Water vapor mass transfer resistance

Since the water vapor pressure increases exponentially with the temperature, the drying rates during freeze-drying processes, which vary in a similar way, are strongly dependent on the sublimation front temperatures that is, on the shelf temperature. 

Otherwise, as already emphasized in Section 3.3.3, for a given formulation, the water vapor mass transfer resistance values are directly dependent on the pore morphology of the freeze-dried zone, and, consequently, on the ice crystal morphology, that is to say on the freezing conditions and on the thickness of the freeze-dried layer which increases throughout the sublimation step. 

This relationship for the Knudsen regime (low total gas pressure, low sublimation temperature) has been validated by Hottot et al. (2005) who compared with freezedrying experiments conducted with a model BSA formulation with annealing treatment (cf. Fig. 3.19). In the case ofa standard freeze-drying cyde, this successful validation represents a positive result due to numerous assumptions involved in the model and to quite large uncertainties in the corresponding modd parameter values. 

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Fig. 3.19 Water vapor mass transfer resistance versus freeze-dried layer thickness. Continuous line = model prediction, according to Hottot et al. (2005).

Equation 3.3 shows that Kis inversely proportional to the water vapor mass transfer resistance, previously defined by Eq. 3.1. 

J modei. respectively, plotted in Fig. 3.20, show pretty good agreement. These data clearly show that the water vapor mass transfer resistance decreases as the nucleation temperature increases, so that the drying process can be accelerated by increasing the nucleation temperature. This tendency is in logical agreement with the ice crystal size estimation obtained both from image analysis and from the model prediction. 

The pressure rise analysis method (PRA method), recently proposed by Chouvenc et al. (2004a), derived from the MTM method originally developed by Milton et al. (1997) and next modified by Obert (2001), appears to be a very promising noninvasive control method. It is a rapid, simple to implement and averaging tool that requires a freeze-dryer equipped with an external condenser and a very fast closing separating valve. The values of the main freeze-drying parameters, such as the temperature of the sublimation front, T , the resistance to water vapor mass transfer of the dried layer, J p, and the overall heat transfer coefficient, could be... 

If the freeze-drying process is controlled by water vapor mass transfer through the freeze-dried layer - for example in the case of dried cake of very low permeability or for very favorable heat transfer conditions - the drying rate for one vial is proportional to the mean driving force P - Pc) - that is, to the water vapor pressure difference between the sublimation front interface and the drying chamber - and controlled by the freeze-dried cake mass transfer resistance, denoted J p, which is inversely proportional to the dried layer permeability defined by Darcy s law. Inthiscaseitholds ... 

Fig. 3.15 Resistance to water vapor mass transfer as a function of dry iayer thickness with and...

Rp water (solvent) vapor mass transfer resistance Pa m s hg ... 

To illustrate the system behavior, the ternary mixture 1 = iso-propanol, 2 = water, and 3 = air is considered here. In order to obtain an algebraic solution, both the dif-fusivities of iso-propanol in air and iso-propanol in water vapor were assumed to be approximately the same, which is not far from reality. The liquid phase mass transfer resistance was negligibly small, as will be shown below. The phase equilibrium constants K/,c and Kjrs were calculated with activity coefficients from van Laar s equation. Water vapor diffuses 2.7-fold faster in the inert gas air than iso-propanol. The ratio of the respective mass transfer coefficients kj3 equals the ratio of the respective diffusivities to the power of 2/3rd according to standard convective mass transfer equations Sh =J Re, Sc). 

The evaporation velocity at ambient pressure and, say, 60 °C, which corresponds to a mole fraction in the sweep gas of about 30 % water vapor, is about uuq = IO-5 m s-i. This results in Kj , = 0.904 which is rather close to 1, so that the effect of the liquid phase mass transfer resistance on the selectivity of an open distillation process with a free gas-liquid interface in most cases can be ignored. If, however, kuq becomes very small (as in the pervaporation process described in the next example), Kuq might become very small and thus reduce the selectivity of the open distillation process practically down to zero. 

The mass transfer resistance at a liquid-vapor interface results from two resistances, the liquid boundary layer and the gas boundary layer. In conditions involving water and sparingly soluble gases, such as occurs here, the liquid-phase resistance is almost always predominant [71]. For this reason, equation (16) involves only k, the mass transfer coefficient across the liquid boundary, and a, which is the gas bubble surface area per unit volume of liquid. Often, as here, those factors cannot be estimated individually, so k is treated as a single parameter. 

The effect of product temperature and mass transfer resistance on sublimation rate may be mathematically illustrated by expressing the sublimation rate per vial, dm/dt, in terms of the driving force for transport of water vapor from the ice-vapor interface to the chamber, Pq - Po,... 

At high enough qualities and mass fluxes, however, it would be expected that the nucleate boiling would be suppressed and the heat transfer would be by forced convection, analogous to that for the evaporation for pure fluids. Shock [282] considered heat and mass transfer in annular flow evaporation of ethanol water mixtures in a vertical tube. He obtained numerical solutions of the turbulent transport equations and carried out calculations with mass transfer resistance calculated in both phases and with mass transfer resistance omitted in one or both phases. The results for interfacial concentration as a function of distance are illustrated in Fig. 15.112. These results show that the liquid phase mass transfer resistance is likely to be small and that the main resistance is in the vapor phase. A similar conclusion was reached in recent work by Zhang et al. [283] these latter authors show that mass transfer effects would not have a large effect on forced convective evaporation, particularly if account is taken of the enhancement of the gas mass transfer coefficient as a result of interfacial waves. 

A modification of the previous model was proposed by Obert (2001) who considered also the desorption of the bound water during the primary drying, which can contribute to the increase in the total pressure, and the thermal inertia of the glass wall of the vial. The temperature at the bottom of the vial and the thickness of the frozen layer should be known in order to use this algorithm, but they are only guessed in the proposed procedure. The overall heat transfer coefficient is expressed adopting the heat and mass transfer steady-state hypothesis, while a non-linear regression analysis is carried out in order to estimate the vapor pressure at the interface, the mass transfer resistance in the dried product and the desorption rate. 

The variable pZ is taken to represent the water vapor pressure in the drying chamber (the external mass transfer resistance is taken to be insignificant) and its value is determined by the design and the operational temperature of the ice condenser. Thus, Ah may be changed by changes in pZ (pZ could be changed by changes in the temperature of the ice... 

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