Unsteady State Approaches

A model [15] based on the schematic shown in Fig. 9-19 was used to analyze the heating of the polymer. The model used an unsteady-state approach with an effective thermal diffusivity (aeffective)-... 

There are two basic approaches to measuring the thermal conductivity of materials. The most basic approadi k a steady state a q>roach based on Fourier s definition of thermal conductivity. Steady state me uements often require a long time to reach equilibrium and, as a result, several unsteady state approaches have been developed that are based on the rate of change of temperature of a material suddenly subjected to a new thermal environment. 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

In this approach, heat transfer to a spherical particle by conduction through the surrounding fluid has been the prime consideration. In many practical situations the flow of heat from the surface to the internal parts of the particle is of importance. For example, if the particle is a poor conductor then the rate at which the particulate material reaches some desired average temperature may be limited by conduction inside the particle rather than by conduction to the outside surface of the particle. This problem involves unsteady state transfer of heat which is considered in Section 9.3.5. 

Figures 9.17-9.19 clearly show that, as the Biot number approaches zero, the temperature becomes uniform within the solid, and the lumped capacity method may be used for calculating the unsteady-state heating of the particles, as discussed in section (2). The charts are applicable for Fourier numbers greater than about 0.2.

The general approach to the solution of unsteady-state problems is illustrated in Example 2.15. Batch distillation is a further example of an unsteady-state material balance (see Volume 2, Chapter 11). 

Equation based programs in which the entire process is described by a set of differential equations, and the equations solved simultaneously not stepwise, as in the sequential approach. Equation based programs can simulate the unsteady-state operation of processes and equipment. 

With the exception of this method, all the methods described solve the stage equations for the steady-state design conditions. In an operating column other conditions will exist at start-up, and the column will approach the design steady-state conditions after a period of time. The stage material balance equations can be written in a finite difference form, and procedures for the solution of these equations will model the unsteady-state behaviour of the column. 

Propagation problems. These problems are concerned with predicting the subsequent behavior of a system from a knowledge of the initial state. For this reason they are often called the transient (time-varying) or unsteady-state phenomena. Chemical engineering examples include the transient state of chemical reactions (kinetics), the propagation of pressure waves in a fluid, transient behavior of an adsorption column, and the rate of approach to equilibrium of a packed distillation column. 

As discussed before, in a strict sense, there is always some degree of dependence between the sample data. An alternative approach is to make use of the covariance matrix of the constraint residuals to eliminate the dependence between sample data (or the influence of unsteady-state behavior of the process during sampling periods). This is the basis of the so-called indirect approach. 

Since fluid shear rates vary enormously across the radius of a capillary tube, this type of instrument is perhaps not well suited to the quantitative study of thixotropy. For this purpose, rotational instruments with a very small clearance between the cup and bob are usually excellent. They enable the determination of hysteresis loops on a shear-stress-shear-rate diagram, the shapes of which may be taken as quantitative measures of the degree of thixotropy (G3). Since the applicability of such loops to equipment design has not yet been shown, and since even their theoretical value is disputed by other rheologists (L4), they are not discussed here. These factors tend to indicate that the experimental study of flow of thixotropic materials in pipes might constitute the most direct approach to this problem, since theoretical work on thixotropy appears to be reasonably far from application. Preliminary estimates of the experimental approach may be taken from the one paper available on flow of thixotropic fluids in pipes (A4). In addition, a recent contribution by Schultz-Grunow (S6) has presented an empirical procedure for correlation of unsteady state flow phenomena in rotational viscometers which can perhaps be extended to this problem in pipe lines. 

In the emulsion phase/packet model, it is perceived that the resistance to heat transfer lies in a relatively thick emulsion layer adjacent to the heating surface. This approach employs an analogy between a fluidized bed and a liquid medium, which considers the emulsion phase/packets to be the continuous phase. Differences in the various emulsion phase models primarily depend on the way the packet is defined. The presence of the maxima in the h-U curve is attributed to the simultaneous effect of an increase in the frequency of packet replacement and an increase in the fraction of time for which the heat transfer surface is covered by bubbles/voids. This unsteady-state model reaches its limit when the particle thermal time constant is smaller than the particle contact time determined by the replacement rate for small particles. In this case, the heat transfer process can be approximated by a steady-state process. Mickley and Fairbanks (1955) treated the packet as a continuum phase and first recognized the significant role of particle heat transfer since the volumetric heat capacity of the particle is 1,000-fold that of the gas at atmospheric conditions. The transient heat conduction equations are solved for a packet of emulsion swept up to the wall by bubble-induced circulation. The model of Mickley and Fairbanks (1955) is introduced in the following discussion. 

In conclusion to this section, research in the RTD area is always active and the initial concepts of Danckwerts are gradually being completed and extended. The population balance approach provides a theoretical framework for this generalization. However, in spite of the efforts of several authors, simple procedures, easy to use by practitioners, would still be welcome in the field of unsteady state systems (variable volumes and flow rates), multiple inlet/outlet reactors, variable density mixtures, systems in which the mass-flowrate is not conserved, etc... On the other hand, the promising "generalized reaction time distribution" approach could be developed if suitable experimental methods were available for its determination. 

An accurate design of unsteady-state catalytic processes requires knowledge about the catalyst behavior and reaction kinetics under unsteady-state conditions, unsteady-state mass and heat transfer processes in the catalyst particle and along the catalyst bed, and dynamic phenomena in the catalytic reactor. New approaches for reactor modeling and optimization become necessary. Together, these topics form a wide area of research that has been continuously developed since the 1960s. 

The general approach to the solution of unsteady-state problems is illustrated in Example 2.15. 

For a binary system, under conditions of small mass transfer fluxes, the unsteady-state diffusion equations may be solved to give the fractional approach to equilibrium F defined by (see Clift et al., 1978)... 

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