Theta methods

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,... 

An alternative is to use the tridiagonal method for calculating compositions, but to calculate the new temperatures directly, without iterating on the bubble-point equation, These new temperatures are approximate but as long as the internal compositions are properly corrected during each column trial, the temperature profile will continue to move toward the solution. This is the basis of the theta method of Holland (7, 9, 26). With either alternative, the energy balances are used to find the total flow rates. 

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. 

In the theta method, the total distillate rate and reflux ratio are all that can be specified and the theta function [Eq. (4.50)] is written to find a value of theta such that the sum of the corrected component distillate rates equals the total distillate rate ... 

The Kb method. For updating the tray temperatures, the theta method relies on the Kb method. The Kb method takes advantage of the near-linear dependence of the logarithm of the K-values and the relative volatilities on temperature over short temperature spans. Relative volatilities (a s) are calculated with respect to a base component K-value, K.bj k, at the stage temperature of the current column trial, Tjk. The base component is usually a middle boiler or a hypothetical component, The K-value of the base component for the next trial, Kbjk + 1( is calculated using a form of the bubble-point equation unique to the Kb method ... 

Next trial. With these new total flow rates and using the new temperatures, the component flow rates can be recalculated using the tridiagonal method and a new column trial can be initiated. Note again that the theta method is different from other methods in that the total product rateB and reflux ratio are specified while the condenser and reboiler duties are calculated. 

Set initial stage temperatures, Tfs, and vapor rates, V - s. Initial liquid rates are found using the tridiagonal method for the total material balances. The theta method requires the distillate rate, sidestream product rates, and reflux ratio to be specified. 

The theta method has found many applications. The FRAKB routines of FLOWTRAN use the theta method, Portions of the theta method, such as the Kb method for temperatures, appear in other rigorous methods. Poor initial temperature and flow rate estimates do not greatly hinder the approach to solution and the calculation is relatively rapid. It has been shown to work well for the systems that the BP methods are meant for, i.e., narrow- and middle-boiling ranges, ideal or nearly ideal mixtures. It has solved columns with as many as 200 stages without great difficulty. 

Compositions or flows calculated by component material balances in the theta method and 2N Newton-Raphson method. Sees. 4.2.5 and 4.2.6. 

A modified Thiele-Geddes method, programmed for an IBM 370-155, was used to perform the calculations needed to size each required column. Experimental activity coefficient data were used to allow for nonideal liquid phase behavior while energy balances, using estimated enthalpy data, were used to correct for non-constant molal overflow. The Theta Method was used for convergence, and all plate efficiencies were assumed to be 100%. (See Reference 7 for additional calculational details and a program listing.)... 

The code can treat transient problems by means a two-point "theta-method" time-stepping algorithm. The dynamic algorithm is also useful in nonlinear problems, in which the final fluid state may be approached dynamically from an estimated initial state. 

DEVELOPMENT AND APPLICATION OF THE THETA METHOD OF CONVERGENCE TO CONVENTIONAL DISTILLATION COLUMNS... 

The calculation procedures [the 0 method, Kb method, and constant composition method] developed in Chap. 2 for conventional distillation columns are applied to complex distillation columns in Sec. 3-1. For solving problems involving systems of columns interconnected by recycle streams, a variation of the theta method, called the capital 0 method of convergence is presented in Secs. 3-2 and 3-3. For the case where the terminal flow rates are specified, the capital 0 method is used to pick a set of corrected component-flow rates which satisfy the component-material balances enclosing each column and the specified values of the terminal rates simultaneously. For the case where other specifications are made in lieu of the terminal rates, sets of corrected terminal rates which satisfy the material and energy balances enclosing each column as well as the equilibrium relationships of the terminal streams are found by use of the capital 0 method of convergence as described in Chap. 7. 

APPLICATION OF THETA METHOD OF CONVERGENCE TO COMPLEX COLUMNS 89... 

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