## Lesson 13. PLANCK’S EQUATION AND RELATED PROBLEMS

Module 2. Food freezing

Lesson 13

PLANCK’S EQUATION AND RELATED PROBLEMS

13.1 Introduction

Freezing times are basic design criteria for freezing systems and represent the residence time for the food product within the freezing system required to achieve the desired level of freezing. The most widely accepted definition of freezing time is the time required to reduce the product temperature from some initial magnitude to an established final temperature at the slowest cooling location. An alternative definition changes the endpoint to the mass average enthalpy equivalent to the desired final temperature for the product. Freezing-time calculations are completed as a first step in the design of a food freezing system. The freezing time establishes the residence time for the product in the system. The final product temperature is established as the magnitude needed to maintain optimum product quality during storage. For a continuous freezing system, the resident time is dependent on the rate of product moves through the system and on the length of the system. More specific characteristics of the design will depend on the type of freezing system being considered.

13.2 Freezing Time Equation (Plank’s Equation)

The most straight forward expression available for computing freezing time was derived by plank. The equation utilized, for computation purpose be derived for various geometries of product. By reference to fig. given below, the case of one-dimensional freezing of a product slab can be illustrated the three basic equations utilized in a the derivation account for the first expression is the basic heat-conduction equation for the frozen product region which has a variable thickness x as follow:

where P and R are constants that depend on product geometry ( Table 13.1).

The limitations to Planck’s equation for estimation of freezing times for foods are numerous and have been discussed by Heldman and Singh (1981) and Ramaswami and Tung (1981). One of the concerns is selection of a latent heat magnitude (L) and an appropriate value for the thermal conductivity (k). In addition, the basic equation does not account for the time required for removal of sensible heat from unfrozen product above the initial freezing temperature or for removal of frozen product sensible heat. There have been numerous attempts to modify Planck’s equation or develop alternative expressions.

The modifications made in the expression by number of scientists.

Limitations of Planck’s Equation

1. Use of equation requires assumption of some latent heat value and doesn’t consider the gradual removal of latent heat.
2. The equation utilized only the initial freezing point and neglects the time required to remove sensible heat above the initial freezing point.
3. Constant thermal conductivity is assumed for the frozen portion. In fact thermal conductivity of the frozen region is temperature dependent and hence variable.
4. Density values for frozen foods are difficult to measure.
5. The initial and final temperature is not accounted for in the equation.

Even with these limitations, Planck’s equation becomes most popular method for freezing time prediction.

Assumptions of Planck’s Equation

1. Freezing starts with all water in the food unfrozen but at its freezing point and loss of sensible heat is ignored.
2. Heat transfer takes place sufficiently slowly for steady state conditions to operate.
3. The freezing front maintains a similar shape to that of the food.
4. There is single freezing point.
5. The density of food doesn’t change.
6. The thermal conductivity and specific heat of the food are constant when unfrozen and then change to a different constant value when the food is frozen.