Reactors boundary conditions

Core damage and containment performance was assessed for accident sequences, component failure, human error, and containment failure modes relative to the design and operational characteristics of the various reactor and containment types. The IPEs were compared to standards for quality probabilistic risk assessment. Methods, data, boundary conditions, and assumptions are considered to understand the differences and similarities observed. 

A solution to Equation (8.12) together with its boundary conditions gives a r, z) at every point in the reactor. An analytical solution is possible for the special case of a first-order reaction, but the resulting infinite series is cumbersome to evaluate. In practice, numerical methods are necessary. 

There is a general trend toward structured packings and monoliths, particularly in demanding applications such as automotive catalytic converters. In principle, the steady-state performance of such reactors can be modeled using Equations (9.1) and (9.3). However, the parameter estimates in Figures 9.1 and 9.2 and Equations (9.6)-(9.7) were developed for random packings, and even the boundary condition of Equation (9.4) may be inappropriate for monoliths or structured packings. Also, at least for automotive catalytic converters. 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find = 0, which is the definition of a closed system. See... 

These boundary conditions are really quite marvelous. Equation (9.16) predicts a discontinuity in concentration at the inlet to the reactor so that ain a Q+) if D >0. This may seem counterintuitive until the behavior of a CSTR is recalled. At the inlet to a CSTR, the concentration goes immediately from to The axial dispersion model behaves as a CSTR in the limit as T) — 00. It behaves as a piston flow reactor, which has no inlet discontinuity, when D = 0. For intermediate values of D, an inlet discontinuity in concentrations exists but is intermediate in size. The concentration n(O-l-) results from backmixing between entering material and material downstream in the reactor. For a reactant, a(O-l-) [Pg.332]

Traditionally, an average Sherwood number has been determined for different catalytic fixed-bed reactors assuming constant concentration or constant flux on the catalyst surface. In reality, the boundary condition on the surface has neither a constant concentration nor a constant flux. In addition, the Sh-number will vary locally around the catalyst particles and in time since mass transfer depends on both flow and concentration boundary layers. When external mass transfer becomes important at a high reaction rate, the concentration on the particle surface varies and affects both the reaction rate and selectivity, and consequently, the traditional models fail to predict this outcome. 

Reactor wall thermal boundary conditions can have a strong effect on the gas flow and thus the deposition. Here, for example, we indicate how cooling the reactor walls can enhance deposition uniformity. We consider the results of three simulations comparing the effects of two different wall boundary conditions. Figure 4 shows how the ratio of the computed susceptor heat flux to the onedimensional heat flux varies with the disk radius for the different conditions (the Nusselt number Nu is a dimensionless surface heat flux). In two cases the reactor walls are held at 300 K (0 = 0), and in one case the walls are insulated ( 0/ r —... 

The other boundary conditions are relatively simple. The temperature and species composition far from the disk (the reactor inlet) are specified. The radial and circumferential velocities are zero far from the disk a boundary condition is not required for the axial velocity at large x. The radial velocity on the disk is zero, the circumferential velocity is determined from the spinning rate W = Q, and the disk temperature is specified. 

No slip Is used as the velocity boundary conditions at all walls. Actually there Is a finite normal velocity at the deposition surface, but It Is Insignificant In the case of dilute reactants. The Inlet flow Is assumed to be Polseullle flow while zero stresses are specified at the reactor exit. The boundary conditions for the temperature play a central role in CVD reactor behavior. Here we employ Idealized boundary conditions In the absence of detailed heat transfer modelling of an actual reactor. Two wall conditions will be considered (1) adiabatic side walls, l.e. dT/dn = 0, and (11) fixed side wall temperatures corresponding to cooled reactor walls. For the reactive species, no net normal flux Is specified on nonreacting surfaces. At substrate surface, the flux of the Tth species equals the rate of reaction of 1 In n surface reactions, l.e. 

In the first two cases the Navier-Stokes equation can be applied, in the second case with modified boundary conditions. The computationally most difficult case is the transition flow regime, which, however, might be encountered in micro-reactor systems. Clearly, the defined ranges of Knudsen numbers are not rigid rather they vary from case to case. However, the numbers given above are guidelines applicable to many situations encoimtered in practice. 

The boundary conditions determine the form of balance equation for the inlet and outlet sections. These require special consideration as to whether diffusion fluxes can cross the boundaries in any particular physical situation. The physical situation of closed ends is considered here. This would be the case if a smaller pipe were used to transport the fluid in and out of the reactor, as shown in Figs. 4.13 and 4.14. 

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. 

According to the boundary conditions, the concentration profile for A must change with a discontinuity at the reactor entrance, as shown in Fig. 4.16. 

The boundary conditions are satisfied by Cq = 1 for a step change in feed concentration at the inlet, and by the condition that at the outlet C n+i = C n, which sets the concentration gradient to zero. The reactor is divided into 8 equal-sized segments. 

Application of the Balzhinimaev model requires assumptions about the reactor and its operation so that the necessary heat and material balances can be constructed and the initial and boundary conditions formulated. Intraparticle dynamics are usually neglected by introducing a mean effectiveness factor however, transport between the particle and the gas phase is considered. This means that two heat balances are required. A material balance is needed for each reactive species (S02, 02) and the product (SO3), but only in the gas phase. Kinetic expressions for the Balzhinimaev model are given in Table IV. 

Alternative methods of estimating Q)L are based on the response of the reactor to an ideal pulse input. For example, equation 11.1.39 may be used to calculate the mean residence time and its variance. Levenspiel and Bischoff (9) indicate that for the boundary conditions cited,... 

Heat Transfer to the Containing Wall. Heat transfer between the container wall and the reactor contents enters into the design analysis as a boundary condition on the differential or difference equation describing energy conservation. If the heat flux through the reactor wall is designated as qw, the heat transfer coefficient at the wall is defined as... 

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