# Mathematical Modelling of Heating of Food

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• Topic: Maxwell's equations, Heat, Microwave oven
• Pages : 7 (1476 words )
• Published : February 1, 2010

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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.

• 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...