In this chapter, the optimization analysis based on the new thermo-ecological criterion (ECOP) first performed by Ust et al.  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 . In the shown model, is the rate of absorbed heat from the heat source at temperature to generator, and are, respectively, the heat rejection rates from the condenser and absorber to the heat sinks at temperatures and and is the heat input rate from the cooling space at temperature 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 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 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 state 3 is condensed to state 4 by removing 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 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 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:
Absorption refrigeration systems operate between three temperature levels, if, or four temperature levels when. In this chapter, by taking, 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
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, , and are, respectively, the heat-transfer areas of the generator, evaporator, condenser and absorber, and 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 .
The absorption refrigeration system does not exchange heat with other external reservoirs except for the three heat reservoirs at temperatures, and, so the total heat-transfer area between the cycle system and the external heat reservoirs is given by the relationships:
The rate of heat leakage from the heat sink at temperature to the cold reservoir at temperature was first provided by Bejan  and it is given as:
where 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:
On the basis of the second law of thermodynamics, for an internally irreversible cycle, so that. If the internal irreversibility is neglected, the cycle is endoreversible and so. The second law of thermodynamics for an irreversible three-heat-source cycle requires that:
The coefficient of performance of the irreversible three-heat-source absorption refrigerator is:
From Eq. (6), it is expressed as:
The first is the coefficient of performance of the irreversible three-heat-source absorption refrigeration cycle without heat leak losses.
By investigating similar reasoning, the heat-transfer areas of the generator and of condenser and absorber are given respectively by:
Combining Eqs. (8), (12), (15) and (20), the coefficient of performance of the irreversible three-heat-source refrigerator as a function of the temperatures, and of the working fluid in the generator, evaporator, condenser and absorber is obtained:
where the parameter
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:
The specific entropy production rate of the irreversible three-heat-source absorption refrigerator is:
Using Eq. (1) is rewritten as:
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 , the new thermo-ecological objective function called ecological coefficient of performance (
where 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 (, and) 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, and that maximize the ECOP function for given and values. Therefore, Eq. (31) can be maximized (or optimized) with respect to, and. The optimization is carried out analytically.
For the sake of convenience, let
Then Eq. (31) is rewritten as:
Starting from Eq. (35), the extremal conditions:
From Eqs (44), it is derived as:
Therefore Eqs. (45) and (46) are rewritten as:
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
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:
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 (), normalized 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, , , , , , , and 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 results from minimum heat consumption so that minimum environmental pollution. The minimum environmental pollution is also achieved by maximizing the. Although the optimal performance conditions and 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 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 (). 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 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.
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.