## 1. Introduction

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, _{3}/H_{2}O and LiBr/H_{2}O are used as the working substances, and these substances abide by ozone depletion regulations, since they do not consist of chloroﬂuorocarbons. 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

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:

Absorption refrigeration systems operate between three temperature levels, if

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:

where

The absorption refrigeration system does not exchange heat with other external reservoirs except for the three heat reservoirs at temperatures

where

The rate of heat leakage

where

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:

On the basis of the second law of thermodynamics,

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

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

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

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

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

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

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

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

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

where the parameter

represents the heat leakage coefficient and its dimension is w/(Km^{2})

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

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

Using Eq. (1)

or

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

where

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:

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

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

where

## 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 (

For the sake of convenience, let

Then Eq. (31) is rewritten as:

where

and

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

give respectively:

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

From Eqs (44), it is derived as:

where

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

where

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

where

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:

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:

where

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:

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

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

## 4. Comparison with COP criterion

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

## 5. Conclusion

This chapter presented an analytical method developed to achieve the performance optimization of irreversible three-heat-source absorption refrigeration models having finite-rate 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.

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