Analytical Solutions

Abstract

Analytical expressions for the freezing time can be derived for some simple, idealised situations. Although usually not directly applicable to practical cases, they are often used to verify numerical models such as finite difference or finite element to ensure that they are correctly implemented. In this chapter, the heat conduction equation is presented, followed by some analytical solutions. Plank’s equation for the freezing of a slab is derived. Although its underlying assumption of zero sensible heat is unrealistic, it gives valuable insight into the freezing process and will be used to develop more accurate formulas. The importance of the Biot number is stressed: it measures the ratio of internal to external resistance to heat transfer, and can guide the technologist in making an industrial freezing process more efficient. Analytical expressions for the freezing shape factor (the ratio of the freezing time of a slab to that of another shape with the same minimum dimension) are plotted for the rectangular rod, the finite cylinder and the rectangular brick.

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-1-4939-0557-7_8

CAUTION

• Older publications in this field often define the Biot number as 2hR/k instead of hR/k. Make sure that the correct interpretation is applied.