Batch reactors with variable volume

Batch Reactors with Variable Volume Although variable volume batch reactors are seldom encountered because they are usually solid steel containers, we will develop the concentrations as a function of conversion because (1) they have been u.sed to collect reaction data for gas-phase reactions, and (2) the development of the equations that express volume as a function of conversion will facilitate analyzing flow systems with variable volumetric flow rates. 

For the batch reactor with variable volume (constant pressure), the gas-phase reaction is... 

The above derivation is for data taken in a constant-volume batch reactor, with t denoting the residence time. The derivation for flow reactors is closely analogous, with space-time as the corresponding variable. 

The design equation for gaseous variable-volume batch reactors was derived under two assumptions (i) AU the species are gaseous, and (ii) the mixture behaves as an ideal gas. In some operations, one or more of the species (especially heavier products generated by the reaction) may be saturated vapor. In this case, any additional amount generated will be in a condensed phase (liquid). While the ideal gas relation provides a reasonable approximation for the volume of species in the vapor phase, it cannot be applied for dieir volume in the liquid phase. Below, we modify the design equations for a variable-volume batch reactor with saturated vapors. 

Flow Reactors with Variable Volumetric Flow Rate. An expression similar to Equation (3-38) for a variable-volume batch reactor exists for a variable-volume flow system. To derive the concentrations of each species in terms of conversion for a variable-volume flow system, we shall use the relationships for the total concentration. The total concentration. Cj-. at any point in the reactor is the total molar flow rate. Fj, divided by volumetric flow rate v [cf. Equation (3-27). In the gas phase, the total concentration is also found from the gas law. Cr = P/ZRT. Equating these two relationships gives... 

ERA The reaction was performed in a batch reactor and in the gas phase. 10% N2 was introduced in the reactor at 2 atm and 450°C. After 50 min, the pressure was 3.3 atm. The reaction is irreversible and of first order. Calculate the specific reaction rate constant. If the same reaction would be performed in a piston system with variable volume, how volume changes keeping pressure constant at 2 atm and considering the same conversion as before Calculate the initial concentration ... 

Equation 3.93 can be compared with Equ. 3.91 for the true steady state in a CSTR. The two are formally identical. The difference is in the real meaning of the factor D in Equ. 3.90 it represents a dilution effect in the case of a chemostat. In the case of a semicontinuous stirred vessel with variable volume, the factor D x in Equ. 3.93 describes the decrease in cell concentration due to the change of the volume in the reactor (Dunn and Mor, 1975). Similar equations can be formulated for all other types of fed-batch processes using the appropriate modification of Equ. 3.86. 

P5-32], What live things are wrong with this solution The reaction in P5-10, is carried out in a variable volume, constant-pressure batch reactor with pure A initially. If it takes 2 hours for the volume to decrease by a factor of 2 [i.e.. from 2 dm to I dm ) when the initial concentration A is 1.0 mol/dnv . what is the specific reaction rale constant ... 

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. 

A variable-volume batch reactor is a constant-pressure (piston-like) closed tank. On the other hand, a variable-pressure tank is a constant-volume batch reactor (Fogler, 1999). Thus, in batch reactors, the expansion factor is used only in the case of a constant-pressure tank whereas and not in a constant-volume tank, even if the reaction is realized with a change in the total moles. However, in continuous-flow reactors, the expansion factor should be always considered. In the following section and for the continuous-flow reactors, the volume V can be replaced by the volumetric flow rate Q, and the moles N by the molar flow rate F in all equations. 

The fed batch reactor (FBR) is a reactor where fresh nutrients are added to replace those already used. The rate of the feed flow u may be variable, and there is no outlet flowrate from the fermenter. As a consequence of feeding, the reactor volume changes with respect to time. Figure 11-22 illustrates a simple fed-batch reactor. The balance equations are ... 

Batch Reactor. In a batch reactor there are no inlet or outlet streams In = Out = 0. The total feed is charged into the reactor at the beginning and no withdrawal is made until the desired conversion level has been reached. Hence a reaction process occurring in a batch reactor is an unsteady one. All variables change with time. In addition, we assume that it is a perfectly mixed batch reactor, so that the concentrations of the reaction components, reactants or products are the same over the whole reactor volume. This assumption allows us to consider applying the mole balance equation across the whole reactor. With the term reactor we mean the space where the reaction(s) take place. For liquid phase reactions the reactor volume is smaller than the size of the physical reactor. It is the volume of the liquid phase, where the reaction ) take(s) place. 

An ideal batch reactor is a perfectly stirred tank of constant volume with no mass transfer from or to the outside. There is a single residence time, which is simply the duration of the reaction. Generally, a batch reactor is operated isothermally and therefore the reaction temperature may be considered as an independent variable. 

Three forms of the reactor operator, R(Y), are shown in Figure 3. These are generally differential operators which operate on each monomer and polymer species to describe the effects of accumulation and the physical processes which move material in and out of the reactor or reactor element. The concentration of a specific species is given by the variable Y. In a simple batch reactor, the reactor operator, RB, is merely defined as the rate of accumulation of a certain species with time per unit volume of reactor—i.e., the rate of change of concentration of the species. 

Another variable-volume situation, which occurs much less frequently, is in batch reactors where volume changes with time. Examples of this situation are the combustion chamber Of the internal-combustion engine and the expanding gases within the breech and barrel of a firearm as it is fired. 

Rewrite the design equation in terms of the measured variabte. When there is a net increase or decrease in the totai number of moles in a gas phase reaction, the reaction order may be determined from experiments performed with a constant-volume batch reactor by monitoring the total pressure as a function of time. The total pressure data should not be converted to conversion and then analyzed as conversion-time data just because the design equations are written in terms of the variable conversions. Rather, transform the design equation to the measured variable, which in this case is pressure. Consequently, we need to express the concentration in terms of total pressure and then substitute for the concemtation of A in Equation (E5-I.1),... 

We continue the analysis of gaseous, variable-volume batch reactors and consider now chemical reactions with two reactants of the general form... 

This latter technique of Himmelblau, Jones, and Bischoff (H-J-B) has proved to be efficient in various practical situations with few, scattered, data available for complex reaction kinetic schemes (see Ex. 1.6.2-1). Recent extensions of the basic ideas are given by Eakman, Tang, and Gay [48,49, 50]. It should be pointed out, however, that the problem has been cast into one of linear regression at the expense of statistical rigor. The independent variables , X jp, do not fulfill one of the basic requirements of linear regression that the Xi p have to be free of experimental error. In fact, the X p are functions of the dependent variables C/tf) and this may lead to estimates for the parameters that are erroneous. This problem will be discussed further in Chapter 2, when the estimation of parameters in rate equations for catalytic reactions will be treated. Finally, all of the methods have been phrased in terms of batch reactor data, but it should be recognized that the same formulas apply to plug flow and constant volume systems, as will be shown later in this book. 

Reactors with complete mixing may be subdivided into batch and continuous types. In a batch type reactor with complete mixing the composition is uniform throughout the reactor. Consequently, the continuity equation may be written for the entire contents, not only over a volume element The composition varies with time, however, so that a first-order ordinary differential equation is obtained, with time as variable. The form of this equation is analogous with that for the... 

When the rate of reaction is given and a feed is to be converted to a value of, say x, Eq. 9.1-2 permits the required reactor volume V to be determined. This is one of the design problems that can be solved by means of Eq. 9.1 -2. Both aspects— kinetic analysis and design calculations—are illustrated further in this chapter. Note that Eq. 9.1-2 does not contain the residence time explicitly, in contrast with the corresponding equation for the batch reactor. E/f o> s expressed here in hr m /kmol 4—often called space time—is a true measure of the residence time only when there is no expansion or contraction due to a change in number of moles or other conditions. Using residence time as a variable offers no advantage since it is not directly measurable—in contrast with V/F q. 

Consider the fed-batch reactor in Figure 12.3. Initially, the reactor is charged with an aqueous volume, V 0, containing E. coli cells (referred to as biomass) in concentration, X 0). Then, an aqueous solutiot of sucrose (referred to as the substrate i.e., the substance being acted on) at a concentration, S/(g/L), is fed to the reactor at a variable flow rate, F[t) (g/hr). The reactor holdup, V /, contains E. coli cells in concentration, X f) (g/L), penicillin product in concentration, P r (g/L), and sucrose in concentration, S f (g/L). Using Monod kinetics, the specific growth rate of the cell mass (g cell growth/g cell) is... 

Steady state. We would, however, like to demonstrate that it is still possible to achieve the specific concentration associated with that steady state. Certainly, this will require a special operating regime to achieve (which shall be detailed in the following sections), but this will always be with the intention that the reactor is operated under bateh conditions—that is, with a distinct cycle time where the state variables of the batch reactor (volume, concentration, etc.) do, in fact, vary for the duration of this period. 

For kinetic purposes, the variation of the concentration or the pressure with reaction time is accompanied in a batch reactor. In a continuous or open system, the pressure is constant and the concentrations or molar flows of the reactants and products are accompanied in the course of the reaction or along the reactor. The time is substituted by an equivalent variable called space time. The space time takes into account the inlet volumetric flow of the reactants and the volume of the reactor, and, thus, has unit of time, here designated by ... 

Monitoring of large-scale fed-batch manufacture of baker s yeast was also possible with the electronic nose [33]. The cultivation took place in a 200-m3 bubble-column reactor. The monitoring procedure is complicated by the large phase variation and circulation times in the bioreactor. On the 200-m3 scale, ethanol and biomass were predicted but with lower accuracy than in the laboratory (10%). The data was compensated for increasing reactor liquid volume and aeration rate during the fed-batch cycle, simply by including these variables in the inputs to the ANN. 

Winyl polymerization as a rule is sensitive to a number of reaction variables, notably temperature, initiator concentration, monomer concentration, and concentration of additives or impurities of high activity in chain transfer or inhibition. In detailed studies of a vinyl polymerization reaction, especially in the case of development of a practical process suitable for production, it is often desirable to isolate the several variables involved and ascertain the effect of each. This is difficult with the conventional batch polymerization technique, because the temperature variations due to the highly exothermic nature of vinyl polymerization frequently overshadow the effect of other variables. In a continuous polymerization process, on the other hand, the reaction can be carried out under very closely controlled conditions. The effect of an individual variable can be established accurately. In addition, compared to a batch process, a continuous process normally gives a much greater throughput per unit volume of reactor capacity and usually requires less labor. 

The experimental measurements produced concentration-time plots of ethylene oxide and ethylene glycol in the liquid phase, as shown in Figure 8.18. The physical picture of this reaction/reactor system is most closely approximated by the plug-flow gas phase, well-mixed batch liquid phase. The appropriate relationships to model this system are given in equations (8-176) to (8-178), (8-183), and (8-188). The bubble volume is variable, and the nature of the variation changes with the extent of conversion (i.e., concentration of glycol in the liquid phase), however, the pure oxide gas phase allows yg = l. The modified equations specific to this reactor are then... 

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