Convection numerical solution

The finite element results obtained for various values of (3 are compared with the analytical solution in Figure 2.27. As can be seen using a value of /3 = 0.5 a stable numerical solution is obtained. However, this solution is over-damped and inaccurate. Therefore the main problem is to find a value of upwinding parameter that eliminates oscillations without generating over-damped results. To illustrate this concept let us consider the following convection-diffusion equation... 

Morton, K. W., 1996. Numerical Solution of Convection Diffusion Problems, Chapman Hall, London. 

An attempt has been made by Johnson and co-workers to relate such theoretical results with experimental data for the absorption of a single carbon dioxide bubble into aqueous solutions of monoethanolamine, determined under forced convection conditions over a Reynolds number range from 30 to 220. The numerical results were found to be much higher than the measured values for noncirculating bubbles. The numerical solutions indicate that the mass-transfer rate should be independent of Peclet number, whereas the experimentally measured rates increase gradually with increasing Peclet number. The discrepancy is attributed to the experimental technique, where-... 

Fig. 10. Numerical solutions of the forced-convection mass-transfer equation for the case of irreversible first-order chemical reaction [after Johnson et al. (J4)] (Solid lines— rigid spheres dashed lines—circulating gas bubbles).

The convective diffusion equations presented above have been used to model tablet dissolution in flowing fluids and the penetration of targeted macro-molecular drugs into solid tumors [5], In comparison with the nonequilibrium thermodynamics approach described below, the convective diffusion equations have the advantage of theoretical rigor. However, their mathematical complexity dictates a numerical solution in all but the simplest cases. 

In the general case, when arbitrary interaction profiles prevail, the particle deposition rate must be obtained by solving the complete transport equations. The first numerical solution of the complete convective diffusional transport equations, including London-van der Waals attraction, gravity, Brownian diffusion and the complete hydrodynamical interactions, was obtained for a spherical collector [89]. Soon after, numerical solutions were obtained for a panoplea of other collector geometries... 

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 numerical jet model [9-11] is based on the numerical solution of the time-dependent, compressible flow conservation equations for total mass, energy, momentum, and chemical species number densities, with appropriate in-flow/outfiow open-boundary conditions and an ideal gas equation of state. In the reactive simulations, multispecies temperature-dependent diffusion and thermal conduction processes [11, 12] are calculated explicitly using central difference approximations and coupled to chemical kinetics and convection using timestep-splitting techniques [13]. Global models for hydrogen [14] and propane chemistry [15] have been used in the 3D, time-dependent reactive jet simulations. Extensive comparisons with laboratory experiments have been reported for non-reactive jets [9, 16] validation of the reactive/diffusive models is discussed in [14]. 

In Fig. 10.4 the sphere diameter, terminal velocity, and temperature difference each appear in only one dimensionless group. The effect of natural convection on is smaller at Pr = 10 because the region over which the buoyancy force acts is much thinner than for Pr = 1. As Pr oo the effect should disappear altogether. For Pr = 0, numerical solutions (W7) show effects about 50% larger than for Pr = 1. 

Koutecky obtained the numerical solution for the catalytic current at a DME by rigorous solution of the convective-diffusion equations by the expanding-plane model with =0 [199]. Subsequently, an approximate analytical solution was obtained which holds over the whole range of k j with an error of not more than 1% [188]. The equations are... 

We have assumed above that the rate constant of the homogeneous reaction, fe2, is infinitely large so that the boundary between the A2 and X-dominated zones is clearly defined. When k2 is smaller, this boundary will be less clear. The modified convective-diffusion equations are difficult to solve and numerical solution is necessary. This has been performed at the RRDE making the assumptions that [221, 222]... 

First, the role of system design on the details of convection and solute segregation in industrial-scale crystal growth systems has not been adequately studied. This deficiency is mostly because numerical simulations of the three-dimensional, weakly turbulent convection present in these systems are at the very limit of what is computationally feasible today. New developments in computational power may lift this limitation. Also, the extensive use of applied magnetic fields to control the intensity of the convection actually makes the calculations much more feasible. 

As indicated in Table 7.10, only in the last decade have models considered all three phenomena of heat transfer, fluid flow, and hydrate dissociation kinetics. The rightmost column in Table 7.10 indicates whether the model has an exact solution (analytical) or an approximate (numerical) solution. Analytic models can be used to show the mechanisms for dissociation. For example, a thorough analytical study (Hong and Pooladi-Danish, 2005) suggested that (1) convective heat transfer was not important, (2) in order for kinetics to be important, the kinetic rate constant would have to be reduced by more than 2-3 orders of magnitude, and (3) fluid flow will almost never control hydrate dissociation rates. Instead conductive heat flow controls hydrate dissociation. 

Stone, H. L., and Brian, P. L. T. (1963). Numerical solution of convective transport problems. American Institute of Chemical Engineering Journal, 9(5), 681-688. 

The overall clearance of hemofiltration is more difficult to calculate than the diffusive clearance or the HF clearance, as it combines diffusive and convective transfers. An approximate equation for this clearance, obtained from an exact numerical solution has been given by Jaffrin et al. [12] as... 

Temperature profiles can be determined from the transient heat conduction equation or, in integral models, by assuming some functional form of the temperature profile a priori. With the former, numerical solution of partial differential equations is required. With the latter, the problem is reduced to a set of coupled ordinary differential equations, but numerical solution is still required. The following equations embody a simple heat transfer limited pyrolysis model for a noncharring polymer that is opaque to thermal radiation and has a density that does not depend on temperature. For simplicity, surface regression (which gives rise to convective terms) is not explicitly included. 

Stagnant Flow situation considered in the discussion Fluid at 0f numerical solution of laminar natural Temp. T convection. 

NUMERICAL SOLUTION OF THE NATURAL CONVECTIVE BOUNDARY LAYER EQUATIONS... 

A numerical solution to the laminar boundary layer equations for natural convection can be obtained using basically the same method as applied to forced convection in Chapter 3. Because the details are similar to those given in Chapter 3, they will not be repeated here. 

This is Fick s second diffusion equation [242], an adaptation to diffusion of the heat transfer equation of Fourier [253]. Technically, it is a second-order parabolic partial differential equation (pde). In fact, it will mostly be only the skeleton of the actual equation one needs to solve there will usually be such complications as convection (solution moving) and chemical reactions taking place in the solution, which will cause concentration changes in addition to diffusion itself. Numerical solution may then be the only way we can get numbers from such equations - hence digital simulation. 

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