A generalized analysis for cascading single fluid vapor compression refrigeration cycles using an entropy generation minimization method

International Journal of Refrigeration 23 (2000) 353±365

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A generalized analysis for cascading single ¯uid vapor compression refrigeration cycles using an entropy generation minimization method Eric B. Rattsa,*, J. Steven Brownb a The University of Michigan-Dearborn, 4901 Evergreen Rd., Dearborn, MI 48128, USA The Catholic University of America, 620 Michigan Ave., N.E., Washington, DC 20064, USA

b

Received 31 May 1999; received in revised form 8 August 1999; accepted 18 November 1999

Abstract This paper focuses on cascading an ideal vapor compression cycle and determining the optimal intermediate temperatures based on the entropy generation minimization method. Only superheating and throttle losses of the cycle are considered since they are inherent to the ideal vapor compression refrigeration cycle. The second law equations have been developed in terms of speci®c heats and temperature ratios with the intent of reducing involved property modeling. Also the entropy generation was expressed in terms of a single independent variable and minimized to develop an advanced rule for selecting optimum intermediate temperatures. Results for a cascade system operating between reduced temperatures of 0.684 and 0.981 with R-134a as the working ¯uid are presented. The approximate method presented here predicted the optimum intermediate reduced temperature for a two-stage system to be 0.88, a di€erence of 2% from the optimum. The method presented was a much better predictor of the optimum temperature than the geometric mean method which was 0.82, a di€erence of 5% from the optimum. The entropy generation distribution of the optimum solution was investigated. For a two-stage system, the lower stage and higher stage entropy generation was 44% and 56%, respectively. In comparison to the single stage, the two-stage reduced losses by 78%. # 2000 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Refrigerating system; Compression system; Cascade system; Entropy; Optimization; Calculation; Process

Analyse des cycles frigori®ques en cascade avec compression de vapeur d'un seul frigorigeÁne utilisant une meÂthode de minimisation de la geÂneÂration d'entropie ReÂsume Les auteurs de cet article examinent un cycle aÁ compression de vapeur ideÂal et deÂterminent les tempeÂratures intermeÂdiaires ideÂales utilisant la meÂthode de la minimisation de la geÂneÂration d'entropie. Ils examinent seulement les pertes dues aÁ la surchau€e et aÁ la deÂtente, puisque ces pheÂnomeÁnes sont inheÂrents au cycle frigori®que aÁ compression de vapeur. A®n de simpli®er la modeÂlisation, on a deÂveloppe des eÂquations fondeÂes sur le deuxieÁme principe de la thermodynamique

* Corresponding author. Tel.: +1-313-593-4969; fax: +1-313-593-3851. E-mail address: [email protected] (Eric B. Ratts). 0140-7007/00/$20.00 # 2000 Elsevier Science Ltd and IIR. All rights reserved. PII: S0140-7007(99)00070-5

354

E.B. Ratts, J.S. Brown / International Journal of Refrigeration 23 (2000) 353±365

pour les chaleurs massiques et les rapports de tempeÂrature. La geÂneÂration d'entropie a eÂte exprimeÂe comme variable indeÂpendante unique qui a eÂte minimiseÂe a®n de deÂvelopper une reÁgle permettant de seÂlectionner des tempeÂratures intermeÂdiaires optimales. On preÂsente les reÂsultats pour un systeÁme aÁ cascade fonctionnant avec des « tempeÂratures reÂduites » de 0,684 et 0,981 et utilisant le R134a comme ¯uide frigorigeÁne. La meÂthode preÂsenteÂe ici a permis de preÂvoir une « tempeÂrature intermeÂdiaire optimale reÂduite » de 0,82, cette valeur repreÂsentant une variation de 5 % par rapport aÁ la valeur optimale. On a eÂgalement eÂtudie la distribution de la geÂneÂration d'entropie de la solution optimale. Pour un systeÁme bieÂtageÂ, la geÂneÂration d'entropie aÁ l'eÂtage infeÂrieur et aÁ l'eÂtage supeÂrieur eÂtaient de 44 % et 56 % respectivement. Le systeÁme bieÂtage (par rapport au systeÁme monoeÂtageÂ) a permis de reÂduire les pertes de 78 %. # 2000 Elsevier Science Ltd and IIR. All rights reserved. Mots cleÂs: SysteÁme frigori®que ; SysteÁme aÁ compression ; SysteÁme en cascade ; Entropie ; Optimisation ; Calcul ; ProceÂdeÂ

Nomenclature cf cg cr;P cP dsP h : m mr R: S s T Tc Tr ˆ T=Tc v x

B speci®c heat at saturation of a liquid phase (kJ kgÿ1 Kÿ1), Eq. (44) speci®c heat at saturation of a gaseous phase (kJ kgÿ1 Kÿ1), Eq. (19) speci®c heat ratio at constant pressure. Eq. (15) speci®c heat at constant pressure (kJ kgÿ1 Kÿ1) change in speci®c entropy at constant pressure (kJ kgÿ1 Kÿ1) speci®c enthalpy (kJ kgÿ1) mass ¯owrate (kJ kgÿ1 hÿ1) mass ¯owrate ratio, Eq. (14) gas constant (kJ kgÿ1 Kÿ1) entropy transfer rate (kJ kgÿ1 Kÿ1 hÿ1) speci®c entropy (kJ kgÿ1 Kÿ1) temperature (K) critical temperature (K) reduced temperature speci®c volume (m3 kgÿ1) quality

Greek letters  constant. Eq. (34) temperature ratio. Eq. (23)

1. Introduction Fig. 1 is a state diagram for the ideal vapor compression cycle which demonstrates three of the irreversibilities present in a typical vapor compression refrigeration cycle. The losses shown are the isenthalpic throttling process CD and the heat transfer processes (referred to as
T-losses) at TH and TL . The ®rst
Tloss considered comes about because of heat transfer from the low-temperature reservoir to the refrigerant across a ®nite temperature di€erence
TL . The other


s
T  Subscripts A, B, ... ave f, g gen i, iÿ1, i+1 j max P

volumetric coecient of thermal expansion (Kÿ1) change in entropy (kJ kgÿ1 Kÿ1) temperature di€erence (K) temperature di€erence ratio, Eq. (13) reference to corner states of cycle average saturated liquid and saturated vapor states, respectively generation reference to temperature level or cascade stage reference to cascade stage maximum pressure

Superscripts (t) reference to change in entropy related to the throttling loss. (s) reference to change in entropy related to the superheat loss.


T-loss shown comes about because of heat transfer from the refrigerant to the high-temperature reservoir across a ®nite temperature di€erence. The latter
T-loss is made up of two parts. The ®rst part is due to sensible heat transfer from the superheated refrigerant vapor to the high-temperature reservoir across a ®nite average temperature di€erence
TH;1 . This loss is a direct result of the isentropic compression of the refrigerant from the evaporator exit. The second part is due to latent heat transfer from the refrigerant to the high-temperature reservoir across the ®nite temperature di€erence
TH;2 .

E.B. Ratts, J.S. Brown / International Journal of Refrigeration 23 (2000) 353±365

In Fig. 1, the compression process AB is assumed to be adiabatic and reversible. Common losses found in an actual compression process such as frictional losses, losses due to thermal mixing, and heat transfer losses are all neglected. The processes BC and DA are both assumed to be internally reversible, that is, frictional losses that occur in an actual system are neglected. Any other losses, in addition to
T-losses and the throttling loss, will reduce the thermodynamic performance of the cycle. By analyzing losses in the cycle, the source and magnitude of the irreversibilities can be identi®ed with the intent of reducing them. There have been steady state studies [1±4], as well as cycling studies [2±9], which have identi®ed and quanti®ed losses in system components. The overall system eciency can be improved if these individual losses are reduced. In a typical single-stage vapor compression system, the compressor, the expansion device, the condenser, and the evaporator generate 46, 18, 18, and 9% of the total losses, respectively [1]. Improving the cycle requires reducing these irreversibilities. In this paper, compressor losses are not considered. Both the condenser and evaporator losses are largely due to imperfect heat exchanger losses that are also not considered in this paper. These losses arise primarily due to the limited sizes of the heat exchangers. If size were not a constraint, these losses could be made small. The remaining losses are the throttling loss and the
T-loss associated with the sensible heat transfer in the condenser. These two remaining losses are considered inherent to the vapor compression refrigeration cycle and must be minimized. The cycle thermal eciency of a vapor compression refrigeration cycle can be improved by a technique referred to as cascading. Fig. 2 shows the improvement of cascading two vapor compression refrigeration cycles with respect to the original single-stage cycle identi®ed by the corner states AJGIA. The ®rst stage of the cascade is identi®ed by the corner states ABCDA and the second stage by EFGHE. One improvement that can be

Fig. 1. Single stage vapor compression refrigeration cycle. Fig. 1. Cycle frigori®que monoeÂtage aÁ compression de vapeur.

355

Fig. 2. Two-stage, single-¯uid vapor compression refrigeration cycle. Fig. 2. Cycle frigori®que bieÂtage aÁ compression de vapeur d'un seul frigorigeÁne.

realized in cascade systems is the reduced superheat that results in the high temperature condenser leading to a reduced
T-loss. In addition, a second improvement is the increased area under path DI. This area represents an increase in refrigeration e€ect. The obvious expenses for such gains are the additional components needed, the increased complexity, and the increased system cost. GoÈktun [10] presented a study on the coecient of performance (COP) of a two-stage cascade refrigeration system. He considered the e€ects of heat transfer losses and internal irreversibilities to derive an equation for the COP considering these losses. Prasad [11] investigated a two-stage, single ¯uid, cascade vapor compression refrigeration system with ¯ash intercooling. He obtained a formula for determining the optimum interstage pressure for a R-12 system based on maximizing COP. He neglected the superheat losses because they are relatively small. Zubair et al. [12] investigated a twostage, single ¯uid, cascade vapor compression refrigeration system by both ®rst and second law analysis. They showed that the optimum inter-stage pressure for twostages is very close to the saturation pressure corresponding to the arithmetic mean of the refrigerant condensation and evaporation temperatures. This paper revisits, focuses, and highlights, the fundamental thermodynamic aspects of improving the eciency of the ideal vapor compression refrigeration cycle. This paper considers the two fundamental losses encountered in the ideal vapor compression refrigeration cycle. In addition to being a fundamental study of cycle cascading, this paper takes a unique perspective on the cycle solely based on the thermodynamic property, entropy. Using the entropy generation minimization method, the optimal cascade system is revisited. In the following section there will be an introduction to the ideal multistage cycle. Following this, the individual losses due to superheating and throttling will be

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E.B. Ratts, J.S. Brown / International Journal of Refrigeration 23 (2000) 353±365

considered and their e€ects on system optimization will be discussed. In addition, an advanced rule on cascade intermediate temperatures is developed. 2. Ideal multistage cycle Let us ®rst consider a multistage refrigeration cycle where there are no irreversibilities, the reversed Carnot cycle. It is assumed that isothermal compression is possible, and follows the isentropic compression process. The ®nal result is a cascade of reversed Carnot cycles shown in Fig. 3. From an energy perspective, the purpose of the device is to transfer energy by heat interactions from a low-temperature thermal reservoir to a high-temperature thermal reservoir with the minimum amount of work input. Since entropy is transferred with heat transfers, an alternative perspective is the device transfers entropy from an entropy source (low-temperature thermal reservoir) to an entropy sink (hightemperature thermal reservoir) without generating entropy. However, in real systems there will always be some entropy generation. Let us consider within this multiple staged cycle the cascade of two Carnot cycles as shown in Fig. 4, and in particular, consider the reversible stage operating between thermal reservoirs i ÿ 1 and i, which will be

referred to as stage i. The entropic capacity of this stage : is Siÿ1 and is given by ÿ
 : :  Siÿ1 ˆ miÿ1 sfg;iÿ1 ÿ sf;i ÿ sf;iÿ1

…1†

: where m is the mass ¯owrate, sfg is the entropy of vaporization, and sf is the saturated liquid speci®c entropy. The entropic capacity of a refrigeration stage is the entropy transfer rate from the low-temperature reservoir to the stage's refrigerant, similar to the refrigeration capacity. The entropy rejection at i is ÿ
 : :  Si ˆ miÿ1 sfg;i ‡ sg;iÿ1 ÿ sg;i

…2†

where sg is the saturated vapor speci®c entropy. The entropy rejection of a refrigeration stage is the entropy transfer rate from the refrigerant to the high-temperature reservoir accompanying the stage's heat rejection from the refrigerant to the high-temperature reservoir. Applying an entropy balance to stage i results in : : : Sgen;i ˆ Si ÿ Siÿ1

…3†

: where S is the entropy generation rate. If Eqs. (1) Ð (3) are combined, the resulting entropy generation for stage i is zero. This is an expected result since all the processes are reversible, which makes the entire cycle reversible. An entropy balance from BCD at stage i to BCD at stage i ‡ 1 results in the mass ¯ow rate ratio between stages: : mi sfg;i ‡
sg;i ˆ : miÿ1 sfg;i‡1 ‡
sg;i‡1

…4†

where
Sg;i ˆ sg;iÿ1 ÿ sg;i

Fig. 3. Cascading of multiple reversed Carnot cycles. Fig. 3. Cycles de Carnot inverseÂs en cascade multiples.

…5†

It can be shown that
sg;i is small in comparison to sfg;i . Therefore, the mass ¯owrate scales with the entropy of vaporization. As the temperature increases, and the refrigerant's entropy of vaporization decreases, more mass ¯ow is required at each higher stage in order to transfer the entropy. Note that for the Carnot multistage system, there is not a thermodynamic optimum set of temperatures. However, from a practical standpoint it may be advantageous to scale the temperatures so that the compression ratios of the di€erent stages are similar. 3. Superheat losses

Fig. 4. Cascading of reversed Carnot cycles i and i ‡ 1. Fig. 4. Cycles de Carnot inverseÂs en cascade inverseÂs i et i + l.

Ideally we would like to incorporate a serial combination of an isentropic compression process AB and an isothermal compression process BC as shown in Fig. 4;

E.B. Ratts, J.S. Brown / International Journal of Refrigeration 23 (2000) 353±365

however, from a practical viewpoint isothermal compression is dicult to realize. Therefore, let us consider the ®rst irreversibility that comes about in the vapor compression refrigeration cycle, namely, the isentropic compression process. Fig. 5 shows the staging process with an isentropic compression process. A closer view of stage i is shown in Fig. 6. The entropic capacity is again : given by Eq. (1). The entropy rejection, Si , consists of both sensible entropy transfer and latent entropy transfer. The sensible entropy transfer, that is the entropy transferred along with the sensible heat transfer, occurs along process CD. The latent entropy transfer, that is the entropy transferred along with the latent heat transfer, occurs along process DE. The entire two step process, in the ideal case, is isobaric. The entropy rejection is thus " # … sg;iÿ1 : TdsP : Si ˆ miÿ1 sfg;i ‡ Ti sg;i "  # … sg;iÿ1  T ÿ Ti : ˆ miÿ1 sfg;i ‡
sg;i ‡ dsP …6† Ti sg;i

Fig. 5. Cascading of multiple vapor compression refrigeration cycles. Fig. 5. Cycles en cascade multiple aÁ compression de vapeur multiple.

357

where dsP refers to the entropy change along an isobar. Applying an entropy balance from i ÿ 1 to i using Eq. (3) results in an expression for the entropy generation, namely :  … sg;iÿ1  Sgen;i T ÿ Ti dsP ˆ : miÿ1 Ti sg;i

…7†

The integral in Eq. (7) is proportional to the shaded area in Fig. 6, that is, area BCDB. For each stage there is a similar area which represents the superheat irreversibility. In this paper we discuss the thermodynamic losses in terms of entropy generation instead of exergy destruction. Either method will give the same overall ®nal result. However, some insights gained are di€erent. We have chosen entropy generation because it demonstrates the cascading e€ect of losses. In cascade refrigeration systems, entropy is transferred over a series of increasing temperatures. Each succeeding temperature level receives entropy from the previous temperature level. The succeeding temperature level does not distinguish between entropy that was generated at the previous level and entropy that was transferred to the previous level from an even lower temperature level. As entropy is generated at each level, as Eq. (7) suggests, the entropy is ampli®ed by a corresponding entropy generation multiplier. Thus, reducing entropy generation at lower temperature levels has a larger impact than reducing the same amount of entropy generation at higher temperature levels. The vapor compression refrigeration cycle operating between temperatures T1 and Tn , including multiple stages as shown in Fig. 5, can be optimized by selecting optimum temperatures T2 , T3 , . . ., Tiÿ1 , Ti , Ti‡1 , . . ., Tnÿ1 . As was stated earlier, the uniqueness of this paper involves selecting optimum temperatures based on minimization of entropy generation. An approach similar to Haywood [13] is taken to determine the optimum staging temperatures. That is, consider all temperatures as ®xed except for Ti , and then determine the optimum intermediate temperature for Ti between Tiÿ1 and Ti‡1 . The following analysis identi®es the optimum temperature for stage i. The entropy generation between the temperatures Tiÿ1 and Ti‡1 is i‡1 X : : : Sgen;j ˆ Sgen;i ‡ Sgen;i‡1

…8†

jˆi

Substituting Eq. (7) into (8) yields : Sgen;j

i‡1 P

Fig. 6. Cascading of two vapor compression refrigeration cycles. Fig. 6. Deux cycles frigori®ques aÁ compression de vapeur en cascade.

jˆi

: miÿ1

 … sg;iÿ1  T ÿ Ti ˆ dsP Ti sg;i  : … sg;i  mi T ÿ Ti‡1 ‡ : dsP miÿ1 sg;i‡1 Ti‡1

…9†

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E.B. Ratts, J.S. Brown / International Journal of Refrigeration 23 (2000) 353±365

By de®nition, the speci®c heat at constant pressure cP is   ds cP ˆ T …10† dT P Substituting Eq. (10) into Eq. (9) and rearranging gives : Sgen;j

i‡1 P

iˆi ˆ : miÿ1 cP;ave;i

 … Ti ‡
Ti;max  T ÿ Ti dTP Ti T Ti   … : mi cP;ave;i‡1 Ti‡1 ‡
Ti‡1;max T ÿ Ti‡1 ‡ : dTP miÿ1 cP;ave;i Ti‡1 Ti‡1 T …11†

where
Ti;max is the temperature di€erence between TC and TB for stage i (see Fig. 6). The speci®c heat is assumed to be approximately constant with temperature since the temperature di€erence is relatively small. Since the temperature di€erence is small, then its magnitude is small with respect to the absolute temperature squared, that is, Ti T is approximately given by T2i . With this assumption, Eq. (11) reduces to : Sgen;j

i‡1 P jˆi

ˆ : miÿ1 cP;ave;i

… i;max 0

i di ‡ mr;i cr;P;i

… i‡1;max 0

i‡1 di‡1

…12†

where the variable transformations are given by T ÿ Ti
Ti ˆ Ti Ti : mi mr;i ˆ : miÿ1 cP;ave;i‡1 cr;P;i ˆ cP;ave;i i ˆ

…13†

…19†

Note that this variable is di€erent from the saturated vapor speci®c heat at constant pressure. Integrating along the saturated vapor line, the loss for one stage is … Tiÿ1   … Tiÿ1 dsg 1
sg;i ˆ cg dT dT ˆ ÿ T dT Ti T   i Ti ˆ cg;ave;iÿ1 ln …20† Tiÿ1 where cg;ave;iÿ1 is the average speci®c heat at saturation of a gaseous phase. Expanding Eq. (20) in Taylor's series expansion, and neglecting third order terms gives   1
sg;i ˆ cg;ave;iÿ1 … iÿ1 ÿ 1† ÿ … iÿ1 ÿ 1†2 …21† 2 and

…15†

where i ˆ Ti‡1 =Ti

: Sgen;j

…16†

It can be shown, using Eq. (11) and using the fact that T is of the same order of magnitude as Ti , that …17†

and

iÿ1 ˆ Ti =Tiÿ1

: Sgen;j

iP ‡1


2 ÿ
2 1 ÿ 1 ˆ
sg;i ‡mr;i
sg;i‡1 : 2cP;ave;i 2cP;ave;i‡1 miÿ1 …18†

…22†

…23†

On closer investigation of Eq. (18), it can be seen that
sg;i and
sg;i‡1 are not independent of one other. In fact, they are dependent variables. For a single ¯uid, the change in entropy between sg;iÿ1 and sg;i‡1 is ®xed since Tiÿ1 and Ti‡1 are ®xed. The single ¯uid constraint is
sg;i ‡
sg;i‡1 ˆ
s…s† ˆ constant

…24†

Combining Eq. (21), (22), and (24) results in   1 cg;ave;iÿ1 … iÿ1 ÿ 1† ÿ … iÿ1 ÿ 1†2 ‡ cg;ave;i 2   1 2 … i ÿ 1† ÿ … i ÿ 1† 2

Substituting Eq. (17) into (16) results in jˆi

dsg dT

…14†

i‡1 P


sg;i ˆ cP;ave;i i;max

cg ˆ ÿT

  1
sg;i‡1 ˆ cg;ave;i … i ÿ 1† ÿ … i ÿ 1†2 2

Integrating Eq. (12) yields 1 1 jˆi 2 ˆ 2 ‡ mr;i cr;P;i i‡1;max : miÿ1 cP;ave;i 2 i;max 2

An interesting result of Eq. (18) is that the superheat losses scale with the square of the change in saturated vapor entropy,
sg;i . Losses per unit mass are larger near the critical temperature since the entropy change per unit temperature is larger near the critical point. The change in entropy from the saturated vapor state at Tiÿ1 to the saturated vapor state at Ti can be found by integrating along the saturated vapor line. Firstly, the speci®c heat at saturation of a gaseous phase, cg , is de®ned as

ˆ
s…s†

…25†

E.B. Ratts, J.S. Brown / International Journal of Refrigeration 23 (2000) 353±365

Substituting Eqs. (21) and (24) into Eq. (18) gives i‡1 P jˆi

: Sgen;j

¯owrate ratio between stages. Substituting Eq. (24) into (29) and rearranging results in : : miÿ1
sg;i ˆ mi
sg;i‡1

1

ˆ : 2cP;ave;i miÿ1   2 1 cg;ave;iÿ1 … iÿ1 ÿ 1† ÿ … iÿ1 ÿ 1†2 2 ( 1 ‡ mr;i
s…s† ÿ cg;ave;iÿ1 2cP;ave;i‡1 " #)2 1 2 … iÿ1 ÿ 1† ÿ … iÿ1 ÿ 1† 2

…26†

The entropy generation is only a function of the intermediate temperature ratio, iÿ1 . Thus, there is an optimum temperature ratio that minimizes the entropy generation. Graphically, the optimum temperature ratio is the one that minimizes the total area given by the sum of the area BCDB at Ti and the area BCDB at Ti‡1 (see Fig. 6). An alternative constraint is iÿ1 i ˆ ˆ constant

…27†

Combining Eqs. (18), (21), (22), and (27) gives i‡1 P

: Sgen;j

 2 cg;ave;iÿ1 1 ˆ … iÿ1 ÿ 1† ÿ … iÿ1 ÿ 1†2 : 2 miÿ1 2cP;ave;i 2 cg;ave;i ‡ mr;i 2cP;ave;i‡1 "   2 #2 1 ÿ1 ÿ ÿ1 iÿ1 2 iÿ1

jˆi

…28†

mr;i
s…s† 1 ‡ mr;i

: Sgen;j

iP ‡1

: miÿ1

…29†

This states that the sensible entropy transfer per unit mass at each stage is weighted with respect to the mass

…30†

This equation states that the sensible entropy transfer should be evenly distributed among the stages. To further understand this result consider the following ideal series of processes. If a steady ¯ow of a perfect gas experiences the consecutive steady-state processes of (i) isentropic compression, (ii) isobaric cooling, (iii) isentropic compression, and (iv) isobaric cooling with a ®xed overall pressure ratio and ®xed compressor inlet temperature ratio, then there is an optimum intermediate pressure that will minimize the net work or the net entropy generated. This is the well-known geometric mean solution for the optimal intermediate temperature of a cascade system. As already known, intercooling reduces the net work. The entropy equation not only identi®es the optimum pressure, but also shows the change in entropy of the two cooling processes should be equal to minimize the external irreversibilities. This is exactly the result of Eq. (30). The cascading stages are intercooling the refrigerant between the compression processes. The intercooling process reduces the compressor work. To minimize the irreversibilities of superheating, the intercooling sensible entropy transfers should be equal for each stage. Assuming the speci®c heats are constant along the entire temperature range and neglecting higher order terms, Eq. (28) can be rewritten as

jˆi

The optimum temperature can be determined by minimizing the entropy generation with respect to iÿ1 . This requires knowing speci®c heats through thermodynamic tables, and implementing a minimization algorithm. Assumptions are now made to understand the distribution of entropy generation among the various stages and to determine the optimum intermediate temperature. Reconsider Eq. (18), assuming that the speci®c heats are constant and uniform over the temperature range. Substituting Eq. (24) into Eq. (18), taking the derivative with respect to
sg;i , and setting equal to zero, the result is
sg;i ˆ

359

"  2 # c2g;ave 2 ˆ … iÿ1 ÿ 1† ‡mr;i ÿ1 iÿ1 2cP;ave

…31†

The mass ¯owrate ratio is dependent on the intermediate temperature. The dependence is determined by an entropy balance and the ¯uid's equation of state. An entropy balance between stages i and i ‡ 1 results in : : mi sfg;i ‡
sg;i Sgen;i ÿ
ˆ ‡ : : miÿ1 sfg;i‡1 ‡
sg;i‡1 miÿ1 sfg;i‡1 ‡
sg;i‡1

…32†

The ®rst term on the right-hand side is the mass ¯owrate ratio for the reversible case, Eq. (4). The second term is the necessary increase in the mass ¯owrate ratio to transfer the generated entropy. The irreversibilities generate entropy thus requiring more mass ¯owrate to reject the entropy at Ti‡1 . This second term has a secondary e€ect and is assumed to be negligible in determining the mass ¯owrate ratio.
sg;i and
sg;i‡1 can also be neglected with respect to the entropy of vaporization. Therefore the mass ¯owrate ratio reduces

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to the ratio of entropy of vaporization. The entropy of vaporization's temperature dependence can be evaluated by the Watson correlation for nonassociating liquids [14].   hfg;i‡1 1 ÿ Tr;i‡1 0:38 ˆ hfg;i 1 ÿ Tr;i


0:38 0 ÿ 1 ÿ Tr;i Tr;i

where

Tr;0 sfg;0 0 ˆ ÿ
0:38 1 ÿ Tr;0

…34†

Substituting Eq. (34) into the simpli®ed mass ¯owrate ratio and then substituting into Eq. (31) results in : Sgen;j

iP ‡1 jˆi

: miÿ1

  c ÿ iÿ1 0:38 c ÿ

    1 c ÿ iÿ1 ÿ0:62 c ÿ 0:62 iÿ1 ˆ0 ÿ … ÿ iÿ1 †2 2 c ÿ c ÿ

…33†

where Tr is the reduced temperature. Eq. (33) can be rewritten as sfg;i ˆ

iÿ1 … iÿ1 ÿ 1† ÿ … ÿ iÿ1 † 2

" c2g;ave ˆ … iÿ1 ÿ 1†2 2cP;ave    2 # c ÿ iÿ1 0:38 ‡ ÿ1 iÿ1 c ÿ iÿ1

…35†

where c …ˆ Tc =Tiÿ1 † is the nondimensional critical temperature. Taking the derivative of this equation with respect to iÿ1 and setting equal to zero yields

…36† For the entire cascade system, this equation needs to be written for each stage. The set of nonlinear algebraic equations are then solved simultaneously to obtain the optimum temperature distribution for the superheat irreversibility. If it is assumed the mass ¯owrate ratio is a weak function of iÿ1 in Eq. (35) and set the derivative equal to zero yields 0 ˆ 3iÿ1 … iÿ1 ÿ 1† ÿ mr;i … ÿ iÿ1 †

…37†

where the trivial solution to this equation is ˆ 2iÿ1

and

mr;i ˆ 1

…38†

This equation is the geometric mean solution commonly used in estimating staging temperatures, but is incorrect in the sense that it does not account for the change in the mass ¯owrate. Fig. 7 shows the comparison between the optimum solution using exact equations of state that do not require the constant speci®c heat assumption and the approximate optimum solution predicted by Eq. (36). The ®gure shows the results for a single ¯uid (R-134a is chosen for illustrative purposes) cascade system with 2,

Fig. 7. Comparison of the approximate optimum solution to the optimum solution for cascading vapor compression refrigeration systems considering only superheat losses. Fig. 7. Comparaison de la solution optimale approximative avec la solution optimale pour des systeÁmes frigori®ques en cascade aÁ compression de vapeur, en ne consideÂrant que les pertes dues aÁ la surchau€e.

E.B. Ratts, J.S. Brown / International Journal of Refrigeration 23 (2000) 353±365

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3, 4, and 6 stages. The reduced refrigeration temperature is 0.684 and the reduced maximum temperature is 0.981. The data symbols identify the predicted intermediate stage temperatures. The optimum intermediate temperature predicted for the two-stage system is 0.90, and the approximate optimum intermediate temperature predicted by Eq. (36) is 0.88, a di€erence of 2%. Note a 2% error is a temperature di€erence of 19 C. The geometric mean solution, Eq. (38), is 0.82, a di€erence of 9% corresponding to a temperature di€erence of 58 C. The arithmetic average solution is 0.83, a di€erence of 8%. The optimum temperature is closer to the higher temperature. Since the slope of the saturated vapor line increases with temperature and the mass ¯owrate is larger at the higher stages, the intermediate temperature must be closer to the higher temperature to evenly distribute the entropy generation. The approximate optimum intermediate solution overpredicted at lower temperatures and underpredicted at higher temperatures. This is related to the constant speci®c heat assumption. The maximum error of the approximate solution is 4%, and it occurs at the smallest intermediate temperature in the six-stage system. Fig. 8 presents the total entropy generated for the solution shown in Fig. 7. The entropy generated is nondimensionalized with respect to the entropic capacity. The maximum error of the approximate solution overpredicted by a maximum of 9%. Table 1 presents the exact optimum solution and the distribution of entropy generated among the stages. The single stage system generates 0.672 kJ Kÿ1 hÿ1 of entropy due to the sensible heat transfer in the condenser. If a cascade system is used, the entropy generated decreases signi®cantly. By adding one stage, the total entropy generated decreases to 0.262 kJ Kÿ1 hÿ1 for a decrease of 61%. From one stage to the next, the decrease is 29, 21, and 28%, respectively. Another result is the evenly distributed entropy generation among the stages.

4. Throttling losses

Table 1 Entropy generation distribution for cascading vapor compression refrigeration stages considering only superheat lossesa

Fig. 8. Comparaison de la geÂneÂration d'entropie optimale approximative pour les systeÁmes frigori®ques en cascade aÁ compression de vapeur en ne consideÂrant que les pertes dues aÁ la surchau€e.

Tableau 1 Distribution de la geÂneÂration d'entropie pour les eÂtages frigori®ques en cascade avec compression de vapeur, en ne consideÂrant que les pertes dues aÁ la surchau€ea

Another loss to consider comes about because of the throttling process. This loss can be thought of as a mechanical dissipation process that generates entropy and thus reduces the cycle's thermal performance. To begin, consider a staging system with an irreversible expansion process, whereas all of the other processes are reversible. Fig. 9 shows two stages where both reversible and irreversible expansion processes are indicated on the diagram. The expansion process can be thought of as a process where ``entropy-starved'' ¯uid is returned to the lower temperature reservoir so that it can absorb more entropy. Before the expansion process, the entropy of the ¯uid at state D maintains its lowest value that it will ever have in that stage. Prior to state D, the entropy of

Fig. 8. Comparison of the approximate to the optimum entropy generated for cascading vapor compression refrigeration systems considering only superheat losses.

No of stages Entropy generation per stage (kJ Kÿ1 hÿ1)

1 2 3 4 6 a

1

2

3

4

5

0.672 0.128 0.059 0.036 0.017

0.134 0.064 0.062 0.038 0.039 0.033 0.021 0.017 0.019 0.016 0.016

Tr;L ˆ 0:684; Tr;H =Tr;L ˆ 1:43; R-134a.

6

Total 0.672 0.262 0.185 0.146 0.105

Fig. 9. Cascading of two vapor compression refrigeration cycles considering only superheat losses. Fig. 9. Deux cycles frigori®ques en cascade aÁ compression de vapeur en examinant les pertes dues aÁ la surchau€e seulement.

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the ¯uid is at its maximum value at state B. The ¯uid at state B arrived at this point by absorbing entropy from the lower temperature cycle. Process BD is where entropy is rejected from the lower temperature stage to the higher temperature stage. Thus, the ¯uid at state D is available to absorb more entropy from the lower stage. However, the only way for the ¯uid to absorb entropy without at the same time generating more entropy is to lower its temperature before receiving the entropy. The only way to reduce the ¯uid's temperature without changing its entropy is to reject energy through a non-entropic process, that is, through an adiabatic work transfer. In returning the ¯uid to the lower temperature it is important to minimize the amount of entropy generated, otherwise it will reduce the entropic capacity of the cycle. Most vapor compression systems use a throttle, an entropy generating device, that ultimately reduces refrigeration capacity. The throttling process DF is totally irreversible, and thus the change in entropy of the ¯uid between D and F is simply the entropy generation per unit mass ¯ow rate, or : Sgen;i t† ˆ sF;iÿ1 ÿ sD;i ˆ
s…iÿ1 : miÿ1

hD ÿ hE Tiÿ1

…40†

The enthalpy at state D is the saturated liquid enthalpy at Ti , namely, hf;i . The enthalpy at E can be determined from the temperature Tiÿ1 and the entropy of the ¯uid at state D, that is hE ˆ hf;iÿ1 ‡ xE;iÿ1 hfg;iÿ1

…41†

The quality of the mixture at E, namely, xE;iÿ1 , is known since Tiÿ1 and the saturated liquid entropy at Ti are both known. Thus, xE;iÿ1

sf;i ÿ sf;iÿ1 ˆ sg;iÿ1 ÿ sf;iÿ1

cf ˆ T

…42†

If Eqs. (41) and (42) are substituted into Eq. (40), it can be shown that ÿ
hD ÿ hE ˆ hf;i ÿ hf;iÿ1 ÿ Tiÿ1 sf;i ÿ sf;iÿ1 …43† The entropy change on the right-hand side of Eq. (43) can be determined by integrating the Gibbsian equation

dsf dT

…44†

Integrating Eq. (44), expanding in a Taylor series, and neglecting third order terms yields   1 sf;i ÿ sf;iÿ1 ˆ cf;ave;iÿ1 … iÿ1 ÿ 1† ÿ … iÿ1 ÿ 1†2 …45† 2 The enthalpy change on the right side of Eq. (43) is approximately hf;i ÿ hf;iÿ1 ˆ cf;ave;iÿ1 …Ti ÿ Tiÿ1 †

…46†

Combining Eqs. (40), (43), (45), and (46) results in 1 t†
s…iÿ1 ˆ cf;ave;iÿ1 … iÿ1 ÿ 1†2 2

…39†

State E is the point at the lower temperature where the ¯uid has the same amount of entropy as it does at state D (i.e. an isentropic expansion process). Therefore, the entropy change along the isotherm Tiÿ1 from E to F can be determined by
s…t† iÿ1 ˆ

along the saturated liquid line assuming that the change in the saturated liquid speci®c volume is negligible. The speci®c heat at saturation of a liquid phase, cf , is de®ned as

…47†

Similarly it follows that 1
s…i t† ˆ cf;ave;i … i ÿ 1†2 2

…48†

We will proceed in a similar fashion as we did in the superheat loss case to ®nd the optimum temperature for stage i. Consider two stages in which the minimum and maximum temperatures of the thermal reservoirs, i.e. Tiÿ1 and Ti‡1 , respectively, are ®xed and the temperature of the middle thermal reservoir, i.e. Ti , is allowed to vary. The entropy generation can be shown to be : Sgen;j

i‡1 P

1 1 ˆ cf;ave;iÿ1 … iÿ1 ÿ 1†2 ‡mr;i cf;ave;i … i ÿ 1†2 : 2 2 miÿ1

jˆi

…49† If the constraint equation given in Eq. (27) is applied to Eq. (49), the result is : Sgen;j

iP ‡1

 2 1 1 ˆ cf;ave;iÿ1 … iÿ1 ÿ 1†2 ‡mr;i cf;ave;i ÿ1 : 2 2 iÿ1 miÿ1

jˆi

…50† Once again the optimum temperature can be determined by minimizing the entropy generation with respect to iÿ1 using a minimization algorithm. To investigate the entropy generation distribution among the various stages, some assumptions need to be

E.B. Ratts, J.S. Brown / International Journal of Refrigeration 23 (2000) 353±365

made. Namely, the speci®c heats are assumed constant. With this assumption the form of the equation is exactly like Eq. (31) except for the speci®c heats. The optimum solution is therefore the same, Eq. (36). Thus, the entropy generation scales with the mass ¯owrate ratio and the entropy generation should be equally distributed among the stages. Fig. 10 shows the comparison between the optimum solution using exact equations of state not assuming constant speci®c heats and the approximate optimum solution predicted by Eq. (36) for a single ¯uid (again R-134a is chosen for illustrative purposes) cascade system with 2, 3, 4, and 6 stages. The reduced refrigeration temperature is again taken to be 0.684 and the reduced maximum temperature is again taken to be 0.981. The exact optimum intermediate temperature for the twostage system is 0.86, and the approximate optimum intermediate temperature predicted by Eq. (36) is 0.88, a di€erence of 2%. Recall the geometric solution of 0.82 is a di€erence of 5%, and the arithmetic solution of 0.83 is a di€erence of 3%. The approximate solution overpredicted the solution in all cases. The error is again related to the constant speci®c heat assumption. Table 2 presents the exact optimum solution and the distribution of entropy generated among the stages. The single stage system generates 94.51 kJ Kÿ1 hÿ1 of entropy due to the throttling process. If a cascade system is used, the entropy generated decreases signi®cantly. By adding one additional stage, the total entropy generated decreases to 20.96 kJ Kÿ1 hÿ1 for a decrease of 78%. From one stage to the next, the decrease is 39, 25, and

363

Table 2 Entropy generation distribution for cascading vapor compression stages considering only throttling lossesa Tableau 2 Distribution de la geÂneÂration d'entropie pour les eÂtages en cascade aÁ compression de vapeur en examinant les pertes dues aÁ la deÂtentea No. of Entropy generation per stage (kJ Kÿ1 hÿ1) stages 1 2 3 4 5 6

Total

1 2 3 4 6

94.51 20.96 12.69 9.55 6.86

94.51 9.31 3.51 1.86 0.78 a

11.66 4.03 2.05 0.84

5.15 2.53 0.97

3.11 1.14

1.37

1.77

Tr;L ˆ 0:684; Tr;H =Tr;L ˆ 1:43; R-134a.

28%, respectively. An interesting result is the distribution of the entropy generated. More of the entropy is generated at the higher temperatures. This is believed to be a result of the cascading e€ect of entropy generation. For example, for the two-stage cascade system the entropy generated at the lower stage is 9.31 kJ Kÿ1 hÿ1, or 44% of the total entropy generated, and at the higher stage it is 11.66 kJ Kÿ1 hÿ1, or 56% of the total entropy generated. For the four stage cascade system the entropy generation for each successive stage, in terms of percent of the total entropy generated, is 19, 21, 27, and 33%, respectively. Another result of importance is the relative magnitudes of the superheat losses and the throttling losses. Comparing the losses in a single stage system, the superheat losses are less than one percent of the throttling losses. For all cases, it is less than 3%. Thus, the throttling losses dominate the superheat losses when determining the optimum temperature distribution. 5. Combined losses

Fig. 10. Comparison of the approximate optimum solution for cascading vapor compression refrigeration systems considering only throttling losses. Fig. 10. Comparaison de la solution optimale approximative et la solution optimale pour un cycle frigori®que aÁ compression de vapeur en cascade en ne consideÂrant que les pertes dues aÁ la deÂtente.

The last step in the analysis is to consider the combination of both superheat and throttling losses and then to identify the optimum temperature distribution based on these two e€ects. It can be shown that the throttling losses are at least an order of magnitude larger than the superheat losses. Therefore the losses and temperature distribution for the optimum system is governed by the throttling losses. The optimum solution is very close to the optimum solution considering only throttling losses. Table 3 shows the optimum solution. As expected the throttling losses dominate. In fact, the entropy generation from the throttling processes is the same as that given in Table 2. The superheat losses are small compared with the total losses; in fact, they never exceed 4% of the total. The superheat losses have little, if any e€ect on the temperature distribution; however, the temperature

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Table 3 Entropy generation distribution for cascading vapor compression refrigeration stages considering combined superheat and throttling lossesa Tableau 3 Distribution de la geÂneÂration d'entropie pour les eÂtages frigori®ques en cascade avec compression de vapeur, en ne consideÂrant que les pertes combineÂes dues aÁ la surchau€e et aÁ la deÂtentea Entropy generation per stage (kJ Kÿ1 hÿ1) No. of 1 stages 1

2

3

5

4

5

6

Total

Total 95.824 95.824 94.511 ab bc 1.313 Total 9.672 11.686 21.357 a 0.074 0.032 b 9.597 11.653 Total 3.610 4.144 5.452 13.206 a 0.038 0.025 0.201 b 3.571 4.119 5.251 Total 1.900 2.136 2.626 3.295 9.958 a 0.026 0.009 0.027 0.133 b 1.874 2.127 2.600 3.162 Total 0.794 0.866 1.010 1.212 1.455 1.838 7.174 a 0.015 0.004 0.004 0.008 0.025 0.070 b 0.779 0.862 1.006 1.204 1.430 1.768

2 3 4 6

a b c

Fig. 11. Entropy generation versus the number of stages and versus the temperature ratio. The entropy generation is expressed as the ratio of entropy generation to the entropic capacity. The temperature ratio is expressed as the ratio of the highest temperature to the refrigeration temperature. Fig. 11. GeÂneÂration d'entropie en relation du nombre d'eÂtages et du rapport thermique. La geÂneÂration d'entropie est exprimeÂe comme le rapport entre la geÂneÂration d'entropie et la capacite entropique. Le rapport thermique est exprime comme la relation entre la tempeÂrature la plus eÂleveÂe et la tempeÂrature frigori®que.

Tr;L ˆ 0:684; Tr;H =Tr;L ˆ 1:43; R-134a. Superheat losses. Throttle losses.

distribution has an interesting e€ect on the superheat loss distribution. In the two-stage cascade system, there is more entropy generated at the lower stage than at the upper stage. In the three-stage cascade system, the superheat losses pass through a minimum as the temperature is increased, with most of the superheat loss being generated at the highest temperature. The superheat loss distributions have the same characteristics in the four and six stage cascade systems as well. The maximum temperature ratio's e€ect on entropy generation is shown in Fig. 11 for various numbers of stages. The entropy generation is nondimensionalized with respect to the entropic capacity, and the shading of the plot indicates the relative amounts of entropy generated. As expected, the entropy generation is greater with smaller numbers of stages and with larger temperature spans. The shaded area can be thought of as an envelope where staging may not be bene®cial for the given temperature span. A signi®cant result shown on Fig. 11 is the steep negative slope of the entropy generation curve with an increasing number of stages. Fig. 12 shows two di€erent planes that have been cut through the data of Fig. 11. Notice that the surface becomes relatively ¯at with more than two stages for temperature spans from 1.05 to 1.45.

Fig. 12. Entropy generation versus the number of stages and versus the temperature ratio. The entropy generation is expressed as the ratio of entropy generation to the entropic capacity. The temperature ratio is expressed as the ratio of the highest temperature to the refrigeration temperature. Fig. 12. GeÂneÂration d'entropie en fonction du nombre d'eÂtages et en fonction du rapport des tempeÂratures. La geÂneÂration d'entropie est exprimeÂe comme le rapport entre la geÂneÂration d'entropie et l'entropie. Le rapport des tempeÂratures est exprime comme la relation entre la tempeÂrature la plus eÂleveÂe et la tempeÂrature la plus basse.

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365

6. Conclusions

References

In this paper, we have provided new insight into the cascading of single-¯uid vapor compression systems based on the use of entropy generation. The paper has strengthened an alternative method of analysis that is normally avoided due to the obscurity of entropy. We have provided insight into, and the quanti®cation of, two seemingly di€erent losses, namely, mechanical dissipation and thermal dissipation. Indirectly, we have also provided a method for system optimization that does not require involved property modeling. Finally, we have suggested a way to quantify the improvement that can be realized in a vapor compression refrigeration system's thermal eciency without much diculty. The optimum temperature distribution for a cascade system was determined by minimizing entropy generation at each stage of the system. The equations for entropy generation were developed in terms of a single independent variable, the intermediate temperature ratio, and in terms of speci®c heats. Through the intermediate temperature ratio and through equations for the speci®c heats, the optimum temperature distribution was found. The development in terms of a single independent variable also provided a way to characterize the distribution of the losses. The distributions of the entropy generation for the superheat losses and for the throttling losses were different. The superheat losses were more evenly distributed among the various stages. On the other hand, the throttling losses increased with increasing temperature. The magnitudes of the throttling losses were far greater than were the magnitudes of the superheat losses, such that the throttling processes set the optimum temperature distribution. Based on this optimum temperature distribution, the superheat losses reached a minimum value at the intermediate stage.

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