Batch reactor solving

Alcorn and Sullivan (1992) faced some specific and difficult problems in connection with coal slurry hydrogenation experiments. Solving these with the falling basket reactor, they also solved the general problem of batch reactors, that is, a good definition of initial conditions. The essence of their... 

The extension to multiple reactions is done by writing Equation (3.1) (or the more complicated versions of Equation (3.1) that will soon be developed) for each of the N components. The component reaction rates are found from Equation (2.7) in exactly the same ways as in a batch reactor. The result is an initial value problem consisting of N simultaneous, first-order ODEs that can be solved using your favorite ODE solver. The same kind of problem was solved in Chapter 2, but the independent variable is now z rather than t. 

Example 4.2 used the method of false transients to solve a steady-state reactor design problem. The method can also be used to find the equilibrium concentrations resulting from a set of batch chemical reactions. To do this, formulate the ODEs for a batch reactor and integrate until the concentrations stop changing. This is illustrated in Problem 4.6(b). Section 11.1.1 shows how the method of false transients can be used to determine physical or chemical equilibria in multiphase systems. 

The accumulation term is zero for steady-state processes. The accumulation term is needed for batch reactors and to solve steady-state problems by the method of false transients. 

A relatively simple example of a confounded reactor is a nonisothermal batch reactor where the assumption of perfect mixing is reasonable but the temperature varies with time or axial position. The experimental data are fit to a model using Equation (7.8), but the model now requires a heat balance to be solved simultaneously with the component balances. For a batch reactor. 

In this paper we formulated and solved the time optimal problem for a batch reactor in its final stage for isothermal and nonisothermal policies. The effect of initiator concentration, initiator half-life and activation energy on optimum temperature and optimum time was studied. It was shown that the optimum isothermal policy was influenced by two factors the equilibrium monomer concentration, and the dead end polymerization caused by the depletion of the initiator. When values determine optimum temperature, a faster initiator or higher initiator concentration should be used to reduce reaction time. 

There are a variety of limiting forms of equation 8.0.3 that are appropriate for use with different types of reactors and different modes of operation. For stirred tanks the reactor contents are uniform in temperature and composition throughout, and it is possible to write the energy balance over the entire reactor. In the case of a batch reactor, only the first two terms need be retained. For continuous flow systems operating at steady state, the accumulation term disappears. For adiabatic operation in the absence of shaft work effects the energy transfer term is omitted. For the case of semibatch operation it may be necessary to retain all four terms. For tubular flow reactors neither the composition nor the temperature need be independent of position, and the energy balance must be written on a differential element of reactor volume. The resultant differential equation must then be solved in conjunction with the differential equation describing the material balance on the differential element. 

For the case where all of the series reactions obey first-order irreversible kinetics, equations 5.3.4, 5.3.6, 5.3.9, and 5.3.10 describe the variations of the species concentrations with time in an isothermal well-mixed batch reactor. For series reactions where the kinetics do not obey simple first-order or pseudo first-order kinetics, the rate expressions can seldom be solved in closed form, and it is necessary to resort to numerical methods to determine the time dependence of various species concentrations. Irrespective of the particular reaction rate expressions involved, there will be a specific time... 

For constant fluid density the design equations for plug flow and batch reactors are mathematically identical in form with the space time and the holding time playing comparable roles (see Chapter 8). Consequently it is necessary to consider only the batch reactor case. The pertinent rate equations were solved previously in Section 5.3.1.1 to give the following results. 

In general, when designing a batch reactor, it will be necessary to solve simultaneously one form of the material balance equation and one form of the energy balance equation (equations 10.2.1 and 10.2.5 or equations derived therefrom). Since the reaction rate depends both on temperature and extent of reaction, closed form solutions can be obtained only when the system is isothermal. One must normally employ numerical methods of solution when dealing with nonisothermal systems. 

If the batch reactor operation is both nonadiabatic and nonisothermal, the complete energy balance of equation 12.3-16 must be used together with the iiaterial balance of equation 2.2-4. These constitute a set of two simultaneous, nonlincmr, first-flijer ordinary differential equations with T and fA as dependent variables and I as Iidependent variable. The two boundary conditions are T = T0 and fA = fAo (usually 0) at I = 0. These two equations usually must be solved by a numerical procedure. (See problem 12-9, which may be solved using the E-Z Solve software.)... 

The material balance for a batch reactor may be used to develop a differential equation which may be solved for the cA(t) profile (see equation 3.4-1) ... 

The operating parameter for the CSTR reactor is the liquid flow rate Q, which sets the residence time of the liquid through the ratio Q/VL and finally the conversion. From a production viewpoint, the (residence) time required to achieve a given conversion of S (or outlet concentration of S) is obtained by solving the set of Eqs. (33) and (34). The characteristics of the reactor kLa and VL must be known. In general, whereas VL is easily determined in a batch reactor, it is not in a CSTR. Rather, VL=fiLVR will be used, which requires knowledge of the liquid hold-up L. Correlations provide kLo (see below) and L characteristics for the different reactor types [3]. 

Ridelhoover and Seagrave [57] studied the behaviour of these same reactions in a semi-batch reactor. Here, feed is pumped into the reactor while chemical reaction is occurring. After the reactor is filled, the reaction mixture is assumed to remain at constant volume for a period of time the reactor is then emptied to a specified level and the cycle of operation is repeated. In some respects, this can be regarded as providing mixing effects similcir to those obtained with a recycle reactor. Circumstances could be chosen so that the operational procedure could be characterised by two independent parameters the rate coefficients were specified separately. It was found that, with certain combinations of operational variables, it was possible to obtain yields of B higher than those expected from the ideal reactor types. It was necessary to use numerical procedures to solve the equations derived from material balances. 

To solve the batch-reactor mass-balance equation, we write... 

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... 

Therefore, to find the behavior of a PFTR for kinetics that we have solved in a batch reactor, all we have to do is make the transformation tbatch —> tpFTR- The solution for the th-order irreversible reaction fi om Chapter 2 is... 

For the batch reactor we saw in the previous chapter that by switching from Ca as the composition variable to fractional conversion X, we could easily write the differential equation to be solved for compositions versus time. We prefer to use concentration units whenever possible, but, if the density is a function of composition, concentrations become cumbersome variables, and we must switch to another designation of density such as the fractional conversion X. 

Note that setting one of the terms on the left side of the equation equal to zero yields either the batch reactor equation or the steady-state PFTR equation. However, in general we must solve the partial differential equation because the concentration is a function of both position and time in the reactor. We will consider transients in tubular reactors in more detail in Chapter 8 in connection with the effects of axial dispersion in altering the perfect plug-flow approximation. 

The molar flow rate of a species in a flow reactor is Fj = vCj. The batch reactor is a closed system in which v = 0. The volumetric flow rate is ti, while thelinear velocity in a tubular reactor is u, We usually assume that the density of the fluid in the reactor does not change with conversion or position in the reactor (the constant-density reactor) because the equations for a constant-density reactor are easier to solve. 

We can calculate the value of Tmax and Cb max just as we did previously for the PFTR and batch reactors, by setting dCsIdx = 0 and solving for Cb and t,... 

Next consider the response of a PFTR with steady flow to a pulse injected at f = 0. Wc could obtain this by solving the transient PFTR equation written earher in this chapter, but we can see the solution simply by following the pulse down the reactor. (This is identical to the transformation we made in transforrning the batch reactor equations to the PFTR equations.) The S(0) pulse moves without broadening because we assumed perfect plug flow, so at position z the pulse passes at time z/u and the pulse exits the reactor at time T = L/u. Thus for a perfect PFTR the RTD is given by... 

The composition in each drop now changes with the time that the drop has been in the CSTR in the same way as if the drop were in a batch reactor. However, since the drops are in a CSTR, the time each drop stays in the reactor is given by the previous expression for p(t), and the overall conversion for any reaction is given by solving the preceding equation. 

We can of course write the six mass balances on each species and solve them in batch or continuous reactors to find the species concentrations as a function of residence time [A](t), [CH3 ](t), [CH4](t), etc. The mass-balance equations in a PFTR (or in a batch reactor by replacing r by t) are... 

Industrially relevant consecutive-competitive reaction schemes on metal catalysts were considered hydrogenation of citral, xylose and lactose. The first case study is relevant for perfumery industry, while the latter ones are used for the production of sweeteners. The catalysts deactivate during the process. The yields of the desired products are steered by mass transfer conditions and the concentration fronts move inside the particles due to catalyst deactivation. The reaction-deactivation-diffusion model was solved and the model was used to predict the behaviours of semi-batch reactors. Depending on the hydrogen concentration level on the catalyst surface, the product distribution can be steered towards isomerization or hydrogenation products. The tool developed in this work can be used for simulation and optimization of stirred tanks in laboratory and industrial scale. 

Analogous to the batch reactor with Michaelis-Menten kinetics, this equation for the residence time of the PFR can be solved directly when the kinetic constants, the inlet concentration of substrate and the desired conversion are known. 

We have seen that the basic P model has the form of a first-order partial differential Eq. (22) describing each narrow slice as a little batch reactor being transported through the reactor at constant speed. This equation was so elementary that it could be solved at sight in Eq. (30). When we added a longitudinal dispersion term governed by Fick s law and took the steady state, Eq. (40), we had a second-order o.d.e. with controversial boundary conditions. This is the model with ( ) = c(z)lcm and Pe = vLID, Da = kL/v,... 

For this study, mass transfer and surface diffusions coefficients were estimated for each species from single solute batch reactor data by utilizing the multicomponent rate equations for each solute. A numerical procedure was employed to solve the single solute rate equations, and this was coupled with a parameter estimation procedure to estimate the mass transfer and surface diffusion coefficients (20). The program uses the principal axis method of Brent (21) for finding the minimum of a function, and searches for parameter values of mass transfer and surface diffusion coefficients that will minimize the sum of the square of the difference between experimental and computed values of adsorption rates. The mass transfer and surface coefficients estimated for each solute are shown in Table 2. These estimated coefficients were tested with other single solute rate experiments with different initial concentrations and different amounts of adsorbent and were found to predict... 

Control and monitoring of the chemical reactor play a central role in this procedure, especially when batch operations are considered because of the intrinsic unsteady behavior and the nonlinear dynamics of the batch reactor. In order to meet such requirements, the following fundamental problems must be solved ... 

A more quantitative analysis of the batch reactor is obtained by means of mathematical modeling. The mathematical model of the ideal batch reactor consists of mass and energy balances, which provide a set of ordinary differential equations that, in most cases, have to be solved numerically. Analytical integration is, however, still possible in isothermal systems and with reference to simple reaction schemes and rate expressions, so that some general assessments of the reactor behavior can be formulated when basic kinetic schemes are considered. This is the case of the discussion in the coming Sect. 2.3.1, whereas nonisothermal operations and energy balances are addressed in Sect. 2.3.2. 

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