Transportable moisture limit

B. Heat Transport Limited Moisture Uptake in Spherical Coordinates... 

We now repeat the derivation of the steady-state heat transport limited moisture uptake model for the system described by VanCampen et al. [17], The experimental geometry is shown in Figure 9, and the coordinate system of choice is spherical. It will be assumed that only conduction and radiation contribute significantly to heat transport (convective heat transport is negligible), and since radiative flux is assumed to be independent of position, the steady-state solution for the temperature profile is derived as if it were a pure conductive heat transport problem. We have already solved this problem in Section m.B, and the derivation is summarized below. At steady state we have already shown (in spherical coordinates) that... 

The solutions for moisture uptake presented in this section are based on the experimental condition of a pure water vapor atmosphere. In the next section a derivation of moisture uptake equations is based on both heat and mass transport that are characteristic of moisture uptake in air. The final section of this chapter presents the results of studies where heat transport is unimportant and mass transport dominates the process. Thus, we will have a collection of solutions covering models that are (1) heat transport limited, (2) mass transport limited, (3) heat and mass transport limited, and (4) mass transport limited with a moving boundary for the uptake of water by water-soluble substances. 

The basic assumption for a mass transport limited model is that diffusion of water vapor thorugh air provides the major resistance to moisture sorption on hygroscopic materials. The boundary conditions for the mass transport limited sorption model are that at the surface of the condensed film the partial pressure of water is given by the vapor pressure above a saturated solution of the salt (Ps) and at the edge of the diffusion boundary layer the vapor pressure is experimentally fixed to be Pc. The problem involves setting up a mass balance and solving the differential equation according to the boundary conditions (see Fig. 10). 

Figure 10 Moisture uptakes in rectangular coordinates for the mass transport limited model.

A. Mass Transport Limited Moisture Uptake in One Dimension... 

Since it is assumed that the only limiting resistance to moisture uptake is mass transport resistance, the basis for the model is contained with Eq. (39). It is assumed that the system is at steady state and that rectangular coordinates (uptake in one dimension) are appropriate. Since the system is at steady state and we are dealing with transport in one direction, the flux into a volume element must be equal to the flux out of that element. This condition is expressed as... 

By integrating Eq. (46) and applying the boundary conditions, the solution for the total moisture uptake limited by mass transport is found. In the solution shown in Eq. (47) the vapor pressures have been converted to relative humidities and it has been assumed that the partial pressure of water is much less than the total pressure. Under these conditions, Eq. (47) is the solution for mass transport resistance in spherical coordinates. As with transport in rectangular coordinates, the important variables are the partial pressures of the chamber and above the solid surface and the distance between the solid surface and chamber wall. 

It is very easy to see that Eqs. (48) and (49) have similar form and yet they represent mass transport and heat transport resistance, respectively. What is needed is a connection between the two transport resistances, and this is accomplished by combining the two equations to eliminate RHS. The resulting combined mass-heat transport limited rate moisture uptake (W mh) is given by... 

The moisture uptake models we have discussed have been concerned with pure components. The deliquescing material could be a drug substance or an excipient material. In pharmaceuticals, however, mixtures of materials are also important. One possible situation involves mixing nondeliquescing and deliquescing materials that are formed into a porous tablet or powder blend. The obvious question is, Do the models for pure components apply to porous heterogeneous materials For pure components we have assumed that the mass and heat limiting transport... 

The major function of cutin is to serve as the structural component of the outer barrier of plants. As the major component of the cuticle it plays a major role in the interaction of the plant with its environment. Development of the cuticle is thought to be responsible for the ability of plants to move onto land where the cuticle limits diffusion of moisture and thus prevents desiccation [141]. The plant cuticle controls the exchange of matter between leaf and atmosphere. The transport properties of the cuticle strongly influences the loss of water and solutes from the leaf interior as well as uptake of nonvolatile chemicals from the atmosphere to the leaf surface. In the absence of stomata the cuticle controls gas exchange. The cuticle as a transport-limiting barrier is important in its physiological and ecological functions. The diffusion across plant cuticle follows basic laws of passive diffusion across lipophylic membranes [142]. Isolated cuticular membranes have been used to study this permeability and the results obtained appear to be valid... 

Van Campen et al. [31] developed models describing the rate of moisture uptake above RH0 that consider both the mass transport of water to the solid substance and the heat transfer away from the surface. For the special case of an environment consisting of pure water vapor (i.e., initial vacuum conditions), the Van Campen et al. model is greatly simplified since vapor diffusion need not be considered. Here, only the rate at which heat is transported away from the surface is assumed to be an important factor in limiting the sorption rate, W. For this special case, an expression was derived to express the rate of moisture uptake solely as a function of RHj, the relative humidity of the environment, and RH0. 

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