10.6.1 Principles of optimization for food thermal processing
The kinetic parameters (D and z) of microorganisms, enzymes and quality factors of foods are different and this fact is exploited to optimize thermal processes for the elimination of the microbial hazards and retention of nutritional and sensory qualities of foods. An optimal thermal process may be defined as the minimum treatment required to achieve commercial sterility because heating cost and product quality losses increase if the process time is prolonged. The procedure of optimization can be summarized in the following four steps:
1 objective functions and decision variables
2 mathematical models
For thermal processing of foods, the objectives for optimization include maximum average quality retention, surface quality retention and minimum process time that meets the required lethality value at the can center or the coldest spot in the can. The factors affecting these optimal objectives are numerous including food thermal properties, can size and shape, retort temperature, kinetic parameters of quality factors (D and z), desired lethality value and so on, but the decision variables that can be optimized for a given packaged food are usually only retort temperature for the constant retort temperature (CRT) processing or retort temperature profile for variable retort temperature (VRT) processing. This means that the optimization of CRT thermal processing belongs to a single variable optimization while that of VRT thermal processing belongs to a multiple variable optimization since the VRT functions usually have more than two parameters. The second step is to develop mathematical models describing the relationships between decision variable(s) and objectives. Constraints are necessary for some optimization problems, which can be a range of decision variables such as retort temperature or (and) additional objectives. For example, to obtain the maximum quality retention, the desired lethality value and (or) the maximum process time can be used as constraints in searching for the optimal retort temperature. The use of searching techniques for optimization is to assure that the process of optimization is efficient and robust. Different searching techniques are available for the optimization of thermal processing, the details of some of which are given in the following section.
One of the earliest mathematical treatments of the optimization process for thiamin destruction versus sterilization, in cylindrical cans of conduction-heating product, was proposed by Teixeira et al. (1969). A finite difference method was used to determine the temperature distribution and the corresponding thia-min destruction, employing first-order degradation kinetics. In the same year, Hayakawa (1969) extended the concept by using a different mathematical technique involving dimensional analysis and the concept of a mass-average sterilizing value. This was subsequently extended (Hayakawa, 1971) to estimate the mass average value for a physical, chemical or biological change resulting from thermal processing. This work led to formulae which could be used to compute values for nutrient retention which were then intended to be used with standard manual procedures.
Barreiro-Mendez et al. (1977) derived models for the loss of nutrients during heating and cooling in cylindrical containers using analytical equations. These equations gave the percentage nutrient retention and experimental results obtained using an analogue system of 6% maize starch and 1.75% carboxymethyl cellulose and these were in good agreement with the predicted results.
Hayakawa (1977) used a computer model to estimate the percentage of thiamin retention in carrot puree, pea puree, pork puree and spinach, and compared the results with experimental determinations. For processes at 115.6°C the results were within ±3%; however, at the higher temperature of 121.1°C for 60min, the differences varied between 10 and 16%. Spinach gave the worst comparative results, the predictions being up to 16% less than the experimental results.
Lenz and Lund (1977) used a method of lethality calculation which made use of a new dimensionless group, the lethality/Fourier number L where:
kr a where a is the thermal diffusivity, x is the fraction of constituent retained (ratio of concentration at any time t to the initial concentration), kr is the rate constant at the reference temperature Tr and a is the container radius (i.e. half thickness). This was derived by combining the first-order kinetic equation and the Arrhenius temperature relationship and substituting the time from the Fourier number at/a2. The latter is obtained from the unsteady state heat transfer equation solution for a finite cylinder and cooling is included by solving the equation for the appropriate boundary conditions at the end of heating.
Thijssen and Kochen (1980) developed a method of process calculation which eliminated the use of tabulated data and interpolation. The model used was based on the following equation [10.18]:
dv where C is the concentration of a specified component at time t, C0 is the concentration of the specified component at time 0, V is the volume of the pack for averaging purposes and k is a temperature-dependent kinetic factor. For a uniform initial product temperature T0, a constant temperature of the heating medium, Th, and a constant temperature of the cooling medium, Tc, the reduction in a heat-labile component is a function of five dimensionless groups: Fourier number, Biot number, residual temperature ratio and two groups related to kinetic factors. The method used the analytical solutions for the heat transfer equation for sphere, cylinder and rectangular bodies, and also other geometrical shapes.
Castillo et al. (1980) extended the method of Barreiro-Mendez et al. (1977) to deal with rectangular retortable pouches of food. The interesting point which emerges from the use of this model is that the predicted and experimental temperatures at the end of heating were in good agreement. However at the end of cooling differences of up to 16% were observed, probably owing to the assumption of a very high heat transfer coefficient at the surface of the pouch. The predicted thiamin retention after the thermal processing was in good agreement with the experimental results.
Fig. 10.6 Diagram of t-T relationship for microbial destruction, F, and cooking, C.
Fig. 10.6 Diagram of t-T relationship for microbial destruction, F, and cooking, C.
10.6.3 Searching techniques
The choice of processing conditions may be determined from a plot of log time versus temperature (Holdsworth, 1997), on which are drawn two straight lines representing constant lethality (F) and cook (C) values, as illustrated in Fig. 10.6. These lines divide the plot into four regions: the line F1OF2 marks the boundary between processes which give adequate sterilization and those which do not, while C1OC2 marks the boundary between adequate and inadequate cooking. Idealized graphs like this are useful for determining the suitability of various combinations of temperature and time. It should be noted that the graphs are based on instantaneous heating followed by instantaneous cooling of the product, and in particular to thin films of product. Under more realistic conditions it is necessary to include the effects of heat transfer and dimensions of the object being processed. When this is done the straight lines in Fig. 10.6 become curved, and the regions have different boundaries.
According to the type of objective function, optimization problems can be divided into two categories: linear optimization and non-linear optimization problems. Linear programming is a useful tool to deal with the linear optimization problems. Although this technique is considered to be of a limited value because of its assumptions of linearity and infinite divisibility, the technique is very flexible. Moreover, linear programming has the ability to deal with large numbers of constraints in an efficient way, so this technique is very useful for analyzing and optimizing large systems (Saguy, 1983). Many searching techniques have been developed for non-linearity problems including single and multiple variable searching techniques. The former includes the grid search, Fibonacci technique, golden section method, quadratic techniques and Powell method for the objective function with only one decision variable, while the latter includes alternating variable search techniques, Pattern search, Powell's search, evolutionary operation and response surface analysis, and gradient methods (Saguy, 1983).
Saguy and Karel (1979) used an elegant multi-iterative mathematical technique to optimize thiamin retention in pea puree in a 303 x 406 can and pork puree in a 401 x 411 can. The method produced a variable temperature heating profile which optimized the nutrient retention. A constant heating temperature regime was shown to be almost as good as the theoretically derived profile.
Hildenbrand (1980) developed a two-part approach to solving the problem of optimal temperature control. In the first part, the unsteady-state equation for heat transfer into a finite cylinder was solved using Green's functions. In the second part, a method to ensure that the container received the calculated temperature profile was determined. While the approach seems interesting, no further development appears to have taken place. Nardkarni and Hatton (1985) examined the previous work and considered that the methods were not sufficiently rigorous to obtain the best optimization results. These workers used the minimum principle of optimal control theory to obtain optimal solutions. Again simple heating and cooling profiles were better than complex heating profiles.
Banga et al. (1991) developed an optimization algorithm, integrated control random search (ICRS), for three objective functions: maximum overall nutrient retention, maximum retention of a quality factor at the surface of the food and the minimum process time. They concluded that the use of a variable temperature profile was advantageous for preserving optimum surface quality.
Artificial intelligence techniques for modeling and optimization With the rapid development of computer technology and software, artificial intelligence technologies such as artificial neural networks (ANNs) and genetic algorithms have been found to have certain advantages over conventional methods in dealing with system modeling and optimization problems especially for those situations involving non-linear and complicated mathematical approaches.
The neural network is a collection of interconnecting computational elements which mimics neurons in biological systems. It has the capability of relating the input and output parameters without any prior knowledge of the relationship between them. Genetic algorithms (GAs) are a combinatorial optimization technique. They search for an optimal value of a complex objective function by simulation of the biological evolutionary process, based on crossover and mutation as in genetics. An optimal value can be searched for in parallel with a multipoint search procedure. In addition, GAs can use ANN models as that guiding function. This makes it possible to develop a comprehensive optimal control technique using both ANN and GAs.
ANN and GA as individual functions have been widely applied for different areas but as a combination procedure have only been reported in recent years. Chen (2001) evaluated the application of both ANN and GA for modeling and optimization of thermal processing including constant and variable retort thermal processing. The results showed that it was reliable to use an ANN model for modeling of thermal processing and use a GA-ANN based optimization method for optimization of thermal processing.
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