مقایسه کارآیی بین دو روش مختلف مدل سازی عددی برای پیش بینی دمای رب گوجه فرنگی در طی فرآیند پاستوریزاسیون.

Introduction: Tomato (Solanum lycopersicum) is the second most important horticultural crops next to potato, with an estimated world production of over 120 million tons per year. Iran has always been among the top tomato producing countries in the world due to the diverse topography and climactic conditions prevailing in different parts of Iran. Tomato paste is one of the most important tomato products originates from tomato juice in which water is removed by evaporation and thermal processing plays a key role in the successful production process (Singh and Headman, 2014). Since it can guarantee safety and extend shelf life of the product. Although several emerging technologies such as ohmic heating, microwave heating, and non-thermal processing techniques such as pulsed electric fields and high pressure processing, have been developed for food preservation, Conventional thermal processing (pasteurization and sterilization) is still widely used in the Iranian food industry. However, the challenges of accurately determining both optimal operating conditions and developing a control system for industrial pasteurization process to prevent either under- or overprocessing are significant (Chen and Ramaswamy, 2007). Mathematical modeling and simulation is one of the most commonly used methods to gain a better understanding the process. Modeling can provide insights into complex processes, shorten the design cycle and optimize the process as a function of various variables at lower cost and time. Different mathematical methods for solving heat conduction problems have been proposed, but numerical methods are more useful, especially when such problems cannot be handled by the exact analysis because of nonlinearities, complex geometries, and complicated boundary conditions (Incropera and De Witt, 1990). Among the numerical methods developed so far, finite difference and finite element techniques have been widely used to analyze heat transfer phenomena in cylindrical cans of food. Although many researchers have tried to develop mathematical models based on numerical methods for predicting temperatures in tomato products during thermal processing, such as (Bichier et al., 1995), (Nicolai et al., 1998), (Tattiyakul et al., 2002) and (Plazl et al., 2006), there is not much research available on the subject of thermal processing of tomato paste in Iran.
In this study, the efficiency of two different modeling approaches (finite difference vs. finite element) for predicting temperature of tomato paste during pasteurization process were compared. The developed model allows to reduce energy consumption during operations while high-quality products are also produced in a short time.
Material and method: Experiments were run with batches of 400 g of tomato paste (pH = 4.1 and 28°Brix) in cylindrical cans (211×400) and hot water was used as the heating medium. The chemical analysis of tomato paste sample (Brix, pH salt, moisture, fat and protein) was performed in the first step and the thermal properties of the tomato paste product including thermal conductivity, specific heat and density were determined based on the sample chemical composition and structures model.
Temperature changes at various positions in the container were checked with a data logger (Testo, Germany) coupled with computer and thermocouples type-K (at 2 min intervals). The cylindrical can was immersed in a vertical position in the water bath and the temperature recording was started. After finishing the heating time, the can was cooled in another water bath (20ºC). The data were used to validate the developed model.
In the next step, 2D heat transfer model was developed in a cylindrical can by using the numerical solution of the Fourier second law with two different methods. 1) Finite difference (explicit scheme) and finite element methods. Computer simulation is done using MATLAB R2009a software (Math works, Inc., Natick, MA, USA) and COMSOL Multiphysic, Ver. 4.0. Finally, in order to evaluate the best model, two criteria, coefficient of determination (R2) and root mean squared error (RMSE) were used.
Result and discussion: The results showed that, by placing the sample in the bath, the surface temperature rises rapidly, while the temperature in the center is much slower. In addition, as can be expected, increasing hot water temperature enhanced the heating rate considerably due to the larger temperature gradient between the center and surface of the can at the higher temperatures. The models have been verified by comparing results with two analytical solutions and validated against experimental data. The statistical analysis results showed that the finite element model developed by COMSOL software can predicted temperature more accurate than finite difference model and may be more useful. The reason for this difference between the results of two numerical methods can be attributed to the consideration of a layer of air-steam mixture on the top of the can (head space) in the finite element method which increases the accuracy of the model in temperature prediction. After validation, the developed model was used to determine the cold spot location of the tomato paste can. Results also showed that the cold point was a stationary point and located at the radial center at a height of 60% of the can height from the bottom (Tattiyakul et al., 2002). Two simulations were conducted at two different head space volume (6 and 12% of total can height) to determine the importance of head space volume on cold point location. Results showed that there was no significant difference in the location and temperature of the cold spot in two simulations (Khakbaz Heshmati et al., 2014).
Conclusion: In this study, the pasteurization process of tomato paste (Brix=28) is investigated by two different numerical methods (finite difference& finite element). The results were compared with experimental data and it was found that the predicted temperature by finite element model is more accurate than finite difference method. Although it is generally believed that the coldest point for a solid product will be at the geometric center of the cylinder, our results indicated that the slowest cooling point was located at a height of 60% of the can height from the bottom. The developed model can predict temperature in tomato paste with different degree of concentration (brix) or different thermal processing conditions and with minor modifications, the model may be used to design and control the process of industrial pasteurization for various solid products. In addition, the results of this study is expected to be a significant contribution for further optimization studies.