Analytical solution for heat transfer problem in a cross-flow plate heat exchanger

Cross-flow plate heat exchangers are used for plenty of applications in industrial and domestic sectors, and the analysis of heat transfer is a key for the evaluation of their performance. There are challenges for an analytical study because heat transfer in each channel is governed by a partial differential equation coupled with temperature fields in the adjacent channels representing a three-dimensional problem. The problem is even more complicated in the turbulent regime as the effective thermal diffusivity varies within the channel cross-section. In the present study, a separate set of governing equations and boundary conditions are considered for each part of the heat exchanger. Appropriate profiles for the flow velocity and thermal diffusivity are substituted into the governing equations of the channels. The resulting partial differential equations in the channels are solved using the separation of variables method. There remains an unknown boundary condition linked to the temperature field on the plate surface which is considered to be in the form of a two-variable series function whose coefficients are calculated by applying energy balance between the two sides of the plate. The obtained solution provides explicit expressions for the temperature fields in the plate and channels. A scaling analysis is conducted showing that the model is valid when Peclet numbers in both hot and cold channels are not small. The results are compared with empirical data and a numerical model, and the accuracy of the derived heat transfer coefficients is investigated. In harmony with the scaling analysis, a close agreement is observed in both laminar and turbulent flows for moderate and high Prandtl numbers and when Peclet number is greater than 1000 in the case of turbulent flow. (C) 2020 Elsevier Ltd. All rights reserved.