Convective problems

Only a small number of solutions for the laminar forced convection problem and experimental investigations are available in the literature with some variations in the associated thermophysical properties. To the authors knowledge, for example, no experimental study is available to clarify the effect of the Prandtl number on the heat transfer in micro-channels with different duct geometries. 

The important reason for the quasi-steady-state approach arises from the difficulty in obtaining a solution to the transient convection problem for two-phase situations. 

This equation has been discussed by Nelson and Pasamehmetoglu (1992) relative to the application of the quasi-steady-state model for the convection problem. 

Combining hindered diffusion theory with the diffusion/convection problem in the model pore, Trinh et al. [399] showed how the effective transport coefficients depend upon the ratio of the solute to pore size. Figure 28 shows that as the ratio of solute to pore size approaches unity, the effective mobility function becomes very steep, thus indicating that the resolution in the separation will be enhanced for molecules with size close to the size of the pore. Similar results were found for the effective dispersion, and the implications for the separation of various sizes of molecules were discussed by Trinh et al. [399]. 

The function P can be computed from either an analytical or a numerical representation of the flow field. In such a way, a 3-D convection problem is essentially reduced to a mapping between two-dimensional Poincare sections. In order to analyze the growth of interfacial area in a spatially periodic mixer, the initial distri-... 

We see that the models which best reproduce the location of all the six data points are the tracks which do not fit the solar location. The models whose convection is calibrated on the 2D simulation make a poor job, as the FST models and other models with efficient convection do therefore this result can not be inputed to the fact that we employ local convection models. A possibility is that we are in front of an opacity problem, more that in front of a convection problem. Actually we would be inclined to say that opacities are not a problem (we have shown this in Montalban et al. (2004), by comparing models computed with Heiter et al (2002) or with AH97 model atmospheres), but something can still be badly wrong, as implied by the recent redetermination of solar metallicity (Asplund et al., 2004). A further possibility is that the inefficient convection in PMS requires the introduction of a second parameter -linked to the stellar rotation and magnetic field, as we have suggested in the past (Ventura et al., 1998 D Antona et al., 2000), but this remains to be worked out. 

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. 

The breakthrough curve for different values of the Courant number is given in Figure E7.3.1. A lower Courant number, less than 1, adds more numerical diffusion to the solution. If the Courant number is greater than 1, the solution is unstable. This Cou > 1 solution is not shown in Figure E7.3.1 because it dwarfs the actual solution. Thus, for a purely convective problem, the Courant number needs to be close to 1, but not greater than 1, for an accurate solution. In addition to the value of the Courant number, the amount of numerical diffusion depends on the value of the term UAz, which is the topic discussed next. 

Central differences are applied to diffusion problems, and upwind differences are applied to convective problems, but most cases have both diffusion and convection. This conundrum led Spaulding (1972) to develop exponential differences, which combines both central and upwind differences in an analytical solution of steady, one-dimensional convection and diffusion. Consider a control volume of length Ax, in a flow fleld of velocity U, and transporting a compound, C, at steady state with a diffusion coefficient, D. Then, the governing equation inside of the control volume is a simphflcation of Equation (2.14) ... 

Thus far diffusion in nonflow systems has been discussed. We now turn our attention to forced-convection problems. Only steady state problems are considered here, and it is assumed that they can all be described by the differential equation [see Eq. (50)]... 

The exact treatment yields expressions which have the same form as the expressions given above only the numerical factors are different. The more detailed theory for the diffusion-convection problem between plane walls was developed by Furry, Jones, and Onsager (F10) and that for the column constructed from two concentric cylinders by Furry and Jones (Fll). Recently more attention has been given to the r61e of the temperature dependence of the transport coefficients in column operation (B9, S15). 

Solution of forced convection problems for flow in non-Newtonian systems. 

The electric field is strongest at the center of the conducting spot and vanishes on the symmetry cell boundary at the membrane. A circulation is expected to result with a typical vortex size4 Rb- The appropriate electro-convective problem will be formulated in 6.5. 

This example is motivated by a natural-convection problem (Fig. 3.13) where the body-force term is caused by slight density variations (often caused by temperature variations). Using the so-called Boussinesq approximation, the flow may be considered incompressible, but with the buoyant forces depending on slight density variations. 

TDFRS allows for experiments on a micro- to mesoscopic length scale with short subsecond diffusion time constants, which eliminate almost all convection problems. There is no permanent bleaching of the dye as in related forced Rayleigh scattering experiments with photochromic markers [29, 30] and no chemical modification of the polymer. Furthermore, the perturbations are extremely weak, and the solution stays close to thermal equilibrium. 

To demonstrate Pawlowski s matrix transformation technique, an example will be used in which a forced convection problem, where a fluid with a viscosity p, a density p, a specific heat Cp and a thermal conductivity k, is forced past a surface with a characteristic size D at an average speed u. The temperature difference between the fluid and the surface is described by AT = Tf — Ts and the resulting heat transfer coefficient is defined by h. 

Note that the above approximation is a first order approximation. If we were to use a central difference, we would increase the order, but contrary to what is expected, this choice will adversely affect the accuracy and stability of the solution due to the fact the information is forced to travel in a direction that is not supported by the physics of the problem. How convective problems are dealt will be discussed in more detail later in this chapter. The following sections will present steady state, transient and moving boundary problems with examples and applications. 

In this Section so far, ADM is used to solve theoretical generalized models in the forms of ordinary differential equations). For diffusion-convection problems, the distributions along the axial direction of the packed bed electrode were neglected in certain cases, and mass transfer in the three dimensional electrodes were characterized by an average coefficient kh... 

We will discuss the solution of steady-state and unsteady-state heat conduction problems in this chapter, using the finite-difference method.. The finite-difference method comprises the replacement of the governing equations and corresponding boundary conditions by a set of algebraic equations. The discussion here is not meant to be exhaustive in its mathematical rigor. The basics are presented, and the solution of the finite-difference equations by numerical methods are discussed. The solution of convection problems using the finite-difference method is discussed in a later chapter. 

In problems of heat convection, the most complex equations to solve are the fluid flow equations. Often times, the governing equations for the fluid flow are the Navier-Stokes equations. It is useful, therefore, to study a model equation that has similar characteristics to the Navier-Stokes equations. This model equation has to be time-dependent and include both convection and diffusion terms. The viscous Burgers equation is an appropriate model equation. In the first few sections of this chapter, several important numerical schemes for the Burgers equation will be discussed. A simple physical heat convection problem is solved as a demonstration. 

Nakayama, A., A Unified Theory for Non-Darcy Free, Forced, and Mixed Convection Problems Associated with a Horizontal Line Heat Source in a Porous Medium , J. Heat Transfer, Vol. 116, pp. 508-513, 1994. 

Huang and Ozisik [33] solved the inverse forced convection problem that is formulated as follows ... 

While Huang and Ozisik solved the spacewise variation of wall heat flux for laminar forced convection problem, Silva Neto and Ozisik [57] used the conjugate gradient method and the adjoint equation simultaneously to solve for the timewise-varying strength of a two plane heat source. 

Thomas K. Sherwood and C. E. Reed, Applied Mathematics in Chemical Engineering, McGraw-Hill, New York, 1939 William R. Marshall and Robert L. Pigford, The Application of Differential Equations to Chemical Engineering Problems, University of Delaware, Newark, 1947 A. B. Newman, Temperature Distribution in Internally Heated Cylinders, Trans. AlChE 24,44-53 (1930) T. B. Drew, Mathematical Attacks on Forced Convection Problems A Review, Trans. AlChE 26,26-79 (1931) Arvind Varma, Some Historical Notes on the Use of Mathematics in Chemical Engineering, pp. 353-387 in W. F. Furter, ed., A Century of Chemical Engineering [17]. 

We may summarize our introductory remarks very simply. Heat transfer may take place by one or more of three modes conduction, convection, and radiation. It has been noted that the physical mechanism of convection is related to the heat conduction through the thin layer of fluid adjacent to the heat-transfer surface. In both conduction and convection Fourier s law is applicable, although fluid mechanics must be brought into play in the convection problem in order to establish the temperature gradient. 

The discussion and analyses of Chap. 5 have shown how forced-convection heat transfer may be calculated for several cases of practical interest the problems considered, however, were those which could be solved in an analytical fashion. In this way, the principles of the convection process and their relation to fluid dynamics were demonstrated, with primary emphasis being devoted to a clear understanding of physical mechanism. Regrettably, it is not always possible to obtain analytical solutions to convection problems, and the individual is forced to resort to experimental methods to obtain design information, as well as to secure the more elusive data which increase the physical understanding of the heat-transfer processes. 

The calculation of laminar heat-transfer coefficients is frequently complicated by the presence of natural-convection effects which are superimposed on the forced-convection effects. The treatment of combined forced- and free-convection problems is discussed in Chap. 7. 

Even though the fluid motion is the result of density variations, these variations are quite small, and a satisfactory solution to the problem may be obtained by assuming incompressible flow, that is, p = constant. To effect a solution of the equation of motion, we use the integral method of analysis similar to that used in the forced-convection problem of Chap. 5. Detailed boundary-layer analyses have been presented in Refs. 13, 27, and 32. 

As in the integral analysis for forced-convection problems, we assume that the velocity profiles have geometrically similar shapes at various x distances along the plate. For the free-convection problem, we now assume that the velocity may be represented as a polynomial function of y multiplied by some arbitrary function of x. Thus,... 

The foregoing analysis of free-convection heat transfer on a vertical flat plate is the simplest case that may be treated mathematically, and it has served to introduce the new dimensionless variable, the Grashof number, which is important in all free-convection problems. But as in some forced-convection problems, experimental measurements must be relied upon to obtain relations for heat transfer in other circumstances. These circumstances are usually those in which it is difficult to predict temperature and velocity profiles analytically. Turbulent free convection is an important example, just as is turbulent forced convection, of a problem area in which experimental data are necessary however, the problem is more acute with free-convection flow systems than with forced-convection systems because the velocities are usually so small that they are very difficult to measure. Despite the experimental difficulties, velocity measurements have been performed using hydrogen-bubble techniques [26], hot-wire anemometry [28], and quartz-fiber anemometers. Temperature field measurements have been obtained through the use of the Zehnder-Mach interferometer. The laser anemometer [29] is particularly useful for free-convection measurements because it does not disturb the flow field. 

A number of references treat the various theoretical and empirical aspects of free-convection problems. One of the most extensive discussions is given by Gebhart [13], and the interested reader may wish to consult this reference for additional information. 

The characteristic dimension to be used in the Nusselt and Grashof numbers depends on the geometry of the problem. For a vertical plate it is the height of the plate L for a horizontal cylinder it is the diamter d and so forth. Experimental data for free-convection problems appear in a number of references, with some conflicting results. The purpose of the sections that follow is to give these results in a summary form that may be easily used for calculation purposes. The functional form of Eq. (7-25) is used for many of these prer sentations, with the values of the constants C and m specified for each case. 

We treat this problem as one with constant heat flux on the surface. Since we do not know the surface temperature, we must make an estimate for determining 7 and the air properties. An approximate value of h for free-convection problems is 10 W/m2 DC, and so, approximately,... 

Most free-convection data are collected under laboratory conditions in still air, still water, etc. A practical free-convection problem might not be so fortunate and the boundary layer could have a slightly added forced-convection effect. In addition, real surfaces in practice are seldom isothermal or constant heat flux so the correlations developed from laboratory data for these conditions may not strictly apply. The net result, of course, is that the engineer must realize that calculated values of the heat-transfer coefficient can vary 25 percent from what will actually be experienced. 

Why is an analytical solution of a free-convection problem more involved than its forced-convection counterpart ... 

Raithby, G. D., and K. G. T.. Hollands A General Method of Obtaining Approximate Solutions to Laminar and Turbulent Free Convection Problems, Advances in Heat Transfer, Academic Press, New York, 1974. 

Since ratiation heat-transfer problems are often very closely associated with convection problems, and the total heat transfer by both convection and radiation is often the objective of an analysis, it is worthwhile to put both processes on a common basis by defining a radiation heat-transfer coefficient ft, as... 

As in the many convection problems we have encountered previously, the solution to a complicated problem of this sort is frequently obtained by appealing to carefully controlled measurements in search of an empirical relationship to predict evaporation rates. 

The upstream propagating mode properties are indicated by the dotted line in Fig. 2.25 and this component of the solution can be seen in the solution shown in Figs. 2.34 and 2.35. Here, we demonstrate these aspects further with respect to the case shown in Fig. 2.34, for which the freestream vortices travel with the freestream speed, c = Uoo, for the pure convection problem. In Fig. 2.30, the gap between successive vortices has been taken as a = IOOtt and that corresponds to a time scale given by wq = = 0.02. 

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