thermodynamic analysis of irreversible heat-transformers

Makara J. Technol. 19/2 (2015), 90-96 doi: 10.7454/mst.v19i2.3039

August 2015 | Vol. 19 | No. 2 90

Thermodynamic Analysis of Irreversible Heat-transformers

Niccolo` Giannetti1*, Arnas1,2, Andrea Rocchetti3, and Kiyoshi Saito1

1. Department of Applied Mechanics and Aerospace Engineering, Waseda University, 3-4-1 Okubo, 4 Shinjuku-ku, Tokyo 169-8555, Japan

2. Department of Mechanical Engineering, Faculty of Engineering, Universitas Indonesia, Depok 16424, Indonesia 3. Department of Industrial Engineering of Florence, Via Santa Marta 3, Firenze 50139, Italy

*e-mail: [email protected]

Abstract

Absorption heat transformers extend the possibilities for efficient and environment-friendly energy conversion processes. Based on a general thermodynamic model of three-thermal cycles with finite thermal capacity of the heat sources, this paper is intent upon analyzing and optimizing the performance of absorption heat transformers, by including the influence of irreversibility in the analytical expression of the system efficiency. Dimensionless parameters for an overall optimization are defined and a first screening is performed to clarify their influence. Dependence on the main factors is highlighted to suggest how to change them in order to enhance the whole system performance. Under this point of view, the analysis evaluates coefficient of performance (COP) improvements and can be used to perform existing plant diagnostics, besides predicting the system performance. The use of this criterion is exemplified for specific heat transformers data from literature. This approach identifies the limitations imposed to the physical processes by accounting for the inevitable dissipation due to their constrained duration and intensity, and constitutes a general thermodynamic criterion for the optimization of three-thermal irreversible systems.

Abstrak

Analisis Termodinamika Ireversibel Heat-transformator. Penyerapan panas transformator memperluas kemungkinan adanya proses konversi energi yang efisien dan ramah lingkungan. Berdasarkan model termodinamika umum siklus three-thermal dengan kapasitas termal yang terbatas dari sumber panas, tulisan ini bertujuan untuk menganalisis dan mengoptimalkan kinerja penyerapan panas transformator, dengan memasukkan pengaruh ireversibilitas dalam ekspresi sistem analitis efisiensi. Parameter berdimensi untuk optimasi secara menyeluruh dan yang pertama dilakukan adalah penyaringan untuk memperjelas pengaruh masing-masing. Ketergantungan pada faktor utama menyoroti dan menunjukkan bagaimana mengubah mereka untuk meningkatkan kinerja sistem secara keseluruhan. Melalui sudut pandang ini, analisis evaluasi perbaikan COP dapat digunakan untuk melakukan diagnosis tanaman yang ada, selain memprediksi kinerja sistem secara keseluruhan. Pendekatan ini mengidentifikasi keterbatasan dalam proses fisik, yang berkontribusi untuk pembuangan tidak terelakkan terkendala lamanya penyinaran dan intensitas, merupakan sebuah kriteria termodinamika umum untuk optimalisasi sistem three-thermal ireversibel. Keywords: dimensionless parameters, efficiency improvement, heat transformers, irreversibility, three-thermal systems 1. Introduction Thermodynamic principles constitute the basis of the technical development towards a rational use of energy. In fact the use of the first principle of thermodynamics will suffice to determine the plant COP at given operating conditions. On the other hand, in order to evaluate COP improvements and perform plant diagnostics, besides predicting the energy conversion system performance, an accurate evaluation of irreversibility, namely the use

of the second principle of thermodynamics, becomes essential. Since every real process occurring as part of an energy conversion system is associated to an unavoidable degradation of the earliest amount of energy, the formulation of optimization criteria should identify the limitations imposed to the physical processes by accounting for the inevitable dissipation due to their constrained duration and intensity, indicate the most relevant parameters and how to change them.


Makara J. Technol. August 2015 | Vol. 19 | No. 2

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Absorption heat transformers have unique characteristics if compared to other energy conversion systems, which is to deliver part of the earliest amount of heat input, from the generator and evaporator temperature level to a higher temperature. These systems are being used increasingly to produce steam from medium temperature waste heat, so that the latter can be reintroduced in the industrial process and reused. A considerable amount of literature has been published about theoretical and experimental studies on different configurations and applications [1-3]. However, the increased complexity of these devices is often prohibitive with regards to a comprehensive modelling of all the details at play, making the calculation very difficult or impossible, and the physical content obscure. An absorption system, in its simplest arrangement, transfers heat between three temperature levels, but more often between four thermal sources with finite heat capacity [4-5]. Three-thermal system has been originally modelled as the combined cycle of an endo-reversible two-heat sources engine driving an endo-reversible two-heat sources refrigerator/ heat pump [6]. In addition, [7] applies entropy production analysis with an analytic irreversible thermodynamic model. By applying thermodynamic analysis to absorption chillers [8] have shown the necessity of accounting for internal dissipation, and defined the concept of Process Average Temperature (PAT). This paper follows [9] and presents a characterization of a three thermal sources absorption heat transformer with finite thermal capacity of the heat sources that considers the influence of both heat and mass transfer irreversibility of the refrigerant and the absorptive solution on the cycle efficiency. 2. Eksperiment Thermodynamic model. The model [10] is exemplified and the analysis is generalized to include both heat and mass transfer phenomena. Considering a single-stage absorption heat transformer application case (Fig. 1), steady cyclic-operability is assumed neglecting the effects of potential and kinetic energy of the refrigerant. Since the circulation pump processes saturated liquid solution, its electrical power consumption is disregarded. Also, heat leaks to the surrounding are considered to be ineffective. With the assumptions stated above, the Coefficient of Performance (COP) is defined as,

A

G E

QCOP

Q Q=

+ (1)

The algebraic form of the first law states,

G E A CQ Q Q Q+ = + (2)

Being defined as a state function, entropy variations of the refrigerant and the solution performing a closed cycle are null and, neglecting other thermal exchanges, internal irreversibility are transferred outside the cycle to the surroundings through the heat exchangers.

( ) ( )F RC RA SA RG SG RES S S S S S S∆ = ∆ + ∆ + ∆ − ∆ + ∆ − ∆ (3) The entropy variation experienced by the pure refrigerant is given by Eq.(4).

o o

R R R i ip

dh vS m s m dp

T T

∂ ∆ = ∆ = − ∂ ∫ ∫

(4) In general,

o

ip

vdp l p

T

∂ = ∆ ∂ ∫

(5) Where l= βv for a liquid, and l=R/p for a perfect gas [11]. This term accounts for internal irreversibility related to pressure change and includes pressure drops contributions. On the other hand, inside the generator and the absorber the terms related to the entropy variation of the aqueous LiBr solution are expressed from Eq.(6) [12].

2

, ,

ln ln1o o o

i i i

S S

T p XLiBrH O

S T p Xsolp X p X

S m s

a ah vm dT dp R dX

T T T M

∆ = ∆ =

−∂ ∂− +∂ ∂∫ ∫ ∫

(6) The circulation ratio is introduced according to Eq.(7), and the refrigerant and solution mass is related to the concentration difference experienced in the absorber and generator.

( ); H LR

S H

X Xmf f

m X

−= = (7)

Combining Eq.(3) and Eq.(7), considering the thermal power exchanged by the fluid streams of the heat exchangers through their respective enthalpy variations, and overlooking pressure losses Eq.(8) is obtained.

Absorber (A)

Generator (G) Evaporator (E)

Condenser (C)

QA

QG Q

E

QC

Wp

TLi

TMi

THi T

Ho

TMo

TLo

Figure 1. Absorption System Schematic

Giannetti, et al.


92

( )( )

( ) ( )( )

RA SAF A

RA SA

RE RG SG RCEG C

RCRE RG SG

f s sS Q

f h h

f s f s s sQ Q Q

hf h f h h

∆ + ∆∆ =

∆ + ∆

∆ + ∆ + ∆ ∆− + +∆∆ + ∆ + ∆

(8)

Terms of the fractions are evaluated referring to the physical state of the refrigerant and the solution, and the characteristics of the heat exchangers. Temperature-entropy diagrams are able to describe thermodynamic transformations in terms of both the first and second laws. In a previous paper, [13] have presented the use of T-s diagram for aqueous LiBr cycles and a real heat transformer cycle (from [14]) is represented in Fig. 2. Additional saturation curves at different solution concentrations are included to extend the depiction to both refrigerant and solution behaviors in absorption cycles. Considering an approximate cycle of the heat temperature boosting cycle (the black lines in Fig. 2), generator and evaporator temperatures are assumed to be close enough to be equal and the vapor generation/absorption processes are split into a constant concentration part embodying temperature changes to reach the equilibrium temperature at the generator/absorber, a constant temperature one representing the release/absorption of the heat of absorption, and an isobaric segment to cool down to its saturation temperature the superheated vapor once separated from the solution at its equilibrium. By defining proper dimensionless parameters characterizing the transformations followed by the refrigerant and the liquid solution, Eq.(9) can be written.

( )G E CAF GE A C

Ei Ai Ci

Q Q QQS

T T T

+∆ = − Θ + Θ + Θ (9)

Where, ( )

( )( )

( )

Ei RE RG SG

GE

RE RG SG

Ai RA SAA

RA SA

Ci RCC

RC

T f s f s s

f h f h h

T f s s

f h h

T s

h

∆ + ∆ + ∆ Θ = ∆ + ∆ + ∆

∆ + ∆Θ =

∆ + ∆∆Θ =

∆

(10)

These dimensionless parameters depend on temperature, concentration and thermodynamic properties of the refrigerant and the solution within the heat exchangers. Further introducing the definition of heat exchangers effectiveness by [15] (Eq. 11), yields to a new expression of the entropy variation (Eq. 12).

( ) ( )( ) ( )( ) ( )

min

min

min

GE GE p Mi Ei GE MGE

A A p Ai Hi A HA

C C p Ci Li C LC

Q mc T T k T

Q mc T T k T

Q mc T T k T

ε

ε

ε

= − = ∆

= − = ∆

= − = ∆

(11)

GE M GE C C CA H AF

Ai Ei Ci

k T k Tk TS

T T T

∆ Θ ∆ Θ∆ Θ∆ = − + (12)

Dimensionless parameters (tH, tM and tL), defined as in Eq.(13), are used to represent temperature levels and temperature differences at the heat exchangers

1

1

1

HiH H H HH

Hi Ai Hi Hi H H

m MiM M MM

Mi Ei Mi Mi M M

LiL L L LL

Li Ci Li Li L L

TT T T tt

T T T T T t

T TT T tt

T T T T T t

TT T T tt

T T T T T t

∆ ∆ ∆= → = =+ ∆ +

∆∆ ∆= → = =− ∆ −

∆ ∆ ∆= → = =+ ∆ +

(13)

Finally, a dimensionless expression (Eq. 14) is obtained.

'' '1 1 1H M L

H M L

t t tG

t t tθ θφ φ= − ++ − +

(14)

Where,

'

''

F

C C

GE GE

C C

A A

C C

SG

k

k

k

k

k

θ

θ

φ

φ

∆=Θ

Θ=Θ

Θ=Θ

(15)

The COP of the system can be expressed as a function of the dimensionless parameters defined above.

1A EG C Lt

E E MG G

Q Q Q Q tCOP

Q Q Q Q tφ+ −= = = −

+ + (16)

Where,

C Lit

GE Mi

k T

k Tφ = (17)

And using the dimensionless expression of the entropy variation G, it is possible to generalize the heat transformer

3’ TEi TGi

T

TA

TC

Xi Yi

54% 52% 0% 0% 52% 54%

s

1

1’ 2

3

4

5

9 8’’

8

12’

12

TLi

TMi

THi

THo

TMo

TLo

Evaporator (E)

Absorber (A)

Condenser (C)

Generator (G) 4’

4’’

4’’’

8’

Figure 2. T-s Diagram of an Absorption Heat Transformer



93

transformer efficiency as a function of either tH and tM (Eq. 18) or tH andtL (Eq. 19).

( ) ( ) ( )( )( )( )( ) ( ) ( )

' ''

' ''

1 1 1 11

1 1 1 1 1M H H M H M

tM M H M H H M

COP

t t t t G t t

t G t t t t t tθ θ

θ θ

φ φφ

φ φ

=− + + − − + −

−− − + + + − −

(18)

( ) ( ) ( ) ( ) ( )( )( ) ( ) ( )

' ''

''

1 1 1 11

1 1 1 1L H L H L L H

tH L H L L H

COP

t G t t t t t t

G t t t t t tθ θ

θ

φ φφ

φ

=− + + + + + +

−− + + + + + +

(19)

3. Results and Discussion The following analysis is based on input experimental data from literature and is intended to explore the results and the potential of the present thermodynamic approach. Table 1 contains measured values of the working parameters from real Li/Br heat transformer. Table 2 shows the values of the dimensionless parameters previously defined and calculated from the equations expressed in the appendix A. Table 2 makes evidence that the present analysis method tends to slightly overestimate the cycle COP and this could be mainly related to the relatively important impact of heat loss and the assumption of negligible work of the circulation pump. Since the analytical interpretation of equations 18 and 19 is complex, a graphical approach is more convenient and understandable. Fig. 3 represents COP curves as a function of either tH (red lines) and tM (black lines), for different values of tL, being other parameters set constant as calculated from [14] in Table 2. These graphs make evidence of the occurrence of a constantly increasing COP with reference to tH and tM, but decreasing with tL. Dashed lines are obtained for the literature value of the secondary variable considered in each graph and the markers represent the operative condition of the real system from [14]. By comparing the actual efficiency with the performance maps obtained, it is possible to perform system diagnostic and show how to improve the overall system efficiency. Considering data from [14], increasing the temperature difference at the high and intermediate temperature heat exchangers, for the same low temperature conditions, or, ceteris paribus, decreasing the temperature difference at the condenser, will bring the system to higher first law efficiency. As an instance, Fig. 3 shows the same system operating in two different conditions, respectively test1 (filled markers) and test2 (empty markers) from [14], making evidence of the fact that an increase of the temperature difference at either the absorber or the generator/evaporator increase the efficiency. However, the values of the temperature parameters are required to positive and limited to those giving positive values of G, standing for the total irreveribilities. Fig. 4 (a) and (b) show

Table 1. Parameters from Literature Experimental Data [14]

Parameter Test1 [14] Test2 [14] THi[K] 359.75 359.25

TGi[K] 347.93 352.53

TGo[K] 351.00 355.60

TMi[K] 359.75 359.25

TCi[K] 318.19 322.76

TCo[K] 297.85 296.35

TAi[K] 381.14 382.22

TAo[K] 377.90 378.99

TLi[K] 294.65 292.85

TEi[K] 344.76 346.26

TEo[K] 353.25 352.45

KA[kW·K-1] 0.03 0.03

KGE[kW·K-1] 0.03 0.03

KC[kW·K-1] 0.03 0.02

XH[-] 0.54 0.54

XL[-] 0.52 0.52

COP[-] 0.27 0.32

tL

0.04

0.11

0.27

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

COP

t

Figure 3. COP Curves as a Function of either th or tM for Different Values oftL

represent COP curves as a function of either tL (red lines) and tM (black lines), respectively, for different values of φt or tH. It can be observed that decreasing either tL or tM is beneficial when tH and φt, and accordingly G, are fixed. Fig. 5 (a) and (b) consider COP curves as a function of either φθ’ (red lines) or φθ’’ (black lines), respectively, for different values of tH or tM. It can be observed that either increasing φθ’’ or decreasing φθ’ is beneficial when G is fixed. In case the dimensionless parameter representing the irreversibility of the cycle G and the temperature parameter defined for the high temperature thermal

Giannetti, et al.


94

source tH are used as variables, COP shows a relentless decreasing trend when the first is increased (Fig. 6). Whereas the opposite trend is shown with respect to tH. Furthermore, if the COP value is fixed, the main dimensionless parameters for system design can be related as follows.

( ) ( )( )( )

( )( )

'' '1 1

1 1

1

1H M M H

H M

M

t M

t t t tG

t t

COP t

COP tθ θφ φ

φ− +

=+ −

−−

− −− (20)

( ) ( )( ) ( )

'' '1 1

1 1 1H L L H

H L

t L

t L

t t t tG

t t

t

COP tθ θφ φ φ

φ+ +

=+ +

−− −

+ (21)

0.16

tH 0.57

0.08

0.26

0.36

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

COP

t

0.38

φt=0.32

0.49

1.57

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6

COP

t (a) (b)

Figure 4. COP Curves as a Function of Either tl or tM for Different Values of φt (a) and (b)tH

0.01

0.02

0.19

0.22

φ 0.29

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6

COP

φ

tM

0.03

0.06

0.170.24

0.30

0.030.06

0.24

0.33

0.33

0.17

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6

COP

φ (a) (b)

Figure. 5.COP Curves as a Function of φ for Different Values of tH (a) or tM (b)

Table 2. Dimensionless Parameters Calculated from Literature Data

ΘE ΘAC ΘG ∆SF tH tM tL φt φθ' φθ'' G COP Test1 [14] 0.88 0.84 1.05 0.005 0.06 0.15 0.13 0.81 0.96 1.31 0.021 0.280 Test2 [14] 0.89 0.83 1.05 0.002 0.06 0.16 0.16 0.64 1.19 1.47 0.010 0.326

0.11

0.21

0.30

G 0.01

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

COP

tH

tH

0.19

0.08

0.16

0.04

0.12

0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.225

0 0.05 0.1 0.15 0.2 0.25 0.3

G l

t

Figure 6. COP Curves as a Function of tH for Different Values of G

Figure 7. G Curves as a Function oftL (Red Lines) and tM (Black Lines) for Different Values of tH



95

Fig. 7, 8 and 9 display the influence of the main dimensionless parameters on the dimensionless function G at constant COP (fixed at value calculated in Table 2 from the data used as a reference [14]) as analytically expressed in Eq. (20) and Eq.(21). By observing Fig. 8, the dimensionless parameter G, which combines first and second principles of thermodynamics, defines a range limitation of the dimensionless temperature parameters tL and tM for a fixed value of the system COP, tH and φt. These practicable ranges narrow down for lower tH. Moreover, with respect to the same parameters, a maximum value of G can be associated to defined values of tL and tM, and those values depends on the dimensionless heat conductance parameter φt (Fig. 8), and on COP (Fig. 9), but not on the value of tH. The operative conditions of the real system described by [14] are plotted in Fig. 7, where it is obvious that the system is designed for a low irreversibility operability, and that test2 bring the system to a less irreversible condition and to a higher COP (as shown in Fig. 3).

φt0.32

0.65

1.05

0.49

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

G l

t

Figure 8. G Curves as a Function of Either tL (Red Lines) and tM (Black Lines) for Different Values of φt

0.15

COP 0

0.39

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.05 0.1 0.15 0.2 0.25 0.3

G s

t

Figure 9. G Curves as a Function of Either tL (Red Lines) and tM (Black Lines) for Different Values of COP

4. Conclusions Based on a general thermodynamic model of three-thermal cycles with finite thermal capacity of the heat sources, and on an analytic thermodynamic approach for the absorption cycle, this paper analyses significant solutions of the performance of heat transformers by considering the influence of internal irreversibility due to temperature and concentration variations of the refrigerant and the solution. These contributions are included in the analytical expression of the system efficiency. The use of this criterion is exemplified for specific heat transformers data from literature. Dimen-sionless parameters for an overall optimization are defined and a first parametric analysis is performed to clarify their influence. Dependence on the main elements is highlighted to suggest how to change them in order to enhance the overall system performance. Under this point of view, the analysis evaluates COP improvements and can be used to perform existing plant diagnostics, besides predicting the system performance. In overcoming the limit of the endo-reversible cycle and capturing the fundamental physical phenomena involved, leaving flexibility in generalizing results to other absorption devices, this model provides a suitable predictive and diagnostic tool. Namely, if pressure losses in the heat exchangers are negligible, QGE, QA and QE can be calculated once concentration, temperatures and thermodynamic properties of the fluids are available. By acting on the heat exchangers temperature differences, represented by the corresponding parameters tL, tM and tH, it is possible to increase the COP of existing systems. Comparing experimental data from literature with the performance suggested by this analysis possible improvement of the system thermodynamic efficiency are pointed out. The dimensionless parameter G stands for the effect of internal irreversibility of the cycle and shows significant impact on the global performance. This approach identifies the limitations imposed to the physical processes constituting the cycle, and constitutes a general thermodynamic criterion for the optimization of three-thermal irreversible systems.

Appendix A.1. Enthalpy and Entropy Considering the entropy and enthalpy difference terms related singularly to the transformations constituting the cycle and appearing in Eq.(9), the analytical expressions of those variations is presented in the following. As a consequence the dimensionless groups defined in Eq.(10) can be developed and calculated, once temperature, concentration and thermodynamic properties of the refrigerant and the solution are defined thorough the cycle. In particular, for the vapor generation process from the refrigerant point of view entropy and enthalpy variations are calculated, respectively, as in Eq.s(A1) and (A2).

Giannetti, et al.


96

,ln abs GGoRG pl

Gi Go

iTs c

T T∆ = + (A1)

( ) ,RG pl Go Gi abs Gh c T T i∆ = − + (A2)

While, from the solution point of view, the vapor generation process is modelled as follows.

2

( , )

( , )

ln lnln

H

L

X H O LiBrGoSG pS pGi XGi X

Gi S pGi XGi

a aTs c R dX

T M

−∆ = +

∫

(A3)

Where, for the estimation of water and lithium-bromide activities a, the calculation procedure presented by [16] is used. The specific enthalpy variation of the generation and absorption processes also refers to [16], in which the molar enthalpy of pure water and lithium-bromide are combined considering the enthalpy excess, as described by [17]. Similarly, with regard to the entropy variation of the absorption process from the solution standpoint Eq. A4 is employed.

2

( , )

( , )

ln lnln

L

H

X H O LiBrAiSA pS pAo XAo X

Ao S pAo XAo

a aTs c R dX

T M

−∆ = +

∫ (A4)

And, for the vapour absorption process from the refrigerant point of view entropy and enthalpy variations are calculated, respectively, as in Eq.s(A5) and (A6).

,ln abs AAi

RA pv

Ao Ao

iTs c

T T∆ = + (A5)

( ) ,RA pv Ai Ao abs Ah c T T i∆ = − + (A6)

Condensation and evaporation of the refrigerant are represented by Eq.s(A7)-(A10).

2

3 1'

ln lnRC Ci

RC pl pv

Ci

r T Ts c c

T T T∆ = + + (A7)

( ) ( )2 3 1'RC RC pl pv Cih r c T T c T T∆ = + − + − (A8)

3' 4 '

4 3 4

ln lnRERE pl pv

r T Ts c c

T T T∆ = + + (A9)

( ) ( )3' 3 4 ' 4RE RE pl pvh r c T T c T T∆ = + − + − (A10)

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thermodynamic analysis of irreversible heat-transformers

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