Material balance expression

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" ... 

Most real situations are, however, such that conditions change with respect to time. Under these circumstances, a steady-state material balance is inappropriate and must be replaced by a dynamic or unsteady-state material balance, expressed as... 

Although one can probably find exceptions, most equilibrium calculations involving flue gas slurries are performed with temperature as a known variable. With temperature known, the numerical values of the appropriate equilibrium constants can be immediately calculated. The remaining unknown variables to be determined are the activities, activity coefficients, molalities, and the gas phase partial pressures. The equations used to determine these variables are formulated from among the equilibrium expressions presented in Table 1, the expressions for the activity coefficients, ionic strength, material balance expressions, and the electroneutrality balance. Although there are occasionally exceptions, the solution sequence generally is an iterative or cyclic sequence. 

The starting point for all design is the material balance expressed for any reactant (or product). Thus, as illustrated in Fig. 4.2, we have... 

By using these ratios and the relationships (3.18) and (3.19), we can alter the material balance expressions and the corresponding solutions of the reactor models if we use other rate expressions. It should be noted that in practice, in fixed-beds and slurries of porous particles, the external area of the particle and thus the parameters au and ac are used, respectively. 

Residue curve (RCM) and distillation curve (DCM) maps are today standard tools for designing distillation systems dealing with nonideal mixtures involving azeotropes. A residue curve characterizes the evolution of the liquid composition in a vessel during a batchwise distillation experiment. The whole compositional space may be spanned by residue curves considering different initial mixture compositions. For nonreactive mixtures the RCM is obtained by solving the component dynamic material balance expressed by the following differential equation ... 

The objective function (7) in accordance with the general purpose of MEIS that was mentioned in the introduction, i.e., finding the state with extreme value of the system property of interest to a researcher, in this case determines the extreme concentration of the given set of substances. Equality (8) represents a material balance. Expression (9) represents the region of thermodynamic attainability from point y. It is obvious that in Dt(y) the inequalities are satisfied G(xeq) < G(x) < G(y), where xeq—the final equilibrium point. Inequalities (10) are used to set the constraints on macroscopic, including irreversible, kinetics. Presence of this constraint makes up principal difference of the model (7)-( 12) from previous modifications of parametric MEISs. The choice of equations for the calculation of individual terms under the sign of sum in the right-hand side of equality (11) depends on the properties of the considered system. 

Solve for reactor volume using the material-balance expression. The material balance for the continuous-flow stirred-tank reactor may now be used to calculate the reactor volume required for the isomerization. Inserting the first-order rate expression into the material balance,... 

Develop the necessary material-balance expressions for a single reactor, and find its volume. 

The concentration data given can be used to determine net rates of reaction by the material-balance expressions. These rates must be analyzed in terms of the stoichiometry to get the individual rates of reaction. Thus, for the trimer,... 

The equations describing total column operation include vapor-liquid equilibrium relations. Equation 5.12 component balances in the rectifying and stripping sections, Equations 5.13 and 5.14 feed stage component balance. Equation 5.15 feed stage energy balance and overall material balance, expressed as Equations 5.16 and 5.17 and overall column component balance. Equation 5.18 ... 

The overall material balance for a binary system, wherein a pure solid component is crystallized by cooling or evaporation techniques, is derived and rearmnged in the following section to provide maximum solute yield and total suspended solids information. The approach described below is also useful for deriving material balance expressions for binary systems (hat forat hydrates and adducts. 

This system of nonlinear equations is readily reduced to one equation in one unknown (say Vt) in the following manner. First observe that the total material balance expression (a dependent equation) may be obtained by summing each member of the third expression of Eq. (1-26) over all components to give... 

This then produces a material balance expressed in terms of F t)... 

From the overall material balances expressed in Eqns (6) and (7)> making use of the additivity of resistances... 

The equation is simple to use but requires a computation of the exit liquid composition and y at both ends of the column. This can be avoided by use of an equation developed by Colburn which incorporates the equilibrium relationship y mx and the material balance expression Li i - x) = Gu(y - y,) to eliminate the need for y. This equation is for absorption ... 

The munber of theoretical and reactive stages is determined from the distillation line and from the intersection of the distillation line and chemical equilibrium manifold (GEM) and represents the boimdary of the forward and backward reactions) (Giessler et al., 1999). Since there are multiple pairs of X and product composition that satisfy the mass balance, the method sets one of the product composition as reference point and solves for the other two (for a 3-component system) by using material balance expressions. Thus, two of the components compositions and X lie on the same line of mass balance (LMB) in the diagram and allow the estimation of the ratio D/B at a certain reboil ratio only by exploring the ratio of the line segments (figure 3.1f>). 

The equalities (9) and (13) represent a material balance. Expressions (10) and (11) determine the region of thermodynamic attainability from the point y. Equation (11) is used to specify constraints on macroscopic kinetics. The choice of equations for calculation of certain terms under the sign of sum in the right-hand side of equality (12) depends on the properties of the considered system. 

This equilibrium ratio must not be confused with the buffer ratio of equation (3-7), which is CJC, the ratio of the analytical concentrations, which may or may not be close to the ratio of the species at equilibrium. The equilibrium ratio is given by H/Kg. A material balance expression for this ratio can be found from the complete... 

To construct a pH problem-solving method we proceed just as with the ratios and find the material balance expression for the fractions just derived from the equilibrium condition equation. From the derivation of equation (3-2) we have (C is the total of HX and X )... 

The material balance expressions for asphaltene and resin molecules are... 

Flash approach. The fugacity equalities, material balance expressions, and contraints define the problem. 

Also, by the material balance expressions, Eqs. 6 to 15, it can be shown that... 

The nonrandom factors T should obey the following material balance expressions ... 

Develop the necessary material-balance expressions for a single reactor, and find its volume. For a single reactor, the mass-balance design equation is... 

Looking back to Packie s work, it is known that for a given separation requirement expressed as (5-95) Gap, a given material balance expressed in terms of ASTM 50 volume percent temperature difference and a given number of trays in the separation section, there is a vdue, called F, which is the product of the number of actual trays and the volumetric reflux ratio in the section. Thus, a minimum allowable reflux falling from draw trays can be calculated. This is not minimum reflux in the sense that infinite plates are required for the separation. It is minimum allowable operating reflux for the specified number of trays and the required separation. 

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