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## Lesson 15. NEUMANN’S PROBLEMS AND TAO SOLUTION

**Module 2. Food freezing**

**Lesson 15**

**NEUMANN’S PROBLEMS AND TAO SOLUTION**

**15.1 Introduction**

The kinetic of freezing has been widely studied mathematically defined as a heat conduction process with phase change. The estimation of freezing times requires analysis of conductive heat flow through frozen and unfrozen layers, in addition to the heat transfer from the sample to the environment. The most common analysis are based on either Plank’s or Neumann’s Models, which are exact solutions valid for unidirectional freezing under the assumption of an isothermal phase change. While Plank’s model assumes quasi-steady state heat transfer, the Neumann’s model is more generally applicable and is based on unsteady state condition through frozen and unfrozen layers.

**15.2 Neumann's Model**

Under conditions of unidirectional freezing, it has been established that the position of the freezing front varies with time as follows:

x = c √ t

Where x is the position of the ice front, t is time and c is a kinetic constant. This equation is valid under the assumption of a planar freezing front. The equations following from Neumann’s model are listed in Table 14.1. They can be solved analytically to result in a relationship similar to above.

X = 2 δ√ a_{1} t

_{1}is the thermal diffusivity of the frozen zone, and δ represents the Neumann's dimensionless number.

**15.3 Tao Solution**

Numerous attempts have been made to improve freezing time capability using analytical equations. These equations or approaches are also given by Tao (1974). In general all these approaches and solutions have been satisfactory for conditions closely related to defined experimental conditions. In addition to analytical methods, numerical procedure have been developed to predict freezing time.