Mass balance liquid phase differential reactor

When (a) there are no external mass-transfer resistances (such as gas-liquid, liquid solid, etc.), (b) catalysts are all effectively wetted, (< ) there is no radial or axial dispersion in the liquid phase, (d) a gaseous reactant takes part in the reaction and its concentration in the liquid film is uniform and in excess, (e) reaction occurs only at the liquid-solid interface, (/) no condensation or vaporization of the reactant occurs, and (g) the heat effects are negligible, i.e., there is an isothermal operation, then a differential balance on such an ideal plug-flow trickle-bed reactor would give... 

To derive the overall kinetics of a gas/liquid-phase reaction it is required to consider a volume element at the gas/liquid interface and to set up mass balances including the mass transport processes and the catalytic reaction. These balances are either differential in time (batch reactor) or in location (continuous operation). By making suitable assumptions on the hydrodynamics and, hence, the interfacial mass transfer rates, in both phases the concentration of the reactants and products can be calculated by integration of the respective differential equations either as a function of reaction time (batch reactor) or of location (continuously operated reactor). In continuous operation, certain simplifications in setting up the balances are possible if one or all of the phases are well mixed, as in continuously stirred tank reactor, hereby the mathematical treatment is significantly simplified. 

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. 

The final form of the differential thermal energy balance for a generic liquid-phase plug-flow reactor that operates at high-mass and high-heat-transfer Peclet numbers allows one to predict temperature as a function of the average residence time r = V/q ... 

Figure 7.22 illustrates the numerical solution of concentrations in the liquid phase of a tank reactor. The simulation also gives the concentration profiles in the liquid film, as shown in Figure 7.22b. The algebraic equation system describing the gas- and liquid-phase mass balances is solved by the Newton-Raphson method, whereas the differential equation system that describes the liquid film mass balances is solved using orthogonal collocation. To guarantee a reliable solution of the mass balances, the mass balance equations have been solved as a function of the reactor volume. The solution of the mass balances for the reactor volume, Vr, has been used as an initial estimate for the solution for the reactor volume, Vr -F A Vr. The simulations show an interesting phenomenon at a certain reactor volume, the concentration of the intermediate product, monochloro-p-cresol, passes a maximum. When the reactor volume—or the residence time— is increased, more and more of the final product, dichloro-p-cresol, is formed (Figure 7.22a). This shows that mixed reactions in gas-liquid systems behave in a manner similar to mixed reactions in homogeneous reactions (Section 3.8) [11,12].

The steady-state model consists of a set of coupled ordinary differential and algebraic equations. The simulation is obtained by integrating simultaneously the mass-balance equations for the gas and liquid phases in the axial direction of the reactor using a fourth-order Runge-Kutta method. The heat balance is used only for the simulation of the industrial reactors. The solid phase algebraic equations are solved between integration steps with the Newton-Raphson method. Physical properties and mass-transfer coefficients are also updated in every integration step. 

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