Variables in equations

For typical conditions in the chemical industry, the effect of pressure on liquid-liquid equilibria is negligible and therefore in this monograph pressure is not considered as a variable in Equation (2). 

We note here that the qiiantnm levels denoted by the capital indices I and F may contain numerous energy eigenstates, i.e. are highly degenerate, and refer to chapter A3.4 for a more detailed discussion of these equations. The integration variable in equation (A3.13.9) is a = 7 j / Ic T. 

The mean value of the von Mises stress ean be approximated by substituting in the mean values of eaeh variable in equation 4.78 to give ... 

We ean use a Monte Carlo simulation of the random variables in equation 4.83 to determine the likely mean and standard deviation of the loading stress, assuming that this will be a Normal distribution too. Exeept for the load, F, whieh is modelled by a 2-parameter Weibull distribution, the remaining variables are eharaeterized by the Normal distribution. The 3-parameter Weibull distribution ean be used to model... 

If the state and control variables in equations (9.4) and (9.5) are squared, then the performance index become quadratic. The advantage of a quadratic performance index is that for a linear system it has a mathematical solution that yields a linear control law of the form... 

An equation expressing the rate in tenns of measurable and/or desirable quantities may now be developed. Based on experimental evidence, the rate of reaction is a function of the concentration of tlie components present in die reaction mixture, tcniperaturc, pressure, and catalyst variables. In equation form. 

The combination of fundamental variables in equation (l.23) that leads to the variable we call G turns out to be very useful. We will see later that AG for a reversible constant temperature and pressure process is equal to any work other than pressure-volume work that occurs in the process. When only pressure-volume work occurs in a reversible process at constant temperature and pressure, AG = 0. Thus AG provides a criterion for determining if a process is reversible. Again, since G is a combination of extensive state functions... 

Each of the variables in equation 1.7 may be expressed in terms of mass, length, and time. Thus, dimensionally ... 

Semibatch Model "GASPP". The kinetics for a semibatch reactor are the simpler to model, in spite of the experimental challenges of operating a semibatch gas phase polymerization. Monomer is added continuously as needed to maintain a constant operating pressure, but nothing is removed from the reactor. All catalyst particles have the same age. Equations 3-11 are solved algebraically to supply the variables in equation 5, at the desired operating conditions. The polymerization flux, N, is summed over three-minute intervals from the startup to the desired residence time, t, in hours ... 

In implicit estimation rather than minimizing a weighted sum of squares of the residuals in the response variables, we minimize a suitable implicit function of the measured variables dictated by the model equations. Namely, if we substitute the actual measured variables in Equation 2.8, an error term arises always even if the mathematical model is exact. 

The partial derivatives with respect to the state variables in Equation 6.67a that are needed in the above ODEs are given next... 

Note that we use the letters C, D to represent the variables in equation 44-64 to avoid confusion with our usage of A to mean absorbance. 

The variables in equation 44-68a are again not separable. While we could formally split equation 44-68a into the sum of two variances ... 

Kinetic analysis usually employs concentration as the independent variable in equations that express the relationships between the parameter being measured and initial concentrations of the components. Such is the case with simultaneous determinations based on the use of the classical least-squares method but not for nonlinear multicomponent analyses. However, the problem is simplified if the measured parameter is used as the independent variable also, this method resolves for the concentration of the components of interest being measured as a function of a measurable quantity. This model, which can be used to fit data that are far from linear, has been used for the resolution of mixtures of protocatechuic... 

The other variables in Equations 3.32-3.34 are either known values, such as the equilibrium constants K and reaction coefficients v, or, in the case of the activity coefficients y, yj and activities aw, a, values that can be considered to be known. In practice, the model updates the activity coefficients and activities during the numerical solution so that their values have been accurately determined by the time the iterative procedure is complete. 

Many of the variables in Equation 9-27 can be estimated or assumed. These variables are regrouped to result in the following form ... 

Relations (fl)-(g) define the MINLP problem. It is important to note that the relations between the binary and continuous variables in Equation (/) are linear. It is possible to impose the desired relations nonlinearly. For example, one could replace Cl by Cl 71 everywhere Cl appears. Then if 71 = 0, Cl does not appear, and if 71 = 1, Cl does appear. Alternatively, one could replace Cl by the conditional expression (if 71 = 1 then Cl else 0). Both these alternatives create nonlinear models that are very difficult to solve and should be avoided if possible. 

Changing variables in equation 3.15, using equation 2.6, and dropping the subscripts rx, equation 3.15 can be written as... 

The variables in Equation A-5 refer to the solution phase, represents the moles of B on the oxide surface, then... 

Correlations based on dimensional analysis with the above variables in equation 3-10 would allow mass transfer rates to be easily predicted, e. g. in scaling-up lab results to full-scale or for changes in the liquid properties. However, no correlations have been developed with this complexity. 

In Equation 14 it has been assumed that the density of the solution, p, is approximately equal to the density of solvent, p0 this approximation improves as the solution becomes more dilute. Note that , defined by Equation 15, varies from 0 at rb to 1 at rm. When the variables in Equation 13 are separated, the equation can be integrated between = 0 and = to give... 

Making transformations here we used equation (7.1.3) and the fact that git as well as p depend on r = n — r[ only. (It permits to change the integration variable in equation (7.1.4).)... 

Figure H1.1.5 Empirical models that are used to predict the complete flow curve of non-Newtonian fluids or portions of the complete curve. In the full-curve models, K is a constant with time as its dimension and m is a dimensionless constant. See text for definition of other variables in equations.

Figure 8 shows the effect of p on the dimensionless temperature profile at x =0. Since p is not an explicit variable in equation (8), a single curve describes the dimensionless temperature profile in terms of X. The actual temperature rise is lower as p decreases and the adsorbent takes longer time to cool down. 

SOLUTION The units on the variables in Equation 5.12 are as follows bound = fmol/g, free = nM, and KV) = nM. Substitution of these units into Equation 5.12 sets up the solution. The nM concentration unit cancels across all parts of the equation to leave only fmol/g. The units on Umax are fmol/g, the number of moles of binding sites per gram of membrane protein. 

Inclusion of the dose sizes in Equation 7.22 is necessary only if the two doses, D0route and /)0IV, are not equal. If all variables in Equation 7.21 are already known, AUC for an orally delivered drug can be quickly calculated with Equation 7.23. 

Once F has been determined for a drug, only one variable in Equation 7.21 remains undiscussed kah X kah may be found through a In Cp versus time plot (Figure 7.15). The... 

The new variables in equation 2, pm, pp, pa and Wa(t), are the densities of the monomer, polymer, and aqueous or solvent components and the mass of solvent or aqueous phase. Collecting terms with the assumption that no change in volume occurs on mixing yields equation 3. 

The other variables in equations (2)-(6) are defined in the nomenclature. The series of differential equations above are solved simultaneously with material balances on the initiator and emulsifier concentrations in the reactors ... 

The remaining state variables in Equation (3.27) display a similar behavior. The fast and slow dynamics are thus not associated with any distinct subsets of the state variables, which is consistent with the statement that the model of the process under consideration is a nonstandard singularly perturbed system of equations. 

In Rousseeuw et al. [43], it is proposed to use the MCD estimates for the center p and the scatter matrix I of the joint (x, y) variables in Equation 6.11 to Equation 6.13. The resulting estimates are called MCD-regression estimates. They inherit the breakdown value of the MCD estimator. To obtain a better efficiency, the reweighed MCD estimates are used in Equation 6.11 to Equation 6.13 and followed by a regression reweighing step. For any lit 0 = (/j(, /i7)7, denote the corresponding -dimensional residuals by r.(0) = yi- Brx. - Then the residual distance of the ith case is defined as... 

We have so far assumed that the nuclei are at rest in fixed positions with respect to each other. When molecular motion, — which in polymers will be chiefly group rotation —, takes place, the variables in equation (2) become functions of time1. If we assume for our present discussion that r is constant and only 6 varies with time (as would be true, for example, for the interaction of protons attached to the same carbon atom or to the same benzene ring), then the time averaged local field will be given... 

One may be concerned about taking the log of a dimensional variable in Equation (1.28). Formally the values of pt° are defined based on a specific reference concentration c0 = 1 Molar and the equation for chemical potential is... 

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