Open-closed boundary condition

Using die results firm Example 19-3 for t and of, estimate values for Pe, fa- both open and closed boundary conditions, as far as possible, firm (a) E(6) and (b) the variance. 

For both closed end and open end boundary conditions, solve the... 

We should note one restriction on the E curve—that the fluid only enters and only leaves the vessel one time. This means that there should be no flow or diffusion or upflow eddies at the entrance or at the vessel exit. We call this the closed vessel boundary condition. Where elements of fluid can cross the vessel boundary more than one time we call this the open vessel boundary condition. 

Bischoff and Levenspiel (1962) have shown that as long as the measurements are taken at least two or three particle diameters into the bed, then the open vessel boundary conditions hold closely. This is the case here because the measurements are made 15 cm into the bed. As a result this experiment corresponds to a one-shot input to an open vessel for which Eq. 12 holds. Thus... 

Boundary Conditions There are two cases that we need to consider boundary conditions for closed vessels and open vessels. In the case of closed-closed vessels we assume that there is no dispersion or radial variation in concentration either upstream (closed) or downstream (closed) of the reaction section, hence tliis is a closed-closed vessel. In an open vessel, dispersion occurs both upstream (open) and downstream (open) of the reaction section hence this is an open-open vessel. These two cases are shown in Figure 14-8, where fluctuations in concentration due to dispersion are superimposed on the plug-flow velocity profile. A closed-open vessel boundary condition is one in which there is no dispersion in the entrance section but there is dispersion in the reaction and exit sections. 

In general, the overall balance for the mass transport streams (Eqs. 6.23 and 6.24) at the column inlet and outlet has to be fulfilled. In Eq. 6.92 the closed boundary condition is obtained by setting the dispersion coefficient outside the column equal to zero. In open systems, the column stretches to infinity and in these limits concentration changes are zero. 

The solution to the model Equation 3.329 depends on the boundary conditions defined at 3 = 0 (vessel inlet) and 3 = 1 (vessel outlet). The boundary (inlet or outlet) of a vessel is defined as closed if the dispersion (axial mixing) begins (at the inlet) or terminates (at the outlet) at the boundary and no dispersion occurs outside the boundary. On the contrary, a boundary is defined as open if the dispersion begins or terminates at a location outside the boundary. Thus, there are four possible boundary condihons, namely, open (inlet)-open (outlet), open (inlet)-closed (outlet), closed (inlet)-open (outlet) and closed (inlet)-closed (outlet). Of these four boundary conditions, the closed-closed boundary condition (called the Danckwarts boundary condition) is regarded as the most appropriate representation of the realistic condition. The Danckwarts closed-closed boundary condition is discussed here. 

Figure 4.10.57 Experimental set-up for open vessel boundary condition (a) and F functions for a large extent of dispersion (high deviation from plug flow) for an open vessel (b) and closed vessel (c) for different values of 6o = uL/Dax-Adapted from Levenspiel (1999).

Figure 3. Floquet band structure for a threefold cyclic barrier (a) in the plane wave case after using Eq. (A.l 1) to fold the band onto the interval —I < and (b) in the presence of a threefold potential barrier. Open circles in case (b) mark the eigenvalues at = 0, 1, consistent with periodic boundary conditions. Closed circles mark those at consistent with sign-changing...

Two types of boundary conditions are considered, the closed vessel and the open vessel. The closed vessel (Figure 8-36) is one in which the inlet and outlet streams are completely mixed and dispersion occurs between the terminals. Piston flow prevails in both inlet and outlet piping. For this type of system, the analytic expression for the E-curve is not available. However, van der Laan [22] determined its mean and variance as... 

Comparison of Solutions of Continuity Equation for DPF. The results for the three cases discussed above, together with those for the case with closed-open boundary conditions, are summarized in Table 19.7. Included in Table 19.7 are the boundary conditions, the expression for C(0) at z = 1, and the mean and variance of C(0). For large values of PeL, the solutions are not very different, but for small values, the results differ considerably. 

Since the values fj of the open end configuration do not fall within the yields of the CSTR and PFR, that boundary condition is not valid. The PFR and CSTR profiles are compared on the figure with the closed end result at Pe = 2. 

Fortunately, for small extents of dispersion numerous simplifications and approximations in the analysis of tracer curves are possible. First, the shape of the tracer curve is insensitive to the boundary condition imposed on the vessel, whether closed or open (see above Eq. 11.1). So for both closed and open vessels... 

Let us consider two types of boundary conditions either the flow is undisturbed as it passes the entrance and exit boundaries (we call this the open b.c.), or you have plug flow outside the vessel up to the boundaries (we call this the closed b.c.). This leads to four combinations of boundary conditions, closed-closed, open-open, and mixed. Figure 13.7 illustrates the closed and open extremes, whose RTD curves are designated as E c and E. ... 

In all cases you can evaluate D/wL from the parameters of the tracer curves however, each curve has its own mathematics. Let us look at the tracer curves for closed and for the open boundary conditions. 

We will not discuss the equations and curves for the open-closed or closed-open boundary conditions. These can be found in Levenspiel (1996). 

Although for a closed vessel the boundary conditions are different from the case of an open pipe (see Section 2.3.5 below), we will assume that the above methods of data treatment apply because, as will subsequently be verified, the dispersion number in this Example is small. 

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. 

The solution to this equation (unlike the partial differential equation 2.14) has been shown not to depend on the precise formulation of the inlet and outlet conditions, i.e. whether they are open or closed 051. In the following derivation, however, the reaction vessel is considered to be closed , i.e. it is connected at the inlet and outlet by piping in which plug flow occurs and, in general, there is a flow discontinuity at both inlet and outlet. The boundary conditions to be used will be those which properly apply to a closed vessel. (See Section 2.3.5 regarding the significance of the boundary conditions for open and closed systems.)... 

Figure 8 The singlet-triplet gap, Est, near the neutral-ionic transition of (36) in the restricted basis as a function of T = A — U/2. Open and closed symbols refer to boundary conditions with and without crossovers, respectively, and the stars as joint N —> oo extrapolations based on both[97].

Dispersion. A discussion of the boundary conditions for closed-closed, open-open, closed-open, and open-closed vessels can be found in... 

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