Structure of the transient cooling of a slab

Abstract

This paper reviews solutions to the classical problem of a slab of homogeneous material (conductivity λ, density ϱ, specific heat c), initially at temperature t i throughout and at time t = 0 subjected to a step change of temperature at its exposed face. The roles of the dimensionless time variables ζ 0 = λt ϱcy 2 and ζ 2 = h 2t λϱc are discussed ( y is the depth below the exposed surface, h is the surface heat transfer coefficient). At large depths of b ( = hy λ ) , a thermal disturbance is propagated at a rate determined mainly by ζ 0, but for smaller values of b it travels relatively slower. The temperature anywhere in a slab, thickness X, insulated on its rear surface is initially independent of X and at the exposed surface depends on ζ 2 alone. After some interval of ζ 0, explainable in terms of the rate of propagation o the thermal signal, temperature everywhere falls exponentially. Values for temperature at the front and back surfaces are given in terms of ζ 2 and B = hX λ . Values of ζ 0 are given relating to the time at which the surface temperature of a finite thickness slab starts to fall more quickly than that of an infinitely thick slab. Values of ζ 0 are also given relating to the time at which exponential cooling is established. Approximate polynomial forms are given for cooling in its early and later stages. The response time for thick and thin slabs is discussed.