Schmidt numerical solution

If we significantly reduce the dimension of the Z axis, then we transform the three-dimensional cooling problem into an unsteady state and monodimensional problem. Figures 3.50 and 3.51 show the results of the simulations oriented to demonstrate this fact. We can notice that all curves present the same tendency as the analytical solution or Schmidt numerical solution of the monodimensional cooling problem ... 

A similar development was provided by Tribollet and Newman for electro-hydrodynamic impedance. The use of look-up tables facilitates regression of models to experimental data that take full accoimt of the influence of a finite Schmidt number on the convective-diffusion impedance. Use of only the first term in equation (11.97) yields a numerical solution for an infinite Schmidt number. Tribollet and Newman report use of the first two terms in equation (11.97) The low level of stocheistic noise in experimental data justifies use of the three-term expansion reported here. 

Numerical solution of the mass transfer equation begins at a small nonzero value of z = Zstart, uot at the inlet where Cp, x, y,z = 0) = Ca, miet for all values of x and y. This is achieved by invoking an asymptotically exact analytical solution for the molar density of reactant A from laminar mass transfer boundary layer theory in the limit of very large Schmidt and Peclet numbers. The boundary layer starting profile is valid under the following condition ... 

Jameson A, Schmidt W, Turkel E. Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta timestepping schemes. In AIAA 14th fluid and plasma dynamics conference, Palo Alto, California. AIAA Paper 81-1259 1981. 

It is evident from these discussions that population balance equations are important in the description of dispersed-phase systems. However, they are still of limited use because of difficulties in obtaining solutions. In addition to the numerical approaches, solution of the scalar problem has been via the generation of moment equations directly from the population balance equation (H2, H17, R6, S23, S24). This approach has limitations. Ramkrishna and co-workers (H2, R2, R6) presented solutions of the population balance equation using the method of weighted residuals. Trial functions used were problem-specific polynomials generated by the Gram-Schmidt orthogonalization process. Their approach shows promise for future applications. 

Equation (2-37) may be solved numerically for different values of the Schmidt number. One important consequence of this solution is that for Sc > 0.6, results for the surface concentration gradient may be correlated by... 

Maiiy numerical and series solutions for the laminar boundary layer model of mass transfer are available for situations such as flow in conduits under conditions of fully developed or developing concentration or velocity profiles. Skelland provides a particularly good summary of these results. The laminar boundary layer tno l has been extended to predict the effects of high mass transfer flux on the mass transfer coefficient frm a flat plate. The results of this work are shown in Fig. 2.4-2 and. in contrast to the other theories, indicate a Schmidt number dependence of the correction factor. 

Byrne,G.D.(1992) The solution of a co-polymerization problem. In Byrne,G.D. and Schiesser,W.E. (Eds.), Recent developments in numerical methods and software for ODEs/DAEs/PDEs. World Scientific, River Edge, NJ, 137-197. Engelmann,U., Schmidt-Naake,G., Maier,J., Seifert,P. (1993) Einhufi der Mikromischung auf die radikalische Polymerisation im diskontinuierlichen Prozefi. Preprint, TU Dresden, NM-TC-8-93. 

The condition expressed by Equation 8.2 is usually fulfilled and channel flow occurs in the laminar range. This allows the problem to be treated analytically and the solutions presented in this chapter rely on this assumption. Accounting for the simultaneous development of the velocity profile compared with concentration/temperature fields requires numerical evaluation, and the importance of this effect is measured by Prandtl s number for heat transfer or by Schmidt s number for mass transfer ... 

---