Optimal performance of an endoreversible three-mass-reservoir chemical pump with diffusive mass transfer law
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The COP χ and the rate of energy pumping R of the chemical pump are
χ=(µLΔN2)/(µHΔΔN1)                                           (5)
Σ=(µLΔN2)/ζ                                       (6)
Combining Eqs. (1)–(6) gives
Χ=(µL(µ12))/(µH23))                                   (7)
Σ=µL13){(µ23)[h1(exp(µH)/kT-exp(µ1)/kT))]-1+(µ13) [h2(exp(µ2)/kT-exp(µL)/kT))]-1+(µ1-µ2) [h3(exp(µO)/kT-exp(µ3)/kT))]-1}-1                                  (8)
Now, the problem is to determine the optimal rate of energy pumping of the chemical pump for a given COP. Therefore, one can introduce a Lagrangian function L = R + kv, where k is the Lagrangian multiplier. One has:
L=µL13)({(µ23)[h1(exp(µH)/kT-exp(µ1)/kT))]-1+(µ13) [h2(exp(µ2)/kT-exp(µL)/kT))]-1+(µ12) [h3(exp(µO)/kT-exp(µ3)/kT))]-1}-1+λ[µH23)]-1)         (9)
From the Euler–Lagrange equations δL/δl1 = 0, δL/δl2 = 0 and δL/δl3 = 0, one can find that the following equations must be satisfied: