Boundary conditions dispersion coefficient determination

The criterion for the validity of Equation 8-141 is Npg 1.0. A rough rule-of-thumb is Npg > 10. If this condition is not satisfied, the correct equation depends on the boundary conditions at the inlet and outlet. A procedure for determining dispersion coefficient Dg [ is as follows ... 

Now, if we are determining the dispersion coefficient through the use of a pulse tracer cloud, the boundary conditions are those of a Dirac delta ... 

The exact formulation of the inlet and outlet boundary conditions becomes important only if the dispersion number (DjuL) is large (> 0.01). Fortunately, when DjuL is small (< 0.01) and the C-curve approximates to a normal Gaussian distribution, differences in behaviour between open and closed types of boundary condition are not significant. Also, for small dispersion numbers DjuL it has been shown rather surprisingly that we do not need to have ideal pulse injection in order to obtain dispersion coefficients from C-curves. A tracer pulse of any arbitrary shape is introduced at any convenient point upstream and the concentration measured over a period of time at both inlet and outlet of a reaction vessel whose dispersion characteristics are to be determined, as in Fig. 2.18. The means 7in and fout and the variances and out for each of the C-curves are found. 

These coefficients can be evaluated for any biopolymer by taking an arbitrary chain length and determining the allowed values of frequencies for any set of boundary conditions. In our calculations we have assumed that the ends of the chain are fixed and we determine the frequency of the modes that permit an odd number of half wavelengths to be present on the chain. The eigenvectors for these frequencies are determined from the dispersion curves for an infinite chain. To make these calculations more compatible with experiment we have determined the absorption cross section which can be related to the Einstein coefficient by the following expression... 

The model is referred to as a dispersion model, and the value of the dispersion coefficient De is determined empirically based on correlations or experimental data. In a case where Eq. (19-21) is converted to dimensionless variables, the coefficient of the second derivative is referred to as the Peclet number (Pe = uL/De), where L is the reactor length and u is the linear velocity. For plug flow, De = 0 (Pe ) while for a CSTR, De = oo (Pe = 0). To solve Eq. (19-21), one initial condition and two boundary conditions are needed. The closed-ends boundary conditions are uC0 = (uC — DedC/dL)L=o and (dC/BL)i = i = 0 (e.g., see Wen and Fan, Models for Flow Systems in Chemical Reactors, Marcel Dekker, 1975). Figure 19-2 shows the performance of a tubular reactor with dispersion compared to that of a plug flow reactor. 

The boundary conditions (5.19) to (5.22) give us four equations in which the expressions for and are inserted through the relations (5.7), (5.8), (5.11), (5.17), and (5.18). The resulting four equations have nontrivial solutions for A to D, if the determinant of coefficients oi A io D is zero. In the case considered here, a film surrounded at both sides by the same medium, it appears that zero equating the determinant gives two relations between w, K, and the system variables the so-called dispersion relation " ... 

The system of Eqs. (68)-(70) are uniform differential equations of the 8th order. Its solution includes unknown coefficients which can be found from the boundary conditions of Eqs. (71)-(72). The nontrivial solution of such a system exists only when the characteristic determinant is zero similar to the case in the previous section. The latter condition is the dispersive ratio connecting the wave vector k, and the frequency CO, while the rest of the parameters are fixed. At the state of neutral stabihty (co=0) the... 

The problem to be solved in this paragraph is to determine the rate of spread of the chromatogram under the following conditions. The gas and liquid phases flow in the annular space between two coaxial cylinders of radii ro and r2, the interface being a cylinder with the same axis and radius rx (0 r0 < r < r2). Both phases may be in motion with linear velocity a function of radial distance from the axis, r, and the solute diffuses in both phases with a diffusion coefficient which may also be a function of r. At equilibrium the concentration of solute in the liquid, c2, is a constant multiple of that in the gas, ci(c2 = acj) and at any instant the rate of transfer across the interface is proportional to the distance from equilibrium there, i.e. the value of (c2 - aci). The dispersion of the solute is due to three processes (i) the combined effect of diffusion and convection in the gas phase, (ii) the finite rate of transfer at the interface, (iii) the combined effect of diffusion and convection in the liquid phase. In what follows the equations will often be in sets of five, labelled (a),..., (e) the differential equations expression the three processes (i), (ii) (iii) above are always (b), (c) and (d), respectively equations (a) and (e) represent the condition that there is no flow over the boundaries at r = r0 and r = r2. 

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