Geddes equation

The distribution of components for the distillate and the bottoms is given by the Hengstebeck-Geddes equation [124,125, 126] ... 

The theta method. 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. The main independent variable of the method is a convergence promoter, theta (or 6). The convergence promoter 0 is used to force an overall component and total material balance and to adjust the compositions on each stage. These new compositions are then used to calculate new stage temperatures by an approximation of the dew- or bubble-point equation called the Kb method. The power of the Kb method is that it directly calculates a new temperature without the sort of failures that occur when iteratively solving the bubble- or dew-point equations. 

Calculate the constants Ac and Be in the Geddes equation, Equation 6.27.1, using a specified recovery and relative volatility for the light and heavy key components. There should be one equation for the light key component and another equation for the heavy key component. Then, solve the two equations for Ac and Be. 

A, B, = correlation constants in Hengstebeck-Geddes equation b = number of moles of a component in the bottoms B = bottoms flow rate moles/hr d = number of moles of a component in the distillate D = distillate flow rate moles/hr F = feed flow rate moles/hr f = number of moles of a component in the feed... 

This simplified procedure is based on the assumption that the concentrations of all the components in the distillate are constant if the reference component concentration in the distillate is constant. If the concentrations of the nonreference components change, the authors (Diwekar and Madhavan, 1991) recalculate the concentrations based on the Hengstebeck-Geddes equation (Hengstebeck, 1946 Geddes, 1958). The authors also describe a procedure for the constant reflux case, where all the component compositions in the distillate change as the distillation progresses. 

Edminster (Ref. 4) has presented a modified absorption factor method that determines the molal quantities for each component as a fraction of their values in the distillate and bottoms in a manner somewhat similar to the Thiele and Geddes equations. The geometric mean of the absorption and stripping factors at the ends of the section under consideration is employed, and empirical correction terms are applied to these averages. 

The constant Ci in the Hengstebeck-Geddes equation is equivalent to the minimum number of plates, Nmin, in the Fenske equation. At this stage, the variable reflux operating mode has Gi and R, the constant reflux has and Ci, and the optimal reflux has, Ci, and R as unknowns. Summation of distillate compositions can be used to obtain Gi for variable reflux and for both constant reflux and optimal reflux operation, and the FUG equations to obtain R for variable reflux and Gi for both constant reflux and optimal reflux operations. The optimal reflux mode of operation has an additional unknown, R, which is calculated using the concept of optimizing the Hamiltonian, formulated using the different optimal control methods. 

Hengstebeck and Geddes (1958) have shown that the Fenske equation can be written in the form ... 

Chang (1980) gives a computer program, based on the Geddes-Hengstebeck equation, for the estimation of component distributions. 

The method starts with an assumption of the column temperature and flow profiles. The stage equations are then solved to determine the stage component compositions and the results used to revise the temperature profiles for subsequent trial calculations. Efficient convergence procedures have been developed for the Thiele-Geddes method. The so-called theta method , described by Lyster et al. (1959) and Holland (1963), is recommended. The Thiele-Geddes method can be used for the solution of complex distillation problems,... 

Iteration solutions were first proposed by Thiele and Geddes (Tl) in 1933. In this method, all temperatures and flows must be estimated before the solution can begin. The solution is broken into three parts first, solution of the mass-balance equations under the estimated flows and temperatures second, correction of the temperatures and third, correction of the flows. Assuming values for all temperatures and flows reduces the set of mass-balance equations shown in Table I to a linear set of equations which can be solved for the compositions at each point. Because the starting assumptions are completely arbitrary, the compositions will undoubtedly be wrong (the liquid and vapor mole-fractions will not sum to unity), and better values of temperature and flows must then be obtained for use in the next iteration. 

The equilibrium melting temperature, T°m, can be obtained from data for crystals of finite thickness using the Thompson-Gibbs equation. The melting point of crystalline polymers with a well-defined crystal thickness (/c) can be measured and the data extrapolated to 41 = 0 using the Thompson Gibbs equation (Gedde 1995) ... 

Equation (3.26) is adapted to nonspherical particles by multiplying D by a dissymmetry factor (Geddes, 1949). 

The classic papers by Lewis and Matheson [Ind. Eng. Chem., 24, 496 (1932)] and Thiele and Geddes [Ind. Eng. Chem., 25, 290 (1933)] represent the first attempts at solving the MESH equations for multicomponent systems numerically (the graphical methods for binary systems discussed earlier had already been developed by Pon-chon, by Savarit, and by McCabe and Thiele). At that time the computer had yet to be invented, and since modeling a column could require hundreds, possibly thousands, of equations, it was necessary to divide the MESH equations into smaller subsets if hand calculations were to be feasible. Despite their essential simplicity and appeal, stage-to-stage calculation procedures are not used now as often as they used to be. 

The Peng-Robinson equation of state was used to estimate K values and enthalpy departures [as opposed to the De Priester charts used in Example 1 and by Seader (ibid.) who solved this problem by using the Thiele-Geddes (op. 

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