Derivation of effectiveness-NTU method for heat exchangers with heat leak

A powerful and useful method for heat exchanger analysis is the effectiveness-NTU method. The equations for this technique presented in textbooks, however, are limited to the case where all of the heat transfer occurs between the two fluid streams. In an application of interest to us, cryogenic heat exchangers, we wish to consider a heat leak term. Thus, we have derived equations for the {var_epsilon}-NTU method with heat leak involved. The cases to be studied include evaporators, condensers, and counter-flow, with heat leak both in and out