Semi-analytical solution

Although semi-analytical solutions are available in some cases [5], these are cumbersome and it is more usual to employ a numerical method. A simple example is presented below which illustrates the solution of the design equation for a batch reactor operated isothermally the adiabatic operation of the same system is then examined. 

Eqs. 3-4 are amenable to semi-analytical solution techniques because of the linear form. The use of more complex kinetic models (e.g., intraaggregate diffusion) has not been attempted, in part because the above models have proved adequate to describe the available data sets, and in part because of a limited understanding of the geometry of the soil/bentonite matrix (gel formation and the resulting diffusion geometry). 

Boundary effects on the electrophoretic migration of a particle with ion cloud of arbitrary thickness were also investigated by Zydney [46] for the case of a spherical particle of radius a in a concentric spherical cavity of radius d. Based on Henry s [19] method, a semi-analytic solution has been developed for the particle mobility, which is valid for all double layer thicknesses and all particle/pore sizes. Two integrals in the mobility expression must be evaluated numerically to obtain the particle velocity except for the case of infinite Ka. The first-order correction to the electrophoretic mobility is 0(A3) for thin double layer, whereas it becomes 0(A) for thick double layer. Here the parameter A is the ratio of the particle-to-cavity radii. The boundary effect becomes more significant because the fluid velocity decays as r l when the double layer spans the entire cavity. The stronger A dependence of the first order correction for thick double layer than that obtained by Ennis and Andersion [45] results from the fact that the double layers overlap in... 

A semi-analytical solution to these equations was derived by Dixon and Cresswell (16), who then matched the fluid phase temperature profile to the one-phase model profile to obtain explicit relations between the parameters of the two models. 

The mathematical problem presented by the model requires the simultaneous solution of equations (2) to (7). The starting point is a semi-analytical solution of equation (8) [I], previously only applied to single component systems, that satisfies the boundary conditions in equation (3) (4). 

Baukal, C. E., and Gebhart, B. "A Review of Semi-Analytic Solutions for Flame Impingement Heat Transfer." International Journal of Heat and Mass Transfer 39, no. 14 (1996) 2989-3002. 

Han, G., loannidis, M., and Dusseault, M.B. 2002. Semi-analytical solutions for the effect of well shut-down on rock stability. Proc Can Int Petrol Conf, Calgary, June Paper 2002-50,9 p. 

The general problem (163)-(166) can be solved numerically. Here we will present a semi-analytical solution in the limit of small/ / = —e, e [Pg.48]

For simple irreversible reactions a (semi) analytical solution of the continuity and energy equations is possible. Douglas and Eagleton [9] published solutions for zero-, first-, and second-order reactions, both with a constant and varying... 

Transfonnation processes are very important for the treatment of many organic contaminants with permeable reactive barriers. The components and their products undergo sorption reactions. This sorption reactions may be either in equilibrium or nonequilibrium. Desintegration of tetrachloroethene or trichloroethylene are two of many examples, that can be treated with this permeable reactive barriers. Khandelwal and Rabideau (1999) developed analytical and semi-analytical solutions for this problem. They consider the sorption reaction with a nonequdibrium model. We verified the numerical RF-RTM model with their analytical solution. 

SEMI-ANALYTICAL SOLUTION FOR PREDICTION OF WATER FLOW RATE INTO A LARGE CAVERN WITH HORSESHOE SECTION... 

FLAG 2D is used to validate whether the simplified linear relationship of equation (4) is reasonable for the horseshoed cross-section, and to calculate the constants of S and C. Then, the semi-analytical solution for water inflow prediction formula would... 

Thus, for this specific project, the semi-analytical solution for water inflow prediction in a subsea rock cavern with horseshoe-shaped section is established as equation (5), which will be easily and conveniently used to predict the water inflow into cavern during further excavation of this cavern or nearby caverns with similar sections and sizes and locations in the future. 

According to equation (5), water inflow rate data monitored during the 65 days after the whole cavern and gallery are fully excavated are used for model calibration and validation, as shown in Figure 6. It could be found that the calculated water inflow rate approximately equates to the monitored water inflow rate, which means the results got from the semi-analytical solution are acceptable and reliable. 

It should be pointed out that the presented approach offers an alternative for estimating the water inflow rate and other hydraulic parameters. At the early stage of the project, on the basis of limited geological survey data, the semi-analytical solution can offer a preliminary evaluation of water inflow. With more data collected during the continued construction, the semi-analytical solution could be utilized to further back analyze the hydrogeological parameters such as hydraulic conductivities, which will provide a more accurate estimation of water inflow in future. 

The theory for most systems involving coupled chemical reactions is rather compUcated. Analytical approximations are available only for a limited number of relatively simple processes. Semi-analytical solutions based on infinite series, integral equations, tabulated... 

A fully analytical solution to this equation is not possible, but a semi-analytical solution has been given by Douglas and Eagleton (1962). However, numerical solution is quite straightforward, as shown below, and appears to be the method of choice. The objective is to establish the temperature and concentration profiles in the reactor. For this purpose, we express V as the product (where is the cross-sectional area and... 

Gottifredi and Froment [1997] presented a straightforward and accurate semi-analytical solution for the concentration profiles inside a catalyst particle in the presence of coke formation. They applied the solution to the butene dehydrogenation dealt with here and obtained an excellent agreement with the profiles shown in Fig. 5.3.3.A-3. The method significantly simplifies and reduces the computational effort involved in reactor simulation and kinetic analysis. 

Dixon and Cresswell [12] gave a semi-analytical solution to these equations. If we confine ourselves to radial heat transfer in this discussion (i.e. set = 0), the fluid phase... 

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