Reactant concentration profile

FIGURE 9.5 Reactant concentration profiles for a thermal runaway in the packed-bed reactor of Examples 9.1 and 9.2 Ti = 704K. 

Schematic representation of reactant concentration profiles in various global rate regimes. I External mass transfer limits rate. II Pore diffusion limits rate. Ill Both mass transfer effects are present. IV Mass transfer has no influence on rate.

Figure 7-7 Reactant concentration profiles in directionx, which is perpendicular to the flow direction z expected for flow over porous catalyst pellets in a packed bed or sluny reactor. External mass transfer and pore diffusion produce the reactant concentration profiles shown.

Figure 7-9 Reactant concentration profiles around and within a porous catalyst pellet for the cases of reaction control, external mass transfer control, and pore difliision control. Each of these situations leads to different reaclion rate expressions.

Figure 7-10 Reactant concentration profiles around a catalyst pellet for reaction control (A > k") and for external mass transfer control ( tni k )-...

Figure 7-11 Reactant concentration profiles within a porous catalyst pellet for situations where surface reaction controls aid vdiere pore difiusion controls the reactions.

Figure 7-12 Reactant concentration profiles down a single catalyst pore of length and diameter dp with a catalytic reaction occurring on the walls of the pore. The concentration is Cm t the pore mouth, x = (, and the gradient is zero at the end of the pore because the end is assumed to be unreactive and there is therefore no flux of reactant through the end.

Figure 7-20 Sketch of tube wall reactor with a porous catalyst film of thickness t on walls. Expected reactant concentration profiles with reaction-limited, mass-transfer-limited, and pore-diffusion-limited reaction.

Figure >-13 Reactant concentration profiles Ca (ji) within a product film Cs for afilm transforming from Bs to Cs. The profile of Ihe difiusion-liniited migration of A through the film remains a straight line so the growth rate (proportional to the gradient) decreases as the film thickens, leading to the parabohc law of film growth.

Now the reaction occurs by A migrating from the gas in the center of the tube, through the faUing Hquid film, and into the porous catalyst, where it reacts with B from the Hquid phase to form products that must then migrate out of the catalyst and into the flowing gas or Hquid streams to escape the reactor. The reactant concentration profile across the reactor Ca(R) now is modified because A reacts in the porous catalyst rather than within the Hquid film discussed in the previous section, as shown in Figure 12-1 3. 

C. Mass Transport and Reactant Concentration Profiles through the Rod. 182... 

B. Derivation of Equations for Reactant Concentration Profile THROUGH Carbon Rods Depending upon Type of Mass Transport... 

Fig. 7.10. Reactant concentration profiles wthin a catalyst slab for an irreversible first-order reaction, at different values of the Thiele modulus.

Figure 12-17 (a) Trickle bed reactor (b) reactant concentration profile. 

In addition to the possibility of a nommiform wafer thickness in the radical direction, the thickness of the wafers can vary down the length of the boat reactor. We want to obtain an analytical solution of the silicon deposition rate and reactant concentration profile for the simplified version of the LPCVD reactor just discussed. Analytical solutions of this type are important in that an engineer can rapidly gain an imderstanding of the important parameters and their sensitivities without making a number of runs on the computer. 

The reactant concentration profile and deposition thickness along the length of the reactor are shown schematically in Figme 12-23 for the case of small values of the Thiele modulus (t — 1). 

When a number of such CSTRs are employed in series, the concentration profile is step-shaped if the abscissa is the total residence time or the stage number as indicated by a typical reactant concentration profile in Fig. 7-2c. 

Often the global reaction rate of heterogeneous catalytic reactions is affected by the diffusion in the pore and the external mass-transfer rate of the reactants and the products. When the diffusion in the pores is not fast, a reactant concentration profile develops in the interior of the particle, resulting in a different reaction rate at different radial locations inside the catalytic pelet. To relate the global reaction rate to various concentration profiles that may develop, a kinetic effectiveness factor is defined [1, 3,4,7, 8] by... 

Figure 7.18 Solids and reactant concentration profiles for gas solid reaction with a large value of the diffusional modulus. No interphase transport limits. [After C.Y. Wen, Ind. Eng. Chem., 60, 34, with permission of the American Chemical Society, (1968).]...

Figure 16-1 Effect of the intrapellet Damkohler number on dimensionless reactant concentration profiles for one-dimensional diffusion and pseudo-homogeneous zeroth-order chemical kinetics in porous catalyst with rectangular symmetry.

Results from the previous section in this chapter illustrate how and when interpellet axial dispersion plays an important role in the design of packed catalytic tubular reactors. When diffusion is important, more sophisticated numerical techniques are required to solve second-order ODEs with split boundary conditions to predict non-ideal reactor performance. Tubular reactor performance is nonideal when the mass transfer Peclet number is small enough such that interpellet axial dispersion cannot be neglected. The objectives of this section are to understand the correlations for effective axial dispersion coefficients in packed beds and porous media and calculate the mass transfer Peclet number based on axial dispersion. Before one can make predictions about the ideal vs. non-ideal performance of tubular reactors, steady-state mass balances with and without axial dispersion must be solved and the reactant concentration profiles from both solutions must be compared. If the difference between these profiles with and without interpellet axial dispersion is indistinguishable, then the reactor operates ideally. 

Even though temperature and reactant concentration profiles are solved for onedimensional diffusion in the thinnest dimension of the pellet (i.e., or radially for long cylinders and spheres), the equations in this section are applicable to multidimensional diffusion and conduction. The coordinate direction denoted by n in (27-26) is not important, because... 

Two coupled ODEs must be solved to calculate temperature and reactant concentration profiles within a catalytic pellet that exhibits rectangular symmetry. The primary contribution to diffusion occurs in the thinnest dimension of the catalyst (i.e., the x direction). Hence, the mass ttansfer equation with one-dimensional diffusion and nth-order irreversible chemical reaction reduces to... 

Figure 27-1 Effect of the thermal energy generation paramete on dimensionless reactant concentration profiles as one travels inward toward the center of a porous catalyst with rectangular symmetry. The chemical kinetics are first-order and irreversible, and the reaction is exothermic. All parameters are defined in Table 27-4. The specific entries for P = 0.6 and = 1.0 are provided in Table 27-6.

The reactant concentration profile jjf representing the probabihty that site i is occupied by a reactant molecule and that the occupation of the other sites (either by reactant or product molecules) is described by the set of parameters cri- cri i, ai+i - aN. 

Thus the reactant concentration profile is found to be nothing else than the Laplace transform of the residence time distribution. 

In this type of reactor, as the name implies, the flow is laminar. In other words, the radial concentration profile is parabolic and not uniform as in a PFR. This is because there is hardly any mixing and the reactant concentration profile closely matches the velocity profile (lowest near the wall to highest at the center). Thus each element of fluid flowing through the reactor is completely independent of the other elements, so that the fluid as a whole tends to behave as a macrofluid. In essence, therefore. Equation 13.10 would be valid for this case also. Integrated... 

The first condition indicates that there is no flux at the center of the catalyst particle. This condition also says that the reactant concentration profile is symmetrical at the center. This situation is commonly referred to as the symmetry condition. The second boundary condition corresponds to high velocity at the boundary since the reactant concentration at the surface is taken equal to that of the bulk surrounding the particle, which is taken to be invariant in the present problem. 

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