Coefficients, in equations

Equation (2.45) represents the weighted residual statement of the original differential equation. Theoretically, this equation provides a system of m simultaneous linear equations, with coefficients Q , i = 1,... m, as unknowns, that can be solved to obtain the unknown coefficients in Equation (2.41). Therefore, the required approximation (i.e. the discrete solution) of the field variable becomes detemfined. 

This form of the virial equation usually does not represent the properties of the gas as well as does equation (A3.3), but it is often more useful since it can be solved explicitly for Vm. The coefficients in equations (A3.3) and (A3.5) are related through... 

If p is not high, terms beyond the second and third virial coefficients in equation (A3.3) and (A3.5) are usually small and can be neglected. This is fortunate, since experimental data are usually not accurate enough to give reliable values for the higher order terms. At low pressures, equation (A3.5) is often used and truncated after the second virial coefficient so that... 

At high flowrates when chum flow sets in, the pressure drop undergoes a sudden increase and the coefficient in equation 8.57 increases ... 

The discrepancy between the coefficients in equations 11.45 and 11,46 is attributable to the fact that the effect of the curvature of the pipe wall has not been taken into account in applying the equation for flow over a plane surface to flow through a pipe. In addition, it takes no account of the existence of the laminar sub-layer at the walls. 

The fugacity coefficients in Equation (7.29) can be calculated from pressure-volume-temperature data for the mixture or from generahzed correlations. It is frequently possible to assume ideal gas behavior so that = 1 for each component. Then Equation (7.29) becomes... 

Data compiled by Hanks (1983) show that, contrary to earlier notions, the m coefficient in Equation 1 may be affected by the genotype. A relatively greater net gain of carbon for the same rate of transpiration under stress may be reflected in the m coefficient (Equation 1), in the ratio between assimilation and transpiration (assimilation ratio) or in the agronomic index, WUE (Equation 2). 

Experimental polymer rheology data obtained in a capillary rheometer at different temperatures is used to determine the unknown coefficients in Equations 11 - 12. Multiple linear regression is used for parameter estimation. The values of these coefficients for three different polymers is shown in Table I. The polymer rheology is shown in Figures 2 - 4. 

While other programs require modification of the actual code in changing the polymer, spectra, or model, only changes in the user database is required here. Changes in the program since a brief report (22) in 1985 include improvement of the menu structure, added utilities for spectral manipulations, institution of demo spectra and database. Inclusion of Markov statistics, and automation for generation of the coefficients in Equation 1. Current limitations are that only three models (Bernoul llan, and first- and second-order Markov) can be applied, and manual input Is required for the N. A. S. L.. 

The optimum coefficients in Equation 2 are Eq -37.8363 and E =-0.5685 au. Eq is very close to the value -37.8366 found for the pure graphite clusters. The value of E corresponds to a contribution of 47.1 kcal/mol to the total energy of Cg 2Hg for each bonded H-atom (the hydrogen atom has an energy of -0.4935 au in the oasis set used). Alternatively, the parameter values can be interpreted as 83.3 kcal/mol per C-C bond, 88.7 kcal/mol per C-H bond. 

In practice, the condensate will not flow smoothly from tube to tube, Figure 12.42b, and the factor of (Nry 1/4 applied to the single tube coefficient in equation 12.49 is considered to be too conservative. Based on results from commercial exchangers, Kern (1950)... 

NMR Self-Diffusion of Desmopressin. The NMR-diffusion technique (3,10) offers a convenient way to measure the translational self-diffusion coefficient of molecules in solution and in isotropic liquid crystalline phases. The technique is nonperturbing, in that it does not require the addition of foreign probe molecules or the creation of a concentration-gradient in the sample it is direct in that it does not involve any model dependent assumptions. Obstruction by objects much smaller than the molecular root-mean-square displacement during A (approx 1 pm), lead to a reduced apparent diffusion coefficient in equation (1) (10). Thus, the NMR-diffusion technique offers a fruitful way to study molecular interactions in liquids (11) and the phase structure of liquid crystalline phases (11,12). 

The activity coefficient in Equation 6.28 (y, = a,/xj) can be considered to be the ratio of the effective to actual concentration. Substituting Equation 6.28 into Equation 6.20 gives ... 

The coefficients a and b in Equation 23.10 can be correlated as functions of the saturation temperature difference across the turbine9. In fact, the coefficients in Equation 23.10 are related to the pressure drop across the turbine. However, in the model, the pressure drop is replaced by its equivalent saturation temperature difference. Use of temperature difference allows easier interface to utility calculations with process heating and cooling demands. Thus ... 

Lower-case letters are used for the coefficients in equations that represent a substituent constant as a function of other substituent constants. The difference between pure and composite parameters is that the former represent a single effect while the latter represent a mixture of two or more. The percent composition of these parameters is given by ... 

The coefficients in equation 3.47 are the corrected values given by Skelland (1967). 

There is evidence in the work reported in Chapter 5 on sedimentation 5) to suggest that where the particles are free to adjust their orientations with respect to one another and to the fluid, as in sedimentation and fluidisation, the equations for pressure drop in fixed beds overestimate the values where the particles can choose their orientation. A value of 3.36 rather than 5 for the Carman-Kozeny constant is in closer accord with experimental data. The coefficient in equation 6.3 then takes on the higher value of 0.0089. The experimental evidence is limited to a few measurements however and equation 6.3, with its possible inaccuracies, is used here. 

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