Convection first problem

The first step in solving convective diffusion problems is the derivation of the velocity profile. In this case, the flow arriving with velocity v is modified by... 

The present section reviews the concepts behind the Generalized Integral Transform Technique (GITT) [35-40] as an example of a hybrid method in convective heat transfer applications. The GITT adds to tiie available simulation tools, either as a companion in co-validation tasks, or as an alternative approach for analytically oriented users. We first illustrate the application of this method in the full transformation of a typical convection-diffusion problem, until an ordinary differential system is obtained for the transformed potentials. Then, the more recently introduced strategy of... 

The LSMs are thus generally best suited solving convection dominated problems. However, in reactor engineering a few successful attempts have been made applying this method to solve convection-diffusion problems, by reformulating the second order derivatives into a set of two first order derivative equations. 

The first problem considered is the classic problem of Rayleigh-Benard convection -namely the instability that is due to buoyancy forces in a quiescent fluid layer that is heated... 

The first problem that we consider involves the same rapid evaporation that was used as the key example in Section 3.1. We recall that at intermediate temperatures, the evaporation rate depends on both diffusion and convection up the tube. 

The first order derivative in Equation (2.80) corresponds to the convection in a field problem and the examples shown in Figure 2.26 illustraTes the ina bility of the standard Galerkin method to produce meaningful results for convection-dominated equations. As described in the previous section to resolve this difficulty, in the solution of hyperbolic (convection-dominated) equations, upwind-ing or Petrov-Galerkin methods are employed. To demonstrate the application of upwinding we consider the case where only the weight function applied to the first-order derivative in the weak variational statement of the problem, represented by Equation (2.82), is modified. 

In many of the applications of heat transfer in process plants, one or more of the mechanisms of heat transfer may be involved. In the majority of heat exchangers heat passes through a series of different intervening layers before reaching the second fluid (Figure 9.1). These layers may be of different thicknesses and of different thermal conductivities. The problem of transferring heat to crude oil in the primary furnace before it enters the first distillation column may be considered as an example. The heat from the flames passes by radiation and convection to the pipes in the furnace, by conduction through the... 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

This problem may look at first to be the same geometry as for diffusion in a single pore, but this situation is quite drSerent hr the single pore we had reaction controlled by diffusion down the pore, while in the tube wall reactor we have convection of reactants down the tube. 

Brenner (B6) pointed out that similar problems arise in obtaining Eq. (3-44) as in the low Re approximation for fluid flow. The neglected convection terms dominate far from the particle, since the ratio of convective to diffusive terms is 0[Pe(r/a)]. An asymptotic solution to Eq. (3-39) with Pe 0 was therefore obtained by the matching procedure of Proudman and Pearson discussed above. Brenner s result for the first term in a series expansion for Sh may be written ... 

The solution of Example 7.3 will be compared with an analytical solution of a diffusive front moving at velocity U, with D = 1/2U Az. First, we must derive the analytical solution. This problem is similar to Example 2.10, with these exceptions (1) convection must be added through a moving coordinate system, similar to that described in developing equation (2.36), and (2) a diffusion gradient will develop in both the +z-and -z-directions. 

Problems in forced convection are solved in two steps first one solves the equation of motion to obtain the velocity distribution, and then one puts the expression for v back into the diffusion equation [usually as given in Eq. (50)]. An illustration of this type of problem is that of the absorption of a gas by a liquid film flowing down a... 

Equations 15 and 16 were first obtained by Kegeles and his co-workers (16, 20, 21) and they are valid only for cells with sector-shaped center-pieces. Because of convective disturbances in the solution column of the ultracentfifuge cell during the transient state when nonsector-shaped cells are used, it is customary to do Archibald experiments in cells with sectorshaped centerpieces which avoid this problem. For details on the Archibald method—pitfalls, extrapolation to zero time—one should consult the papers of LaBar (17) and Fujita et al. (22). 

Actual measurements of reduced gaseous diffusion across certain compressed surfactant monolayers, at an air/water interface, have been reported in detail by numerous investigators in the past (ref. 115-120). For such measurements to be completely trustworthy, it is first necessary to eliminate convection in the bulk (aqueous) phase, since the monolayer can reduce the rate of gas absorption by reducing convection at the surface, and this has a greater effect than a diffusion barrier (ref. 116). The problem of... 

The balance equations used to model polymer processes have, for the most part, first order derivatives in time, related with transient problems, and first and second order derivatives in space, related with convection and diffusive problems, respectively. Let us take the heat equation over an infinite domain as... 

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. 

Problem definition requires specification of the initial state of the system and boundary conditions, which are mathematical constraints describing the physical situation at the boundaries. These may be thermal energy, momentum, or other types of restrictions at the geometric boundaries. The system is determined when one boundary condition is known for each first partial derivative, two boundary conditions for each second partial derivative, and so on. In a plate heated from ambient temperature to 1200°F, the temperature distribution in the plate is determined by the heat equation 8T/dt = a V2T. The initial condition is T = 60°F at / = 0, all over the plate. The boundary conditions indicate how heat is applied to the plate at the various edges y = 0, 0[Pg.86]

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