Unsteady heat transfer in ducts with time-varying inlet temperature and participating walls

An analysis is made of unsteady laminar heat transfer in a duct with periodically varying inlet temperature and time- and space-dependent wall temperature. The wall temperature variation is not specified in advance, but rather, is dynamically determined by a balance of the heat-transfer rate and the energy storage. In the analytical formulation, the commonly used quasi-steady assumption is lifted in favor of the local application of the energy equation, the solution of which leads to an eigenvalue problem in which the eigenvalues are complex (real and imaginary parts). The series expansion properties of the corresponding complex eigenfunctions, which are essential to the solution, are verified. Numerical results are obtained for the time and space dependence of the wall and bulk temperatures and of the Nusselt number. These results identify conditions under which the wall temperature variations are negligible and the instantaneous Nusselt number is virtually time-independent. In addition, numerical information on the overall performance of the duct as a heat exchanger is provided by means of the ratio of the outgoing to the incoming energy flux. It is shown that for a wide range of operating conditions, the overall performance can be described by a single curve. For comparison purposes, results for the overall performance are also derived using the quasi-steady model. Quasi-steady results are evaluated for both spatially uniform and spatially varying heat-transfer coefficients. It was found that, for a range of operating conditions, the quasi-steady model is able to give accurate performance predictions, especially where it is used in conjunction with spatially varying heat-transfer coefficients.