Batch reactors differential

THEORETICAL MODEL FOR ELECTROCHEMICAL BATCH REACTOR Differential mass balance for Ag(II)... 

The differential reactor is the second from the left. To the right, various ways are shown to prepare feed for the differential reactor. These feeding methods finally lead to the recycle reactor concept. A basic misunderstanding about the differential reactor is widespread. This is the belief that a differential reactor is a short reactor fed with various large quantities of feed to generate various small conversions. In reality, such a system is a short integral reactor used to extrapolate to initial rates. This method is similar to that used in batch reactor experiments to estimate... 

The most important characteristic of an ideal batch reactor is that the contents are perfectly mixed. Corresponding to this assumption, the component balances are ordinary differential equations. The reactor operates at constant mass between filling and discharge steps that are assumed to be fast compared with reaction half-lives and the batch reaction times. Chapter 1 made the further assumption of constant mass density, so that the working volume of the reactor was constant, but Chapter 2 relaxes this assumption. 

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. 

Suppose we perform an organic synthesis in a batch reactor where the desired molecule is the intermediate and not the end product. It is then very important that we know how long we should let the reaction run to obtain the highest yield of the intermediate. Setting the differential d[I]/dt in Eq. (99) equal to zero and substituting Eq. (102) into Eq. (99) we find the time, at which the maximum is reached - and by inserting Wx in Eq. (102) the corresponding optimal concentration of the intermediate ... 

Differential equations Batch reactor with first-order kinetics. Analytical or numerical solution with analytical or numerical parameter optimisation (least squares or likelihood). Batch reactor with complex kinetics. Numerical integration and parameter optimisation (least squares or likelihood). 

A simulation model needs to be developed for each reactor compartment within each time interval. An ideal-batch reactor has neither inflow nor outflow of reactants or products while the reaction is carried out. Assuming the reaction mixture is perfectly mixed within each reactor compartment, there is no variation in the rate of reaction throughout the reactor volume. The design equation for a batch reactor in differential form is from Chapter 5 ... 

Thus within each time interval, the batch reactor can be modeled using Equation 5.39. This differential form of the design equation reflects the fundamental dynamics of a... 

Thus, the design equations for a batch reactor for the optimization of a temporal superstructure can be based on differential or algebraic equations. 

In reality, it is the discreteness of tasks that differentiates batch processes from their continuous counterparts. To illustrate, Fig. 1.1a shows a typical batch reactor with all the tasks comprising the entire batch reaction. On the other hand, Fig. 1.1b depicts a typical continuous reactor at steady-state. The discreteness of tasks in Fig. 1.1a is evident, which is not the case in Fig. 1.1b. Consequently, it is fair to deem batch processes distributed in time , whilst continuous processes, at steady-state, are frozen in 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. 

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) ... 

Heat and material balances of a batch reactor are derived in Section 2.6.2. In the present instance, the differential heat balance is Heat of reaction + Heat tranfer = Sensible heat gain or... 

A set of first order differential equations descriptive of the molar concentrations for polymeric species is given. Species, less than size n, are saturated and, therefore, accmulate only within the batch reactor and do not participate in branch/cross-llnk reactions. Molecules greater in size than n are unsaturated and will experience the reaction described by branch formation. 

Where the composition within the reactor is uniform (independent of position), the accounting may be made over the whole reactor. Where the composition is not uniform, it must be made over a differential element of volume and then integrated across the whole reactor for the appropriate flow and concentration conditions. For the various reactor types this equation simplifies one way or another, and the resultant expression when integrated gives the basic performance equation for that type of unit. Thus, in the batch reactor the first two terms are zero in the steady-state flow reactor the fourth term disappears for the semibatch reactor all four terms may have to be considered. 

Differential (flow) reactor Integral (plug flow) reactor Mixed flow reactor Batch reactor for both gas and solid... 

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. 

Kinetic data are frequently acquired in continuous reactors rather than batch reactors. These data permit one to determine whether a process has come to steady state and to examine activation and deactivation processes. These data are analyzed in a similar fashion to that discussed previously for the batch reactor, but now the process variables such as reactant flow rate (mean reactor residence time) are varied, and the composition will not be a function of time after the reactor has come to steady state. Steady-state reactors can be used to obtain rates in a differential mode by maintaining conversions small. In this configuration it is particularly straightforward to vary parameters individually to find rates. One must of course wait until the reactor has come to steady state after any changes in feed or process conditions. 

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... 

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. 

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-... 

The rates of liquid-phase reactions can generally be obtained by measuring the time-dependent concentrations of reactants and/or products in a constant-volume batch reactor. From experimental data, the reaction kinetics can be analyzed either by the integration method or by the differential method ... 

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,... 

If the compositions vary with position in the reactor, which is the case with a tubular reactor, a differential element of volume SV, must be used, and the equation integrated at a later stage. Otherwise, if the compositions are uniform, e.g. a well-mixed batch reactor or a continuous stirred-tank reactor, then the size of the volume element is immaterial it may conveniently be unit volume (1 m3) or it may be the whole reactor. Similarly, if the compositions are changing with time as in a batch reactor, the material balance must be made over a differential element of time. Otherwise for a tubular or a continuous stirred-tank reactor operating in a steady state, where compositions do not vary with time, the time interval used is immaterial and may conveniently be unit time (1 s). Bearing in mind these considerations the general material balance may be written ... 

In another example of differential heating, a two-phase water/chloroform system (1 1 by volume) was heated in a microwave batch reactor (MBR)65. About 40 s after commencement, the temperatures of the aqueous and organic phases were 105 and 48°C, respectively, because of the differences in the dielectric properties of each solvent. A sizeable differential could be maintained for several minutes before cooling was begun. Differential heating is particularly advantageous for Hofmann eliminations. In a typical example,... 

For the determination of reaction parameters, as well as for the assessment of thermal safety, several thermokinetic methods have been developed such as differential scanning calorimetry (DSC), differential thermal analysis (DTA), accelerating rate calorimetry (ARC) and reaction calorimetry. Here, the discussion will be restricted to reaction calorimeters which resemble the later production-scale reactors of the corresponding industrial processes (batch or semi-batch reactors). We shall not discuss thermal analysis devices such as DSC or other micro-calorimetric devices which differ significantly from the production-scale reactor. 

Equations 7.4 and 7.5 form a system of differential equations for which no analytical solution is known. Thus, the description of the behavior of the semi-batch reactor with time requires the use of numerical methods for the integration of the differential equations. Usually, it is convenient to use parameters which are more process-related to describe the material balance. One is the stoichiometric ratio between the two reactants A and B ... 

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. 

Theories are not used directly, as in the discussion presented in Sect. 3.1, but allow building a mathematical model that describes an experiment in the unambiguous language of mathematics, in terms of variables, constants, and parameters. As an example, when considering the identification of kinetic parameters of chemical reactions from isothermal experiments performed in batch reactors, the relevant equations of mass conservation (presented in Sect. 2.3.1) give a set of ordinary differential equations in the general form... 

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. 

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