If we assume that the ice is thin enough so that the temperature gradient can be considered as uniform from the upper to the lower surface, we can derive at once a very simple solution. . .. 1
4.1 Introduction
The quotation cited above appeared in the classic text Heat Conduction with Engineering and Geological Applications, which still serves as a basic reference book in this field of research. In particular, this statement was made in connection with justifying when the unsteady-state freezing of water-saturated soil could be assumed to be quasi-steady-state. However, an appropriate rejoinder to the quote above would be: "How thin is thin?" The solution to the quasi-state-state problem indeed is "very simple". However, leaping to the conclusion that "the ice is thin enough" is intuitive. Alternatively, scaling analysis can be used to develop a quantitative criterion for assessing the applicability of the quasi-steady-state approximation. This is considered in Section 4.7 for this freezing problem involving heat transfer with phase change.
In this chapter we consider the application of scaling analysis to heat transfer. The organization of this chapter is the same as that used in Chapter 3. To understand fully the material in this chapter, it is necessary first to read Chapters 1 and 2. However, since some readers might be interested primarily in heat transfer rather than fluid dynamics, the first few examples are developed in the same detail as was done in Chapter 3; that is, it will be possible to understand how to apply scaling analysis to heat transfer without necessarily thoroughly understanding the material in Chapter 3. However, it will clearly be necessary to understand some aspects of fluid dynamics when convective heat transfer is considered. Note that in this chapter we again use the ordering symbols โข(1) and โข(1) introduced in Chapter 2.