Distributed parameter systems such as tubular reactors and heat exchangers often can be modeled as a set of lumped parameter equations. In this case an alternative (approximate) physical interpretation of the process is used to obtain an ODE model directly rather than by converting a PDE model to ODE form by means of a lumping method such as finite differences. As an example, consider a single concentric-tube heat exchanger with energy exchange between two liquid streams flowing in opposite directions, as shown in Figure. We might model this process as if it were three small, perfectly stirred tanks with heat exchange. If the mass flow rates w1 and w2 and the inlet temperatures T1 and T2 are known functions of time, derive transfer function expressions for the exit temperatures T1 and T2 in terms of the inlet temperature T1. Assume that all liquid properties (p1. p2, Cp1, Cp2) are constant, that the area for heat exchange in each stage is A, that the overall heat transfer coefficient U is the same in each stage, and (hat the wall between (he two liquids has negligible thermalcapacitance.

T2, w2 T, U T7 T5 T3 T1 w2 T2 TA T5 Tg