# Solution equation

Table 3 shows results obtained from a five-component, isothermal flash calculation. In this system there are two condensable components (acetone and benzene) and three noncondensable components (hydrogen, carbon monoxide, and methane). Henry s constants for each of the noncondensables were obtained from Equations (18-22) the simplifying assumption for dilute solutions [Equation (17)] was also used for each of the noncondensables. Activity coefficients for both condensable components were calculated with the UNIQUAC equation. For that calculation, all liquid-phase composition variables are on a solute-free basis the only required binary parameters are those for the acetone-benzene system. While no experimental data are available for comparison, the calculated results are probably reliable because all simplifying assumptions are reasonable the [c.61]

Selectivity In chromatography, selectivity is defined as the ratio of the capacity factors for two solutes (equation 12.11). In capillary electrophoresis, the analogous expression for selectivity is [c.601]

The second term allows for solvation, which effectively increases the volume fraction of the particles to a larger value than that calculated on the basis of dry solute. Equation (9.18) shows how this can be quantified. [c.597]

To obviate the tedious graphical iategration, a simplified design procedure was developed on the basis of Colburn s analytical solution, equation 56. Substitution of the ratio presents no problem because this ratio stays fairly constant ia the tower at the low coaceatratioas for which Figure 12 is [c.31]

A simple equation for the fugacity of a species in an ideal solution follows from equation 190. Written for the special case of species / in an ideal solution, equation 160 becomes equation 195 [c.497]

Eor partially retained solutes, equation 10 becomes [c.300]

Diffusion coefficients decrease roughly in proportion to the square root of molecular weight, are widely tabulated for aqueous solutions, or may be estimated from the Stokes Einstein equation (3). Ignoring boundary layer effects for the moment, and by assuming that diffusion within the membrane is analogous to that in free solution, equation 1 can be integrated across a homogeneous membrane of thickness dto yield the following equation, where S represents the dimensionless solute partition coefficient, ie, the ratio of solute concentration in external solution to that at the membrane surface, and represents solute diffusion within the membrane and is assumed independent of solute concentration. [c.31]

A key feature of this model is that no data for mixtures are required to apply the regular-solution equations because the solubiHty parameters are evaluated from pure-component data. Results based on these equations should be treated as only quaHtative. However, mixtures of nonpolar or slightly polar, nonassociating chemicals, can sometimes be modeled adequately (1,3,18). AppHcations of this model have been limited to hydrocarbons (qv) and a few gases associated with petroleum (qv) and natural gas (see Gas, natural) processiag, such as N2, H2, CO2, and H2S. Values for 5 and H can be found ia many references (1—3,7). [c.237]

This equation is the basis for development of expressions for all other thermodynamic properties of an ideal solution. Equations (4-60) and (4-61), apphed to an ideal solution with replaced by Gj, can be written [c.520]

Selectivity In chromatography, selectivity is defined as the ratio of the capacity factors for two solutes (equation 12.11). In capillary electrophoresis, the analogous expression for selectivity is [c.601]

Note that if Bn is zero, then T13 and T23 are also zero, so Equation (5.81) reduces to the specially orthotropic plate solution. Equation (5.65), if D11 =D22- Because Tn, T12, and T22 are functions of both m and n, no simple conclusion can be drawn about the value of n at buckling as could be done for specially orthotropic laminated plates where n was determined to be one. Instead, Equation (5.81) is a complicated function of both m and n. At this point, recall the discussion in Section 3.5.3 about the difference between finding a minimum of a function of discrete variables versus a function of continuous variables. We have already seen that plates buckle with a small number of buckles. Consequently, the lowest buckling load must be found in Equation (5.81) by a searching procedure due to Jones involving integer values of m and n [5-20] and not by equating to zero the first partial derivatives of N with respect to m and n. [c.308]

Note that if B g and 825 are zero, then and T23 are also zero, so Equation (5.92) reduces to the specially orthotropic plate solution. Equation (5.65). The character of Equation (5.92) is the same as that of Equation (5.81) for antisymmetric cross-ply laminated plates, so the remarks on finding the buckling load in Section 5.4.3 are equally applicable here. [c.313]

Normalized maximum deflections (at the center) for graphite-epoxy unsymmetrically cross-ply laminated rectangular plates with uniform transverse loading are shown in Figure 5-37. The results are obtained by summing exact deflection solutions, Equation (5.49), for each component of the Fourier sine series expansion for a uniform transverse load, Equation (5.26). The plate aspect ratio is 3, a value for which the results are the most strikingly different from the baseline results of a laminate with all 0° layers in the x-direction (in which a = 3b) and the unsymmetric laminate with bending-extension coupling ignored (Bjj = o) hence, an orthotropic laminate or a specially orthotropic laminate. The unsymmetric laminate is stiffer than the all-0°-layer laminate (i.e., less center deflection occurs for the unsymmetric laminate) and more flexible than the orthotropic laminate. [c.324]

Note that if B, is zero, then 1,3 and T23 are also zero, so Equation (5.81) reduces to the specially orthotropic plate solution. Equation (5.65), if D.,1 = D22- Because T.,., T. 2, and T22 are functions of both m and n, no simple conclusion can be drawn about the vaiue of n at buckiing as could be done for specially orthotropic laminated piates where n was determined to be one. Instead, Equation (5.81) is a complicated function of both m and n. At this point, recall the discussion in Section 3.5.3 about the difference between finding a minimum of a function of discrete variables versus a function of continuous variables. We have already seen that plates buckle with a small number of buckles. Consequently, the lowest buckling load must be found in Equation (5.81) by a searching procedure due to Jones involving integer values of m and n [5-20] and not by equating to zero the first partial derivatives of with respect to m and n. [c.308]

Note that if B. g and B26 are zero, then T.,3 and T23 are also zero, so Equation (5.92) reduces to the specially orthotropic plate solution. Equation (5.65). The character of Equation (5.92) is the same as that of Equation (5.81) for antisymmetric cross-ply laminated plates, so the remarks on finding the buckling load in Section 5.4.3 are equally applicable here. [c.313]

Normalized maximum deflections (at the center) for graphite-epoxy unsymmetrically cross-ply laminated rectangular plates with uniform transverse loading are shown in Figure 5-37. The results are obtained by summing exact deflection solutions, Equation (5.49), for each component of the Fourier sine series expansion for a uniform transverse load, Equation (5.26). The plate aspect ratio is 3, a value for which the results are the most strikingly different from the baseline results of a laminate with all 0° layers in the x-direction (in which a = 3b) and the unsymmetric laminate with bending-extension coupling ignored (Bjj = o) hence, an orthotropic laminate or a specially orthotropic laminate. The unsymmetric laminate is stiffer than the all-0°-layer laminate (i.e., less center deflection occurs for the unsymmetric laminate) and more flexible than the orthotropic laminate. [c.324]

Diffusivity of the liquid light key component is calculated by the dilute solution equation of Wilke-Chang [243]. [c.43]

Solution of Simultaneous Thermodynamic Equations [c.3]

For such components, as the composition of the solution approaches that of the pure liquid, the fugacity becomes equal to the mole fraction multiplied by the standard-state fugacity. In this case,the standard-state fugacity for component i is the fugacity of pure liquid i at system temperature T. In many cases all the components in a liquid mixture are condensable and Equation (13) is therefore used for all components in this case, since all components are treated alike, the normalization of activity coefficients is said to follow the symmetric convention. [c.18]

In a binary liquid solution containing one noncondensable and one condensable component, it is customary to refer to the first as the solute and to the second as the solvent. Equation (13) is used for the normalization of the solvent s activity coefficient but Equation (14) is used for the solute. Since the normalizations for the two components are not the same, they are said to follow the unsymmetric convention. The standard-state fugacity of the solvent is the fugacity of the pure liquid. The standard-state fugacity of the solute is Henry s constant. [c.19]

[c.71]

This reduces the calculation at each step to solution of a set of linear equations. The program description and listing are given in Appendix H. [c.99]

Many different manipulations of these equations have been used to obtain solutions. As discussed by King (1971), many of the older approaches work in terms of V/L, which has the disadvantage of being unbounded and which, in the classical implementation, leads to poorly convergent iterative calculations. A preferable arrangement of this equation system for solution is based on the ratio V/F, which must lie between 0 and 1. If we substitute in Equation (7-1) for L from Equation (7-2) and for y from Equation (7-4), and then divide by F, we obtain [c.113]

For mixtures of nonassociating or weakly associating compounds, fugacity coefficients are evaluated using Equation (3-lOb). For mixtures including one or more strongly associating compound, fugacity coefficients are Calculated from Equation (3-13) with true mole fractions, 2, determined from a first-order iterative solution of material balance and equilibrium equations. (See discussion on chemical theory in Appendix A.) [c.299]

Assuming that the phase split operates at 40 bar and 40 C, a rigorous solution of the phase equilibrium using the Soave-Redlich-Kwong equation of state and the recycle equations using flowsheet simulation software gives a composition of the reactor effluent given in Table 4.4. [c.114]

Solution First, calculate the theoretical oxygen demand from the equation that represents the overall oxidation of the acetone [c.309]

A] = b/a (equation (A3.4.145)) is stationary and not [A ] itself This suggests d[A ]/dt < d[A]/dt as a more appropriate fomuilation of quasi-stationarity. Furthemiore, the general stationary state solution (equation (A3.4.144)) for the Lindemaim mechanism contams cases that are not usually retained in the Bodenstein quasi-steady-state solution. [c.791]

Wlien describing the interactions between two charged flat plates in an electrolyte solution, equation (C2.6.6) cannot be solved analytically, so in the general case a numerical solution will have to be used. Several equations are available, however, to describe the behaviour in a number of limiting cases (see [41] for a detailed discussion). Here we present two limiting cases for the interactions between two charged spheres, surrounded by their counterions and added electrolyte, which will be referred to in further sections. This pair interaction is always repulsive in the theory discussed here. [c.2678]

In liquid metal solutions Z is normally of the order of 10, and so this equation gives values of Ks(a+B) which are close to that predicted by the random solution equation. But if it is assumed that the solute atom, for example oxygen, has a significantly lower co-ordination number of metallic atoms than is found in the bulk of die alloy, dieii Z in the ratio of the activity coefficients of die solutes in the quasi-chemical equation above must be correspondingly decreased to the appropriate value. For example, Jacobs and Alcock (1972) showed that much of the experimental data for oxygen solutions in biiiaty liquid metal alloys could be accounted for by the assumption that die oxygen atom is four co-ordinated in diese solutions. [c.355]

Note that the terra e/D is the relative roughness from Figure 2-11. The solution of the above equation is trial and error. Colebrook [6] also proposed a direct solution equation that is reported [7] to have [c.68]

It is strictly for convenience that certain conventions have been adopted in the choice of a standard-state fugacity. These conventions, in turn, result from two important considerations (a) the necessity for an unambiguous thermodynamic treatment of noncondensable components in liquid solutions, and (b) the relation between activity coefficients given by the Gibbs-Duhem equation. The first of these considerations leads to a normalization for activity coefficients for nonoondensable components which is different from that used for condensable components, and the second leads to the definition and use of adjusted or pressure-independent activity coefficients. These considerations and their consequences are discussed in the following paragraphs. [c.17]

However, if the liquid solution contains a noncondensable component, the normalization shown in Equation (13) cannot be applied to that component since a pure, supercritical liquid is a physical impossibility. Sometimes it is convenient to introduce the concept of a pure, hypothetical supercritical liquid and to evaluate its properties by extrapolation provided that the component in question is not excessively above its critical temperature, this concept is useful, as discussed later. We refer to those hypothetical liquids as condensable components whenever they follow the convention of Equation (13). However, for a highly supercritical component (e.g., H2 or N2 at room temperature) the concept of a hypothetical liquid is of little use since the extrapolation of pure-liquid properties in this case is so excessive as to lose physical significance. [c.18]

More general forms of the Gibbs-Duhem equation have been derived to allow for variations in temperature or pressure (or both) but these are not useful for our purposes since they are not easily integrated. Equation (16) is satisfied by various simple algebraic forms relating an y to x well-ltnown examples are the Margules and van Laar equations but many others exist. The particular relation used in this work, the UNIQUAC equation, while significantly different from the equations of Margules and van Laar, is also a solution to the Gibbs-Duhem differential equation. [c.20]

When a condensable solute is present, the activity coefficient of a solvent is given by Equation (15) provided that all composition variables (x, 9, and ) are taicen on an (all) solute-free basis. Composition variables 9 and 4 are automatically on a solute-free basis by setting q = q = r = 0 for every solute. [c.57]

See pages that mention the term

**Solution equation**:

**[c.104] [c.584] [c.476] [c.556] [c.476] [c.59] [c.319] [c.321] [c.319] [c.321] [c.462] [c.4] [c.34] [c.212] [c.9] [c.45]**

Advanced control engineering (2001) -- [ c.239 ]