Tubular boundary conditions

The zero slope boundary condition at = 0 assumes S5anmetry with respect to the centerline. The mathematics are then entirely analogous to those for the tubular geometries considered previously. Applying the method of lines gives... 

Again the entrance and exit boundary conditions must be considered. Thus the two boundary conditions at Z = 0 and Z = L are used for solution, as shown in Fig. 4.15. Note, that these boundary conditions refer to the inner side of the tubular reactor. A discontinuity in concentration at Z = 0 is apparent in Fig. 4.16. 

Optimization of a distributed parameter system can be posed in various ways. An example is a packed, tubular reactor with radial diffusion. Assume a single reversible reaction takes place. To set up the problem as a nonlinear programming problem, write the appropriate balances (constraints) including initial and boundary conditions using the following notation ... 

We used the wall temperature in the boundary condition, and this may be different from the coolant temperature T. There may be temperature variations across the wall as well as through the coolant. These are described through the overall heat transfer coefficient U, but in practice all these effects must be considered for a detailed description of the wall-cooled tubular reactor. 

That notorious pair, the Danckwerts boundary conditions for the tubular reactor, provides a good illustration of boundary conditions arising from nature. Much ink has been spilt over these, particularly the exit condition that Danckwerts based on his (perfectly correct, but intuitive) engineering insight. If we take the steady-state case of the simplest distributed example given previously but make the flux depend on dispersion as well as on convection, then, because there is only one-space dimension,/= vAc — DA dddz), where D is a dispersion coefficient. Then, as the assumption of steady state eliminates... 

Different problems are modeled by two-point boundary value differential equations in which the values of the state variables are predetermined at both endpoints of the independent variable. These endpoints may involve a starting and ending time for a time-dependent process or for a space-dependent process, the boundary conditions may apply at the entrance and at the exit of a tubular reactor, or at the beginning and end of a counter-current process, or they may involve parameters of a distributed process with recycle, etc. Boundary value problems (BVPs) are treated in Chapter 5. 

Develop the model equations for a countercurrent cooling jacket and the same tubular reactor. This will lead to several coupled boundary value problems with boundary conditions at l = 0 and l = Lt. 

Axial symmetric representation is typically considered for single tubular fuel cell modeling because it allows one to design, mesh and compute only a portion of the cell taking advantage of the symmetry condition. The partial differential equations are written in cylindrical coordinates and solved in two dimensions (radial and axial). This assumption is possible when there are no significant circumferential variations in the boundary conditions. Such a hypothesis is reasonable when the... 

Figure 7.11 shows the air velocity field at the closed end of a 30 cm long tubular cell is presented. The boundary conditions set to obtain this result are presented in Table 7.3. 

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. 

Howard Brenner Let me give a simple example of this, that derives from the generalized Taylor dispersion theory references cited in my previous comments. Think of a tubular reactor in which one has a Poiseuille flow, together with a chemical reaction occurring on the walls. One can certainly write down all the relevant differential equations and boundary conditions and solve them numerically. However, the real essence of the macrophysics is that if one examines the average velocity with which the reactive species moves down the tube, this speed is greater than that of the carrier fluid because the solute is destroyed in the slower-moving fluid streamlines near the wall. Consequently, the only reactive solute molecules that make it... 

Thirty years later, Gerhard Damkohler (1937) in his historic paper, summarized various reactor models and formulated the two-dimensional CDR model for tubular reactors in complete generality, allowing for finite mixing both in the radial and axial directions. In this paper, Damkohler used the flux-type boundary condition at the inlet and also replaced the assumption of plug flow with parabolic velocity profile, which is typical of laminar flow in tubes. 

As our first application, we consider the classical Taylor-Aris problem (Aris, 1956 Taylor, 1953) that illustrates dispersion due to transverse velocity gradients and molecular diffusion in laminar flow tubular reactors. In the traditional reaction engineering literature, dispersion effects are described by the axial dispersion model with Danckwerts boundary conditions (Froment and Bischoff, 1990 Levenspiel, 1999 Wen and Fan, 1975). Here, we show that the inconsistencies associated with the traditional parabolic form of the dispersion model can be removed by expressing the averaged model in a hyperbolic form. We also analyze the hyperbolic model and show that it has a much larger range of validity than the standard parabolic model. 

We now compare the solution of the hyperbolic model with that of the parabolic model used widely in the literature to describe dispersion in tubular reactors. The parabolic model with Danckwerts boundary conditions (in dimensionless form) is given by... 

The boundary conditions at the wall, on the other hand, influence the performance of the reactor critically, and should be determined as accurately as possible. For the equations of concentration (or conversion) the condition at the wall is that the flux of material normal to the wall is zero, which requires that the directional derivative of concentration normal to the wall be zero. For the tubular reactor, with cylindrical symmetry, the condition is expressed by the equation... 

Consider steady state plug flow in a tubular reactor.[14] The governing equation and boundary conditions for dimensionless concentration are ... 

Ordinary differential equations govern systems that vary either with time or space, but not both. Examples are equations that govern the dynamics of a CSTR or the steady state of mbular reactors. Both the dynamics of a CSTR and the steady state of a plug-flow reactor are governed by first-order ordinary differential equations with prescribed initial conditions. The steady-state tubular reactors with axial dispersion are governed by a second-order differential equation with the boundary conditions spec-... 

This example uses the open-vessel boundary conditions where an inlet (upstream) section and an outlet (downstream) section are added to a tubular reactor where dispersion occurs but no reaction. 

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