Equations batch concentrations

We want this concentration to be achieved at the end of the fed-batch interval when t = tfuii = VfuulQin- Equate the concentrations in Equations (14.4) and (14.5) and solve for Qi . The solution is numerical. 

Equation (20-70) is the unsteady-state component mass balance for fed-batch concentration at constant retentate volume. Integration yields the equations for concentration and yield in Table 20-19. 

Quantitative Treatment, Plug Flow or Batch Reactor. In Chapter 3 we developed the equations relating concentration with time for all components of the unimolecular-type reactions... 

Equations 5-146, 5-149, and 5-152 are first order differential equations. The concentration profiles of A, B, C, and the volume V of the batch using Equation 5-137 is simulated with respect to time using the Runge-Kutta fourth order numerical method. 

Our concern in the last chapter was to get a feel for the way in which the reaction rate varies with composition and temperature. Here we wish to sec how the composition varies in time as the reaction proceeds isothermally in a batch reactor. When we come to discuss different types of reactor we shall have to deal with variations of temperature and hence of the rate constants, but here they will be assumed to be constant throughout the reaction. The reaction rate will depend only on the composition, but this of course will vary during the reaction and we shall have to solve differential equations. Sometimes we shall work with the extent of reaction, sometimes with concentrations of reactants or products. No apology is made for this variety of approach since it is important that the student be versatile with the use of different variables and develop an eye for those that will give the simplest form of a solution. The use of the extent is a routine matter, useful for avoiding mistakes in complex situations, but in simpler cases it is often possible to write down the differential equations for concentrations by inspection. From Sec. 5.2 onward, the rate constants will be denoted by lower case fc s with a variety of suffixes, the concentrations by or the lower case letter corresponding to the species. 

Calculation of Conversion and Reactor Size. The most important use of kinetics for the engineer is in the calculation of reactor size. In the batch-reactor calculations, the typical kinetic equations involving concentration terms can be used without alteration. In this type of calculation it is usually desirable to calculate either the length of time per batch for a reactor of a given size or the amount of initial charge needed if the production rate is specified for a given degree of conversion. 

A quantitative estimate of the concentration of any solute (microsolute or macrosolute) in the solution remaining in the nth stage after discontinuous dictfiltration may be developed as follows. Focus on Figure 6.4.9(c). Consider vessel 1, containing a solute concentration C/oL in a solution volume Vfo. If we carry out the batch concentration for time ti to reduce the solution volume to V/r (<< Vfo), then the retentate concentration in solution volume V/fi is (by equation (6.4.101))... 

A common altemative used where possible is the diafUtration mode with a crossflow UF membrane unit and concentrate recycle (Figure 7.2.5(e)). Here the solution concentration and viscosity are not allowed to increase due to the continuous addition of buffer replacing the permeate volume lost Equations developed in Section 6.4.2.1 for well-stirred UF cells having continuous diafUtration may be used here with appropriate care since we can treat the crossflow UF device as a blackbox far the purpose of an overall process mass balance and solute selectivity analysis. Similarly, the equations developed in Section 6.4.2.1 for a batch concentration process may be utilized here to determine various quantities, such as the yield of macrosolute, retentate concentration, etc. 

In TBP extraction, the yeUowcake is dissolved ia nitric acid and extracted with tributyl phosphate ia a kerosene or hexane diluent. The uranyl ion forms the mixed complex U02(N02)2(TBP)2 which is extracted iato the diluent. The purified uranium is then back-extracted iato nitric acid or water, and concentrated. The uranyl nitrate solution is evaporated to uranyl nitrate hexahydrate [13520-83-7], U02(N02)2 6H20. The uranyl nitrate hexahydrate is dehydrated and denitrated duting a pyrolysis step to form uranium trioxide [1344-58-7], UO, as shown ia equation 10. The pyrolysis is most often carried out ia either a batch reactor (Fig. 2) or a fluidized-bed denitrator (Fig. 3). The UO is reduced with hydrogen to uranium dioxide [1344-57-6], UO2 (eq. 11), and converted to uranium tetrafluoride [10049-14-6], UF, with HF at elevated temperatures (eq. 12). The UF can be either reduced to uranium metal or fluotinated to uranium hexafluoride [7783-81-5], UF, for isotope enrichment. The chemistry and operating conditions of the TBP refining process, and conversion to UO, UO2, and ultimately UF have been discussed ia detail (40). 

Equations 5-110, 5-112, 5-113, and 5-114 are first order differential equations and the Runge-Kutta fourth order numerical method is used to determine the concentrations of A, B, C, and D, with time, with a time increment h = At = 0.5 min for a period of 10 minutes. The computer program BATCH57 determines the concentration profiles at an interval of 0.5 min for 10 minutes. Table 5-6 gives the results of the computer program and Figure 5-16 shows the concentration profiles of A, B, C, and D from the start of the batch reaction to the final time of 10 minutes. 

Once the microbiocide is selected, a method of application should be considered. The chemical can be introduced to the system by either batch treatment, continuous treatment or by a combination of both. For batch treatment, NACE provides an equation given below. This equation can be used to determine the concentration of chemical at any time during the eight hour period. The equation is... 

The distribution coefficient can be determined by batch experiments in which a small known quantity of resin is shaken with a solution containing a known concentration of the solute, followed by analysis of the two phases after equilibrium has been attained. The separation factor, a, is used as a measure of the chromatographic separation possible and is given by the equation,... 

There is no cell removal from the batch vessel and the cell propagation rate is proportional to specific growth rate, /jl (h 1), using the differential growth equation the cell concentration with respect to the time is ... 

The effect of substrate concentration on specific growth rate (/i) in a batch culture is related to the time and p,max the relation is known as the Monod rate equation. The cell density (pcell) increases linearly in the exponential phase. When substrate (S) is depleted, the specific growth rate (/a) decreases. The Monod equation is described in the following equation ... 

Equation (1.24) is arguably the most important result in chemical reaction engineering. It shows that the concentration of a reactant being consumed by a first-order batch reaction decreases exponentially. Dividing through by Oq gives the fraction unreacted,... 

There are two uses for Equation (2.36). The first is to calculate the concentration of components at the end of a batch reaction cycle or at the outlet of a flow reactor. These equations are used for components that do not affect the reaction rate. They are valid for batch and flow systems of arbitrary complexity if the circumflexes in Equation (2.36) are retained. Whether or not there are spatial variations within the reactor makes no difference when d and b are averages over the entire reactor or over the exiting flow stream. All reactors satisfy global stoichiometry. 

Find the batch reaction time that maximizes the concentration of component B in Problem 2.10. You may begin with the solution of Problem 2.10 or with Equation (2.23). 

Compare these results with those of Equation (2.22) for the same reactions in a batch reactor. The CSTR solutions do not require special forms when some of the rate constants are equal. A plot of outlet concentrations versus t is qualitatively similar to the behavior shown in Figure 2.2, and i can be chosen to maximize bout or Cout- However, the best values for t are different in a CSTR than in a PFR. For the normal case of bi = 0, the t that maximizes bout is a root-mean, t ix = rather than the log-mean of... 

In principle, Equation (7.28) is determined by equating the rates of the forward and reverse reactions. In practice, the usual method for determining Kkinetic is to run batch reactions to completion. If different starting concentrations give the same value for Kkinetic, the functional form for Equation (7.28) is justified. Values for chemical equilibrium constants are routinely reported in the literature for specific reactions but are seldom compiled because they are hard to generalize. 

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