Unsteady-State Heating and Cooling of Solid Objects

Abstract

If a hot object is plunged into cold water it cools, but not instantaneously. Two factors govern the cooling rate of the object:

• the film resistance at the surface of the object, characterized by the h value for that situation; and

• The rate of heat flow out of the interior of the object. The governing differential equation for this conduction process is

  $$\frac{\partial T_{s}}{\partial t} = \alpha \left ( \frac{\partial^{2}T_{s}}{\partial x^{2}} + \frac{\partial^{2}T_{s}}{\partial y^{2}} + \frac{\partial^{2}T_{s}}{\partial z^{2}}\right )$$

  ((11.1))

  where

  $$\alpha = \frac{k_{s}}{\rho_{s}C_{s}},\;\;\;\;\textup{thermal\;diffusivity}\;[\textup{m}^{2}/\textup{s}]$$

  and

  $$T_{s} = \textup{temperature\;at\;any\;point\;in\;the\;object}\;\;[\textup{K}]$$