Graphical integrated numerical

GINA graphical integrated numerical analysis H Henry... 

Gowayed, Y. and Barrowski, L., pcGINA , A pc-based Graphical Integrated Numerical Analysis of composite materials to calculate the properties of fabric... 

Figure Bl.11.4. Hydrogen-decoupled 100.6 MHz C NMR spectrum of paracetamol. Both graphical and numerical peak integrals are shown.

The integration can be carried out graphically or numerically using a computer. For illustrative purposes the graphical procedure is shown in Figure 5. In this plot of vapor enthalpy or FQ vs Hquid temperature (T or T, the curved line is the equiHbtium curve for the system. For the air—water system, it is the 100% saturation line taken direcdy from the humidity diagram (see Fig. 3). 

To integrate Eq. (11-3), and AT must be known as functions of Q. For some problems, varies strongly and nonlinearly throughout the exchanger. In these cases, it is necessary to evaluate and AT at several intermediate values and numerically or graphically integrate. For many practical cases, it is possible to calculate a constant mean overall coefficient from Eq. (11-2) and define a corresponding mean value of AT,n, such that... 

In situations in which one cannot assume that Hl and Hql I e constant, these terms must be incorporated inside the integrals in Eqs. (14-24) and (14-25). and the integrals must be evaluated graphically or numerically (by using Simpsons nile, for example). In the normal case involving stripping without chemical reactions, the hquid-phase resistance will dominate, making it preferable to use Eq. (14-25) in conjunction with the relation Hl — Hql. 

Equation 1 is normally integrated by graphical or numerical means utilizing the overall material balance and the saturated air enthalpy curve. 

In the first one, the desorption rates and the corresponding desorbed amounts at a set of particular temperatures are extracted from the output data. These pairs of values are then substituted into the Arrhenius equation, and from their temperature dependence its parameters are estimated. This is the most general treatment, for which a more empirical knowledge of the time-temperature dependence is sufficient, and which in principle does not presume a constancy of the parameters in the Arrhenius equation. It requires, however, a graphical or numerical integration of experimental data and in some cases their differentiation as well, which inherently brings about some loss of information and accuracy, The reliability of the temperature estimate throughout the whole experiment with this... 

Due to the inherent uncertainty of the Langmuir model and difficulties in solving the transcendental equation (41), probably the most accurate treatment in the near-equilibrium cases is a numerical or graphical integration of the expression... 

This expression will give the point value of the Stanton number and hence of the heat transfer coefficient. The mean value over the whole surface is obtained by integration. No general expression for the mean coefficient can be obtained and a graphical or numerical integration must be carried out after the insertion of the appropriate values of the constants. 

Mean values may be obtained by graphical or numerical integration. 

The number of transfer units is obtained by graphical or numerical integration of equations 11.101, 11.103 or 11.104. 

If this is done at several points along the condensation curve the area required can be determined by graphical or numerical integration of the expression ... 

Table 11,1.1 contains a workup of the data in terms of the above analysis. In the more general case one should be sure to use appropriate averaging techniques or graphical integration to determine both F(t) and T. When there is an abundance of data, plot it, draw a smooth curve, and integrate graphically instead of using the strictly numerical procedure employed above. 

As an alternative to this numerical procedure, graphical integration of the terms in equation B could be employed. 

In Figure 3.11, we exclude Ihe use of differential methods with a BR, as described in Section 3.4.1.1.1, This is because such methods require differentiation of experimental ct(t) data, either graphically or numerically, and differentiation, as opposed to integration, of data can magnify the errors. 

If analytic equations for the heat capacities of reactants and products are unavailable, we still can carry out the integration required by Equation (4.78) by graphical or numerical methods. In essence, we replace Equation (4.79) by the expression... 

It can be seen from Equation (7.70) that to calculate AG at any temperature and pressure we need to know values of AH and AS at standard conditions (P= 100 kPa, T = 298 K), the value of ACp as a function of temperature at the standard pressure, and the value of AEj- as a function of pressure at each temperature T. Thus, the temperature dependence of AC/> and the temperature and pressure dependence of AVj-are needed. If such data are available in the form of empirical equations, the required integrations can be carried out analytically. If the data are available in tabular form, graphical or numerical integration can be used. If the data are not available, an approximate result can be obtained by assuming ACp and AVp are constant... 

The ratio of the fugacity/2 at the pressure P2 to the fugacity/i at the pressure Pj can be obtained by graphical or numerical integration, as indicated by the area between the two vertical lines under the isotherm for the real gas in Figure 10.6. However, as Pi approaches zero, the area becomes infinite. Hence, this direct method is not suitable for determining absolute values of the fugacity of a real gas. 

The integration indicated by Equation (11.21) then is carried out in two steps. From approximately 20 K up, graphical or numerical methods can be used (see Appendix A). However, below 20 K, few data are available. Therefore, it is customary to rely on the Debye equation in this region. 

The integral in Equation (19.63) is usually evaluated graphically or numerically. Although both I — g and m2 go to zero as m2 — 0, the ratio has a finite limit. Such a finite limit is not apparent from a plot of (1 — g)/m2 against m2 for electrolytes, as can be observed from Figure 19.11. But, if the integral in Equation (19.63) is transformed to... 

Where there is an analytical expression for the equilibrium such as Eq. (8.3), then Eq. (8.15) may be integrated directly. Otherwise it is necessary to perform the integration numerically or graphically. 

Figure 5.6 displays these performance equations and shows that the space-time needed for any particular duty can always be found by numerical or graphical integration. However, for certain simple kinetic forms analytic integration is possible—and convenient. To do this, insert the kinetic expression for in Eq. 

The values of the work terms for numerous samples, calculated from equations (12.10), (12.11) and (12.12), show excellent agreement with the corresponding work terms obtained by graphical integration of the areas in the intrusion-extrusion cycles. 

One of the limitations in the use of the compressibility equation of state to describe the behavior of gases is that the compressibility factor is not constant. Therefore, mathematical manipulations cannot be made directly but must be accomplished through graphical or numerical techniques. Most of the other commonly used equations of state were devised so that the coefficients which correct the ideal gas law for nonideality may be assumed constant. This permits the equations to be used in mathematical calculations involving differentiation or integration. 

This integral may be evaluated either by splitting into partial fractions, or by graphical or numerical means. Using the method of partial fractions, we obtain after some fairly lengthy manipulation ... 

Substituting the expressions for partial pressures into the rate equation the mass of catalyst required may be determined by direct numerical (or graphical) integration. Thus ... 

Unfortunately, temperature (Th2o) is not a simple function of and Hair-HjO- Hence, we must perform the integration of Equation 5.33, either graphically or numerically. The following example illustrates both approaches to solving for Ntu. ... 

---