Graphical Integrations

Langmuir-Blodgett flint deposited on copper. From graphic integration of the data of Ref. 144. See Ref. 158. 

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... 

The graphical integration method is based on graphical presentation of the average flow profile. For a circular duct, the cross-section is virtually divided into several concentric ring elements. The spatial mean velocity of such an element is determined as an arithmetical mean of local velocities along the circumference of the corresponding radius. For a circular cross-section the flow rate can be expressed as... 

Plot curve of 1/ (yj - Xj) versus xj see Figure 8-38, graphical integration by Simpson s rule. 

For a binary mixture the values of x and y can be obtained from the equilibrium curve. Select values of x and read the corresponding value of y from the equilibrium curve. Tabulate values of l/(y - x), and plot versus X], resulting in a graphical integration of the function dx (y - x) [130] between x and xj. This system would have no column internals and no reflux. 

Figure 8-38. Graphical integration of Rayleigh or similar equation by Simpson s Rule, for Example 8-14.

Graphical integration shows the area under the curve. Figure 8-38A, to be 15.764. Appl)dng this to ... 

Figure 8-38A. Graphical integration for boil-up rate of batch distillation for Example 8-15. Used by permission, Treybal, R. E., Cftem. Eng. Oct. 5 (1970), p. 95.

It is usually necessary to graphically integrate the first terms of the above equations, although some problems do allow for mathematical treatment. 

This compares with the value from graphical integration of 6,27 and is a good check. 

Note that the graphical integration is never exact and hence the correction often makes little difference except for cases of curved equilibrium lines. 

Figure 9-72. Graphical integration number of transfer units for Example 9-11.

An alternate method to determining the number of transfer units is the graphical integration of dy/(y - y). The procedure is basically the same as for absorbers, that is ... 

Graphically integrate, by plotting l/h -h vs. t, reading (h -h) firom the operating-equilibrium line plot for various values of temperature. See Figure 9-125. 

Figure 9-125. Graphical integration to determine number of transfer units.

A planned purchasing schedule in dollars graphically integrated with time. 

GINA graphical integrated ICM injection-compression molding... 

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... 

The three areas are found by graphical integration. The thermodynamic consistency test consists of comparing the sum of the three areas [left-hand side of Eq. (81)] with the right-hand side of Eq. (81). The three areas depend upon equilibrium data for the composition range x2 = 0 to x2 = x2. However, the right-hand side of Eq. (81) depends only on equilibrium data at the upper limit x2 = x2. The comparison indicated by Eq. (81) should be made for several values of x2 up to and including the critical composition. 

T2 = 200 K2). A graph such as the one shown in Figure 4.3 can be used to extrapolate Cp.m/T to 0 Kelvin. Values can be taken from the graph and included in the graphical integration. 

If one wishes to determine activities for component 2 with a Henry s law standard state, the lower limit of the right-hand integral of equation (6.177) goes to 0 as a 2 — 0. Thus, it would appear that a graphical integration could be performed by obtaining the area starting from an abscissa of In 7r, 1 = 0 to the... 

Equation (7.53) can be used to obtain 7 from <5, while equation (7.54) gives from 7 . What must be known is the relationship between (f> and m [for equation (7.53)] or between 7 and m [for equation (7.54)]. At higher concentrations, graphical integrations are usually used at low m where the... 

Figure A1.5 Graphical integrations using (a) the trapezoidal rule and (b) Simpson s rule.

Euler s theorem 612 exact differentials 604-5 extensive variables 598 graphical integrations 613-15 Simpson s rule 614-15 trapezoidal rule 613-14 inexact differentials 604-5 intensive variables 598 line integrals 605-8... 

Figure 3.9 Graphical integration of Gibbs-Duhem equation.

The use of Figure 11.40 will slightly overestimate the number of stages and a more accurate estimate would be made by graphical integration of equation 11.104 but this is not justified in view of the uncertainty in the prediction of the transfer unit height. Molecular weights SO2 = 64, H20 = 18, air = 29... 

From experimentally determined values of the various concentrations, or from the value of a single measured concentration and the reaction stoichiometry, calculate the value of ipiCi) at the times corresponding to these measurements. In some cases it may be necessary to resort to graphical integration to determine (Cf). 

Illustrations 3.2 and 3.3 are examples of the use of the graphical integral method for the analysis of kinetic data. 

ILLUSTRATION 3.2 USE OF A GRAPHICAL INTEGRAL METHOD FOR DETERMINING THE RATE CONSTANT FOR A CLASS II SECOND-ORDER REACTION... 

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