**ECOP Criterion for Irreversible Three-Heat-Source Absorption Refrigerators**

Paiguy Armand Ngouateu Wouagfack and Réné Tchinda

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51547

## **1. Introduction**

444 Thermodynamics – Fundamentals and Its Application in Science

The basic part of my work in turbulent mixing was done in the period of my Ph. D. thesis at the Polytechnic University of Bucarest Romania. Fruitful experiments and special discussions were done togheter with the scientific team of the Institute of Applied Ecology,

But my entire acknowledgement points to Ph. D. Prof. Eng. Stefan Savulescu, from Politechnic University of Bucarest. Without the special vortexation installation constructed by him, the mixing flow experiments could not be possible. Due to his great experience in the turbulence field, and his fundamental observations, advices and suggestions, my work in turbulent mixing have an important sense and the great challenge of unifying the

Abell, L. Martha.& Braselton, P. James. (2005). *Maple by Example. 3rd edition.* Elsevier

Dimotakis, P. E., Miake-Lye, R. C., and Papantoniou, D. A. (1983). Structure and Dynamics

Ionescu, A. (2002). *The structural stability of biological oscillators. Analytical contributions,* 

Ionescu, A. (2010). *Recent trends in computational modeling of excitable media dynamics. New computational challengs in fluid dynamics analysis,* Lambert Academic Press , ISBN 978-3-

Ionescu, A. & Costescu, M. (2008). The influence of parameters on the phaseportrait in the mixing model. *Int. J. of Computers, Communications and Control*, vol III, Suppl. issue:

Ionescu, A. (2009). Recent challenges in turbulence: computational features of turbulent mixing. Recent advances in Continuum Mechanics. *Proceedings of the 4th IASME/WSEAS International Conference, (Mathematics and Computers in Science and Engineering).* ISSN

Ionescu, A. & Coman, D. (2011). New approaches in computational dynamics of mixing

Ottino, J.M. (1989). *The kinematics of mixing: stretching, chaos and transport,* Cambridge

Savulescu, St. N. (1998) Applications of multiple flows in a vortex tube closed at one end,

*Internal Reports, CCTE, IAE (Institute of Applied Ecology)* Bucarest, Romania

Academic Press, ISBN 0-12- 088526-3, San Diego, California

InTech, Ph.D. thesis, Politechnic University of Bucarest, Romania

of Round Turbulent Jets. *Phys. Fluids* 26, ISSN 3185-3192

Proceedings of IJCCCC2008, pp. 333-337, ISSN 1841-9836

8383-9316-2 Saarbrucken, Germany

1790-2769, Cambridge U.K., 24-26 feb 2009

University Press. ISBN .0-521-36878-2, UK

flow. In *http://dx.doi.org/10.1016/j.amc.2011.03.142*

**Acknowledgement** 

Bucarest. I am grateful to them.

turbulence theory takes shape.

**5. References** 

In this chapter, the optimization analysis based on the new thermo-ecological criterion (ECOP) first performed by Ust et al. [1] for the heat engines is extended to an irreversible three-heat-source absorption refrigerator. The thermo-ecological objective function ECOP is optimized with respect to the temperatures of the working fluid. The maximum ECOP and the corresponding optimal temperatures of the working fluid, coefficient of performance, specific cooling load, specific entropy generation rate and heat-transfer surface areas in the exchangers are then derived analytically. Comparative analysis with the COP criterion is carried out to prove the utility of the ecological coefficient of performance criterion.

## **2. Thermodynamics analysis**

The main components of an absorption refrigeration system are a generator, an absorber, a condenser and an evaporator as shown schematically in Fig. 1 [2]. In the shown model, . *QH* is the rate of absorbed heat from the heat source at temperature *HT* to generator, . *QC* and . *QA* are, respectively, the heat rejection rates from the condenser and absorber to the heat sinks at temperatures *CT* and *AT* and . *QL* is the heat input rate from the cooling space at temperature *<sup>L</sup> T* to the evaporator. In absorption refrigeration systems, usually NH3/H2O and LiBr/H2O are used as the working substances, and these substances abide by ozone depletion regulations, since they do not consist of chlorouorocarbons. In Fig. 1, the liquid rich solution at state 1 is pressurized to state 1' with a pump. In the generator, the working fluid is concentrated to state 3 by evaporating the working medium by means of . *QH* heat rate input. The weak solution at state 2 passes through the expansion valve into the absorber with a pressure reduction (2–2'). In the condenser, the working fluid at

© 2012 Ngouateu Wouagfack and Tchinda, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

state 3 is condensed to state 4 by removing *QC* heat rate. The condensed working fluid at state 4 is then throttled by a valve and enters the evaporator at state 4'. The liquid working fluid is evaporated due to heat transfer rate . *QL* from the cooling space to the working fluid (4'–5). Finally, the vaporized working fluid is absorbed by the weak solution in the absorber, and by means of . *QA* heat rate release in the absorber, state 1 is reached.

.

Work input required by the solution pump in the system is negligible relative to the energy input to the generator and is often neglected for the purpose of analysis. Under such assumption, the equation for the first law of thermodynamics is written as:

$$
\dot{Q}\_H + \dot{Q}\_L - \dot{Q}\_C - \dot{Q}\_A = 0 \tag{1}
$$

ECOP Criterion for Irreversible Three-Heat-Source Absorption Refrigerators 447

**Figure 1.** Schematic diagram of absorption refrigeration system [2]

**Figure 2.** Considered irreversible absorption refrigeration model and its T–S diagram.

Absorption refrigeration systems operate between three temperature levels, if *<sup>A</sup> <sup>C</sup> T T* = , or four temperature levels when *<sup>A</sup> <sup>C</sup> T T* ¹ . In this chapter, by taking *<sup>A</sup> <sup>C</sup> T T* = , the cycle of the working fluid consists of three irreversible isothermal and three irreversible adiabatic processes. The temperatures of the working fluid in the three isothermal processes are different from those of the external heat reservoirs so that heat is transferred under a finite temperature difference, as shown in Fig. 2 where

$$
\dot{Q}\_O = \dot{Q}\_C + \dot{Q}\_A \tag{2}
$$

<sup>1</sup> *T* and 2 *T* are, respectively, the temperatures of the working fluid in the generator and evaporator. It is assumed that the working fluid in the condenser and absorber has the same temperature 3 *T* [2]. . *QLC* is the heat leak from the heat sink to the cooled space.

The heat exchanges between the working fluid and heat reservoirs obey a linear heat transfer law, so that the heat-transfer equations in the generator, evaporator, condenser and absorber are, respectively, expressed as follows:

$$
\dot{Q}\_H = \mathcal{U}\_H A\_H \left( T\_H - T\_1 \right) \tag{3}
$$

$$\dot{\mathbf{Q}}\_L = \mathbf{U}\_L \mathbf{A}\_L \left(\mathbf{T}\_L - \mathbf{T}\_2\right) \tag{4}$$

$$\dot{Q}\_O = \mathcal{U}\_O \left( A\_A + A\_C \right) \left( T\_3 - T\_O \right) \tag{5}$$

where *AH* , *AL* , *AC* and *AA* are, respectively, the heat-transfer areas of the generator, evaporator, condenser and absorber, *UH* and *UL* are, respectively, the overall heat-transfer coefficients of the generator and evaporator, and it is assumed that the condenser and absorber have the same overall heat-transfer coefficient *UO* [2].

**Figure 1.** Schematic diagram of absorption refrigeration system [2]

working fluid is evaporated due to heat transfer rate

.

state 4 is then throttled by a valve and enters the evaporator at state 4'. The liquid

working fluid (4'–5). Finally, the vaporized working fluid is absorbed by the weak

Work input required by the solution pump in the system is negligible relative to the energy input to the generator and is often neglected for the purpose of analysis. Under such

Absorption refrigeration systems operate between three temperature levels, if *<sup>A</sup> <sup>C</sup> T T* = , or four temperature levels when *<sup>A</sup> <sup>C</sup> T T* ¹ . In this chapter, by taking *<sup>A</sup> <sup>C</sup> T T* = , the cycle of the working fluid consists of three irreversible isothermal and three irreversible adiabatic processes. The temperatures of the working fluid in the three isothermal processes are different from those of the external heat reservoirs so that heat is transferred under a finite

. ..

<sup>1</sup> *T* and 2 *T* are, respectively, the temperatures of the working fluid in the generator and evaporator. It is assumed that the working fluid in the condenser and absorber has the same

The heat exchanges between the working fluid and heat reservoirs obey a linear heat transfer law, so that the heat-transfer equations in the generator, evaporator, condenser and

where *AH* , *AL* , *AC* and *AA* are, respectively, the heat-transfer areas of the generator, evaporator, condenser and absorber, *UH* and *UL* are, respectively, the overall heat-transfer coefficients of the generator and evaporator, and it is assumed that the condenser and

*QLC* is the heat leak from the heat sink to the cooled space.

. ...

assumption, the equation for the first law of thermodynamics is written as:

.

*QC* heat rate. The condensed working fluid at

*QA* heat rate release in the absorber, state 1 is

0 *Q QQQ H LC A* +-- = (1)

*Q QQ O CA* = + (2)

*Q UA T T <sup>H</sup>* = - *HH H* 1 (3)

*Q UA T T <sup>L</sup>* = - *LL L* 2 (4)

*Q UA AT T <sup>O</sup>* = +- *OA C O* <sup>3</sup> (5)

*QL* from the cooling space to the

.

state 3 is condensed to state 4 by removing

solution in the absorber, and by means of

temperature difference, as shown in Fig. 2 where

.

absorber are, respectively, expressed as follows:

( ) .

( ) .

( )( ) .

absorber have the same overall heat-transfer coefficient *UO* [2].

reached.

temperature 3 *T* [2].

**Figure 2.** Considered irreversible absorption refrigeration model and its T–S diagram.

.

The absorption refrigeration system does not exchange heat with other external reservoirs except for the three heat reservoirs at temperatures *HT* , *<sup>L</sup> T* and *OT* , so the total heat-transfer area between the cycle system and the external heat reservoirs is given by the relationships:

$$A = A\_H + A\_L + A\_O \tag{6}$$

ECOP Criterion for Irreversible Three-Heat-Source Absorption Refrigerators 449

( ) ( )



( )( ) ( )( )

( )( ) ( )( )

2 2

1 3

*H H O O*

(13)

(12)

(14)

(17)

(18)

The coefficient of performance of the irreversible three-heat-source absorption refrigerator is:

*QQ Q <sup>Q</sup> COP*

1 *L*

=

*<sup>A</sup> <sup>A</sup>*

.

.

.

.

expressed as a function of *T*<sup>1</sup> , *T*2 and *T*3 for a given total heat-transfer areas :

*<sup>A</sup> <sup>A</sup>*

*<sup>A</sup> <sup>A</sup>*

( )( ) ( )( )

( )( )

*L*

*H*

*O*

*L*

*<sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup>*

( ) ( )

*L L*

*Q T T IT T IT T <sup>Q</sup>*

*Q IT T T T IT T <sup>Q</sup>*

The first is the coefficient of performance of the irreversible three-heat-source absorption

Substituting Eqs. (15) and (16) into Eq. (14), the heat-transfer area of the evaporator is

*H H O O*

By investigating similar reasoning, the heat-transfer areas of the generator and of condenser

<sup>=</sup> -- -- + + -- - -

*H L L O L L*


*Q UT T UT T Q UT T UT T Q Q*

( )

21 3

13 2

31 2

1 32

1 3 2 2 31 2 2

21 3 1 31 2 1 13 2 2 13 23

*H HH H L LO O*

*U T T IT T T U IT T T T T U T IT T T T U T IT T T T* <sup>=</sup> - - -- + + -- --

*L LL L*

*U T IT T T T U IT T T T T U T T T IT T U T T T IT T*

1 1 32 3 1 32

( )

. .

. .

+ +

. . . . . .. <sup>1</sup> *L LC <sup>L</sup> LC H HL*

*Q QQ* æ ö <sup>ç</sup> <sup>÷</sup> - <sup>ç</sup> <sup>÷</sup> <sup>ç</sup> <sup>÷</sup> = =- <sup>÷</sup> <sup>ç</sup> <sup>÷</sup> <sup>ç</sup> <sup>÷</sup> <sup>ç</sup> <sup>÷</sup> çç ÷÷ è ø

> *H O L L*

*A A*

From Eq. (6), it is expressed as:

Using Eqs. (3)-(5), Eq. (13) is rewritten as:

1

Combining Eqs. (1) and (11), the following ratios are derived:

( )

( )

refrigeration cycle without heat leak losses.

1

( )( )

1

*L*

and absorber are given respectively by:

*H*

*L*

=

where

$$A\_{\bullet} = A\_{\bullet} + A\_{A}.\tag{7}$$

The rate of heat leakage *QLC* from the heat sink at temperature *OT* to the cold reservoir at temperature *<sup>L</sup> T* was first provided by Bejan [3] and it is given as:

$$
\dot{Q}\_{LC} = K\_{LC} \left( T\_O - T\_L \right) \tag{8}
$$

where *KLC* is the heat leak coefficient.

Real absorption refrigerators are complex devices and suffer from a series of irreversibilities. Besides the irreversibility of finite rate heat transfer which is considered in the endoreversible cycle models and the heat leak from the heat sink to the cooled space, there also exist other sources of irreversibility. The internal irreversibilities that result from friction, mass transfer and other working fluid dissipations are an another main source of irreversibility, which can decrease the coefficient of performance and the cooling load of absorption refrigerators. The total effect of the internal irreversibilities on the working fluid can be characterized in terms of entropy production. An irreversibility factor is introduced to describe these internal irreversibilities:

$$I = \frac{\Delta S\_3}{\Delta S\_1 + \Delta S\_2} \tag{9}$$

On the basis of the second law of thermodynamics, 312 D >D +D *SSS* for an internally irreversible cycle, so that 1 *I* > . If the internal irreversibility is neglected, the cycle is endoreversible and so *I* =1 . The second law of thermodynamics for an irreversible threeheat-source cycle requires that:

$$\oint \frac{\delta \dot{Q}}{T} = \frac{\dot{Q}\_H}{T\_1} + \frac{\dot{Q}\_L}{T\_2} - \frac{\dot{Q}\_O}{T\_3} \le 0 \tag{10}$$

From Eq. (9), the inequality in Eq. (10) is written as:

$$\frac{\dot{Q}\_H}{T\_1} + \frac{\dot{Q}\_L}{T\_2} - \frac{\dot{Q}\_O}{IT\_3} = 0\tag{11}$$

The coefficient of performance of the irreversible three-heat-source absorption refrigerator is:

$$\dot{\mathbf{COP}} = \frac{\dot{\mathbf{Q}}\_L - \dot{\mathbf{Q}}\_{LC}}{\dot{\mathbf{Q}}\_H} = \frac{\dot{\mathbf{Q}}\_L}{\dot{\mathbf{Q}}\_H} \left( 1 - \frac{\dot{\mathbf{Q}}\_{LC}}{\dot{\mathbf{Q}}\_L} \right) \tag{12}$$

From Eq. (6), it is expressed as:

448 Thermodynamics – Fundamentals and Its Application in Science

.

temperature *<sup>L</sup> T* was first provided by Bejan [3] and it is given as:

( ) .

where

The rate of heat leakage

where *KLC* is the heat leak coefficient.

to describe these internal irreversibilities:

heat-source cycle requires that:

<sup>3</sup>

From Eq. (9), the inequality in Eq. (10) is written as:

The absorption refrigeration system does not exchange heat with other external reservoirs except for the three heat reservoirs at temperatures *HT* , *<sup>L</sup> T* and *OT* , so the total heat-transfer area between the cycle system and the external heat reservoirs is given by the relationships:

*A* = ++ *A AA H LO* (6)

. *AOCA* = + *A A* (7)

Real absorption refrigerators are complex devices and suffer from a series of irreversibilities. Besides the irreversibility of finite rate heat transfer which is considered in the endoreversible cycle models and the heat leak from the heat sink to the cooled space, there also exist other sources of irreversibility. The internal irreversibilities that result from friction, mass transfer and other working fluid dissipations are an another main source of irreversibility, which can decrease the coefficient of performance and the cooling load of absorption refrigerators. The total effect of the internal irreversibilities on the working fluid can be characterized in terms of entropy production. An irreversibility factor is introduced

1 2

*S S*

On the basis of the second law of thermodynamics, 312 D >D +D *SSS* for an internally irreversible cycle, so that 1 *I* > . If the internal irreversibility is neglected, the cycle is endoreversible and so *I* =1 . The second law of thermodynamics for an irreversible three-

. . . .

. . .

12 3 <sup>0</sup> *Q Q H L QO*

1 23 <sup>0</sup> *<sup>Q</sup> Q Q H L QO TTTT*

*<sup>S</sup> <sup>I</sup>*

*QLC* from the heat sink at temperature *OT* to the cold reservoir at

*Q KTT LC* = - *LC O L* (8)

<sup>D</sup> <sup>=</sup> D +D (9)

*<sup>d</sup>* ò = +-£ (10)

*T T IT* +- = (11)

$$A\_L = \frac{A}{1 + \frac{A\_H}{A\_L} + \frac{A\_O}{A\_L}}\tag{13}$$

Using Eqs. (3)-(5), Eq. (13) is rewritten as:

$$A\_L = \frac{A}{1 + \frac{\dot{Q}\_H}{\dot{Q}\_L} \frac{\mathcal{U}\_L \left(T\_L - T\_2\right)}{\mathcal{U}\_H \left(T\_H - T\_1\right)} + \frac{\dot{Q}\_O}{\dot{Q}\_L} \frac{\mathcal{U}\_L \left(T\_L - T\_2\right)}{\mathcal{U}\_O \left(T\_3 - T\_O\right)}}\tag{14}$$

Combining Eqs. (1) and (11), the following ratios are derived:

$$\frac{\dot{Q}\_L}{\dot{Q}\_H} = \frac{T\_2 \left(T\_1 - IT\_3\right)}{T\_1 \left(IT\_3 - T\_2\right)}\tag{15}$$

$$\frac{\dot{Q}\_O}{\dot{Q}\_L} = \frac{IT\_3\left(T\_1 - T\_2\right)}{\left(T\_1 - IT\_3\right)T\_2} \tag{16}$$

The first is the coefficient of performance of the irreversible three-heat-source absorption refrigeration cycle without heat leak losses.

Substituting Eqs. (15) and (16) into Eq. (14), the heat-transfer area of the evaporator is expressed as a function of *T*<sup>1</sup> , *T*2 and *T*3 for a given total heat-transfer areas :

$$A\_{L} = \frac{A}{1 + \frac{\mathcal{U}\_{L}T\_{1}\left(IT\_{3} - T\_{2}\right)\left(T\_{L} - T\_{2}\right)}{\mathcal{U}\_{H}\left(T\_{H} - T\_{1}\right)\left(T\_{1} - IT\_{3}\right)T\_{2}} + \frac{\mathcal{U}\_{L}IT\_{3}\left(T\_{1} - T\_{2}\right)\left(T\_{L} - T\_{2}\right)}{\mathcal{U}\_{O}\left(T\_{3} - T\_{O}\right)\left(T\_{1} - IT\_{3}\right)T\_{2}}}\tag{17}$$

By investigating similar reasoning, the heat-transfer areas of the generator and of condenser and absorber are given respectively by:

$$A\_H = \frac{A}{1 + \frac{\mathcal{U}\_H T\_2 \left(T\_1 - IT\_3\right) \left(T\_H - T\_1\right)}{\mathcal{U}\_L T\_1 \left(IT\_3 - T\_2\right) \left(T\_L - T\_2\right)} + \frac{\mathcal{U}\_H IT\_3 \left(T\_1 - T\_2\right) \left(T\_H - T\_1\right)}{\mathcal{U}\_O T\_1 \left(IT\_3 - T\_2\right) \left(T\_3 - T\_O\right)}\tag{18}$$

and

$$A\_O = \frac{A}{1 + \frac{\mathcal{U}\_O T\_1 \left(T\_3 - T\_2\right) \left(T\_3 - T\_O\right)}{\mathcal{U}\_H T\_3 \left(T\_1 - T\_2\right) \left(T\_H - T\_1\right)} + \frac{\mathcal{U}\_O T\_2 \left(T\_1 - T\mathcal{U}\_3\right) \left(T\_3 - T\_O\right)}{\mathcal{U}\_L T\_3 \left(T\_1 - T\_2\right) \left(T\_L - T\_2\right)}\tag{19}$$

Substituting Eq. (17) into Eq. (4):

$$\dot{Q}\_{L} = \frac{A}{\frac{1}{\mathcal{U}\_{L}\left(T\_{L} - T\_{2}\right)} + \frac{T\_{1}\left(IT\_{3} - T\_{2}\right)}{\mathcal{U}\_{H}\left(T\_{H} - T\_{1}\right)\left(T\_{1} - IT\_{3}\right)T\_{2}} + \frac{IT\_{3}\left(T\_{1} - T\_{2}\right)}{\mathcal{U}\_{O}\left(T\_{3} - T\_{O}\right)\left(T\_{1} - IT\_{3}\right)T\_{2}}}\tag{20}$$

Combining Eqs. (8), (12), (15) and (20), the coefficient of performance of the irreversible three-heat-source refrigerator as a function of the temperatures 1 *T* , 2 *T* and 3 *T* of the working fluid in the generator, evaporator, condenser and absorber is obtained:

$$\text{COP} = \frac{T\_2 \left(T\_1 - IT\_3\right)}{T\_1 \left(IT\_3 - T\_2\right)} \left| 1 - \xi \left(T\_O - T\_L\right) \frac{1}{\left[\mathcal{U}\_L \left(T\_L - T\_2\right)\right.} + \frac{T\_1 \left(IT\_3 - T\_2\right)}{\mathcal{U}\_H \left(T\_H - T\_1\right) \left(T\_1 - IT\_3\right)T\_2} + \frac{IT\_3 \left(T\_1 - T\_2\right)}{\mathcal{U}\_O \left(T\_3 - T\_O\right) \left(T\_1 - IT\_3\right)T\_2} \right| \right| \tag{21}$$

where the parameter

$$
\xi = \frac{K\_{LC}}{A} \tag{22}
$$

ECOP Criterion for Irreversible Three-Heat-Source Absorption Refrigerators 451

è øè ø (26)

( ) ( )( )

13 2 13 2 31 2 21 3 2 1 1 32 3 1 32

(25)

1

(27)

(28)

(30)

( ) ( )( )

. . . .

*L*

. 11 11 11 *LC H L <sup>L</sup> LO OH O L*

> . . . 11 11 *<sup>H</sup> L LC*

*<sup>Q</sup> Q Q <sup>Q</sup> <sup>s</sup> T T A T T A T TA <sup>Q</sup>*

æ öæ ö æ ö çç ç ÷÷ ÷ =- +- + è øè ø è ø

> *OH OL <sup>Q</sup> Q Q <sup>s</sup> TTA TT A* æ öæ ö ç ç ÷ ÷ - =- +-

( ) ( )

*L O L L H H O O T IT T T IT T IT T T <sup>s</sup> T T*

*r*

*e*

is the coefficient of performance for reversible three-heat-source refrigerator.

coefficient of performance (*ECOP)* of an absorption refrigerator is defined as:

*env*

*T*

Substituting Eqs.(8), (15) and (20) into Eq. (25), the specific entropy production rate as a

*T T T T IT U T T U T T T IT T U T T T IT T*

1

*O L*

æ ö ç ÷ ç - ÷ ç ÷ ç ÷÷ è ø = æ ö ç ÷ ç - ÷ ç ÷ ç ÷÷ è ø

*T T*

According to the definition of the general thermo-ecological criterion function for different heat engine models [4-9], a two-heat-source refrigerator [10, 11] and three-heat-source absorption refrigerator [2], the new thermo-ecological objective function called ecological

.. ..

*L LC L LC*

*AT s <sup>T</sup> <sup>s</sup>*

1 1 1 11

*T T T T COP* <sup>=</sup> <sup>é</sup> æ ö <sup>ù</sup> <sup>ê</sup> <sup>ç</sup> <sup>÷</sup> <sup>ú</sup> -+ - <sup>ç</sup> <sup>÷</sup> <sup>ê</sup> <sup>ç</sup> <sup>÷</sup> <sup>ú</sup> <sup>ç</sup> ÷÷ <sup>ê</sup> è ø <sup>ú</sup> <sup>ë</sup> <sup>û</sup>

*OH OL*

When Eq. (21) is put in Eq. (30), the ecological coefficient of performance of the irreversible three-heat-source absorption refrigerator as a function of 1 *T* , 2 *T* and 3 *T* is derived as :

*env*


.

*env*

*QQ QQ ECOP*

1

*O H*

*T T*

<sup>ì</sup> - üï æ öï <sup>é</sup> - -- ù é <sup>ù</sup> <sup>ï</sup> <sup>ç</sup> <sup>÷</sup>

ïï <sup>ê</sup> ú ê <sup>ú</sup> ïï = - - -- <sup>ç</sup> <sup>ç</sup> ֒ <sup>ê</sup> ú ê <sup>+</sup> <sup>+</sup> <sup>ú</sup> <sup>ý</sup> <sup>ç</sup> <sup>÷</sup> ֕ - - -- - - <sup>ï</sup> è øï <sup>ê</sup> ú ê <sup>ú</sup> <sup>ï</sup> ïî <sup>ë</sup> û ë <sup>û</sup> ïïþ

or

where

Putting Eq.(26) into Eq. (29):

<sup>1</sup>

*ECOP*

function of 1 *T* , 2 *T* and 3 *T* is given by :

*<sup>e</sup> <sup>x</sup>*

*O L*

÷

( ) ( )

1 1 <sup>1</sup> <sup>1</sup> *<sup>r</sup>*

represents the heat leakage coefficient and its dimension is w/(Km2)

The specific cooling load of the irreversible three-heat-source refrigerator is deduced as:

$$r = \frac{\dot{Q}\_L - \dot{Q}\_{L\complement}}{A} = \left[ \frac{1}{\mathcal{U}\_L \left( T\_L - T\_2 \right)} + \frac{T\_1 \left( T\_3 - T\_2 \right)}{\mathcal{U}\_H \left( T\_H - T\_1 \right) \left( T\_1 - T T\_3 \right) T\_2} + \frac{\mathcal{U} T\_3 \left( T\_1 - T\_2 \right)}{\mathcal{U}\_O \left( T\_3 - T\_O \right) \left( T\_1 - T T\_3 \right) T\_2} \right]^{-1} - \xi \left( T\_O - T\_L \right) \tag{23}$$

The specific entropy production rate of the irreversible three-heat-source absorption refrigerator is:

$$s = \frac{\dot{\mathcal{Q}}\_O - \dot{\mathcal{Q}}\_{LC}}{A} - \frac{\dot{\mathcal{Q}}\_H}{T\_H} - \frac{\dot{\mathcal{Q}}\_L - \dot{\mathcal{Q}}\_{LC}}{T\_L} \tag{24}$$

Using Eq. (1) *s* is rewritten as:

ECOP Criterion for Irreversible Three-Heat-Source Absorption Refrigerators 451

$$s = \left(\frac{1}{T\_L} - \frac{1}{T\_O}\right)\frac{\dot{\mathcal{Q}}\_{LC}}{A} + \left(\frac{1}{T\_O} - \frac{1}{T\_H}\right)\frac{\dot{\mathcal{Q}}\_H}{\dot{\mathcal{Q}}\_L}\frac{\dot{\mathcal{Q}}\_L}{A} + \left(\frac{1}{T\_O} - \frac{1}{T\_L}\right)\frac{\dot{\mathcal{Q}}\_L}{A} \tag{25}$$

or

450 Thermodynamics – Fundamentals and Its Application in Science

( )( )

*<sup>A</sup> <sup>A</sup>*

1

( )

( ) ( ) ( )

*KLC*

( )

*s*

*s*

represents the heat leakage coefficient and its dimension is w/(Km2)

<sup>1</sup> <sup>1</sup> *O L*

1

*O*

Substituting Eq. (17) into Eq. (4):

.

*L*

( )

where the parameter

*COP T T*

1 *L LC*

Using Eq. (1) *s* is rewritten as:

refrigerator is:

( )( )

( ) ( )( )

<sup>=</sup> - - + + - -- - -

*<sup>A</sup> <sup>Q</sup> T IT T IT T T*

working fluid in the generator, evaporator, condenser and absorber is obtained:

*L L H H O O*

*U T T U T T T IT T U T T T IT T*

Combining Eqs. (8), (12), (15) and (20), the coefficient of performance of the irreversible three-heat-source refrigerator as a function of the temperatures 1 *T* , 2 *T* and 3 *T* of the

21 3 13 2 31 2 13 2 2 1 1 32 3 1 32

(21)

The specific cooling load of the irreversible three-heat-source refrigerator is deduced as:

( ) ( )( )

*r T T <sup>A</sup> U T T U T T T IT T U T T T IT T <sup>x</sup>*

The specific entropy production rate of the irreversible three-heat-source absorption

. . . . . . *O LC H L LC OHL QQ QQ Q TTT*


= = (24)

*Q Q T IT T IT T T*

*LL HH O O*

*A A*

. . 1


*T IT T U T T U T T T IT T U T T T IT T <sup>x</sup>* - ìï <sup>é</sup> - - ùüï <sup>ï</sup> <sup>ê</sup> <sup>ú</sup> <sup>ï</sup> = -- + <sup>ï</sup> <sup>+</sup> <sup>ï</sup> <sup>í</sup> <sup>ê</sup> úý - <sup>ï</sup> - -- - - <sup>ï</sup> <sup>ï</sup> <sup>ê</sup> úï ïî <sup>ë</sup> ûïþ

*T T IT T IT T IT T T*

13 23 21 3 3 31 2 1 31 2 2

*U T IT T T T U T T IT T T U IT T T T T U IT T T T T* <sup>=</sup> -- - - + + - - --

*O O O O H HL L*

13 2 31 2 2 1 1 32 3 1 32

> ( ) ( )( )

*LL HH O O*

13 2 31 2 2 1 1 32 3 1 32

( )( ) ( )( )

> ( ) ( )( )

*<sup>A</sup> <sup>x</sup>* <sup>=</sup> (22)

( )

( )( ) ( )


*O L*

(23)

(19)

(20)

( ) ( )( )

and

$$s = \left(\frac{1}{T\_O} - \frac{1}{T\_H}\right) \frac{\dot{Q}\_H}{A} + \left(\frac{1}{T\_O} - \frac{1}{T\_L}\right) \frac{\dot{Q}\_L - \dot{Q}\_{LC}}{A} \tag{26}$$

Substituting Eqs.(8), (15) and (20) into Eq. (25), the specific entropy production rate as a function of 1 *T* , 2 *T* and 3 *T* is given by :

$$s = \left(\frac{1}{T\_L} - \frac{1}{T\_O}\right) \left| \xi \left(T\_O - T\_L\right) - \left[1 - \frac{\varepsilon\_r T\_1 \left(IT\_3 - T\_2\right)}{T\_2 \left(T\_1 - IT\_3\right)}\right] \frac{1}{\left| \mathcal{U}\_L \left(T\_L - T\_2\right) \right.} + \frac{T\_1 \left(IT\_3 - T\_2\right)}{\mathcal{U}\_H \left(T\_H - T\_1\right) \left(T\_1 - IT\_3\right) T\_2} + \frac{IT\_3 \left(T\_1 - T\_2\right)}{\mathcal{U}\_O \left(T\_3 - T\_O\right) \left(T\_1 - IT\_3\right) T\_2}\right]^{-1} \right| \tag{27}$$

where

$$\varepsilon\_r = \frac{\left(1 - \frac{T\_O}{T\_H}\right)}{\left(\frac{T\_O}{T\_L} - 1\right)}\tag{28}$$

is the coefficient of performance for reversible three-heat-source refrigerator.

According to the definition of the general thermo-ecological criterion function for different heat engine models [4-9], a two-heat-source refrigerator [10, 11] and three-heat-source absorption refrigerator [2], the new thermo-ecological objective function called ecological coefficient of performance (*ECOP)* of an absorption refrigerator is defined as:

$$\text{HCOP} = \frac{\dot{Q}\_L - \dot{Q}\_{LC}}{T\_{env}\sigma} = \frac{\dot{Q}\_L - \dot{Q}\_{LC}}{AT\_{env}s} \tag{29}$$

Putting Eq.(26) into Eq. (29):

$$ECOP = \frac{1}{T\_{env} \left[ \frac{1}{T\_O} - \frac{1}{T\_H} + \left( \frac{1}{T\_O} - \frac{1}{T\_L} \right) \frac{1}{COP} \right]} \tag{30}$$

When Eq. (21) is put in Eq. (30), the ecological coefficient of performance of the irreversible three-heat-source absorption refrigerator as a function of 1 *T* , 2 *T* and 3 *T* is derived as :

$$ECOP = \frac{1}{1 - \frac{\varepsilon\_r T\_1 (IT\_3 - T\_2)}{T\_2 \left(T\_1 - IT\_3\right)} \left| 1 - \xi \left(T\_O - T\_1\right) \left| \frac{1}{\mathcal{U}\_L \left(T\_L - T\_2\right)} + \frac{T\_1 \left(IT\_3 - T\_2\right)}{\mathcal{U}\_H \left(T\_H - T\_1\right) \left(T\_1 - IT\_3\right)T\_2} + \frac{IT\_3 \left(T\_1 - T\_2\right)}{\mathcal{U}\_O \left(T\_3 - T\_O\right) \left(T\_1 - IT\_3\right)T\_2} \right|^{-1}}\right|^{-1} \tag{31}$$

ECOP Criterion for Irreversible Three-Heat-Source Absorption Refrigerators 453

*<sup>T</sup>* <sup>=</sup> (32)

*<sup>y</sup> <sup>T</sup>* <sup>=</sup> (33)

( )( ) ( )( )

*<sup>I</sup>* <sup>=</sup> (37)

¶ <sup>=</sup> ¶ (38)

¶ <sup>=</sup> ¶ (39)

¶ <sup>=</sup> ¶ (40)

( ) ( )

2

0

1

(35)

(41)


1 *IT*

2 *IT*

3 *z IT* = (34)

1 1

( )



1

*env O L*

*TT T*

1 1 1

*<sup>y</sup> <sup>y</sup> x y y x T T x U Ty z U T x z x Uz T x*


*L L H H*

*<sup>O</sup> T IT* = (36)

*x*

*y*

*z*

1 1 1

*y z y*


*O E LL H H*

*T T U Ty z Uz T U Tx z x*

1 1

( )

*x*

For the sake of convenience, let

Then Eq. (31) is rewritten as:

1 1

*r*

*e*

*ECOP*

where

and

=

give respectively:

<sup>3</sup>

<sup>3</sup>

( ) ( ) ( )

*x*

. *UO <sup>U</sup>*

<sup>0</sup> *ECOP*

<sup>0</sup> *ECOP*

<sup>0</sup> *ECOP*

( ) ( ) ( )

Starting from Eq. (35), the extremal conditions:

*O E*

where *env T* is the temperature in the environment conditions.

## **3. Performance optimization for a three-heat-source irreversible absorption refrigerator based on ECOP criterion**

The ECOP function given in Eq. (31) is plotted with respect to the working fluid temperatures ( <sup>1</sup> *T* , 2 *T* and 3 *T* ) for different internal irreversibility parameters as shown in Fig. 3(a), (b) and (c). As it can be seen from the figure, there exists a specific 1 *T* , 2 *T* and 3 *T* that maximize the ECOP function for given *I* and *x* values. Therefore, Eq. (31) can be maximized (or optimized) with respect to 1 *T* , 2 *T* and 3 *T* . The optimization is carried out analytically.

**Figure 3.** Variation of the ECOP objective function with respect to 1 *T* (a), 2 *T* (b) and 3 *T* (c) for different *I* values ( 403 *GT K* = , 273 *<sup>L</sup> T K* = , 303 *OT K* = , 290 *env T K* = , 1163 *UG* = <sup>2</sup> *W mK* / , <sup>2326</sup> *UE* <sup>=</sup> <sup>2</sup> *W mK* / , 4650 *UO* <sup>=</sup> <sup>2</sup> *W mK* / , 1082 *KL* <sup>=</sup> *W K*/ , *<sup>A</sup>* <sup>=</sup> <sup>1100</sup> <sup>2</sup> *<sup>m</sup>* )

For the sake of convenience, let

452 Thermodynamics – Fundamentals and Its Application in Science

<sup>1</sup> 1 1

( ) ( ) ( )

*O L*

where *env T* is the temperature in the environment conditions.

**absorption refrigerator based on ECOP criterion** 

**3. Performance optimization for a three-heat-source irreversible** 

with respect to 1 *T* , 2 *T* and 3 *T* . The optimization is carried out analytically.

**Figure 3.** Variation of the ECOP objective function with respect to 1 *T* (a), 2 *T* (b) and 3 *T* (c) for different *I* values ( 403 *GT K* = , 273 *<sup>L</sup> T K* = , 303 *OT K* = , 290 *env T K* = , 1163 *UG* = <sup>2</sup> *W mK* / ,

<sup>2326</sup> *UE* <sup>=</sup> <sup>2</sup> *W mK* / , 4650 *UO* <sup>=</sup> <sup>2</sup> *W mK* / , 1082 *KL* <sup>=</sup> *W K*/ , *<sup>A</sup>* <sup>=</sup> <sup>1100</sup> <sup>2</sup> *<sup>m</sup>* )

*T T*

( )

*<sup>e</sup> <sup>x</sup>*

*r*

*ECOP*

=

( )



1

*env O L*

*T T IT U T T U T T T IT T U T T T IT T*

<sup>ï</sup> - -- + <sup>ï</sup> <sup>+</sup> <sup>ï</sup> <sup>í</sup> <sup>ê</sup> úý - <sup>ï</sup> - -- - - <sup>ï</sup> <sup>ï</sup> <sup>ê</sup> úï ïî <sup>ë</sup> ûïþ

*TT T*

13 2 13 2 31 2 21 3 2 1 1 32 3 1 32

*T IT T T IT T IT T T*

The ECOP function given in Eq. (31) is plotted with respect to the working fluid temperatures ( <sup>1</sup> *T* , 2 *T* and 3 *T* ) for different internal irreversibility parameters as shown in Fig. 3(a), (b) and (c). As it can be seen from the figure, there exists a specific 1 *T* , 2 *T* and 3 *T* that maximize the ECOP function for given *I* and *x* values. Therefore, Eq. (31) can be maximized (or optimized)

1 1


( ) ( )( )

*LL HH O O*

( ) ( )( ) 1


(31)

$$\mathbf{x} = \frac{\mathbf{I}\mathbf{T}\_3}{T\_1} \tag{32}$$

$$y = \frac{\Pi T\_3}{T\_2} \tag{33}$$

$$z = \text{I}\mathbf{T}\_3\tag{34}$$

Then Eq. (31) is rewritten as:

$$ECOP = \frac{\frac{1}{T\_{mv}\left(T\_O^{-1} - T\_L^{-1}\right)}}{1 - \frac{\varepsilon\_r(y-1)}{1-\chi} \left[1 - \xi\left(T\_O - T\_E\right)\left|\frac{y}{\mathcal{U}\_L\left(T\_L y - z\right)} + \frac{\mathbf{x}\left(y-1\right)}{\mathcal{U}\_H\left(T\_H x - z\right)\left(1-\mathbf{x}\right)} + \frac{y-\mathbf{x}}{\mathcal{U}\left(z-T\right)\left(1-\mathbf{x}\right)}\right|}\right]^{-1} \tag{35}$$

where

$$T = IT\_{\odot} \tag{36}$$

and

$$
\mathcal{U} = \frac{\mathcal{U}\_O}{I}.\tag{37}
$$

Starting from Eq. (35), the extremal conditions:

$$\frac{\partial \text{ECOP}}{\partial \mathbf{x}} = \mathbf{0} \tag{38}$$

$$\frac{\partial \text{ECOP}}{\partial y} = 0 \tag{39}$$

$$\frac{\partial \text{ECOP}}{\partial \mathbf{z}} = \mathbf{0} \tag{40}$$

give respectively:

$$\frac{1}{\xi \left(T\_O - T\_E\right)} - \frac{y}{\mathcal{U}\_L \left(T\_L y - z\right)} - \frac{1}{\mathcal{U} \left(z - T\right)} - \frac{z \left(y - 1\right)}{\mathcal{U}\_H \left(T\_H x - z\right)^2} = 0\tag{41}$$

$$\frac{1}{\xi \left(T\_O - T\_E\right)} - \frac{y}{\mathcal{U}\_L \left(T\_L y - z\right)} - \frac{1}{\mathcal{U} \left(z - T\right)} - \frac{z \left(y - 1\right)}{\mathcal{U}\_L \left(T\_L y - z\right)^2} = 0\tag{42}$$

$$\frac{y\left(1-x\right)}{\mathcal{U}\_{L}\left(T\_{L}y-z\right)^{2}} + \frac{x\left(y-1\right)}{\mathcal{U}\_{H}\left(T\_{H}x-z\right)^{2}} - \frac{y-x}{\mathcal{U}\left(z-T\right)^{2}} = 0\tag{43}$$

Combining Eqs (41)-(43), the following general relation is found:

$$
\sqrt{\mathcal{U}\_H} \left( T\_H \mathbf{x} - z \right) = \sqrt{\mathcal{U}\_L} \left( T\_L \mathbf{y} - z \right) = \sqrt{\mathcal{U}} \left( z - T \right) \tag{44}
$$

From Eqs (44), it is derived as:

$$x = \frac{\left(1 + b\_1\right)z}{T\_H} - \frac{b\_1 T}{T\_H} \tag{45}$$

ECOP Criterion for Irreversible Three-Heat-Source Absorption Refrigerators 455

= + (52)

*<sup>T</sup>* <sup>=</sup> (53)


<sup>+</sup> <sup>=</sup> <sup>+</sup> (55)

<sup>+</sup> <sup>=</sup> + + (56)

<sup>+</sup> <sup>=</sup> + (58)

(51)

(57)

2

1 *<sup>O</sup> L*

<sup>÷</sup> <sup>ç</sup> <sup>÷</sup> è ø

1

*<sup>b</sup> <sup>T</sup> <sup>d</sup> U T <sup>x</sup>* <sup>+</sup> æ ö <sup>ç</sup> <sup>÷</sup> = - <sup>ç</sup> <sup>÷</sup> <sup>ç</sup> <sup>÷</sup>

> *H <sup>T</sup> x BD B T*

> > *L T y D*

<sup>1</sup> 1 *b b <sup>B</sup> b*

1 1 *<sup>b</sup> <sup>B</sup>*

Using Eqs. (49), (52) and (53) with Eqs.(32)-(34), the corresponding optimal temperatures of the working fluid in the three isothermal processes when the ecological coefficient of

> <sup>1</sup> <sup>1</sup> *<sup>H</sup> D b T T*

> > *D b T T*

2

*bDB*

<sup>2</sup> <sup>1</sup> *<sup>L</sup>*

<sup>2</sup> <sup>1</sup> *<sup>O</sup> D b T T*

( ) max 1 1 1

*TT T* - - <sup>=</sup> -

*env O L*

Substituting Eqs. (56)-(58) into Eqs. (21), (23), (27) and (31) the maximum *ECOP* function and the corresponding optimal coefficient of performance, optimal specific cooling load and

*b*

*D b* <sup>+</sup> <sup>=</sup> <sup>+</sup>

*b*

1

1

( )<sup>2</sup>

1 ( )

2 1

<sup>1</sup>

performance is a maximum, are, respectively, determined by:

( )( ) \* <sup>2</sup>

( ) \* <sup>2</sup>

\* <sup>2</sup>

1

2

3

optimal specific entropy generation rate are derived, respectively, as:

*ECOP*

Therefore Eqs. (45) and (46) are rewritten as:

where

$$y = \frac{\left(1 + b\_2\right)z}{T\_L} - \frac{b\_2 T}{T\_L} \tag{46}$$

where

$$b\_1 = \sqrt{\frac{U}{U\_H}}\tag{47}$$

$$b\_2 = \sqrt{\frac{U}{U\_L}}\tag{48}$$

$$\text{When Eqs. (45) and (46) are substituted into Eq. (43):}$$

$$z = T \frac{D + b\_2}{1 + b\_2} \tag{49}$$

where

$$D = \frac{1 + \sqrt{d\_1 \left[ 1 - \frac{T\_L \left( 1 - d\_1 \right)}{T} \right]}}{1 - d\_1} \tag{50}$$

ECOP Criterion for Irreversible Three-Heat-Source Absorption Refrigerators 455

$$d\_1 = \xi \frac{\left(1 + b\_2\right)^2}{\mathcal{U}} \left(\frac{T\_O}{T\_L} - 1\right) \tag{51}$$

Therefore Eqs. (45) and (46) are rewritten as:

$$\mathbf{x} = \frac{T}{T\_H} \mathcal{B}\_1 \left( D + \mathcal{B} \right) \tag{52}$$

$$y = \frac{T}{T\_L}D\tag{53}$$

where

454 Thermodynamics – Fundamentals and Its Application in Science

( )

From Eqs (44), it is derived as:

where

where

( ) ( ) ( )

( )

Combining Eqs (41)-(43), the following general relation is found:

( <sup>1</sup>) <sup>1</sup> <sup>1</sup>

( <sup>2</sup> ) <sup>2</sup> <sup>1</sup>

1

2

When Eqs. (45) and (46) are substituted into Eq. (43):

<sup>2</sup>

1 1 1

*y z y*


( )

*U T x z U Ty z U z T H H* ( ) -= -= - *L L* ( ) ( ) (44)

*H H b z b T*

*L L b z b T*

*H*

*L*

<sup>2</sup> 1 *D b z T*

1

*d*

1

1 1

*D*

*b*

1

*d*

( )<sup>1</sup>

1

*<sup>L</sup> T d*

*T*

*<sup>y</sup> T T*

*<sup>U</sup> <sup>b</sup>*

*<sup>U</sup> <sup>b</sup>*

*T T*

*y x xy y x U Ty z U T x z Uz T*

( ) ( ) 2 22


*O E LL L L*

1 1

*x*

*L L H H*

*T T U Ty z Uz T U Ty z x*

( ) ( )

2

0

<sup>+</sup> = - (45)

<sup>+</sup> = - (46)

*<sup>U</sup>* <sup>=</sup> (47)

*<sup>U</sup>* <sup>=</sup> (48)

<sup>+</sup> <sup>=</sup> <sup>+</sup> (49)

<sup>é</sup> - <sup>ù</sup> <sup>ê</sup> <sup>ú</sup> + -ê <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> <sup>û</sup> <sup>=</sup> - (50)

0

(43)

(42)

$$B = \frac{b\_2 - b\_1}{1 + b\_1} \tag{54}$$

$$B\_1 = \frac{1 + b\_1}{1 + b\_2} \tag{55}$$

Using Eqs. (49), (52) and (53) with Eqs.(32)-(34), the corresponding optimal temperatures of the working fluid in the three isothermal processes when the ecological coefficient of performance is a maximum, are, respectively, determined by:

$$T\_1^\* = T\_H \frac{D + b\_2}{(1 + b\_1)(D + B)}\tag{56}$$

$$T\_2 \stackrel{\*}{=} T\_1 \frac{D + b\_2}{D\left(1 + b\_2\right)}\tag{57}$$

$$T\_3 \stackrel{\*}{=} T\_O \frac{D + b\_2}{1 + b\_2} \tag{58}$$

Substituting Eqs. (56)-(58) into Eqs. (21), (23), (27) and (31) the maximum *ECOP* function and the corresponding optimal coefficient of performance, optimal specific cooling load and optimal specific entropy generation rate are derived, respectively, as:

$$ECOP\_{\max} = \frac{1}{T\_{env} \left(T\_O^{-1} - T\_L^{-1}\right)}$$

$$\times \frac{1}{1 - \frac{\varepsilon\_r T\_H \left(T D - T\_L\right)}{\left[T\_H - B\_1 \left(D + B\right)T\right]T\_L} \left[1 - \xi \left(T\_O - T\_L\right) \frac{T\_H D - B\_1^{-2} \left(D + B\right) \left(T\_L + BDT\right)}{\mathcal{U}^\* B\_1^{-2} \left(D - 1\right) \left[T\_H - B\_1 \left(D + B\right)T\right]T\_L}\right]^{-1}}\tag{59}$$

$$\text{COP}^\* = \frac{\left[T\_H - \text{B}\_1 (D + \text{B})T\right]T\_L}{T\_H \left(\text{TD} - T\_L\right)} \left[1 - \xi \left(T\_O - T\_L\right) \frac{T\_H D - \text{B}\_1^{-2} \left(D + \text{B}\right) \left(T\_L + BDT\right)}{\text{L}^\* \text{B}\_1^{-2} \left(D - 1\right) \left[T\_H - \text{B}\_1 \left(D + \text{B}\right)T\right]T\_L}\right] \tag{60}$$

$$\dot{r}^\* = \frac{\mathbf{U}^\* \mathbf{B}\_1^{-2} \left(\mathbf{D} - \mathbf{1}\right) \left[T\_H - \mathbf{B}\_1 \left(\mathbf{D} + \mathbf{B}\right)T\right] T\_H}{T\_H \mathbf{D} - \mathbf{B}\_1^{-2} \left(\mathbf{D} + \mathbf{B}\right) \left(T\_L + \mathbf{B} \mathbf{D} T\right)} - \xi \left(T\_O - T\_L\right) \tag{61}$$

$$\boldsymbol{s}^\* = \left| \frac{1}{T\_L} - \frac{1}{T\_O} \right| \left| \xi \left( \boldsymbol{T}\_O - \boldsymbol{T}\_L \right) - \left[ 1 - \frac{\boldsymbol{\varepsilon}\_r \boldsymbol{T}\_H \left( \boldsymbol{\mathrm{TD}} - \boldsymbol{T}\_L \right)}{\left[ \boldsymbol{T}\_H - \boldsymbol{\mathcal{B}}\_1 \left( \boldsymbol{D} + \boldsymbol{\mathcal{B}} \right) \boldsymbol{T}\_L \right]} \left| \frac{\left[ \boldsymbol{\mathrm{T}}^\star \boldsymbol{\mathcal{B}}\_1 \left( \boldsymbol{D} - \boldsymbol{1} \right) \right] \boldsymbol{T}\_H - \boldsymbol{\mathcal{B}}\_1 \left( \boldsymbol{D} + \boldsymbol{\mathcal{B}} \right) \boldsymbol{T}\_L}{\left[ \boldsymbol{T}\_H \boldsymbol{\mathcal{D}} - \boldsymbol{\mathcal{B}}\_1 \left( \boldsymbol{D} + \boldsymbol{\mathcal{B}} \right) \right] \boldsymbol{T}\_L + \boldsymbol{\mathcal{B}} \boldsymbol{D} \boldsymbol{T}} \right| \tag{62}$$

where

$$\boldsymbol{U}^\* = \frac{\boldsymbol{U}}{\left(1 + b\_1\right)^2} \tag{63}$$

ECOP Criterion for Irreversible Three-Heat-Source Absorption Refrigerators 457

*ECOP ECOP*

<sup>1</sup> *<sup>T</sup>* , \*

<sup>2</sup> *<sup>T</sup>* , \*

max

*ECOP* ), normalized COP (

<sup>3</sup> *<sup>T</sup>* , \* *<sup>A</sup>*<sup>1</sup> , \* *<sup>A</sup>*<sup>2</sup> , \* *<sup>A</sup>*<sup>3</sup> , \* *<sup>r</sup>* ,

Obviously, this relation is independent of the heat leak and the temperatures of the external

*COP* ) and the specific cooling load (r) with respect to the specific entropy

generation rate (s) are demonstrated. One interesting observation from this figure is that maximum of the ECOP and COP coincides although their functional forms are different: the coefficient of performance gives information about the necessary heat rate input in order to produce certain amount of cooling load and the ecological coefficient of performance gives information about the entropy generation rate or loss rate of availability in order to produce certain amount of cooling load. The maximum ECOP and COP conditions give the same amount of cooling load and entropy generation rate. It is

*<sup>s</sup>* and \* *COP COP* <sup>=</sup> max at the maximum ECOP and maximum COP are same. Getting the same performance at maximum ECOP and COP conditions is an expected and logical result. Since, for a certain cooling load the maximum *COP* results from minimum heat consumption so that minimum environmental pollution. The minimum environmental pollution is also achieved by maximizing the *ECOP* . Although the optimal performance conditions *ECOP* and *COP* criteria are same, their impact on the system design performance is different. The coefficient of performance is used to evaluate the performance and the efficiency of systems. This method only takes into account the first law of thermodynamics which is concerned only with the conversion of energy, and therefore, can not show how or where irreversibilities in a system or process occur. Also, when different sources and forms of energy are involved within a system, the *COP* criterion of a system doesn't describe its performance from the view point of the energy quality involved. This factor is taken into account by the second law of thermodynamics characterized by the entropy production which appears in the ecological coefficient of performance criterion ( *ECOP* ). This aspect is of major importance today since that with the requirement of a rigorous management of our energy resources, one should have brought to be interested more and more in the second principle of thermodynamics, because degradations of energy, in other words the entropy productions, are equivalent to consumption of energy resources. For this important reason, the *ECOP* criterion can enhance the system performance of the absorption refrigerators by reducing the irreversible losses in the system. A better understanding of the second law of thermodynamics reveals that the ecological coefficient of performance optimization is an

heat reservoirs.

*COP COP*

\*

max

**4. Comparison with COP criterion** 

In Fig.4, the variation of the normalized ECOP (

also seen analytically that the performance parameters \*

important technique in achieving better operating conditions.

From Eqs. (17)-(19) and (56)-(58), it is found that , when the three-heat-source absorption refrigerator is operated in the state of maximum ecological coefficient of performance, the relations between the heat-transfer areas of the heat exchangers and the total heat-transfer area are determined by:

$$A\_H \stackrel{\*}{=} A \frac{b\_1}{1+b\_1} \frac{B\_1 \,^2(D+B)(TD-T\_L)}{T\_HD - B\_1 \,^2(D+B)(T\_L+BDT)}\tag{64}$$

$$A\_L \stackrel{\*}{=} A \frac{b\_2}{1+b\_1} \frac{DB\_1 \left[T\_H - B\_1 T \left(D + \mathcal{B}\right)\right]}{T\_H D - B\_1^2 \left(D + \mathcal{B}\right) \left(T\_L + BDT\right)}\tag{65}$$

$$A\_O \stackrel{\*}{=} A \frac{1}{1+b\_1} \frac{B\_1 \left| T\_H D - B\_1 T\_L \left( D + B \right) \right|}{T\_H D - B\_1^{-2} \left( D + B \right) \left( T\_L + BDT \right)} \tag{66}$$

From Equations (64)-(66), a concise optimum relation for the distribution of the heat-transfer areas is obtained as:

$$
\sqrt{\mathcal{U}\_H} A\_H \stackrel{\*}{+} \sqrt{\mathcal{U}\_L} A\_L \stackrel{\*}{=} \sqrt{\mathcal{U}} A\_O \stackrel{\*}{\tag{67}} \tag{67}
$$

Obviously, this relation is independent of the heat leak and the temperatures of the external heat reservoirs.

#### **4. Comparison with COP criterion**

456 Thermodynamics – Fundamentals and Its Application in Science

( )

*COP T T*

( ) ( )

\* 2

1 1 \* 2 1

( ) ( )

( )( )

*H*

*L*

*O*

*A A*

*A A*

*A A*

\* 1 1

<sup>1</sup> <sup>1</sup> 1

1 1 \* 2

<sup>1</sup> <sup>1</sup> 1

1 1 \*

1 1

1 1

1 1

*O L*

*e*

( )

*s T T*

area are determined by:

where

areas is obtained as:

*x*

( ) ( ) ( )( )

1

*T TD T T D B D B T BDT*

1 1 1

*rH L H L O L H L H L*


*H L H L O L*

*T B D BTT T D B D B T BDT*

1 *H H*

é ù - + ì ü ï ï ê ú - ++ <sup>=</sup> ë û í ý - - - ï ï --+ é ù

*T T T B D BTT UB D T B D BTT*

*x*

\* 1 1

1

*H L*

1 1 \*

*e*

*x*

*UB D T B D BTT r T T T D B D B T BDT*

( )

*T T T B D BTT T D B D B T BDT*

*<sup>U</sup> <sup>U</sup>*

=

*L O H L H L*

\*

<sup>1</sup> 1 1 <sup>1</sup> *rH L H L*

( )

+

From Eqs. (17)-(19) and (56)-(58), it is found that , when the three-heat-source absorption refrigerator is operated in the state of maximum ecological coefficient of performance, the relations between the heat-transfer areas of the heat exchangers and the total heat-transfer

2

2

2

2

<sup>é</sup> - + <sup>ù</sup> <sup>ê</sup> <sup>ú</sup> <sup>=</sup> <sup>ë</sup> <sup>û</sup>

From Equations (64)-(66), a concise optimum relation for the distribution of the heat-transfer

\* \*\* *U A U A UA HH LL O* + = (67)

*H L*

ê ú - + <sup>=</sup> ë û

*H*

*b DB T B T D B*

*H L*

*b T D B D B T BDT*

*H L*

*b T D B D B T BDT*

*B T D BT D B*

*b B D B TD T*

*H L*

*b T D B D B T BDT*

*b*

ì ü ï ï <sup>é</sup> ù é <sup>ù</sup> é ù æ ö <sup>ê</sup> - ú ê --+ <sup>ú</sup> <sup>ç</sup> ÷ï ï <sup>ï</sup> ê ú <sup>÷</sup> <sup>ë</sup> û ï = - - -- <sup>ç</sup> ÷í ý <sup>ê</sup> ú ê <sup>ú</sup> <sup>ç</sup> <sup>ç</sup> ÷÷ï ï <sup>ê</sup> é ù - + ú ê - ++ <sup>ú</sup> è øï ï <sup>ê</sup> ê ú ú ê <sup>ú</sup> <sup>ë</sup> <sup>û</sup> ï ï î þë û ë <sup>û</sup>

\* 2

( ) ( ) ( )( )

*H L H L*

*T TD T UB D T B D BTT*

1 1

2 <sup>1</sup> 1

\* 2

( )( ) ( )

\* 2

( )( )

é ù

( ) ( )( )

( ) ( )( )

*L*

<sup>+</sup> - ++ (65)

<sup>+</sup> - ++ (66)

+ - <sup>=</sup> <sup>+</sup> - ++ (64)

() ( )

2

1 1

1

*x* --+ <sup>é</sup> <sup>ù</sup> ê ú <sup>=</sup> ë û - - - ++ (61)

*T TD T UB D T B D BTT*

ê ú î þ ë û

() ( )

*O L*

2

() ( )

( )( )

2 1

1

1

(59)

(60)

(62)

(63)

In Fig.4, the variation of the normalized ECOP ( max *ECOP ECOP ECOP* ), normalized COP (

max *COP COP COP* ) and the specific cooling load (r) with respect to the specific entropy

generation rate (s) are demonstrated. One interesting observation from this figure is that maximum of the ECOP and COP coincides although their functional forms are different: the coefficient of performance gives information about the necessary heat rate input in order to produce certain amount of cooling load and the ecological coefficient of performance gives information about the entropy generation rate or loss rate of availability in order to produce certain amount of cooling load. The maximum ECOP and COP conditions give the same amount of cooling load and entropy generation rate. It is

also seen analytically that the performance parameters \* <sup>1</sup> *<sup>T</sup>* , \* <sup>2</sup> *<sup>T</sup>* , \* <sup>3</sup> *<sup>T</sup>* , \* *<sup>A</sup>*<sup>1</sup> , \* *<sup>A</sup>*<sup>2</sup> , \* *<sup>A</sup>*<sup>3</sup> , \* *<sup>r</sup>* ,

\* *<sup>s</sup>* and \* *COP COP* <sup>=</sup> max at the maximum ECOP and maximum COP are same. Getting the same performance at maximum ECOP and COP conditions is an expected and logical result. Since, for a certain cooling load the maximum *COP* results from minimum heat consumption so that minimum environmental pollution. The minimum environmental pollution is also achieved by maximizing the *ECOP* . Although the optimal performance conditions *ECOP* and *COP* criteria are same, their impact on the system design performance is different. The coefficient of performance is used to evaluate the performance and the efficiency of systems. This method only takes into account the first law of thermodynamics which is concerned only with the conversion of energy, and therefore, can not show how or where irreversibilities in a system or process occur. Also, when different sources and forms of energy are involved within a system, the *COP* criterion of a system doesn't describe its performance from the view point of the energy quality involved. This factor is taken into account by the second law of thermodynamics characterized by the entropy production which appears in the ecological coefficient of performance criterion ( *ECOP* ). This aspect is of major importance today since that with the requirement of a rigorous management of our energy resources, one should have brought to be interested more and more in the second principle of thermodynamics, because degradations of energy, in other words the entropy productions, are equivalent to consumption of energy resources. For this important reason, the *ECOP* criterion can enhance the system performance of the absorption refrigerators by reducing the irreversible losses in the system. A better understanding of the second law of thermodynamics reveals that the ecological coefficient of performance optimization is an important technique in achieving better operating conditions.

ECOP Criterion for Irreversible Three-Heat-Source Absorption Refrigerators 459

**Author details** 

Réné Tchinda

**6. References** 

Istanbul. (2004).

151 (2005).

Paiguy Armand Ngouateu Wouagfack

*L2MSP, Department of Physics, University of Dschang, Dschang , Cameroon* 

International Journal of Refrigeration. 34, 1008-1015 (2011).

of Heat Transfer. 32, 1631-1639 (1989).

Thermal Science. 45 (1), 94–101 (2006).

Energy. 82 (1), 23–39 (2005).

83 (6), 558–572 (2006).

4713–4721 (2006).

(2007).

*LISIE, University Institute of Technology Fotso Victor, University of Dschang, Bandjoun, Cameroon* 

[1] Ust, Y.: Ecological performance analysis and optimization of powergeneration systems, Ph.D. Thesis Progress Report, Yildiz Technical University,

[2] Ngouateu, Wouagfack, P. A., Tchinda, R.: Performance optimization of three-heatsource irreversible refrigerator based on a new thermo-ecological criterion.

[3] Bejan, A.: Theory of heat transfer-irreversible refrigeration plant. International Journal

[4] Ust, Y., Sahin, B., Sogut, O. S.: Performance analysis and optimization of an irreversible Dual cycle based on ecological coefficient of performance (ECOP) criterion. Applied

[5] Ust, Y., Sahin, Kodal, A.: Ecological coefficient of performance (ECOP) optimization for generalized irreversible Carnot heat engines. Journal of the Energy Institute. 78 (3), 145–

[6] Ust, Y., Sahin,B., Kodal, A.: Performance analysis of an irreversible Brayton heat engine based on ecological coefficient of performance criterion. International Journal of

[7] Ust, Y., Sogut, O. S., Sahin, B., Durmayaz, A.: Ecological coefficient of performance (ECOP) optimization for an irreversible Brayton heat engine with variable-temperature

[8] Ust, Y., Sahin, B., Kodal, A., Akcay, I. H.: Ecological coefficient of performance analysis and optimization of an irreversible regenerative Brayton heat engine. Applied Energy.

[9] Sogut, O. S., Ust, Y., Sahin, B.: The effects of intercooling and regeneration on the thermo-ecological performance analysis of an irreversible-closed Brayton heat engine with variable-temperature thermal reservoirs. Journal of Physics D: Applied Physics. 39,

[10] Ust, Y., Sahin, B.: Performance optimization of irreversible refrigerators based on a new thermo-ecological criterion. International Journal of Refrigeration. 30, 527–534

thermal reservoirs. Journal of the Energy Institute. 79 (1), 47–52 (2006).

**Figure 4.** Variation of the normalized *ECOP* , normalized *COP* and the specific cooling load with respect to the specific entropy generation rate ( 403 *GT K* = , 273 *<sup>L</sup> T K* = , 303 *OT K* = , 290 *env T K* = , 1163 *UG* = <sup>2</sup> *W mK* / , 2326 *UE* = <sup>2</sup> *W mK* / , 4650 *UO* = <sup>2</sup> *W mK* / , 1082 *KL* = *W K*/ , *A* = 1100 <sup>2</sup> *m* )

#### **5. Conclusion**

This chapter presented an analytical method developed to achieve the performance optimization of irreversible three-heat-source absorption refrigeration models having finiterate of heat transfer, heat leakage and internal irreversibility based on an objective function named ecological coefficient of performance (ECOP). The optimization procedure consists in defining the objective function ECOP in term of the temperatures of the working fluid in the generator, evaporator, condenser and absorber and using extremal conditions to determine analytically the maximum ECOP and the corresponding optimal design parameters. It also established comparative analyses with the COP criterion and shown that the performance parameters at the maximum ECOP and maximum COP are same. The three-heat-source absorption refrigerator cycles are the simplified models of the absorption refrigerators, but the four-heat-source absorption refrigerators cycles are closer to the real absorption refrigerators.

## **Author details**

458 Thermodynamics – Fundamentals and Its Application in Science

<sup>2</sup> *m* )

**5. Conclusion** 

refrigerators.

**Figure 4.** Variation of the normalized *ECOP* , normalized *COP* and the specific cooling load with respect

1163 *UG* = <sup>2</sup> *W mK* / , 2326 *UE* = <sup>2</sup> *W mK* / , 4650 *UO* = <sup>2</sup> *W mK* / , 1082 *KL* = *W K*/ , *A* = 1100

This chapter presented an analytical method developed to achieve the performance optimization of irreversible three-heat-source absorption refrigeration models having finiterate of heat transfer, heat leakage and internal irreversibility based on an objective function named ecological coefficient of performance (ECOP). The optimization procedure consists in defining the objective function ECOP in term of the temperatures of the working fluid in the generator, evaporator, condenser and absorber and using extremal conditions to determine analytically the maximum ECOP and the corresponding optimal design parameters. It also established comparative analyses with the COP criterion and shown that the performance parameters at the maximum ECOP and maximum COP are same. The three-heat-source absorption refrigerator cycles are the simplified models of the absorption refrigerators, but the four-heat-source absorption refrigerators cycles are closer to the real absorption

to the specific entropy generation rate ( 403 *GT K* = , 273 *<sup>L</sup> T K* = , 303 *OT K* = , 290 *env T K* = ,

Paiguy Armand Ngouateu Wouagfack *L2MSP, Department of Physics, University of Dschang, Dschang , Cameroon* 

Réné Tchinda

*LISIE, University Institute of Technology Fotso Victor, University of Dschang, Bandjoun, Cameroon* 

## **6. References**

	- [11] Ust, Y.: Performance analysis and optimization of irreversible air refrigeration cycles based on ecological coefficient of performance criterion. Applied Thermal Engineering. 29, 47–55 (2009).

**Section 6** 

**Thermodynamics in Diverse Areas** 

**Thermodynamics in Diverse Areas** 

460 Thermodynamics – Fundamentals and Its Application in Science

29, 47–55 (2009).

[11] Ust, Y.: Performance analysis and optimization of irreversible air refrigeration cycles based on ecological coefficient of performance criterion. Applied Thermal Engineering.

**Chapter 18** 

© 2012 Măluţan and Vilda, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Thermodynamics of Microarray Hybridization** 

Microarrays make the use of hybridization properties of nucleic acids to monitor Deoxyribonucleic acid (DNA) or Ribonucleic acid (RNA) abundance on a genomic scale in different types of cells. The hybridization process takes place between surface-bound DNA sequences - the probes, and the DNA or RNA sequences in solution - the targets. Hybridization is the process of combining complementary, single-stranded nucleic acids into a single molecule. Nucleotides will bind to their complement under normal conditions, so two perfectly complementary strands will bind to each other readily. Conversely, due to the different geometries of the nucleotides, a single inconsistency between the two strands

In oligonucleotide microarrays hundreds of thousands of oligonucleotides are synthesized *in situ* by means of photochemical reaction and mask technology. Probe design in these microarrays is based on complementarity to the selected gene or an expressed sequence tag (EST) reference sequence. An important component in designing an oligonucleotide array is

The dynamics of the hybridization process underlying genomic expression is complex as thermodynamic factors influencing molecular interaction are still fields of important research [1] and their effects are not taken into account in the estimation of genetic

Many techniques have been developed to identify trends in the expression levels inferred from DNA microarray data, and recently the attention was devoted to methods to obtain accurate expression levels from raw data on the underlying principles of the thermodynamics and hybridization kinetics. The development of DNA chips for rapidly

and reproduction in any medium, provided the original work is properly cited.

ensuring that each probe binds to its target with high specificity.

expression by the algorithms currently in use.

Raul Măluţan and Pedro Gómez Vilda

http://dx.doi.org/10.5772/51624

will prevent them from binding.

**2. State of the art** 

**1. Introduction** 

Additional information is available at the end of the chapter
