Properties of systems are frequently dependent on, or are functions of, more than one independent variable. Modeling of such systems leads to a partial differential equation [4]. Temperature within a rod, for example, may vary radially as well as axially. Similarly, concentration of a species within a system may depend on the location as well as vary with time. Figure 4.5 shows batch drying of a polymer film cast on a surface. The solvent present in the polymer diffuses through the film to the surface, where it is carried away by an air sweep.
Figure 4.5 Drying of polymer film.
The concentration of the solvent within the film is a function of time as well as distance from the surface. Equation 4.12 is the fundamental equation3 for governing the solvent mass transport within the film, a partial differential equation that is first order with respect to time t and second order with respect to location x.
3. This is called the Fick’s second law equation.
DA is the diffusivity of solvent A in the polymer film, which depends on the properties of the system.
The solution of this (and other partial differential equations) requires an appropriate number of specifications (boundary and initial conditions) depending on orders with respect to the independent variables.
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