Solidification of Couette Flow in an Annulus with Inner Cylinder Rotation

Abstract

A semi-analytic solution is presented for the solidi cation of laminar Couette ow within a one-dimensional annular region with a rotating inner cylinder and stationary outer cylinder. Viscous dissipation in the liquid is taken into account. The inner cylinder is maintained under adiabatic conditions while the outer cylinder is convectively cooled. Analytic expressions for the dimensionless quasi-steady temperature distribution in the solid and liquid regions, Nusselt number at the solid-liquid interface, dimensionless power and torque per unit length, and dimensionless steady-state freeze front location are derived as a function of liquid-to-solid thermal conductivity ratio, Brinkman number, annulus radius ratio, and Stefan number, which is assumed to be small (<0.1) but nonvanishing. The dimensionless instantaneous solid-liquid interface location is determined using numerical integration. In addition, the Brinkman number is related to the Reynolds number, and expressions for the factor increase in transition Reynolds and Brinkman numbers are derived. It is observed that an in nitely large Biot number, corresponding to the isothermal cooling limit, increases the ow stability in the liquid region as the transition Reynolds increases by a factor of up to 11. However, this stability increase comes at the expense of a factor increase in the power per unit length of up to 2.4. NomenclatureBi = Biot number Br = Brinkman number c = speci c heat, J ¢ kg ¡1 ¢ K ¡1 h = convective heat transfer coef cient, W ¢ m ¡2 ¢ K ¡1 h s f = latent heat of fusion, J ¢ kg ¡1 k = thermal conductivity, W ¢ m ¡1 ¢ K ¡1 Nu = Nusselt number P = power, W P 0 = power per unit length, W ¢ m ¡1 P ¤ = dimensionless power per unit length Pr = Prandtl number Re = Reynolds number R i = inner radius of annulus, m R o = outer radius of annulus, m r = radial coordinate location, m Ste = Stefan number T = temperature, K = = torque, kg ¢ m 2 ¢ s ¡2 = 0 = torque per unit length, kg ¢ m ¢ s ¡2 t = time, s U = dimensionless velocity component in azimuthal direction u = velocity component in azimuthal direction, m ¢ s ¡1 ®= thermal diffusivity, m 2 ¢ s ¡1 = annulus radius ratio°= liquid-to-solid thermal conductivity ratio 1 = dimensionless solid-liquid interface location ± = solid-liquid interface location, ḿ = dimensionless radial coordinate location µ = dimensionless temperature ¹ = dynamic viscosity, kg ¢ m ¡1 ¢ s ¡1 º = kinematic viscosity, m 2 ¢ s ¡1 ¿ = dimensionless time Á = dimensionless parameter of Eq. (26) Ã = solid-to-liquid thermal diffusivity ratio ! = angular speed of inner cylinder, 1 ¢ s ¡1 Subscripts = liquid region m = solid-liquid interface conditions s = solid region ss = steady-state conditions tr = transition from laminar to turbulent ow 1 = far-eld or freestream conditions