
NaNO3 + KNO3 46/54 526.2 0.4675 0.475 1.6
541.5 0.465 0.472 1.5
557.5 0.4579 0.469 2.4
572.4 0.4543 0.466 2.6
588 0.4484 0.463 3.3
NaNO3 + KNO3 50/50 497.5 0.462 0.485 5
509.4 0.46 0.483 5
545.3 0.457 0.476 4.2
573.7 0.446 0.471 5.6
592.2 0.446 0.467 4.7
NaNO3 + KNO3 75/25 546.4 0.479 0.505 5.4
569.6 0.471 0.501 6.4
587.3 0.465 0.497 6.9
592.8 0.47 0.496 5.5
Li2CO3 + Na2CO3 53.3/46.7 792 0.527 0.831 57.7
1062 0.573 0.786 37.2
Li2CO3 + K2CO3 62/38 795 0.542 0.848 56.5
1072 0.55 0.802 45.8
Li2CO3 + Na2CO3 43.5/31.5/25 679 0.568 0.779 37.1
+ K2CO3 1030 0.612 0.725 18.5
28
tal difficulties, such as improper accounting of the effects of convection. The
nitrate salts in Table 7 demonstrate relatively constant error across tempera-
ture, suggesting that the difference is related to the experimental uncertainties
in values used in the prediction. However, the carbonate salts tend to decrease
in error very rapidly with temperature, casting doubt on the validity of the
experimental results.
More recently, work has been done utilizing the transient hot-wire method to
thermal conductivity. Zhang and Fujii[45] conducted experiments wherein they
were able to measure the thermal conductivty NaNO3 with an alumina sputtered
platinum wire. Their results were within expected uncertainty of the suggested
correlation, giving validity to their other results. The sputtered alumina coating,
which has been confirmed to have a negligible effect on measurement accuracy
[46], prevents reaction between the salt and the hot wire and prevents current
leakage through the conductive molten salt. Zhang and Fujii’s experiments
also included measurement of the same Li2CO3 + Na2CO3 eutectic as Otsubo,
and their results provide an average thermal conductivity of 0.876 W(mk)−1
between 837 and 967 K. This compares much more favorably to the theory, which
predicts an average thermal conductivity of 0.812 W(mk)−1 between those same
temperatures.
Finally, the Rayleigh scattering method measures thermal diffusivity, and
then thermal conductivity is calculated from this. The transient hot-wire method
measures thermal conductivity directly. This, along with the more recent re-
sults via Zhang and Fujii, suggests that the hot-wire technique may be more
accurate, however, further work is required to validate the method for a variety
of eutectics. Despite this, there is some evidence to suggest Eq. (22) is a valid
method for predicting the thermal conductivity of salt mixtures.
29
6. Economic Analysis
As the cost of the system is an important parameter in the feasibility of
the latent heat storage systems (LHTESS), a thorough cost analysis should be
performed prior to any experimental analysis. Current cost analyses of LHTESS
are based on experimental or fictional values of PCMs [47, 48, 49, 50], however
as previously mentioned experimental values are only available for a fraction of
the possible PCMs and thus many potential cost-effective PCMs are overlooked.
Using the cost method described below a comparison of theoretical encapsulated
LHTES system costs using the predicted PCM properties can be performed. For
cost analysis procedure of other LHTES systems readers are directed to [48].
The direct cost of an encapsulated LHTES system is made of three major
components; the cost of encapsulation, the cost of the tank and the cost of the
storage material. The installed cost is assumed to be twice the direct cost. The
cost of encapsulation is based on costs produced by Nithyanandam et al.[47] for
the cost of encapsulating a PCM in a shell based on the size of the capsule using
a fluidised bed coating method. We extend the validity of Nithyanandam et al.’s
estimation by using the following equation, as it takes into account the size of
the capsule and the cost of the shell material. This allows a more thorough cost
analysis to be performed for various shell materials.
CE = (ms ∗ Cs) +rcap
0.005
0.3∗ Cpro ∗mp (23)
where CE is the cost of encapsulation, ms is the mass of the total required shell
material, Cs is the cost of the encapsulation material, rcap is the capsule radius,
Cpro is the processing cost, and mp is the mass of the PCM to be encapsulated.
The estimation of the cost of the tank is based on previous research[51, 52, 53]
on the actual cost of storage tanks for a two-tank molten salt system. The cost
of the tank is broken into three main costs, namely; the tank material, the
30
Table 8: Properties of the NaCl + KCl + BaCl2 eutectic from the literature and predictedvia Eqs. (7), (15), and (13). Note that † is calculated via Eq. (13) but uses the literaturecomposition.
Property Theory Literature
Composition (%mol) 40.1/33.2/26.7 34/39.3/26.7 [1]
Latent Heat of Fusion (J/g) 233 221 [1]
Solid Density (g/cm3) 2.96 3.01†Heat Capacity (J/gK) – 0.63
insulation and the foundation:
Ct = [ρTMht(π(rt + w)2 − πr2)]CTM + πr2tCf + 2πrthtCi (24)
where ρTM , ht, rt, w, and CTM are the density of the tank material, the height,
radius, thickness of the storage tank, and the cost of the tank material respec-
tively. Cf and Ci are the cost of the foundations ($1210/m2) and insulation
($235/m2) respectively. Using the design methodology described in[49], the size
of the storage tank and the mass of the storage materials for the theoretical and
measured PCM is calculated.
The cost estimation for the storage material is based on the bulk price of
the PCM and HTF multiplied by the mass of each. As most storage systems
require large quantities of material the assumption that bulk prices can be used
is valid. However it must be noted that additional costs may be associated due
to transportation and further processing which is not explored here. The cost
of some common PCMs and HTFs are shown in [54] and [50].
Table 8 lists the properties determined by the methods described previously,
as well as experimental results from the literature. The total cost, as per the
method described previously, is 33.26 $/kWh for the costs based on theoretical
properties, and 33.66 $/kWh for costs based on experimental properties. This
indicates excellent agreement between the theory and the literature.
31
7. Conclusions
In this paper, several methods for evaluating properties of eutectic salts
have been critically evaluated. The melting point and concentration can be
effectively evaluated using several methods, although some disagreement exists
between experimental work and the theory. Of these, the geometric method
proposed by Martynova and Susarev, and modified here and by Trunin, yields
good results despite requiring little experimental data.
The density of molten salt mixtures at the melting point can be predicted
by Eq. (13), which is simply a mass mixing model.
The thermal conductivity of molten salts is difficult to predict accurately.
This stems from large experimental uncertainties in the properties of the in-
dividual salts. Recent work has described theoretical predictions of single salt
thermal conductivities. These predictions fall within the experimental uncer-
tainty, for the most part, and using these theoretical predictions and a mixing
model, Eq. (22), yields results close to experiments for salt mixtures. Recent
research has also been undertaken to understand the applicability of the tran-
sient hot-wire method to measure directly the thermal conductivity of molten
salts. The few results from this technique correlate well to the aforementioned
predictions.
The last property whose prediction has been reviewed is the latent heat of fu-
sion. Two methods were evaluated, entropy or enthalpy balance. Both yielded
predictions which deviated less than 10% for salts that did not contain fluo-
rine. For fluorine salts, enthalpy balance predictions required precise enthalpy
of mixing data to be accurate. This enthalpy of mixing data can be difficult to
obtain, or can be incorrect, making this a difficult method by which to predict
properties of large numbers of eutectics.
Finally, a the properties of the eutectic NaCl + KCl + BaCl2 were pre-
32
dicted and used to perform an economic analysis. This analysis suggests that
using the predicted properties to estimate the cost of the eutectic yields a cost
which deviates only 2.5% from the cost estimated from experimentally measured
properties. This is in excellent agreement.
In conclusion, the following gaps have been identified: Firstly, the measure-
ment of the latent heat has uncertain correlation to the theory, especially for
a few key salt species, such as common fluoride mixtures. Secondly, there is
little reliable data to corroborate a mixing model for the thermal conducitvity
of multi-component eutectics. This is partially due to experimental difficulties
in measuring this property. However, despite these difficulties, component con-
centration and density for n-component mixtures can be predicted, with some
degree of accuracy.
8. Acknowledgements
We would like to thank the Australian Solar Thermal Research Initiative and
the Australian Renewable Energy Agency for funding the research contained
herein.
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”There was always something thatyou had to do before you could do
the thing you wanted to doand even then you might get it wrong”
Terry Prachett[2]
41
Chapter 3
Experimental Verification ofEutectic Composition Predic-tion
The following paper is published in the January 2018 edition ofSolar Energy Materials and Solar Cells. In this paper we adress theissue of the prediction of eutectic composition and the latent heat offusion. Our experiments indicate that our approximations for thesemeasures are valid. In addition, the modified predictions for ternaryeutectic compostion are shown to produce excellent mixtures.
This paper completes the solution to Research Question 1 byproviding further experimental evidence that the modified methodsdiscussed in the previous chapter are reliable. In particular, fol-lowing Chapter 2 there were concerns that the modified methodsfor composition and latent heat prediction were unproven. In thiswork, evidence is provided which supports these methods. Follow-ing this chapter, the properties of any binary or ternary salt mixturecan be completed.
Finally, note that the prediction of density and thermal conduc-tivity are not further investigated. This was because the methodsdiscussed in Chapter 2 were sufficiently supported by the literatureto not warrent further work.
42
Experimental verification of theoretically estimatedcomposition and enthalpy of fusion of eutectic salt
mixtures
Ralf Rauda,, Stuart Bella, Kyla Adamsa, Rodrigo Limaa, Geoffrey Willa,Theodore A. Steinberga
aQueensland University of Technology, 2 George St, Brisbane QLD 4000, Australia
Abstract
The design of latent heat thermal energy storage systems is critically depen-
dent on the properties of the chosen Phase Change Material (PCM). Currently
published data on the thermodynamic properties of eutectic salts contains con-
tradictory data, impeding the selection of PCMs.
In this work, previously elaborated theoretical methods for predicting the
composition and enthalpy of fusion for eutectic salt mixtures is experimen-
tally validated for three ternary eutectic mixtures: LiNO3 + NaNO3 + KNO3,
Li2CO3 + Na2CO3 + K2CO3, and NaCl + Na2CO3 + Na2SO4. For each
combination of salts, the eutectic composition suggested by published sources
and the composition predicted by the theory are created. The latent heat of
fusion, melting temperature, and congruency of the melting peak of these six
mixtures are examined via differential scanning calorimetry. In addition, scan-
ning electron microscopy and energy dispersive spectroscopy are used to verify
the expected eutectic structure. Analysis of these six mixtures confirms that
the composition can be predicted to within 8%. The enthalpy of fusion can
be predicted to within 10% of experimental results. In addition, three binary
eutectics (NaCl + Na2SO4, KCl + K2CO3, and NaCl + Na2CO3) are created
Email address: [email protected] (Ralf Raud)
Preprint submitted to Elsevier December 7, 2017
43
and the theoretical enthalpy of fusion is experimentally verified.
1. Introduction
In order to reduce the total cost of Concentrated Solar Thermal Power, ev-
ery sub-system must be optimized for cost. In particular, the Thermal Energy
Storage (TES) system must be designed to reduce cost while still storing en-
ergy in the correct temperature range [1]. To achieve these goals, selecting
the optimal storage medium is imperative. Prior research has investigated eu-
tectic salt mixtures as storage media in Latent Heat Thermal Energy Storage
Systems (LHTESS). As a subset of TES, LHTESS use the phase change of stor-
age media to store energy in narrow temperature ranges with high volumetric
energy density [2]. This increases the efficiency and cost effectiveness of TES
systems. Eutectic salt mixtures are good candidates for storage media as there
is a large possible combination space of mixtures, they have large latent heats,
they are available in industrial quantities, and present excellent value. Thus, for
every temperature range there are multiple possible mixtures [3]; from which
the most cost effective option can be determined and selected for the applica-
tion. Unfortunately, this selection often requires extensive and time consuming
experimental testing.
The design of LHTESS are further complicated by published data on the
storage media that are often contradictory. For example, two reports for the
latent heat of the salt eutectic NaCl + MgCl2 [4, 5], 430 J(g)−1 and 328 J(g)−1
respectively, contradict each other. Further testing of this eutectic yields 204
J(g)−1 [6]. Thus, the selection process is made difficult by the lack of reliable
data for thermodynamic properties of the storage media. The impractical ex-
perimental task of blanket testing motivates the creation of a theoretical model
which can accurately asses these thermodynamic properties.
44
In this study, a method for predicting the latent heat and a method for
predicting the composition are experimentally verified. Additionally, the latent
heat of several previously studied eutectics is experimentally verified to cor-
roborate previous results in the literature. Experimental work was conducted
via Differential Scanning Calorimetry (DSC) to measure the melting tempera-
ture and the occurence of a single phase change. Scanning Electron Microscopy
(SEM) and Energy Dispersive Spectroscopy (EDS) were employed to examine
eutectic microstructures on a microscopic scale. Analysis of these results reveal
that prediction of the composition and latent heat is possible.
2. Experimental Methods
To prepare eutectics, anhydrous powders of the constituent salts1 are dehy-
drated at 180 °C. The powders are then mixed according to the mass percent
recommended by Factsage [7] for binary eutectics, and from various sources for
ternary eutectics [8, 9, 3]. The powder mixture is placed into a 99.7% alumina
crucible and placed in a furnace at a temperature 50 °C above the melting tem-
perature reported in the literature for the eutectic composition. This tempera-
ture is chosen as it is below the decomposition temperature of the constituent
salts, but provides a temperature differential large enough to provide the energy
to melt the mixture.
The mixture is periodically checked until it is fully molten, typically 24 hours.
Then the crucible is removed from the furnace and the molten salt is poured
onto a clean metal surface to cool and solidify. At least three independent
DSC samples were tested in sizes ranging from 8 to 18 mg. These samples are
removed from the greater melt and placed into 99.7% DSC alumina crucibles
for DSC testing. These DSC crucibles are then sealed using a thin cap of wet
1sources and purity of these are listed in Table 1
45
Salt Purity (wt%) Source
NaCl >99.9 Univar
KCl >99.0 Chem-Supply Ltd
Na2SO4 >99.0 Chem-Supply Ltd
Na2CO3 >99.0 Chem-Supply Ltd
Li2CO3 >99.0 Alfa Aesar
K2CO3 >99.0 United Laboratories Inc.
NaNO3 >99.0 United Laboratories Inc.
LiNO3 >99.0 Chem-Supply Ltd
KNO3 >99.0 Chem-Supply Ltd
Table 1: Single salt source and purity
potters clay. Clay lids are used to prevent sample creep, and were confirmed to
not have any features visible by the DSC once baked above 450 °C. For samples
less than 4 mg, or with a melting point below 500 °C, no clay lid was used, as
samples did not creep out of the crucible.
The clay lid is allowed to dry for thirty minutes, and then the samples are
placed into a Seteram Labsys 13 DSC, with nitrogen used as a purge gas. The
samples are heated to 60 degrees below their melting point, and held isothermal
for 45 minutes to sinter the clay. The samples are then ramped at 10 °C/min
to a temperature approximately 10 degrees after the completion of the melting
process, then immediately brought back down to 30 °C at the fastest cooling
rate available (20 °C/min). This experimental procedure reduces the amount
of time the sample is molten, reducing creep, whilst still providing the linear
regions before and after the melting feature.
To calculate latent heat, the onset and completion temperatures must be
calculated. This is done by fitting linear equations to the regions before and after
the melting feature. The points where the data deviates from the fitted line are
chosen as the melting onset and completion temperatures. A line joining these
points acts as a baseline for the integration of the DSC curve. This integrated
value is divided by the sample size and multiplied by the correlation factor
46
to determine the latent heat. The temperature at which the melting features
of the DSC curve disappear is calculated for the purposes of determining the
latent heat. However, this depends strongly on sample size[10] and is thusly not
reported. Larger samples take longer to melt, as the temperature distribution
inside the sample is not as homogeneous as smaller samples. Thus, more time is
required to provide the energy for the phase transition. The correlation factor
is determined via the melting of 99.999% pure aluminum samples, both before
and after testing of salt samples. Drift in this correlation factor is included in
the calculation of experimental uncertainty.
Uncertainty in the DSC measurements is a combination of the standard
deviation of the individual experiments, the drift in the correlation factor, and
the uncertainty of the mass measurements. This is calculated via standard
equations and is uniformly ±8% across all samples.
SEM and EDS analysis is undertaken with a Zeiss Sigma Variable Pressure
Field Emission SEM and an Oxford XMax 50 Silicon Drift EDS detector. From
the melt, samples are selected which are large enough to polish opposing faces
flat and parallel. Polishing is conducted with 500 grit SiO2 sandpaper. As solid
salts are electrical insulators, the electrons fired at the sample take time to dis-
perse. Occasionally, the sample may become locally overcharged, leading to an
overloaded detector. To reduce this overcharging, samples are partially covered
with carbon tape and electron backscatter and EDS imagery is conducted close
to the tape.
3. Theory
Previously described methods for predicting latent heat and for predicting
the composition of a ternary eutectic are presented in the following section.
The method for prediction of composition is chosen as it requires less exper-
47
imental data and has been shown to be accurate for several multi-component
mixtures [11]. Reciprocal mixtures consist of all possible combinations of an-
ions and cations in the definition. For example, KCl and NaF can combine to
form a quaternary eutectic KF+NaF+KCl+NaCl. The latter mixture is recip-
rocal, while the former is not. Reciprocal fluoride mixtures have been shown to
have large heats of mixing, thus this work ignores them as possible candidates.
Combining fluoride salt species leads to dramatically reduced latent heats in
comparison to the individual components, lessening the effectiveness of combi-
nations. In addition, previous work has predicted the latent heat [12, 13] for
reciprocal fluoride mixtures.
3.1. Latent Heat
As storing energy is the primary objective of interest in LHTESS, the amount
of energy that can be stored is the primary property of any potential PCM. Thus,
predicting this property is extremely valuable. Kosa et al. [14] developed an
equation to estimate the enthalpy of fusion for binary eutectics. Their results
were based on the assumption that the entropy of the mixture was the same
as the entropy of the individual components, plus an additional term for the
entropy due to mixing of differentiable species. This assumption generalizes to
the assumption that the heat of mixing is zero. Several authors disagree, finding
this assumption to be over-generalized[15]. However, we will proceed with this
assumption as it will be shown later to be almost exclusively accurate when
fluoride mixtures are disregarded. Presented here is an equation expanded to
n-component eutectics, in order to compare more possible mixtures:
∆Hfus = Tfus ·n∑
i
xi · (∆Sf,i −R ∗ ln(ai · xi)) (1)
where Tfus is the melting temperature of the eutectic, Ti is the melting tem-
perature, ∆Sf,i =∆Hf,i
Tiis the entropy of fusion, xi is the molar concentration,
48
R is the universal gas constant, and ai is the activity coefficient, all for the ith
component. Originally Kosa et al. found the best agreement by assuming the
activity is unity. Further analysis with more mixtures finds this assumption can
lead to poor agreement. The activity coefficient is determined via an application
of Raoult’s law, given a known composition:
ai =1
xi· exp(
∆Hf,i
R · Ti− ∆Hf,i
R · Tfus) (2)
Substitution of Eq. (2) into Eq. (1) leads directly to a latent heat determined
via:
∆Hfus =n∑
i
xi ·∆Hi (3)
3.2. Composition and Melting Point
Martynova and Susarev [16] proposed using triangular Gibbs diagrams to
compute the ternary eutectic composition. Their work is advantageous as it
requires far less data than other geometric or thermodynamic methods; only the
composition and melting point of the binary eutectics. However, this method
retains similar accuracy to more in depth predictions.
Their method begins by computing the intersection of the two most “stable”
folds, where the folds join the binary eutectic point on a side to the opposite
corner. The stability of the fold is determined by computing the magnitude of
offset of the corresponding “secant,” where the secant joins the other two binary
eutectic points, as shown in Fig. 1. The offset of a secant is relative to the line
which joins the center of the two sides. The offset (Ai,ji ) is calculated with:
Ai,ji =
2.3
xi,jjln
(Tix
i,ji
T i,jfus
)(4)
49
where xi,ji is the concentration of component i in the binary eutectic of i and j,
and T i,jfus is the melting temperature of said eutectic.
A B
C
Binary Eutectic Point
secant
fold
Figure 1: Diagram of terms used in the composition calculation method [15].
The “stability” (P (i)) of fold i is calculated as:
P (i) = |(Ai,jj −Ai,k
k )(Aj,kj +Aj,k
k )| (5)
For systems wherein the stability of the three elements is similar; i.e, when
|P (x)− P (z)| ≤ 1.5, the following equations are used to determine the compo-
nent concentrations of the ternary eutectic [15]:
xx =xx,zx xy,zz
xx,zx xy,zz + xx,zz xy,zy + xx,zz xy,zz
xy =xx,zz xy,zy
xx,zx xy,zz + xx,zz xy,zy + xx,zz xy,zz
xz =xx,zz xy,zz
xx,zx xy,zz + xx,zz xy,zy + xx,zz xy,zz
(6)
where (xyz) is the rotation of (ijk) such that P (x) < P (y) < P (z).
50
For all other systems, a different set of equations is used:
∆xx,y,zx = (Tx − T y,zfus)
(∆xx,yx xy,zy
Tx − Ty+
∆xx,zx xy,zz
Tx − Tz
)
∆xx,yx = |0.5− xx,yx |
∆xx,zx = |0.5− xx,zx |
xx = |0.5−∆xx,y,zi |
xy = (1− xx)xy,zy
xz = (1− xi)xy,zz
(7)
where (xyz) is the rotation of (ijk) such that P (x) < P (y) < P (z).
In addition, when the element x is the element with the median melting
point, Eq. (7) is modified as such:
xy = |0.5−∆xx,y,zx |,
xx = (1− xx)xx,zx ,
xz = (1− xx)xx,zz .
(8)
4. Results and Analysis
4.1. Enthalpy of Fusion of Binary Eutectics
To validate Eq. (3), the enthalpies of fusion of several binary eutectics are
calculated and compared with values reported in literature reviews, as shown in
Table 2. However, the difficulty in accurate experimental measurement of the
latent heat of these eutectics must be noted [17, 6]. These difficulties arise from
salt and container interactions and salt creep, which can damage instruments.
Some of data tabulated here is of limited use; for the LiF + LiOH mixture,
no method was described for the experimental determination of the latent heat.
51
Additional references cite this value as ”estimated” [4]. For the xF+yBr mix-
tures, DTA was used to determine the latent heat. No uncertainty analysis or
precision is cited [18], which in addition to being a method superseded by mod-
ern measurement techniques, casts doubt on the validity of these values. For the
KCl+ZnCl2, NaCl+CaCl2, and Na2CO3+K2CO3 mixtures, the latent heats are
all estimated [4]. The KCl+MnCl2 mixture’s latent heat value is reported in a
patent [5], which is not peer reviewed. Thus, Eq. (3) agrees with the literature,
but has better agreement data from more reliable sources.
To better establish the validity of Eq. (3), experimental measurements of
three binary mixtures are undertaken (Table 3). The concentrations for these
mixtures are suggested to be eutectic, and are taken from Factsage [7]. When
testing was undertaken, these mixtures did not have published values for the
latent heat. The experimental measurements of latent heat corroborate the
validity of Eq. (3), as the results are within the calculated uncertainty. The
composite DSC curves of these mixtures are shown in Fig. 2. Only the NaCl
+ Na2CO3 mixture displays a smooth single peak as expected. The KCl +
K2CO3 mixture is unusually broad, and the NaCl + Na2SO4 mixture has a
broad peak. These are characteristic of off-eutectic mixtures. This suggests
that the concentrations suggested by the literature as eutectic are not precise.
52
Eutectic Concentration Literature ∆H ∆H via Eq. (3) Relative Error
%mol J(g)−1 J(g)−1 %
LiF+LiOH 20/80 870 914 -4.8%
KF+KBr 40/60 315 287 9.9%
NaF+NaBr 27/73 360 326 10.6%
KCl+ZnCl2 68.3/31.7 218 226 -3.4%
NaCl+CaCl2 47.2/52.8 281 326 -13.9%
Na2CO3+K2CO3 56.6/43.4 163 183 -11.2%
KCl+MnCl2 36/64 236 271 -13%
KF+KCL 55/45 407 428 -4.9%
NaF+NaCl 33.5/66.5 572 565 1.3%
LiNO3+KNO3 42/58 170 182 -6.7%
LiNO3+NaNO3 54.2/45.8 265 273 -2.9%
LiNO3+NaCl 85/15 369 387 -4.7%
NaNO3+KNO3 54/46 117 132 -11.5%
LiNO3+KNO3 43/57 178 185 -3.7%
Li2CO3+K2CO3 62.4/37.6 351 346 -1.4%
Li2CO3+K2CO3 50.2/49.8 295 [6] 310 -4.7%
NaCl + Na2CO3 55.3/44.7 283 [19] 294 -3.7%
Table 2: A comparison of literature and predicted latent heats for various binary eutectics.
From [3] unless otherwise noted.
Eutectic Concentration Experimental ∆H ∆H via Eq. (3) Relative Error
%mol J(g)−1 J(g)−1 %
NaCl + Na2SO4 53.3/46.7 266 266 0%
KCl + K2CO3 62.4/37.6 239 273 -12%
NaCl + Na2CO3 55.3/44.7 282 294 -4.1%
Table 3: Experimental and predicted latent heats for tested salt mixtures.53
0
5
10
15
20
25
30
35
40
45
615 625 635 645 655
Hea
t Flo
w (m
J/m
g)
Temperature (°C)
Figure 2: Heat flow vs temperature for the three binary mixtures tested (NaCl + Na2SO4
dotted, KCl + K2CO3 dashed, NaCl + Na2CO3 solid)
4.2. Component Concentration
To verify that Eqs. (6) and (7) can be used to determine the composition
of the ternary eutectic via the binary eutectic temperatures and compositions,
both the predicted composition and the composition suggested by the literature
are synthesized and compared. Table 4 outlines the differences in experimental
and literature values for the eutectics studied here. For each mixture, DSC is
used to measure the latent heat of fusion, which is compared to the predicted
value.
DSC is also used to corroborate evidence of a eutectic microstructure [20].
A visual analysis of the DSC curve can provide evidence of multiple melting
peaks. This indicates that a sample does not uniformly melt. Additionally,
shoulders can be detected by comparing the shape of the melting feature to
known pure samples with a single phase change. Shoulders in the melting feature
can be attributed to the liquidus temperature being very close to the eutectic
temperature, indicating slight hypo- or hyper-eutecticity. A lack of multiple
54
peaks or shoulders indicates that the sample is eutectic to within experimental
uncertainty. In this article, the lack of shoulders in the heat flow diagram
indicates that all phase changes occur within approximately 10 degrees of the
melting temperature.
In addition, SEM and EDS are used to examine the structure and elemental
dispersion of the mixture. The structure is compared to the accepted eutectic
microstructure [21]. This microstructure is characterized by heavily intermixed
single species structures on the order of 2 µm in width. The shape of these
structures depends on the direction of heat flow upon solidification; highly di-
rectional cooling can cause dendritic tendrils which are lamellar. Slow and
uniform cooling rates yield circular microstructures. However, the character-
istic width remains the same. Off-eutectic microstructures are characterized
by localized large homogeneous conglomerations of elements surrounded by a
eutectic microstructure. Slight off-eutecicity is characterized primarly by EDS
results which suggest the elemental distribution is not homogeneous.
Components Literature Composition Reference Predicted Composition
(%mol) (%mol)
NaCl + Na2CO3 + Na2SO4 51.9/24.1/24.1 [9] 51.3/31.2/17.5
Li2CO3 + Na2CO3 + K2CO3 43.5/31.5/25.0 [8] 40.4/34.9/24.7
LiNO3 + NaNO3 + KNO3 37.4/18.2/44.3 [22] 31/26.8/42.2
Table 4: Predicted and reported compositions for ternary eutectic mixtures.
4.2.1. Ternary Nitrate
Roget et al. [9] studied the LiNO3 + NaNO3 + KNO3 ternary nitrate mix-
ture and determined the composition with the highest latent heat (37.4/18.2/44.3,
respectively). They tested a mixture suggested by Bergman and Nogoev [23]
as well as three mixtures close to the suggested eutectic point. The mixture
37.4/18.2/44.3 had the highest latent heat, 155 J(g)−1, of their tested mix-
55
tures, suggesting that it is the eutectic composition. Using Eq. (6), and the
binary eutectic points provided by Factsage [7], yields a suggested composi-
tion of 31/26.8/42.2. Despite these two compositions differing substantially in
the concentration of sodium nitrate, the latent heat predicted by Eq. (3) only
changes from 189 J(g)−1 to 194 J(g)−1. The latent heat for the literature com-
position, as measured in this study, is 156 J(g)−1, which is almost precisely the
value measured by Roget et al. This presents the largest failure of Eq. (3), with
an error of 18%. As shown in Fig. 3, the measured heat flow of the literature
composition again matches what is described by Roget et al. In addition, the
melting point and DSC curve of the predicted mixture closely matches the mix-
ture from the literature. The latent heat of the predicted mixture is measured
as 144 J(g)−1. This compares favorably with Roget et al.’s mixture.
SEM and EDS of the literature (Fig. 4) and predicted (Fig. 5) ternary
nitrate compositions show both mixtures to be highly segregated. The clusters
of each type of salt are significantly larger than expected for a eutectic mixture,
on the order of 10 µm in width. Both mixtures exhibit similar cluster sizing and
distribution, suggesting that they are similarly off-eutectic. This implies that
the melting range of both mixtures is similar, which is verified by DSC testing.
In the case of the ternary nitrate mixture, Eq. (6) describes a mixture with
almost the maximum latent heat as well as a similarly narrow melting range as
the eutectic mixture. However, the assumption by Kosa et al. [14] that the heat
of mixing is zero fails in this case, with Eq. (3) overpredicting the latent heat
by 25%. This suggests that for nitrates, a heat of mixing term in Eq. (3) must
be included.
56
0
2
4
6
8
10
12
100.00 110.00 120.00 130.00 140.00 150.00 160.00
Hea
t Flo
w (m
J/m
g)
Temperature (°C)
Figure 3: Heat flow versus temperature for the ternary nitrate mixture; literature (solid line)
and predicted (dotted line) compositions.
100μm Na
N K
a) b)
c) d)
Figure 4: SEM and EDS of the literature composition of the ternary nitrate mixture [a)
electron backscatter b) sodium c) potassium d) nitrogen.]
57
50μm
a) b)
c) d)
Na
N K
Figure 5: SEM and EDS of the predicted composition of the ternary nitrate mixture [a)
electron backscatter b) sodium c) potassium d) nitrogen.]
4.2.2. Ternary Carbonate
The DSC scans of the composition suggested by the literature for the ternary
mixture of sodium, lithium, and potassium carbonate, shown in Fig. 6, suggest
a single phase change in the expected range. The latent heat is calculated to
be 288 J(g)−1, which compares favorably with the measured enthalpy of 260
J(g)−1. Fig. 7 demonstrates that this mixture is close to eutectic. This par-
ticular mixture, while created in the same controlled manner as other verified
eutectics, demonstrates some variability in the distribution of sodium and potas-
sium elements. This variability suggests localized clusters of single salt species,
which is associated with hypo- or hyper-eutecticity [21]. Unfortunately, lithium
cannot be detected by the EDS2, thus, a strictly correct assertion of eutectic
composition cannot be demonstrably proven. However, the EDS and SEM mi-
2This is due to the low energy nature of lithium’s characteristic x-rays.
58
crographs cover the same area, so overlaying EDS results onto the backscatter
image can show if certain detected elements are localized. This method reveals
that the darkest areas (and thus the lightest element) do not line up with any
of the detected elements, strongly suggesting that lithium is interspersed evenly
with the sodium and potassium.
0
4
8
12
16
20
24
28
32
370.00 380.00 390.00 400.00 410.00 420.00
Hea
t Flo
w (m
J/m
g)
Temperature (°C)
Figure 6: Heat flow versus temperature for the ternary carbonate mixture; literature (dotted
line) and predicted (solid line) compositions.
59
50μm Na
C K
a) b)
c) d)
Figure 7: SEM and EDS of the literature composition of the ternary carbonate mixture [a)
electron backscatter b) sodium c) potassium d) carbon.]
Fig. 8 demonstrates that the predicted eutectic composition of the mixture
has a eutectic microstructure similar to the literature composition. EDS scans
support this assertion, as distribution of the detectable anions is fairly even.
Using the predicted composition with Eq. (3) yields a predicted latent heat
of 279 J(g)−1. This compares favorably with the measured latent heat of the
mixture (275 J(g)−1). In addition, this mixture appears to complete melting
in the same temperature range as the eutectic, despite the higher temperature
peak and shoulder suggesting less uniform melting. (Fig. 6).
60
10μm Na
C K
a) b)
c) d)
Figure 8: SEM and EDS of the predicted composition of the ternary carbonate mixture [a)
electron backscatter b) sodium c) potassium d) carbon.]
The measured latent heat of the predicted composition and the literature
composition are well within experimental uncertainty. This, coupled with the
phase change occurring in the same temperature range, suggests that both com-
positions could be used interchangeably in large heat storage applications, with
little to no adverse effects.
4.2.3. Ternary Sodium
The composition cited in the literature [22] for a ternary eutectic of sodium
chloride, sulfate, and carbonate does not have the expected eutectic microstruc-
ture or heat flow. The latent heat is predicted to be 272 J(g)−1, but DSC
experiments suggest a latent heat of only 219 J(g)−1. This large variation is
attributed to incomplete mixing and a clearly off-euctectic mixture. A compar-
ison of the melting curves (Fig. 9) shows that the literature composition also
exhibits a shoulder in the DSC curve and has a slightly lower melting point.
61
However, the total area is significantly less than that of the predicted mix-
ture, despite the curves being corrected for sample size, and the peak of the
literature composition is significantly broader. This suggests that the samples
tested displayed more variance in melting curves. Finally, examination of the
microstructure reveals this mixture has large unmixed sodium chloride crystals
in the mixture, as shown in Fig. 10.
0
5
10
15
20
25
30
595 605 615 625 635 645
Hea
t Flo
w (m
J/m
g)
Temperature (°C)
Figure 9: Heat flow versus temperature for the ternary sodium mixture; literature (solid line)
and predicted (dotted line) compositions.
Fig. 11 demonstrates that the predicted composition of the ternary sodium
system has evenly distributed sulfur and chloride, suggesting that the anions
are also evenly distributed. Carbon is difficult to isolate, as the environment
can easily contaminate the sample. However, the even distribution of two of the
three anions suggests that the mixture is of eutectic or close to eutectic composi-
tion. In addition, the microstructure is composed of small dendritic tendrils on
the order of one µm in width, which is characteristic of eutectic microstructures
[24]. Note also that on the larger scale, large agglomerations of chloride evident
62
in the literature composition do not exist. This assertion is supported by DSC
scans, which indicate a single phase change in the expected range, as shown in
Fig. 9. A minor shoulder exists in the individual samples as well as the average
of the samples, around 620 °C, suggesting that a perfect eutectic structure does
not exist, but no clear secondary peak is visible. Regardless, Eq. (7) predicted
a composition which is so close to the eutectic point to make differentiation dif-
ficult. The latent heat is measured at 252 J(g)−1, as compared to a prediction
of 273 J(g)−1. This corroborates the validity of Eq. (3), as the error is within
experimental uncertainty.
250μm Cl
CS
a) b)
c) d)
Figure 10: SEM and EDS of the literature composition of the ternary sodium mixture [a)
electron backscatter b) chloride c) sulfur d) carbon.]
63
Cl
S
a)
c)
e)
b)
100μm 10μm
d)
f)
Cl
S
Figure 11: SEM and EDS of the predicted composition of the ternary sodium mixture [a,b)
scanning electron microscope c,d) chloride e,f) sulfur.]
For the ternary sodium mixture, Eq. (7) yields a mixture which demon-
strates more homogeneous melting and a higher latent heat than the mixture
suggested by the literature. Eq. (3) again predicts the latent heat to within
experimental uncertainty.
5. Conclusion
In this study, experimental results for the latent heat of fusion for binary
eutectics are compared to a theoretical prediction (Eq. (3)) and have been shown
to accurately predict the latent heat of binary eutectics. This is confirmed with
64
experimental measurements of three mixtures, two of which do not have reported
results in the literature. The method generally over-predicts the latent heat by
approximately 6%, which is inside the experimental uncertainty of the methods
employed to measure the latent heat.
In addition, an asymmetric geometric method to predict the composition
of the ternary eutectic from binary eutectic points is evaluated. This method
(Eqs. (6) and (7)) was first proposed by Martynova and Susarev and applied
to systems with common fluoride anions. This method successfully predicts
the composition of other salt species, and common cation systems, to within
8%. Additionally, the predicted compositions have latent heats within 10% of
the eutectic mixtures, and have similar phase change behaviors. Of the three
ternary compositions tested here, two had predicted compositions with superior
latent heats. The prediction of latent heat and composition can therefore be
used to evaluate the properties of salt mixtures. These predictive methods
can drastically reduce the amount of experimental testing which is required for
evaluating PCM candidates. Future work can readily determine and validate
optimal PCM candidates, allowing for cost optimization to be readily conducted.
Finally, a ternary mixture has been identified which has superior latent heat
and melting homogeneity than the mixture identified in the literature as the
eutectic point. The ternary mixture of NaCl + Na2CO3 + Na2SO4 in the com-
position 51.3/31.2/17.5 demonstrates homogenous melting with a latent heat of
252 Jg−1. This mixture has a lower melting temperature than similar sodium
chloride based binary salts, with similar latent heat. This makes it a good
potential PCM candidate for further investigation.
65
6. Acknowledgments
We would like to thank ASTRI and ARENA for funding the research con-
tained herein.
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68
Chapter 4
Design Optimization Methodfor Latent Heat Thermal En-ergy Storage Systems
This chapter has been published in the September 2017 edition ofEnergy. This paper sought to address the deficiency in estimatingthe cost of the containment vessel. As previously elaborated on, thecontainment vessel cost is an important factor in the selection ofsalts. In this paper, tube and shell and tube and fin containmentvessels are considered. An analytic solution is presented which iscomputationally cheap, easing the burden of optimization.
This solution addresses Research Question 2. Without properheat exchanger design optimization, total system cost cannot be es-timated with any confidence. This is due to containment vessel costbeing the majority of the cost of the system. However, the solutionpresented here solves this issue and allows for selection to continue.This method will be utilized later; combined with the properties ofa large list of salts, the total system cost can be estimated and themost cost effective solution selected.
69
Design optimization method for tube and fin latent heat thermalenergy storage systems
Ralf Rauda,, Michael E. Cholettea, Soheila Riahib, Frank Brunob, Wasim Samanb, Geoffrey Willa,Theodore A. Steinberga
aQueensland University of Technology, 2 George St, Brisbane QLD 4000, AustraliabBarbara Hardy Institute, University of South Australia, Mawson Lakes Campus, SA 5095, Australia
Abstract
The search for optimal phase change materials (PCMs) for latent heat thermal energy storage
systems (LHTESS) focuses almost exclusively on the properties of the PCM. This neglects the
significant contribution of the cost of the containment vessel on the total cost of the LHTESS.
Thus, to accurately assess the thermoeconomic performance of various PCMs, the relationship
between the cost of the containment vessel and the properties of the PCM must be understood.
This paper presents an analytical method for optimizing the design of tube and shell or tube
and fin containment vessels for the least cost, subject to geometric and performance constraints.
One of the key performance constraints is the time to charge, whose evaluation typically requires
computationally-expensive simulations which are unsuitable for optimization algorithms. To en-
able efficient optimization, a novel closed-form approximation of the charge time is developed and
validated through numerical simulations.
Subsequently, the optimization methodology is used to investigate the relationship between the
optimal vessel geometry and configuration, fin properties, PCM properties, and cost of two PCMs.
For these PCMs, the use of high thermal conductivity fins is shown to dramatically reduce the
total cost of the system. The new methodology is an efficient way to compare the thermoeconomic
performance of PCMs as latent heat storage solutions, allowing for the accurate assessment of
PCMs.
Preprint submitted to Elsevier January 14, 2018
70
common abbreviationsPCM phase change materialPV photovoltaic
LHTESS latent heat thermal energy storage systemHTF heat transfer fluidSte Stefan numberRe Reynolds numberPr Prandtl number
material properties∆H latent heat of fusionρs PCM densityλs PCM thermal conductivityρm containment metal densityλm containment metal thermal conductivitySt containment metal yield strengthρf fin densityλf fin thermal conductivitycHTFp specific heat of the HTFρHTF HTF densityP0 HTF pressure
design variablesS total storageCo linear corrosion allowanceN number of tubesL length of tubeR outer radius of tuber inner radius of tube
v(f,s) volume fraction (fins, PCM)ws fin half pitch` width of PCM
∆T temperature difference between PCM and HTFP(m,f,PCM) Price per kg (metal, fins, PCM)
calculated∆He effective latent heatρe effective PCM densityλe effective PCM thermal conductivityCF cylinder factorNEF non-equilibrium factortm time to meltQ heat flowv HTF velocity
Hr(x, t) Required remaining energy to melt (at position x and time t)etc.
x distance into the tubeN number of infinitesimal elementse natural exponentz the parameters modified in the optimization (`, r, L, etc.)
a ∆Heρe`2
2λe∆T0· CF ·NEF
b 2((R+`)2−R2)λe
`2r2cHTFp ·ρHTF ·vHTF ·CF ·NEF
71
1. Introduction
As climate change accelerates, the need for research into renewable energy has become critical.
Electrical generation creates more than a quarter of worldwide emissions [1]. In order to reduce
emissions, renewable energy sources that can drive large power grids are being investigated. Solar
energy in particular has attracted intensive research over the last few decades. This research has
led to the widespread adoption of photovoltaic (PV) power sources, which poses problems for large
grids. PV currently lacks cost-effective storage technologies; this negatively impacts the ability
to provide consistent power [2], and leads to large stresses and inefficiencies in baseload power
generation. Large scale adoption of rooftop PV has caused rolling blackouts and brownouts in
interconnected power systems. Concentrated solar thermal power mitigates these disadvantages as
it has storage and controlled output built into the paradigm. In particular, the ability to cheaply
store thermal energy has driven significant investment into the research of thermal energy storage
systems. Several reviews [3, 4, 5] conclude that latent heat thermal energy storage could be more
cost effective than directly storing electricity.
Latent heat thermal energy storage systems (LHTESS) use the phase change of the storage
medium to store large amounts of thermal energy over small temperature ranges. The discharge of
these systems provides consistent output temperatures, which is important for the efficient operation
of heat engines. Several reviews have examined a large range of phase change materials (PCM) as
possible storage mediums. In particular, Khare et al. [5] conducted a study in which the properties
of numerous known metal alloy PCMs were compared. However, this study did not take into
account the cost of the containment material, which is greater than half the cost of the storage
system [6, 7], compared to the PCM, which usually is less than 35% of the total cost. Wen et al. [8]
showed that an increase in thermal conductivity from 1 W/mK to 2 W/mK can reduce the amount
of containment material required by nearly 30%.
As the PCM is a smaller component of the overall cost, changing the PCM to a more expensive,
but higher thermal conductivity option, could reduce the total system cost by reducing the cost of
the required containment vessel. This cost reduction balances decreased containment vessel cost
with increased PCM cost. This balancing requires a systematic optimization to achieve the lowest
72
possible total system cost, a problem which has been articulated for some time [10]. Recently,
Rathgeber et al. [9] demonstrated that the storage system cost is not primarily a function of the
PCM cost, but rather the containment vessel. This further demonstrates that optimized design is
necessary. To date, no rigorous optimization has been undertaken. While limited parametric studies
have been conducted [10, 11, 12, 13, 14, 15], these studies neglect the interaction effects between
design variables (such as thermal conductivity and density), making the resulting configuration
suboptimal.
The following studies have attempted to optimize the containment vessel design, but fall short
of achieving a rigorous optimization. Esen et al. [10] first articulated the problem of the coupled
design optimization and PCM properties forcing a joint analysis for efficient selection. They develop
a numerical enthalpy based model which considers heat flow in both the radial and axial directions.
This model is later validated experimentally [16]. A parametric study is conducted which optimizes
the design for four low temperature PCMs. This study only varies one variable at a time, fixing
the others. This type of parametric analysis leaves large holes in the design space, thus the optimal
designs are not rigorous. Bechiri [17] develop an analytic solution to the tube and shell LHTESS
thermal behavior problem. This analytic solutions promises to be easily optimized, but because it
contains exotic functions, must be solved iteratively. Their solution closely matches experimental
results, but neglecting the addition of fins, and an iterative solution, makes it less than optimal for
design optimization. This led them to conduct a parametric analysis to optimize design for paraffin
based LHTESS. Tehrani et al. [15] explored the feasibility of tube and shell heat exchangers for
LHTES in the temperature range of 286-565 °C. They sought to determine the optimal parameters
for the design while minimizing the heat transfer area. This optimization was conducted with a
numerical model developed by Esen et al. and using a time step of approximately 0.01 seconds. To
determine optimal properties, 36 simulations were performed over a range of parameters. Unfortu-
nately, only 12 simulations were performed for each length. For each length, L/D ratios from 10 to
100 were covered and similarly large ranges of Rr0
. Again, this porous exploration yields important
general trends in regards to system performance, but neglects to fully optimize the design variables.
Campos-Celador et al. [18] develop an analytic solution for the heat transfer behavior of a finned
73
plate LHTESS. This analytic solution collapsed each cell to a single node, drastically reducing the
computation time as compared to a strictly numerical solution. Campos-Celador et al. recognized
that optimization requires solutions which are computationally cheap; they succeeded in creating
a model which agreed with numerical experiments and experimental data. Their model is limited,
however, to rectangular enclosures which are not space efficient when the most expensive compo-
nent is the containment material. Nithyanandam and Pitchumani [11] have conducted a rigorous
optimization. Their work studies the geometries of encapsulated PCMs and PCMs with embedded
heat pipes. For both geometries, their solution was a numerical model based on balanced heat flow
equations discretized into a numerical grid. Their heat flow model takes into account all of the heat
transfer processes which occur in the system: convective heat transfer between the heat transfer
fluid (HTF) and the tank filler, the radial thermal conduction into the wall of the encapsulated
PCMs, and the conduction and convection inside the PCM. Once discretized, the behavior of the
LHTESS is iteratively solved using a non-dimensional time step of 5 · 10−5. Campos-Celador et al.
[18] have shown that iterative numerical solutions are extremely resource intensive, especially for
the fine grid structures that Nithyanandam et al. [11] uses. In addition, Nithyanandam’s optimiza-
tion is not entirely rigorous; they perform parametric studies by varying one variable at a time. In
addition, their search driven optimization uses advanced search space functions with parameters
which require careful fine tuning. Without a doubt, their results are accurate for the particular
cases studied, but applying these results to a wider selection of PCMs would be difficult.
In addition to the aforementioned shortcomings, none of the studies mentioned so far attempt
to optimize fin and tube geometries. Optimizing tube and fin geometries is extremely important
because several studies have found that fins dramatically increase the heat transfer over tube and
shell geometries [13, 14, 19, 20]. Few studies have attempted to optimize fin and tube geometries
[13, 14]. Almsater et al. [13] investigated the effectiveness of finning tubes via a numerical model.
Their numerical model is based on the effective heat capacity method and utilizes a network of
resistors to determine the heat flow. Their results bear up the large increases in effectiveness and
charging capacity that can be obtained with the use of fins. However, their numerical model consists
of approximately 60,000 cells. As it will be shown later, this suggests computation times on the order
74
of days for each simulation. Hubner et al. [14] consider the economic impact of longitudinal tree
branching fins in a tube and shell geometry. They perform an optimization to minimize the levelized
cost of electricity of a LHTESS coupled with a steam generator. Their optimization is unique in
several ways. Firstly, they consider multiple longitudinal fin geometries. Secondly, they discard
the circular outer perimeter simplification present in many analyses. In addition, they consider
multiple differing fin and tube materials. However, they assume a fixed temperature HTF and fixed
tube length. Both together present an unrealistic scenario with extremely poor effectiveness. This
simplification will fail to account for the complex association between cost and PCM properties
such as density and thermal conductivity. Finally, this analysis, like many before, uses an iterative
solution which makes precise cost estimates difficult to ascertain.
Thus, no design methodology has been presented which can accurately optimize the performance
of a PCM heat exchanger with reasonable computational requirements. In a review of the known
design methodologies Castell et al. [21] concluded that the effectiveness-NTU method showed the
greatest promise of the available methods. Amin et al. [22] and Tay et al. [23] have developed
empirical correlations which relate the effectiveness to the mass flow rate of the HTF. This correla-
tion is accurate for all design cases of tube in shell PCMs, and has a consistent systematic error of
approximately 11%. Thus, this correlation can be used to analytically optimize design constraints.
However, the effect of radial or longitudinal finned tubes is not completely characterized. To be
precise: Using this P factor, the PCM system can be designed and optimized using the e-NTU
method developed. Further analysis is needed to develop a more generalized equation [for the P
factor] [19].
Thus, the effectiveness-NTU method cannot compare PCMs with different thermal conduc-
tivities across all geometries. In addition, the characterization of tube and fin geometries via the
effectiveness-NTU method requires experimental correlations which do not exist for all design cases.
The use of numerical methods also has limitations; these methods take time to develop for specific
geometries and require large amounts of computational power to accurately solve problems [18].
Attempting to use these methods to optimize heat exchanger design for large numbers of potential
PCM candidates would prove time and resource intensive.
75
In this paper, a method to optimize the design of a PCM heat exchanger for least cost is
presented. This method relies on a novel closed-form analytic expression to determine the melting
time of a PCM in a tube and fin or tube in shell heat exchanger. This analytic expression, which
depends wholly on the geometry of the heat exchanger and the properties of the materials, is used
as a non-linear constraint to minimize the volume of the fins and or tubes which comprise the heat
exchanger, to determine the least cost design for the given PCM and performance targets. This
optimization is computationally cheap, rigorous, and precise. This method allows for PCMs to be
compared based on the total cost of the system, rather than simply the properties of the PCM
itself, yielding a more accurate comparison.
The paper is organized as follows: First, the optimization problem statement is given. Subse-
quently, A novel closed-form solution for the time of melting of the PCM in the heat exchanger is
derived. This analytic melt time method is validated with CFD results. Finally, the optimization
method is used to compare the cost performance of two PCMs for use in Solar Thermal Energy Stor-
age. These two PCMs are compared under a variety of different conditions, and the relationships
between their properties and the cost of the system is discussed.
2. Optimization Problem Statement
To determine the least cost design of the containment vessel, the primary cost is assumed to
be the cost of the metal and the cost of the PCM. The cost is assumed to be primarily a function
of the cost of the raw materials of metal and the PCM, as previous studies have shown that the
additional costs, such as manufacturing, are relatively constant and small [7]. Other studies have
found the manufacturing costs to be large [14], but proportional to the material cost and consistent
across different geometries and configurations. Thus, the material costs can be inflated to account
for this discrepancy.
The general geometry used is the same as depicted by Esen et al. [10]1 The outer containment
shell is assumed to have a minimum thickness equal to the corrosion tolerance (Co). Thus the cost
1Fig. 1, Esen et al., Sol Ener 62 (1998)
76
is determined to be:
Cost =
(vf ρf S
vsρs∆HPf
)†+[(NLπ(R2 − r2) + 2Co · Lπ(R+ `)N0.5
+ 2Co · πN(R+ `)2)· ρm · Pm
]‡+
(S
∆HPPCM
)?(1)
where v is the volume fraction, ρ is the density, S is the total stored energy, ∆H is the latent
heat, P is the cost per kilogram, N is the number of tubes, L is the length of tubes, R is the outer
radius of the tube, r is the inner radius of the tube, ` is the width of the PCM, N is the number
of tubes, and the subscripts f , m, and PCM refer to the fins, containment material, and PCM
respectively. For clarity, the cost of the fins has been denoted with †, the cost of the tubes and shell
with ‡, and the cost of the PCM with ?. For a finless system, the total cost is given by:
Cost =(N · Lπ(R2 − r2) + 2Co · Lπ(R+ `)N0.5
+2Co · πN(R+ `)2)· ρm · Pm +
S
∆HPPCM (2)
The optimization problem can be stated as follows;
minz
Cost(z)
such that:
P (z) ≥ P0 (3)
QPCMin (z) = QHTFout (z) (4)
vHTF = vHTFmin (5)
Melt(z) ≤Meltmax (6)
†The volume fraction of both the fins and PCM are known, so the ratio of these relates the volume of the PCMto the volume of the fins. The PCM volume is fixed given a specific storage size S, so the volume of PCM multipliedby the volume fraction ratio, multiplied by the density of the fins gives the mass of the fins.
‡The cost of the containment vessel is given by multiplying the number of tubes times the volume of each tube,then adding two terms for the outer containment vessel. All three terms are then multiplied by the density todetermine the mass.
77
where z = (`, L, r) if cost is given by Eq. (2) or z = (`, L, r, vs, vf ) for cost given by Eq. (1).
The constraint Eq. (3) requires that yield strength of the tube, P (z) be greater than the oper-
tating pressure of the heat transfer fluid, P0. Barlow’s formula is used to determine the necessary
thickness of the HTF tubes:
R = r · St
St− P0+ Co (7)
where P0 is the pressure of the HTF and St is the yield strength of the containment material
multiplied by a safety factor.
Eqs. (4) and (5) require that the HTF be able to expell heat faster than the PCM can accept it.
This assures that the limiting heat flow condition will be the amount of heat that can be transferred
into the PCM. The HTF velocity is set via the Dittus-Boelter correlation (assuming turbulent flow):
QPCMin (z) =((R+ `)2 −R2)vsρs∆H
tm(z)
QHTFout (z) = 0.023Re0.8Pr0.333λHTF∆T0 (8)
where tm(z) is the time to melt, Re is the Reynolds number, Pr is the Prandtl number, and ∆T0
is the difference between inlet temperature of the HTF and the PCM melting temperature. Thus
constraints (4) and (5) implicitly set the velocity due to the presence of the Reynolds number in
QHTFout .
Eq. (6) constrains the time to melt, as recent reviews by the Sunshot Initiative suggest that
charging and discharge time is a primary consideration in the development of thermal energy storage
for solar thermal power. This constraint can be constructed by numerical or CFD methods, however,
as previously discussed these are time and resource intensive. Instead, an analytic expression is
developed for the time of melting. The constraint can be modified for the time of solidification,
but the PCMs of interest to this study have significantly reduced thermal conductivity in the liquid
state. This causes the melting process to take longer.
78
3. Time of Melting Constraint
Bauer [24] developed an analytic equation to determine the time of solidification for a PCM in fin
and tube arrangements with isothermal walls. These results assume the PCM starts at the melting
temperature, the tube wall is isothermal, and the contribution of the specific heat is negligible.
Bauer’s results are based on the thermophysical properties of the PCM and the geometry of the
containment design:
tm = NEF · CF · ∆Heρe`2
2λe∆T(9)
where tm is the time to melt. The other variables are given by:
NEF = 0.8
(`
R
)1/4 (ws`
)2 vfλfvsλs
+ 1
CF = ln
(`
R+ 1
)(R
`+ 1
)2
−(
1
2+R
`
)
ρe = vsρs + vfρf
∆He = ∆Hvsρsρe
λe = vsλs + vfλf (10)
where ` is the length of the fin, R is the radius of the pipe, ws is the half-pitch between the fins, vs
is the volume fraction of PCM, vf is the volume fraction of the fin, λs is the thermal conductivity
of the PCM, λf is the thermal conductivity of the fin, ∆H is the latent heat of fusion of the PCM,
∆T is the temperature difference between the PCM and the isothermal wall, ρs is the density of
the PCM, ρf is the density of the fins.
Fig. 1 visually demonstrates these variables. Bauer has verified the accuracy of his analytic
expressions with numerical methods for a large range of input parameters. The numerical methods
were verified with comparison to experimental data. These results were accurate for Stefan numbers
< 0.1, requiring that ∆T be ≈ 5°C, however, as will be shown later, calculations with Ste< 0.25
produce consistently precise results.
79
Based on previous experiments [25] relating the times of melting and solidification of PCMs,
neglecting the influence of convection2 in the PCM on the result (as Bauer does), the relationship
between the melting and solidification times is dominated by the relationship of the thermal con-
ductivity and densities of the liquid and solid phases. In this paper, the liquid thermal conductivity
and liquid density are used in Bauer’s analytic results to yield the melting time. These properties
are assumed to be constant as the temperature of the PCM does not vary significantly. In addition,
the properties of the HTF are also assumed to be constant with temperature. In the case of the
HTF used in Section 4 this is clearly the case [26].
L
HTF
r
R
PCM
2ws2wf
`
Figure 1: Geometry of heat exchanger with variables included.
In the following derivation Bauer’s results will be extended to heat exchangers with non-
isothermal walls. The objective is to describe how the melting proceeds along the tube. In order to
determine the time of melting at the exit of the tube, a model of the HTF temperature is required.
To develop this model, a number of assumptions must be made. First, the heat flow is assumed to
be independent of the melt fraction [27, 28]. Second, the PCM is assumed to start at the melting
temperature, but solid. Then define x is as the distance into the tube (x = 0 is the inlet). As
the HTF moves through the tube, energy is transferred from the HTF to the PCM, reducing the
difference in temperature between the HTF and the immediately adjacent PCM.
2This assumption is discussed further in Section 3.2
80
In order to utilize Bauer’s results, the length of the tube L is split into discrete elements with
length ∆x, as shown in Fig. 2. Thus, each element at position x can be assumed to be an annulus
with an isothermal wall. Then, Eqs. (9),(10) can be used to find the “time to melt” for that element
tm(x, t). However, this “time to melt” is simply an approximation of the remaining time for the
element to completely melt, as the temperature of the HTF at a point is not constant in time.
Instead, the heat transfer rate from the HTF to the PCM is related to the temperature difference
(∆T (x)). Assuming that all of the heat lost by the HTF is transferred into the PCM, then the
energy flowing into each PCM element reduces the temperature of the corresponding HTF element.
Letting z = (x, 0), QPCMin in Eq. (8) describes the heat flow into a single PCM element:
t m(x,0
)
t m(x,a
)
L L8∆x6∆x4∆x2∆x08∆x6∆x4∆x2∆x0
Figure 2: Discretization of tube into elements with unique time to melt and length ∆x, at time t = 0 and t = a.
QPCMin =Total Energy
Time to melt=
((R+ `)2 −R2)ρs∆He · dxtm(x, 0)
= r2cHTFp · ρHTF · vHTF · d∆T (x, 0) (11)
81
rearranging yields the temperature profile of the HTF:
d∆T (x, 0)
dx=
((R+ `)2 −R2)ρs∆He
r2cHTFp · ρHTF · vHTF · tm(x, 0)(12)
at the beginning of the charging process (t = 0).
Since tm(x, t) depends on ∆T (x, t), via Eq. (9), (12) is found to be a constant coefficient
differential equation in x, which has the solution:
∆T (x, 0) = ∆T0 · e−bx
b =2((R+ `)2 −R2)λe
`2r2cHTFp · ρHTF · vHTF · CF ·NEF(13)
Rearranging (9) and (13) the time of melting at t = 0 can be found;
tm(x, 0) = a · ebx (14)
a =∆Heρe`
2
2λe∆T0· CF ·NEF (15)
where a is simply the time for the first element, x = 0 to melt.3 Note that this expression only
describes the “instantaneous” time to melt, assuming that the HTF temperature profile stays
constant. This is not true, as once the first element completes melting, the HTF temperature
profile will change.
Assuming that the PCM starts at the melting temperature (i.e sensible heat is negligible), the
melting at x = 0 is clearly independent of what happens further down the tube. Thus, prior to the
melting of the first element, the “instantaneous” time remaining to melt for the elements further
3This is the time of melting (Eq (9)) expressed using the inlet temperature differential (∆T0), which is thetemperature differential at x = 0.
82
down the tube can be expressed as:
tm(x, t) = a · ebx − t ∀ t ≤ a (16)
After the melting is complete at x = 0, the sensible heat stored is neglected by assumption,
implying that there is no transfer of energy out of the HTF into the PCM. Thus, the temperature
gradient of the HTF will start to move. Fig. 3 shows that, for t > a, if the sensible heat is neglected
the temperature differential will not change as the HTF crosses melted elements.
∆x
∆T
(x,t
)
L2∆x
∆T0
t ≤ aa < t ≤ a+ ∆t1
a+ ∆t1 < t ≤ a+ ∆t1 + ∆t2
Figure 3: Temperature differential as a function of x, for t ≤ a (solid line). After the first element has melted, ∆T (x, t)
=∆T0 in the range 0 < x ≤ ∆x, and then begins to decrease again. The dotted and dashed lines demonstrate the
change in the temperature differential as a function of x and for the times as listed in the legend.
Because the optimization is based on the time to melt of the entire PCM, the time taken for the
last element to melt must be determined. However, the melting of the last element is complicated
by the fact that after t = a the HTF temperature at x = L changes constantly with time. Instead,
the focus will be on the progression of melting through the tube. Clearly, once the first element
melts, the subsequent elements will complete melting in order. The time when the “melt front”
reaches the end of the tube yields the time of melting.
To describe the progression of the melt front, the time ∆t1 to move the melt front from x = 0
83
to the element at ∆x is described. Assuming a left continuous partition4, Eq. (16) can be written
as:
tm(x, a) =Hr(x, a)
∆T (x, a)· ρe`
2
2λe· CF ·NEF
Hr(x, a) = ∆He(1− e−bx) (17)
∆T = ∆T0e−bx
This is developed by understanding that for t ≤ a the time remaining to melt is proportional
to the amount of energy required to melt5. Thus, the amount of energy required to complete the
melting of any element ∆x, at time t = a, is given by Hr(x, a), called the “heat required.” For the
melt front to move from x = 0 to x = ∆x, some amount of time is required for the HTF to deliver
this heat. This can be computed by substituting Eq. (17) into Eq. (9) in place of ∆He yielding:
∆t1 = a · (1− e−b·∆x) (18)
where ∆t1 is the amount of time required for the melt front to move from x = 0 to x = ∆x. To
understand how the melt front moves with time, the additional time (∆t2) for the next element(2∆x)
to complete melting must be computed. To determine this, the amount of heat remaining at
t = a + ∆t1 must be computed. Note that at t = a, the amount of heat required to complete
melting of 2∆x can be given by Eq. (17).
By Eq. (11), the heat flow rate into element 2∆x can be given by:
Qin(2∆x, t) =Hr(2∆x, a)
tm(2∆x, a)∀ t ∈ (a, a+ t1] (19)
4∆T (x, t) for each element is assumed to be ∆T (x, t) at the left edge of the element5Consider how Eq. (9) is a linear function of the latent heat. Thus as the latent heat decreases, the time
decreases in a linear manner. In this case, the assumption is that a partially melted PCM is equivalent to a PCMwith proportionally less latent heat, at least in regards to the time required to complete the melting process.
84
Recall that for a < t < a + ∆t1 the melt front has moved by one element, thus the new
temperature profile is given by:
∆T (x, t) = ∆T0e−b(x−∆x) ∀ t ∈ (a, a+ t1], x ≥ ∆x (20)
The heat required to complete the melting of element 2∆x at t = a+ ∆t1 is given by the heat
flow rate (Eq. (19)), utilizing the new temperature profile (Eq. (20)), and multiplied by the time
elapsed (∆t1) and can be computed:
Hr(2∆x, a+ t1) = ∆He −∆He · e−b·2∆x − ∆t1tm(2∆x, a)
·Hr(2∆x, a) (21)
Using a similar development as used for Eq. (18), ∆t2 can be evaluated:
∆t2 = a− (a · e−b·2∆x + a(1− e−b∆x)e−b∆x)
∆t2 = a · (1− e−b·∆x) (22)
Note that Eqs. (18) and (22) are identical and the time for the melt front to pass between any
two elements is constant. To determine the time taken for the melt front to go a distance L, let
∆x = L/N , and the time to melt at a distance L can be written as:
Melt(L) = limN→∞
a · N(1− e−b LN ) (23)
where Melt(L) is the time for the melt front to pass to a distance L. Eq. (23) simplifies to:
Melt(L) = a · b · L+ a (24)
via an application of L’Hopital’s rule. This shows that the melt front moves linearly as a function
of time. Thus, the constraint Eq (6) is fulfilled by use of Eq. (24).
85
3.1. CFD Validation
In order to validate Eq. (24), numerical modeling of the melting of a low temperature PCM
with pure conduction heat transfer is performed using ANSYS FLUENT [29]. The time dependent
Navier-Stokes equations are solved including the latent heat transfer and phase change at solid-
liquid interface, and zero velocity in the solid region as introduced in a previous work by Riahi et
al. [30].
The following simplifications are assumed: i) density is constant ii) same density for the solid
and liquid phases, iii) the natural convection flow is ignored, incompressible and inviscid flow, iv)
internal thermal radiation is negligible. These assumptions were used in a model developed in
previous work by Riahi et al. [30], using the solidification and melting option in FLUENT[29].
The model, and the assumptions, were validated with experimental data from a study by Jones
et al. [31] and readers are referred to these studies for further details. The same algorithm,
schemes, and discretization methods implemented in the validated model are used in this study.
The convergence thresholds for all calculations were at least 10−4 for the continuity equation and
10−6 for other equations. A two dimensional symmetric grid with 17500 cells is generated from
half of the geometry with higher resolution at the boundaries. The time step was set to 0.1 seconds
to capture details in the temperature fields in a feasible computation time. The properties of the
geometry, PCM, and HTF, used in this correlation are outlined in Table 1. The initial temperature
is set to one °C below the melting temperature in order for the PCM to start in the solid phase.
As shown in Table 1, the error6 is consistent across several geometries and HTF velocities.
This systematic error is consistent across variable geometries, suggesting that it has little effect on
comparisons. In addition, the systematic error is consistent with Bauer’s sensitivity analysis. As
these CFD tests take on the order of 29-450 hours to complete with standard desktop computing
power, evaluation of the melt time using CFD would be intractable for optimization, as a large
number of such simulations would be needed. Thus, the analytic solution enables rapid evaluation
and optimization with a modest systematic approximation error.
6Error is calculated as the difference between the estimated time and the simulation time divided by the simulationtime. It is not expressed in absolute value.
86
Inlet Temperature Differential 30 °CHTF specific heat 4180 J/kgK
HTF density 1000 kg/m3
PCM latent heat 248 J/gPCM thermal conductivity (l) 0.146 W/mK
PCM density 769 kg/m3
Test # 1 r 0.0025 m ` 0.0175 m L 0.5m vHTF 0.1 m/s 12.8% errorTest # 2 r 0.0025 m ` 0.0175 m L 0.5m vHTF 0.01 m/s 11.8% errorTest # 3 r 0.0025 m ` 0.0175 m L 0.25m vHTF 0.01 m/s 10.9% errorTest # 4 r 0.005 m ` 0.035 m L 0.5m vHTF 0.01 m/s 13.6% error
Table 1: Constants used for the CFD correlation. The wall thickness is set to 0.001 m.
3.2. Natural Convection
The assumption that natural convection does not play a large role is not uncommon [10, 14, 32],
but engenders some limitations. In certain orientations, the effect of natural convection for tube
and shell geometries is large [23, 33], but not in others. Additionally, the effect of natural convection
in systems with long and densely packed fins is small [19, 34].
In particular, Tay et al. have studied the effect of natural convection in finned tube geometries
[19]. They found that neglecting natural convection did not effect the accuracy of their CFD
simulation. To account for the effect of natural convection in vertically orientated tube and shell
configurations, Tay et al. [23] used an “effective thermal conductivity”. The effective thermal
conductivity is just the thermal conductivity multiplied by a constant factor. This was successfully
applied to a number of geometries, with an uncertainty of 8.2%.
To determine if this effective thermal conductivity is a viable means of accounting for natural
convection, Tay et al.’s example is followed, and Eq. (24) is compared to the results of Lacroix’s
[27] simulations and experiments. Lacroix developed a computational model using the standard
heat balance equations and included the effect of natural convection. Their results were extremely
accurate when compared against experiments, and they proceeded to simulate the behavior of
numerous geometries. In this study, an effective thermal conductivity factor of 1.33 is utilized and
the melt time model (Eq. (24)) is compared to Lacroix’s results. With this factor, an uncertainty of
12% is achieved. This compares favorably with Tay et al.’s results. If required, an effective thermal
87
Inlet Temperature Differential 30 °CHTF pressure 120 bar
HTF specific heat 2000 J/kgKHTF density 150 kg/m3
Inconel Aluminum NaCl + Na2CO3 NaCl + Na2SO4
Price ($/kg) 7.5 1.5 0.22 0.11Density(kg/m3) 7780 2560 1898 2020
Thermal Conductivity (W/mK) 22 225 0.60 0.494Latent Heat (J/g) - - 283 266
Table 2: Constants used for the optimization[36]
conductivity can be substituted for λe in Eq. (24). However, ignoring natural convection gives
more consistent uncertainty, as seen in the previous section, and thus is better for comparison of
similar PCMs.
4. Comparison of Eutectic Salt PCMs
The optimization formulation developed above will now be used to compare the economic per-
formance of two systems with different eutectic salt PCMs. This optimization is performed via
the Matlab function “fmincon.” No upper bound is set. The lower bound is set to 0.1mm for each
variable to prevent divide by zero errors. The function and constraint tolerances are set to 1e-8.
The properties of these PCMs are listed in Table 2. The two PCMs are referred to by their differing
anion. This investigation focuses on the potential of a supercritical CO2 Brayton cycle heat engine,
thus, sCO2 operating at 120 bar is chosen as the heat transfer fluid. The charging inlet temperature
is set to 660 °C. Both potential PCMs, referred to herein by the differing anion, have melting points
around 630 °C. Thus, the difference between the PCM and the HTF is set to 30 °C. Inconel 601 is
chosen as the tube and containment materials for its high fatigue strength at high temperature, low
creep, and corrosion resistance. The yield strength listed in Table 2 includes a 0.72 safety factor.
The corrosion allowance is an estimate determined from a recent review [35]. The effect of the
corrosion allowance on the cost is elaborated on later. The time to melt is set at 6 hours based on
SunShot targets.
88
29
30
31
32
33
34
35
36
37
10 100 1000 10000 100000 1000000
Cost
($\k
Wh)
Total Storage (kWh)
Figure 4: Optimal cost per kWh as a function of total stored energy for two PCM systems in the tube and shell
geometry, with the Na2CO3 PCM cost shown as a dotted line and the Na2SO4 PCM cost shown as a solid line.
Fig. 4 shows the relationship between cost per kWh and the total desired storage. Clearly,
at small storage capacities, the external containment cost has a large impact on the total cost.
However, this effect is quickly reduced as the storage capacity grows. As shown in Fig. 4 the cost
of the CO3 PCM system is less, regardless of storage capacity. This implies that the 20% higher
thermal conductivity of the CO3 PCM provides a benefit which outweights the fact that its cost
is double that of the other PCM. Important to note is that the energy density, in Jm−3, for both
PCMs is nearly identical.
89
31
32
33
34
35
36
37
38
10 100 1000 10000 100000 1000000
Cost
($\k
Wh)
Total Storage (kWh)
Figure 5: Optimal cost per kWh as a function of total stored energy for two PCM systems in the tube and fin
geometry with Inconel 601 fins, with the Na2CO3 PCM cost shown as a dotted line and the Na2SO4 PCM cost
shown as a solid line.
Fig. 5 demonstrates that the superiority of CO3 carries over to a tube and fin configuration
wherein the fin is made of the same material of the tube. For this optimization, note that the
analytic expressions by Bauer require that ws
` < 0.5 and vf > 0.01, so this solution does not
terminate in the finless system. Nevertheless, of note is that the cost actually increases for this
configuration, but this cost increase is not proportionally the same between the two PCMs. The
CO3 system experiences a cost increase of approximately 5%, while the SO4 system cost only
increases by 1.4%, on average. This is the influence of the thermal conductivity on the cost; the
SO4 system benefits from the effectively increased thermal conductivity significantly more than the
CO3 system.
90
6
8
10
12
14
16
18
20
10 100 1000 10000 100000 1000000
Cost
($\k
Wh)
Total Storage (kWh)
Figure 6: Optimal cost per kWh as a function of total stored energy for two PCM systems in the tube and fin
geometry with aluminum fins, with the Na2CO3 PCM cost shown as a dotted line and the Na2SO4 PCM cost shown
as a solid line.
Fig. 6 shows the cost per kWh for a tube and fin system with Inconel tubes and Aluminum
fins7. The cost is less than 35% of the cost of the finless system. This is due to the influence of the
cheap thermal conductivity enhancement of the aluminum. The fins, with a thermal conductivity
500 times greater, greatly enhance the heat transfer. This leads to a smaller required number of
HTF tubes, which form the bulk of the total cost of the system. The most interesting result from
this series of optimizations is the reversal of the more cost effective system. For this geometry the
SO4 system is more cost effective, by as much as 13%. This is a much greater cost advantage than
7Aluminum fins are used as an example. Clearly, in this situation, they would melt. However, other similarmaterials may exist, or slightly lower temperatures may be selected
91
Tube in Shellr ` L N Total Volume $
(m3)CO3 0.0021 0.0147 0.1732 32600 7.24 30.603SO4 0.0021 0.0136 0.1589 40100 7.31 32.273
Inconel Finsr ` L ws N Total Volume $
(m3)CO3 0.00256 0.01784 0.19509 0.00751 20700 7.25 32.12SO4 0.00254 0.0176 0.19231 0.00595 21600 7.29 32.661
Aluminum Finsr ` L ws N Total Volume $
(m3)CO3 0.00318 0.05358 0.72778 0.00134 880 7.33 10.83SO4 0.00321 0.05386 0.72785 0.00132 871 7.34 9.54
Table 3: Optimum parameters based on geometry for 1 MWh of storage. All units are in (m).
the CO3 system has in the finless configuration. Again, this is the effect of a significantly cheaper
PCM when the thermal conductivity advantage is reduced.
Table 3 enumerates the design parameters for different PCMs and configurations with one MWh
of storage. The results show that the optimum ` is similar for the two different PCMs for each
configuration. The differences in cost are made up for in large differing L values for tube and
shell configurations. In contrast, for finned configurations the fin pitch appears to be the dominant
factor in optimizing the geometry for cost. This demonstrates that, even for geometries designed
to increase the effective thermal conductivity, the thermal conductivity difference between the
two PCMs has a large impact on the optimum parameters. However, when the effective thermal
conductivity is dramatically increased by the aluminum fins, the thermal conductivity difference
of the two PCMs is much less influential. This can be seen in the parameters for aluminum fins,
which are very close. For this configuration, the Na2SO4 PCM is more cost effective because it is
cheaper.
92
02468
10121416
0.5 1.5 2.5 3.5
Affo
rdab
le C
ost (
$/kg
)
Corrosion Allowance (mm)
b)
02468
1012
0.5 1.5 2.5 3.5
Cost
($\k
Wh)
Corrosion Allowance (mm)
a)
Figure 7: Effect of corrosion allowance on optimal cost (a) and the equivalent optimal cost of the containment
material to maintain the same sytem cost as a fuction of corrosion allowance (b)
Fig. 7 demonstrates the effect of the linear corrosion allowance on the cost of the system.
This linear corrosion allowance is the variable “Co” in Eqs (1) and (2). The corrosion allowance
increases the thickness of the tubes and outer shell to account for losses due to corrosion. For this
comparison, the SO4 system with 104 kWh of storage was used in conjunction with the aluminum fin
configuration. Clearly, the dominant cost variable in this optimization is the effect of the corrosion
allowance, as varying this parameter from one to three millimeters increases the cost per kWh by
84%. However, Fig. 7(b) is a more interesting result. This line relates the linear corrosion rate to a
cost of metal such that the total system cost remains at $8.87 per kWh. Here it is demonstrated that
a tube material which costs 14.42$/kg yields the same total system cost as a material which costs
5.07$/kg, if the more expensive material has a third the corrosion allowance. This cost relationship
demonstrates that expensive and exotic containment materials can be cost effective if they reduce
the corrosion rate. In addition, while uncharacterized, the effect on the thermophysical properties
of the corrosion products is undoubtedly greater, and worse for system performance, with a larger
corrosion allowance. Thus, the investigation of containment materials should focus on solutions
which present as little corrosion as possible, with lower cost is a secondary consideration.
93
5. Conclusions
A new method is presented to optimize for least cost the geometry of a tube in shell, and tube
and fin, heat exchangers for latent heat thermal energy storage systems. This optimization is based
on analytic solutions for the melting time of a PCM in the aforementioned geometries, and gives
specific insight into the effect of the PCM’s thermophysical properties on the cost of the system.
This optimization, because it is based on analytic expressions, is very fast to compute, allowing for
easy integration into a system wide parameter optimization. This is in contrast to optimizations
presented previously, which are based on CFD or numerical methods, both of which take signif-
icantly longer to compute. However, this optimization is limited by two important assumptions.
First, the neglect of natural convection implies that the results for tube and shell containment are
not rigorous for all arrangements. The effect of convection on these results is a subject for futher
study. Secondly, the assumption that the PCM starts at the melting temperature limits the ap-
plicability of these results to storage systems which include a significant amount of sensible heat
storage. Again, the results contained herein can be expanded, but this is a subject for further study.
A comparison was undertaken between two potential eutectic salt PCMs for use in a LHTESS in
a concentrated solar thermal power plant. This comparison showed that for tube and shell geome-
tries, the dominant property is the thermal conductivity of the PCM, with a 20% larger thermal
conductivity being enough advantage to overcome a 100% price difference. However, for tube and
fin geometries with large thermal conductivity enhancement, the optimum geometry is based on
the energy density, with the thermal conductivity of the PCM playing a smaller role. The effect
of the corrosion allowance on the cost of the system is discussed. The cost of the tube material is
found to play a large role, and because of the uncertain effects of corrosion products on PCM per-
formance, expensive containment materials with large costs are suggested to be further researched,
in preference to cheaper containment materials with potentially less corrosion resistance.
94
6. Acknowledgments
The authors would like to thank acknowledge the support of the Australian Government for this
study, through the Australian Renewable Energy Agency (ARENA) and within the framework of
the Australian Solar Thermal Research Initiative (ASTRI).
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Chapter 5
Thermal Conductivity Enhance-ment via Fins or Suspension
This conference paper was presented at the AustralianHeat and Mass Transfer Conference in Brisbane duringthe winter of 2016. As thermal conductivity enhance-ment via nanoparticle suspension is an active area ofinterest, the question of design priorities must be ad-dressed. In the following paper we determine that highthermal conductivity fins are preferred over nanoparticlesuspension.
In particular, untill this point the thesis has ignoredthe potential to enhance the PCM thermal conductiv-ity via particle suspension. This is an active area ofresearch, and must be addressed to validate the conclu-sions which follow in the final chapter. Specifically, ifenchanced PCMs are ignored the question remains as totheir cost effectiveness over other, un-enhanced PCMs,and their cost effectiveness when compared to the tubeand fin heat exchanger configuration. In this conferencepaper, we show that enhanced PCMs are not cost effec-tive. This allows us to continue ignoring them.
100
Analysis of the Economic Effect of Nanoparticle Suspension In Phase Change Materials for Latent Heat Thermal Energy
Storage Ralf Raud1, a *, Michael E. Cholette1,b, Geoffrey Will1,c,
and Theodore A. Steinberg1,d 1Queensland University of Technology, 2 George Street, Brisbane, Queensland Australia,
4000 [email protected], [email protected], [email protected], [email protected]
Keywords: Latent Heat Thermal Energy Storage Design, enhanced phase change materials, PCM
Abstract. Low thermal conductivity is a common issue which prevents many phase change materials (PCM) from being used in latent heat thermal energy storage systems (LHTESS). This low thermal conductivity requires significantly greater heat exchange area, which further increases the prevalence of corrosion and the cost of the containment vessel. To improve the thermal conductivity, studies have been undertaken which measure the properties of PCMs with suspended nanoparticles. Using a method, described elsewhere, for optimizing the design of the heat exchanger for least cost, the effect of nanoparticle dispersion in the storage material is characterized from a total system cost perspective. This analysis describes the effect of the combination of system performance characteristics and storage material property enhancement on the total cost of the system. Thus, the optimum enhancement of storage material by nanoparticle dispersion is described.
Introduction Salt mixtures and eutectics have been considered as possible storage medium for thermal energy going back to the 1960’s [1]. Vast amounts of effort and research have gone into studying these PCMs, partially due to the overwhelming number of possible combinations, which enable the storage temperature to be picked precisely for the application. In addition, salts are plentiful and cheap, as most are used in various industrial settings, such as for fertilizer, the manufacturing of metals, and for food processing [2]. However, the widespread adoption of salts as thermal energy storage medium has been limited by the generally low thermal conductivity. To combat low thermal conductivity, heat exchanger area must be increased, which increases the cost of the system as a whole. Recent work has been leaning towards the use of molten metals as PCMs; due to the significantly higher thermal conductivity they offer [3]. In addition, much work has been done to measure the effect of nanoparticle suspension on the properties of PCMs [4]. Suspension of expanded graphite (EG) in many different types of PCMs, such as salts, acids, and water, has been shown to dramatically increase the thermal conductivity. Studies have shown that by suspending EG in paraffin wax in 10 wt%, the thermal conductivity of the PCM is increased by 400% [5]. In this paper the effect of suspending EG in a one to one mixture of KNO3 and NaNO3 is studied. Recent work has characterized the latent heat, thermal conductivity, and apparent density of this PCM, which allows for the total system cost to be approximated.
101
Theory
The system geometry is assumed to be tube and shell or tube and fin. The system cost is assumed to be taken as the cost of the salt and the cost of the material comprising the heat exchanger, as studies have shown these components make up the majority of the cost [6, 7].
To optimize the design of the heat exchanger, an input HTF temperature and desired discharging time are specified. Then, utilizing Bauer’s [8] time for solidification of a bulk PCM in a tube in shell geometry, these results are expanded upon to find that the time for solidification to be complete (ts) a distance L away from the inlet is given by:
𝑡𝑡𝑠𝑠 = 𝑎𝑎 ∙ 𝑏𝑏 ∙ 𝐿𝐿 + 𝑎𝑎 (1)
The derivation of Eq. (1) is given elsewhere. The variables a and b are given by:
𝑎𝑎 =𝛥𝛥𝛥𝛥𝑒𝑒𝜌𝜌𝑒𝑒ℓ2
2𝜆𝜆𝑒𝑒𝛥𝛥𝛥𝛥𝑜𝑜∙ 𝐶𝐶𝐶𝐶 ∙ 𝑁𝑁𝑁𝑁𝐶𝐶
(2)
𝑏𝑏 =2((𝑅𝑅 + ℓ)2 − 𝑅𝑅2)𝜆𝜆𝑒𝑒ℓ2𝑟𝑟2𝜌𝜌𝐻𝐻𝐻𝐻𝐻𝐻𝑣𝑣𝐻𝐻𝐻𝐻𝐻𝐻𝑐𝑐𝑝𝑝𝐻𝐻𝐻𝐻𝐻𝐻
∙ 𝐶𝐶𝐶𝐶 ∙ 𝑁𝑁𝑁𝑁𝐶𝐶 (3)
and where ΔHe is the latent heat of fusion of the PCM, ρe is the density of the eutectic, ℓ is the thickness of the PCM, λe is the thermal conductivity of the PCM, ΔT0 is the difference in temperature between the PCM melting temperature and the HTF input temperature, CF is the cylinder factor, and NEF is the non-equlibirum factor, both are specified by Bauer [9]. Note that for the tube and shell geometry, NEF is set to one. For tube and fin geometries NEF is calculated via the formula given by Bauer. In addition, R is the outer radius of the tube, r is the inner radius, and ρHTF, vHTF, and cp
HTF are the density, velocity and specific heat of the HTF respectivly. Eq. (1) is used to constrain the time for solidification. Barlow’s formula is used to ensure the pipe can withstand the pressure of the HTF, and the Dittus-Boelter correlation is used to set the HTF velocity such that the HTF can absorb the required amount of energy from the PCM. A numerical optimization minimizing the volume of the containment material is performed via MATLAB’s "fmincon" function. The optimization based on using supercritical CO2 as the heat transfer fluid, Inconel 600 as the heat exchanger material, and various times and input temperatures for the heat transfer fluid. In addition, there is a corrosion tolerance designed into the optimization to account for corrosion of the tube walls by the PCM. As the corrosive effects of the EG suspension have not been studied, we assume that the corrosion rate does not change with respect to EG concentration. In addition, further degradation of the PCM is ignored as commercial PCMs of similar composition have been used with little issues, and EG suspensions have been shown to be consistent over long periods of time. The properties of these materials, as used in the simulation, are presented in Table 1.
102
Table 1 Properties of Materials as Used in the Optimization
HTF pressure 120 bar [9] HTF specific heat 2000 J/gK HTF density 150 kg/m3 Inconel 600 Aluminum Price ($/kg) 7.5 [10] 1.5 [10] Thermal Conductivity (W/mK) 22 [11] 225 [12] Density (kg/m3) 7780 [11] 2560 [13]
Inconel 600 is chosen because of its high creep resistance, its resistance to corrosion, and its high strength at high temperatures. Supercritical CO2 is chosen as the HTF because it forms the basis for high efficiency thermal cycles currently being investigated for concentrated solar thermal power plants [14]. The properties of the PCM are taken from Xiao et al. [15] as this study has all three properties of the PCM necessary for the optimization, and all the experiments were performed on the same samples. This consistency is difficult to find in the literature for salt PCMs, so only one salt is analyzed in this study. The properties as taken from Xiao et al. are given in Table 2.
Table 2 Properties of PCM [15]. 0.75$/kg is used as the cost of the PCM [16].
wt% of EG ΔHe
(J/g)
λe
(W/mK)
ρe
(g/cm3)
0 122 0.75 1.88 5 109 2 1.73
10 101 4 1.74 20 90 6.5 1.69
Results and Analysis For the first analysis, a total storage of 1000 kWh is chosen and the effect of discharge time and input temperature differential on the system cost is tabulated. In addition, since mass producing a PCM with suspended EG has an unknown cost, the cost of the suspension and the EG is assumed to be the same as the cost of the PCM it replaces. This assumption is nearly the best case and the analysis can begin by considering the best possible outcome.
As shown in Fig. 1, the total cost decreases rapidly with increasing EG content, but the 20 wt% PCM suffers from increased cost across the possible simulations.
103
Figure 1. (a) Total LHTESS cost vs wt% EG for target times of 2, 4, and 6 hours and an input
temperature differential of 30 K. (b) Total LHTESS cost vs wt% EG for input temperature differentials of 10, 20, 30, and 40 K with a target charging time of 4 hours.
This analysis is naïve, however. Note that:
System Cost = Cost of Heat Exchanger + Cost of PCM + Cost of EG
= ‘’ + kg of PCM ·$/kg + kg of EG ·$/kg (4)
To better understand the economic benefit of the suspended EG, the marginal benefit of the cost of the EG must be considered. To that effect, by applying Eq. (4) to the data from Fig. 1, and fixing the system cost, the maximum cost of the suspension and the EG can be found. This analysis, as shown in Fig. 2, will give a better idea of how cost effective the addition of the EG is.
Figure 2. Maximum cost per kg of EG for the total system cost to be equal to the cost of 0 %wt EG, as a function of wt% of EG and (a) target discharge time (hr) or (b) input HTF
temperature differential (K).
Similarly to the analysis in Fig.1, for Fig. 2a the input HTF differential is 30 K, and for Fig. 2b the discharge time is 4 hours. Fig 2 illustrates that for fast discharging at low temperature differentials, EG can be extremely expensive in comparison to the pure PCM but still have a large cost savings. In fact, the 5 wt% EG PCM can have EG costs approaching 10 times the cost of the PCM but still be more cost effective than the pure PCM. This indicates that small concentrations of EG can be cost effective, but larger concentrations, or for systems with
30.00
35.00
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0 10 20
Syst
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/kW
h)
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(a) (b)
104
longer discharge times, EG suspention is not cost effective. For the target discharge time of 6 hours, the 10 wt% PCM has a cost savings of 14%. This large savings is called into question however, when considering that if the EG suspension costs twice as much as the PCM, the savings is halved. In addition, with recent cost estimates of EG approaching $12/kg[17], no EG PCM presents cost savings. This sensitivity to price makes the suspension a risky avenue to increase thermal conductivity.
Figure 3. System cost of least cost EG PCM vs pure PCM with tube and aluminum fin
geometry, as a function of (a) input temperature differential (K) or (b) discharge time (hr). Fig. 3 demonstrates the futility of suspended EG particles in this particular PCM. A tube and aluminum fin geometry outperforms the cheapest EG system by a consistent margin. This analysis assumes the cost of the EG suspension is the same as the cost of the PCM, which implies that the aluminum fins would outperform the EG suspension in all imaginable use cases. Conclusion
A numerical optimization based on an analytic solution to the behaivor of PCMs in tube and shell heat exchangers was performed on a PCM and suspensions of various concentrations of EG in that PCM. This optimization determined the least cost heat exchanger design, which allowed for the total system cost to be calculated for a variety of conditions. These conditions were compared and the following conclusions were drawn. The 10 wt% EG suspension was the cheapest EG suspension solution. Cost savings of up to 30% could be realized in certain situations. However, the maximum cost for the EG to be cost effective was small in comparison to the PCM cost for most use cases. Only for low HTF temperature differentials and short discharge times was EG suspension cost effective. In addition, a tube and aluminum fin geometry was shown to outperform EG suspension in all reasonable use cases.
Acknowledgments
We would like to thank ASTRI and ARENA for funding the research contained herein.
35
37
39
41
43
45
47
10 20 30 40
Syst
em C
ost
Input Temperature Differential
10 wt% EG
Al fins
35363738394041424344
2 3 4 5 6Sy
stem
Cos
t Discharge Time
10 wt% EG
Al fins
(a) (b)
105
References [1] Telkes, M. Composition of matter for the storage of heat, US Patent US 2677664 A, 1954 [2] Kenisarin, M.M, High-temperature phase change materials for thermal energy storage, Renewable and Sustainable Energy Reviews 2010 [3] Khare, S., Dell’Amico, M., Knight, C., and McGarry, S., Selection of materials for high temperature latent heat energy storage, Solar Energy Materials and Solar Cells, 2012 [4] Kibria, M.A, Anisur, M.R, Mahfuz, M.H, Saidur, R., and Metselaar, I.H.S.C., A review on thermopphyscial properties of nanoparticle dispersed phase change materials, 2015 [5] Xiao, X., Zhang, P., Meng., Z., and Li, M., Thermal Conductivity of Eutectic Nitrates and Nitrates/Expanded Graphite Composite as Phase Change Materials, Journal of nanoscience and nanotechnology, 2015 [6] Jacob, R., Saman, W., Belusko, M., and Bruno, F. Techno-Economic Analysis of Phase Change Material Thermal Energy Storage Systems in High Temperature Concentrated Solar Power Plants, APSRC 2014 [7] Nandi, B.R., Bandyopadhyay, S. and Banerjee, R., Analysis of high temperature thermal energy storage for solar power plant, Sustainable Energy Technologies (ICSET), 2012 IEEE [8] Bauer, T., Approximate analytical solutions for the solidification of PCMs in fin geometries using effective thermophysical properties. International Journal of Heat and Mass Transfer 2011 [9] Gupta, S., Developing 1-d heat transfer correlations for supercritical water and carbon dioxide in vertical tubes, Masters Thesis, University of Ontario Institute of Technology, 2014 [10] Information on www.metalprices.com [11] Inconel 601 (UNS N06601) Information Sheet, Special Metals 2005 [12] Powell, R. W., Ho, C. Y., and Liley, P. E., Thermal Conductivity of Selected Materials, NIST 1966 [13] Narender, K., Rao, A.S.M., Rao, K.G.K. and Krishna, N.G., Temperature Dependence of Density and Thermal Expansion of Wrought Aluminum Alloys 7041, 7075 and 7095 by Gamma Ray Attenuation Method, Journal of Modern Physics, 2013 [14] Liu, M., Tay, N.H.S., Belusko, M. and Bruno, F., Investigation of cascaded shell and tube latent heat storage systems for solar tower power plants. Energy Procedia 2015.
106
[15] Xiao, X., Zhang, P. and Li, M., Thermal characterization of nitrates and nitrates/expanded graphite mixture phase change materials for solar energy storage. Energy Conversion and Management 2013 [16] Nithyanandam, K., Pitchumani, R., Design of a latent heat thermal energy storage system with embedded heat pipes. Applied Energy 2014 [17] Fukushima, H., Drzal, L.T., Rook, B.P. and Rich, M.J., Thermal conductivity of exfoliated graphite nanocomposites. Journal of thermal analysis and calorimetry 2006
107
”But how awful would that be?How terrible to live surrounded
by the stark,sharp,
hollownessof things that simply were enough?”
Patrick Rothfuss[1]
108
Chapter 6
Optimized Salt Selection
The preliminary research conducted previously is com-bined to answer the central question; ”What salt do Iuse?” This paper entails the final product of the work,and thus shares its title.
For this optimization, the ASTRI target of a super-critical CO2 Brayton cycle heat engine is used, as well asthe sunshot target of $15/kWh. Tube and shell and tubeand fin heat exchanger configurations are investigated,and PCM enhancement via nanoparticle suspension areignored.
109
Optimized Salt Selection for Solar Thermal Latent HeatEnergy Storage
Ralf Rauda,, Stuart Bella, Geoffrey Willa, Theodore A. Steinberga
aQueensland University of Technology, 2 George St, Brisbane QLD 4000, Australia
Abstract
For Solar Thermal Power to be cost effective as a baseload generation paradigm,
the cost and efficiency of every subsystem must be improved. To reduce the cost
of thermal storage and increase the temperature, appropriate storage materials
must be identified for study. Unfortunately, there is an enormous search space
of salt mixtures which must be analyzed for cost effectiveness.
To lessen the experimental burden, this article combines previously vali-
dated theories for estimating the properties of mixtures with an analytic design
optimization method to better estimate the true cost of each candidate. Five
hundred and sixty three binary, ternary, and quaternary mixtures are analyzed.
Five mixtures have been identified as the prime candidates for a cascaded la-
tent heat thermal energy storage system for a supercritical CO2 brayton cycle
generator. This system is cheaper than the SunShot cost targets, bringing the
solar thermal paradigm into cost competition with fossil fuels.
1. Introduction
At the time of writing, the last several years have set consecutive records
for the hottest years in recorded history. The influence of man-made climate
change is now undeniable, and to mitigate the negative effects an immediate
Email address: [email protected] (Ralf Raud)
Preprint submitted to Elsevier December 30, 2017
110
transition to renewable energy sources is necessary.
With this driving force, a large amount of investment has flowed into solar
energy research. The objective of much of this research is reducing the cost and
increasing the efficiency of photovoltaic (PV) solar panels. However, despite PV
reaching cost parity with fossil fuels, it remains a distinct disadvantage for grid
management services. Indeed PV over and under generation has caused serious
brownouts in several locations around the world. These places have implemented
legislation which limits the growing penetration of PV. This disadvantage for the
PV paradigm is due to battery storage focusing on amp density and lightweight
solutions, niether of which is necessary for large fixed installations, thus, directly
storing overproduced electricity is an expensive proposition.
In contrast, currently operating solar thermal power plants, while not yet
achieving cost parity with fossil fuels, can store thermal energy on the order
of 1/3 of the cost of batteries. This advantage allows for the solar thermal
paradigm to act much like traditional baseload generators; it decouples the
incident solar radiation from the output electricity. This assists in mitigating
grid instability rather than contributing to it. Indeed, some of the currently
operational solar thermal plants have produced consistent power continuously
for a week or more.
However, for solar thermal to achieve cost parity, large cost savings must be
realized in every major subsystem. In particular, to increase efficiency, the op-
erating temperature must be increased. For the thermal energy storage subsys-
tem, molten salts have been the subject of intense investigation as heat storage
media. A number of highly successful reviews cover the basics of using salts
as phase change media (PCM) in latent heat thermal energy storage systems
(LHTESs), a paradigm which stores the thermal energy in the phase change
between liquid and solid [1, 2, 3].
111
Few studies go in depth in determining the cost effectiveness of various salts.
The U.S National Renewable Energy Laboratory [4] has conducted a rigorous
comparison of 48 individual mixtures. Their experimental studies revealed that
the property predictions for these mixtures were not accurate. NASA [5] has
conducted further studies, with an eye on thermal storage for space applications.
They also tested few mixtures, and many of the properties they used were vague
estimates which have since been experimentally disproven. More recently Myers
and Goswami [2] conducted a rigorous comparison of binary chloride mixtures.
While they tested more than one hundred mixtures, they neglected to include
the cost of the containment vessel.
The domination of the cost of the containment vessel, and the understand-
ing that selection of salts requires this cost estimate, is a well understood phe-
nomenon [6, 7, 8, 9, 10]. Esen et al. [8] first articulated that PCM selection
is a coupled problem involving the PCM properties and an optimized storage
design. They developed a model which estimated the heat flow of the system,
and performed limited parametric studies. Thier results, as well as results more
recently from Nithyanandam et al. [11] and others [12] show clearly that the
least cost containment vessel designs still cost more than the salt. Thus, the
optimization of the containment design is an important part of selecting the
correct PCM.
In this paper, previously proven methods to predict the properties of binary,
ternary, and quaternary eutectic salt mixtures are combined with an efficient
optimization method for the design of the containment vessel. The cost of
each candidate is estimated, and the five best mixtures for a next generation
cascaded LHTESS supplying heat to a supercritical CO2 brayton power cycle
are determined. The effect of storage unit geometry and the corrosion allowance
of the design are investigated. Conclusions are drawn regarding the minimum
112
targets for effective storage, including the corrosion allowances, the targets for
high thermal conductivity fins, and the benefits of finned tube geometries.
2. Estimation of Salt Mixture Properties
In order to optimize the design of the containment vessel for each mixture,
the properties of this mixture must be known. The density and thermal con-
ductivity can be calculated using a simple mixing model [10]. The latent heat
of fusion and component concentration require more involved calculation, which
is described in the following sections.
2.1. Latent Heat
As storing energy is the primary objective of interest in LHTESS, the amount
of energy that can be stored is the primary property of any potential PCM.
Thus, predicting this property is extremely valuable. Previous results have
demonstrated the efficacy of the following equation in the case of mixtures
without fluoride ions:
∆Hfus =n∑
i
xi ·∆Hi (1)
where ∆Hfus is the heat of fusion of the eutectic, xi is the molar concentration
of component i, and ∆Hi is the molar latent heat of component i. This equation
fails, however, for some mixtures [10, 13], particularly for mixtures containing
fluoride ions. For these, a heat of mixing term is introduced. This heat of mixing
is determined from several sources, including Beilmann et al. [14], Hong and
Kleppa [15], and Misra et al. [16]. Table 1 elaborates on the fitting parameters
for these heats of mixing. The equation to predict latent heat becomes:
∆Hfus = −∆Hmix(1, ..., i, ...n) +n∑
i
∆Hi · xi (2)
113
where ∆Hmix(1, ..., i, ...n) is the heat of mixing which depends on all of the
components in the mixture:
∆Hmix(1, ..., i, ...n) =
n∑
i,j
∆Hi,jmix · (xi + xj) (3)
Table 1: Approximations of heats of mixing for Eq.(3) via [1, 17, 16, 2]
Eutectic Heat of Mixing
kj/mol
LiF + KF 9
LiF + NaF 3.5
NaF + MgF 9
CaF +MgF 5
LiF + MgF 17
KCl+NaCl 8.5
KNO3 + NaNO3 1.5
2.2. Quaternary Composition and Melting Point
Martynova and Susarev [18] proposed using triangular Gibbs diagrams to
compute the ternary eutectic composition. Their work, and the follow up re-
search by Trunin et al. [19] has been further validated against literature [10] and
experimental work [13]. In order to evaluate more mixtures, quaternary mix-
tures will be considered in this study. The method for predicting quaternary
composition will be elaborated here.
The method begins by computing the most stable pair of components, as
determined via the following equation:
P (i, j) = |(Ai,ji +Ai,jj )(Aj,kj −Ai,ki )|+ |(Ai,ji +Ai,jj )(Aj,lj −Ai,li )| (4)
114
where Ai,ji is calculated by:
Ai,ji =2.3
xi,jjln
(Tix
i,ji
T i,jfus
)(5)
where xi,ji is the concentration of component i in the binary eutectic of i and j,
and T i,jfus is the melting temperature of said eutectic.
Let x, y be the most stable pair of components, and without loss of generality
let w, z be the remaining components. The calculation procceeds by drawing a
new gibbs triangle with the verticies on the eutectic of x, y and on the compo-
nents w and z. The following system of equations is solved to determine the
composition:
xx −xx,yxxx,yy
= 0
xx,y,zz
xx,y,zx+xx,y,zz
xx,y,zx− 2xz = 0
xx,y,ww
xx,y,wx+xx,y,ww
xx,y,wx− 2xw = 0
xx + xy + xz + xw = 1
(6)
The melting temperature for the quaternary eutectic is calculated via a sim-
ilar geometric method as for the ternary:
T i,j,kfus = Tγ −m‡ · (1− xγ)
m‡ =Tγ − Tα,γfus
xα,γα
xβxβ + xα
+Tγ − T β,γfus
xβ,γβ
xαxβ + xα
(7)
where (α, β, γ) is the rotation of (ij, l, k) such that Tα > Tβ > Tγ . Note that
this differs slightly as instead of three single salt components, there is a binary
on one vertex of the Gibbs triangle. Thus, the melting points of the ternary
eutectics T i,j,lfus and T i,j,kfus are required.
115
Previous analysis has shown that these predictions are correct to within some
small uncertainty [10, 13].
2.3. Predicted mixture properties
The data for salt properties used to generate Table A1 was taken from many
sources. The data on individual salts was taken primarily from Kensiarin [1]
and Janz et al. [20, 21, 22, 23, 24, 25, 26, 27, 28] as well as the CRC Handbook
for Chemisty and Physcis [29]. For the thermal conductivity of individual salts,
the estimation by Gheribi et al. [30] was used. For binary eutectic salt concen-
trations and melting points, Factsage [31] was used primarily, with further data
provided by Janz et al., Freidina and Fray [32], and Gittins and Tuttle [33].
Costing for individual salts was provided by an analysis of available public
quotes for large quantities in excess of 1000 kg. Manufacturers where chosen
based on the advertising material indicating a consistent quality of material,
in addition to the manufacturers being large and established. This data was
compared to the last ICS published bulk price guidelines [34] and found to be
well within the same order of magnitude.
Mixtures containing Rb, Cs, and I, were removed from further costing anal-
ysis as the salts were found to be unavailable for reasonable prices (<$50/kg).
Table 2 details the data used here.
116
PPPPPPPPPPPPPCation
AntionF Cl Br SO4 NO3 CO3
Li 10 0.5 3.42 1 0.5 2.2
Na 0.73 0.1 1.8 0.1 0.36 0.22
K 0.2 0.5 2.5 0.65 0.36 0.72
Mg 1 0.18 1 0.4 0.8 0.2
Ca 0.5 0.259 3.2 0.8 0.38 0.1
Sr 1.1 0.672 1 1.195 1.335 0.65
Ba 1.4 0.72 1 0.4 0.625 0.35
Table 2: Bulk salt prices, in USD/kg
The full list of mixtures is described in the Appendix. A small selection is
described in Table 3 where the mixtures are grouped into three sets of mixtures
for a five PCM cascaded LHTESS. The groups correspond to the most cost
effective mixtures under various conditions, which will be discussed in Section
4.
117
Mixture Molar Cost Melting Density Thermal Latent Heat Group
Concentration Point Conductivity
($/kg) (K) (g/cm3) (W/mK) (J/g)
CaCl2 + NaCl 0.521/0.479 0.207 777 2.04 0.51 328 a,b,c
KF + MgF2 + CaF2 0.409/0.319/0.272 0.543 803 2.58 0.68 591 a,b
LiCl + NaCl 0.72/0.28 0.36 827 1.58 0.68 439 a,c
KF + MgF2 + CaF2 0.582/0.085/ 0.5 871 2.71 0.62 453 a
+SrF2 0.23/0.103
NaF + KF + CaF2 0.306/0.464/ 0.58 910 2.7 0.72 475 a,b
+SrF2 0.097/0.133
NaF + NaCl + Na2SO4 0.181/0.367/ 0.2 848 2.03 0.65 304 b
+Na2CO3 0.155/0.297
MgCl2 + MgF2 0.78/0.22 0.308 887 1.87 0.32 529 b
LiCl + NaCl + MgCl2 0.518/0.26/0.222 0.28 802 1.65 0.58 447 c
NaCl + Na2SO4 + Na2CO3 0.482/0.187/0.332 0.147 879 1.99 0.56 264 c
NaCl + Na2SO4 0.533/0.467 0.1 899 2.02 0.5 266 c
Table 3: Small Selection of Mixture properties
3. Calculation of Minimum System Cost
Once the properties of each mixture are calculated, the design optimization
can begin. As this can be a very computationally involved procedure for the
number of mixtures tested, an analytic solution was developed [9]. This solution
optimizes the geometry of the design vessel for minimal cost based on the prop-
erties of the mixture, the heat transfer fluid (HTF), and the target performance
characteristics.
A standard nonlinear optimization is performed with the objective function
118
given as:
Total System Cost = Cost of + Cost of + Cost of + Cost of
Per MWh Salt Fins Tube Shell
where the geometry is varied to find the minimum cost. The shell is assumed
to be a vessel which contains the entire system, made of the tube material,
with thickness equal to the corrosion allowance. The geometry of the fins is
determined by the pitch and thickness1, with the length fixed as the width of
the salt. The geometry of the tube is determined by an inner radius, length,
and the thickness of the salt layer. Finally, a fixed amount of salt must be
contained: the amount of salt required to store one MWh of energy. This opti-
mization ignores sensible energy and assumes that the entire salt passes through
a phase change. Storage, and thus the thermal conductivity and density while
molten, is targeted for optimization as the salts always have significantly lower
thermal conductivity while molten. Sunshot targets require that the storage
be completed in less than six hours. The estimated time to melt, along with
ensuring the HTF can deliver the required energy, and the requirement that
the tubes withstand the pressure of the HTF, all form the constraint functions.
1The fin thickness tends almost exclusively to the pre-defined minimum. Further studiestake this into account by varying the minimum fin thickness
119
PCM1 PCM2
TH
TF
~ 6
50 C
PCM3 PCM4 PCM5 TH
TF ~
500 C
L
HTF
r
R
PCM
2ws2wf
`
Figure 1: A schematic outline of the cascaded system. The HTF enters from the left, decreasesin temperature by approximately 30 °C as it travels through each PCM, then exits on the right.Each PCM consists of several tubes which are finned or not as the configuration demands,with the geometry as shown [9].
For this selection optimization, the parameters listed in Table 4 have been
chosen. These parameters were chosen based on the use of a supercritical CO2
brayton power cycle (between 450 °C and 600 °C) and a cascaded storage setup
consisting of five phase change temperatures. This implies the approximately
30 °C temperature drop assumed as part of the optimization parameters. This
cascaded setup was chosen for the optimization based on efficiency improvements
that have been demonstrated by Xu and Zhao [35]. The HTF is supercritical
CO2, the tubes are vertical with the hot-side on top. The diameter is fixed for
each PCM. The tube material properties are based on a nickel-superalloy due to
thier strength at the high targeted storage temperature. Regular stainless steels
give sub-par corrosion and strength performance at this level, necessitating the
120
use of superalloys.
The fin materials are based on the properties of representative metals and
ceramics at the targeted storage temperatures. The properties are based on
commonly used metals for heat exchangers, and high-thermal conductivity ce-
ramics.
Inlet Temperature Differential 30 °C
HTF pressure 120 bar
HTF specific heat 2000 J/kgK
HTF density 150 kg/m3
Tube Material Metal fins Ceramic fins
Price ($/kg) 7.5 1.5 1.5
Density(kg/m3) 7780 2560 3600
Thermal Conductivity (W/mK) 22 225 20
Table 4: Constants used for the optimization [13]
4. Results and Analysis
In the following section, each mixture’s properties is combined with the op-
timization model to estimate the cost of a LHTESS utilizing that mixture. The
most cost effective mixtures are selected for further analysis, in which param-
eters such as corrosion allowance, fin thickness, and fin material are varied to
understand the effect on the cost estimate. Dependant on several conditions,
the best mixtures for a five PCM cascaded LHTESS are presented as groups
“a”, “b”, and “c” in Table 3.
These cascaded groups were chosen based on the individual mixtures having
a heat exchanger cost minimization performed, then the five lowest cost mixtures
approximately 30 °C apart were chosen. Note that because the interaction
between the salt and the container or fin materials is unknown, the sensitivitiy
121
analysis performed is a general assessment to determine how sensitive the cost is
to changes in the parameters. Note also that the corrosion allowance is a linear
thickness which is in addition to the safety factor from the pipe wall thickness.
Corrosion is assumed to be uniform when rates are discussed.
Prior to the analyzing the cost, the uncertainty in estimating the cost is
discussed.
4.1. Uncertainty analysis of costing
Based on previous analyses [10], the uncertainty of the mixture properties
can be estimated. The latent heat of fusion is generally overestimated by 6%, but
can be overestimated by as much as 10%. In few cases is it under-estimated.
Thus, the uncertainty is assumed to be a maximum of +-10%. The density
is predicted to within 3.5% in almost every case. The uncertianty of thermal
conductivity is more difficult to estimate. Only a few studies have measured the
thermal conductivity of salt mixtures, and they generally report that mixture
thermal conductivity follows a simple mixing model. However, the theoretical
values for single salts have stated uncertainty of +-20%. This is, however, much
larger than the analysis suggests. For the costing uncertainty, we assume that
the thermal conductivity is overestimated by 10%.
To understand how the uncertainty of the material properties impacts the
uncertainty of the costing analysis, the worst case properties were used to per-
form an optimization. The worst case material property is that all three proper-
ties are overestimated. A less dense, less energetic, and less thermally conductive
salt will perform worse when compared to a salt of the same cost. The costing
of the salts is kept fixed when estimating the uncertainty.
For all salts tested here, the average increase in cost associated with the
worst case properties is 13.2% for tube and shell geometries and 10.5% for tube
and fin geometries. The standard deviation of these increases is 0.5% and 0.2%
122
respectively. Based on these results, a 13% uncertainty should applied to the
cost estimates.
4.2. Tube and shell cost analysis
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
0 200 400 600 800 1000 1200 1400 1600 1800
Cost
per
MW
h ($
/MW
h)
Melting Point
Figure 2: Cost per MWh for tube and shell geometries. Binary mixtures are circles, ternaryare triangles, and quaternary mixtures are squares. Solid points are with a two mm corrosionallowance, and empty points are with one mm corrosion allowance.
Fig. 2 shows the cost of the PCM and containment vessel as a function of
the melting point. Each point represents a different mixture. Few mixtures
fall below the Sunshot target of $15,000 per MWh, denoted as the dotted line.
Most can be disregarded immediately. In addition, the few mixtures which fall
123
below the target only do so with a corrosion allowance of one mm. For a twenty
year lifetime, this implies a yearly corrosion rate of 50 µm. Fluoride mixtures
have been demonstrated to be effective for long lifespans, but they must be
of extremely high purity to do so. Slightly impure fluoride mixtures contain
moisture which starts an HF accelerated corrosion chain reaction. This chain
reaction leads to extremely high corrosion rates and the release of a hazardous
material. The best mixtures for tube and shell geometries have been identified
as group “a” in Table 3. The costing estimation leaves approximately $0.51 per
kg for the required purification of the flouride salts. If this is insufficeint, the
tube and shell geometry will not meet the cost targets. Table 5 outlines how
the corrosion allowance affects the cost estimate.
Table 5: The average least system cost is presented for a variety of conditions, based on themixtures outlined as groups a, b, and c.
Average Cost of PCM ($/MWh)
group Co=0 Co=1 Co=2 Co=3
a 4661 13672 22876 31820
b 3934 6357 8568 10671
c 3520 6648 9461 12109
group “a” uses tube and shell geometry,
while b and c are finned tubes with metal fins
of thickness 0.1 mm. The corrosion allowance (Co)
(in mm) is varied to determine sensitivity.
Average Cost of PCM ($/MWh)
group Co=0 Co=1 Co=2 Co=3
b 4753 10731 15764 20309
c 4555 12260 18687 24444
124
In this comparison, ceramic fins of thickness
0.1 mm are utilized.
Average Cost of PCM ($/MWh)
group Co=0 Co=1 Co=2 Co=3
b 5511 8015 10106 12111
c 5480 8684 11315 13817
In this comparison, metal fins of thickness
1 mm are utilized.
Average Cost of PCM ($/MWh)
group wf = 0.1 0.5 1 2
b 6357 7117 8015 9623
c 6648 7582 8684 10642
In this comparison, the corrosion allowance
is fixed at 1 mm and metal fin thickness (wf )
is varied from 0.1 mm to 2 mm.
Average Cost of PCM ($/MWh)
group wf = 0.1 0.5 1 2
b 10731 11548 12734 14998
c 12260 13261 14726 17480
In this comparison, the corrosion allowance
is fixed at 1 mm and ceramic fin thickness (wf )
is varied from 0.1 mm to 2 mm.
125
4.3. Finned tube cost analysis
0
10000
20000
30000
40000
50000
60000
0 200 400 600 800 1000 1200 1400 1600 1800
Cost
per
MW
h ($
/MW
h)
Melting Point (K)
Figure 3: Cost per MWh for metal finned tube geometries. Binary mixtures are diamonds,ternary are triangles, and quaternary mixtures are squares. The corrosion allowance is onemm.
Fig. 3 shows the cost of the tested mixtures, as a function of the melting
point, in the finned tube configuration. The parameters in this figure include
a one mm corrosion allowance with 0.1 mm metal fins, which reflects nearly
the most optimal conditions studied here. A one mm corrosion allowance was
chosen as standards for pressure vessels often list a minimum around this value
126
[36, 37, 38]. Metals fins thinner than 0.1 mm would be unlikely to survive
the conditions, as despite being non-structural, the choice of fin material is not
dependent on its strength, but its thermal conductivity and corrosion resistance.
The addition of metal fins greatly reduces the final cost. However, this anal-
ysis does not consider the cost of manufacturing the fins, merely the additional
cost of the material. This additional manufacturing cost may be offset by the
smaller number of tubes and welds required. The marginal cost of manufacture
is beyond the scope of the current study, but should be included when final
determinations are made.
As fluorides are highly corrosive unless extremely pure, two sets of mixtures
have been identified as the prime candidates for finned tube configurations.
Group “b” contains fluoride mixtures, while group “c” does not. The cost
estimation with various conditionals is presented in Table 5. On average, across
the conditions tested here, group “c” is 9% more expensive than group “b.”
Thus, if corrosion due to fluorides proves to be insurmountable, options exist
which are marginally more expensive but still within target. In addition, if tube
materials and tank conditions are found which minimize corrosion, group “c” is
often the best group identified.
Fin thickness plays a significant role in the costing of the optimal geometries.
The effect of fin thickness is studied in depth. For metal fins, a fin thickness of
one mm versus 0.1 mm increases the cost by 20 %. This cost increase can be
attributed to smaller fins more effectivenly penetrating the salt layer. Of note is
that finned tube configurations with 1 mm thick fins are still more cost effective
than tube and shell configurations.
Ceramic fins were investigated due to the high corrosion resistance of sin-
tered alumina. The properties chosen were averages of available alumina ceram-
ics. Despite the significantly lower thermal conductivity of the ceramic versus
127
the metal, ceramic fins of 0.1 mm thickness present large cost savings when
compared to tube and shell configurations. Ceramic fins are more sensitive to
fin thickness, with one mm thick fins presenting little cost benefit over tube and
shell configurations.
5. Conclusion
The properties of several hundred salt mixtures have been estimated. Using
this data, the salt and containment vessel cost can be estimated. As the cost
of these two components forms the majority of the cost of the system, this is
an accurate way to compare the cost effectiveness of various mixtures as PCMs
for latent heat thermal energy storage systems. The comparison yields several
interesting insights:
� For tube and shell geometries, almost exclusively Fluoride salts meet the
threshold for cost effectiveness. To meet the system cost target, a further
0.51 $/kg is left for purification, which is neccessary for the corrosion to
be kept to controllable levels.
� Finned tube geometries dramatically increase the scope of possible salt
mixtures meeting the cost targets. In addition, this geometry is highly
resilient to more difficult constraints, such as lower thermal conductivity
fins, or increased tube corrosion targets. This geometry can be success-
fully used with a combination of higher corrosion rate, and relatively low
thermal conductivity fins.
� The least cost cascaded PCM candidates have been determined for three
conditions:
– Tube and Shell Geometries
– Finned Tube Geometries
128
– Finned Tube Geometries without Fluoride PCMs
� Each of these sets of PCMs is estimated to cost less than the Sunshot
targets.
� The binary mixture of NaCl + CalCl2 is highly cost effective across a large
number of use cases and conditionals, and should be a priority for future
investigation.
Future work must accurately characterize the melting points of the predicted
ternary and quaternary mixtures, as well as verifying the other properties. With
this information, an accurate assesment and more detailed system costing can
be accomplished. However, this work has narrowed down the search space to a
small set of mixtures.
6. Acknowledgments
We would like to thank ASTRI and ARENA for funding the research con-
tained herein.
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Chapter 7
Conclusion
The research questions which underpin this work are:
1. First, due to conflicting data in the literature and a large num-ber of untested mixtures, a theoretical basis must be deter-mined for accurately assessing the thermal properties of thePCMs.
2. Second, a systematic method to determine the containmentvessel cost must be developed, as a large part of the system costdepends on the properties of the PCM influencing the designof the containment vessel.
3. With these two issues resolved, the combination space of binary,ternary, and quaternary salt mixtures should be catalogued andthe total system cost calculated. From here, the most optimalPCMs can be selected for a cascaded system with specific prop-erties.
A comprehensive review (Chapter 2) has identified theoreticalmeans to predict mixture properties. These theories have been ex-perimentally verified in Chapter 3, where data suggests that theestimations are within experimental uncertainty. Furthermore, thepredicted ternary mixture of NaCl + Na2SO4 + Na2CO3 is shown tohave a significantly higher latent heat and a more homogeneous DSCcurve. This, coupled with the postiive results from other ternarymixtures, suggests that the predicted concentrations are valuablefor further study. This answers research question number one.
To answer question two, a new tool is developed; an analyticmethod to determine the least cost of heat exchanger. This toolis powerful as its analytic nature makes it computationally cheap,in contrast to other methods which are highly computationally in-tensive, making optimization difficult. However, this analytic tool
135
relies on assumptions, such as no convection, and has geometric con-straints, which makes its use limited. Future work can expand thegeometries studied and can evaluate the effect of natural convection.This would make the analytic results extremely versatile and a valu-able result across many fields. Despite the limitations however, thistool provides a solution to a problem which has dogged the field fora number of years.
Finally, this thesis succeeds in its goal of identifying the primarycandidates for solar thermal latent heat energy storage media. Thisis a success with regards to research question number three. In gen-eral, the cost target is met under a variety of conditions, and theselection of the mixtures is resilient to changes in cost estimates orhigher than anticipated corrosion. In particular, the set of fluoridemixtures identified in Chapter 6 as group “b” can possibly undercutthe cost target by 50% under the optimal conditions. Under lessthan optimal conditions, the use of ceramic fins is shown to reducethe system cost when compared to a tube and shell configuration. Afinal group of PCMs is suggested in the case that suitable fin mate-rial cannot be found and the tube and shell configuration adopted.This group still meets the cost targets set by SunShot.
Future work in this space is important and the following pointsillustrate some of the avenues which can be explored.
Firstly, Chapter 6 does not present a fully comprehensive analysisof the combination space. This is due to a lack of single salt data anda lack of binary eutectic points. If further eutectics are identified,the tools exist to analyze thier properties. Furthermore, the analysisis limited to the cycle identified. Again, the tools exist to effectivelyestimate the system cost for power cycles of different temperatures,or systems where a different HTF is required.
One potential PCM candidate, NaCl + CaCl2, is of interest as itappears in all three groups identified in Chapter 6. This PCM is in-teresting as it fits in a temperature range relatively unpopulated bycheap and effective mixtures (around 775 K), and is so far relativelyunstudied. Future work for any high temperature storage must con-sider this mixture because of its cost-effectiveness compared to othermixtures at the same temperature.
In addition, with further information on corrosion the appropri-ate group from Table 6.3 can be selected for futher study. Theprecise properties of the mixtures must be verified for an accurate
136
analysis of costing for industrial application.
137
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140
Appendix
Table 1: Predicted Properties of Salt Mixtures
Mixture Molar Cost Melting Density Thermal Latent Heat
Concentration Point Conductivity
($/kg) (K) (g/cm3) (W/mK) (J/g)
KNO3 + LiNO3 0.576/0.424 0.4 398 1.98 0.51 169
Ca(NO3)2 + KNO3 0.345/0.655 0.366 418 1.45 0.43 104
Ca(NO3)2 + NaNO3 0.3/0.7 0.366 433 1.43 0.52 168
LiNO3 + NaNO3 0.537/0.463 0.423 468 1.91 0.57 265
LiBr + LiNO3 0.25/0.75 1.306 493 2.06 0.57 326
KNO3 + NaNO3 0.51/0.49 0.36 496 1.96 0.48 123
LiCl + LiNO3 0.126/0.874 0.5 516 1.78 0.62 376
LiF + LiNO3 0.047/0.953 0.657 523 1.79 0.64 384
LiNO3 + Li2SO4 0.975/0.025 0.518 525 1.8 0.59 363
Li2CO3 + LiNO3 0.993/0.007 2.188 527 2 1.02 508
NaBr + NaNO3 0.1/0.9 0.498 554 2 0.52 184
NaNO3 + Na2SO4 0.948/0.052 0.343 559 1.95 0.53 176
Ba(NO3)2 + KNO3 0.13/0.87 0.406 563 1.57 0.39 109
Ba(NO3)2 + NaNO3 0.065/0.935 0.39 567 1.7 0.51 181
NaCl + NaNO3 0.066/0.934 0.35 571 1.91 0.53 188
NaF + NaNO3 0.035/0.965 0.365 576 1.91 0.55 186
KF + KNO3 0.086/0.914 0.354 580 1.91 0.44 104
Na2CO3 + NaNO3 0.02/0.98 0.357 580 1.91 0.53 177
KCl + KNO3 0.095/0.905 0.367 581 1.89 0.42 102
KBr + KNO3 0.07/0.93 0.489 594 1.92 0.4 96
KNO3 + K2SO4 0.985/0.015 0.365 595 1.89 0.41 90
K2CO3 + KNO3 0.37/0.63 0.492 599 1.98 0.45 130
KBr + LiBr 0.4/0.6 2.981 602 2.57 0.43 209
KCl + LiCl 0.408/0.592 0.5 626 1.7 0.66 381
KCl + MgCl2 0.698/0.302 0.386 696 1.74 0.39 389
LiBr + LiF 0.76/0.24 3.987 721 2.54 0.75 275
MgCl2 + NaCl 0.431/0.569 0.144 732 1.75 0.41 467
CaCl2 + LiCl 0.35/0.65 0.359 748 1.94 0.66 321
LiCl + Li2SO4 0.626/0.374 0.804 750 1.92 0.73 214
K2CO3 + Li2CO3 0.38/0.62 1.409 758 2 0.8 345
141
LiCl + SrCl2 0.643/0.357 0.616 765 2.49 0.64 205
KF + LiF 0.49/0.51 3.309 765 2.1 1.13 622
LiCl + LiF 0.7/0.3 2.473 773 1.64 1 546
Li2CO3 + Na2CO3 0.52/0.48 1.072 775 2.04 0.83 313
CaCl2 + NaCl 0.521/0.479 0.207 777 2.04 0.51 328
LiBr + NaBr 0.75/0.25 2.961 782 2.55 0.45 218
LiBr + LiCl 0.625/0.375 2.758 783 2.33 0.58 251
BaCl2 + LiCl 0.249/0.751 0.636 787 2.74 0.66 205
LiCl + Li2CO3 0.688/0.312 1.25 790 1.7 0.82 457
K2SO4 + Li2SO4 0.189/0.811 0.906 795 2.13 0.61 119
MgCl2 + SrCl2 0.462/0.538 0.505 808 2.52 0.32 222
LiF + Li2SO4 0.389/0.611 2.175 816 2.11 1.02 209
LiCl + NaCl 0.72/0.28 0.36 827 1.58 0.68 439
BaCl2 + MgCl2 0.429/0.571 0.516 832 2.8 0.28 219
NaCl + SrCl2 0.484/0.516 0.525 838 2.59 0.48 200
Li2CO3 + Li2SO4 0.4/0.6 1.371 838 2.05 0.79 215
LiCl + MgCl2 0.7/0.3 0.343 844 1.62 0.58 435
KBr + KF 0.595/0.405 1.926 854 2.21 0.43 288
KCl + SrCl2 0.573/0.427 0.605 857 2.41 0.43 200
Li2SO4 + Na2SO4 0.619/0.381 0.601 866 2.16 0.6 120
CaCl2 + KCl 0.25/0.75 0.42 873 1.8 0.44 320
KCl + KF 0.55/0.45 0.383 879 1.8 0.52 413
BaCl2 + CaCl2 0.359/0.641 0.495 881 2.8 0.43 162
Li2CO3 + LiF 0.52/0.48 4.109 886 1.88 1.23 639
MgCl2 + MgF2 0.78/0.22 0.308 887 1.87 0.32 529
CaCl2 + MgCl2 0.474/0.526 0.22 890 1.93 0.33 351
NaCl + Na2SO4 0.533/0.467 0.1 899 2.02 0.5 266
NaBr + NaF 0.72/0.28 1.653 900 2.4 0.53 329
KCl + K2CO3 0.624/0.376 0.616 904 1.82 0.45 273
NaCl + Na2CO3 0.553/0.447 0.171 905 1.9 0.58 294
CaCl2 + CaCO3 0.3/0.7 0.151 908 2.05 0.47 120
CaCl2 + CaF2 0.813/0.187 0.293 910 2.24 0.49 271
KBr + NaBr 0.513/0.487 2.184 916 2.3 0.31 233
BaCl2 + KCl 0.445/0.555 0.652 922 2.83 0.4 162
LiF + NaF 0.61/0.39 5.285 922 2.03 1.31 890
BaCl2 + NaCl 0.399/0.601 0.536 924 2.87 0.46 197
KCl + NaCl 0.494/0.506 0.322 930 1.61 0.47 378
CaCl2 + SrCl2 0.584/0.416 0.467 935 2.49 0.44 177
LiF + NaCl 0.42/0.58 2.507 950 1.69 0.93 618
NaCl + NaF 0.67/0.33 0.265 952 1.76 0.67 564
CaCl2 + MgF2 0.872/0.128 0.316 956 2.17 0.49 305
142
K2CO3 + KF 0.4/0.6 0.519 958 2 0.56 320
KCl + K2SO4 0.74/0.26 0.568 963 1.78 0.39 289
Na2CO3 + NaF 0.603/0.397 0.325 967 2.06 0.78 295
K2CO3 + Na2CO3 0.415/0.585 0.46 982 2.01 0.58 183
KCl + LiF 0.78/0.22 1.349 985 1.59 0.65 414
KBr + KCl 0.65/0.35 1.996 990 1.99 0.32 250
KF + NaF 0.603/0.397 0.371 991 2.04 0.75 600
LiF + MgF2 0.671/0.329 5.133 1003 2.32 1.23 886
NaBr + NaCl 0.75/0.25 1.529 1013 2.23 0.38 291
SrCl2 + SrF2 0.89/0.11 0.71 1014 2.91 0.4 114
KF + SrF2 0.78/0.225 0.546 1027 2.76 0.56 399
BaSO4 + Li2SO4 0.093/0.908 0.893 1029 1.68 0.62 100
NaF + Na2SO4 0.267/0.733 0.161 1030 2.13 0.6 226
CaF2 + LiF 0.208/0.792 5.806 1040 2.26 1.29 750
LiF + SrF2 0.815/0.185 5.34 1044 2.97 1.28 614
CaF2 + KF 0.148/0.852 0.257 1055 2.11 0.59 483
KF + MgF2 0.873/0.127 0.308 1056 2.06 0.6 565
KF + K2SO4 0.845/0.155 0.36 1065 1.97 0.54 402
CaF2 + NaF 0.314/0.686 0.624 1087 2.39 0.84 604
CaCO3 + CaF2 0.735/0.365 0.212 1098 2.18 0.56 148
Na2CO3 + Na2SO4 0.64/0.36 0.168 1099 2.03 0.56 165
K2SO4 + Na2SO4 0.258/0.742 0.264 1107 2.06 0.41 179
MgF2 + NaF 0.224/0.776 0.811 1108 2.24 0.89 804
BaCl2 + SrCl2 0.312/0.688 0.69 1122 2.93 0.37 93
NaF + SrF2 0.739/0.261 0.92 1139 3.01 0.81 502
MgF2 + SrF2 0.5/0.5 1.067 1151 3.49 0.57 462
CaSO4 + Na2SO4 0.39/0.61 0.366 1163 1.28 0.42 179
K2CO3 + K2SO4 0.95/0.05 0.716 1178 1.9 0.46 203
CaF2 + MgF2 0.46/0.54 0.742 1247 2.64 0.65 633
CaF2 + SrF2 0.623/0.377 0.796 1640 3.04 0.51 305
LiNO3 + RbNO3 0.31/0.69 417 2.54 0.44 75
CsNO3 + LiNO3 0.42/0.58 430 2.81 0.46 144
NaNO3 + RbNO3 0.43/0.57 433 2.47 0.45 69
CsNO3 + NaNO3 0.48/0.52 451 2.81 0.42 99
Ca(NO3)2 + CsNO3 0.575/0.425 461 2.17 0.36 93
CsNO3 + KNO3 0.4/0.6 489 2.62 0.36 78
CsI + LiI 0.338/0.662 490 3.5 0.3 48
LiI + RbI 0.625/0.375 525 3.33 0.3 51
LiBr + RbBr 0.59/0.41 535 2.97 0.43 168
CsNO3 + RbNO3 0.2/0.8 559 2.63 0.32 41
KI + LiI 0.366/0.634 559 3.09 0.33 61
143
KNO3 + RbNO3 0.31/0.69 560 2.38 0.36 43
CsBr + LiBr 0.377/0.623 560 3.27 0.41 148
NaI + NaNO3 0.14/0.86 563 2.13 0.5 173
CsF + CsNO3 0.314/0.686 574 3.16 0.31 84
Ba(NO3)2 + CsNO3 0.22/0.78 578 2.51 0.26 91
LiCl + RbCl 0.578/0.422 586 2.29 0.61 268
CsCl + LiCl 0.42/0.58 599 2.74 0.59 197
CsCl + CsNO3 0.301/0.699 624 2.93 0.27 80
LiCl + LiI 0.36/0.64 644 2.96 0.51 63
CsBr + CsNO3 0.0789/0.9211 679 2.86 0.25 73
LiF + LiI 0.166/0.834 686 3.12 0.56 39
LiBr + LiI 0.364/0.636 691 3.01 0.4 55
CsI + NaI 0.515/0.485 701 3.25 0.2 118
CsF + CsI 0.466/0.534 704 3.61 0.25 112
CsBr + CsF 0.515/0.485 709 3.68 0.31 124
CsCl + CsF 0.504/0.496 712 3.47 0.34 131
MgCl2 + RbCl 0.261/0.739 720 2.33 0.31 253
LiI + NaI 0.79/0.21 723 3.08 0.33 36
CsBr + NaBr 0.587/0.413 738 3.22 0.29 148
LiF + RbF 0.47/0.53 740 2.79 1.03 391
CsF + LiF 0.607/0.393 753 3.74 0.85 232
CsCl + NaCl 0.65/0.35 759 2.77 0.39 178
CsCl + MgCl2 0.807/0.193 764 2.81 0.28 161
RbF + RbI 0.41/0.59 766 3.04 0.31 141
NaBr + RbBr 0.447/0.553 770 2.79 0.31 179
CsCl + CsI 0.52/0.48 775 3.17 0.22 106
NaI + RbI 0.497/0.503 778 2.99 0.21 126
RbCl + SrCl2 0.736/0.264 783 2.59 0.36 167
RbBr + RbF 0.557/0.443 791 2.9 0.38 177
CsI + KI 0.614/0.386 809 3.08 0.17 110
BaCl2 + CsCl 0.181/0.819 812 3.02 0.3 111
KF + KI 0.328/0.672 816 2.51 0.37 198
NaCl + RbCl 0.442/0.558 823 2.2 0.42 276
RbCl + RbF 0.536/0.464 823 2.6 0.41 219
RbCl + RbI 0.415/0.585 833 2.82 0.23 131
CsCl + SrCl2 0.856/0.144 833 2.89 0.3 119
CsBr + KBr 0.6/0.4 844 3.03 0.25 139
NaCl + NaI 0.4/0.6 846 2.59 0.37 225
CaCl2 + RbCl 0.187/0.813 849 2.34 0.35 207
CsI + RbI 0.56/0.44 852 3.13 0.14 99
KI + NaI 0.412/0.588 853 2.7 0.24 152
144
CsBr + CsI 0.528/0.472 860 3.26 0.18 103
NaF + NaI 0.185/0.815 869 2.77 0.39 196
KCl + KI 0.399/0.601 872 2.32 0.3 193
CaCl2 + CsCl 0.091/0.909 872 2.8 0.29 129
BaCl2 + RbCl 0.186/0.814 876 2.65 0.33 163
CsBr + CsCl 0.585/0.415 882 3.11 0.24 115
CsF + NaF 0.76/0.24 883 3.64 0.5 195
CsBr + RbBr 0.626/0.374 888 3.11 0.23 121
CsCl + KCl 0.625/0.375 889 2.57 0.33 170
RbBr + RbI 0.485/0.515 890 2.87 0.2 120
Cs2SO4 + Li2SO4 0.1/0.9 893 2.23 0.62 89
CsF + KF 0.57/0.43 898 3.37 0.46 225
CsCl + RbCl 0.8257/0.1743 905 2.74 0.28 131
KI + RbI 0.26/0.74 911 2.82 0.17 113
NaBr + NaI 0.313/0.687 918 2.68 0.27 181
KBr + KI 0.33/0.67 936 2.39 0.24 163
NaF + RbF 0.33/0.67 943 2.66 0.65 338
KBr + RbBr 0.25/0.75 962 2.6 0.25 155
RbBr + RbCl 0.6/0.4 967 2.56 0.27 159
CsF + RbF 0.98/0.02 973 3.64 0.32 144
KCl + RbCl 0.02/0.98 993 2.24 0.3 199
KF + RbF 0.12/0.88 1059 2.59 0.46 266
K2SO4 + KNO3 + K2CO3 0.542/0.288/0.169 0.586 63 2.39 0.45 179
KCl + K2SO4 + KNO3 0.671/0.236/0.093 0.542 169 2.2 0.52 264
KF + KCl + KNO3 0.331/0.404/0.265 0.373 371 2.08 0.59 273
KCl + KBr + KNO3 0.312/0.58/0.108 1.765 382 2.4 0.42 227
LiNO3 + NaNO3 + KNO3 0.283/0.244/0.474 0.386 389 2 0.52 161
LiF + Li2SO4 + LiNO3 0.281/0.442/0.277 1.717 400 2.16 0.98 254
NaCl + Na2SO4 + NaNO3 0.408/0.358/0.234 0.166 404 2.21 0.58 244
KF + KBr + KNO3 0.298/0.438/0.264 1.381 406 2.37 0.5 218
NaNO3 + KNO3 + CaNO3 0.576/0.278/0.146 0.363 463 1.73 0.5 141
LiCl + Li2SO4 + LiNO3 0.109/0.135/0.756 0.596 491 1.88 0.63 320
NaNO3 + KNO3 + Ba(NO3)2 0.467/0.487/0.046 0.379 492 1.82 0.47 129
LiCl + LiBr + LiNO3 0.013/0.247/0.74 1.301 492 2.06 0.58 327
LiF + LiBr + LiNO3 0.002/0.249/0.748 1.313 493 2.06 0.58 327
LiCl + RbCl + CaCl2 0.01/0.007/0.983 1.046 499 2.32 0.51 253
LiF + LiCl + LiNO3 0.031/0.122/0.847 0.608 512 1.78 0.65 384
KF + K2SO4 + KNO3 0.08/0.069/0.851 0.38 513 1.98 0.44 113
LiCl + LiNO3 + Li2CO3 0.121/0.839/0.04 0.57 516 1.79 0.63 382
Li2SO4 + LiNO3 + Li2CO3 0.024/0.007/0.97 2.147 522 2.01 1.01 493
LiF + LiNO3 + Li2CO3 0.027/0.007/0.966 2.263 523 2.01 1.04 513
145
Na2SO4 + NaNO3 + Na2CO3 0.292/0.189/0.519 0.202 523 2.24 0.62 167
NaCl + NaBr + NaNO3 0.13/0.087/0.783 0.468 543 2 0.53 207
NaF + NaBr + NaNO3 0.005/0.099/0.895 0.498 554 2 0.52 186
NaF + Na2SO4 + NaNO3 0.03/0.05/0.92 0.347 558 1.96 0.55 183
NaF + NaCl + NaNO3 0.021/0.065/0.914 0.354 570 1.91 0.55 193
NaCl + NaNO3 + Na2CO3 0.065/0.923/0.012 0.349 572 1.91 0.54 188
NaF + NaNO3 + Na2CO3 0.035/0.945/0.019 0.362 578 1.92 0.55 185
KF + KNO3 + K2CO3 0.064/0.682/0.253 0.449 580 1.97 0.46 129
KCl + KNO3 + K2CO3 0.064/0.613/0.323 0.484 581 1.98 0.45 135
LiCl + KCl + BaCl2 0.491/0.338/0.171 0.596 604 2.54 0.62 248
LiBr + NaBr + KBr 0.545/0.091/0.364 2.87 607 2.58 0.43 213
LiCl + KCl + CaCl2 0.521/0.359/0.12 0.448 611 1.83 0.64 354
LiCl + KCl + SrCl2 0.482/0.332/0.185 0.568 612 2.23 0.62 272
LiCl + NaCl + KCl 0.498/0.159/0.343 0.434 635 1.71 0.65 390
LiF + LiCl + Li2CO3 0.356/0.259/0.385 3.293 638 1.9 1.17 589
LiCl + Li2SO4 + Li2CO3 0.326/0.405/0.27 1.217 659 2.03 0.81 251
LiCl + SrCl2 + BaCl2 0.358/0.442/0.2 0.667 660 3.02 0.55 132
KCl + MgCl2 + BaCl2 0.457/0.198/0.345 0.578 672 2.79 0.39 209
Li2CO3 + Na2CO3 + K2CO3 0.404/0.349/0.247 0.974 678 2.09 0.77 279
KCl + MgCl2 + SrCl2 0.525/0.227/0.248 0.499 686 2.24 0.4 276
NaCl + KCl + MgCl2 0.06/0.656/0.284 0.373 687 1.75 0.4 389
LiCl + CaCl2 + SrCl2 0.46/0.315/0.225 0.474 687 2.39 0.61 229
LiCl + NaCl + BaCl2 0.297/0.423/0.281 0.531 690 2.83 0.58 225
NaCl + MgCl2 + CaCl2 0.568/0.227/0.205 0.169 690 1.9 0.47 407
LiCl + MgCl2 + CaCl2 0.239/0.4/0.361 0.253 692 1.96 0.44 358
LiF + LiCl + Li2SO4 0.204/0.476/0.32 1.618 694 1.97 0.92 278
NaCl + MgCl2 + SrCl2 0.354/0.268/0.378 0.442 703 2.45 0.42 261
NaCl + MgCl2 + BaCl2 0.422/0.32/0.257 0.428 705 2.64 0.41 274
LiCl + CaCl2 + BaCl2 0.554/0.298/0.147 0.486 711 2.51 0.63 235
KCl + MgCl2 + CaCl2 0.549/0.237/0.214 0.352 714 1.87 0.41 352
LiF + LiCl + LiBr 0.154/0.359/0.487 3.124 719 2.31 0.76 310
LiCl + NaCl + CaCl2 0.391/0.399/0.21 0.264 722 1.88 0.63 380
NaCl + KCl + CaCl2 0.326/0.319/0.355 0.292 729 1.96 0.5 324
LiCl + NaCl + SrCl2 0.464/0.18/0.356 0.563 729 2.5 0.61 220
LiCl + KCl + MgCl2 0.684/0.221/0.096 0.447 731 1.64 0.64 403
LiCl + MgCl2 + SrCl2 0.445/0.191/0.364 0.543 738 2.45 0.53 233
NaCl + CaCl2 + BaCl2 0.391/0.425/0.184 0.389 742 2.57 0.49 239
LiF + NaF + KF 0.447/0.123/0.43 2.989 746 2.13 1.12 645
MgCl2 + CaCl2 + BaCl2 0.256/0.231/0.514 0.561 754 3.02 0.36 164
CaF2 + CaCl2 + CaCO3 0.258/0.223/0.519 0.224 757 2.27 0.52 175
MgCl2 + CaCl2 + SrCl2 0.234/0.211/0.555 0.518 759 2.62 0.39 188
146
LiF + KF + SrF2 0.493/0.474/0.034 3.099 763 2.3 1.11 587
LiF + KF + CaF2 0.496/0.476/0.028 3.167 764 2.14 1.12 612
LiF + KF + MgF2 0.505/0.485/0.01 3.273 765 2.11 1.13 626
LiCl + MgCl2 + BaCl2 0.564/0.249/0.187 0.511 779 2.47 0.55 273
NaCl + CaCl2 + SrCl2 0.262/0.285/0.454 0.489 779 2.58 0.47 192
KCl + CaCl2 + SrCl2 0.525/0.175/0.301 0.533 793 2.33 0.44 223
MgCl2 + SrCl2 + BaCl2 0.435/0.507/0.058 0.524 798 2.61 0.33 209
NaCl + KCl + SrCl2 0.303/0.296/0.401 0.537 801 2.45 0.47 214
LiCl + NaCl + MgCl2 0.518/0.26/0.222 0.28 802 1.65 0.58 447
KF + MgF2 + CaF2 0.409/0.319/0.272 0.543 803 2.58 0.68 591
LiF + CaF2 + SrF2 0.236/0.476/0.288 1.507 813 3.39 0.8 361
LiF + Li2SO4 + Li2CO3 0.274/0.43/0.296 2.182 818 2.05 1.01 295
NaCl + SrCl2 + BaCl2 0.451/0.481/0.068 0.549 829 2.7 0.47 185
KCl + SrCl2 + BaCl2 0.455/0.375/0.17 0.64 833 2.72 0.42 161
KCl + CaCl2 + BaCl2 0.484/0.161/0.355 0.593 837 2.76 0.42 179
KF + KCl + KBr 0.402/0.209/0.389 1.504 845 2.1 0.46 320
NaF + NaCl + Na2CO3 0.214/0.435/0.351 0.241 862 1.96 0.68 356
CaCl2 + SrCl2 + BaCl2 0.439/0.386/0.175 0.547 863 2.78 0.43 146
KF + KCl + K2SO4 0.43/0.525/0.045 0.412 873 1.83 0.51 391
KF + KCl + K2CO3 0.436/0.533/0.031 0.404 873 1.82 0.52 400
KCl + K2SO4 + K2CO3 0.525/0.158/0.317 0.625 877 1.9 0.43 258
NaCl + Na2SO4 + Na2CO3 0.482/0.187/0.332 0.147 879 1.99 0.56 264
NaCl + KCl + BaCl2 0.401/0.332/0.266 0.527 881 2.59 0.45 223
KF + K2SO4 + K2CO3 0.429/0.286/0.286 0.576 886 2.05 0.5 273
LiF + NaF + MgF2 0.509/0.381/0.11 4.18 887 2.2 1.25 856
Li2SO4 + Na2SO4 + K2SO4 0.754/0.182/0.063 0.773 891 2.12 0.61 113
NaF + NaCl + Na2SO4 0.031/0.516/0.452 0.109 893 2.03 0.52 274
LiF + NaF + CaF2 0.532/0.34/0.127 4.037 897 2.25 1.23 759
NaF + CaF2 + SrF2 0.329/0.418/0.253 0.784 900 3.24 0.7 391
LiF + MgF2 + CaF2 0.427/0.209/0.364 2.624 903 2.62 1.03 622
LiF + NaF + SrF2 0.557/0.356/0.086 4.159 903 2.6 1.25 713
LiF + MgF2 + SrF2 0.504/0.247/0.249 3.024 928 3.26 1.06 553
NaF + KF + SrF2 0.339/0.514/0.147 0.586 944 2.66 0.72 489
NaF + KF + CaF2 0.352/0.535/0.112 0.392 949 2.19 0.75 565
NaF + KF + MgF2 0.387/0.587/0.026 0.391 979 2.07 0.76 608
NaF + Na2SO4 + Na2CO3 0.291/0.255/0.453 0.24 988 2.09 0.69 245
NaF + MgF2 + CaF2 0.569/0.164/0.267 0.693 996 2.49 0.84 645
KF + MgF2 + SrF2 0.756/0.11/0.134 0.506 1004 2.57 0.59 481
NaF + NaCl + NaBr 0.012/0.247/0.741 1.525 1008 2.23 0.38 294
NaF + MgF2 + SrF2 0.623/0.18/0.197 0.926 1042 2.96 0.82 578
KF + CaF2 + SrF2 0.833/0.145/0.022 0.295 1046 2.2 0.59 472
147
MgF2 + CaF2 + SrF2 0.351/0.299/0.351 0.919 1078 3.34 0.59 435
LiI + RbI + CsI 0.369/0.278/0.353 318 3.69 0.26 75
LiNO3 + KNO3 + CsNO3 0.21/0.285/0.506 344 2.91 0.4 96
LiI + KI + RbI 0.293/0.184/0.524 347 3.45 0.27 88
LiNO3 + KNO3 + RbNO3 0.134/0.569/0.297 351 2.33 0.45 81
LiI + NaI + KI 0.202/0.054/0.745 365 3.1 0.31 121
LiBr + RbBr + CsBr 0.35/0.243/0.407 368 3.55 0.38 136
LiI + KI + CsI 0.443/0.215/0.342 379 3.52 0.29 75
NaNO3 + KNO3 + CsNO3 0.206/0.214/0.58 390 2.92 0.37 83
LiNO3 + NaNO3 + CsNO3 0.311/0.268/0.42 394 2.83 0.46 125
LiNO3 + RbNO3 + CsNO3 0.276/0.615/0.109 403 2.66 0.42 74
NaNO3 + CsNO3 + CaNO3 0.562/0.186/0.252 407 2.14 0.47 128
LiNO3 + NaNO3 + RbNO3 0.245/0.211/0.544 413 2.47 0.46 89
KCl + RbCl + BaCl2 0.002/0.118/0.879 428 3.65 0.41 85
NaNO3 + KNO3 + RbNO3 0.305/0.291/0.404 428 2.36 0.44 73
LiBr + KBr + RbBr 0.415/0.146/0.439 434 3.04 0.41 169
NaNO3 + CsNO3 + Ba(NO3)2 0.454/0.419/0.128 446 2.47 0.4 112
KNO3 + CsNO3 + CaNO3 0.648/0.15/0.203 452 2.07 0.41 90
NaBr + NaI + NaNO3 0.261/0.573/0.166 454 2.95 0.37 181
CsCl + SrCl2 + BaCl2 0.219/0.538/0.244 458 3.33 0.42 99
KNO3 + CsNO3 + Ba(NO3)2 0.567/0.378/0.055 476 2.5 0.36 84
RbCl + SrCl2 + BaCl2 0.314/0.472/0.214 477 3.18 0.43 118
LiCl + KCl + RbCl 0.424/0.012/0.564 479 2.48 0.57 244
LiBr + KBr + CsBr 0.478/0.209/0.313 489 3.21 0.4 159
NaNO3 + RbNO3 + CsNO3 0.653/0.278/0.069 489 2.37 0.47 102
LiI + NaI + CsI 0.648/0.022/0.331 493 3.49 0.3 50
LiCl + RbCl + CsCl 0.456/0.095/0.449 512 2.88 0.55 183
LiI + NaI + RbI 0.557/0.11/0.334 539 3.29 0.3 62
LiBr + NaBr + RbBr 0.544/0.077/0.378 545 2.95 0.42 174
KNO3 + RbNO3 + CsNO3 0.231/0.615/0.154 546 2.54 0.35 48
LiCl + KCl + CsCl 0.49/0.191/0.319 547 2.61 0.59 227
CsF + CsCl + CsNO3 0.238/0.242/0.52 554 3.19 0.31 90
NaCl + NaI + NaNO3 0.064/0.131/0.805 557 2.13 0.5 184
LiBr + NaBr + CsBr 0.615/0.013/0.372 561 3.27 0.41 149
NaF + NaI + NaNO3 0.002/0.14/0.859 563 2.13 0.5 174
LiCl + CsCl + CaCl2 0.381/0.276/0.343 564 2.59 0.57 218
LiCl + RbCl + BaCl2 0.504/0.368/0.129 575 2.68 0.59 213
LiCl + RbCl + SrCl2 0.553/0.404/0.044 585 2.36 0.61 254
LiCl + NaCl + RbCl 0.565/0.022/0.413 588 2.28 0.61 272
LiCl + CsCl + BaCl2 0.561/0.406/0.033 596 2.8 0.59 189
LiCl + NaCl + CsCl 0.575/0.009/0.416 600 2.74 0.59 199
148
NaCl + RbCl + SrCl2 0.311/0.357/0.332 607 2.67 0.46 199
LiCl + RbCl + MgCl2 0.503/0.367/0.13 608 2.2 0.56 297
LiF + LiCl + LiI 0.087/0.329/0.584 623 2.95 0.61 86
CsCl + CaCl2 + SrCl2 0.457/0.317/0.226 624 2.89 0.4 148
CsCl + CsBr + CsNO3 0.284/0.056/0.66 625 2.96 0.27 81
NaCl + CsCl + SrCl2 0.242/0.449/0.309 626 2.94 0.42 151
LiCl + LiBr + LiI 0.225/0.375/0.4 628 2.86 0.51 111
NaCl + RbCl + BaCl2 0.264/0.333/0.403 633 3.09 0.43 156
LiCl + CsCl + SrCl2 0.602/0.341/0.057 634 2.68 0.6 201
CsF + CsCl + CsI 0.447/0.288/0.266 640 3.6 0.31 120
CsF + CsBr + CsNO3 0.099/0.071/0.83 645 2.96 0.26 77
LiF + LiBr + LiI 0.123/0.319/0.558 656 3.01 0.56 85
CsCl + MgCl2 + CaCl2 0.602/0.144/0.255 675 2.77 0.34 179
NaCl + RbCl + MgCl2 0.232/0.567/0.2 678 2.28 0.37 284
CsF + CsBr + CsI 0.46/0.285/0.255 682 3.67 0.28 117
NaI + KI + CsI 0.319/0.343/0.338 686 3.1 0.22 126
CsF + CsCl + CsBr 0.485/0.214/0.301 694 3.62 0.32 127
RbCl + MgCl2 + SrCl2 0.6/0.185/0.215 699 2.53 0.34 208
LiCl + CsCl + MgCl2 0.198/0.144/0.658 700 2.07 0.33 366
RbCl + MgCl2 + BaCl2 0.618/0.218/0.164 705 2.65 0.32 206
NaCl + CsCl + BaCl2 0.306/0.568/0.126 705 2.95 0.39 159
NaCl + KCl + RbCl 0.735/0.005/0.26 708 2.07 0.51 361
KCl + CsCl + BaCl2 0.376/0.323/0.301 712 2.99 0.39 146
KCl + CsCl + SrCl2 0.377/0.341/0.281 723 2.71 0.4 165
NaCl + CsCl + MgCl2 0.096/0.729/0.174 735 2.8 0.31 173
NaBr + RbBr + CsBr 0.383/0.231/0.386 736 3.11 0.3 154
LiF + NaF + RbF 0.434/0.077/0.489 736 2.77 1.04 409
NaCl + KCl + CsCl 0.337/0.329/0.335 740 2.43 0.44 239
LiF + NaF + CsF 0.365/0.07/0.564 741 3.71 0.87 248
CsCl + MgCl2 + BaCl2 0.756/0.139/0.105 745 2.95 0.29 142
NaCl + CsCl + CaCl2 0.31/0.627/0.063 745 2.76 0.39 178
KCl + RbCl + MgCl2 0.013/0.643/0.343 749 2.24 0.3 274
LiF + RbF + CsF 0.391/0.012/0.596 752 3.73 0.85 234
NaF + KF + CsF 0.459/0.232/0.308 756 3.22 0.75 362
RbCl + MgCl2 + CaCl2 0.251/0.089/0.66 761 2.24 0.42 253
KF + KCl + KI 0.284/0.347/0.369 763 2.32 0.43 255
NaCl + RbCl + CaCl2 0.334/0.542/0.125 767 2.27 0.43 261
KCl + CsCl + MgCl2 0.084/0.739/0.177 770 2.76 0.29 169
RbF + RbCl + RbI 0.399/0.249/0.351 770 2.9 0.35 164
NaBr + KBr + CsBr 0.236/0.248/0.516 771 3.08 0.28 151
KCl + RbCl + SrCl2 0.015/0.725/0.26 784 2.58 0.36 169
149
CsCl + CsBr + CsI 0.338/0.349/0.312 791 3.23 0.22 108
RbF + RbCl + RbBr 0.446/0.221/0.332 791 2.8 0.4 192
LiF + KF + CsF 0.248/0.238/0.513 792 3.51 0.73 248
RbCl + CsCl + MgCl2 0.353/0.522/0.125 792 2.68 0.29 172
CsCl + MgCl2 + SrCl2 0.818/0.084/0.098 793 2.87 0.29 136
CsCl + CaCl2 + BaCl2 0.779/0.078/0.143 795 2.98 0.31 120
RbCl + CaCl2 + BaCl2 0.736/0.169/0.095 802 2.54 0.36 187
RbCl + CaCl2 + SrCl2 0.767/0.136/0.097 811 2.44 0.36 192
NaF + NaCl + NaI 0.114/0.354/0.532 816 2.6 0.45 251
NaI + RbI + CsI 0.216/0.554/0.23 816 3.06 0.17 110
RbCl + CsCl + BaCl2 0.148/0.702/0.15 819 2.95 0.3 121
RbF + RbBr + RbI 0.218/0.274/0.509 830 2.95 0.26 132
LiF + KF + RbF 0.246/0.478/0.277 861 2.42 0.82 425
KCl + CsCl + CaCl2 0.353/0.588/0.059 861 2.58 0.34 174
KBr + RbBr + CsBr 0.157/0.315/0.527 865 3.04 0.24 130
NaCl + RbCl + CsCl 0.082/0.16/0.758 869 2.74 0.3 142
NaCl + NaBr + NaI 0.217/0.245/0.538 874 2.61 0.33 213
RbCl + RbBr + RbI 0.268/0.355/0.377 876 2.78 0.23 134
KF + KBr + KI 0.136/0.285/0.579 881 2.42 0.3 183
NaF + NaBr + NaI 0.109/0.279/0.612 882 2.69 0.36 203
NaF + RbF + CsF 0.238/0.015/0.747 882 3.63 0.5 196
KCl + KBr + KI 0.272/0.24/0.488 885 2.31 0.29 193
RbCl + CsCl + CaCl2 0.169/0.802/0.028 891 2.74 0.29 133
KCl + RbCl + CsCl 0.14/0.15/0.71 896 2.67 0.3 147
KF + RbF + CsF 0.411/0.012/0.577 900 3.38 0.46 221
KI + RbI + CsI 0.193/0.5/0.307 902 2.95 0.16 107
RbCl + CsCl + SrCl2 0.022/0.97/0.008 912 2.79 0.27 122
NaBr + KBr + RbBr 0.108/0.223/0.669 915 2.63 0.27 163
KCl + RbCl + CaCl2 0.019/0.936/0.044 958 2.26 0.31 201
NaI + KI + RbI 0.236/0.526/0.239 971 2.61 0.2 136
NaF + KF + RbF 0.213/0.094/0.692 980 2.61 0.59 321
KF + K2SO4 + KNO3 0.061/0.052/ 0.46 347 2.14 0.49 134
+K2CO3 0.647/0.24
NaCl + Na2SO4 + NaNO3 0.267/0.016/ 0.24 381 2.16 0.65 219
+Na2CO3 0.296/0.42
KCl + K2SO4 + KNO3 0.031/0.525/ 0.58 419 2.2 0.41 181
+K2CO3 0.279/0.164
LiF + LiCl + LiBr 0.002/0.013/ 1.31 491 2.06 0.58 325
+LiNO3 0.246/0.739
LiF + LiCl + Li2SO4 0.027/0.106/ 0.68 493 1.88 0.66 326
+LiNO3 0.131/0.736
150
NaF + Na2SO4 + NaNO3 0.013/0.022/ 0.28 537 2.12 0.64 170
+Na2CO3 0.399/0.566
NaF + NaCl + NaBr 0.004/0.129/ 0.47 549 1.99 0.53 207
+NaNO3 0.087/0.78
LiCl + RbCl + CaCl2 0.01/0.007/ 1.05 553 2.29 0.51 253
+SrCl2 0.982/0.001
NaF + NaCl + NaNO3 0.021/0.064/ 0.35 573 1.91 0.55 193
+Na2CO3 0.904/0.011
LiF + LiCl + LiNO3 0.036/0.085/ 0.77 578 1.77 0.67 396
+Li2CO3 0.796/0.083
NaCl + KCl + MgCl2 0.116/0.735/ 0.41 578 1.84 0.48 357
+CaCl2 0.088/0.061
LiF + KF + MgF2 0.231/0.635/ 1.42 580 2.37 0.93 563
+CaF2 0.074/0.061
LiCl + KCl + MgCl2 0.261/0.301/ 0.33 592 1.87 0.49 378
+CaCl2 0.297/0.141
LiCl + KCl + CaCl2 0.441/0.304/ 0.55 593 2.51 0.61 248
+BaCl2 0.101/0.154
LiCl + KCl + SrCl2 0.413/0.285/ 0.62 595 2.68 0.59 208
+BaCl2 0.159/0.144
LiCl + KCl + CaCl2 0.434/0.299/ 0.52 597 2.24 0.61 268
+SrCl2 0.1/0.167
LiCl + NaCl + KCl 0.447/0.143/ 0.4 613 1.82 0.64 365
+CaCl2 0.308/0.103
LiF + LiCl + Li2SO4 0.139/0.325/ 1.83 618 1.99 0.96 359
+Li2CO3 0.219/0.317
LiCl + Li2SO4 + LiNO3 0.136/0.081/ 0.7 620 1.8 0.65 353
+Li2CO3 0.699/0.083
LiCl + NaCl + KCl 0.418/0.134/ 0.52 625 2.18 0.62 289
+SrCl2 0.288/0.16
LiCl + MgCl2 + CaCl2 0.321/0.366/ 0.37 629 2.17 0.48 318
+SrCl2 0.173/0.14
LiCl + NaCl + SrCl2 0.363/0.141/ 0.63 632 2.97 0.58 161
+BaCl2 0.278/0.218
LiCl + NaCl + CaCl2 0.396/0.154/ 0.51 635 2.5 0.61 224
+SrCl2 0.147/0.304
LiCl + MgCl2 + CaCl2 0.34/0.387/ 0.36 638 2.24 0.47 324
+BaCl2 0.183/0.09
KCl + MgCl2 + CaCl2 0.411/0.249/ 0.47 639 2.48 0.4 261
+BaCl2 0.154/0.187
NaCl + KCl + MgCl2 0.115/0.727/ 0.47 650 2.08 0.46 316
151
+BaCl2 0.087/0.07
LiCl + NaCl + CaCl2 0.354/0.361/ 0.38 651 2.33 0.62 302
+BaCl2 0.191/0.094
NaCl + MgCl2 + CaCl2 0.35/0.274/ 0.39 663 2.57 0.42 274
+BaCl2 0.17/0.206
LiCl + KCl + MgCl2 0.4/0.292/ 0.51 664 2.33 0.54 293
+BaCl2 0.176/0.132
LiCl + CaCl2 + SrCl2 0.446/0.24/ 0.54 667 2.68 0.6 194
+BaCl2 0.195/0.119
NaCl + KCl + MgCl2 0.109/0.691/ 0.48 670 2.03 0.46 309
+SrCl2 0.083/0.117
KCl + MgCl2 + SrCl2 0.376/0.163/ 0.6 672 2.84 0.4 185
+BaCl2 0.178/0.284
KCl + MgCl2 + CaCl2 0.457/0.192/ 0.47 672 2.25 0.42 269
+SrCl2 0.129/0.223
LiCl + KCl + MgCl2 0.619/0.2/ 0.5 675 1.98 0.63 333
+SrCl2 0.086/0.095
LiCl + NaCl + RbCl 0.01/0/ 1.05 681 2.24 0.49 246
+CaCl2 0.007/0.983
NaCl + MgCl2 + SrCl2 0.291/0.221/ 0.52 685 2.78 0.42 206
+BaCl2 0.311/0.178
LiCl + NaCl + MgCl2 0.271/0.276/ 0.23 689 1.84 0.5 409
+CaCl2 0.308/0.146
NaCl + MgCl2 + CaCl2 0.298/0.247/ 0.39 690 2.38 0.43 270
+SrCl2 0.166/0.288
LiCl + MgCl2 + SrCl2 0.389/0.167/ 0.59 696 2.73 0.52 194
+BaCl2 0.317/0.127
NaCl + CaCl2 + SrCl2 0.233/0.253/ 0.53 701 2.79 0.47 171
+BaCl2 0.404/0.11
LiCl + NaCl + MgCl2 0.443/0.235/ 0.45 704 2.39 0.57 307
+BaCl2 0.178/0.143
NaCl + KCl + CaCl2 0.283/0.276/ 0.41 705 2.41 0.49 255
+BaCl2 0.308/0.133
NaCl + KCl + CaCl2 0.208/0.203/ 0.49 723 2.49 0.48 209
+SrCl2 0.227/0.361
LiCl + NaCl + MgCl2 0.4/0.213/ 0.45 732 2.25 0.56 293
+SrCl2 0.161/0.227
LiF + Li2SO4 + LiNO3 0.216/0.339/ 1.83 752 1.99 0.92 311
+Li2CO3 0.212/0.234
LiF + NaF + KF 0.434/0.12/ 2.83 753 2.3 1.1 611
+SrF2 0.417/0.03
152
MgCl2 + CaCl2 + SrCl2 0.337/0.226/ 0.47 756 2.55 0.36 218
+BaCl2 0.392/0.045
LiF + NaF + KF 0.436/0.12/ 2.88 757 2.15 1.11 634
+CaF2 0.419/0.024
LiF + KF + CaF2 0.48/0.461/ 2.98 762 2.33 1.09 578
+SrF2 0.027/0.033
LiF + NaF + KF 0.443/0.122/ 2.96 763 2.12 1.12 649
+MgF2 0.426/0.009
LiF + KF + MgF2 0.488/0.469/ 3.07 764 2.3 1.1 589
+SrF2 0.01/0.033
LiF + NaF + MgF2 0.444/0.298/ 3.28 785 2.43 1.19 755
+CaF2 0.142/0.116
LiCl + NaCl + KCl 0.189/0.1/ 0.43 805 1.66 0.49 374
+MgCl2 0.634/0.076
NaCl + KCl + SrCl2 0.248/0.243/ 0.59 811 2.76 0.45 172
+BaCl2 0.329/0.18
NaF + NaCl + Na2SO4 0.181/0.367/ 0.2 848 2.03 0.65 304
+Na2CO3 0.155/0.297
LiF + MgF2 + CaF2 0.352/0.173/ 2.11 864 3.14 0.94 492
+SrF2 0.301/0.174
KF + KCl + K2SO4 0.417/0.51/ 0.43 866 1.85 0.51 380
+K2CO3 0.044/0.029
KF + MgF2 + CaF2 0.582/0.085/ 0.5 871 2.71 0.62 453
+SrF2 0.23/0.103
LiF + NaF + CaF2 0.492/0.315/ 3.41 881 2.66 1.18 645
+SrF2 0.117/0.076
LiF + NaF + MgF2 0.455/0.34/ 3.27 908 2.75 1.15 673
+SrF2 0.098/0.107
NaF + KF + CaF2 0.306/0.464/ 0.58 910 2.7 0.72 475
+SrF2 0.097/0.133
KCl + CaCl2 + SrCl2 0.319/0.299/ 0.54 931 2.52 0.42 186
+BaCl2 0.263/0.119
NaF + KF + MgF2 0.344/0.523/ 0.41 933 2.21 0.75 572
+CaF2 0.023/0.11
NaF + MgF2 + CaF2 0.482/0.139/ 0.81 968 2.95 0.78 524
+SrF2 0.226/0.152
LiCl + NaCl + KCl 0.22/0.313/ 0.52 1016 2.4 0.49 244
+BaCl2 0.259/0.208
CsCl + CaCl2 + SrCl2 0.136/0.379/ 1372 2.47 0.35 142
+BaCl2 0.334/0.151
NaF + NaCl + NaI 0.019/0.057/ 563 2.12 0.51 187
153
+NaNO3 0.123/0.802
RbCl + CaCl2 + SrCl2 0.204/0.349/ 849 2.72 0.41 155
+BaCl2 0.307/0.139
NaNO3 + KNO3 + CsNO3 0.312/0.446/ 414 2.08 0.46 108
+CaNO3 0.103/0.14
NaCl + RbCl + CaCl2 0.244/0.281/ 641 2.57 0.46 209
+SrCl2 0.215/0.26
LiCl + CsCl + MgCl2 0.307/0.178/ 490 2.35 0.47 304
+CaCl2 0.349/0.165
NaCl + RbCl + MgCl2 0.252/0.289/ 569 2.55 0.42 240
+SrCl2 0.191/0.268
NaCl + RbCl + SrCl2 0.297/0.341/ 761 2.64 0.43 189
+BaCl2 0.317/0.045
LiCl + KCl + RbCl 0.682/0.005/ 824 2.13 0.62 270
+SrCl2 0.23/0.083
LiNO3 + NaNO3 + KNO3 0.12/0.103/ 394 2.28 0.46 87
+RbNO3 0.51/0.267
LiCl + CsCl + CaCl2 0.366/0.265/ 793 2.46 0.52 211
+SrCl2 0.329/0.04
LiCl + CsCl + CaCl2 0.373/0.27/ 563 2.64 0.56 212
+BaCl2 0.335/0.022
LiCl + RbCl + SrCl2 0.484/0.354/ 578 2.7 0.58 205
+BaCl2 0.038/0.124
LiBr + NaBr + KBr 0.408/0.146/ 1072 2.61 0.32 169
+CsBr 0.178/0.268
LiCl + NaCl + CsCl 0.379/0.006/ 627 2.54 0.56 217
+CaCl2 0.274/0.341
KCl + RbCl + MgCl2 0.306/0.416/ 655 2.39 0.39 237
+SrCl2 0.128/0.149
LiNO3 + NaNO3 + KNO3 0.306/0.203/ 333 2.59 0.49 134
+CsNO3 0.27/0.221
NaCl + CsCl + CaCl2 0.215/0.366/ 672 2.79 0.43 171
+SrCl2 0.189/0.229
NaF + NaCl + NaBr 0.049/0.545/ 1006 2.23 0.43 303
+NaI 0.127/0.278
LiBr + NaBr + KBr 0.47/0.067/ 506 2.92 0.42 178
+RbBr 0.137/0.326
LiCl + NaCl + RbCl 0.494/0.019/ 597 2.65 0.58 215
+BaCl2 0.361/0.126
KCl + RbCl + MgCl2 0.265/0.454/ 618 2.56 0.37 231
+BaCl2 0.16/0.12
154
LiCl + KCl + CsCl 0.396/0.154/ 899 2.19 0.46 272
+MgCl2 0.257/0.192
NaCl + CsCl + MgCl2 0.221/0.376/ 569 2.83 0.39 192
+SrCl2 0.167/0.236
LiNO3 + KNO3 + RbNO3 0.127/0.541/ 344 2.42 0.44 80
+CsNO3 0.283/0.05
KF + KCl + KBr 0.162/0.255/ 836 2.29 0.37 228
+KI 0.251/0.332
NaNO3 + KNO3 + CsNO3 0.253/0.423/ 408 2.49 0.42 97
+Ba(NO3)2 0.282/0.041
LiCl + RbCl + MgCl2 0.331/0.413/ 725 2.49 0.47 235
+BaCl2 0.146/0.11
NaCl + CsCl + SrCl2 0.16/0.476/ 344 3.44 0.43 123
+BaCl2 0.08/0.284
LiCl + RbCl + CsCl 0.357/0.064/ 300 2.79 0.6 215
+CaCl2 0.258/0.321
RbCl + MgCl2 + CaCl2 0.544/0.192/ 624 2.67 0.35 210
+BaCl2 0.119/0.144
NaCl + RbCl + MgCl2 0.215/0.492/ 667 2.59 0.38 233
+BaCl2 0.163/0.131
KCl + CsCl + CaCl2 0.257/0.428/ 734 2.78 0.38 151
+SrCl2 0.043/0.272
LiCl + RbCl + MgCl2 0.258/0.445/ 691 2.44 0.45 229
+SrCl2 0.137/0.16
RbCl + MgCl2 + CaCl2 0.534/0.164/ 664 2.53 0.36 211
+SrCl2 0.11/0.191
LiCl + KCl + CsCl 0.49/0.185/ 628 2.54 0.58 224
+SrCl2 0.278/0.047
NaCl + CsCl + CaCl2 0.185/0.45/ 1228 2.43 0.33 166
+BaCl2 0.242/0.123
KCl + RbCl + CaCl2 0.012/0.582/ 798 2.51 0.38 182
+SrCl2 0.198/0.209
RbCl + MgCl2 + SrCl2 0.586/0.18/ 697 2.57 0.34 202
+BaCl2 0.21/0.024
NaCl + KCl + RbCl 0.304/0.01/ 796 2.43 0.42 219
+SrCl2 0.504/0.181
LiCl + NaCl + RbCl 0.317/0.169/ 754 2.12 0.49 304
+MgCl2 0.387/0.128
KCl + CsCl + MgCl2 0.245/0.409/ 716 2.48 0.32 241
+CaCl2 0.305/0.041
LiCl + NaCl + KCl 0.344/0.11/ 616 2.52 0.55 235
155
+CsCl 0.237/0.309
LiCl + NaCl + CsCl 0.54/0.103/ 336 2.86 0.66 220
+SrCl2 0.306/0.051
LiCl + KCl + CsCl 0.475/0.185/ 1091 2.15 0.48 228
+CaCl2 0.309/0.031
KCl + RbCl + CaCl2 0.242/0.558/ 804 2.4 0.38 212
+BaCl2 0.128/0.072
LiCl + KCl + RbCl 0.326/0.009/ 874 2.05 0.43 295
+MgCl2 0.434/0.231
LiCl + KCl + CsCl 0.456/0.188/ 573 2.67 0.57 213
+BaCl2 0.33/0.026
NaCl + RbCl + MgCl2 0.219/0.502/ 619 2.32 0.4 279
+CaCl2 0.166/0.114
KCl + CsCl + CaCl2 0.351/0.488/ 720 2.81 0.38 163
+BaCl2 0.054/0.108
LiCl + NaCl + RbCl 0.541/0.021/ 589 2.34 0.61 257
+SrCl2 0.395/0.043
NaNO3 + KNO3 + RbNO3 0.286/0.272/ 651 2.25 0.39 72
+CsNO3 0.379/0.063
NaCl + CsCl + MgCl2 0.239/0.444/ 694 2.58 0.35 240
+CaCl2 0.27/0.046
LiCl + RbCl + CaCl2 0.251/0.551/ 685 2.53 0.48 209
+BaCl2 0.127/0.071
NaCl + KCl + CsCl 0.225/0.315/ 688 2.58 0.42 212
+CaCl2 0.417/0.044
LiCl + CsCl + SrCl2 0.529/0.383/ 612 2.81 0.58 181
+BaCl2 0.057/0.031
NaCl + KCl + CsCl 0.154/0.307/ 525 2.84 0.43 186
+SrCl2 0.461/0.078
LiF + LiCl + LiBr 0.087/0.202/ 695 2.84 0.58 114
+LiI 0.274/0.437
NaCl + RbCl + CaCl2 0.151/0.625/ 643 2.6 0.41 208
+BaCl2 0.144/0.08
LiBr + NaBr + RbBr 0.249/0.288/ 977 2.78 0.31 160
+CsBr 0.173/0.29
LiCl + NaCl + KCl 0.559/0.022/ 385 2.42 0.65 271
+RbCl 0.012/0.408
NaCl + KCl + RbCl 0.23/0.01/ 1227 1.88 0.29 283
+MgCl2 0.562/0.198
KCl + RbCl + MgCl2 0.013/0.624/ 1054 2.03 0.27 273
+CaCl2 0.333/0.03
156
LiCl + NaCl + CsCl 0.556/0.009/ 601 2.8 0.59 190
+BaCl2 0.403/0.032
LiCl + RbCl + CsCl 0.416/0.087/ 1523 2.1 0.36 166
+BaCl2 0.41/0.088
NaBr + KBr + RbBr 0.344/0.103/ 797 2.99 0.29 158
+CsBr 0.207/0.346
LiCl + CsCl + MgCl2 0.24/0.574/ 750 2.84 0.4 163
+BaCl2 0.106/0.08
NaCl + KCl + RbCl 0.184/0.258/ 923 2.37 0.36 208
+CsCl 0.217/0.342
NaCl + CsCl + MgCl2 0.151/0.642/ 621 2.99 0.35 162
+BaCl2 0.118/0.089
NaF + KF + RbF 0.155/0.347/ 812 3.4 0.56 257
+CsF 0.01/0.488
CsCl + MgCl2 + CaCl2 0.696/0.128/ 588 3.05 0.33 148
+BaCl2 0.079/0.096
LiBr + KBr + RbBr 0.312/0.108/ 1014 2.79 0.29 142
+CsBr 0.217/0.363
KCl + CsCl + MgCl2 0.134/0.707/ 656 3.01 0.33 141
+BaCl2 0.058/0.101
KCl + RbCl + CsCl 0.122/0.13/ 827 2.87 0.32 134
+BaCl2 0.616/0.132
LiF + NaF + KF 0.249/0.16/ 838 2.68 0.86 380
+RbF 0.071/0.52
LiF + KF + RbF 0.293/0.246/ 737 3.5 0.8 267
+CsF 0.009/0.452
LiCl + RbCl + CsCl 0.426/0.089/ 1014 2.37 0.44 199
+MgCl2 0.42/0.066
LiCl + KCl + RbCl 0.419/0.082/ 1023 2.35 0.45 191
+CsCl 0.087/0.413
LiF + NaF + KF 0.232/0.064/ 693 3.59 0.78 261
+CsF 0.223/0.481
RbCl + CsCl + SrCl2 0.144/0.682/ 1012 2.76 0.28 120
+BaCl2 0.029/0.146
NaCl + RbCl + CsCl 0.07/0.138/ 803 2.93 0.32 130
+BaCl2 0.652/0.14
LiNO3 + NaNO3 + RbNO3 0.223/0.192/ 391 2.6 0.45 86
+CsNO3 0.497/0.088
LiCl + NaCl + RbCl 0.435/0.046/ 1206 2.22 0.43 189
+CsCl 0.09/0.429
RbCl + CsCl + CaCl2 0.145/0.685/ 819 2.94 0.31 122
157
+BaCl2 0.024/0.147
LiCl + NaCl + CsCl 0.209/0.111/ 814 2.67 0.4 182
+MgCl2 0.597/0.084
NaCl + KCl + CsCl 0.089/0.077/ 748 2.74 0.32 180
+MgCl2 0.673/0.161
LiCl + CsCl + MgCl2 0.137/0.706/ 695 2.91 0.37 147
+SrCl2 0.073/0.085
CsCl + MgCl2 + CaCl2 0.774/0.08/ 628 3 0.33 140
+SrCl2 0.054/0.093
KCl + CsCl + MgCl2 0.139/0.735/ 672 2.91 0.33 148
+SrCl2 0.06/0.066
RbCl + CsCl + MgCl2 0.283/0.542/ 688 2.89 0.31 154
+BaCl2 0.1/0.075
CsCl + MgCl2 + SrCl2 0.809/0.083/ 660 3 0.31 135
+BaCl2 0.097/0.011
NaCl + RbCl + CsCl 0.236/0.278/ 730 2.69 0.38 186
+CaCl2 0.439/0.046
NaCl + RbCl + CsCl 0.126/0.309/ 655 2.75 0.35 189
+MgCl2 0.456/0.109
RbCl + CsCl + MgCl2 0.215/0.642/ 687 2.89 0.32 146
+SrCl2 0.066/0.077
LiF + NaF + RbF 0.361/0.07/ 746 3.69 0.86 247
+CsF 0.011/0.558
KCl + RbCl + CsCl 0.056/0.333/ 794 2.64 0.3 177
+MgCl2 0.492/0.118
RbCl + CsCl + MgCl2 0.15/0.713/ 827 2.73 0.29 156
+CaCl2 0.112/0.025
KCl + RbCl + CsCl 0.137/0.146/ 836 2.72 0.31 149
+CaCl2 0.693/0.024
RbCl + CsCl + CaCl2 0.164/0.776/ 912 2.73 0.29 132
+SrCl2 0.027/0.033
LiCl + RbCl + CsCl 0.061/0.021/ 464 3.24 0.37 127
+SrCl2 0.911/0.007
NaCl + RbCl + CsCl 0.013/0.022/ 242 3.49 0.38 124
+SrCl2 0.957/0.008
KCl + RbCl + CsCl 0/0.022/ 844 2.86 0.28 122
+SrCl2 0.97/0.008
LiI + NaI + KI 0.514/0.017/ 515 3.35 0.29 68
+CsI 0.207/0.262
LiI + NaI + RbI 0.472/0.016/ 524 3.42 0.27 66
+CsI 0.272/0.241
158
NaI + KI + RbI 0.176/0.189/ 720 3.09 0.19 115
+CsI 0.449/0.186
CsF + CsCl + CsBr 0.344/0.222/ 721 3.49 0.28 117
+CsI 0.229/0.205
CsF + CsCl + CsBr 0.215/0.218/ 784 2.96 0.27 91
+CsNO3 0.097/0.47
LiI + KI + RbI 0.224/0.141/ 872 2.99 0.2 90
+CsI 0.401/0.235
LiI + NaI + KI 0.225/0.231/ 1111 2.62 0.2 101
+RbI 0.142/0.403
159