Optimal Control of Microwave Heating Using Mathematical Models of Medium Complexity

The dynamic optimization of microwave heating of foods is considered. Two classes of problems (maximum temperature uniformity and maximum quality uniformity) are stated here. Results obtained using a deterministic method, an stochastic method and new hybrid method are reported. It is shown that, for the models and food loads considered, it is theoretically possible to achieve a quite uniform final quality distribution using the combined optimal controls of microwave heating and oven (convection) heating. INTRODUCTION Most processes in the food industry are operated in batch or semi-continuous mode. In order to find the best operating policies, robust optimal control (dynamic optimization) methods are needed. For example, thermal sterilization of prepackaged foods, one of the most important preservation techniques, is often carried out in batch retorts. The determination of the optimal retort temperature profile is a complex dynamic optimization problem which has received great attention in the literature (Banga et al, 1991; Silva et al, 1993). In contrast, there is a lack of studies regarding the dynamic optimization of microwave heating of foods. This has been very likely due to the high complexity of this process, and therefore of its mathematical models, which has hindered the application of dynamic optimization methods. In this study, our objective is to obtain the optimal control policies (microwave power and oven temperature) for microwave combination ovens considering process models of medium complexity. The ultimate aim is to improve the final quality of the food by optimally operating the oven, avoiding some of the well-known problems of microwave heating, like cold spots and edges overheating. STATEMENT OF OPTIMAL CONTROL PROBLEMS Here, we present the general mathematical statements for two classes of optimal control problems concerning microwave heating in combination ovens: OCP-I Find the optimal controls (microwave field power P{t} and oven temperature Toven{t}) to achieve maximum temperature uniformity at a ACoFoP IV (Automatic Control of Food & Biological Processes), Göteborg, Sweden, 21-23 September 1998. Correspondence to: Dr. Julio R. Banga, IIM-CSIC. Eduardo Cabello 6, 36208 Vigo, SPAIN. E-mail:[email protected] 2 given final time. OCP-II Find the optimal controls, P{t} and Toven{t}, to achieve maximum uniformity of the final product quality (as measured by the “cook” Cvalue), while ensuring a minimum C-value. The corresponding mathematical statements (e.g. for cylindrical coordinates) are: OCP-I Maximum temperature uniformity Find {} t P and {} t Toven over [ ] f t , t t 0 ∈ to minimize [ ] [ ] } t ; z , r { T min } t ; z , r { T max J f food f food − = (1) subject to ƒ An inequality constraint on the minimum desired temperature to be achieved: [ ] SET min f food T } t ; z , r { T min ≥ (2) ƒ The system model (set of partial differential, ordinary differential and algebraic equality constraints, i.e. the equation for heat conduction with internal generation due to microwaves, its boundary and initial conditions, degradation kinetics, etc.) OCP-II Maximum C-value uniformity (free terminal time) Find {} t P , {} t Toven and f t over [ ] f t , t t 0 ∈ to minimize [ ] } t ; z , r { C max J f Tref = (3) subject to ƒ An inequality constraint on the minimum desired C-value to be achieved: [ ] SET min , Zref f Zref C } t ; z , r { C min ≥ (4) ƒ An inequality constraint on the maximum allowed temperature at the end of the process: [ ] SET max f food T } t ; z , r { T max ≤ (5) ƒ The system model (set of partial, ordinary differential and algebraic equality constraints, as mentioned above) In this latter statement, the C-value (“cook” value) is used as the performance index. The C-value has been used successfully to measure the product quality changes during thermal processing (Ohlsson, 1980; Ohlsson, 1986; Ohlsson, 1988). It is defined as the equivalent number of minutes the product would have to spend at a reference temperature (usually Tref=100°C) to achieve a given effect on the final product quality. Mathematically: dt C t Z / ) T T ( Z ref ref ref ∫ − =