Volume 17 | Number 1 | Spring 2005 | Automation Issue

Mathematical Modeling
By Jianrong Zhang, Ph.D.

Figure 1. Core temperature profile for chicken meat cooking in an oven.

Heat and mass transfer processes are among the most important physical phenomena that occur during food processing. For any further processed operation, the temperature and the moisture content inside the product is important to product quality and safety and is impacted by the length of time the product is processed and its position in the food processing system. Quality attributes, such as microbial load, nutritional value, texture, and organoleptic quality, are also determined by these two variables.

In order to better understand what happens to the product internally during processing, researchers have spent a significant amount of time and effort developing mathematical models that simulate typical thermal processes, such as canning, baking, pasteurization, drying, etc. Mathematical models can predict the time/temperature and moisture content change rate of the product during processing at different conditions; the models also help to eliminate the wasted effort and time spent repeating experimental tests in pilot plants and direct measurements in actual processing operations, thus improving new process development efficiency, saving energy costs, and reducing labor/operation costs.

One of the key tools in mathematical modeling is numerical computation techniques. They allow the simulation environment to quantify the interrelationships of change taking place in the product. Three such methods commonly used are: finite difference method, finite element method, and finite volume method.

The finite difference method is most often used for regular- shaped foods in different thermal processes. It breaks the product into a fixed number of segments and analyzes the changes to each segment as a result of their interrelationship to one another, and the process then recombines them back into a whole. It has been used for the prediction of temperature in the center of canned foods in cylindrical containers.

The finite element method is more dynamic and is quite popular now because of its accuracy in modeling irregular- shaped foods. It breaks the product into very small elements or regions and analyzes the changes to each region as a result of interactions, and the process then integrates these findings over the entire product. It has been used on such products as chicken breasts, broccoli stalks, tuna fish, etc.

The finite volume method is used primarily in commercial CFD (computational fluid dynamics) analysis and combines the flexibility of the finite element method with the execution speed of the finite difference method. It breaks the fluid flow into fixed volumes and analyzes the influence of the process on the boundaries of the volume element.

Mathematical modeling can simulate both batch and continuous processes. It is possible to apply mathematical models to predict continuous processes for several stages, combine the simulation with a control system to monitor the processes at the same time, and change the process settings to ensure proper process conditions and good quality products. For example, the temperature of product entering an oven impacts the cooking time needed to achieve a preset final temperature. If the entering temperature varies because of handling conditions, it becomes difficult to control over- and undercooked conditions. Using captured input temperature data, modeling can allow oven controls to adapt to changing input temperatures, thereby reducing the potential waste of energy and poor product quality due to overcooked products, while also reducing the potential for food safety problems due to undercooked products. Figure 1 shows temperature profiles from mathematical modeling and actual cooking measurements recorded in the lab on raw chicken meat. The average error between the real temperature and simulation temperature is as low as ±1.2 °C.

At Georgia Tech, researchers are using finite element analysis to try and predict the approximate core temperature of poultry products. The goal of the project is to develop a method that allows in-process visual and thermal scanning of product surfaces to be used to estimate core temperature levels to an accuracy of 1 to 3°F. If successful, the system could be used to help control the cooking operation. The research team initially focused its studies on uncoated product, but recently has begun focusing on coated product as well. With coated product, the computations become more complicated because of the internal boundary changes of the coating material. But early efforts are showing good results.

In summary, mathematical modeling is a powerful tool that can be used to improve both the quality and yield of food products, while also improving processing efficiency in plant operations and ensuring the safety of food products. One of its main strengths is its ability to allow measurable variables to be used to determine unmeasurable conditions that are important to processing performance. In such situations, improved control can be exercised over processing operations.

Jianrong Zhang is a postdoctoral fellow in Georgia Tech’s Food Processing Technology Division. Her areas of expertise are food engineering and mathematical modeling and food safety.