Graphical methods determinations

Using the graphical method, determine the roots of the following equations ... 

This equation relates the (instantaneous) copolymer composition with the monomer feed of M and M2. Values for and are usually determined by graphical methods (9,10). Today, with the prevalence of powerful desktop computers, numerical minimisa tion methods are often used (11—14). 

Typical methods are those of F. C. Zevnik and R. L. Buchanan [Chem. Eng. Progi , 59, 70-77 (Feb. 1963)] and J. H. Taylor Eng. 6-Proc. Econ., 2, 259-267, 1977). The former is mainly a graphical method of estimating the cost per functional unit (Cp) based on the capacity, the maximum pressure, the maximum temperature, and the materials of construction. The Taylor method requires the determination of the costliness index, which is dependent on the complexity of the process. A simpler method was suggested by S. R. Timms (M.Phil. thesis, Aston University, England, 1980) to give the battery hmits cost for gas phase processes only in U.S. dollars with a Marshall and Swift index of 1000. The simple equation is... 

MT McMahon, E Oldfield. Determination of order parameters and correlation times m proteins A comparison between Bayesian, Monte Carlo and simple graphical methods. I Biomol NMR 13 133-137, 1999. 

By using vapor-liquid equilibrium data the above integral can be evaluated numerically. A graphical method is also possible, where a plot of l/(y - xj versus Xr is prepared and the area under the curve over the limits between the initial and fmal mole fraction is determined. However, for special cases the integration can be done analytically. If pressure is constant, the temperature change in the still is small, and the vapor-liquid equilibrium values (K-values, defined as K=y/x for each component) are independent from composition, integration of the Rayleigh equation yields ... 

Service Goal. A goal for line service in terms of ton-miles between cutoffs should be selected. This value can initially be determined from Figures 4-84 and 4-85 and later adjusted in accordance with experience. Figure 4-86 shows a graphical method of determining optimum cutoff frequency. 

Figure 4-86, Graphic method of determining optimum frequency of cutoff to give maximum total ton-miles for a particular rig operating under certain drilling conditions [11],...

Currently, a graphical method of casing setting depth determination is used. The method is based on the principle according to which the borehole pressure should always be greater than pore pressure and less than fracture pressure. 

Mansfeld points out that a major limitation of the polarisation resistance is that the factor b b /2-3(b + b ) must be determined in order to evaluate, and has devised a procedure in which this can be achieved by a graphical method. 

Click Coached Problems for a self-study module on determining the rate law graphical method. 

By a brilliant physical intuition B. van der Pol succeeded finally (1920) in establishing his equation (which is given in Section 6.11) but, not having any mathematical theory at his disposal, he determined the nature of Ike solution by the graphical method of isoclines. It became obvious that the problem, which was a real stumbling block for many years, had been finally solved, at least in principle. 

The determination of k from Eq. (3-28), given a series of t, [A], pairs, requires a nonlinear least-squares program for the analysis of the data. A formidable calculation is thus made relatively simple. Otherwise, the graphical method suggested by Eq. (3-27) can be used. If the reaction is followed by an instrumental method, one can easily show that these readings, designated Y, are related to concentrations by... 

In practice, even the determination of a fourth-order correlation function requires a large amount of calculation. However, this procedure may be standardized by a graphical method, which performs the integration in Eq. (1.54) by using properties of Hermite polynomials [37], Without going into details, we give the result... 

White (14) proposed a graphical method of determining the critical Reynolds number at which the fully turbulent flow... 

What are the numerical values for the two time constants in the stagnancy model of Example 15.7 See Figure 15.8, but do not use a graphical method to determine them. 

If he selects the still pressure (which for a binary system will determine the vapour-liquid-equilibrium relationship) and one outlet stream flow-rate, then the outlet compositions can be calculated by simultaneous solution of the mass balance and equilibrium relationships (equations). A graphical method for the simultaneous solution is given in Volume 2, Chapter 11. 

The distillation of binary mixtures is covered thoroughly in Volume 2, Chapter 11, and the discussion in this section is limited to a brief review of the most useful design methods. Though binary systems are usually considered separately, the design methods developed for multicomponent systems (Section 11.6) can obviously also be used for binary systems. With binary mixtures fixing the composition of one component fixes the composition of the other, and iterative procedures are not usually needed to determine the stage and reflux requirements simple graphical methods are normally used. 

Mutual exclusivity can also be tested for by the effects of combinations of two inhibitors on the activity of a target enzyme. The advantage of this approach is that it does not require any special labeling of either compound, and only catalytic quantities of enzyme are required for the studies. There are a number of graphical methods that can be used to determine the effects of inhibitor combinations on enzyme velocity (see Copeland, 2000). The most popular of these was introduced by Yonetani and Theorell (1964) and is based on the following reciprocal equation ... 

The kinetic data below were reported for an enzyme catalyzed reaction of the type E + S ES E + P. Since the data pertain to initial reaction rates, the reverse reaction may be neglected. Use a graphical method to determine the Michaelis constant and Fmax for this system at the enzyme concentration employed. 

Step 4 Separation of Distortions of 1st and 2nd Kind. From Eq. (8.27) the graphical method for the separation of small lattice distortions of the first and the second kind is obvious. It is sketched in Fig. 8.5. In a plot of fo (r/L) vs. r/L the amount of lattice distortions of the second kind is determined from the intercept. Lattice distortions of the first kind are computed from the slope of the observed... 

Bakierowska, A.-M., Trzeszqzynski, J. (2003) Graphical method for the determination of water/gas partition coefficients of volatile organic compounds by a headspace gas chromatography technique. Fluid Phase Equil. 213, 139-146. 

Worked Example 4.15 The isomerization of 1-butene (X) to form frans-2-butene (XI). The equilibrium constants of reaction are given below. Determine the enthalpy of reaction AH using a suitable graphical method. 

Warning if a chemical process comprises several reaction steps, only the progress of the slowest step can be followed kinetically. These graphical methods of determining k are only useful for obtaining the rate-determining step (RDS) of such reactions. Although the reaction may appear kinetically simple, it is wisest to assume otherwise. 

Worked Example 8.21 The rate of hydrolysis for a biological molecule was studied over the temperature range 300-500 K, and the rates found to be as follows. Use a suitable graphical method to determine AH, Aand AG, for the reaction at 298 K. 

Diskind (1959) determined the operating characteristics for centrifugal pumps in parallel and in series using a simple graphical method. 

To determine the concentration using the graphical method draw a horizontal line from A — 0.443 to the least-squares line. Then construct a vertical line from that point to the concentration axis and read the value of the unknown. In this case it appears to be slightly less than 6.0 ppm. Using the least-squares equation we get... 

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