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where R = can radius, L = one-half can height, and fh is the heating rate factor in minutes. This relationship is also useful to determine the heating rate factor, fh, for the same product in a different sized container, since the thermal diffusivity is a combination of physical properties that characterize the product and its ingredient formulation and remains unaffected by different container sizes.

### COMPUTER SIMULATION Numerical Models

Another purpose for obtaining the thermal diffusivity of products from a heat penetration curve is to make use of numerical computer models capable of simulating the heat transfer in canned foods. One of the primary advantages of these models is that once the thermal diffusivity has been determined, the model can be used to predict the product temperature history at any specified location within the can for any set of processing conditions and container size. With the use of such models, it is unnecessary to carry out repeated heat penetration tests in the laboratory or pilot plant in order to determine the heat penetration curve for a different retort temperature or can size. A second advantage of even greater importance is that the retort temperature need not be held constant, but can vary in any prescribed manner throughout the process, and the model will predict the correct product temperature history at the can center. Use of these models has been invaluable for simulating the process conditions experienced in continuous sterilizer systems, in which cans pass from one chamber to another experiencing a changing retort temperature at the can wall as they pass through the system. Another important application of these models is in the rapid evaluation of an unscheduled process deviation, such as when an unexpected drop in retort temperature occurs during the course of the process. The model can quickly predict the product center temperature history in response to such a deviation and calculate the delivered F-value for comparison with the sterilizing value specified for the product (6-8).

The first published numerical computer model for simulating the thermal processing of canned foods made use of a numerical solution by finite differences of the two-dimensional partial differential equation (equation 8) that describes conduction heat transfer in a finite cylinder (9). Temperature in the food product is a function of the retort temperature (TR), initial product temperature (Tz), location within the container (x), thermal diffusivity of the product (a), and time (t) in the case of a conduction-heating food. In practice, a is obtained from the slope of the heat penetration curve (fh) and is readily known. Because heat is applied only at the can surface, temperatures will rise first only in regions near the can walls, and temperatures near the can center will begin to respond only after a considerable time. Mathematically, the temperature (T) is a distributed parameter in that at any point in time (t) during heating, the temperature takes on a different value with location in the can (r,y), in any one location, the temperature changes with time as heat gradually penetrates the product in accordance with the thermal diffusivity (a).

Equation 8 can be written in the form of finite differences for numerical solution by digital computer, as shown in equation 9. The finite differences are discrete increments of time and space defined as small fractions of process time and container height and radius (At, Ah, and Ar, respectively). As a framework for computer iterations, the cylindrical container is imagined to be subdivided into volume elements that appear as layers of concentric rings having rectangular cross-sections, as illustrated in Figure 16 for the upper half of the container. Temperatures nodes are assigned at the corners of each volume element on a vertical plane as shown in Figure 17, where i and j are used to denote the sequence of radial and vertical volume elements, respectively. By assigning appropriate boundary and initial conditions to all the temperature nodes (interior nodes set at initial product temperature and surface nodes

Figure 16. Subdivision of a cylindrical container for application of finite differences for numerical solution of heat conduction equation in a finite cylinder. Source: Ref. 3, reprinted with permission, copyright 1992 by Marcel Dekker, Inc., New York.

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