Graphical integration for Example

The average distillate composition is 0.85 for the key component which is much larger than obtained in simple distillation. 

In 1937, Bogart [41] presented the first analysis of the variable reflux policy for a binary system. The steps involved in the calculation procedure for the variable reflux mode are similar to those in the case of the constant reflux mode however, in the variable reflux case, the reflux ratio is varied instead of the distillate composition at each step. Moreover, the Rayleigh equation, though valid for the variable reflux condition, takes a simplified form. Since the distillate composition remains constant (remember that we are considering binary systems here) throughout the operation, the Rayleigh equation reduces to the following equation. 

Example 4.3 Rework the problem in Example 4.2 for the variable reflux mode. 

Solution Since the distillate composition is held constant throughout the variable reflux mode of operation, the distillate composition xp = xpav = 0.85. For the various iterates of R, we obtain the corresponding values of xb - The value of the amount of product distilled at each xb is also calculated using the Rayleigh equation for the variable reflux condition (Equation 4.9). 

Graphical integration for calculation of batch time for Example 4.3 

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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.

In the integral method, the concentrations of the reactants and products are plotted as a function of die residence time and the integrated forms are used in graphical evaluation. For example, in the case of power law kinetics (10.30) with the reaction order different from unity one arrives at... 

In order to calculate f in a mixture of chosen composition, temperature and pressure, experimental values of F must be available over the whole range of integration. For example, let it be supposed that the mixture in question consists of gases A and B only. At any particular teinperature and pressure, the partial molar volumes Vji and Fj may be determined, as described in 2 14, by measurements of the total volume of the gas in mixtures of several different compositions. A repetition at the same temperature and a lower pressure will give a fresh set of values F and Vb. and so on over the pressure range. If we now pick out the values of F which are appropriate to a particular composition, may be evaluated by graphical integration of (3 58). In this way a set of values of/ and may be... 

Pig. 19-14. Temperature and pressure curves for the graphical integration of Example 19-9. 

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. 

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

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

If the reaction in the cell proceeds to unit extent, then the charge nF corresponding to integral multiples of the Faraday constant is transported through the cell from the left to the right in its graphical representation. Factor n follows from the stoichiometry of the cell reaction (for example n = 2 for reaction c or d). The product nFE is the work expended when the cell reaction proceeds to a unit extent and at thermodynamic equilibrium and is equal to the affinity of this reaction. Thus,... 

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

We are therefore in a position to compare Faradaic and non-Faradaic currents and at 10-5 M they are of comparable magnitude (using Cj 20/xFcm-2). However, ic decays with t1/3 whereas iF increases with t1/6. This is, of course, an important reason for the sampling of current towards the end of drop life, which is used, for example, in the pulse polarography techniques. Pulse techniques are also useful when the integral capacitance, Ci, varies with potential in these cases graphical elimination of ic can be difficult. 

Graphical Assessment Using Integrated Equations Directly. Another way to ascertain mechanistic rate laws is to use an integrated form of Eq. (2.7). One way to solve Eq. (2.7) is to conduct a laboratory study and assume that one species is in excess (i.e., B) and therefore, constant. Mass balance relations are also useful. For example [A] -I- [Y] = A0+ Y0 where Y() is the initial concentration of product. One must also specify an initial... 

To calculate the amount of catalyst for a particular case, mass and heat balance have to be considered they can be described by two differential equations one gives the differential CO conversion for a differential mass of catalyst, and the other the associated differential temperature increase. As analytical integration is not possible, numerical methods have to be used for which today a number of computer programs are available with which the calculations can be performed on a powerful PC in the case of shift conversion. Thus the elaborate stepwise and graphical evaluation by hand [592], [609] is history. For the reaction rate r in these equations one of the kinetic expressions discussed above (for example, Eq. 83) together with the function of the temperature dependence of the rate constant has to be used. 

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