Volumetric flow differential reactors

This complex system would be difficult to solve directly. However, the problem is separable by taking advantage of the widely different time scales of conversion and deactivation. For example, typical catalyst contact times for the conversion processes are on the order of seconds, whereas the time on stream for deactivation is on the order of days. [Note Catalyst contact time is defined as the volume of catalyst divided by the total volumetric flow in the reactor at unit conditions, PV/FRT. Catalyst volume here includes the voids and is defined as WJpp — e)]. Therefore, in the scale of catalyst contact time, a is constant and Eq. (1) becomes an ordinary differential equation ... 

St is the total sorbed concentration (M/M), a, is the first-order mass-transfer rate coefficient for compartment i (1/T), / is the mass fraction of the solute sorbed in each site at equilibrium (assumed to be equal for all compartments), Kp is the distribution coefficient (L3 /M), C is the aqueous solute concentration (M/L3), St is the mass sorbed in compartment i with respect to the total mass of the sorbent (M/M), 0 is the volumetric flow rate through the reactor (L3/T), C, is the influent concentration of solute (M/L3), M, is the mass of sorbent in the reactor (M), and V is the aqueous reactor volume (L3). Using the T-PDF, discrete values for the mass-transfer rate coefficients were generated for the NK compartments. The median value of the mass-transfer rate coefficient within each compartment was chosen as the representative value. The resulting system of ordinary differential equations was solved numerically using a 4th-order Runge-Kutta integration technique. 

In the differential flow reactor, the residence time is short so that the conversion remains small, usually a few per cent. This can be achieved in short beds and/or with high feed flow rates. Since the conversion is small all pellets operate approximately under the same conditions and variation of the volumetric flow rate can be neglected. The observed or apparent conversion rate follows directly from the measured inlet and outlet concentrations via a material balance over the bed ... 

The exit volumetric flow rate from a differential packed bed containing 10 g of catalyst was maintained at 300 dmVmin for each run. The partial pressures of and CO were determined at the entrance to the reactor, and the methane concentration was measured at the reactor exit. 

The solution to this problem requires an analysis of multiple gas-phase reactions in a differential plug-flow tubular reactor. Two different solution strategies are described here. In both cases, it is important to write mass balances in terms of molar flow rates and reactor volume. Molar densities and residence time are not appropriate for the convective mass-transfer-rate process because one cannot assume that the total volumetric flow rate is constant in the gas phase, particularly when the total number of moles is not conserved. In each reaction, 2 mol of reactants generates 1 mol of product. Furthermore, an overall mass balance suggests that the volumetric flow rate is constant only when the overall mass density does not change. This is a reasonable assumption for liquid-phase reactors but not for gas-phase problems when the total volume is not restricted. The exception is a constant-volume batch reactor. 

This problem requires an analysis of coupled thermal energy and mass transport in a differential tubular reactor. In other words, the mass and energy balances should be expressed as coupled ordinary differential equations (ODEs). Since 3 mol of reactants produces 1 mol of product, the total number of moles is not conserved. Hence, this problem corresponds to a variable-volume gas-phase flow reactor and it is important to use reactor volume as the independent variable. Don t introduce average residence time because the gas-phase volumetric flow rate is not constant. If heat transfer across the wall of the reactor is neglected in the thermal energy balance for adiabatic operation, it... 

Here, c and c°" are the inlet and exit concentrations of component i, respectively. Variables Q and V str Ihe volumetric flow rate and CSTR vessel volume, respectively. Similarly, the equation for a constant volume batch reactor follows the differential equation ... 

Note that v A is volumetric flow rate [L T ] and A dz is dV, the differential volume of catalyst [L ]. V is the fluid volume of the catalyst mass, which is the true volume of the reactor. However, the chemical processing industry generally identifies V as the weight W of the solid-supported catalyst loaded in the reactor. Using W instead of V complicates our analysis for a fixed-bed reactor. Therefore, we will identify V as the true volume for a fixed-bed reactor. That volume is... 

The flow model was taken from cin unpublished report described In the Appendix. The reactor Is considered to consist of a set of annuli each containing cin equal portion of the volumetric flow. The axial distance Is broken Into subdivisions whose size is dependent upon the stability analysis of the numerical technique. The physical properties at the entrance of each annular section are considered constant in the radial direction. At each axial point, these properties are calculated and the radial coordinate Is resubdlvlded into new annuli. The heat balance is then obtained from the differential equation in cylindrical coordinates transformed Into a difference equation of trldlagonal form which was solved by the method of L.H. Thomas (Reference 6). 

Wama and Salmi (1996) developed dynamic models for three-phase slurry and trickle-bed reactors operating in nonisothermal conditions. The model equations for the gas, liquid, and catalyst phases consisted of parabolic partial differential equations (PDF) and ODEs expressed in terms of volumetric flow rates for concurrent and countercurrent flow. Oxidation of SO2 was chosen for the slurry reactor simulation. 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

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