Mass balance differential reactor

The fed-batch reactor model is commonly built using classical thermal and mass balance differential equations [11], Under isothermal conditions, the material balance for each measured component in the H KR of epichlorohydrin with continuous water addition is expressed by one of the following equations (Eqs. 12-17). These equations can be solved using an appropriate solver package (Lsoda, Ddassl [12]) with a connected optimization module for parameter estimation. 

As an alternative to deriving Eq. (8-2) from a dynamic mass balance, one could simply postulate a first-order differential equation to be valid (empirical modeling). Then it would be necessary to estimate values for T and K so that the postulated model described the reactor s dynamic response. The advantage of the physical model over the empirical model is that the physical model gives insight into how reactor parameters affec t the v ues of T, and which in turn affects the dynamic response of the reac tor. 

In the above reactions, I signifies an initiator molecule, Rq the chain-initiating species, M a monomer molecule, R, a radical of chain length n, Pn a polymer molecule of chain length n, and f the initiator efficiency. The usual approximations for long chains and radical quasi-steady state (rate of initiation equals rate of termination) (2-6) are applied. Also applied is the assumption that the initiation step is much faster than initiator decomposition. ,1) With these assumptions, the monomer mass balance for a batch reactor is given by the following differential equation. 

In the present study, the UASB reactor was modeled in terms of the dispersed plug flow and the Monod type of rate equations to constmct the differential mass balance equations fcs- the anaerobic biodegradation of single and multiple substrates components of the volatile fetty acids. 

Steady-state reactors with ideal flow pattern. In an ideal isothermal tubular pZi/g-yZovv reactor (PFR) there is no axial mixing and there are no radial concentration or velocity gradients (see also Section 5.4.3). The tubular PFR can be operated as an integral reactor or as a differential reactor. The terms integral and differential concern the observed conversions and yields. The differential mode of reactor operation can be achieved by using a shallow bed of catalyst particles. The mass-balance equation (see Table 5.4-3) can then be replaced with finite differences ... 

Stirred tank reactor (ST/ ). The differential mass balance referred to the azo-dye converted by bacteria (assuming unstructured model for the biophase, i.e., that it is characterized only by cell mass or concentration X) yields... 

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 growth of biomass in the reactor is assumed to follow Monod kinetics with a first-order death rate. A mass balance on the biomass in the reactor yields the following differential equation (assuming that no biomass enters the reactor in the feed) ... 

An unsteady state mass balance over the typical reactor element lying between x and x + 5x gives rise to the partial differential equation... 

We can therefore replace dt by dz/u in all of the preceding differential equations for the mass balance in the batch reactor and use these equations to describe reactions during flow through a pipe. This reactor is called the plug-flow tubular reactor, which is the most important continuous reactor encountered in the chemical industry. 

Note that this problem is even easier than for a batch reactor because for the CSTR we just have to solve an algebraic equation rather than a differential equation For second-order kinetics, r = kC, the CSTR mass-balance equation becomes... 

In this chapter we consider the performance of isothermal batch and continuous reactors with multiple reactions. Recall that for a single reaction the single differential equation describing the mass balance for batch or PETR was always separable and the algebraic equation for the CSTR was a simple polynomial. In contrast to single-reaction systems, the mathematics of solving for performance rapidly becomes so complex that analytical solutions are not possible. We will first consider simple multiple-reaction systems where analytical solutions are possible. Then we will discuss more complex systems where we can only obtain numerical solutions. 

These come from simple application of the mass-balance equation dCj jdx = Yli which you should verify. Differential equations always need initial or boundary conditions, and for the batch reactor these are the initial concentrations of A, B, and C. For this system, the feed may be expected to be pure A, so Ca = Cao and Cbo Cco 0-... 

A complete description of the reactor bed involves the six differential equations that describe the catalyst, gas, and thermal well temperatures, CO and C02 concentrations, and gas velocity. These are the continuity equation, three energy balances, and two component mass balances. The following equations are written in dimensional quantities and are general for packed bed analyses. Systems without a thermal well can be treated simply by letting hts, hlg, and R0 equal zero and by eliminating the thermal well energy equation. Adiabatic conditions are simulated by setting hm and hvg equal to zero. 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

In this chapter we will keep the description of transport simpler than Fick s law, which would eventually lead to partial differential equations and thus to rather complex models. Instead of allowing the concentration of a chemical to change continuously in space, we assume that the concentration distribution exhibits some coarse structure. As an extreme, but often sufficient, approximation we go back to the example of phenanthrene in a lake and ask whether it would be adequate to describe the mass balance of phenanthrene by using just the average concentration in the lake, a value calculated by dividing the total phenanthrene mass in the lake by the lake volume. If the measured concentration in the lake at any location or depth would not deviate too much from the mean (say, less than 20%), then it may be reasonable to replace the complex three-dimensional concentration distribution of phenanthrene (which would never be adequately known anyway) by just one value, the average lake concentration. In other words, in this approach we would describe the lake as a well-mixed reactor and could then use the fairly simple mathematical equations which we have introduced in Section 12.4 (see Fig 12.7). The model which results from such an approach is called a one-box model. 

In this section we develop a dynamic model from the same basis and assumptions as the steady-state model developed earlier. The model will include the necessarily unsteady-state dynamic terms, giving a set of initial value differential equations that describe the dynamic behavior of the system. Both the heat and coke capacitances are taken into consideration, while the vapor phase capacitances in both the dense and bubble phase are assumed negligible and therefore the corresponding mass-balance equations are assumed to be at pseudosteady state. This last assumption will be relaxed in the next subsection where the chemisorption capacities of gas oil and gasoline on the surface of the catalyst will be accounted for, albeit in a simple manner. In addition, the heat and mass capacities of the bubble phases are assumed to be negligible and thus the bubble phases of both the reactor and regenerator are assumed to be in a pseudosteady state. Based on these assumptions, the dynamics of the system are controlled by the thermal and coke dynamics in the dense phases of the reactor and of the regenerator. 

In a tubular reactor, the reactants are fed in at one end and the products withdrawn from the other. If we consider the reactor operated at steady state, the composition of the fluid varies inside the reactor volume along the flow path. Therefore, the mass balance must be established for a differential element of volume dV. We assume the flow as ideal plug flow, that is, that there is no back mixing along the reactor axis. Hence, this type of reactor is often referred to as Plug Flow Reactor (PFR). 

This equation represents the mass balance of A in the differential volume dV. In order to obtain the overall mass balance of the reactor, we must integrate this expression over the reactor volume, by taking into account that the reaction rate is a function of the local concentration ... 

The effectiveness of the proposed approach has been tested in simulation by considering a jacketed batch reactor in which the phenol-formaldehyde reaction presented in Chap. 2 takes place. The complete system of differential equations given by the 13 mass balances presented in Sect. 2.4 has been simulated in the MATLAB/SIMULINK environment. 

Given any complex system of heterogeneous catalytic first order reactions the mass balance on a differential volume element of the reactor at the height h yields the following system of differential equations for the j-th reaction component i) for the bubble phase... 

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. 

The dependent variable y is most frequently the reaction rate independent variables are the concentration or pressure of reaction components, temperature and time. If in some cases the so-called integral data (reactant concentrations or conversion versus time variable) arc to be treated, a differential kinetic equation obtained by the combination of a rate equation with the mass balance equation 1 or 3 for the given type of reactor is used. The differential equation is integrated numerically, and the values obtained arc compared with experimental data. 

The macroscopic approach, in which it is not taken into account what happens inside the cell in detail, but only an overall view of the system is described. In fact, the system is considered as a black box from the fluid dynamic point of view and then, it is assumed that the cell behaves a mixed tank reactor (the values of the variables only depend on time and not on the position since only one value of every variable describes all positions). This assumption allows simplifying directly all the set of partial differential equations to an easier set of differential equations, one for each model species. For the case of a continuous-operation electrochemical cell, the mass balances take the form shown in (4.5), where [.S, ]... 

The concentration of any of these species depends on the total concentration of dissolved aluminum and on the pH, and this makes the system complex from the mathematical point of view and consequently, difficult to solve. To simplify the calculations, mass balances were applied only to a unique aluminum species (the total dissolved aluminum, TDA, instead of the several species considered) and to hydroxyl and protons. For each time step (of the differential equations-solving method), the different aluminum species and the resulting proton and hydroxyl concentration in each zone were recalculated using a pseudoequilibrium approach. To do this, the equilibrium equations (4.64)-(4.71), and the charge (4.72), the aluminum (4.73), and inorganic carbon (IC) balances (4.74) were considered in each zone (anodic, cathodic, and chemical), and a nonlinear iterative procedure (based on an optimization method) was applied to satisfy simultaneously all the equilibrium constants. In these equations (4.64)-(4.74), subindex z stands for the three zones in which the electrochemical reactor is divided (anodic, cathodic, and chemical). 

Recycle reactors working at high internal recycle ratios approximate differential conditions on the catalyst bed quite well and therefore can be treated as a CSTR, where the production rate of each organic reactant j can be calculated from a mass balance by ... 

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) ... 

These basic rate models were Incorporated Into a differential mass balance In a tubular, plug-flow reaction. This gives a set of coupled, non-llnear differential equations which, when Integrated, will provide a simulation model. This model corresponds to the Integral reactor data provided by experimentation. A material balance Is written for each of the four components In our system ... 

Consider first the tubular reactor. From the material balance (Table 3.5.1), it is clear that in order to solve the mass balance the functional form of the rate expression must be provided because the reactor outlet is the integral result of reaction over the volume of the reactor. However, if only initial reaction rate data were required, then a tubular reactor could be used by noticing that if the differentials are replaced by deltas, then ... 

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. 

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