### Mathematical Modelling of Heating of Food Essay

MATHEMATICAL MODELLING OF MICROWAVE HEATING OF FOOD By M . SHIVA KUMAR ( 04AG1008 ) Under the guidance of PROF. SURESH PRASAD ABSTRACT • The use of microwaves in the food industry is attributed to the lower time needed to increase the temperature of foodstuffs compared to the traditional heating methods. •However, the heating is not uniform and the products show hot and cold spots. • In order to analyze the behaviour of foods heated by microwave oven, a mathematical method was developed solving the unsteady state heat transfer differential equations.. It takes into account variable thermal and electromagnetic properties. •The numerical solution was developed using an implicit finite difference method in one dimensional system (slab). • It allows predicting temperature profiles. • The model will later be validated with experiments and data on apple fruit. ADVANTAGES OF MICROWAVE HEATING • Does not involve a conduction or convection medium • Food material is heated directly due to agitation of the polar molecules contained – namely water • Reduced drying time • Improved final quality of the dried products • Better rehydration DRAWBACKS OF MICROWAVE HEATING Inherent non-uniformity of the electromagnetic field within a Microwave cavity. • Excessive temperatures along the edges and corners of products may lead to overheating and irreversible drying-out resulting in possible scorching and development of off-flavors. • Final product temperature in MW drying is difficult to control, compared to that in hot-air drying in which product temperature never rises beyond air temperature. • Limited amount of water is available during the final stages of drying processes, hence the material temperature can easily rise to a level that causes scorching

TEMPERATURE PROFILE PREDICTION A mathematical model is proposed to predict temperatures during microwave food heating, taking into account thermal and dielectric properties of the food material in question. The following assumptions are made • Uniform initial temperature within product to be heated • Temperature dependent thermal and dielectric properties • Volume changes during heating are considered negligible • Convective boundary conditions • The incident electric field is considered to be normal to the material surface • Mass transfer is of minor relevance

ENERGY BALANCE •This equation gives us microscopic energy balance per unit volume of the food sample •The first term on the left side denotes net heat absorbed per unit time per unit volume •The first term on the right hand represents diffusive energy •The Q term represents Microwave heat generation INITIAL AND BOUNDARY CONDITIONS • The first condition assumes uniform product temperature initially • The second condition assumes that no heat transfer takes place at central plane of the slab ( L= Half thickness of slab ; x = 0 signifies central section of the slab ) The third condition comes from assumption of convective boundary conditions • The conversion of electromagnetic energy into the heat energy is governed by the equation. Q ? 2?? 0? ” fE 2 A simple method to determine E is given ahead. It involves the use of equation and the following assumptions: • During initial or short periods of intense heating, thermal diffusion and surface heat losses can be minimal and hence neglected. • In such situations, the heat conduction in food is very small compared to the rate of volumetric heating. Power absorption in the food may be uniform or might vary spatially • Now by dropping the convection and diffusion terms in equation we get where Q is a function of location r. For a given location r, if the absorbed microwave power density does not vary with time, the rate of temperature rise at the location is constant, giving rise to linear temperature rise with time. • Combining the two equations we get the value of E as a function of temperature gradient. This temperature gradient can be obtained from data of water heating.

We can do so for apple since it has high moisture content (85%), under the assumption that it containing 85% water will show properties similar to water. • Thus by putting gradient value we can calculate E electric field intensity, which we can put back in equation to get Q, internal volumetric heat generation of Microwave energy • Note that the dielectric factors vary with temperature and must be replaced with appropriate value for corresponding temperature PREDICTIVE EQUATIONS FOR DIELECTRIC FACTORS OF APPLE AT 2. 45 GHz AS A FUNCTION OF TEMPERATURE • Dielectric constant =64. 63 – 0. 0524T – 0. 0005831T2 Dielectric loss factor = 17. 23 – 0. 2407T + 0. 001348T2 • Temperature here is given in 0C. VARIATION OF DIELECTRIC FACTORS WITH FREQUENCY Dielectric constant (? ?) and loss factor (? ?) of ‘ Golden Delicious’ (GD) apple ( ____ ), ‘Red Delicious’ (RD) apple ( ….. ) Cherry ( _ _ _ ) at 20°C as a function of frequency. NUMERICAL MODEL FOR FOOD MICROWAVE HEATING • The aim of this work is to make a model to predict the temperature profile of a product with the initial parameters and values. • A few initial values will be used to simulate conditions and generate or predict values further in the profile of the product. This will be achieved by numerical solution of the governing partial differential equation of heat transfer by using finite difference scheme (Crank Nicholson) to convert differential equation to a system of linear equations by replacing the derivatives in the equation as well as the boundary conditions by their finite difference approximations. • This resultant system of linear equations is solved by iterative schemes. Crank Nicholson Finite Difference Scheme for Food Microwave Heating • The Crank-Nicholson method provides an alternative implicit scheme that is second-order accurate in both space and time.

To do this, develop difference approximations at the midpoint of the time increment. The following equations are used to replace the derivatives by finite difference approximations. (1) • (2) (3) The subscripts and terms of the previously shown equations signify thus • i denotes node position; • n, the time interval, • ? x space increment (radial or axial) such that x = i ? x • ? t, the time step, such that t = n ? t, with • i = 0 (center), • b (boundary); • n stands for time t, while (n + 1) corresponds to time (t + ? t). SOLUTION OF LINEARIZED EQUATIONS The above set of equations is solved by iteration schemes • One such iteration scheme suitable for solution of Crank Nicholson equations is Gauss Seidel iteration. • The complete solution will be codified in a program. EXPERIMENTAL VERIFICATION The iterative solutions will be compared with experimental data for verification of model and determining its feasibility EXPERIMENTAL PARAMETERS •The product which will be used in the experiment will be Apple of Red Delicious variety. • It will be bought / procured from the local market. •The geometry which will be used for the experiment will be an infinite slab. Testing will be done for the heat transfer in one direction for an infinite slab. • The experimental sample will have a thickness of 1cm. The thickness of slab is a very important factor because the sample thickness has to be less than penetration depth at the associated frequency which is about 2 cm. at 2. 45 GHz. • As it is for infinite slab geometry, length of sample will be kept twice of thickness. • No pretreatment is required. • Fiber optic sensors embedded in sample will be used to measure temperature at specific points and reading will be obtained on connected computer. SUMMARY The model has been formulated taking all factors into consideration • First some initial values will be put into the model • Then further values of temperature in sample will be predicted using the model • These will be verified experimentally • The accuracy of the model will be known from comparison of predicted and experimental values Bibliography • Y. Soysal, S. Oztekin and O. Eren. Microwave Drying of Parsley: Modelling, Kinetics, and Energy Aspects Biosystem Engineering Journal . Volume 93, Issue 4, April 2006, Pages 403-413 • M. E. C. Oliveira and A. S. Franca . Microwave heating of foodstuffs. Journal of Food Engineering .

Volume 53, Issue 4, August 2002, Pages 347-359 • Ashim K. Datta, Ramaswamy C. Anantheswaran (2002), Handbook of Microwave Technology for Food Applications • Metaxas, A. C. and Meredtith, R. J. 1983. Industrial Microwave Heating. Peter Peregrinus Ltd. , Herts England • • • • L. A. Campanone and N. E. Zaritzky. Mathematical analysis of microwave heating process . Journal of Food Engineering . Volume 69, Issue 3, August 2005, Pages 359-368 James M. Hill and Timothy R. Marchant . Modelling microwave heating. Applied Mathematical Modelling Journal . Volume 20, Issue 1, January 1996, Pages 3-15 Y. E. Lin,R. C. Anantheswaranat & V.

M. Purib(1995). Finite Element Analysis of Microwave Heating of Solid Food. Joumal of Food Engineering. Volume 25 . Pages 85-112 Henk H. J. Van Remmen, Carina T. Ponne, Herry H. Nijhuis, Paul V. Bartels, and Piet J. A. M. Kerkhof (1996) . Microwave Heating Distributions in Slabs, Spheres and Cylinders with Relation to Food Processing. Journal of Food Science. Volume 61, No. 6, 1996. Pages 1105-111 O. Siphaioglu and S. A. Barringer (2003) (2003). Dielectric Properties of Vegetables and Fruits as a Function of Temperature, Ash, and Moisture Content. Journal of Food Science. Volume 68. Pages – 234 – 239. • • •

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