Published in:
Computer Methods In Applied Mechanics and Engineering,
Vol. 148, 1997, pp. 105-125
A
COMPUTATIONAL MODEL FOR FINITE ELEMENT ANALYSIS OF THE FREEZE-DRYING PROCESS
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.
Abstract
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.
Introduction
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.