Solidification of a low peclet number fluid flow in a round pipe with the boundary condition of the third kind

This paper presents a theoretical investigation on the solidification of a low Peclet number fluid flow in the thermal entrance region of a round pipe. The velocity is assumed laminar and fully developed throughout the pipe and the fluid temperature is taken to be uniform at X = ?∞. The pipe wall is adiabatic at X ≤ 0 and is cooled convectively at X ≥ 0. The exact solution in the liquid and solid phase are obtained by using the method of separation of variables and constructing two sets of orthonormal functions from the nonorthogonal eigenfunctious at X = ± 0. The solutions including temperature distributions, liquid-solid interfaces and Nusselt numbers for the parameters, Bi = 20, 4, 0.4, 0.04, Pe = 1, 3, 5, 10, 20, 30, ∞ and the superheat ratios, ξ = (1 ? θ_f)/θ_f, θ_f= 0.1, 0.2, …, 0.9 are presented in this paper. The length of ice-free zone x_f corresponding to Pe = ∞ are in excellent agreement with the existing solution.