Distillate Problem

Converse and Gross (1963), Converse and Huber (1965), Murty et al. (1980), Diwekar et al. (1987), Mujtaba (1989), Logsdon (1990) and Logsdon et al. (1990) considered an optimisation problem which maximises the amount of distillate 

Linear bounds on reflux ratio (inequality constraints) 

Kerkhof and Vissers (1978) combined the minimum time and the maximum distillate problems into an economic profit function P to be maximised. 

Both the amount of distillate and the time of operation are to be established. The only constraint arises from the required purity of the distillate product. Mathematically, the problem can be written as  

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MicroMENTOR is an educational package for solving distillation problems and includes MCCABE, PONCH, and BATCH for the MaCabe-Thiele, Ponchon-Savarit, and Batch binary distillations (11). The commercially available distillation software packages have been surveyed (15). Por reactive distillation, ASPEN software (16) is weU-known and widely adopted. 

Computer solutions entail setting up component equiUbrium and component mass and enthalpy balances around each theoretical stage and specifying the required design variables as well as solving the large number of simultaneous equations required. The expHcit solution to these equations remains too complex for present methods. Studies to solve the mathematical problem by algorithm or iterational methods have been successflil and, with a few exceptions, the most complex distillation problems can be solved. 

Example This equation is obtained in distillation problems, among others, in which the number of theoretical plates is required. If the relative volatility is assumed to be constant, the plates are theoretically perfect, and the molal liquid and vapor rates are constant, then a material balance around the nth plate of the enriching section yields a Riccati difference equation. 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

Gallun, S. E. and C. D. Holland, Solve More Distillation Problems, Part 5 Hydrocarbon Processing 137, (1976). 

Holland, C. D., G. P. Pendon, and S. E. Gallun, Solve More Distillation Problems, Hydrocarbon Processing June, 101 (1975). 

The vapor composition at the top of the condenser (Y,i) is different from that at the bottom (Y, ). The condenser may be compared to a fractional distillation problem in reverse. Butane, having a higher boiling point, will condense out faster than the propane, although both are condensing at the same time. Thus, the vapor and liquid mol fractions from the top to the bottom of the condenser tube bundle are always changing. Proceed as follows The vapor at the top has the same composition as the gas leaving the evaporator. Therefore, Y,. = Y,. 

The principle of the perfectly-mixed stirred tank has been discussed previously in Sec. 1.2.2, and this provides essential building block for modelling applications. In this section, the concept is applied to tank type reactor systems and stagewise mass transfer applications, such that the resulting model equations often appear in the form of linked sets of first-order difference differential equations. Solution by digital simulation works well for small problems, in which the number of equations are relatively small and where the problem is not compounded by stiffness or by the need for iterative procedures. For these reasons, the dynamic modelling of the continuous distillation columns in this section is intended only as a demonstration of method, rather than as a realistic attempt at solution. For the solution of complex distillation problems, the reader is referred to commercial dynamic simulation packages. 

For any particular distillation problem equation 11.28 will have only one real root k between 0 and 1... 

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,... 

HENGSTEBECK, R. J. (1946) Trans. Am. Inst. Chem. Eng. 42, 309. Simplified method for solving multicomponent distillation problems. 

Two other problems that fit this category are calculating the number of stages in a multicomponent distillation problem, and obtaining the material balance when complicated recycle operations occur. 

Logsdon, J. S. and L. T. Biegler. Accurate Determination of Optimal Reflux Policies for the Maximum Distillate Problem in Batch Distillation. Ind Eng Chem Res 32 (4) 692-700 (1993). 

This method is one of the most important concepts in chemical engineering and is an invaluable tool for the solution of distillation problems. The assumption of constant molar overflow is not limiting since in very few systems do the molar heats of vaporisation differ by more than 10 per cent. The method does have limitations, however, and should not be employed when the relative volatility is less than 1.3 or greater than 5, when the reflux ratio is less than 1.1 times the minimum, or when more than twenty-five theoretical trays are required(13). In these circumstances, the Ponchon-Savarit method described in Section 11.5 should be used. 

These basic methods were the inspiration for scores of papers, ranging from simplifications based on various assumptions to shortened rigorous mathematical derivations applicable to more or less specific distillation problems. Among these may be mentioned those of Fenske (19). Dodge and Huffman (18), Smoker (53), Jenny (27), and Underwood (56). 

Another evident trend is the application of mechanical computers for the rapid calculation of distillation problems. At least two papers have already appeared, one by Weil (57) on the use of an electronic computer, and the other by Rose and Williams (47) on the application of the IBM card punch technique. If a short rigorous method is not forthcoming, machine calculations may be the answer to accurate design of fractionating equipment. 

The calculational base consists of equilibrium relations and material and energy balances. Equilibrium data for many binary systems are available as tabulations of x vs. y at constant temperature or pressure or in graphical form as on Figure 13.4. Often they can be extended to other pressures or temperatures or expressed in mathematical form as explained in Section 13.1. Sources of equilibrium data are listed in the references. Graphical calculation of distillation problems often is the most convenient... 

In the usual distillation problem, the operating pressure, the feed composition and thermal condition, and the desired product compositions are specified. Then the relations between the reflux rates and the number of trays above and below the feed can be found by solution of the material and energy balance equations together with a vapor-liquid equilibrium relation, which may be written in the general form... 

Estimates must be made of V/L at the top and bottom and the feed zone. In distillation problems, assumption of constant molal overflow in each zone probably is within the accuracy of the method. In stripping or absorption columns, first iteration evaluations of the amounts of stripping or absorption will provide improved estimates of V/L at the key points in the columns. 

A distillation problem is worked out by this method by Edmister [Pet. Eng., 128-142 (Sept. 1948)]. The method is developed there. 

Until the advent of computers, multicomponent distillation problems were solved manually by making tray-by-tray calculations of heat and material balances and vapor-liquid equilibria. Even a partially complete solution of such a problem required a week or more of steady work with a mechanical desk calculator. The alternatives were approximate methods such as those mentioned in Sections 13.7 and 13.8 and pseudobinary analysis. Approximate methods still are used to provide feed data to iterative computer procedures or to provide results for exploratory studies. 

I am indebted to M. J. E. Golay for allowing me to see his contribution to the Amsterdam Gas Chromatography Symposium whilst it was still in typescript, and to J. W. Westhaver for correspondence on his work. Fruitful discussions of the distillation problem were had with N. Macleod of the Chemical Engineering Department, Edinburgh University. The formulae in 3 were checked by Mr. Kung-You Lee, who also did the calculation for the figure. 

Komatsu H, Holland CD. A new method of convergence for solving reacting distillation problems. J Chem Eng Japan 1977 4 292-297. 

Application to Batch Distillation Maximum Distillate Problem... 

Referring to Simple Model (with zero column holdup) presented in section 5.5.1, a maximum distillate problem can be formulated as ... 

The Hamiltonian, the adjoint equations and the optimal reflux ratio correlation will be same as those in Equations P.10-P.13 (Diwekar, 1992). However, note that the final conditions (stopping criteria) for the minimum time and the maximum distillate problems are different. The stopping criterion for the minimum time problem is when (D, xq) is achieved, while the stopping criterion for the maximum distillate problem is when t, xo) is achieved. See Coward (1967) for an example problem. 

Logsdon and Biegler (1993) considered a binary separation of cyclohexane-toluene mixture in a conventional batch distillation column. Maximum distillate problem was considered to maximise the amount of distillate with cyclohexane purity of 0.998 molefraction. The input data for the problem is given in Table 5.7. 

Note that for a fixed operation time, t in Equation 9.1, the profit will increase with the increase in the distillate amount and a maximum profit optimisation problem will translate into a maximum distillate optimisation problem (Mujtaba and Macchietto, 1993 Diwekar, 1992). However, for any reaction scheme (some presented in Table 9.1) where one of the reaction products is the lightest in the mixture (and therefore suitable for distillation) the maximum conversion of the limiting reactant will always produce the highest achievable amount of distillate for a given purity and vice versa. This is true for reversible or irreversible reaction scheme and is already explained in the introduction section. Note for batch reactive distillation the maximum conversion problem and the maximum distillate problem can be interchangeably used in the maximum profit problem for fixed batch time. For non-reactive distillation system, of course, the maximum distillate problem has to be solved. 

Walsh et al. (1995) considered an industrial batch reactive distillation problem originally presented by Leversund et al. (1993) as a case study. A condensation polymerisation reaction between a dibasic aromatic acid (R1) and two glycols (R2, R3) was considered. The reaction products were a polymer product (P) and water... 

Mujtaba (1997) used the maximum distillate problem to compare the performances of the two types of distillation columns (CBD and continuous). With the amount of initial charge and the feed flow rate fixed in a continuous column, the operation time (pass time) also becomes fixed. The performance measure using maximum distillate problem allows fixing of the operation time. Other types of optimisation problems such as minimum time or maximum profit problems (presented in the previous chapters) are not suitable for the purpose of comparing the performances of... 

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