Convection general discussion

Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. Journal of Fluid Mechanics 5, 113-133. 

Diffusion problems in systems involving forced and free convection are good illustrations of the importance of presenting all three of the equations of change as a prelude to a general discussion of diffusion. Only a handful of idealized problems of this type have been solved analytically. Since they are, however, of considerable importance in chemical engineering it is worth while to make some general remarks about them. 

Eckert, E.R.G. and Soehnghen. E., "Interferometric Studies on the Stability and Transition to Turbulence of a Free-Convection Boundary Layer , Proc. of the General Discussion on Heat Transfer, pp. 321-323. ASME-1ME. London. 1951. 

So far we presented some general discussions on boiling. Now we turn our attention to the physical mechanisms involved in pool boiling, that is, the boiling of stationary fluids. In pool boiling, the fluid is not forced to flow by a mover such as a pump, and any motion of the Iluid is due to natural convection currents and Ihe motion of the bubbles under the influence of buoyancy. 

Prior to a general discussion on specifics of thermophysical properties and forced convective heat transfer at critical and supercritical pressures, it is important to define special terms and expressions used at these conditions. For a better understanding of these terms and expressions, their definitions are listed below together with complementary (Figs. A3.1—A3.4). 

Basara B, Alajbegovic A, Beader D (2004) Simulation of single- and two-phase flows on sliding unstructured meshes using finite volume method. Int J Numer Meth 45 1137-1159 Batchelor GK (1959) SmaU-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. J Ruid Mech... 

In the previous section we discussed wall functions, which are used to reduce the number of cells. However, we must be aware that this is an approximation that, if the flow near the boundary is important, can be rather crude. In many internal flows—where all boundaries are either walls, symmetry planes, inlets, or outlets—the boundary layer may not be that important, as the flow field is often pressure determined. However, when we are predicting heat transfer, it is generally not a good idea to use wall functions, because the convective heat transfer at the walls may be inaccurately predicted. The reason is that convective heat transfer is extremely sensitive to the near-wall flow and temperature field. 

Because the mechanisms governing mass transfer are similar to those involved in both heat transfer by conduction and convection and in momentum transfer (fluid flow), quantitative relations exist between the three processes, and these are discussed in Chapter 12. There is generally more published information available on heat transfer than on mass transfer, and these relationships often therefore provide a useful means of estimating mass transfer coefficients. 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

The transfer of heat by radiation in general can be said to occur simultaneously with heat transfer by convection and conduction. Transfer by radiation tends to become more important than that by the other two mechanisms as the temperature increases. It is useful to gain an appreciation of the basic definitions of the energy flux terms, the surface property terms and their relationships while discussing radiative heat transfer. With this objective, reference may be made to Table 3.4 in which these are presented. 

With this equation, we can now discuss a generalized mass balance equation. We still use Figure 1 to show the derivation. Based on Eq. (5), the net contribution by diffusion and convection now becomes... 

Membrane transport represents a major application of mass transport theory in the pharmaceutical sciences [4], Since convection is not generally involved, we will use Fick s first and second laws to find flux and concentration across membranes in this section. We begin with the discussion of steady diffusion across a thin film and a membrane with or without aqueous diffusion resistance, followed by steady diffusion across the skin, and conclude this section with unsteady membrane diffusion and membrane diffusion with reaction. 

Some transition times calculated for this type of free convection, following a concentration step in 0.05 M CuS04 solution with excess H2S04, are given in Table IV. It can be seen that the transition times (to a flux 1°() in excess of the steady-state flux) vary appreciably along the plate also in forced convection (which is discussed below) the transition times are generally shorter, except at very low flow rates. 

As a general observation on free-convection limiting currents, it should be remembered that in solutions where the driving force is small, several experimenters (B9, F3) have found it difficult to obtain satisfactory limiting-current plateaus. This question has been discussed in Section IV,D. Caution is justified in interpreting results at low Ra values. 

The advection—diffusion equation with a source term can be solved by CFD algorithms in general. Patankar provided an excellent introduction to numerical fluid flow and heat transfer. Oran and Boris discussed numerical solutions of diffusion—convection problems with chemical reactions. Since fuel cells feature an aspect ratio of the order of 100, 0(100), the upwind scheme for the flow-field solution is applicable and proves to be very effective. Unstructured meshes are commonly employed in commercial CFD codes. 

Cleland and Wilhelm (C18) used a finite-difference technique which could be used for nonlinear reactions, but they limited their study to a first-order reaction. Experiments were also performed to test the results of the theory. In a small reaction tube, the two checked quite well. In a large tube there were differences which were explained by consideration of natural convection effects which were due to the fact that completely isothermal conditions were not maintained. This seems to be the only experimental data in the literature to date, and shows another area in which more work is needed. The preceding discussion considered only isothermal conditions except for Chambre (C12) who presented a general method for nonisothermal reactors. 

Here we review some of the correlations of convective mass transfer. We will find that many reactors are controlled by mass transfer processes so this topic is essential in describing many chemical reactors. This discussion will necessarily be very brief and qualitative, and we win summarize material that most students have encountered in previous courses in mass transfer. Our goal is to write down some of the simple correlations so we can work examples. The assumptions in and validity of particular expressions should of course be checked if one is interested in serious estimations for particular reactor problems. We will only consider here the mass transfer correlations for gases because for liquids the correlations are more comphcated and cannot be easily generalized. 

In general, it can be very difficult to determine the nature of the boundary terms. A specific result in an exactly solvable case is discussed in Section IV.A.2. Equation (55) is the Gallavotti-Cohen FT derived in the context of deterministic Anosov systems [28]. In that case, Sp stands for the so-called phase space compression factor. It has been experimentally tested by Ciliberto and co-workers in Rayleigh-Bemard convection [52] and turbulent flows [53]. Similar relations have also been tested in athermal systems, for example, in fluidized granular media [54] or the case of two-level systems in fluorescent diamond defects excited by light [55]. 

Of the three general categories of transport processes, heat transport gets the most attention for several reasons. First, unlike momentum transfer, it occurs in both the liquid and solid states of a material. Second, it is important not only in the processing and production of materials, but in their application and use. Ultimately, the thermal properties of a material may be the most influential design parameters in selecting a material for a specific application. In the description of heat transport properties, let us limit ourselves to conduction as the primary means of transfer, while recognizing that for some processes, convection or radiation may play a more important role. Finally, we will limit the discussion here to theoretical and empirical correlations and trends in heat transport properties. Tabulated values of thermal conductivities for a variety of materials can be found in Appendix 5. 

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