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B. Tech. Petrochemical Engineering

Multicomponent DistillationUniversity College of Engineering (A) KakinadaDept. of Petroleum Engineering & Petrochemical EngineeringThe Method of Convergence&The KB method Jawaharlal Nehru Technological University KakinadaProf. K. V. RaoProgramme DirectorPetroleum CoursesJNTUKThe Method of ConvergenceTheta method of convergence is recommended for solving problems involving any type of distillation column; provided that the mixtures do not deviate too widely from ideal solutions.

For such columns, the method is one of the fastest known methods.

This method has been primarily applied to the Thiele-Geddes equations but a form of the theta method equation has also been applied to the equations of the Lewis-Matheson method. To achieve fast convergence of the problems dealing with the simulation of distillation columns, the method combined with Kb method was proposed by Holland. 2The main independent variable of the method is a convergence promoter, theta (or ).

The convergence promoter is used to force an overall component and total material balance and to adjust the composition on each stage.

In the iterative procedure, the improved sets of liquid mole fractions on each plate in a column required for the calculation of a new temperature profile are obtained using the corrected product compositions (top and bottom).

The corrected product rates are used as weight factors in the calculation of mole fractions. 3The corrected terminal rates are selected such that they are both in overall component material balance and in agreement with the specified value of D, that is

The overall component material balance and

ii.The criterion of the sum of the top product compositions, which is equal to unity.

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24These two conditions must satisfied simultaneously by suitable choice of the multiplier , which is defined by

The subscripts co and ca are the corrected and calculated values of a variable, respectively.

Elimination of (bi / di)ca from Eqs (a) and (b) yields the formula for (di)co

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a

b

5Since the specified values of (di)co are to have a sum equal to the specified value of D, the desired value of is that > 0 that makes g() = 0, where

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56In the determination of by Newtons method, the following formula for the first derivate, g ( )

After the desired value of has been obtained, (di)co may be computed by use of Eqa (b). Since Newtons method converges to the positive root of g(), provided =0 is taken to be the first trail value.

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Figure 2: Geometrical Representation of the function g() in the neighbourhood of the positive root .8The corrected mole fractions for the liquid and vapour phases are computed as follows

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79The KB methodRobinson and Gilliland pointed out that if the relative values of the Ki are Pi independent of temperature, the trial and error calculations are avoided in the determination of bubble-point and dew-point temperatures. The ratio Ki / Kb is called the relative volatility i of component I with respect to component b, that is

Where and Ki are Kb evaluated at the same temperature and pressure. Component b may or may not be a member of the given mixture under consideration.

810When the xi and pressure Pare given and it is desired to determine the bubble-point temperature, the formula needed may be developed as follows

Summation of the members of above Eq (8) over all components i, followed by rearrangement yields

Since i are independent of temperature, they may be computed buy use of the values of Kb and Ki evaluated at any arbitrary value of T and at the specified pressure.

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911After Kb has evaluated by use of Eq (9), the desired bubble-point temperature is found from the known relationship between Kb and T.

If the yi are known instead of the xi s, then the desired formula for the determination of the dew-point temperature is found by rearranging the Eq (9) to the following form

and then summing over all components to obtain

(10)

(11)12This Eq (4) is used to determine the dew-point temperature in a manner analogous to that described to determine the bubble point temperature.

Many families of compounds are characterized by the fact that their vapor pressures may be approximated by the Clausius-Clapeyron equation, and by the fact that their latent heats of vaporization are approximately equal.

The logarithm of the vapor pressures of the members of such families of compounds falls on parallel lines when plotted against the reciprocal of the absolute temperature. For any two members i and b of such a mixtures, it is readily shown that i is independent of temperature.13Although there exists many systems whose i are very nearly constant and Eqs (9) and (11) are applicable for the determination of the Bubble-point and Dew-point temperatures, respectively, the greatest use of these relationships lies in their application in the iterative procedures for solving Multicomponent distillation problems.

This method replaces the bubble point method in some of the P.W.ams. The new temperature profile for the next iteration is calculated on the basis of the corrected liquid mole fractions and the temperature profile of the previous iteration by using the KB method. 14

This method is considered as a modified bubble point method. For any plate, Eq (9) may be applied as follows

Where, j,i = Kj,i / Kj,b, the relative volatility of component i at the temperature of plate j , and Kj,b is the K value of base component (in the calculations , heavy key component is considered as the base component. As the value of Kj,b is function of temperature .

The new temperature of each plate can be calculated using K equation rewritten for T.

(12)

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Thank You16