Variable-Volume Batch Reactors

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. 

The component balance for a variable-volume but otherwise ideal batch reactor can be written using moles rather than concentrations ... 

This is the piston flow analog of the variable-volume batch reactor. Equation (2.30). 

Solution The obvious way to solve this problem is to choose a pressure, calculate Oq using the ideal gas law, and then conduct a batch reaction at constant T and P. Equation (7.38) gives the reaction rate. Any reasonable values for n and kfCm. be used. Since there is a change in the number of moles upon reaction, a variable-volume reactor is needed. A straightforward but messy approach uses the methodology of Section 2.6 and solves component balances in terms of the number of moles, Na, Nb, and Nc-... 

Table 26.1 shows the effects of the two main design variables. Specifically, the results of the batch simulations for the same system as described above are given for different in-loop reactor (catalyst) volumes, recirculation rates. As would be expected increasing the catalyst volume decreases the hypochlorite concentration at all points and times through the process. Increasing the recirculation rate also appears to have a... 

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. 

If the volume C of a batch reactor depends on conversion or time, then the derivations of all of the previous equations are incorrect. We could find V(Ca) and integrate the mass-balance equation as before, but it is usually more convenient to use a different variable such as the fractional conversion X. We finally write Ca= Na / V and then substitute for Na (X) and V(X) to find Ca(X). 

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. 

For a variable volume system, / R / 0 by analogy to batch reactor. 

One important issue of the fed-batch operation is the variable volume of material in the reactor and its effect on heat transfer area. If jacket cooling is used, the heat transfer area covered by the liquid in the reactor will be proportional to the volume of the liquid at any point in time. However, if the reaction liquid is circulated through an external heat exchanger, the full heat transfer area is available throughout the batch cycle. 

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. 

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. 

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. 

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. 

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. 

Figure 1-3 shows two different types of batch reactors used for gas-phase reactions. Reactor A is a constant-volume (variable-pressure) reactor and Reactor B is a constant-pressure (variable-volume) reactor. At time r = 0, the reactants are injected into the reactor and the reaction is initiated. To see clearly the different forms the mole balance will take for eadi type of reactor, consider the follovring examples, in which the gas-phase decomposition of dimethyl ether is taking place to form methane, hydrogen, and carbon monoxide ... 

For batch-reactor systems in which the volume varies while the reaction is proceeding, the volume may usually be expressed either as a function of time alone or of conversion alone, for either adiabatic or isothermal reactors. Consequently, the variables of the differential equation (2-6) can be separated in one of the following ways ... 

Equation (3-38) applies only to a variable-volume batch reactor. If the reactor is a rigid steel container of constant volume, then of course V = Vq. For a constant-volume container, V = Vq. and Equation (3-38) can be used to calculate the pressure inside the reactor as a function of temperature and conversion. 

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 the species in terms of conversion for a variable-volume flow system, we shall use the relationships for the total concentration. The total concentration at any point in the reactor is... 

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),... 

Liquid Phase. For liquid-phase reactions in which there is no volume change, concentration is the preferred variable. The mole balances are shown in Table 4-5 in terms of concentration for the four reactor types we have been discussing. We see from Table 4-5 that we have only to specify the parameter values for the system (CAo,Uo,etc.) and for the rate law (i.e., ifcyv. .3) to solve the coupled ordiaaiy differential equations for either PFR, PBR, or batch reactors or to solve the coupled algebraic equations for a CSTR. 

Batch stirred tank reactor. Low-volume products flexibility to produce numerous grades (batch-to-batch variability) and homogenous (narrow MWD) or segregated (broad MWD). 

For gas-phase variable-volume batch reactors, like the one shown schematically in Figure 6.2, F/f(T) varies during the operation. Assuming ideal gas behavior, the... 

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