Model of Mass and Heat Transfer during Vacuum Freeze-Drying for Cornea

Cornea is the important apparatus of organism, which has complex cell structure. Heat and mass transfer and thermal parameters during vacuum freeze-drying of keeping corneal activity are studied. The freeze-drying cornea experiments were operated in the homemade vacuum freeze dryer. Pressure of the freeze-drying box was about 50 Pa and temperature was about −10◦C by controlled, and operating like this could guarantee survival ratio of the corneal endothelium over the grafting normal. Theory analyzing of corneal freeze-drying, mathematical model of describing heat and mass transfer during vacuum freeze-drying of cornea was established. The analogy computation for the freeze-drying of cornea was made by using finite-element computational software. When pressure of the freeze-drying box was about 50 Pa and temperature was about −10◦C, time of double-side drying was 170min. In this paper, a moving-grid finiteelement method was used. The sublimation interface was tracked continuously. The finite-element mesh is moved continuously such that the interface position always coincides with an element node. Computational precision was guaranteed. The computational results were agreed with the experimental results. It proved that the mathematical model was reasonable. The finite-element software is adapted for calculating the heat and mass transfer of corneal freeze-drying.


Introduction
The storage of isolated freeze-drying biological tissue is a significant field in the application of vacuum freeze-drying technology and studies for its mass and heat transfer theory under low-temperature, low-pressure conditions which have already become a hot frontal topic in theoretical researches.The vacuum freeze-drying of cornea freeze-dry is the freeze-drying for isolated biological tissue which has to remain active afterwards.5 The semidry layer is a porous region due to the back of sublimation interface, dry-material, and adsorbed-water solid phase were inside the region.The gas consists of vapor and permanent gases, which were considered to maintain thermal equilibrium with the solid phase.
6 Surface and interface temperature stay unchanged.7 The frozen zone is assumed to be uniform and has uniform thermal conductivity, density and specific heat, also it contains small amount of dissolved gases.
8 Changes of the overall size of cornea are out of question.
9 The total pressure of the drying chamber was controlled by appropriate-size vacuum pump and required devices most of the permanent gases in the chamber was considered leak in .

Mass Transfer Equation
Continuity equations of dry-layer can be expressed as follows: C * sw is the density of water in solid phase when water-gas equilibrium was achieved.Equations 2.4 and 2.5 are the mass-trans rate formula of binary mixture pass through the dry zone, based on diffusion equation Evans et al. 16 and viscous flow formula D Arcy .In the formula, the main vapor diffusion is passing through the dry layer, and escaping by way of Knudsen diffusion and the corresponding total pressure gradient; also with no consideration of surface diffusion and thermal diffusion model as such diffusion is not significant 17 .It has been proved that the role of viscous flow is not that important for N w and N in 18 .Knudsen diffusion is the most important one in the case of low/permanent gas nonexist.However, main diffusion controls the rate when they do exist.Therefore, 2.4 and 2.5 can be simplified as

2.8
By 2.1 and 2.4 , we obtain: By 2.2 and 2.5 , we obtain

2.10
The work in 17 has proved that permanent gas got a very small change rate; 2.9 could hence be solved without 2.10 .Therefore, the following manipulating only includes finite element formulation for 2.9 .
In the 2D symmetric space, 2.9 can be expressed as

2.11
Under low pressure, the concentration of water vapor and permanent gases can be expressed by using the ideal gas law: 2.12 So, 2.11 can be described by the following equation: and assuming that

Heat Transfer Equation
In a 2D symmetric space, energy equation for dry-layer is

2.16
Frozen layer energy equation is 2.17

Concentration Equation
Moisture concentration change in the dry layer versus time can be described as

Initial and Boundary Condition
When t 0,

2.20
Equation 2.20 specified the boundary value of pressure and temperature.The concentration equation is an issue relating to initial value, with no consideration of any boundary conditions.

The Trajectory of the Sublimation Interface
During the process of drying, the sublimation latent heat was released at the sublimation interface, and the sublimation heat was added as boundary condition when solving a series of control equation.It, hence, becomes necessary to track the location of the interface and impose boundary conditions on the interface spot.The movement of the interface is affirmed by exam the equilibrium of heat at the interface.In the case of noninternal heat source, the heat originates from the dried layer equals to the total heat both from the absorption and the heat flowed to the frozen layer.The heat balance equation can be expressed as follows: The velocity of interface ν n perpendicular to the interface itself, and ν n could conclude from 3.2 in order to obtain the new interface location.Numerous numerical calculation techniques were used to solve the problem of moving interface, in the paper, however, the finite-grid-element method has been used 13 , where the sublimation interface was tracked consecutively and the interface was assumed to be boundary condition of the move.The finite grid would be redivided when changes occur in the sublimation interface, by doing this, to fix the interface location always at the unit node.In other words, instead of stay in the unit, the interface location would always be on the outline or node of the unit, and so the boundary conditions would be added on the contour line.Therefore, the equation of both dry and frozen layers can be solved together with the combination of interface boundary conditions.

Finite-Element Formulations
Variational equation within the finite equation defines the formulation 2.15 and 2.18 as 1 2 Freezed district (II) Dried district (I)

Results and Discussion of the Frozen Model
Figure 1 is an illustration of a sided schematic lyophilized.At the beginning of calculation, the initial dry layer thickness needs to be determined, and the equation would be unsolved when the freeze starts and the thickness of dried layer is zero.Similarly, it is required to determine the residual value of the frozen layer when the frozen layer thickness trends to be zero 19 .As the test results tell, the initial thickness of frozen layer would be 6 × 10 −6 m, and for residual layer thickness, the value should be 1 × 10 −6 m as shown in Table 1.

The Relationship between Lyophilized Chamber Pressure and Drying Time
In the process of freeze the cornea, radiation heat is the dominating way of heating.During the radiation heating, sublimation starts at the frozen material surface, which gradually forms an interface between the dried layer and frozen layer as shown in Figure 1 .The heat passes through the porous dried layer by thermal conductivity and spreads to the sublimation interface, adsorbed by the ice sublimation process simultaneously.The vapor emerged from the diffusion spread along the opposite direction of heat from the sublimation interface to the dry layer.Also, the sublimation interface moves inwards gradually until the frozen water disappears and the sublimation come to an end.So, the moving rate of the interface could stand for the lyophilized rate.
The speed of the freeze-dry process depends on the sublimation of ice and the diffusion rate of vapor, the former indicator is also determined by the intensity of heat transfer to the sublimation interface, hence, the drying process is controlled by tow mechanisms.Filled with vapor and small amount of air, the gas thermal conductivity rate inside the dried layer arises with the pressure increase.Therefore, when the pressure rises in the freezedry chamber, the effective thermal conductivity of the dried layer would arise and provide heat to the sublimation interface which accelerates the process.The driving force of vapor is the differential pressure of the sublimation and dried layer.The vapor pressure on the sublimation interface can be assumed equal to the saturation pressure under the interface temperature.When the temperature of the lyophilized chamber arises, the partial pressure of the dried layer increases which slows down the vapor diffusion rate control by the dried layer differential pressure of mass transfer.Clearly, the pressure of lyophilized chamber impact oppositely to the ice crystal simulation rate and water vapor diffusion rate, that is, to double impact the lyophilize process both boost and retard the process.Apparently, there should be a certain pressure that optimizes the freeze-drying rate 22, 23 .During the cornea lyophilized process, there also exists an optimum value for the chamber, which can both ensure the endothelial cell survival chance and taking into account the lyophilized rate.
Figure 2 shows the lyophilized cornea freeze time under the condition of the pressure 20 Pa, 50 Pa, 80 Pa, 110 Pa.As can be concluded from the figure, the time for freeze dry the cornea basically equals to the pressure when it is below 50 Pa, then as the pressure increases, the time needed increases accordingly.The simulation results and the experimental test afterwards conclude that the chamber pressure should be over 50 Pa if more than 80% of the survival rate is required.

The Relationship between Temperature of Sublimation Interface and Time of Drying
In the process of freeze-drying, as the ice is sublimating, the interface between the porous dried layer and the frozen layer will be continuously moving backward, inducing the increase  of water vapor diffusion resistance at the sublimation interface and the rise of interface temperature 24-27 , as shown in Figure 3.When the temperature of condenser's surface is constant, the interface temperature would be the main parameter to determine water vapor diffusion intensity.In previous studies, the interface temperature was assumed to be a constant value 28, 29 , some experts believe that the assumption will decrease the simulation analysis precision, and they assume the sublimation interface temperature in the process of freeze-drying is gradually rising.They replace the constant interface temperature in the previous studies with the linear expression between the sublimation interface temperature and the location, thereby improving the mass transfer model in freeze-drying process.Figure 3 shows that when the sublimation interface of freeze-dried material moves 30 cm, its temperature just increases by 6 • C, and corneal thickness is around 0.6 mm, so in this simulation, it is viable to assume the interface temperature is constant.
The temperature of sublimation interface, the temperature of condenser surface, and the pressure of freeze-drying chamber are correlated.If the temperature of sublimation interface is lower, the sublimation interface saturation pressure is lower accordingly, which requires the lower freeze-drying chamber pressure and lower condenser surface temperature.The relationship among them indicates that the saturation pressure of sublimation interface > the pressure of freeze-drying chamber > the saturation pressure of condenser surface.During the stages of a freeze-drying process, when the condenser surface temperature is −40 • C, it affects little on shortening the time of freeze-drying.When the condenser surface temperature declines from −40 • C to −100 • C, the time of freeze-drying reduces just by 4%.The influence tends to be similar to the results of Millman's study on skim milk freeze-drying 30 .Too low temperature of the condenser surface does not greatly improve the differential pressure of water vapor diffusion, which is determined by the exponential function between the saturation pressure and temperature.Instead, if the sublimation interface temperature would be increased dramatically, which increases the differential pressure of water vapor diffusion the saturated pressure is 260 Pa, when it is −10 • C , the time of freezing can be reduced distinctly.
During the process of corneal freeze-drying, sublimation interface temperature cannot be in −30 • C to −60 • C, within which biological cells are at risk.If the temperature of the sublimation interface is below −60 • C, it will not only reduce the temperature of the condenser surface, but also can make the time of freeze-drying much longer causing the increase of the damaged cornea chance.The experiment indicates that the corneal endothelial survival chance can achieve the standard value if the sublimation temperature is fixed around −10 • C.

The Relationship between Differential Surface Temperature and the Time of Drying
Figure 4 shows the computational results of double-sided freeze-dried model of Figure 1.In Figure 4, the pressure in the freeze-drying chamber is 50 Pa, and the surface temperatures are −10 • C, 0 • C, and 10 • C, respectively.Three curves are basic coincident, confirming that the variation of surface temperature is from −10 • C to 10 • C, it has little a influence on the corneal freeze-drying process.Diffusion rate constant of the main body Constant only related to the structure of the porous media, which can produce relative Knudsen air permeability C 2 :

Simulative Calculation Performed when the
Constant only related to the structure of the porous media, which is the ratio of the main body diffusion rate to free gas diffusion rate, no dimension K w : Knudsen diffusion coefficient, K w C 1 RT/M w , where M w is molecular weight of water K in : Knudsen diffusion coefficient, K in C 1 RT/M in , where M in is molecular weight of permanent gas K mx : The average Knudsen diffusion coefficient of binary gas mixture, y w K in y in K w D w,in : Diffusion coefficient of free gas in binary gas mixture with water vapor and permanent gas R: Gas constant kJ/mole•K M: Molecular weight kg/kg mole P : Total pressure in the already dried layer Pa Interface velocity m/s x: Space coordinates through the thickness direction y: Space coordinates through the warp direction Y i : The number of moles in the component i in gas phase μ mx : The viscosity of binary gas mixture with water vapor and permanent gas in porous already dried layer kg/m•s ΔH v : The vaporization heat of adsorbed water kJ/kg ΔH s : Latent heat of sublimation of ice kJ/kg .

Figure 2 :
Figure 2: The calculated sublimation interface positions versus time at various chamber pressures.

Figure 3 :
Figure 3: Relationship between temperature and position of sublimation interface.

Figure 4 :
Figure 4: The calculated sublimation interface positions versus time at various surface temperatures.

Figure 5 :1Figure 6 :
Figure 5: Relationship between position of sublimation interface and time.

2 Mathematical
P w : Partial vapor pressure Pa P 0 w : Initial partial vapor pressure Pa P in : Partial pressure of permanent gas Pa P 0 in : Initial partial pressure of permanent gas Pa q: Rate of heat flow kW/m Initial temperature K T ∞ : Outside temperature K T int f : Interface temperature K ν:

ε:
The volume of space in each unit volume of material; ρ: Density, kg/m 3 .Subscript i: Components In: Permanent gas w: Water e: Effective value I: Dried area II: Freezing area n: Quadrature component x: X direction component y: Y direction component.