Reactor, batch steady state equations

Applying the steady state equations for the free radicals H, CH3, and C2H 5, the rate of formation of ethylene C2H4 for a constant volume batch reactor is ... 

The plug-flow tubular reactor (PFTR) rqnesents an extreme case. In it tho e is no back-mixing, and this model reactor is oftm consid ed as the spatial resolution of the temporal chemical process, inasmuch as the steady-state equations for the PFTR correspond to the time-depoident equations fm the wdl stirred batch reactor. Contonporary studies of PFTR date frmn 1956 ffilous and... 

The copolymer equation represents the composition ratio of the copolymer produced at the given instant and is rewritten in Fig. 3.46. The final correlation between composition ratio, F1/F2, and the feed ratio, x = fi/f2 = [Mi]/[M2], can be reorganized, as also given in Fig. 3.46. For a reaction in a continuous flow reactor in steady state, one can thus predict the composition of the copolymer from the monomer feed ratio. In a batch reaction, in contrast, the composition changes continuously. 

Equations 11 and 12 are only valid if the volumetric growth rate of particles is the same in both reactors a condition which would not hold true if the conversion were high or if the temperatures differ. Graphs of these size distributions are shown in Figure 3. They are all broader than the distributions one would expect in latex produced by batch reaction. The particle size distributions shown in Figure 3 are based on the assumption that steady-state particle generation can be achieved in the CSTR systems. Consequences of transients or limit-cycle behavior will be discussed later in this paper. 

Equations (1.1) to (1.3) are diflerent ways of expressing the overall mass balance for a flow system with variable inventory. In steady-state flow, the derivatives vanish, the total mass in the system is constant, and the overall mass balance simply states that input equals output. In batch systems, the flow terms are zero, the time derivative is zero, and the total mass in the system remains constant. We will return to the general form of Equation (1.3) when unsteady reactors are treated in Chapter 14. Until then, the overall mass balance merely serves as a consistency check on more detailed component balances that apply to individual substances. 

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. 

This equation predicts the intensity of segregation to decay with time in the batch reactor. It is also applicable to a steady-state plug flow system, where t is the residence 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. 

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. 

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. 

We can also obtain these expressions from the energy-balance equation for the steady-state PFTR by simply transforming dzju dt with A,/ V replacing Pw/At. The solutions of these equations for the batch reactor are mathematically identical to those in the PFTR, although the physical interpretations are quite different. 

There is of course an equation for each species, subject to conservation of total mass given by the continuily equation, and we implicitly used conservation of total mass as stoichiometric constraints, Njo — Nj)lvj are equal for all 7 in a batch (closed) system or (Fjo — Fj)/vj are equal for aU j in a steady-state continuous reactor. 

Textbooks state that the pseudo-steady-state approximation will be valid if the concentration of a species is small. However, one then proceeds by setting its time derivative equal to zero (]/t/f = 0) in the batch reactor equation, not by setting the concentration (CH3CO ) equal to zero. This logic is not obvious from the batch reactor equations because setting the derivative of a concentration equal to zero is not the same as setting its concentration equal to zero. 

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 these equations it is understood that CA may be (a) the concentration of A at a particular time in a batch reactor, (b) the local concentration in a tubular reactor operating in a steady state, or (c) the concentration in a stirred-tank reactor, possibly one of a series, also in a steady state. Let St be an interval of time which is sufficiently short for the concentration of A not to change appreciably in the case of the batch reactor the length of the time interval is not important for the flow reactors because they are each in a steady state. Per unit volume of reaction mixture, the moles of A transformed into P is thus 9LAP6t, and the total amount reacted (9lAP + 3tAQ)St. The relative yield under the circumstances may be called the instantaneous or point yield will change (a) with time in the batch reactor, or (b) with position in the tubular reactor. 

Most catalytic cycles contain several steps that are intricately related. Although the rate equations of individual steps are usually simple, the overall rate equation is often very complex. Moreover, measuring the concentrations of catalytic intermediates is a difficult task. One way of overcoming these difficulties is by assuming that the concentration of the catalytic intermediates remains constant throughout the reaction - at a steady-state concentration (see Figure 2.9a). This is a reasonable assumption even in batch reactors, because in every cycle the catalytic intermediates are formed and consumed continuously. It certainly applies to continuous processes,... 

Applying the rate expressions to Equations 1-222, 1-223, 1-224, 1-225 and 1-226, and using the steady state approximation for CH3, C2H5, and H, for a constant volume batch reactor yields ... 

Equation (19-22) indicates that, for a nominal 90 percent conversion, an ideal CSTR will need nearly 4 times the residence time (or volume) of a PFR. This result is also worth bearing in mind when batch reactor experiments are converted to a battery of ideal CSTRs in series in the field. The performance of a completely mixed batch reactor and a steady-state PFR having the same residence time is the same [Eqs. (19-5) and (19-19)]. At a given residence time, if a batch reactor provides a nominal 90 percent conversion for a first-order reaction, a single ideal CSTR will only provide a conversion of 70 percent. The above discussion addresses conversion. Product selectivity in complex reaction networks may be profoundly affected by dispersion. This aspect has been addressed from the standpoint of parallel and consecutive reaction networks in Sec. 7. 

Only the case of a batch reactor will be considered here. The transposition of equations to deal with steady-state continuous reactors is done by replacing mole numbers (mole) by molar flows (moles-1). 

The CSTR operator, Rc, has an identical term to describe accumulation under transient operation. The algebraic sum of the two other terms indicates the difference of in-flow and out-flow of that species. This operator also describes semibatch or semicontinuous operation in cases where the volume can be assumed to be essentially constant. In the more general case of variable volume, V must be included within the differential accumulation term. At steady state, it is a difference equation of the same form as the differential equation for a batch reactor. 

Continuous Stirred Tank Reactors. (CSTR). The first analysis of continuous reactors for polymerization was by Denbigh (14). He treated the same mechanisms in a CSTR that Gee and Melville (21) had treated in a batch reactor. The problem is simpler in a steady state CSTR since the equation for each dead and live specie is an algebraic rather than a differential equation. These are solved sequentially. The PSSA is not needed. He predicted a narrower molecular weight distribution for a continuous chain polymerization than for the same polymerization carried... 

The force distribution that gives minimum lost work is uniform over the length of the steady-state reactor or over time in a batch reactor. Since the affinity A < 0 for a spontaneous reaction, A is a negative constant. Equation (8.195) indicates that the force is close to constant when we have... 

For multiple reactions occurring in either a semibatch or batch reactor, Equation (9-18) can be generalized in the same manner as the steady-state energy balance, to give... 

In an ideal continuous stirred tank reactor, composition and temperature are uniform throughout just as in the ideal batch reactor. But this reactor also has a continuous feed of reactants and a continuous withdrawal of products and unconverted reactants, and the effluent composition and temperature are the same as those in the tank (Fig. 7-fb). A CSTR can be operated under transient conditions (due to variation in feed composition, temperature, cooling rate, etc., with time), or it can be operated under steady-state conditions. In this section we limit the discussion to isothermal conditions. This eliminates the need to consider energy balance equations, and due to the uniform composition the component material balances are simple ordinary differential equations with time as the independent variable ... 

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