Showing papers on "Stefan number published in 1970"

Numerical Solution To Two-phase Stefan Problems By The Heat Balance Integral Method

Abstract: Caldwell and Chiu [1, 2] have used a simple front-fixed method, namely, the Heat Balance Integral Method (HBIM), to solve one-phase solidification problems. In this paper the method is extended to two-phase Stefan problems which arise when the effect of sub-cooling is taken into consideration. Numerical results for the two-phase cylindrical problem are obtained for a range of sub-divisions (n), sub-cooling parameters (0), Stefan number (a) and ratio of conductivities (/ r AT ~\ ;>o (i) Z>0 (2) Ti-Tj, r = R(t), t > 0, and 7\ = T,, r = a, Z > 0 T2 = To, r>5(0, t > 0, and T2 = 7>, r = R(t), t>Q and for the moving boundary and the solid-liquid interface, we have (3) < 2 , , _ -i \A2 % = Lpi (4) dr dr dt where /3 — 0, 1 or 2 depending on the geometry of the problem, and represents solidification process in planes, infinite cylinders and spheres. Heat balance integral method Since we have to solve two partial differential equations in the two-phase problem, we have to employ two sets of sub-divisions, one for the solid region and one for the liquid region (in our formulation we use R and S respectively for this purpose). In each region, we sub-divide the temperature range into n equal intervals and use a linear approximation profile at each interval as in the case of one-phase problems (see Figure 1). The above set of partial differential equations can then be reduced to a system of ordinary differential equations. — ~dt Transactions on Engineering Sciences vol 5, © 1994 WIT Press, www.witpress.com, ISSN 1743-3533