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Passage DEM - Discrete Element Method

Published in:

Computer Methods In Applied Mechanics and Engineering,

Vol. 148, 1997, pp. 105-125


W. J. Mascarenhas and H. U. Akay*

Technalysis Inc., 7120 Waldemar Drive, Indianapolis, Indiana 46268

M. J. Pikal

Eli Lilly and Co., Lilly Corporate Center, Indianapolis, In 46285

* Also, Dept. of Mechanical Engineering, Purdue School of Engineering and Technology, IUPUI, Indianapolis, Indiana.


A brief overview of the freeze-drying process is given, followed by presentation of the governing equations and the finite element formulation in two-dimensional axisymmetric space. The model calculates the time-wise variation of the partial pressure of water vapor, the temperature, and the concentration of sorbed water. An Arbitrary Lagrangian - Eulerian method is used to accurately model the sublimation front of the freeze-drying process. Both the primary and secondary drying stages of the process are modeled. Several examples are presented that validate the model and demonstrate representative applications of such calculations.


Freeze-Drying, or "lyophilization", is a drying process where the solution, normally aqueous, is first frozen, thereby converting most of the water to ice, and the ice is removed by sublimation at low temperature and low pressure during the "primary drying" stage of the process. Sublimation occurs at the interface between the frozen and dry material and starts at the top of the material. The interface moves through the material and starts at the top of the material. The interface moves through the dry material until only a dried porous material remains at the end of primary drying. Water vapor flows out of the material through the pores of the material and is then collected on a condenser operating at very low temperatures (i.e., approximately minus sixty-five degrees Celsius). At the end of the primary drying stage, all ice has been removed. Normally, a significant quantity of water remains associated with the solute phase and does not freeze. This "unfrozen water" is removed by desorption in the "secondary drying" stage of the process, usually employing temperatures above ambient. Secondary drying is continued until the residual water content decreases to the desired moisture content. In reality some sorbed water is removed in the primary drying stage. Thus the two stages occur concurrently.

Since freeze drying is a low temperature process, the process is often used in pharmaceutical and food industry to dry materials which offer degradation or other "loss of quality" during high temperature drying. In the pharmaceutical industry, the solution is normally filled into glass vials, the vials are placed on temperature controlled shelves in a large vacuum chamber, and the shelf temperature is lowered to freeze the product. After complete solidification, the pressure in the chamber is lowered to initiate rapid sublimation. Process times are often quite long, and since commercial freeze drying plants are expensive, process costs are relatively high. However, since the commercial value of a batch may approach $1,000,000, maintaining product quality is normally the most important concern. Although a higher ice temperature produces a shorter, more economical process, excessive ice temperatures may result in severe loss of product quality and rejection of the batch [1, 2]. Product temperature control is critical to preserve product quality and yet minimize process time. However, product temperature it normally controlled directly. Rather, the shelf temperature and chamber pressure are controlled to control heat and mass transfer, such that the optimum product temperature profile with time is obtained. In practice, the appropriate shelf temperature and chamber pressure conditions are frequently established empirically in a "trial-and-error" experimental approach. Theoretical modeling studies, which are predictive, have considerable potential to guide the experimental studies, thereby decreasing development time and insuring the design of a process which is optimal and robust.

A number of freeze-drying models have been published in the literature [3 - 11] to describe the freeze-drying process. The "sublimation" model of Liapis and Litchfield [11] was seen to be more accurate than the "uniformly retreating ice front" model of King [5]. The sublimation model was then improved upon [12 -14] by including the removal of bound water in equations. This model is commonly known as the "sorption-sublimation model".

Tang, et al. [15] extended the dynamic, one-dimensional model described in [14] to a two-dimensional freeze-drying in a vial. Ferguson, et al. [16] have presented a two-dimensional model based upon the uikov system of partial differential equations. Various numerical methods can be used to solve the governing equations. Liapis and coworkers used a one-dimensional method which immobilizes the moving interface by rewriting the equations in terms of normalized co-ordinates [17]. The method of orthogonal collocation [18 - 20] was then used to solve the equations. Ferguson et al. [16] used the finite element method to solve the governing equations. However only one-dimensional examples have been presented in the paper. In all one-dimensional models, the geometry of the material must be rectangular and also the shape of the interface cannot be a curve.

The two-dimensional model presented in this paper overcomes these disadvantages. The material geometry and the interface can be of any arbitrary shape. The finite-element method with an arbitrary Langrangian-Eulerian (ALE) scheme for tracking the sublimation front and a two-step rational Runge-Kutta (RRK) integration scheme for unsteady calculations is used throughout the simulation of the process. The two-dimensional model developed in this paper is very general and can be used to study a variety of freeze-drying processes.

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