JDept. o

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nsidered by the present models, namely the effects of pressure drop on heat

2011 Elsevier Ltd. All rights reserved.

. . . . . .. . . . .. . . . . .. . . . . .ropertieon . . .al cond. . . . . .and ge. . . . . .

4.2.2. Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3724.3. Stream evolution models (SEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

4.3.1. Aspen plate fin exchanger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

4.3.2. GENIUS, by Linde AG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3734.4. Summary. Features for cryogenic applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Corresponding author.

Cryogenics 51 (2011) 366379

Contents lists available at ScienceDirect

CryogenicsE-mail address: julio.pacio@ntnu.no (J.C. Pacio).3.2. Coil wound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3693.3. Plate-fin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3693.4. Perforated plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3703.5. Regenerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

4. Heat exchanger models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3704.1. Lumped parameters models (LPM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

4.1.1. Mean temperature difference (MTD). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3714.1.2. Other efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

4.2. Distributed parameters models (DPM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3724.2.1. Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372Contents

1. Introduction . . . . . . . . . . . . . . . . .2. Challenging features for modeling

2.1. Complex processes . . . . . .2.2. Non-negligible effects . . . .

2.2.1. Changes in fluid p2.2.2. Flow maldistributi2.2.3. Longitudinal therm2.2.4. Heat-in-leakage. .

3. Cryogenic heat exchangers. Types3.1. Concentric tubes . . . . . . . .0011-2275/$ - see front matter 2011 Elsevier Ltd. Adoi:10.1016/j.cryogenics.2011.04.005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369uction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369ometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369transfer and the existence of partial owmixing. These two effects are highly relevant for two-phase owand multi-component applications, as in LNG processes.points stand out as not coeywords:eat exchangerhermal hydraulic

These formulations fail to take all relevant effects into account.A general discussion on the performance of the reviewed models is presented. In general, more effects

are included in the framework of numerical solution of discretized energy balance equation. Two mainle online 20 April 2011In this work a systematic review of the state of art and challenges in modeling cryogenic heat exchang-

ers is presented. They include lumped parameters, distributed parameters and stream-evolution models.Article history:Received 10 November 2010Received in revised form 3 April 2011Accepted 12 April 2011Availab

Heat exchangers are the main components in cryogenic processes. Thermo-economic considerations setthe need for high-effectiveness equipment and accurate models. This situation is challenging due to thecomplex operating conditions and the fact that some physical effects, such as changes in uid properties,ow maldistribution, axial conduction and heat leakage, cannot be neglected.f Energy and Process Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

i c l e i n f o a b s t r a c tulio Cesar Pacio , Carlos Alberto DoraoA review on heat exchanger thermal hydraulic models for cryogenic applicationswReviejournal homepage: www.elsevier .com/locate /cryogenicsll rights reserved.

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1. Introduction

Heat exchangers (HEs) are the main components in cryogenicprocesses. In air separation units and Liquefaction of Natural Gas(LNG) plants, they represent 2030% of the investment costs [38].

spepracti[10] c97% t

pwco

Carnot cycle [20, ch. 10] can be computed according to (1). Giventhe usually low values of Tc, the mechanical-to-thermal power ra-tio W/Q given by (1) is relatively high. For this reason, the refriger-ation capacity should be kept at a minimum, emphasizing therequirements for high HE performance.

Q Tc

DPM distributed parameters modelHE heat exchangerHTC heat transfer coefcientLHC longitudinal heat conductionLMTD logarithmic mean temperature differenceLPM lumped parameters modelsMTD mean temperature differenceNTU number of thermal units []PFHE plate-n heat exchangerPPHE perforated-plate heat exchangerSEM stream evolution model

Greek symbolsDT local temperature difference (K)DTm effective mean temperature difference (K)DTlm logarithmic mean temperature difference (K)g heat exchanger efciency ()/ general scalar function of one or more variablese heat exchanger effectiveness ()

Physical variables_m mass ow rate (kg s1)

A area (m2)C heat capacity rate (W K1)C heat capacity rate ratio ()cp specic heat capacity at constant pressure (J kg1 K1)dA differential area (m2)dQ differential heat transfered (W)dT differential temperature (K)En heat exchanger new effectiveness ()F correction factor for LMTD ()P temperature effectiveness ()Q heat transfered (W)R heat capacity rate ratio ()T temperature (K)U overall heat transfer coefcient (W m2 K1)W mechanical power (W)

Subscriptsc refers to the lower temperature level (hot)h refers to the higher temperature level (cold)max maximum valuemin minimum value

yogeroduction rate. The refrigeration capacity has to be increased, tures of cryogenic HE, however, make their formulation a challeng-

ith toncef remfor modications in the process to achieve the desired liquid large range of sizes and different processes [106]. The special fea-produced if e < 85%.An important consequence of HE under-performance is the

need

ing than for high-temperature applications. The design of tradi-tional HEs, such as shell-and-tube, is rather well established for ae) departs from the ideal value of 100% to a morecal one of 96.5%. In the case of liquefaction of helium, Atreyalculated that 12% less liquid is obtained if e is reduced fromo 95%, and Barron [13], Barron [15] stated that no liquid is

and large amounts of additional power input are required. This setsthe need for high-effectiveness heat exchangers, in the order ofe > 90% [116]. This situation is reected in the requirements forthe accuracy of the models used for design, that are more demand-ensitive to the HE performance. For example, Kanoglu et al. [53]redicted a reduction of 22% in the production of liquid if the HEffectiveness (

In short, if the HE has low performance, the production is reducedThermodynamic considerations make cryogenic processes veryIn addition, their performance affects the sizing and design of othermajor equipments, namely compressors and their power drivers. W Th Tc 15. Other effects reported in literature . . . . . . . . . . . . . . . . . . . . . . . . . . .5.1. Changes in fluid properties. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1.1. Specific heat capacity . . . . . . . . . . . . . . . . . . . . . . . . .5.1.2. Heat transfer coefficient (HTC) . . . . . . . . . . . . . . . . .

5.2. Flow maldistribution in parallel channels . . . . . . . . . . . . . . . .5.3. Longitudinal heat conduction (LHC). . . . . . . . . . . . . . . . . . . . .5.4. Heat exchange with the surroundings (heat leakage). . . . . . .5.5. Combined effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6. Discussion on the state-of-the-art. . . . . . . . . . . . . . . . . . . . . . . . . . . .7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Nomenclature

Abbreviations and AcronymsCWHE coil-wound heat exchanger

J.C. Pacio, C.A. Dorao / Crhe corresponding increase in power input, which is a majorrn in cryogenics. The minimum ideal power input per unitoved heat in a refrigeration circuit, given by the inverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

W nondimensional effective mean temperature difference()

nics 51 (2011) 366379 367ing task. Usually this situation results in a particular design for thegiven application, since sizing is not as straightforward as for moretraditional geometries.

phase presents some challenges to model heat transfer andpressure drop. These are usually modeled on the base of empir-ical correlations, which predict them with a large degree ofuncertainty.

Multi-component mixtures. Two important mixtures playimportant roles in cryogenics: air and natural gas. Pre-treatednatural gas to be processed in a cryogenic plant is a mixtureof several hydrocarbons and some minor contents of nitrogen.In addition, roughly 95% of the LNG plants use a multi-compo-nent mixture as a refrigerant [114]. Condensation and evapora-tions of mixtures is a challenging scenario to model, given theeffects of mass resistances on heat transfer.

ogenics 51 (2011) 366379This work intends to review the present models for cryogenicHE design. These models fail to consider some physical effects,such as ow distribution and axial thermal conduction, that canusually be neglected for other applications, but might dominatethe performance in the case of cryogenic applications. This sce-nario has been noticed by several authors, and the available litera-ture concerning these effects is reviewed in this work.

The structure of this work is as follows. Section 2 discusses thechallenges in the modeling of cryogenic HE. The different equip-ment and geometries are presented in Section 3. Section 4 reviewsthe present available models for HE design. Section 5 deals withrelevant effects reported in the reviewed literature. A critical dis-cussion on the state-of-the-art on this subject is presented in Sec-tion 6. The conclusions of this work are summarized in Section 7.

2. Challenging features for modeling

As discussed in Section 1, cryogenic HE have large efciencyrequirements. This makes their modeling a challenging issue.Two particular features are most demanding. First, the thermalprocesses used in cryogenic engineering present some complexelements, such as simultaneous heat exchange between multiplestreams. Second, some physical effects that are usually neglectedfor high-temperature applications cannot be disregarded in thecase of cryogenics. These two features are studied in detail in thefollowing subsections.

2.1. Complex processes

A process for producing refrigeration at liqueed gas tempera-tures always involves some equipment at ambient temperaturein which the gas is compressed and heat is rejected to a coolant[37, ch. 6]. In some cases, such as the Linde cycle, the working uidis self-refrigerated after an expansion. In other cases, the use of arefrigerant is necessary, which needs to be cooled as well.

The complexity of the processes is more marked in the case ofLiqueed Natural Gas (LNG) production. In this case, the aim foroverall economic optimization resulted in the development ofcomplex liquefaction processes that represent roughly half of theplant capital costs [34]. A comprehensive review on LNG technolo-gies was presented by Brendeng and Hetland [19]. Around 80% ofthe installed capacity uses a propane-precooled mixed refrigerantprocess, licensed by Air Products and Chemicals, Inc. (APCI) [11].Other cycles include optimized cascade, single mixed refrigerantand natural gas expanders.

As a consequence of the mentioned processes, the cryogenicHEs operate in challenging conditions. These include:

Large temperature range. Starting from ambient temperature,cryogenic uids need to be cooled to temperatures as low as110 K (LNG), 77 K (nitrogen), 20 K (hydrogen) or 4 K (helium)to achieve liquid state at atmospheric pressure. In some refrig-eration cycles, this is done in a cascade process. However, inother cases, as the single mixed refrigerant process, the entirerange is covered in one single HE. This leads to problems suchas large temperature gradients inside the exchanger, and thechanges in uid properties.

Multiple streams. In LNG processes, both the natural gas andhigh-pressure refrigerant need to be cooled. This process pre-sents some economical advantages if it is performed withinthe same HE. For this reason, plate-n and coil-wound HE areused in these processes with three or four simultaneous streams.

Boiling and condensation. Cryogenic processes present conden-

368 J.C. Pacio, C.A. Dorao / Crysation of the working uid, and evaporation of the low-pressurerefrigerants to achieve high heat transfer rates. The change of Flow injection and removal. During the liquefaction of naturalgas, heavier hydrocarbons are separated to meet the specica-tions. This is performed within the main cryogenic HE wherethe liquefaction takes place, resulting in localized changes inmass ow rates and composition of the streams.

2.2. Non-negligible effects

Traditional heat exchanger models neglect some effects [99, ch.3.2] since they are not relevant for the typical required engineeringaccuracy. However, the high-effectiveness requirements for cryo-genic HEs make necessary to take these effects into account. Theyinclude: changes in uid properties, heat exchange with the sur-rounding (heat leakage), longitudinal thermal conduction in thewall, and ow maldistribution.

The relative importance of these effects is summarized in Fig. 1.For low-efciency applications, all of them can be neglected. Forhigher effectiveness requirements, they all need to be considered,in increasing order of accuracy: changes in uid properties, owmaldistribution, longitudinal conduction and heat leakage. Theseconsiderations depend on the particular operating conditions andcover relatively wide and approximate ranges.

Some of these effects have been addressed in literature for rat-ing the performance of HE in simple processes involving single-phase ow. The studies reported in literature are reviewed in Sec-tion 5. However, no reports were found on applications to complexprocesses as those used for production of LNG. The following sub-sections present some details of these effects.

2.2.1. Changes in uid propertiesBarron [14] stated that the main distinctive feature of cryogenic

heat transfer is that all constants become variables. For single-phaseow, the main effect is given by changes in the specic heat capac-ity. In the case of two-phase ow, this is accompanied by largevariations in the heat transfer coefcient, as well as density andviscosity. For the analysis of high-temperature HEs, Kays and Lon-don [55] suggested the use of a constant value, corresponding tothe physical properties evaluated at a mean temperature. However,Fig. 1. Effects to be considered for a given design effectiveness.

this approach is not applicable for cryogenic applications. Soyars[101] stated that the use of this approximation for simulation ofHE with helium below 15 K leads to mispredictions of 30100%of the refrigeration heat load that can be achieved. Oonk and Hus-tvedt [74] found underpredictions of up to 12% on performance ofhelium HE between 4 and 20 K using this approach.

2.2.2. Flow maldistributionIn many scenarios, the ow distribution can deviate from design

tions are based in concentric-tubes geometries. A simple tube-in-

(CWHE), also known as GiauqueHampson HE is widely used. Lay-

J.C. Pacio, C.A. Dorao / Cryogeconditions, usually homogeneous. Mueller and Chiou [68] pre-sented a comprehensive review on causes of maldistribution. Theyinclude mechanical issues such as fouling, fabrication tolerances,bypass and poor header performance, two-phase instabilities,and heat-transfer induced as a consequence of changes in viscosityor density. Flow maldistribution results in a deterioration of per-formance of single-phase HE [26,36,51], although this effect is onlyrelevant for high efciency equipment such as those used in cryo-genics [68]. The effect on two-phase ow systems is more compli-cated, and both a reduction [79,102] and an increase [2,111] inperformance have been observed.

2.2.3. Longitudinal thermal conductionThis effect reduces the local temperature difference between

the working uids and the separating wall, deteriorating the heattransfer. In the extreme case of innite thermal conductivity, theperformance of a balanced counterow HE is reduced roughly byhalf, matching the co-current ow case [46, ch. 6]. This effect ismore signicant in small systems with short conduction lengths,such as perforated-plate, than in the large coil-wound (CWHE)and plate-n (PFHE) heat exchangers.

2.2.4. Heat-in-leakageSince cryogenic processes operate at much lower temperature

than ambient, cryogenic equipment exchange heat with the sur-roundings. The development of multilayer insulations in the1960s, with apparent thermal conductivity as low as 10100 lW/(mK) [109] has reduced the heat-leakage to a practical minimum.However, when high-effectiveness equipment is required, this ef-fect has to be considered.

3. Cryogenic heat exchangers. Types and geometries

Several types of heat exchangers have been used in cryogenictechnology through its more-than-one-century-long history. Theywere described in detail by Barron [14,15]. In general, the selectionof the geometry depends on the application. The decision shouldconsider the operating pressure, mass ow rates, total heat duty,as well as operation and investment costs. In this section, the dif-ferent geometries are reviewed from a modeling and designperspective.

To a broad extent, HE can be divided into two categories: recu-perators and regenerators [99]. Both types are used for cryogenicapplications. In a recuperator heat is transferred between the uidsFig. 2. Types of cryogenic heat exchangers.ers of tubes are coiled around a central mandrel, which providesmechanical stability. Several tube-side streams can be used inthe different layers, two or three being common in LNG processes,exchanging heat with a common shell-side stream. This featurerepresents a major comparative advantage to the more traditionalshell-and-tube exchangers. They can be constructed in very largeunits, only limited in size by transportation issues [65]. The maindisadvantage is that they are proprietary and expensive equip-ment, only manufactured by APCI and Linde Group [105].

The Collins-type HE [29] presents a similar geometrical arrange-ment. Similarly, successive layers of tubes are coiled around a cen-tral mandrel, and the low pressure stream has an helical ow-patharound the tubes. The differences between both geometries arenot large, they include the ow distribution devices and somemechanical support.

A major challenge in these last two geometries is the ow dis-tribution on both tube and shell sides. For this reason, both tubespacing and length are kept almost constants.

3.3. Plate-ntube exchanger was used by Linde in 1895 for the rst-time lique-faction of air [89]. The efciency can be improved with the use of awire spacer that keeps the distance between tubes roughly con-stant. For more complex processes that involve multiple streamsa multiple-concentric-tubes HE can be used. In this last geometry,several high-pressure streams ow inside parallel smaller tubes lo-cated inside a larger enclosing tube that contains the low-pressurestream.

Fromamodel perspective, these geometries donot presentmajorchallenges. Since each stream ows in a single channel, no mixingneeds to be considered. In addition, they are one-dimensionalequipment in pure co-current or counter-current owarrangement.

3.2. Coil wound

For large scale applications, the coil-wound heat exchangerthrough a separating wall. In a regenerator, both uids alternatelyoccupy the same physical space and thermal energy is stored andreleased from a matrix buffer material.

The different types of recuperators used for cryogenic applica-tions are summarized in Fig. 2, including tubular, plate-n and per-forated-plate HE. In particular, the rst type can be subdivided intosimple concentric-tube HE and more complex geometries with amulti-channel arrangement. The above mentioned types andgeometries are studied in the following subsections, and the mainfeatures are summarized in Table 1.

A distinction should be made between HE for small-size andlarge-scale applications. The rst ones require simple equipmentthat are easy to build and maintain and are dominated by theuse of concentric-tube geometries (see Section 3.1), perforated-plate HE (see Section 3.4) and regenerators (see Section 3.5). Onthe other hand, the design of large-scale HE is more focused onminimizing the unit cost. Following this objective the selection ofmain HE for large cryogenic processes has been dominated bythe coil-wound and plate-n geometries [34]. They are studied inSections 3.2 and 3.3, respectively.

3.1. Concentric tubes

Simple tubular HE used for small scale and laboratory applica-

nics 51 (2011) 366379 369This type of HE consists of sets of layers of corrugated plates(usually made of aluminum) which serve as ns, and separating

nduction. Sources: [1,8,15,60].

FHE PPHE Regenerators

ultiuid, low cost High HTC Good for only one working uidensitive to transients Limited in size Flow mixingarge Small Small001000 up to 6000 up to 6500edium High Lowin efciency HTC and LHC Matrix heat capacity

ogenics 51 (2011) 366379thin metal sheets. This conguration results in small ow passagesand large extended surface area, which makes plate-n heatexchangers (PFHE) very compact equipment. In addition, theycan handle up to ten different streams in the same unit and verylow temperature differences can be achieved. This results in lowercapital and operation costs compared to traditional shell-and-tube-type HE [61]. For this reason they are used in severalindustries, covering large ranges of temperature and pressure,and many manufacturers produce them around the world, groupedin the Brazed Aluminium Plate-Fin Heat Exchanger ManufacturersAssociation [4].

The main comparative advantages of PFHE over CWHE are thehigh compactness and lower cost per unit refrigeration duty [60].However, PFHE are more limited in size and operating pressure[31].

A key issue in modeling and design of PFHE is the n efciency.Extensive research is focused on the specication of the n geom-etry. Fin types include plain, triangular, perforated, serrated,among others.

3.4. Perforated plate

This type of HE has found extensive use for small scale refriger-ators. A comprehensive review of the history and applications ofperforated-plate HE (PPHE) is given by Venkatarathnam andSarangi [116]. The geometry consists of several parallel perforatedplates separated by spacers.

In this conguration, heat transfer in two directions needs to beconsidered. The main heat exchange between streams occurs later-ally through the high-conductivity plates, usually copper or alumi-num, acting as ns. Longitudinal heat conduction is avoided tosome extent by the use of a relatively low-conductivity materialfor the spacers, such as stainless steel.

The periodic disruption of the ow when passing through theorices provides high non-equilibrium heat transfer coefcient.At the same time this effect produces a relatively large pressuredrop.

The consideration of longitudinal heat conduction is critical forthe design of PPHE. Other main challenges for design are a properestimation of the heat transfer coefcient, and the performance ofthe plates acting as ns.

Table 1Summary of HE geometries. HTC = heat transfer coefcient. LHC = longitudinal heat co

CWHE P

Advantage(s) Multiuid, robust to transients MDisadvantage(s) High cost SScale Large LHeating surface density (m2/m3) 50150 3Cost per unit duty Very high MModeling challenges Flow distribution F

370 J.C. Pacio, C.A. Dorao / Cry3.5. Regenerators

Regenerators present a design that is conceptually different tothe above mentioned geometries. In a regenerator both uids occu-py the same space alternately and the transfered heat is temporar-ily stored in a packing material, called the matrix. Therefore theyrun in a cyclic operation, storing and releasing energy from the ma-trix. A review on the historical development of regenerators andapplications in cryogenics can be found in the book by Ackermann[1]. They are widely used for small-scale single-phase gaspurposes.

The main advantage of regenerators is their extremely high areadensity, reaching up to 6500 m2/m3 [1]. The periodic ow reversalprovides a self-cleaning mechanism. In spite of the inclusion of aswitching device, they are generally simple to construct, resultingin a relatively low-cost component. An important disadvantage isthe occurrence of some mixing between streams, since they alter-nately occupy the same physical space.

Considering that regenerators operate in a periodic mode, a keyparameter for the transient modeling is the heat capacity of thematrix material. In addition, given the usually high operating fre-quencies the switching devices requires special considerations.

4. Heat exchanger models

This section deals with the present models used for thermal-hydraulic design of HEs. A complete analysis of a heat exchangermust consider mechanical and corrosion effects, and other issuesas fabrication and shipping procedures should be taken into ac-count. However, these effects exceed the scope of this review, lim-ited to thermal-hydraulic models.

Heat exchanger thermal-hydraulic modeling involves the solu-tion of two problems: rating and sizing [99]. Rating consists ofevaluating the performance of an existing HE. Since all the relevantinformation is given (geometry, ow conditions), detailed modelscan be used. Sizing refers to the opposite problem, that is to selectthe proper HE geometry, ow arrangement and size to meet thespecied performance within some given constraints. The geome-try is still unknown, and for that reason simpler models arerequired.

A possible classication of the present HE models used for siz-ing is presented in Fig. 3. They can be grouped in three main cate-gories: lumped parameters, distributed parameters and stream-evolution, which are further explained in detail in the followingsubsections and summarized in Section 4.4 and Table 2.

4.1. Lumped parameters models (LPM)

These models represent the basic design theory for HE and canbe found in most textbooks [52,58,99]. They are based on the fol-lowing energy balances for two single-phase streams:

CcdTc UdATh Tc 2ChdTh UdATh Tc 3Fig. 3. Summary of models for sizing heat exchangers.

ions

DPM

Zone

yogewhere the heat capacity rate (C), dened as the mass ow rate _mtimes the specic heat (cp) is used.

C _mcp 4Eqs. (2) and (3) are integrated considering the followingassumptions:

Steady-state operating conditions. No heat transfer with the surroundings. Longitudinal heat conduction is negligible. Constant overall heat transfer coefcient. Constant heat capacity.

Following this integration, the HE is represented with twoparameters: one for the physical size and another for thethermal performance. This category includes ve different models[99, ch. 3], namely mean temperature difference (MTD), e NTU,P NTU, W P and P1 P2. The rst two methods are the mostwidely used for cryogenic applications. All these lumped parame-ters models (LPM) yield the same results, since they solve thesame equations under identical assumptions. The only differencebetween them is the selection of the parameters. In general,lumped parameters models are meant to be used for single-phaseow with constant properties.

4.1.1. Mean temperature difference (MTD)The local differential heat ux (dq) between streams is given by

the product of the local temperature difference (DT), the differen-tial area (dA) and an overall heat transfer coefcient (U), that is:

Table 2Review of HE models. Evaluation of interesting effects for cryogenic and LNG applicat

Effect LPM

Single-phase ow Two-phase ow Flow mixing (partial) Flow mixing (complete) Changes in uid properties Multiple streams Multi-component mixture Flow injection/removal Heat-in-leakage Axial thermal conduction Flow maldistribution Effect of pressure drop on heat transfer

J.C. Pacio, C.A. Dorao / Crdq UDT dA 5Integrating (5) and considering a constant heat transfer coefcient(HTC), an effective mean temperature difference (DTm) can be con-sidered as acting through the total area (A)

DTm QUA 6

In the case of co-current or counter-current ow, the integration re-sults in a logarithmic mean temperature differenceDTlm, dened as:

DTlm Th Tchot end Th Tccold endlnTh Tchot end=Th Tccold end7

Since (7) applies to two of the most common ow arrangements,this method is also known as the logarithmic mean temperature dif-ference (LMTD) method. For other geometries, a correction factor F,given by (8), is employed.

F DTmDTlm

! Q UAFDTlm 8e-NTU. This method is widely used and is found in most textbookson heat transfer. Its simplicity makes it very useful for the economicanalysis of heat recovery. However, is of little help for the designengineer [104] if its assumptions are not fullled, which is a com-mon scenario in cryogenic systems. The thermal performance is ex-pressed in a dimensionless way using an effectiveness (e) dened asthe ratio of actual heat duty (Q) and the maximum achievable withthe given inlet conditions (Qmax).

e QQmax

9

The model uses two nondimensional parameters: the ratio of heatcapacity rate of both uids (C) and a number of thermal units(NTU) that relates the HTC and physical size area with the mini-mum heat capacity rate.

NTU UACmin

10

C CminCmax

11

For a given ow arrangement, the three variables e, NTU and C arerelated, and two useful expressions can be obtained. On the onehand, (12) allows to rate the performance of a given HE. On theother hand, (13) gives the required size, in terms of NTU, for the de-sign conditions. Basic forms of the solutions are available for bothEqs. (12) and (13).

e /NTU;C; flow arrangement 12

. : effect considered, : effect not considered.

SEM

s Elements ASPEN GENIUS

nics 51 (2011) 366379 371NTU /e;C ; flow arrangement 13

P-NTU. This method employs two individual parameters for eachstream. The rst parameter, a number of thermal units, is denedfor each stream as:

NTUh UACh ; NTUc UACc

14

The second parameter is a heat capacity rate ratio (R) for eachstream:

Rh ChCc ; Rc CcCh

15

The thermal performance of the HE is evaluated with individualtemperature effectiveness (P) dened as:

Pc QCcDTmax ; Ph Q

ChDTmax16

According to this denition, the temperature effectiveness P isrelated to e as given by (17)

4.3. Stream evolution models (SEM)

4.3.1. Aspen plate n exchanger

etries are cosine or hyperbolic cosine proles, as reported byBassiouny and Martin [16], Bassiouny and Martin [17]. In addi-

ogePc eCminCc ; Ph eCminCh

17

In general, for a given ow arrangement, P depends on the HE sizeand the heat capacity rate ratio, that is:

Pc /cNTUc;Rc; flow arrangment 18Ph /hNTUh;Rh; flow arrangement 19The comparative advantage of the P-NTU method is that, with theuse of individual parameters, it is not necessary to identify thestream with the minimum heat capacity ow rate.

W P. This method, rst proposed by Mueller [67], introducesthe parameterW that represents a nondimensional effective meantemperature difference.

W DTmDTmax

20

This parameter is related to the previous models as:

W eNTU

PcNTUc

PhNTUh

21

The W P method was introduced by Mueller [67] as a simplergraphical representation of the HE performance.

P1 P2. Roetzel and Spang [93] proposed this method as a sim-pler way of graphical representation of the HE performance. It doesnot introduce new parameters, but recommends the use of thetemperature effectiveness of both streams.

4.1.2. Other efcienciesClayton [27] proposed the use of a two new parameters to rate

the HE performance. They are a HE efciency g and a new effective-ness En.

The rst one is the ratio of the actual heat transfer rate to theone that would be achieved if both uids had an innite specicheat. It is always g < 1. Its use is advantageous for small-sizeexchangers, since in this case g? 1, while e? 0, thus reducingthe relative sensitivity to uncertainties. With these considerations,a boiler-condenser system has an efciency of g = 1.

A new effectiveness En is dened as the ratio of the actual heattransfer to the one obtained by direct mixing. According to this def-inition, it can take values larger than unity. In particular, an inniteco-current ow HE would have En = 1, and a balanced innitelylarge counter-current ow case would give En = 2. The use of thisnew effectiveness identies more clearly the balanced heat capac-ity ow rates case as an optimal design, as opposite to the e NTUmethod, which indicates that e is larger when C is lower.

4.2. Distributed parameters models (DPM)

These models are based on dividing the HE in elements of var-iable size and applying a lumped parameters model in each ofthem; the e NTU and MTD being the most common ones. Theapplication of a LPM is then restricted to a small region wherethe assumptions listed above in Section 4.1 are better fullled.They are widely used for applications with complex ow arrange-ments, such as air conditioning systems and heat pumps. This canbe done at two different levels: zones or elements [49]. Optimalresults are obtained using a mixed approach.

4.2.1. ZonesEvaporators and condensers are usually divided in three zones:

single-phase liquid, two-phase and single-phase vapor. Thisapproach applies a lumped parameter model over the whole

372 J.C. Pacio, C.A. Dorao / Crytwo-phase region, where its assumptions (listed in Section 4.1)are not strictly valid. In particular, the assumption of constantHTC is only applicable to a certain extent in scenarios where thetion, this approach does not consider interaction betweenlayers.

This code represents a powerful tool for the modeling of PFHE,ASPENTech [9] offers this commercial software for simulation ofPFHE, that can be integrated into its proprietary process simulator.Some relevant features include:

Pressure drop is evaluated in detail, including the localizedeffects in distributors, headers and nozzles. However, its conse-quences on heat transfer are not considered, since it is com-puted a posteriori.

Flow maldistribution can be considered to a certain extent in alayer-by-layer simulation mode. The evolution of each layer ismodeled individually, given an imposed ow distribution pro-le, that can be either linear or parabolic. However, it shouldbe noticed that the usual ow distribution for plate-type geom-SEM are based on steady-state one dimensional mass, momen-tum and energy balance equations for each individual stream. Thisfeature makes them appropriate for multi-stream heat exchangers,which are often used in cryogenic applications, in particular forLNG processes. The inclusion of the mass balance allows the eval-uation of the individual compositions of vapor and liquid in boilingand condensation of mixtures, and the momentum equation isused to evaluate the pressure drop. While a one-dimensional anal-ysis is simple and fast, heterogeneous behavior through the crosssection is neglected.

These models are usually implemented into proprietary soft-ware, and their key features are related to the correlations usedfor uid properties and heat transfer and pressure drop character-istics. Two proprietary programs are reviewed in this section: As-pen Plate Fin Exchanger

, offered by AspenTech and GENIUS,

developed by Linde AG. Other models, such as the one proposedby Fredheim et al. [39], are designed as user-dened subroutineintegrated into a process simulator.thermal resistance is dominated by the single-phase stream. Forthis reason, this approach has been used by several authors formodeling air-cooled condensers and air-heated evaporators[30,35,69]. Orth et al. [75] further divided each zone into elements.

4.2.2. ElementsThe heat exchanger is divided in elements of some given phys-

ical length, in a geometry-oriented approach. This approach is use-ful for complex geometries like air conditioning and heat pumpsystems with multiple tube passes in several directions [33,59].Since this model is not ow-oriented, the transition between sin-gle- and two-phase is not intrinsically considered, which may leadto some numerical problems. These issues can be solved by reduc-ing the element size to a tube-segment, as proposed by severalauthors [80,112], therefore reducing the size of the transition ele-ment, although this scenario is only conditionally stable. Iu et al.[48] further divided the transition elements into two zones, in amixed geometric- and ow-oriented approach.

nics 51 (2011) 366379with the advantage of its integration to a process simulator.Although it considers several effects, a complete description ofthe HE is not achieved, as discussed in Section 4.4 and Table 2.

developed an tested by Linde AG to model CWHE. Although this

applications. Some important conclusions can be extracted from

opment. However, this author recognized that no reasonably sim-

Kumar and Sarangi [57] noted that the use of averaged constant

yogethis table:

1. All models are capable of modeling complete ow mixing. Thiscan be simply done by using averaged values in a one-dimen-sional analysis. However, ow mixing is not always complete,and a partial mixing analysis would require a two- or three-dimensional model, which none of them considers. In the caseof multi-component mixtures, this situation may lead to massconcentration proles that affect the heat transfer performance.

2. The consequences of pressure drop on heat transfer is neglectedin all cases. The importance of this effect is discussed in Section6.

3. The most advanced models (stream evolution category) are pro-prietary. Furthermore, some of them are not commerciallyavailable.

4. The model given by ASPENTech [9] is the only one that consid-ers ow maldistribution in a layer-by-layer approach. All layersare considered to be identical except for the mass ow rate. Thisallows to evaluate the consequence of a given distribution pro-le and cannot be used for predicting the existence of owmaldistribution.

In summary, present heat exchanger models are not capable ofconsidering all physical effects relevant for cryogenic and LNGapplications.

5. Other effects reported in literatureprogram is not commercially available, it is used by one of the onlytwo manufacturers of CWHE. Some important features include:

Local heat transfer coefcients (HTC) and pressure gradients areused, evaluated with the local uid properties. However, theHTC is only dependent on the local enthalpy and the wall tem-perature, neglecting the effects of changes in pressure (inletvalue is assumed).

Heat losses and heat feed can be considered. To our knowledge,this is the only code that includes heat exchange with the ambi-ent. However, no details in geometrical location of the heatsource/sink can be incorporated. In cryogenic systems, thisexchange occurs in the outer part of the HE, producing temper-ature and densities heterogeneities in the shell-side ow thatlead to some degree of ow mixing.

Cross-sectional averaged temperatures are used. In addition, amean uid velocity is considered. This means that the HE ismodeled at a stream-by-stream level, and not layer-by-layer.

The rst version was released in 1993 and since then large ef-forts were focused on developing and testing correlations againstreported literature and measurements performed in Linde AG lab-oratories. Although this gives an extensive level of validation to thecode, it implies that all the physical effects that are not consideredby the model, are implicitly taken into account by the correlations.

4.4. Summary. Features for cryogenic applications

Table 2 summarizes the HE models introduced in this sectionand their consideration of physical effects relevant for cryogenic4.3.2. GENIUS, by Linde AGThis program, presented by Steinbauer and Hecht [103], was

J.C. Pacio, C.A. Dorao / CrSection 4.4 highlights the limitations of the available models toconsider all relevant effects. This situation has been noticed by sev-eral authors. The present section reviews the available literature onproperties fails to predict the location of points with minimumtemperature difference (pinch points). This pinch points occurwhen the heat capacity ow ratio Ch/Cc varies from a value of lessthan unity to one larger than unity, or vice versa. This situation islikely to occur in cryogenic systems with a close-to-balanced de-sign. In this case, the authors recommend a nite-difference solu-tion. The considered working uids were both normal- and para-hydrogen.

Soyars [101] studied a cryogenic helium HE using discretizede NTU and LMTDmodels. The author noticed that the e NTU ap-proach relies on the estimation of the heat capacity ow. In close-to-balanced design condition, a poor estimation of the heat capac-ity may lead to sections of the HE with a value of e larger thanple procedure is successful when large variations are present.While most published works were limited to co-current and coun-ter-current ow arrangements, Roetzel [91] extended this study tocrossow HE, considering both length and temperature effects.

5.1.1. Specic heat capacityKays and London [55] suggested the use of constant values for

the uid properties in a recuperator, evaluated at the mean tem-perature. This approach is valid when the variation is limited to afactor of 2. However, this is usually not the case in cryogenic sys-tems. Chowdhury and Sarangi [24] studied the case of supercriticalhydrogen in the temperature range 30080 K and 30040 K, wherethe specic heat varies by a factor 4. They observed that the use ofa harmonic mean specic heat gives good results for balanced owHE, but present some deviations for unbalanced, high NTU cases.The same approach (harmonic mean) was proposed by Sahooand Sarangi [94] for the analyisis of regenerators. They observedthat the overall effectiveness is correctly calculated, however thismethod does not satisfactorily predict the temperature proles.improvements to these models. This research has been focused inthe main effects that should not be neglected, as described in Sec-tion 2.2, that is:

1. Changes in uid properties.2. Flow maldistribution in parallel channels.3. Longitudinal thermal conduction.4. Heat exchange with the surroundings (heat leakage).

The following subsections cover the state-of-the-art on these ef-fects. Section 5.5 describes the literature on combination of thesefour effects. Finally, the review described in this section is summa-rized in Tables 38.

5.1. Changes in uid properties

Distributed parameters and stream-evolution models, dis-cussed in Sections 4.2 and 4.3, deal with this effect in a discretizedapproach. Some other approaches have been reported in literature,for example the use of averaged or effective values, and are re-viewed in this section.

For single-phase ow HE design, the most relevant effects aregiven by the specic heat capacity and the overall heat transfercoefcient (HTC). Most research has been focused on their separateeffect. The combination of both effects has been studied by Roetzel[90]. He recommended the use of suitable averaged values basedon the calculation at two points. Following this work, Peters [77]included the variation of HTC in length due to laminar ow devel-

nics 51 (2011) 366379 373unity. This situation is not physical, and leads to an erroneous siz-ing of the HE. For this reason, the author recomended the use of adiscretized LMTD method.

ogeTable 3Summary of available literature on the effects of variations in specic heat capacity.

374 J.C. Pacio, C.A. Dorao / Cry5.1.2. Heat transfer coefcient (HTC)This effect has been studied as early as 1933 by Colburn and de

du Pont [28]. They presented an analytical solution for the counter-current ow case where the overall HTC varies linearly with thetemperature of one of the uids, keeping all other uid properties

Method Observations

Use cp at mean temperature Up to a factor 2 variationHarmonic mean (recuperators) Up to a factor 4 variation. MisleHarmonic mean (regenerators) Predicts well e, but the temperaDistributed parameters LMTD preferred over e NTUFinite-difference Identies pinch points

Table 4Summary of available literature on the effects of variations in heat transfer coefcient.

Method Observations

Review (1977) Widely studied for singleLogarithmic mean of U DT For linear functions U(Tc,Arithmetic mean of U DT For linear functions U(Tc,Four-points integration For polynomial or powerNon-linear energy equations General functions U(T,DT

Table 5Summary of available literature on the effects of ow maldistribution.

Method Observation

Divide the HE in two sections, step prole Flow mixinHeat transfer coefcient depends on uid velocity 2% change iAxial dispersion Transient aPlate condensers Optimum nTransverse heat conduction Similar effeEqual pressure drop constraint Secondary m

Table 6Summary of available literature on the effects of longitudinal heat conduction.

Method Observation

Review (1994) Most relevanPioneer work, non-dimensional Maximum eConduction in outer wall Less severeRegenerators (conduction in uid) Only signicRegenerators (conduction in matrix) Gives optim

Table 7Summary of available literature on the effect of heat exchange to the surroundings.

Method Observations

Heat exchange to only one stream Some graphical results. For cryogeHeat exchange to both streams Counter-current ow arrangementUniform heat source Analytical solution, useful as a guidP-NTU method If Cc < Ch, an optimum NTU exists

Table 8Summary of available literature on combined effects.

Method Observations

Flow maldistribution and LHC Tend to eliminateHeat exchange to surroundings at one end, and LHC An optimum valuChanges in uid properties, LHC and heat exchange to

surroundingsComputing their10%nics 51 (2011) 366379constant. This approach uses a logarithmic mean combined func-tion of HTC and temperature difference. This means that the aver-age HTC is calculated on the basis of the extreme values.

Gardner and Taborek [41] presented an up-to-date review of HEmodels with variable HTC. For counter-current ow, the model

Reference(s)

[55]ading results for unbalanced, high NTU cases [26]ture proles are incorrect [94]

[101][57]

Reference(s)

-phase [41]Th) in counter-ow arrangement [28]Th) in multipass ow arrangements [18,40,110]-law functions U(DT), 1% uncertainty [100]) [54]

s Reference(s)

g improves performance [36]n e compared to [36] [82]nalysis [92,95]umber of plates [8487]ct to ow mixing in [36] [51]aldistribution further reduces performance [76]

s Reference(s)

t in counter-current [98]ffect for balanced, high NTU. [56]than in inner wall [115]ant for high reduced length HE [96]um value for charging time [32]

Reference(s)

nics, heat exchange to the cold uid is more severe [12,24,78,97]is less affected [6,5]eline [72,3]

[6264]

Reference(s)

each other for high NTU, and augment each other for low NTU [21,81]e for NTU is found to exist [44,71]effect simultaneously or separated gives similar results, within [43,72]

er temperature gradients. Finally, they recommended the use ofthe model given by Kroeger if the conduction in the inner wall is

yogegiven by Colburn and de du Pont [28] is recommended. The use ofan arithmetic mean of product of HTC and temperature differencehave been suggested by Bowman et al. [18] for a general multipassow arrangement, assuming a linear dependence of the HTC ontemperature. Its use has been found to be satisfactory by laterauthors [40,110] within an accuracy range of 10%. For generalfunctional dependencies of the HTC, the method of Kao [54], basedon solving simultaneous non-linear energy balance equations, isrecommended.

Recently, Sharqawy and Zubair [100] presented numerical andanalytical solutions for more complex variations of the HTC. Theystudied polynomial and power-law dependences on the local tem-perature difference and compared the results with experimentaldata available in the literature. These authors recommend theuse of a four-points numerical integration which predicts therequired surface area within a 1% uncertainty.

5.2. Flow maldistribution in parallel channels

Extensive research on this eld was focused on modeling anddesign of distribution headers to improve the ow distribution.However, ow maldistribution can occur for several reasons otherthan header performance, summarized by Mueller and Chiou [68].Since maldistribution occurs in most practical cases, its effect onHE performance must be quantied. In general, most authors agreethat this effect is negligible for low-effectiveness HE.

The effects on high-effectiveness HE, as used in cryogenics, wasrst studied by Fleming [36]. In particular, he studied a counter-ow HE where one side is uniformly distributed and the otherone is not. This latter stream with maldistribution is modeled asa fraction FL of the channels with lower-than-average ow and afraction 1 FL with higher-than-average ow. The model assump-tions include a constant HTC, independent of the velocity, which isreasonably valid only for fully developed laminar ow. He pre-sented results in terms of e and an effective value of NTU, whichcan be reduced to less than half of the design value. The situationis improved if the uniform side is completely and continuouslymixed, as an idealization of shell-and-tube or coil-wound geome-tries. The main conclusion of this work is that, for high-effective-ness cryogenic applications, there is very little to be gained fromincreasing the design NTU of a HE, but rather the ow distributionshould be improved.

Rao and coworkers studied the effect on plate-type HE in sev-eral articles [8487]. They considered the ow distribution prolesuggested by Bassiouny and Martin [16], Bassiouny and Martin[17]. Their study includes the dependence of HTC on velocity,and the analysis of single- and multi-pass HE for single-phase lam-inar ow as well as condensers. Their results indicate that theZ-type exchangers are more severely affected by maldistribution.In addition, they noticed that increasing the number of platesimproves the performance only up to certain optimum point. Fur-ther increase produces a higher extent of maldistribution, whichresult in a reduction of performance.

Roetzel and Ranong [92] introduced an hyperbolic axial disper-sion for the steady-state analysis of HE with ow maldistribution.In this framework, they studied the response to imposed distribu-tion proles. The selected proles were linear and quadratic. Theexistence of back-ow in some channels was also considered. Thismodel was later extended to transient analysis [95].

A cross-ow arrangement was considered by Ranganayakulu[82]. They investigated the inuence of changes in the HTC as aconsequence of the variations in the velocity. According to their re-sults, this inuence is limited to a 2% effect on the effectiveness e,

J.C. Pacio, C.A. Dorao / Crcompared to a simpler model, asumming constant HTC. The dete-rioration of thermal performance was studied considering four dif-ferent ow distribution proles.much higher than in the outer wall.The effects of LHC have also been investigated in regenerators.

Sarangi and Baral [96] considered the axial conduction in the uid.In this case, this effect is relevant because of the eddy thermal con-duction given by the ow through porous media. Solving transitoryenergy balances, the authors concluded that the consequences inperformance are signicant for high reduced-length HE. In theirstudy case, the expected ineffectiveness was doubled by the effectof LHC.

The heat conduction in the matrix was considered by Das andSahoo [32], using a similar procedure. Performing an optimizationbased on the second law of thermodynamics, the authors observedthat, in order to improve the performance, the charging timeJung and Jeong [51] studied the effect of transverse heat con-duction in single-body HE for counter-current single-phase ow.This condition provides some extent of thermal coupling betweenthe different channels. The consequence is similar to that of owmixing, that is, provides a more homogeneous temperature prole,thus reducing the effect of ow maldistribution in performance.This situation is interesting for the design of anisotropic equip-ment, such as perforated-plate HE.

Recently, Pacio and Dorao [76] presented a numerical analysisincluding the momentum equation for two-phase ow. Theyconcluded that the pressure-drop coupling between channels isexpected to produce a secondary maldistribution that furtherreduces the HE performance.

5.3. Longitudinal heat conduction (LHC)

The work by Hennecke [45] indicates that LHC in a single-phaseuid can usually be neglected, except when the Prandtl number isvery low, which is the case for liquid metals. In cryogenic systems,then, LHC is only considered to occur in the wall. The effect of niteLHC is to atten the wall temperature distribution, thus reducingthe performance of a given HE.

A comprehensive review on this subject is given by Shah [98]. Inthis work, the author stated that the effect of LHC is negligible forco-current ow, since in this case the temperature gradient in thewall is small. In addition, this effect is the largest for cross-ow HE,given the two-dimensional proles. For this reason, the researchlist on cross-ow arrangement is long [22,47,83,117,118]. How-ever, this type of exchangers is frequently used for applicationswithin a lower range of effectiveness, usually e < 0.8. The consider-ation of this effect is most relevant for counter-current ow.

In this scenario, the work by Kroeger [56] stands out as a com-prehensive analysis for a wide range of operating conditions. Theeffect of LHC on HE performance is more important with increasingnumber of thermal units, and it is maximum for a balanced opera-tion, that is Ch = Cc. Results were obtained numerically and pre-sented graphically and by means of approximate expressions.These results were later conrmed by Chowdhury and Sarangi[23] and Narayanan and Venkatarathnam [70].

Following the work by Kroeger, Venkatarathnam [113] consid-ered LHC in perforated-plate HE. In this case, an axial conductionparameter is dened as a function of the number of spacers andtheir thermal conductivity.

Venkatarathnam and Narayanan [115] studied the LHC in theouter wall of a tube-in-tube HE. In general, they concluded thatthe degradation of performance is lower as compared to the LHCoccurring in the inner wall. This result is a consequence of the low-

nics 51 (2011) 366379 375should be increased to an optimum value. Cryogenic regenerators,however, are usually designed to operate at high frequency, thusresulting in a short charging period.

deterioration of HE performance tend to eliminate each other inthe regions of higher NTU, but tend to augment each other in the

oge5.4. Heat exchange with the surroundings (heat leakage)

This effect might be benecial if the objective is to cool downhot uids (heat rejection) or heat up a cold stream, such as in rega-sication of LNG. However, in most cryogenic processes, the maingoal is refrigeration below ambient temperature, and in these casesheat leakage has a negative effect.

The physical effect of heat-in-leakage depends on which streamis being heated. On the one hand, if the cold stream receives heatfrom the surroundings, the temperature difference (and heat ex-change) between streams is reduced, and even temperature crossmay occur. On the other hand the opposite situation occurs if heatis transfered from the environment to the warm stream, and theheat duty is increased. However, keeping in mind that the objec-tive is to refrigerate the warm stream, its outlet temperature ishigher than predicted by an adiabatic model. This situation reectsthat the total heat duty (or, in dimensionless form, the HE effec-tiveness e) is not sufcient to rate the performance.

The available literature on this subject is limited to single phaseapplications. Barron [12] studied a counter-current HE assumingconstant uid properties. In this framework, he presented non-dimensional heat transfer equations for the external heat transferto the cold and hot uids separately. His results showed that whenthe environmental temperature is greater than the temperature ofthe inlet hot uid temperature, as in cryogenic applications, theheat transfer to cold uid has a more pronounced effect on theHE performance.

Chowdhury and Sarangi [25] studied a double-pipe for cryo-genic applications from a design perspective. In this geometry, onlythe stream owing in the annulus interacts with the surroundings.They presented analytical solutions and an estimation of thereduction in the effective number of thermal units. This situationwas later studied by Prasad [78] for high-temperature applications.

In some occasions, such as microminiature exchangers withthin insulations layers on both sides, both streams are subjectedto simultaneous external heating. This situation was investigatedby Ameel and Vitharana [6], Ameel [5] in co-current and coun-ter-current ow arrangement. They presented analytical resultsconsidering non-dimensional conductance ratios and concludedthat, when considering this effect, the counter-current arrange-ment gives a higher performance.

Seetharamu et al. [97] incorporated the effect of heat leakage ina tube-in-tube HE as a three-uid exchanger where the thirdstream is ambient air at a constant temperature. Changes in uidproperties were also considered in a nite-element formulation.The author observed reduction of performance in all cases, andtemperature cross when the hot uid is cooled by heat transferto the ambient.

Nellis and Pfotenhauer [73] presented analytical results for acounterow HE with uniformly distributed heat load applied toone or both sides. However, this situation is not representative ofcryogenic HE, that can be better represented by a constant ambienttemperature scenario. Nevertheless, the analytical non-dimen-sional solutions are useful guidelines for the numerical analysisof arbitrary heat load proles. A similar study was performed byAl-Dini and Zubair [3] for co-current ow arrangement.

Recently, Mathew and Hegab [62], Mathew and Hegab [63],Mathew and Hegab [64] studied microchannel HE in co- and coun-ter-current ow with heat-in-leakage. Considering individual tem-perature effectiveness for each uid, they concluded that the cold-side effectiveness is increased,while the hot-side effectiveness is re-duced. For unbalanced ows, the results depend on which uid hasthe lowest heat capacity rate. If this is the warm stream, the effec-

376 J.C. Pacio, C.A. Dorao / Crytiveness is always increased with larger NTU. On the other hand,when the cold uid has the lowest heat capacity ow rate, there isan optimum value of NTU that gives a peakmaximumeffectiveness.regions of lower NTU. Nevertheless, in all cases the performanceis reduced, ranging from a few percent up to 30%.

Mehrabian et al. [66] studied the performance of a plate-HEwith a nite-difference model that includes axial conduction inplates and ow channels and the dependence of viscosity on tem-perature. Their results show that LHC in the plates reduces the per-formance of the exchanger, while LHC in the ow channelsimproves it.

The most extensive research is dedicated to the combination ofLHC and heat leakage. This can be divided in two categories. First,some authors assume constant physical properties and simulatethe HE in a non-dimensional analysis. The second category is basedon discretized energy balances that allow for the variation of uidproperties.

Included in the rst category, the work by Narayanan and Venk-atarathnam [71] investigated the performance of a JouleThomp-son recuperator with heat losses at the cold end due to itsproximity to a low-temperature sink. In this scenario, the bound-ary condition at the wall is changed from adiabatic to conductive.Their results indicate that the hot uid exits at a lower tempera-ture. Gupta and Atrey [44] studied both effects, however withoutdrawing general conclusions on their coupling. Their study con-cluded that increasing NTU can cause more degradation due toheat in leak and an optimum value of NTU is found to exist.

The second category is based in a discretized solution. Nellis[72] applied a nite-difference numerical approach to the model-ing of axial conduction, parasitic heat loads, and property varia-tions in a HE. The accuracy of separately calculating theimportant loss mechanisms in a high-effectiveness HE was foundto give reasonable (within 10%) results as compared with the fullmodel. Ghosh et al. [43] extended a simulation algorithm for mul-tistream plate-n HE presented earlier [42] to include LHC, heatexchange to the surrounding, and variable uid properties.

6. Discussion on the state-of-the-art

Section 5 described the available literature on HE modeling.These can be considered advanced models, since they incorporateeffects that are ignored by current models used for HE design, de-scribed in Section 4. In general, they can be divided in two catego-ries: non-dimensional analysis and discretized equations.

The rst category (non-dimensional analysis) consists of theanalytical or numerical solution of dimensionless energy balances.They include the denition of axial conduction parameters andow distribution proles. Although this analysis is useful in a rststage of design, it is only applicable for simple cases with single-phase ow.5.5. Combined effects

The vast majority of the available literature is focused in onlyone of the four effects mentioned above. There are, however, somereported articles on the analysis of the combination of two or moreof them. In all cases, they include the modeling of longitudinal heatconduction (LHC).

Chiou [21] attempted the combined effects of LHC and owmaldistribution for specic imposed distribution proles. Rang-anayakulu and Seetharamu [81] incorporated the effects of non-uniform inlet temperatures in the modeling of cross-ow HE. Aninteresting observation is that these three (LHC, ow maldistribu-tion and temperature non-uniformities) combined effects on the

nics 51 (2011) 366379On the other hand, the vast majority of these models belong tothe second category, namely the numerical solution of discretizedenergy balances. This approach can easily accommodate for

Flow mixing is neglected by some authors, while others assumea complete and continuous mixing. The practical case is an inter-

yogemediate situation, producing temperature non-uniformities. Thereason why this condition was not considered is because it cannotbe successfully modeled by a one-dimensional formulation. Never-theless, this effect is interesting for large shell-type HE, given thatit affects the performance. In the case of multicomponent mixtures,partial mixing can also produce non-uniformities in the mass con-centration proles.

7. Summary

1. HE are key equipment in cryogenic systems. Thermodynamicand economic considerations set high-efciency requirementswhich result in the need for accurate models. The state of theart on HE modeling for cryogenic applications is reviewed inthis article.

2. Cryogenic systems involve two main challenges for the model-ing of HE: complex processes and non-negligible physicaleffects. The complexity of the processes include large tempera-ture ranges, multiple streams, two-phase ow and the use ofmulticomponent mixtures. The required accuracy is such thatphysical effects like changes in uid properties, axial conduc-tion, ow maldistribution and heat-in-leakage cannot beneglected.

3. The geometries used in cryogenic applications are summarizedin Fig. 2 and Table 1. The selection of the HE type depends onthe particular application. In this work, they were presentedfrom a design-challenges perspective.

4. Present HE models used for design were reviewed. In general,they can be divided in three categories: lumped-parameters,distributed-parameters and stream-evolution. Their ability toconsider relevant effect is summarized in Table 2. While thestream-evolution models are the most advanced, they are pro-prietary and, nevertheless, do not take all physical effects intoaccount.variation in uid properties and heat transfer coefcients. In addi-tion, more effects can be included with the consideration of energybalances in the wall. Following the same trend as in Table 2, moreeffects can be considered within this framework. The computa-tional costs, however, are more demanding and for this reason ad-vanced numerical methods, more efcient than the traditionalnite-differences, will be needed. Examples of advanced methodsused by some authors for the solution of HE include: rst-order -nite-element [81,83,88,97], spectral Galerkin [7,50], and colloca-tion schemes [107,108]. In addition, the least-squares spectralelements method presents some potential advantages for its appli-cation to this problem.

Some general comments on the performance of available mod-els for HE design were presented in Section 4.4 and Table 2, con-cluding that they fail to take into account all relevant effects.This situation is improved with the advanced models presentedin Section 5. However, two points are not considered: rst, the ef-fects of pressure drop on heat transfer and second, partial owmixing. Some remarks on these two points are presented next.

Changes in theworking pressure affect the uid physical proper-ties and consequently, the heat transfer performance. While single-phase uid properties are mainly dependent on the temperature,that is not always the case for two-phase ow. Of special interestfor boiling and condensation applications are the variations in thesaturation temperature. This situation can bring the temperaturecurves closer, thus reducing the total heat duty.

J.C. Pacio, C.A. Dorao / Cr5. Extensive research covering the four above mentioned non-neg-ligible effects is available in the open literature, resulting instate-of-the-art advanced models. In general, they can bedivided into two categories: non-dimensional analysis and dis-cretized energy equations. While most works were focused inonly one of these effects, there are some published reports ontheir combination, indicating that no signicant error is intro-duced by considering them separately.

6. The advanced models include more effects than those used fordesign. Nevertheless, two points are not considered: the effectsof pressure drop on heat transfer and partial ow mixing. Thevariation on operating pressure is particularly important fortwo-phase ow, since it affects the saturation temperature.Flow mixing is neglected by some authors, while others assumea complete a continuous mixing. The practical case is an inter-mediate situation, producing temperature non-uniformities.

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