A framework for modeling internal mass transfer in atmospheric freeze drying
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In modeling of atmospheric freeze drying (AFD), the modeling of the product is often confined to either one-dimensional models or multi-dimensional models of very basic geometries. Such models are not able to handle a product of complex shape, and require simplifying assumptions on the geometry of the product in order to work. In addition, although many models incorporate equations that account for the effects of several physical phenomena, they often assume that the coefficients in the equations are either constant or certain functions of external parameters, such as the ambient temperature. The effects of local variations in the internal structure of the product, such as the effects of inhomogeneity or anisotropy on transport properties, are rarely modeled. The main goal of the work presented in this thesis was to develop a three-dimensional, geometrically flexible framework for modeling AFD, in order to make handling of complicated geometries not only possible, but also straightforward. To this end, the definition of the geometry was made implicit, i.e. any surface or interface is defined as the set of points where a certain geometry defining function (GDF) is zero. The GDFs can be made time-dependent, to account for deformations (shrinkage) as time goes by. Thus, we should in principle be able to model any product in any surroundings with this framework, as long as all surfaces and interfaces can be described by GDFs. Moreover, we can encase an implicitly defined product in a box-shaped, fixed domain. This makes the numerical grid simple and fixed, even if the product itself shrinks with time. Another important goal was to make the framework accommodate quasilinear transport coefficients, i.e. that the coefficients could depend on local conditions, including local moisture concentration. This allows the effects of inhomogeneity to be included. By allowing different coefficients in different directions, the framework also allows anisotropy to be modeled. The emphasis was on modeling the mass transfer inside the product, which was assumed to be caused by vapor diffusion through Darcy’s law. External mass transfer was modeled as convective, with the ambient conditions as given boundary conditions (i.e. not modeled). A mass transfer equation was developed to describe the sublimation of ice and subsequent diffusion of vapor through the product. It was based on the retreating ice front (RIF) assumption. A method of investigating the validity of theRIFassumption was proposed and tested on pieces of cod fillet, using a scanning electron microscope (SEM). It was verified that the equation was mathematically well posed, and a spectral method was chosen to solve it numerically. The method was adapted to the specific equation, and several improvements were made to it, to decrease memory costs and time consumption. To test the framework, simulations were carried out on pieces of cod fillet at -5 ◦C and -10 ◦C. The calculated drying curves showed good agreement with experimental results, and the qualitative properties were found to be satisfactory within certain bounds on the parameters.