Analytical mass transfer solution of longitudinal laminar flow of Happel's free surface model

The mass transfer problem of longitudinal laminar flow of the Happel's free surface model (HFSM) is studied under constant wall flux and constant wall concentration boundary conditions. For different shell void fractions, the analytical solution of Sherwood's number in fully developed region, the strict Graetz analytical solution of Sherwood's number in the developing region and approximate Leveque solution are obtained. Furthermore, the expression of local Sherwood's number for solving the whole range of shell void fraction is obtained by combining Leveque solution with analytical solution in fully developed region. The obtained equation is as accurate as Graetz analytical solution. In the HFSM, the mass transfer coefficient decreases with the increase of shell void fraction. The shell void fraction also affects the entrance region length of mass transfer. If Reynolds's number, Schmidt's number and radius of fiber are constant, the entrance region length will increase with the increase of shell void fraction.