Material balance equations

The material balance in a constant volume chemostat is derived based on cell balance as shown in the following equations. Material balance in a chemostat with recycle, pcell ... 

M equations—Material balance for each component (C equations for each stage) ... 

The batch reactor is modeled by the following equations Material balances in the reactor... 

The simplicity of equilibrium-based theories results from the fact that the coupled hrst-order partial differential equations (material balances) can be recast as two ordinary differential equations (but these are still coupled). The mathematical technique employed is called the method of characteristics. The results are... 

Reaction class Reaction type Basic differential equation Material balance equations Product distribution ... 

The process model equations (material balances of the species present in the reactor) can be written as follows ... 

Together with the equilibrium equations, material balances are used, see Equations 4.33-4.39 ... 

The amounts of each phase and their compositions are calculated by resolving the equations of phase equilibrium and material balance for each component. For this, the partial fugacities of each constituent are determined ... 

Reservoir engineers describe the relationship between the volume of fluids produced, the compressibility of the fluids and the reservoir pressure using material balance techniques. This approach treats the reservoir system like a tank, filled with oil, water, gas, and reservoir rock in the appropriate volumes, but without regard to the distribution of the fluids (i.e. the detailed movement of fluids inside the system). Material balance uses the PVT properties of the fluids described in Section 5.2.6, and accounts for the variations of fluid properties with pressure. The technique is firstly useful in predicting how reservoir pressure will respond to production. Secondly, material balance can be used to reduce uncertainty in volumetries by measuring reservoir pressure and cumulative production during the producing phase of the field life. An example of the simplest material balance equation for an oil reservoir above the bubble point will be shown In the next section. 

The material balance equation relating produced volume of oil (Np stb) to the pressure drop in the reservoir (AP) is given by ... 

In unsteady states the situation is less satisfactory, since stoichiometric constraints need no longer be satisfied by the flux vectors. Consequently differential equations representing material balances can be constructed only for binary mixtures, where the flux relations can be solved explicitly for the flux vectors. This severely limits the scope of work on the dynamical equations and their principal field of applicacion--Che theory of stability of steady states. The formulation of unsteady material and enthalpy balances is discussed in Chapter 12, which also includes a brief digression on stability problems. 

To reduce the material balance conditions (11,1) to differential equations for the composition and pressure, flux relations must be used to relate the vectors to the gradients of the composition and pressure... 

Solution of the material balance equations gives the pressure and... 

Knowing the solution of the material balance equations It is easy to calcu>... 

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. 

When used in the material balance condition this again gives a single differ-ential equation for but it is not the same as the equation obtained... 

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 order to Introduce thermal effects into the theory, the material balance equations developed in this chapter must be supplemented by a further equation representing the condition of enthalpy balance. This matches the extra dependent variable, namely temperature. Care must also be taken to account properly for the temperature dependence of certain parameters In... 

As In the case of the material balance equations, the enthalpy balance can be written in dimensionless form, and this introduces new dimensionless parameters in addition to those listed in Table 11.1. We shall defer consideration of these until Chapter 12, where we shall construct the unsteady state enthalpy and material balances, and reduce them to dimensionless form. 

Equation (12.29) then represents the material balance on species A, while equation (12.30) represents the overall material balance. The reaction rate... 

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). 

Simplified Design Procedure for Linear Equilibrium and Operating Lines. A straight operating line occurs when the concentrations are low such that and remain essentially constant. (The material balance is obtained from equation 35.) In cases where the... 

Equation 16 is the correct material balance expression for calculating the chlorine efficiency of diaphragm cells. Whereas many approximate versions are used (8), the one closest to equation 16 is the "six equation" ... 

If the T and P of a multiphase system are constant, then the quantities capable of change are the iadividual mole numbers of the various chemical species / ia the various phases p. In the absence of chemical reactions, which is assumed here, the may change only by iaterphase mass transfer, and not (because the system is closed) by the transfer of matter across the boundaries of the system. Hence, for phase equUibrium ia a TT-phase system, equation 212 is subject to a set of material balance constraints ... 

Equation 235 is the basic expression of material balance for a closed system in which r chemical reactions occur. It asserts that ia such a system... 

The general criterion of chemical reaction equiUbria is the same as that for phase equiUbria, namely that the total Gibbs energy of a closed system be a minimum at constant, uniform T and P (eq. 212). If the T and P of a siagle-phase, chemically reactive system are constant, then the quantities capable of change are the mole numbers, n. The iadependentiy variable quantities are just the r reaction coordinates, and thus the equiUbrium state is characterized by the rnecessary derivative conditions (and subject to the material balance constraints of equation 235) where j = 1,11,.. ., r ... 

When these half-reactions are summed, there is no net reaction. Thus the material balance of the cell is not altered by overcharge. At open circuit, equation 19 at the negative electrode is the sum of a two-step process, represented by equation 15 and... 

The value functions appearing in equation 3 may be expanded in Taylor series about x and, because the concentration changes effected by a single stage are relatively small, only the first nonvanishing term is retained. When the value of is replaced by its material balance equivalent, ie, equation 4 ... 

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