A Reduced-Boundary-Function Method for Convective Heat Transfer With Axial Heat Conduction and Viscous Dissipation

We introduce a new method of solution for the convective heat transfer under forced laminar flow that is confined by two parallel plates with a distance of 2a or by a circular tube with a radius of a. The advection–conduction equation is first mapped onto the boundary. The original problem of solving the unknown field $T(x,r,t)$ is reduced to seek the solutions of T at the boundary (r = a or r = 0, r is the distance from the centerline shown in Fig. 1), i.e., the boundary functions $Ta(x,t)≡T(x,r=a,t)$ and/or $T0(x,t)≡T(x,r=0,t)$⁠. In this manner, the original problem is significantly simplified by reducing the problem dimensionality from 3 to 2. The unknown field $T(x,r,t)$ can be eventually solved in terms of these boundary functions. The method is applied to the convective heat transfer with uniform wall temperature boundary condition and with heat exchange between flowing fluids and its surroundings that is relevant to the geothermal applications. Analytical solutions are presented and validated for the steady-state problem using the proposed method.