Batch constant-volume models

Similarly, the transport model could be specified generically as batch, constant volume or pressure, CSTR, plug flow, or other standard reactor design, causing a diagram to immediately appear on the screen with prompts for needed parameters and boundary conditions. At a lower lever, a graphics editor/interpreter would be used to create the flowchart and superimpose the numerical values. 

Briggs and Haldane [8] proposed a general mathematical description of enzymatic kinetic reaction. Their model is based on the assumption that after a short initial startup period, the concentration of the enzyme-substrate complex is in a pseudo-steady state (PSS). For a constant volume batch reactor operated at constant temperature T, and pH, the rate expressions and material balances on S, E, ES, and P are... 

Since the rates Rj are functions of concentrations (and of temperature), the mathematical model of an isothermal, constant volume, batch reactor is a system of ordinary differential equations with initial conditions. 

In addition to the basic continuous column model assumptions of equilibrium stages and adiabatic operation, dynamics-related assumptions are made for the batch model. Distefano (1968) assumed constant volume of liquid holdup, negligible vapor holdup, and negligible fluid dynamic lag. Although different solution strategies may be employed, the fundamental model equations are the same. 

Chain transfer to the aluminium alkyl was also deserved. Using the method of moments the authors obtained an equation for the first three moments of active, temporarily deactivated, and dead chains. As a result of a computerized search for the values of constants, based on the model and on the experimental data obtained in a batch reactor (volume = 131), some of the values were found to differ considerably from those published in the literature. 

An irreversible reaction A - B is occurring in an isothermal, constant-volume batch reactor. The concentration has been measured at various times. Find an objective function (i.e., SS ) that is suitable for fitting the model 01 = —ka -. 

These are constitutive kinetics that we need to complete the model for this type of reaction taking place in a constant volume batch reactor. The component equations are ... 

It is instructive to compare the fed-batch model with the model for a constant-volume chemostat (see Equs. 3.90 and 3.91). Equation 6.3a,b is formally identical with Equ. 3.92a,b, but it is important to note that the physical meaning of the terms is not identical. Comparing the term x JV in Equs. 3.92 and 6.3 it can be seen that the origin of this term for the fed batch is the expression x -dV/dt it thus represents a decrease in cell concentration due to the volume change that arises from inlet flow rate On the other hand, —F x in Equ. 6.3, the chemostat, is a washout term expressing the mass flow rate of cells that leave with the outgoing stream. A fed batch can be compared with a constant-volume chemostat whose feed rate is decreasing slowly. 

In modeling a batch reactor, we assume there is no inflow or outflow of material and that the reactor is well mixed. For most liquid-phase reactions, the density change with reaction is usually small and can be neglected (i.e., U = Vo). In addition, for gas-phase reactions in which the batch reactor volume remains constant, we also have V = Vo-... 

For the semi-batch stirred tank reactor, the model was based on the following assumptions the reactor is well agitated, so no concentration differences appear in the bulk of the liquid gas-liquid and liquid-solid mass transfer resistances can prevail and finally, the liquid phase is in batch, while hydrogen is continuously fed into the reactor. The hydrogen pressure is maintained constant. The liquid and gas volumes inside the reactor vessel can be regarded as constant, since the changes of the fluid properties due to reaction are minor. The total pressure of the gas phase (P) as well as the reactor temperature were continuously monitored and stored on a PC. The partial pressure of hydrogen (pnz) was calculated from the vapour pressure of the solvent (pvp) obtained from Antoine s equation (pvpo) and Raoult s law ... 

Figure 4 compares several of these models with respect to the nature of the constants that each uses. The simplest model (linear sorption or Ai ) is the most empirical model and is widely used in contaminant transport models. values are relatively easy to obtain using the batch methods described above. The Aid model requires a single distribution constant, but the Aid value is conditional with respect to a large number of variables. Thus, even if a batch Aid experiment is carefully carried out to avoid introduction of extraneous effects such as precipitation, the Aid value that is obtained is valid only for the particular conditions of the experiment. As Figure 4 shows, the radionuclide concentration, pH, major and minor element composition, rock mineralogy, particle size and solid-surface-area/solution volume ratio must be specified for each Aid value. 

The early kinetic model by Smith and Ewart was based on Harkin s mechanistic understanding of the batch process. The particle population balances were written for a stationary state assuming that the rate of formation of particles with n radicals equals the rate of their disappearance (see equation at the bottom of this page). Where / , is the rate of radical entry into a particle (m /sec) is the rate constant for radical exit (m/sec) S is the particle surface area (m ) ktp is the rate constant for bimolecular termination in the particles (m /sec) and o is the particle volume. According to Smith and Ewart three limiting cases can be identified ... 

The inhibition of the catalyst in the batch reachon was also studied by detailed kinetic analyses using a Langmuir-Hinshelwood model that allows quantification of its nature and extent. The adsorption equilibrium constant for para-methoxyacefophenone exceeds by a factor of af least 6 the adsorption equilibrium constant for any reacfanf, and fhe occupancy of fhe infracrystalline volume of fhe zeolite by para-mefhoxyacetophe-none increases rapidly with conversion, thereby reducing the access of reactants to the catalytic sites and resulting in catalyst deactivation as the conversion increases. ... 

Because the mathematical form of the fed-batch model Equ. 3.92a,b is identical to that of the chemostat Equ. 6.3a,b, it can be concluded that the fed batch will behave analogously. A dynamic steady state will be achieved for sufficiently low flow rates such that the specific growth rate is maintained exactly equal to the dilution rate F JV. This phenomenon has been identified previously, and the dynamic steady state has been termed a quasi-steady state. It is characterized by a constant value of which must exist because jU = Fin/K Since the volume is increasing steadily, must be maintained by a decrease in therefore, ds Jdt is not zero. Computer simulations obtained by solving Equ. 3.92a,b together with Equ. 6.39b numerically show that the phenomena do indeed occur as described. The computer simulation shown in Fig. 3.37 indicates that substrate concentration steadily decreases during the quasi-steady state ju = D, in order to maintain the quasi-steady state as the volume increases (Dunn and Mor, 1975). When the quasi-steady state is achieved, the equality of ju and F JV leads to the following relationship for... 

A semi-batch reactor is more difficult to analyze mathematically because at least one of the reactant or product species enters or leaves the system boundaries, thus specific applications should be modeled [1,5]. However, the most typical application for a semi-batch reactor is the presence of one reactant initially contained in a stirred tank reactor and a second reactant continuously added to the reactor, with no flow out of the reactor. The addition of a gas to participate in a liquid-phase reaction is one of the more common situations involving a semi-batch reactor, especially because the rate of addition of the gas can be controlled to keep its partial pressure essentially constant as well as providing quantitative information about the rate of reaction. In addition, there is frequently little or no change in the volume of the liquid phase. Well-mixed autoclave reactors coupled with gas pressure controllers, mass flow meters and computers can nicely provide continuous, real-time rate data related to heterogeneous catalysts used in such gas/liquid systems [6-8]. Again, it must be emphasized that experiments must be performed and/or calculations made to verify that no heat or mass transfer limitations exist. 

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