Thermal dispersion driven by the spontaneous imbibition process

In this work, we have theoretically analyzed the thermal dispersion process under the influence of the spontaneous imbibition of a liquid trapped in a capillary element, considering the presence of a uniform temperature gradient. The capillary element is represented by a porous medium which is initially found at temperature T₀ and pressure P₀. Suddenly, the lower part of the porous medium touches a liquid reservoir at temperature T_l and pressure P₀. This contact between both phases, in turn causes spontaneously the imbibition process. Using a one-dimensional formulation of the average conservation laws, we derive the corresponding nondimensional momentum and energy equations. The numerical solutions permit us to evaluate the position and velocity of the imbibition front as well as the temperature profiles and Nusselt numbers. The above results are shown by taking into account the influence of three dimensionless parameters: the ratio of the characteristic thermal time to the characteristic imbibition time, β, the ratio of the hydrostatic head of the imbibed liquid to the characteristic pressure difference for the imbibition front, α, and the ratio of the dispersive thermal diffusivity to the effective thermal diffusivity of the medium, Ω. The predictions show that temperature profiles and the heat transfer process are strongly dependent on thermal dispersion effects, indicating a clear deviation in comparison with the case of Ω = 0 that represents the absence of the thermal dispersion.