Unsteady analog solution

As we learned in this chapter, the formulation of unsteady distributed problems leads to partial differential equations. The solution of these equations is much more involved than that of ordinary differential equations. Among the techniques available, the analytical and computational methods are most frequently referred to. Exact analytical methods such as separation of variables and transform calculus are beyond the scope of the text. However, the method of complex temperature and the use of charts based on exact analytical solutions, being useful for some practical problems, are respectively discussed in Sections 3.4 and 3.6. Among approximate analytical methods, the integral method, already introduced in Sections 2.4 and 3.1, is further discussed in Section 3.5. The analog solution technique is also briefly treated in Section 3.7. 

The model proposed above is analogous to a continuous, unsteady state filtration process, and therefore may be called "Filtration Model". In this model, the concentration of the filtrate, viz. the concentration of the solute remained in the treated solid is one s major concern. This is given by the rate of Step 3, which may be expressed by an equation similar to Pick s Law including a transmission coefficient D for the porous medium, viz. the P.S.Z. and the concentration difference Aw across the P.S.Z. as the driving force, and the thickness of the P.S.Z. as the distance Ax. 

Unsteady state diffusion processes are of considerable importance in chemical engineering problems such as the rate of drying of a solid (H14), the rate of absorption or desorption from a liquid, and the rate of diffusion into or out of a catalyst pellet. Most of these problems are attacked by means of Fick s second law [Eq. (52)] even though the latter may not be strictly applicable as mentioned previously, these problems may generally be solved simply by looking up the solution to the analogous heat-conduction problem in Carslaw and Jaeger (C2). Hence not much space is devoted to these problems here. 

Another approach that has promise for study of turbulence structure is the fluctuating velocity field (FVF) closure, adopted by Deardorff (D3). Using the analog of a MVF closure for turbulent motions of smaller scale than his computational mesh, Deardorff carried out a three-dimensional unsteady solution of Navier-Stokes equations, thereby calculating the structure of the larger-scale eddy motions. While it is likely that calculations of such complexity will remain beyond the reach of most for some time to come, results like Deardorff s should serve as guides for framing closure models. 

As we have seen in the preceding sections, the solution of unsteady conduction problems is, in general, not mathematically simple, and one must usually resort to a number of solution methods to evaluate the unsteady temperature distribution. We have also learned how to obtain solutions by using the available charts for a class of analytical results. In Chapter 4 we will explore the use of numerical computations to evaluate multidimensional and unsteady conduction problems. These computations require approximate difference formulations to represent time and spatial derivatives. Actually there exists a third and hybrid (analog) method that allows us to evaluate the temperature distribution in a conduction problem by using a timewise differential and spacewise difference formulation. This method utilizes electrical circuits to represent unsteady conduction problems. The circuits are selected in such a way that the voltages (representing temperatures) obey the same differential equations as the temperature. 

The extraction of aromatic hydrocarbons present in low concentration in packed beds of soil particles was performed [104] using near critical CO2 as the solvent. The results were compared with an unsteady-state model for the particle phase, assuming a steady state in the fluid phase and negligible variation in particle phase concentration with bed height. Under these assumptions, the loss of solute from the particle phase is analogous to evaporation from a sphere into a quiescent medium and the equation... 

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