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Optimal performance of an endoreversible three-mass-reservoir chemical pump with diffusive mass transfer law
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L)/kT)2)=(exp(µ3)/kT)/(h3(exp(µo)/kT-exp(µ3)/kT)2)=C.         (10)
Substituting Eq. (10) into Eqs. (7) and (8) yields the optimal dimensionless rate of energy pumping Σ*=Σ/(h1µHexp(µH)/kT) and the COP as follows:

Where χ=(exp(µ1)/kT)/(exp(µH)/kT), b1 = h1/h2 and b2 = h1/h3. Eliminating χ=(exp(µ1)/kT)/(exp(µH)/kT) from Eqs. (11) and (12) yields the fundamental optimal relation between the dimensionless rate of energy pumping and the COP of the endoreversible threemass-reservoir chemical pump. It can reveal the Σ-χ characteristic of an endoreversible chemical pump with diffusive mass transfer, as shown by solid line in Fig. 2.
4. Results and discussion
(1) In order to study the characteristics of the three-mass-reservoir endoreversible chemical pump with the diffusive mass transfer law, a numerical example is provided. In the calculations, b1=1.1, b2=1.2, exp(µO)/kT-=2, exp(µO)/kT/ exp(µL)/kT = exp[-(µL¬-µ0)/kT]=e-1, exp(µO)/kT/ exp(µH)/kT = exp[-(µH¬-µ0)/kT]=e-3, and T = 300 K are set. The characteristic curves between the dimensionless rate of energy pumping and the COP of three-mass-reservoir chemical pump with the diffusive mass transfer law and linear mass transfer law are shown in Fig. 2. The curve with diffusive mass transfer law is shown by solid line, while the curve with linear mass transfer law is shown by dashed line. The influence of (µL-µ0)/(kT) on Σ* versus v characteristic with (µH-µ0)/(kT) = 3 is shown in Fig. 3. The influence of (µH-µ0)/(kT) on Σ* versus v characteristic with (µL-µ0)/(kT) = 1 is shown in Fig. 4.

It can be seen clearly from Fig. 2 that the characteristic curve between the rate of energy pumping and the COP is monotonic one. When Σ= 0, one can obtain the reversible coefficient of performance χr of the three-mass-reservoir chemical pump, namely χr=(1-µOH)/(1-µLH) . A real chemical pump device always needs a certain rate of energy pumping. Therefore, the minimum irreversible loss of mass transfer is avoidable. The COP of the cycle for a real chemical pump device is always smaller than χr. Similarly, a real chemical pump cannot be operated in the state of χ= 0, which is the other extreme state, i.e. when infinite output is needed and it is unpractical. On the other hand, it may be found clearly from Fig. 2 that the COP decreases as the rate of energy pumping increases. They are contradictory. In the optimal design of corresponding devices, if one places particular emphasis on the rate of energy pumping, he or she should decrease appropriately the COP. When the COP is considered, one has to sacrifice the rate of energy pumping as a price. Generally, consideration must be given to both.
It can be seen from Fig. 3 that Σ* increases as (µLO)/(kT) increases for the same χ. It can be seen from Fig. 4 that R* decreases as (µHO)/(kT) increases for the same χ.
(2) Owing to the irreversibility of mass transfer, the entropy production rate for the cyclic system of chemical pump is larger than zero. If the environment temperature of the cyclic system is represented by T0, one can obtain the minimum entropy production rate as follows:

Combining Eqs. (10) and (11) with (13), one can obtain the relation between the minimum entropy production rate and the rate of energy pumping or between the minimum entropy production rate and the COP.