Kinetics material balances

The differential material balances contain a large number of physical parameters describing the structure of the porous medium, the physical properties of the gaseous mixture diffusing through it, the kinetics of the chemical reaction and the composition and pressure of the reactant mixture outside the pellet. In such circumstances it Is always valuable to assemble the physical parameters into a smaller number of Independent dimensionless groups, and this Is best done by writing the balance equations themselves in dimensionless form. The relevant equations are (11.20), (11.21), (11.22), (11.23), (11.16) and the expression (11.27) for the effectiveness factor. 

The Michaehs-Menten equation and other similar nonhnear expressions characterize immobihzed enzyme kinetics. Therefore, for a spherical porous carrier particle with enzyme molecules immobilized on its external as well as internal surfaces, material balance of the substrate will result in the following ... 

In general, if a reaction leads to two or more products, and the products are not formed at equal rates, there must be an intermediate to account for the material balance. (The converse, of course, is not necessarily true, for an intermediate may be present at vanishingly low concentrations and yet be kinetically important.)... 

The failure to identify the necessary authigenic silicate phases in sufficient quantities in marine sediments has led oceanographers to consider different approaches. The current models for seawater composition emphasize the dominant role played by the balance between the various inputs and outputs from the ocean. Mass balance calculations have become more important than solubility relationships in explaining oceanic chemistry. The difference between the equilibrium and mass balance points of view is not just a matter of mathematical and chemical formalism. In the equilibrium case, one would expect a very constant composition of the ocean and its sediments over geological time. In the other case, historical variations in the rates of input and removal should be reflected by changes in ocean composition and may be preserved in the sedimentary record. Models that emphasize the role of kinetic and material balance considerations are called kinetic models of seawater. This reasoning was pulled together by Broecker (1971) in a paper called "A kinetic model for the chemical composition of sea water."... 

In integral analysis concentration-versus-time (or equivalently concentration-versus-distance from the inlet of the integral flow reactor) data are known. Kinetic expressions to be determined are incorporated into the differential material balance equations ... 

The following reaction system of industrial importance was studied a liquid reactant A simultaneously reacts with a liquid B and a gas G to give products R and S and by-products lumped as L. For proprietary reasons, the names of the species are not specified. The reaction is carried out in semibatch mode. The amounts of A, B, G, R, and S at the end of experiments leading to identification of the kinetic model were determined analytically while the amount of L was evaluated from the material balance. 

At each temperature the simple Monod kinetic model can be used that can be combined with material balances to arrive at the following unstructured model... 

The Flory principle allows a simple relationship between the rate constants of macromolecular reactions (whose number is infinite) with the corresponding rate constants of elementary reactions. According to this principle all chemically identical reactive centers are kinetically indistinguishable, so that the rate constant of the reaction between any two molecules is proportional to that of the elementary reaction between their reactive centers and to the numbers of these centers in reacting molecules. Therefore, the material balance equations will comprise as kinetic parameters the rate constants of only elementary reactions whose number is normally rather small. 

Application of the Balzhinimaev model requires assumptions about the reactor and its operation so that the necessary heat and material balances can be constructed and the initial and boundary conditions formulated. Intraparticle dynamics are usually neglected by introducing a mean effectiveness factor however, transport between the particle and the gas phase is considered. This means that two heat balances are required. A material balance is needed for each reactive species (S02, 02) and the product (SO3), but only in the gas phase. Kinetic expressions for the Balzhinimaev model are given in Table IV. 

Kapur (K3) has formulated a set of kinetic equations for the agglomerate population and crushed material balance as follows ... 

Since most kinetic studies of this type involve initial rate experiments, it is usually necessary to rederive this expression in terms of the initial concentrations and the initial rate. From material balance considerations... 

The analysis of simultaneous diffusion and chemical reaction in porous catalysts in terms of effective diffusivities is readily extended to geometries other than a sphere. Consider a flat plate of porous catalyst in contact with a reactant on one side, but sealed with an impermeable material along the edges and on the side opposite the reactant. If we assume simple power law kinetics, a reaction in which there is no change in the number of moles on reaction, and an isothermal flat plate, a simple material balance on a differential thickness of the plate leads to the following differential equation... 

Based upon this set of elementary reactions, a series of coupled kinetic differential equations may be derived by taking material balances over the various reaction species, as shown below ... 

A laminar-flow reactor (LFR) is rarely used for kinetic studies, since it involves a flow pattern that is relatively difficult to attain experimentally. However, the model based on laminar flow, a type of tubular flow, may be useful in certain situations, both in the laboratory and on a large scale, in which flow approaches this extreme (at low Re). Such a situation would involve low fluid flow rate, small tube size, and high fluid viscosity, either separately or in combination, as, for example, in the extrusion of high-molecular-weight polymers. Nevertheless, we consider the general features of an LFR at this stage for comparison with features of the other models introduced above. We defer more detailed discussion, including applications of the material balance, to Chapter 16. 

PFR OS integral reactor. In Figure 3.8, the entire vessel indicated from sampling points Sjn to Sout, over which a considerable change in fA or cA would normally occur, could be called an integral PFR. It is possible to obtain values of kinetics parameters by means of such a reactor from the material balance equation 2.4-4 rearranged as... 

Briggs and Haldane (1925) proposed an alternative mathematical description of enzyme kinetics which has proved to be more general. The Briggs-Haldane model is based upon the assumption that, after a short initial startup period, the concentration of the enzyme-substrate complex is in a pseudo-steady state. Derivation of the model is based upon material balances written for each of the four species S, E, ES, and P. 

The number of 144-h intervals in 30 d is 24(30)/144 = 5. During each interval, the reaction follows pseudo-first-order kinetics. Thus, from equation (B) and the material balance, equation 2.2-10, applied to glucose,... 

We can, however, consider the stability of each of the three operating points in Example 14-7 with respect to the inevitable small random fluctuations in operating conditions, including cA, in steady-state operation. Before doing this, we note some features of the rate law as revealed in Figure 14.4. There is a maximum value of (- rA) at cA = 1.166 mol m-3. For cA < 1.166, the rate law represents normal kinetics ( rA) increases as cA increases for cA > 1.166, we have abnormal kinetics (—rA) decreases as cA increases. We also note that (-rA) in equation (C), the rate law, represents the (positive) rate of disappearance of A by reaction within the CSTR, and that (—rA) in equation (D), the material balance, represents the (positive) net rate of appearance of A by flow into and out of the reactor. As noted above, in steady-state operation, these two rates balance. 

To simplify the treatment for an LFR in this chapter, we consider only isothermal, steady-state operation for cylindrical geometry, and for a simple system (A - products) at constant density. After considering uses of an LFR, we develop the material-balance (or continuity) equation for any kinetics, and then apply it to particular cases of power-law kinetics. Finally, we examine the results in relation to the segregated-flow model (SFM) developed in Chapter 13. 

A stoichiometric analysis based on the species expected to be present as reactants and products to determine, among other things, the maximum number of independent material balance (continuity) equations and kinetics rate laws required, and the means to take into account change of density, if appropriate. (A stoichiometric table or spreadsheet may be a useful aid to relate chosen process variables (Fj,ch etc.) to a minimum set of variables as determined by stoichiometry.)... 

The continuity or material-balance equations for A stem from the flow/kinetics scheme shown in Figure 23.8, which corresponds to the representation in Figure 23.7. 

This situation is one involving both a total and a component material balance, combined with a kinetic equation for the rate of decomposition of the waste component. Neglecting density effects, the total material balance equation is... 

The component material balance, when coupled with the heat balance equation and temperature dependence of the kinetic rate coefficient, via the Arrhenius relation, provide the dynamic model for the system. Batch reactor simulation examples are provided by BATCHD, COMPREAC, BATCOM, CASTOR, HYDROL and RELUY. 

Fermentation systems obey the same fundamental mass and energy balance relationships as do chemical reaction systems, but special difficulties arise in biological reactor modelling, owing to uncertainties in the kinetic rate expression and the reaction stoichiometry. In what follows, material balance equations are derived for the total mass, the mass of substrate and the cell mass for the case of the stirred tank bioreactor system (Dunn et ah, 2003). 

The component material balance equation, combined with the reactor energy balance equation and the kinetic rate equation, provide the basic model for the ideal plug-flow tubular reactor. 

It is important to note that Eqs. 5, 8, and 9 were derived entirely from a silicon material balance and the assumption that physical sputtering is the only silicon loss mechanism thus these equations are independent of the kinetic assumptions incorporated into Eqs. 1, 2, and 7. This is an important point because several of these kinetic assumptions are questionable for example, Eq. 2 assumes a radical dominated mechanism for X= 0, but bombardment-induced processes may dominate for small oxide thickness. Moreover, ballistic transport is not included in Eq. 1, but this may be the dominant transport mechanism through the first 40 A of oxide. Finally, the first 40 A of oxide may be annealed by the bombarding ions, so the diffusion coefficient may not be a constant throughout the oxide layer. In spite of these objections, Eq. 2 is a three parameter kinetic model (k, Cs, and D), and it should not be rejected until clear experimental evidence shows that a more complex kinetic scheme is required. 

Watanabe and Ohnishi [39] have proposed another model for the polymer consumption rate (in place of Eq. 2) and have also integrated their model to obtain the time dependence of the oxide thickness. Time dependent oxide thickness measurement in the transient regime is the clearest way to test the kinetic assumptions in these models however, neither model has been subjected to experimental verification in the transient regime. Equation 9 may be used to obtain time dependent oxide thickness estimates from the time dependence of the total thickness loss, but such results have not been published. Hartney et al. [42] have recently used variable angle XPS spectroscopy to determine the time dependence of the oxide thickness for two organosilicon polymers and several etching conditions. They did not present kinetic model fits to their results, nor did they compare their results to time dependent thickness estimates from the material balance (Eq. 9). More research on the transient regime is needed to determine the validity of Eq. 10 or the comparable result for the kinetic model presented by Watanabe and Ohnishi [39]. 

Steady-State Oxide Thickness. The steady-state etching rate (R = S/M) does not contain any of the kinetic parameters thus it does not contain any information about the kinetics of the oxidation process. In contrast, the steady-state oxide thickness is determined by the kinetics of the transport and oxidation processes thus one can learn about these processes by studying the steady-state oxide thickness. The silicon material balance (Eq. 9)... 

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