History Terms in the Energy and Mass Transfer Equations for Particles, Bubbles, and Drops

Abstract The Lagrangian transient equation of motion for spheres includes a history term, which in the most general case, appears as a time integral and is sometimes referred to as the Basset term. This term is the result of vorticity diffusion around particles, bubbles, and drops as they move within viscous fluids. Detailed solutions of the transient energy equation around spheres prove that there are also history terms associated with the heat and mass transfer processes. These terms are analogous but not similar to the corresponding terms in the equations of motion. The effect of the history terms in engineering computations is to transform the energy and mass transfer equations for particles from simply differential equations to integrodifferential equations, which are more complex to solve analytically and numerically. This paper examines the analytical forms of the history terms at zero and finite Reynolds and Peclet numbers for spheres and spheroids and draws inferences about the effect of the shape of the particle on the form of the history terms and the kernels of the integral. Numerical results complement the analysis in cases when analytical solutions (both general and asymptotic) are not possible to obtain. The paper also addresses the mass transfer from spheres in porous media and the significance of all the history terms in the heat and mass transfer processes.