Mathematical Analysis of Parallel Convective Exchangers with General Lateral Boundary Conditions Using Generalised Graetz Modes

We propose a mathematical analysis of parallel convective exchangers for any general but longitudinally invariant domains. We analyse general Dirichlet or Neumann prescribed boundary conditions at the outer solid domain. Our study provides general mathematical expressions for the solution of convection/diffusion problems. Explicit form of generalised solutions along longitudinal coordinate are found from convoluting elementary base Graetz mode with the applied sources at the boundary. In the case of adiabatic zero flux counter-current configuration we recover the longitudinally linearly varying solution associated with the zeroth eigenmode which can be considered as the fully developed behaviour for heat-exchangers. We also provide general expression for the infinite asymptotic behaviour of the solutions which depends on simple parameters such as total convective flux, outer domain perimeter and the applied boundary conditions. Practical considerations associated with the numerical precision of truncated mode decomposition is also analysed in various configurations for illustrating the versatility of the formalism. Numerical quantities of interest are investigated, such as fluid/solid internal and external fluxes.