The optimum performance of absorption cycles with external and internal Irreversibilities

The optimum performance of absorption cycles with external and internal Irreversibilities

R. Fathi^∗ and S. Ouaskit

Laboratoire de physique de la matière Condensée. Université Hassan II- Mohammedia, Faculté des Sciences Ben M’sik, Casablanca-Maroc.

A new model is presented to describe an irreversible absorption refrigerator, in which not only the irreversibilities of heat conduction but also the irreversibilities resulting from the friction, eddy and other irreversible effects inside the working fluid are considered. The influence of these irreversible effects on the performance of an absorption refrigerator with continuos flow is investigated. The analytical expressions of the optimal refrigeration coefficient and the cooling rate of the refrigerator are derived. The predictions of the model are compared with semi-empirical cycle model of single-stage absorption refrigeration machines. The results obtained here can describe the optimal performance of a four-temperature-level absorption refrigeration affected simultaneously by the internal and external irreversibilities and provide the theoretical base s for the optimal design and operation of real absorption refrigerators operating between four temperature level.

I. INTRODUTION

Since the early 1970s there has been increasing concern about the very damaging effects of the emissions of chlor- ofluocarbons in the stratospheric ozone layer. Many countries have been searching more environment friendly refrigerants and new advanced refrigeration cycles. The absorption refrigerators using H₂O-LiBr, NH₃-H₂O, etc as the working substance conform to all of the ozone- preserving regulations because there are no chlorofl- uocarbons. On the other hand, the absorption refrigerators [1-5], which can be driven by ‘low-grade’ heat energy rather than ‘high-grade’ work, are a class of currently developing refrigerator equipment. They can make use of some low-grade’ heat reservoir, such as solar energy, terrestrial heat energy, industrial waste heat and so on .This have a great potential for saving energy and decreasing environmental pollution, so many people have paid great attention to the theoretical analysis and their application.

In the theory of classical thermodynamics, the cyclical model of an absorption refrigerator is treated as reversible four-heat-source cycle, which consists of four reversible adiabatic and four reversible isothermal processes. In order to achieve the coefficient of performance of reversible four- heat-source refrigeration cycles, the isothermal parts of the cycle have to be carried out infinitely slowly because of thermal resistance. This implies that the cooling rate of the resulting refrigerator slowly be zero because an infinite time is taken to draw a finite amount of heat from the cooled space. During the last 20 years, finite-time thermodynamics has been applied successfully to a large number of problems [6-9]. The significance of using the concept of finite-time thermodynamics to investigate thermodynamic cycles has been appreciated; many important results have been obtained.

It is well known that the endoreversible cycle models [10,11] have played an important role in the development of finite-time thermodynamics. They can reveal the effects of the irreversibility of finite-rate heat transfer on the

performance of thermodynamic cycles. However, a real absorption refrigerator is a complex device. There are other sources of irreversibilities besides the irreversibility of finite-rate heat transfer, such as internal dissipation of the cyclic working fluid, irreversibilities caused by mass transfer and so on. According to the theory of classical thermodynamics, the coefficient of performance of a reversible absorption refrigerator operating between four temperature levels is given by:

)]

T / 1 ( ) T / 1 [(

n )]

T / 1 ( ) T / 1 [(

)]

T / 1 ( ) T / 1 [(

n )]

T / 1 ( ) T / 1 [(

c e

a e

g c

g a

r − + −

− +

= −

Ψ (1)

where n=q_c/q_a

The reversible coefficient of performance Ψ_ris important in theory, but it is invariably very far from the coefficient of performance of real absorption refrigerators and hence is of very limited practical value.

The aim of the present study is to perform a direct analysis of the absorption cycle affected by the irreversibility of finite-rate heat transfer between the working fluid and the heat reservoirs ,and the irreversibilities resulting from the friction, eddy, mass transfer and other irreversible effects inside the cyclical working. The predicted performance of the absorption cycle at the maximum cooling load is compared with results obtained by semi-empirical model.

II. AN IRREVERSIBLE ABSORPTION REFRIGERATION CYCLE MODEL

We consider an irreversible absorption refrigeration system which consists of a generator, an absorber, a condenser and an evaporator. These element are combined as four heat sources of the system. The temperatures which are, respectively, T_g, T_a, T_c and T_e as shown in figure 1.

In this figure we note that T₁, T₂, T₃ and T₄ are the

.

temperatures of the four isothermal processes for which working fluid exchanges heat with the heat sources at temperatures T_g, T_a, T_c and T_e, respect-ively, and q_g, q_a, q_c, and q_e are the corresponding rates of heat exchange.

Heat energy q_g is supplied from the heat source at a high temperature T_g to the generator. In this stage, the concentration of the absorbent in the working fluid is increased (rich solution). The weak solu-tion passes thro- ugh the valve into the absorber. The working medium is then condensed in the condenser and subsequently transferred to the evaporator. In such a process, the amount of heat qc is released from the condenser to one heat reservoir at temperature T_c . The liquid working medium is evaporated due to the additional heat q_e from the cooled space at a low temperature T_e to the evaporator. The vapourized working medium is transferred to the absorber where it is absorbed by the weak solution and the amount of heat q_a is released from the absorber from the absorber to the other heat reservoir at temperature T_a . The strong solution produced in the absorber is pumped to the generator. 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 [12].

According to the first law of thermo-dynamics, we have

q

_g +

q

_e=

q

_a+

q

_c (2) Because heat exchange is carried out under a finite temperature difference, the cycle maybe fulfilled at a given finite cycle time. It is assumed that the working fluid in the system flows continuously during the whole cycle time and that heat exchange between the working fluid and heat reservoirs obeys Newton’s law, so that one has

(

^{g 1}

)

g g

g

K A T T

q

= − (3)

(

2 a

)

a a

a

K A T T

q

= − (4)

(

^{3 c}

)

c c

c

K A T T

q

= − (5)

(

e 4

)

e e

e

K A T T

q

= − (6) where T1, T2, T3 and T4 are, respectively, the temperatures of the working substance in the generator, absorber, condenser and evaporator, K_g, K_a, K_c and K_e are,

respectively, the overall heat-transfer coefficients of the generator, absorber, condenser and evaporator, and A_g, A_a, A_c and A_e are, respectively, the heat-transfer areas of the generator, absorber, condenser and evaporator. The total heat-transfer area between the cycle system and the external heat reservoirs is

e c a

g

A A A

A

A = + + +

(7) The irreversibilities for a thermodynamic cycle can be considered as external and internal ones. External irreversibilities arise from temperature differences required to transfer heat between the working fluid and heat sources.

Internal irreversibilities result from the friction, eddy, mass transfer and other irreversible effects inside the working fluid. Because the total effect of the irreversibilities within the working fluid, namely the internal irreversibility, can be characterized in terms of the entropy production inside the cycle [13], we may introduce a parameter

) S S (

) S S I (

e g

c a

∆

∆

∆

∆ +

= +

(8)

to characterize the internal irreversibility , where ∆S_a, Sc

∆ , ∆S_g^and ∆Se are, respectively, the rates of entropy flows flowing out of the working fluid in the isothermal processes T₂ and T₃ and flowing into the working fluid in the isothermal processes T₁ and T₄. Furthermore,

Sa

∆ +∆S_c=∆S_g⁺∆Se+∆S_i, as shown in figure 2, in which ∆S_i is the entropy production which is caused by various internal irreversibilities inside the working fluid. On the basis of the second law of thermodynamics, the entropy production

∆ S

_i is always positive for an internally irreversible cycle, so that ∆S_a+∆S_c>∆S_g⁺∆Se for a real refrigerator, such that I>1. If the internal irreversibility can be neglected, that is, if the cycle is endoreversible [14], then

S

i

∆

^{=0 and} ∆Sa+∆S_c=∆S_g⁺∆Se, thus I=1.

!Sg

!Se

!Sa+!Sc

!Si

cycle

FIG.2 : A schematic diagram of the entropy flows in an irreversible absorption refrigeration cycle.

Using equations (1)-(8), we find that the coefficient of performance and the specific cooling load [15] (i.e. the cooling load per unit total heat-transfer area) of a four temperature-level absorption refrigerator are, respectively, given by

)]

/ 1 ( ) / 1 [(

) / 1 ( ) / 1 (

)]

/ 1 ( ) / 1 [(

) / 1 ( ) / 1 (

3 4

2 4

1 3

1 2

IT T

n IT T

T IT

n T IT

q q

g e

irr

− + −

− +

= −

=

Ψ

⁽⁹⁾

FIG.1 : Schematic diagram of an irreversible absorption refrigeration cycle

Condenser (Tc) Absorber (Ta)

Tg

Ta

Te

Tc

q_g

q_a qc

qe

Generator (Tg)

Cycle

1 L

) 1 t / 1 ( ) 4 T / 1 (

) T T ( K

1 )

T T ( K

1

F )]}

IT / 1 ( ) T / 1 [(

n

)]

IT / 1 ( ) T / 1 ) {[(

T T ( K

1

) T T ( K q

q )

T T ( K q

q

) T T ( K q

q )

T T ( K

1 A

q

]

) (

[

) (

c 3 c a

2 a

3 4

2 4

4 e e

c 3 c e

c a

2 a e

a

1 g g e

g 4

e e e

1

− −

×

+ −

−

×

− +

−

− +

=

+ − + −

+ −

= −

=

−

Π

(10) where:

)]

T / 1 ( ) IT / 1 [(

n )]

T / 1 ( ) IT / 1 [(

L =

₂

−

₁

+

₃

−

₁

1 1

3 1

2 1

g g

)]}]]

T / 1 ( ) IT / 1 [(

n )]

T / 1 (

) IT / 1 ){[(

T T ( K [[

F

−

−

+

−

−

=

For the sake of convenience, let T_a^*=IT_a, T_c^*=IT_c, K_a^*=K_a/I, K_c^*=K_c/I, x=T₄/T₁, y=T₄/(IT₂) and y=T₄/(IT₃).

Equations (9) and (10) may be rewritten as

) z 1 ( n y 1

) x z ( n x y

− +

−

− +

= −

ψ

(11) and

1

* c 4

* c

* a 4

* a

4 g g 4

e e

] [

G ) ) T z / T ( K

) x 1 ( n )

T y / T ( K

x 1

) x / T T ( K

) z 1 ' n y ( 1 ) T T ( K

1

×

−

− + −

− + −

−

− + + −

= − Π

(12)

where:

) x z ( n x y G 1

− +

= −

Using (12) and its external conditions

0 x / ∂ =

∂ Π

^,

∂Π / ∂ y = 0 0

z / ∂ =

∂ Π

^,

∂Π / ∂ T

4 (13) we can prove that for a given value of n, when the temperatures of the working substance in the generator, absorber, condenser and evaporator, are given by

B ) b 1 ( d

B C T C

T

_{1 g} ¹

− +

+

= −

(14)

B ) b 1 ( d

B C T C

T

2 2

1 a

2

+ +

+

= −

(15)

B ) b 1 ( d

B C T C

T

3 3

1 c

3

+ +

+

= −

(16)

c B C T C

T

₄

=

_e

−

¹

+

(17)

respectively, the specific cooling load: attains its maximum, i.e.

}

1

] ][

) (

[ {

1 nz y ) x n 1 ( nz 1

y nz b y )b x 1 (

nz 1 y

n x 1 b C 1

) B C ( T K

3 2

1 1

e e max

− − + +

+ −

− +

+ + −

+ +

= −

Π

(18) The coefficient of performance is given by:

1 e

* a

g

* a 1 1

m

R 1 T / CT ) n 1 (

T T R

] B ) b 1 ( d )[

n 1 1 (

−

 

  + −

×

 





 



 − + + −

Ψ =

(19) Where:

* c

* a 3 3

2

2

( 1 b ) B n [ d ( 1 b ) B ] T / T

d

R = + + + + +

C B ) b 1 ( d T

x T ^{1 1}

g

e + −

=

C B ) b 1 ( d T

y T_* ^{2 2}

a

e + +

=

C B ) b 1 ( d T

z T_* ^{3 3}

c

e + +

=

2 / 1

* a e

2 (K /K )

b =

2 / 1 g e

1 (K /K )

b =

2 / 1

* c e

3 (K /K )

b =

g

* a 2 1

* c

* a 2 3 2

2) n(1 b ) T /T (1 n)(1 b ) T /T b

1 (

C= + + + − + −

g

* a 1

* c

* a 3 2

1 (1 b ) n(1 b )T /T (1 n)(1 b )T /T

C = + + + − + −

* c

* a 3 1 3 2

1 2

1 (1 b )(b b ) n(1 b )(b b )T /T

d = + + + + +

g

* a 2 1 1

* c

* a 2 3 3

2 n(1 b )(b b )T /T (1 n)(1 b)(b b )T /T

d = + − + + − +

g

* a 3 1 1 3

2 2

3 (1 b )(b b ) (1 n)(1 b )(b b )T /T

d = + − + + − +

g

* a 2 2 1

* c

* a 2 2

3

b ) T / T ( 1 n )( b b ) T / T

b ( n [

B = − − + + +

.

III. RESULTAT AND DSISSCUSSION The analysis developed in the preceding section was applied to a practical water-lithium bromide absorption cooling system. As in the work of Bejan [16], the working fluid in the present analysis is assumed to undergo reversible processes. The results for the irreversible absorption refrigeration cycle model have been compared with semi-empirical absorption refrigeration system and found to be in good agreement. Therefore, the conductance used in the present computation are the same as those used by Gommed and Grossman [17] in their computer simulation.

A comparison of the variation of the

Ψ

^{with T}g , T_a and T_c is shown in Figures 3-5. In the practical range of temperatures, there is good trend-wise agreement between the predicted

Ψ

^{and the}

Ψ

obtained by semi-empirical model of the absorption refrigerator. In figures 3-5, curve A gives the variation of the coefficient of performance at the maximum cooling load, which is given by equation (19) at I=1. This

Ψ

is lower than that of the semi- empirical model of the single stage machine given by curve B. This seems to indicate that the externally irreversible machine operating at its maximum cooling load will have larger irreversibilities than the real machine under the same conditions.

Curve D in Figures 3-5 gives the coefficient of performance of the internally irreversible cycle predicted by equation (19) at I=1.05.This curve show that the Ψ^of the single-stage machine with its larger internal irreversibilities is closer to the Ψat the externally irreversible cycle operating at the maximum cooling load.

The curves A and D are together theoretical estimations for the first one (curve A) irreversibility are just external but in the second (cure D) we add the internal irreversibility. In this conditions we can understand easily that B underestimate the experimental reality because the we have surestimate the entropy sources. Now, if we compare curve B and A, we have to pointe out that there is a lower temperature limit T=70°C which correspond to crystallisation of LiBr. So, we have to compare A and B in the interval 75°C≤T≤120°C. For this limits the relative difference is respectively:

21 .

= 0 Ψ

∆Ψ

(For T=75°C) and

= 0 . 15

Ψ

∆Ψ

( For T=120°C).

This implies a maximum difference of about 6%

between curves A and B, which may be acceptable if we compare theoretical model to empirical model with adjustable parameters. We pointe out that the curves A and B have the same behaviour and our model with irrevers-ibility is a good approximation of experimental results.

The Ψ variation of the ideal reversible (Carnot) cycle operating between the same heat reservoirs is shown by curve C in figures 3-5. These values are much larger than the

Ψ

of the corresponding real machine. Also, the variation of the

Ψ

is different of the real machine.

These values can therefore be used as an estimation of the performance limits of semi-empirical absorption refrigerators with given conductances operating at given temperature levels. Similarly, equation (19) may be used to estimate the maximum cooling load of real machines

D A

B C

Coefficient of performance

T_g(°c)

60 70 80 90 100 110 120

0 1 2 3

Ka=Kc=23.98Kw/K K_e=11.93Kw/K Kg=8.48Kw/K T_e=6°c

Ta=20°C and Tc=30°C

FIG.4 : Comparison of coefficient of performance with

semi-empirical results: Ta=20°c and Tc=30°c

60 70 80 90 100 110 120

0 1 2 3

Coefficient of performance

60 70 80 90 100 110 120

0 1 2 3

T_g(°c)

60 70 80 90 100 110 120

Ka=Kc=23.98Kw/K Ke=11.93Kw/K K_g=8.48Kw/K Te=6°c Ta=Tc=20°c

60 70 80 90 100 110 120

0 1 2 3

B A

C

D

FIG.3: Comparison of coefficient of performance (ψ) with Semi-empirical results: Ta=Tc=20°C

Bibliography:

[1] T. Berlitz, P. Satzer, F. Summerer, F. Zeigler and G. Alefeld, "A contribution to the economic perspectives of absorption chillers", Int. Journal of Refrigeration 220, 67 (1999).

[2] A Bejan and J V C.Vargas, Int. J. Heat Mass Transfer 38, 2997 (1995).

[3] Wu Chen, Energy Convers. Manag. 36, 1053 (1995).

[4] J V C. Vargas, M Sokolov and A Bejan, J. Sol. Energy Eng.118, 131 (1996).

[5] J Chen and Schouten, J. A. Energy Convers. Manag.

39, 999 (1998).

[6]Jincan chen, "The influence of multi-irreversibilities on the performance of a heat transformer", J. Phys. D: Appl.

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[8]N. E. Wijeysundera , "Performance of three-heat- reservoir absorption cycles with external and internal irreversibilities", Applied thermal engineering 17, 1151 (1997)

VI. CONCLUSION

We have derived two important expressions were of

Ψ

at the maximum cooling load and the maximum cooling load of the absorption cycle affected by the irreversibility of finite-rate heat transfer and the internal irreversibilities of the working substance. The performance limits predicted by the above expressions were compared with the performance of single-stage machine obtained by semi-empirical model. It is expected that these results may lead to a foundation of deeper investing-ation of real four-temperature-level absor-ption refrigerators.

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[10]N. E. Wijeysundera, "Performance limits of absorption cycles with external heat- transfer irreversibilities", Applied thermal engineering 16,175 (1996)

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J. Phys. D: Appl. Phys. 27, 736 (1994)

[12] W I Eames and S J Aphornatana, Inst. Energy. 66, 29 (1993).

[13] Z Yan and L J Chen, Phys. A: Math. Gen. 28, 6167 (1995).

[14] Z. Yan and J Chen, J. Appl. Phys. 20, 1 (1989).

[15] Wu Chen, Energy Convers. Manag. 36, 7 (1995).

[16] A Bejan "Theory of Heat Transfer- irreversible refrigeration plants", Int. J. Heat Mass Transfer 32, 1631 (1989).

[17]K. Gommed and G. Grossman "Performance analysis of staged absorption heat pump: water-lithium bromide systems", ASHARE Trans. 96, 1590 (1990).

FIG.5: Comparison of coefficient of performance with

semi-empirical results: Ta=30°c and Tc=20°c

Ka=Kc=23.98Kw/K K_e=11.93Kw/K K_g=8.48Kw/K Te=6°c Ta=30°c and Tc=20°c

D A

C

B

Cofficient of performance

T_g(°c)

60 70 80 90 100 110 120

0 1 2 3