HEAT AND MASS TRANSFER APPLIED TO THERMAL FOOD PROCESSING

During thermal food processing, heat must be transferred between a heat source or sink and the inside zone of the food usually through an interface such as a food surface or container wall. External heat transfer between the source or sink and the interface may occur by any heat transfer mechanism (including conduction, convection, radiation, and phase changes). Internal heat transfer from the interface to the inside zone of foods is usually by conduction for solid foods or by conduction and convection for liquid foods. If a microwave is used, heat can also travel to the inside zone by penetrating radiation. Moisture, water vapor, nutrients, and flavor must first travel to the food surface by any of internal mass transfer mechanisms, such as diffusion. Then they must travel from the food surface to the ambient by external mass transfer processes such as con- vective mass transfer. For a series of the mechanisms of external heat transfer, internal heat transfer, internal mass transfer, and external mass transfer, the step with the greatest effect on the rate will be the slowest one, which is the rate- determining step.79

Heat transfer through solid foods is normally modeled by Fourier’s equation of heat conduction, and mass transfer is generally described by Fick’s law of diffusion.80 _{For thermal processes of fluid foods, the conservation of mass,} momentum, and energy in a fluid should be considered together. The continuity equation and Navier–Stokes equations are used to describe fluid flow.59 _{The actual} conditions imposed by the processing equipment are considered as the boundary conditions of the governing equations. Most heat and mass transfer models can only be solved analytically for simple cases. Numerical methods are useful for estimating the thermal behavior of foods under complex but realistic conditions such as variation in initial temperature, nonlinear and nonisotropic thermal prop- erties, irregular-shaped bodies, and time-dependent boundary conditions. In solv- ing the models, the finite difference and finite element methods are widely used. In recent years, the finite volume method was the main computational scheme used in commercial computational fluid dynamics (CFD) software packages. CFD has been increasingly used to simulate thermal processes of foods for analyzing complex flow behavior.27,58

2.3.1 PASTEURIZATION AND STERILIZATION

Pasteurization and sterilization are widely used in the food industry to inactivate microorganisms present in foods for ensuring food safety and extending the shelf life of foods. In aseptic processing, the products are first thermally treated, then carried to a previously sterilized container and sealed under sterile environment conditions. The thermal processing of packed products is carried out in equipment that uses steam or hot water as the heating fluid. The pasteurization and steril- ization techniques are initially used in liquid foods such as milk and fruit juices. Recently, they have also been applied to particulate food products.81

2.3.1.1 Pasteurization and Sterilization of Liquid Foods

The sterilizing process of canned liquid foods is a typical example of fluid flow with heat transfer. The CFD model can thus be used to predict transient flow patterns and temperature profiles in a can filled with liquid foods. For simulating the sterilizing process of canned liquid foods, the energy equation needs to be solved simultaneously with the continuity and momentum equations in a CFD model.63,64,69 _{Continuous sterilization processes of single-phase mixtures such as} milks and fruit juices have become more and more common. The continuous process is called the high-temperature short-time (HTST) sterilization process, which gives the same level of sterility but a reduced quality loss, compared to batch sterilization process. For optimizing the quality of foods during continuous sterilization, the laminar flow of liquid foods in circular pipes with uniform wall temperature can be described by a CFD model.65

2.3.1.2 Pasteurization and Sterilization of Particle–Liquid Foods

Sterilization of canned solid particle foods with a brine solution in a container is a typical liquid–solid thermal process. Blanching of fresh vegetables and sous vide processing of particle foods are also heating practices in a liquid–solid system. In this system, the low-viscosity brine liquid is heated by convection and the solid particle foods by conduction. A heat conduction model can be used to simply determine the temperature distribution in a canned particle body. Meanwhile, the temperature of brine liquid in the heated cans, which is variable with the temperature outside of the cans, can be simply described by the regular regime differential equation:

(2.26)

where the thermal inertia, Φ, which characterizes the temperature lag of the brine liquid from the temperature of heating medium, is experimentally determined by monitoring the temperature of the brine with linearly increasing, holding, and linearly decreasing the temperature of the medium.9,13

For the liquid–solid thermal process, because the heat transfer coefficient of surface convection is normally very large due to good circulation of brine liquid in the container, the effect of the coefficient on the temperature profiles of foods is normally assumed to be negligible. This means that if the coefficient is big enough, the total heat transfer rate is controlled by conduction through the particle food body. For this reason, the heat transfer coefficient can arbitrarily be set at a very high value in a simulation, for example, 5000 W/m2 _K.6,24

2.3.2 DEHYDRATION AND DRYING

Dehydration, or drying, is a unit operation of food thermal processing most com-

monly used for food preservation. Reduction of water in foods during drying can

dT dt

T T

l = m− l

achieve better microbiological preservation and retard many undesirable reactions. Drying can also decrease packaging, handling, storage, and transport costs due to the decrease of food weight. The drying process is mainly characterized by moisture loss of foods. In most cases, the removal of water from a food is achieved by blowing a dry airflow, which transports water from the surface of the product to the airstream. However, spray drying, freeze drying, microwave drying, and other methods are also used for drying some special products. Drying of food materials is normally a complex process involving simultaneously coupled heat and mass transfer in the materials. It is important to know the mechanisms related to the movements of water inside and outside the food.81

2.3.2.1 Air Drying

Air drying is the most popular drying method in the food industry. For a drying process with a small Biot number, a uniform temperature profile in foods can be assumed in simulation. This uniform temperature can be determined by a heat balance between the dried food body and drying medium,82,83 _{or be assumed to be} the air temperature.84 _{The moisture transfer through the foods is normally described} by the differential equation of Fick’s law of diffusion, which is expressed as

(2.27)

The diffusion coefficient is important for the accuracy of model prediction. The diffusion coefficient can be regressed as a function of temperature and concentration by using data in the literature.83 _{Alternatively, the diffusion coefficient can be} determined by Arrhenius’s law as7,82,84

(2.28)

and Ea and D0 are varied during simulation until a reasonable agreement between predicted and experimental results is obtained.

However, for a drying process with a big Biot number, a coupled mass and heat transfer should be taken into account in the simulation. For drying of a composite food system, simulation found that the predicted temperature, mois- ture, and pressure distributions in the composite food system by the coupled model agreed with experimental data. However, there was a big difference between the predicted values by the uncoupled model and experimental data.85

In most cases, it is often assumed that moisture diffuses to the outer boundaries in a liquid form and evaporation takes place only on the surface. The diffusion models do not separate liquid water and water vapor diffusion.7 _{However, in some cases,} inner water evaporation during drying is significant, and therefore, simultaneous heat, water, and vapor diffusion should be considered in simulation.22 _{For example,}

∂ ∂ = ∇ ⋅ ∇ X t D X w w ( ) D D E RT a K = −      0exp

for predicting the drying process of breads, simultaneous heat, water, and vapor diffusion through breads was described by using three governing equations, respec- tively. The three governing equations of heat, moisture, and vapor were connected by the equilibrium of local moisture evaporation and vapor condensation, which is determined by the relationship between saturated vapor pressure and local temper- ature.22 _{Simulations on the drying process of vegetables and fruits using the coupled} heat, water, and vapor diffusion model confirmed that the assumption of an evapo- ration–condensation front in the drying model was valid for drying of porous mois- ture materials with big permeability, such as banana. However, the assumption of an evaporation–condensation front was invalid and more comprehensive analysis was necessary if the permeability of dehydrated foods and vegetables was below 10–19 _m2_.21

On the surface of a food body, external mass transfer is normally assumed to be proportional to the vapor pressure difference between the surface and the drying media.7 _{The surface mass transfer coefficients are affected by the properties} of air, operating conditions, design of the dryer, and the product. Pressure profiles and velocity of heated air above products in an air dryer can be determined by a CFD model.86 _{In this case, the turbulent flow, which is characterized by relatively} high velocity and the presence of many obstacles in the air dryer, can be described by the Chen–Kim k-e model.87

2.3.2.2 Spray Drying

During spray drying, a coupled heat, mass, and pressure transfer phenomenon occurs. The drying of droplets is influenced by external and internal transport phenomena alike. For simulating gas flow in a spray dryer and calculating the trajectories and the course of the atomized particles, CFD is widely used.62 _The k-e turbulence model is used to calculate the gas flow field. The differential equation that describes the diffusion process in spherical particles is then solved simultaneously with equations for external heat and mass transfer.88

2.3.2.3 Microwave Drying

A microwave is used in drying of some heat-sensitive foods.39,48 _{The heat and} moisture transfer during microwaving can be described by Fourier’s equation of heat conduction with inner heat generation and Fick’s law of diffusion, respectively.48 _{In modeling the coupled heat and moisture transfer through} porous materials during microwave-assisted vacuum drying, a combination of liquid water and vapor transfer should be taken into account in the equation of mass transfer. Meanwhile, heat transfer can be described by Fourier’s equation of heat conduction with an inner heat generation term covering latent heat of water evaporation and source heat of microwave power. However, as moisture transfer is caused by the temperature gradient in foods, the equation of moisture transfer can even be simplified into an isothermal equation if the temperature gradient is too small.39

2.3.3 COOKINGAND FRYING

2.3.3.1 Air Convection Cooking

An air convection-heating oven is popular cooking equipment. For predicting transient temperature and moisture distribution in chicken patties of regular shapes in a cooking oven, a coupled heat and mass transfer model was found to give better prediction than that of single heat transfer model.41 _{In some cases, if} it is difficult to find data for the mass diffusivity and mass transfer coefficient, a volumetric moisture loss rate due to evaporation can be experimentally deter- mined and the heat removed due to moisture loss can then be incorporated into Fourier’s equation of heat conduction as inner heat generation.37

With powerful computers available, heating and cooking of solid foods in an industrial convection-type oven can be modeled as a fluid flow and heat transfer problem. CFD offers an efficient and effective tool to analyze the performance of an industrial convection-type oven such as hot-air electrical forced convection ovens. In the CFD models, the electrical heating coils and the fan can be modeled in the momentum equation (the Navier–Stockes equations) as a distributed resis- tance and a distributed body force in the region of the flow domain where the coils and fan are positioned. The value of turbulent viscosity in the momentum equation can be obtained by using the standard and renormalization group version of the κ-ε turbulence model.89,90

2.3.3.2 Microwave Cooking

Microwave-heated and -cooked foods are becoming increasingly popular in the food market and at home. For modeling the microwave heating process, the heat transfer through a solid food body can also be described by Fourier’s equation of heat conduction with inner heat generation due to the microwave energy absorbed by the food components. The microwave power density absorbed at any location in foodstuffs can be derived as a function of dielectric properties and geometry of the food. Meanwhile, heat losses on the surface of the food body by convention and evaporation can be included in the boundary conditions. For simulating micro- wave heating of solid food with rectangular and cylindrical shapes, finite element analysis may be a powerful tool to numerically solve the model.35 _{During microwave} heating, a big moisture loss sometimes occurs. In this case, a coupled heat and mass transfer model should be developed and additional moisture transfer through a solid food body can be modeled by the diffusion equation of Fick’s law.34 _The moisture evaporation rate on the surface can be obtained by using a drying exper- iment and regressed as a function of temperature.91,92

2.3.3.3 Frying

When foods are fried, crust formation is easily observed in many foods. The crust layer increases in thickness as the frying process proceeds, and the interface between the crust and the core region becomes a moving boundary. For a phase

change problem in frying, one side of the interface is crust and the other is the core region. Fourier’s law of heat conduction can be used to describe the heat transfer on both sides of the interface:

(2.29)

(2.30)

The interface between two phases is tracked by Equation 2.8. It should be stressed that the crust and core regions have significantly different thermophysical properties. Because the phase change in foods occurs over a range of temperature, the thermophysical properties of foods experience extreme discontinuities at the phase change temperatures. These discontinuities cause instability in the numer- ical solutions. Alternatively, the enthalpy formulation technique, based on the relationship between enthalpy and temperature, is used to model the phase change problem. One advantage of the enthalpy formulation is that it is not necessary to track the moving interface. Other advantages include the relative stability and simplicity of the method. Using the enthalpy method, Equations 2.29 and 2.30 can be replaced by one single equation as93

(2.31)

During frying, there occurs significant mass transfer as the movement of fat/oil and moisture into or out of the food. A set of mass transfer models based on Fick’s law of diffusion is widely used to describe the moisture and oil/fat movement during frying. Both mass and heat transfer models are coupled for simulating the frying process of foods.56,93

2.4 CHALLENGES IN MODELING HEAT