A. Imbalanced counterflow heat exchangers

Consider the counterflow shown in Figure 1 where \(({\dot {m}c}_{p})_1 > ({\dot {m}c}_{p})_2\) and both fluids can be modeled as ideal gases with constant specific heats. The inlet conditions of the two streams are (\(T_{1}\), \(P_{1})\) and (\(T_{2}\), \(P_{2}\)). The total entropy generation rate of the heat exchanger as a control volume is

\[\label{eq1} \dot {S}_{gen} = (\dot {m}c_{p})_1 \biggl[\ln \frac{T_{1,out}} {T_{1}} - \left(\frac{R}{c_{p}} \right)_{1} \ln \frac{P_{1,out}} {P_{1}} \biggr] + (\dot {mc}_{p})_{2} \biggl[\ln \frac {T_{2,out}}{T_{2}} - \left( \frac{R}{c_{p}} \right)_{2} \ln \frac{P_{2,out}} {P_{2}} \biggr]\tag{1}\]

The entropy generation rate does not vanish in the "ideal limit" of zero pressure drops (\(P_{1,out}\) = \(P_{1}\), \(P_{2,out}\) = \(P_{2}\)) and infinitely large heat transfer area, when \(T_{2,out}\) = \(T_{1}\). In that limit the imbalance (or remnant) entropy generation rate is finite and a function of \(T_{1}\)/\(T_{2}\) and C = (\(\dot{m}c_{p}\))\(_{1}\) / (\(\dot{m}c_{p}\))₂, ref Bejan (1996),

\[\label{eq2} N_{S,imbalance} = \frac{\dot{S}_{gen,imbalance}} {(\dot{m}c_{p})_{2}} = \ln \Biggl\{\frac{T_{1}}{T_{2}}\biggl[1 + \frac{1}{C}\left(\frac{T_{2}}{T_{1}} - 1 \right) \biggr]^{c} \Biggr\} \tag{2}\]