An alternative to heat transfer coefficient: A relevant model of heat transfer between a developed fluid flow and a non-isothermal wall in the transient regime

Abstract

This paper deals with the relevant model that can be proposed for modelling interface heat transfer between a fluid and a wall for thermal boundary conditions varying in space and time. Usually, for a constant and uniform heat transfer (unidirectional steady-state regime), the problem can be solved through the introduction of the notion of a h heat transfer coefficient. This quantity, which is uniform in space and constant in time, links heat flux to a temperature difference (between the wall temperature Tw and an equivalent fluid temperature Tf, where h and Tf both depend on the system geometry) in a linear way.The problem we consider in this work concerns the heat transfer between a dynamically developed steady-state fluid flow and a wall submitted to transient and non-uniform thermal excitations, for instance a steady-state flow over a flat plate submitted to a pulsed and space-reduced heat flux, or a steady-state flow in a duct stimulated by a periodic flux on its outer surface. More generally, we assume that this kind of thermal problem can be described by:-one or several linear partial differential equations with their associated linear boundary and interface conditions;-the coefficients of the homogeneous part of these equations do not depend neither on time nor on the coordinates in the direction parallel to the fluid/solid interface (they may depend on the coordinate in the normal direction);-volume and surface sources (non-homogeneous part of the previous equations) that can depend on space and/or time.We will show that the relevant representation for describing the interfacial heat transfer does not consist in defining a non-uniform and variable heat transfer coefficient h(x,t), as done usually: the corresponding relationship is not really intrinsic because it depends on the thermal boundary conditions. An alternative approach is proposed here. It relies on the introduction of a generalized impedance Z(ω,p), which is a double integral transform of a transfer function z(x,t) in the original space (x)/time (t) domain. This impedance function links heat flux and temperature difference through a convolution product (noted “⊗” here) rather than through a scalar product:Tw(x,t)−Tf(x,t)=z(x,t)⊗φ(x,t)After a presentation of the generic problem, simple cases, with analytical solutions, will be presented for validation, such as a plug flow, in steady-state and transient regimes.To conclude and show the interest of our approach, a comparison between a global approach and a numerical simulation in a more complex and less academic case will be presented.