2.9 Heat Transfer Models
2.9.4 Numerical Solutions for Heat Transfer
Cleland et at. ( 1 987c) stated that numerical methods are accurate and reliable for freezing and thawing time predictions provided that they are formulated and implemented correctly to reduce any truncation and rounding errors. Due to the availabil ity of fast computers it is now possible to reduce numerical errors by going to very small time and space step sizes to solve the heat transfer equations. The two numerical methods commonly used for freezing and thawing models are the finite element and finite difference methods. For irregular shapes finite element method is the most appropriate and for the regular geometric shapes the finite difference solution scheme is usually preferred (Cleland, 1 985).
According to Cleland ( 1 990) the finite difference method can be implemented in several ways including;
• Simple explicit scheme where thermal conductivity and heat capacity are combined and
thermal diffusivity is taken as a function of temperature.
• Explicit solution where thennal conductivity and specific heat are taken as separate functions of temperature.
• Explicit difference formulae based on the enthalpy transformation (equation 2- 1 4)
• Ful l y implicit, multi-time level schemes such as the Crank Nicholson scheme (Crank &
Nicholson, 1 947); and Lees scheme ( Lees, 1 966) etc.
Due to the phase change problem in this work the enthalpy approach is the most appropriate. Implicit schemes for the enthalpy transformation involve iterative solutions at each time step, making the solution more complicated but all ow greater numerical accuracy and therefore larger time steps. With the availability of fast computers explicit schemes can be used successfully with small time step and are favored because they are easier to implement.
A number of papers reported the application of numerical methods for predicting freezing and thawing of food systems. For example Manpperuma & Singh ( 1 988) used an explicit numerical method involving the enthalpy transformation for one, two and three dimensions with the 1
S\
2ndand 3rd kind of boundary conditions to predict thawing and freezing in foods. The predicted results showed good agreement with experiment data for cylindrical, slab shaped and spherical products.
Cleland ( 1 990) reviews numerical methods suitable for predicting freezing and thawing of food products. Most of these methods use equilibrium thermal property data (the process is assumed to be heat transfer limited). However in many food systems freezing involves supercooling (non equilibrium behavior). Therefore it was decided to review the literature and look at the different approaches available to model the freezing behavior of food systems where supercooling occurs.
2.9.5 Freezing Models for Water-enriched Foods:
The freezing of food is not a simple process and involves the complexity of removal of both latent and sensible heat in the food. This removal of latent heat depends on the structure of food. In some food materials the removal of latent heat occurs once the initial freezing point of the food is reached without any supercooling effects. At the initial freezing point a portion of the water within the food crystallize that makes the remaining solution more concentrated which further decreases the freezing point.
In other kinds of water-enriched foods water supercools to many degrees below its i nitial freezing point, but as soon as the ice nuclei are formed the crystallisation process begins and starts releasing the latent heat. This process gives a quick rise in temperature unti l the temperature approaches the initial freezing point.
Cleland et
at.
( 1 982) observed that high rates of cooli ng increased the freezing time up to 25% and they postulated that this may be caused by the supercooling throughout the process of ice crystal growth and prior to the initial ice nucleation. Pham ( 1 989) suggested that unless the physical phenomena of ice crystal nucleation and growth are better understood, numerical methods are unlikely to give better predictions than the simple empirical or semi-empirical equations.Miyawaki ( 1 989) described the freezing process i ncluding
A typical relationship between the enthalpy - temperature is given in Figure 2. 1 1 . Initially the enthalpy during the freezing proceeds along the line a�b�c. The section b�c corresponds to supercooling and heat transfer is simple heat conduction without phase change. Once nucleation occurs the supercooling ceases, and is followed by rapid crystal growth and the enthalpy curve shifts from c�d. This transition happens throughout the sample. The remaining latent heat transfer is limited by the heat removal rates and follows the enthalpy curve from d�e�f until the whole sample is frozen completely. Lastly pure sensible heat transfer occurs following the enthalpy curve follows f�r .
Pham( 1 989) used a similar type of step approach t o model supercooling fol lowed by dendritic growth of ice crystal in one dimensional meat slab. Pham ( 1 989) used the same enthalpy
temperature relationship given in Figure 2 . 1 1 with the inclusion of the process c-d by assuming that the crystallization takes place at a definite nucleation temperature followed by the dendritic growth of crystals.
These models are appropriate in foods in which water is a continuous phase. In such systems nucleation anywhere in the product will initiate ice crystal growth throughout the supercooled water phase. In butter where water is in the form of discontinuous droplets, such models are not likely to be appropriate unless all the droplets begin to freeze at the same level of A model combining conductive heat transfer and the nucleation/growth models reported in Section 2.6 is a promising approach to model the freezing behaviour of butter that has not yet been attempted.
H I' (ii) d e 9 ;/ Temperature a
Figure 2.1 1 Enthalpy - temperature relationship. Pure water: ahdgf', water-enrichedjood without supercooling: ahdejf', jood with supercooling: ahcdegff'.