Balance equations 404 INDEX

If one wants to take into account all three mass-transfer resistances, the mass-balance equation [Equation 14.16] should be given for every layer. From that one can obtain the solution for every layer (i = m for membrane layer, i = g for gel layer, the concentration-polarization layer has no index according to Figure 14.4) ... 

To calculate the buffer index of an aqueous system we begin by deriving the equation for the acidity titration curve for that system. For pure water we will assume the litrant is NaOH. The charge-balance equation is then simply... 

To derive a function for the buffer index of a weak monoprotic acid, HA, we begin as before with equilibrium, mass-balance, and charge-balance equations, and first derive an equation for the titration curve. We are given the following expressions ... 

In these equations, index a runs over all inlets to the system while p runs over all outlets. The energy balance (12.3.2) is a special form of (3.6.3) assuming negligible mass for the boundary and negligible kinetic and potential energy changes across the system. Note we have C material balances (12.3.1) but only one energy balance (12.3.2). 

The solutiOTi of Eq. (13.48) depends on further simplifications. If we assume that the director in the odd layers with i = 0 is unaffected by an external field and only the azimuth in the even layers is changed from Jt to 0, then, for an infinitely thick sample (x —> oo), the free energy is independent of both x and Oi. The corresponding torque balance equation reduces to the form with index (/ -P 1) omitted ... 

The index j in relation (5.65) is converted to the index fc by a purely formal process. The reason follows from the following modification. We now substitute from relation (5.66) into the mass balance equations (5.59). We obtain a set of equations... 

Tab. 9.5. Full 3D set of population balance equations for radical polymerization of vinyl acetate in living and dead chain concentration variables Rn,i,k and P j k- (for indices, see Table 9.4) summation over the branching index k, yielding a 2D formulation of exactly the same form, provides the basis for the TDB classes model.

The balance equations. The discrete model is based on assumptions Al to AlO of section 2.2. Let k be the index of the mixing-cell. The mass balance in the bulk gas and solid phases are respectively... 

A note on index notation is in order for those unfamiliar with its use. Index notation is used to shorten the transport equations that are presented above. Each index (shown as a subscript) can represent one of the three Cartesian directions or one of the chemical components. The index that appears on both sides of the equation in all or most terms is the equation index. If it is x, the equation is a balance for quantities in the x-direction if it is 1, the equation is for chemical component 1. The indices that appear only on one side of the equation in occasional terms are running indices and take on each of the three values x, y, and z or each of the component values 1,2,3,... such that the term in which they appear will have several forms that are additive. In the case of component values, indices of product components may not be equal. To illustrate use of the... 

Here Zc u> Ub u ( teui2/6 ) the stoichiometric coefficients Zc>, Yv (Zg, Yi) are the chemicEil symbols of the reactants (products) involved in the chemical reaction firom adsorbed (Z) and gaseous (V) phases index u enumerates reactions resulting in the production of c component and are the detailed rate constants of direct and reverse reactions. Balance equation for the operator X for reactions of the type (6.1.17) reads... 

The biocatalytic reduction step B in synthetic route B demands more raw materials (mass index S , see equation (5.1)) and generates more waste (environmental factor , see equation (5.2)) as compared to reduction step C (Figure 5.1). Solvents used to perform the extraction of the product from the aqueous phase in reduction step B are denoted as auxiliaries in Figure 5.1. These solvents and the aqueous phase dominate the mass balances as well as the environmental scores in Figure 5.2 (M4, M8). 

Inputs and outputs assessed in mass balancing are shown in Figure 5.3. The software EATOS was used to calculate all mass balances of processes. Outputs of EATOS are the mass index (equation (5.1), mass of raw material per mass of product output), and the environmental factor (equation (5.2), mass of waste output per mass of product output). EATOS also allows the calculation of cost indices (e.g., reference [15]) (equation (5.3), cost of raw material per mass of product output). 

The measurement of the width of the metastable zone is discussed in Section 15.2.4, and typical data are shown in Table 15.2. Provided the actual solution concentration and the corresponding equilibrium saturation concentration at a given temperature are known, the supersaturation may be calculated from equations 15.1-15.3. Data on the solubility for two- and three-component systems have been presented by Seidell and Linkiv22 , Stephen et alS23, > and Broul et a/. 24. Supersaturation concentrations may be determined by measuring a concentration-dependent property of the system such as density or refractive index, preferably in situ on the plant. On industrial plant, both temperature and feedstock concentration can fluctuate, making the assessment of supersaturation difficult. Under these conditions, the use of a mass balance based on feedstock and exit-liquor concentrations and crystal production rates, averaged over a period of time, is usually an adequate approach. 

The main objectives in calibrating the SEC detection system in absolute refractive index and absorption units are the estimation of v and E at the normal flow conditions and the standardization of the measurement errors. The first step in the calibration process is the estimation of the instrument s constants to transform the computer units into absorbances and refractive index units. The Waters AAO UV spectrophotometer displays absorbance units. Therefore, step changes in the instrument s balance and sampling of the signal provide the necessary data for the calibration. The equations obtained are ... 

Reliable estimates of relative humidity are critical for use of the isotope mass-balance method however, humidity data are difficult to interpret and commonly not available for a specific study area. The index-lake method provides a means for checking the accuracy of these data and the validity of their use in isotope hydrology. The equation that describes the steady-state isotopic composition of a lake (35) is... 

The reaction system involved in the case studied is a kind of relatively complex van de vusse reaction, nevertheless the reaction system in real manufacturing process may involve more reaction types and, therefore, is more complex than that one. One can, however, simulate the change of environmental indexes within a reactor by combining traditional reactor mathematical model with the PEI balance, and may also discover the effects of reaction conditions and engineering factors on environmental performance by PEI rate-law expression and/or combinations it with other reaction rate equations as well as other related equations in reactor mathematical models. 

Model. A difference equation for the material balance was obtained from a discrete reactor model which was devised by dividing the annulus into a two dimensional array of cells, each taken to be a well stirred batch reactor. The model supposes that axial motion of the mobile phase and bed rotation occur by instantaneous discontinuous jumps, between cells. Reaction occurs only on the solid surface, and for the reaction type A B + C used in this work, -dn /dt = K n - n n. Linear isotherms, n = BiC, were used, and while dispersion was not explicitly included, it could be simulated by adjusting the number of cells. The balance is given by Eq. 2, where subscript n is the cell index in the axial direction, and subscript m is the index in the circumferential direction. 

The calculated values required to evaluate the performance index are determined from the solution of differential equations describing flow. Pressure and velocity values are obtained by solving the locally volume-averaged equation of continuity and differential momentum balance (Eqs (1) and (2)). For steady-state conditions, and incompressible flow, the equation of continuity becomes... 

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