Mass Balance for the Reactor

A mass balance for the reactor was also included in the simple Langmuir-Hinshelwood model analyzed by Morton and Goodman (29J) with constant activation energies ... 

In a review article on oscillatory reactions (294), Sheintuch discusses the effect of introducing a heat balance for the catalyst rather than a mass balance for the reactor into the differential equation system for a surface reaction with oxidation/reduction cycles. Although the coverage equations alone can yield oscillatory behavior, as was the case for the models discussed in the previous section, Sheintuch s model is discussed in this section because introduction of the heat balance adds qualitatively new features. In this extended system complex, multiple peak behavior and quasiperiodicity was observed as shown in Fig. 8. Sheintuch also investigated the interaction of two oscillators. This work, however, will be treated in detail in Section V, were synchronization and chaos are discussed. 

It. is possible that the volume of fluid in a reactor is a function of time, in which case a total. mass balance for the reactor would have to be written to obtain this time dependence. However, few reactors are operated in this manner, so we will neglect this complication, as well as volume changes that may occur on mixing and reaction, and assume that q = out = Also, since the tank-type reactor i used almost exclusively for liquid-phase reactions, we may safely assume that U = H (since PV RT for liquids). Thus Eqs. 14.1-3 and 14.1-4 can be rewritten as... 

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].

To illustrate the development of a physical model, a simplified treatment of the reactor, shown in Fig. 8-2 is used. It is assumed that the reac tor is operating isothermaUy and that the inlet and exit volumetric flows and densities are the same. There are two components, A and B, in the reactor, and a single first order reaction of A B takes place. The inlet concentration of A, which we shall call Cj, varies with time. A dynamic mass balance for the concentration of A (c ) can be written as follows ... 

Owing to the high computational load, it is tempting to assume rotational symmetry to reduce to 2D simulations. However, the symmetrical axis is a wall in the simulations that allows slip but no transport across it. The flow in bubble columns or bubbling fluidized beds is never steady, but instead oscillates everywhere, including across the center of the reactor. Consequently, a 2D rotational symmetry representation is never accurate for these reactors. A second problem with axis symmetry is that the bubbles formed in a bubbling fluidized bed are simulated as toroids and the mass balance for the bubble will be problematic when the bubble moves in a radial direction. It is also problematic to calculate the void fraction with these models. 

The balances for the reactor liquid are as follows Total mass balance, assuming constant density... 

It is important to note the impact of the reactor on the resulting model. Since the reactor does not use water at all, water mass balances around the reactor are not required. The reactor can also be excluded from the reuse scheduling as the operation of the reactor does not directly affect the reuse of water. However, task scheduling constraints are still required for the reactor as are raw material and product mass balances. 

Definitions for the variables and constants appearing in eqns. 1 and 2 are given in the nomenclature section at the end of this paper. The first of these equations represents a mass balance around the reactor, assuming that it operates in a differential manner. The second equation is a species balance written for the catalyst surface. The rate of elementary reaction j is represented by rj, and v j is the stoichiometric coefficient for component i in reaction j. The relationship of rj to the reactant partial pressures and surface species coverages are given by expressions of the form... 

The catalytic reactor is an example where reaction occurs only at the boundary with a solid phase, but, as long as the solid remains in the reactor and does not change, we did not need to write separate mass balances for the soHd phase because its residence time Tj is infinite. In a moving bed catalytic reactor or in a slurry or fluidized bed catalytic reactor... 

In the ideal batch stirred-tank reactor (BSTR), the fluid concentration is uniform and there are no feed or exit streams. Thus, only the last two terms in the previous equation exist. For a volume element of fluid (VL), the mass balance for the limiting reactant becomes (Smith, 1981 Levenspiel, 1972)... 

Leung s method is given in 6.3.2 below. The method.is an approximate solution to the differential mass and energy balances for the reactor during relief and takes account of both emptying via the relief system and the tempering effect of vapour production due to relief. The method makes use of adiabatic experimental data for the rate of heat release from the runaway reaction (see Annex 2). Nomenclature is given in Annex 10. 

With these parameters we can set-up a mass balance on the system, which is the basis for evaluating the experimental results. The mass balance for the absorption (of any gas, e. g. ozone) in a continuous-flow stirred tank reactor (CFSTR) under the assumption that the gas and liquid phases are ideally mixed (cL = cLe, cG = cGe), are as follows ... 

An additional substrate molecule interacts with the ES complex at high substrate concentration to form the unproductive complex ESI, or in this case ESS. With the respective mass balances for the ideal reactors the integral rate laws of Eqs. (5.19) and (5.20) are obtained. 

The reaction is also influenced by the heat of reaction that develops during the conversion of the reactants, a problem in tubular screening reactors. In micro structures, the heat transport through the walls of the channels is facilitated owing to their small dimensions. The catalysts are deposited on the walls of these micro structures and will thus have the appropriate environment for exothermic reactions by enabling fast quenching of the reaction with near isothermal conditions. Hence also the heat and mass balance in the reactor will be decoupled, which permits the analytical description of the flow in the screening reactor. 

The fraction of Ca(0H)2 reacted at any given time can be calculated by integrating the S02 versus time curve obtained by the recorder on the SO2 analyzer and doing a mass balance in the reactor. As a backup, the reacted solids are analyzed for sulfite and hydroxide using acid/base and iodine titrations. 

The total BEA catalyst amount in the reactor ranged from 91 to 103 g for the different monolith sets. The mass balances for the experiments were in the range of 99 % to 101 %. The water balances for the reported experiments were in the range of 97 % to 101 %, thus giving sufficient confidence in the results of GC- and KF-analyses. Flooding of the column was never observed. 

A differential characteristic which demands a lower degree of standardization is the reaction rate. The rate of a chemical reaction with respect to compound B at a given point is defined as the rate of formation of B in moles per unit time per unit volume. It cannot be measured directly and is determined from the rates of change of some observable quantities such as the amount of substance, concentration, partial pressure, which are subject to measurements. Reaction rates are obtained from observable quantities by use of the conservation equations resulting from the mass balance for the given reactor type. 

Numerous reactions are performed by feeding the reactants continuously to cylindrical tubes, either empty or packed with catalyst, with a length which is 10 to 1000 times larger than the diameter. The mixture of unconverted reactants and reaction products is continuously withdrawn at the reactor exit. Hence, constant concentration profiles of reactants and products, as well as a temperature profile are established between the inlet and the outlet of the tubular reactor, see Fig. 7.1. This requires, in contrast to the batch reactor, the application of the law of conservation of mass over an infinitesimal volume element, dV, of the reactor. In contrast to a batch reactor the existence of a temperature profile does not allow us to consider the mass balances for the reacting components and the energy balance separately. Such a separation can only be performed for isothermal tubular reactors. 

Figure 7.3 also compares the evolution of the concentrations of the intermediate Q and the product in case of two first-order reactions in series in a CSTR with that in a batch or plug flow reactor. For constant density, the mass balance for the reaction components in a CSTR are ... 

Differences in the Responses of the Different Types of Models. The basic differences that exist in the heat and mass balances for the different types of models determine deviations of the responses of types I and II with respect to type III. In a previous work (1) a method was developed to predict these deviations but for conditions of no increase in the radial mean temperature of the reactor (T0 >> Tw). In this work,the method is generalized for any values of T0 and Tw and for any kinetic equation. The proposed method allows the estimation of the error in the radial mean conversions of models I and II with respect to models III. Its validity is verified by comparing the predicted deviations with those calculated from the numerical solution of the two-dimensional models. A similar comparison could have been made with the numerical solution of the one-dimensional models. 

Employing a high recirculating flow rate in this small laboratory reactor, the following assumptions can be used (i) there is a differential conversion per pass in the reactor, (ii) the system is perfectly stirred, (iii) there are no mass transport limitations. Also, it can be assumed that (iv) the chemical reaction occurs only at the solid-liquid interface (Minero et al., 1992) and (v) direct photolysis is neglected (Satuf et al., 2007a). As a result, the mass balance for the species i in the system takes the following form (Cassano and Alfano, 2000) ... 

The mass balance in the reactor is derived under the following assumptions (i) unsteady state operation, (ii) convective laminar Newtonian flow in the axial direction z (the Re)molds number is below the transition regime), (iii) diffusion in the z direction is negligible with respect to convection, (iv) symmetry in the y direction (the lamp length is much larger than the reactor width), and (v) constant physical properties. The local mass balance for a species i in the reactor and the corresponding initial and boundary conditions are... 

Due to their complexity, the model equations will not be derived or presented here. Details can be found elsewhere [Adris, 1994 Abdalla and Elnashaie, 1995]. Basically mass and heat balances arc performed for the dense and bubble phases. It is noted that associated reaction terms need to be included in those equations for the dense phase but not for the bubble phase. Hydrogen permeation, the rate of which follows Equation (10-51b) with n=0.5, is accounted for in the mass balance for the dense phase. Hydrodynamic parameters important to the fluidized bed reactor operation include minimum fluidization velocity, bed porosity at minimum fluidization, average bubble diameter, bubble rising velocity and volume fraction of bubbles in the fluidized bed. The equations used for estimating these and other hydrodynamic parameters are taken from various established sources in the fluidized bed literature and have been given by Abdalla and Elnashaie [1995]. 

For the isothermal tubular plug-flow reactor (PFR) discussed previously, the mass balance for the G gaseous components is... 

If A is the cross-sectional area of the reactor, q the volume flow rate, and x the distance from the inlet of the bed, a mass balance for the component A, over an elementary length of bed gives... 

Equation 5 is meaningful only if the reaction rate rjt is greater than zero. The concentration of Sic in the general case can be found from a mass balance around the reactor using Equation 3. Converting from moles to concentration terms, Equation 3 will yield the following for the concentration of component A ... 

In contrast to a batch reactor, the existence of a temperature profile does not allow us to consider the mass balances for the reacting components and the energy balance separately. Such a separation can only be performed for isothermal tubular reactors. 

Two different modeling approaches are used for simulated moving bed reactors. The first approach combines the model of several batch columns with the mass balances for the external inlet and outlet streams. By periodically changing the boundary conditions the transient behavior of the process is taken into account. The model is based on the SMB model introduced in Chapter 6 and is, therefore, referred to as the SMBR model. The second approach assumes a true counter-current flow of the solid and the liquid phase like the TMBR. Therefore, this approach is called the TMBR model. 

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