Material balance independent

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. 

A third approach is suggested by Hugo s formulation of material balances at the limit of bulk diffusion control, described in Section 11.3. Hugo found expressions for the fluxes by combining the stoichiometric conditions and the Stefan-Maxvell relations, and this led to no inconsistencies since there are only n - 1 independent Stefan-Maxwell relations for the n fluxes. An analogous procedure can be followed when the diffusion is of intermediate type, using the dusty gas model equations in the form (5.10) and (5.11). Equations (5.11), which have the following scalar form ... 

For ease of exposition, let us limit attention to. two independent reactions--the generalization to more reactions is straightforward. Then the material balance equations take the form... 

In section 11.4 Che steady state material balance equations were cast in dimensionless form, therary itancifying a set of independent dimensionless groups which determine ice steady state behavior of the pellet. The same procedure can be applied to the dynamical equations and we will illustrate it by considering the case t f the reaction A - nB at the limit of bulk diffusion control and high permeability, as described by equations (12.29)-(12.31). 

Note that application of a systematic approach enables us to resolve a material-balance system into a number of independent equations equal to the number of unknowns that it needs to solve for. The following steps should be followed with any material-balance system, regardless of complexity ... 

A special case of the above equation applies to a continuous steady-state flow process when all of the rate terms are independent of time and the accumulation term is zero. Thus, the differential material balance for any component i in such a process is given by... 

Perform an overall material balance and the necessary component material balances so as to provide the maximum number of independent equations. In the event the balance is written in differential form, appropriate integration must be carried out over time, and the set of equations solved for the unknowns. 

At steady-state condition for chemostat operation, change of concentration is independent of time. Material balance for the fermentation vessel is ... 

The design equations for a CSTR do not require that the reacting mixture has constant physical properties or that operating conditions such as temperature and pressure be the same for the inlet and outlet environments. It is required, however, that these variables be known. Pressure in a CSTR is usually determined or controlled independently of the extent of reaction. Temperatures can also be set arbitrarily in small, laboratory equipment because of excellent heat transfer at the small scale. It is sometimes possible to predetermine the temperature in industrial-scale reactors for example, if the heat of reaction is small or if the contents are boiling. This chapter considers the case where both Pout and Tout are known. Density and Q ut wiU not be known if they depend on composition. A steady-state material balance gives... 

Solution The equal reactivity assumption says that kp and kc are independent of chain length. The quasi-steady hypothesis gives d R /dt = 0. Applying these to a material balance for growing chains of length / gives... 

A balance equation can be written for each independent component. Not all the components in a material balance will be independent. 

Material-balance problems are particular examples of the general design problem discussed in Chapter 1. The unknowns are compositions or flows, and the relating equations arise from the conservation law and the stoichiometry of the reactions. For any problem to have a unique solution it must be possible to write the same number of independent equations as there are unknowns. 

Consider the general material balance problem where there are Ns streams each containing Nc independent components. Then the number of variables, N, is given by ... 

Number of variables (component flow rates) = 9 Number of independent material balance equations = 3... 

It is often possible to make a material balance round a unit independently of the heat balance. The process temperatures may be set by other process considerations, and the energy balance can then be made separately to determine the energy requirements to maintain the specified temperatures. For other processes the energy input will determine the process stream flows and compositions, and the two balances must be made simultaneously for instance, in flash distillation or partial condensation see also Example 4.1. 

We will first consider the simple case of diffusion of a non-electrolyte. The course of the diffusion (i.e. the dependence of the concentration of the diffusing substance on time and spatial coordinates) cannot be derived directly from Eq. (2.3.18) or Eq. (2.3.19) it is necessary to obtain a differential equation where the dependent variable is the concentration c while the time and the spatial coordinates are independent variables. The derivation is thus based on Eq. (2.2.10) or Eq. (2.2.5), where we set xj> = c and substitute from Eq. (2.3.18) or Eq. (2.3.19) for the fluxes. This yields Fick s second law (in fact, this is only a consequence of Fick s first law respecting the material balance—Eq. 2.2.10), which has the form of a partial differential equation... 

The unit ratio material balance is based on the production of one pound of salable product. This basis is used because it is independent of the plant size and because the use of numbers near one minimizes the possibility of future calculation errors. 

Now the equations derived from Kirchoff s first law are essentially material balances around each of (N — 1) vertices. As an alternative, balances could also be drawn up around groups of such vertices. Is there a special way of grouping the vertices, which will yield a particularly advantageous formulation Also, as we have noted, the selection of cycles is not unique, but the cycles must be independent. How can we generate an independent set of cycles Are some of these independent sets more fundamental than others If so, how many fundamental sets are there To answer these questions we must explore further the properties of a graph. 

There are a variety of limiting forms of equation 8.0.3 that are appropriate for use with different types of reactors and different modes of operation. For stirred tanks the reactor contents are uniform in temperature and composition throughout, and it is possible to write the energy balance over the entire reactor. In the case of a batch reactor, only the first two terms need be retained. For continuous flow systems operating at steady state, the accumulation term disappears. For adiabatic operation in the absence of shaft work effects the energy transfer term is omitted. For the case of semibatch operation it may be necessary to retain all four terms. For tubular flow reactors neither the composition nor the temperature need be independent of position, and the energy balance must be written on a differential element of reactor volume. The resultant differential equation must then be solved in conjunction with the differential equation describing the material balance on the differential element. 

The various energy transfer constraints enter into the analysis primarily as boundary conditions on the difference equations, and we now turn to the generation of the differential equations on which the difference equations are based. Since the equations for the one-dimensional model are readily obtained by omitting or modifying terms in the expressions for the two-dimensional model, we begin by deriving the material balance equations for the latter. For purposes of simplification, it is assumed that only one independent reaction occurs within the system of interest. In cases where multiple reactions are present, one merely adds an appropriate term for each additional independent reaction. 

Since both Xj and X] are independent of z, the relationship between X3 and X j can be approximated by material balance of the coarse particles injected into the bed to serve as the tracer. 

Schwarz s model is a multiradical extension of the Ganguly-Magee model with some additional improvements, to be described later. Schwarz assumes that initially—that is, 10 11 s after the act of energy deposition in water—there appear five species, namely eh, H, OH, H30+, and H2. Their initial yields, indicated by superscript zero, are related by charge conservation and material balance. Thus, there are three independent initial yields, taken to be those of eh, H, and Hr The initial yield of H2 is identified with the unscavengable molecular hydrogen yield. No mechanism of its production is speculated, except that it is not formed by radical recombination. For the gaussian distribution of the radicals, two initial... 

For reaction in a constant-volume BR, with only A present initially, the concentrations of A, B and C as functions of time t are governed by the following material-balance equations for A, B and C, respectively, incorporating the two independent rate... 

Equation 13.5-2 is the segregated-flow model (SFM) with a continuous RTD, E(t). To what extent does it give valid results for the performance of a reactor To answer this question, we apply it first to ideal-reactor models (Chapters 14 to 16), for which we have derived the exact form of E(t), and for which exact performance results can be compared with those obtained independently by material balances. The utility of the SFM lies eventually in its potential use in situations involving nonideal flow, wheic results cannot be predicted a priori, in conjunction with an experimentally measured RTD (Chapters 19 and 20) in this case, confirmation must be done by comparison with experimental results. 

The interpretation of cA(t) comes from the realization that each cylindrical shell passes through the vessel as an independent batch. Thus, cA(/) is obtained by integration of the material balance for a batch reactor (BR). Accordingly, we may rewrite equation 16.2-11, in terms of either cA(x) or fA(x), as... 

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 solution of this set of equations, 18.4-26 (with expression (A) incorporated) to -29, must be coupled with the set of three independent material-balance or continuity equations to determine the concentration profiles of three independent species, and the temperature profile, for either a specified size (V) of reactor or a specified amount of reaction. A nu-merical solution of the coupled differential equations and property relations is required. Equations (A), (B), and (C) in Example 18-6 illustrate forms of the continuity equation. 

A further (independent) material balance around the entire column enables a relation to be established between, for example, pKout and cA our... 

Figure E2.7 shows the process flow chart for a series of two distillation columns, with mass flows and splits defined by j, x2,..., 5. Write the material balances, and show that the process model comprises two independent variables and three degrees of freedom.

Because the rank of the coefficient matrix is three, there are only three independent equations, so Equation (2.14) indicates that there are two degrees of freedom. You can reduce the dimensionality of the set of material balances by substitution of one equation into another and eliminating both variables and equations. 

With 12 variables and 9 independent linear equality constraints, 3 degrees of freedom exist that can be used to maximize profits. Note that we could have added an overall material balance, xn + xl2 + 7 = 8 + x9 + 10, but this would be a redundant equation since it can be derived by adding the material balances. 

Note that since there are two independent variables of both length and time, the defining equation is written in terms of the partial differentials, dC/dt and dC/dZ, whereas at steady state only one independent variable, length, is involved and the ordinary derivative function is used. In reality the above diffusion equation results from a combination of an unsteady-state material balance, based on a small differential element of solid length dZ, combined with Fichs Law of diffusion. 

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. 

One difference between conducting and nonconducting media is that in the former case a charge balance may take the place of one of the material balances. For the same number of solution species, however, the number of independent balance equations is the same in the two cases. 

Assume for the moment that it is the concentration changes for Pj+ and (P A) as solid S reacts that are sought. Material balance equations can be written for P-j+ and (P A) as well as a charge balance but trial and error shows that it is impossible to do so without introducing nR+ and n -. P2 need not appear, however. A set of three independent balances is ... 

Table XI lists the equation numbers and the variables that appear in each equation. The numbers in parentheses are the assigned numbers of each variable. Only five of the material balances are independent, and those selected are designated as Eqs. 1-5 in Table XI. Altogether there are six independent material balances, but because the overall balance for NaOH can be solved directly for F, F is removed from the role of a variable and the number of independent material balances can be reduced to five. The five selected were...

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