Unsteady approximate solution

The experimental evidence discussed so far concerns turbulent free convection at a horizontal surface. Even in the classical example of laminar free convection, namely, at a vertical plate, the analysis of unsteady-state transport is very complicated. Only recently have approximate solutions been... 

There are no solutions for transfer with the generality of the Hadamard-Rybczynski solution for fluid motion. If resistance within the particle is important, solute accumulation makes mass transfer a transient process. Only approximate solutions are available for this situation with internal and external mass transfer resistances included. The following sections consider the resistance in each phase separately, beginning with steady-state transfer in the continuous phase. Section B contains a brief discussion of unsteady mass transfer in the continuous phase under conditions of steady fluid motion. The resistance within the particle is then considered and methods for approximating the overall resistance are presented. Finally, the effect of surface-active agents on external and internal resistance is discussed. 

In Chapters 2 and 3 we have already introduced the concept of penetration depth for an approximate solution of conduction problems (recall Section 2.4.1, and Exs. 2.11 and 3.9). This concept, which we utilized to determine the steady or unsteady penetration depth of heat (or thermal boundary layer) in solids and stagnant fluids, actually applies to all diffusion processes, such as diffusion of momentum, mass, electricity, and neutrons, as well as diffusion (or conduction) of heat It is a convenient tool for an approximate solution of conduction problems and is indispensable for convection problems, which are considerably more complicated than conduction problems. 

Zeng et al. (1993) proposed that the dominant forces leading to bubble detachment could be the unsteady growth force and buoyancy force. In order to derive an accurate detachment criterion from a force balance, all forces should be accurately known. If a mechanism is not known precisely, then approximate expressions, one or two fitted parameters and comparison with experiments might offer a solution. Such fitting procedures have indeed been applied (Klausner et al. 1993 Mei et al. 1995a Helden et al.l995). 

The research on the flow regimes in packed tubes suggests that laminar flow CFD simulations should be reasonable for Re <100 approximately, and turbulent simulations for Re >600, also approximately. Just as RANS models provide steady solutions that are regarded as time averages of the real time-dependent turbulent flow, it may be suggested that CFD simulations in the unsteady laminar inertial range 100 time-averaged picture of the flow field. As with wall functions, comparisons with experimental data and an improved assessment of what information is really needed from the simulations will inform us as to how to proceed in these areas. 

One approach is to approximate the unsteady solution by a quasi-steady solution where Q(t) is a function of time and time is adjusted. This can be approximated by determining the transport time (fD) from the origin to position z by... 

Only a finite difference numerical solution can give exact results for conduction. However, often the following approximation can serve as a suitable estimation. For the unsteady case, assuming a semi-infinite solid under a constant heat flux, the exact solution for the rate of heat conduction is... 

Analytic solutions for flow around and transfer from rigid and fluid spheres are effectively limited to Re < 1 as discussed in Chapter 3. Phenomena occurring at Reynolds numbers beyond this range are discussed in the present chapter. In the absence of analytic results, sources of information include experimental observations, numerical solutions, and boundary-layer approximations. At intermediate Reynolds numbers when flow is steady and axisym-metric, numerical solutions give more information than can be obtained experimentally. Once flow becomes unsteady, complete calculation of the flow field and of the resistance to heat and mass transfer is no longer feasible. Description is then based primarily on experimental results, with additional information from boundary layer theory. 

It is usual in laminar mixing simulations to represent the flow using tracer trajectories. The computation of such flow trajectories in a coaxial mixer is more complex than in traditional stirred tank modelling due to the intrinsic unsteady nature of the problem (evolving topology, flow field known at a discrete number of time steps in a Lagrangian frame of reference). Since the flow solution is periodic, a node-by-node interpolation using a fast Fourier transform of the velocity field has been used, which allowed a time continuous representation of the flow to be obtained. In other words, the velocity at node i was approximated... 

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. 

If an unsteady temperature distribution resembles its ultimate steady limit, we may select X (x) to be the steady solution of the problem. Thus, in view of Eq. (259), an approximate unsteady solution may be written as... 

The steady solution of the problem (corresponding to t - oo) is T — Tw. Any scaling of the steady solution does not appear to resemble the unsteady temperature distribution. This fact suggests a second approach for the construction of approximate profiles, one that requires the selection of polynomials satisfying the boundary conditions. For example, a parabola satisfying the boundary conditions of Eq. (3.114) gives... 

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. 

Consistent with steady-state solution discussed earlier, we will consider the surface reaction (QS) approximation for unsteady behavior. Accordingly, the chemical heat release term is removed from the energy differential equation and placed in the surface boundary condition and a fraction of the absorbed radiant flux (1-/ ) is deposited in the surface layer ... 

INTERNAL AND EXTERNAL MASS-TRANSFER COEFFICIENTS. The overall coefficient depends on the external coefficient and on an effective internal coefficient Diffusion within the particle is actually an unsteady-state process, and the value of decreases with time, as solute molecules must penetrate further and further into the particle to reach adsorption sites. An average coefficient can be used to give an approximate fit to uptake data for spheres ... 

Rate of leaching when diffusion in solid controls. In the case where unsteady-state diffusion in the solid is the controlling resistance in the leaching of the solute by an external solvent, the following approximations can be used. If the average diffusivity Da eff of the solute A is approximately constant, then for extraction in a batch process, unsteady-state mass-transfer equations can be used as discussed in Section 7.1. If the particle is approximately spherical. Fig. 5.3-13 can be used. 

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