A constant-property, one-dimensional plane wall ofwidth

Chapter 5, Problem 5.127

(choose chapter or problem)

A constant-property, one-dimensional plane wall ofwidth 2L, at an initial uniform temperature Ti, isheated convectively (both surfaces) with an ambientfluid at T??T?,1, h?h1. At a later instant in time,t?t1, heating is curtailed, and convective cooling isinitiated. Cooling conditions are characterized byT??T?,2?Ti, h?h2. (a) Write the heat equation as well as the initial andboundary conditions in their dimensionless formfor the heating phase (Phase 1). Express theequations in terms of the dimensionless quanti-ties ?*, x*, Bi1, and Fo, where Bi1is expressed interms of h1.(b) Write the heat equation as well as the initial andboundary conditions in their dimensionless form forthe cooling phase (Phase 2). Express the equationsin terms of the dimensionless quantities ?*, x*, Bi2,Fo1, and FowhereFo1 is the dimensionless timeassociated with t1, and Bi2is expressed in terms ofh2. To be consistent with part (a), express thedimensionless temperature in terms of T??T?,1.(c) Consider a case for which Bi1?10, Bi2?1, andFo1?0.1. Using a finite-difference method with?x*?0.1 and ?Fo?0.001, determine the tran-sient thermal response of the surface (x*?1),midplane (x*?0), and quarter-plane (x*?0.5)of the slab. Plot these three dimensionless temper-atures as a function of dimensionless time over therange 0?Fo?0.5.(d) Determine the minimum dimensionless tempera-ture at the midplane of the wall, and the dimen- sionless time at which this minimum temperatureis achieved.