Systems of Columns

Two approaches exist for solving problems involving systems of columns, the column modular method and the system modular method proposed by Hess.15 In the column modular approach, the equations for each column of a system are solved in succession, and in the system modular approach, the complete set of equations for the system are solved simultaneously. While the system 

Solution of Systems of Columns by Use of the Column Modular Method 

In the column modular approach, the equations for each column are solved by use of the most efficient procedure for each column. After one trial has been made on each column, the terminal flow rates are placed in component-material balance and in agreement with the specified values of the terminal flow rates by use of the capital 0 method for systems. The entire calculational process is repeated until convergence has been achieved. 

Suppose that the total flow rates Lm, D2, WXj and W2 are specified. These specifications in turn fix Vx and B2. The specifications for the reboiled absorber are as follows F, A ,, the thermal condition of F,/ N, and the column pressure. The specifications for the distillation column are as follows N, type of condenser,/, Lj, D, and the common pressure. 

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An external feed of any amount, of one or more components, can be fed to any stage in the system of columns. This stage may be a plate, a reboiler, or a condenser. All external feed compositions, flows, and enthalpies are independent variables that are set in the problem description. A side stream of either vapor or liquid (or both) may be withdrawn from any stage except a reboiler or condenser, and either removed as product or sent to any stage of any column. The flow of each side stream... 

Destination (stage and column) for each side stream that is not a product removed from the system of columns. 

The first step in the basic program is to call subroutine input which reads in the program specifications. Next, the iteration counter, iterat, and a calculational variable, kinven, are set to zero. The amount of each component entering the system of columns is summed up in sumfd(i) where the subscript (i) refers to component. Subscript (j) refers to stage, and subscript (k) refers to column. Also the total feed of all components is summed in totfd. 

The program repeats the entire calculation through statement 140 for each column in turn and then proceeds to the next section in which the calculated amounts of each component in all products leaving the system of columns are summed in prdsum. Each side flow or flow from an end stage is examined to determine whether it goes to some other point in the column systems if not, it is a product. The over-all unbalance for each component is calculated as cunbal, equal to the total input of the component to the column system minus the total output. 

Next, side streams are read as shown in the program, and possible linkages through streams from end stages being sent to other points are read. The liquid stream from the bottom of a column, for example, will be considered a product leaving the system of columns unless it is specified as going to some other point. Since such a stream could be a dependent variable with its amount unknown, it cannot be sent back to the same column. 

NcooLS == total number of intercoolers in system of columns. 

NFEEDS = total number of feed streams entering the system of columns. 

SUMLD = sum of condenser and reboiler loads for entire system of columns... 

The equipment used in HPIEC is different from that used in CIEC. The columns, valves, and pipes are exposed to high pressures. This requires them to be not only corrosion resistant to acids, bases, and other chemical agents, but they also need to be pressure resistant. Because of this, the materials used in their construction are required to be of higher quality than the material used to construct the CIEC devices. The devices, e.g., the column, valves, pipes, sieves, pressure vessels, and pressure gauges used by the authors, were primarily made of special stainless steel. Some junctions were made of polytetrafluoroethylene plastic. Usually the last column of a system of columns was made of glass so as to permit observation of the separation affected. 

Marginal vapor flow is the added vapor flow required in the column because the other species—i.e., those on the list—are present. The vapor flow in a column is an indicator of the cost of purchasing and operating the column. A difficult separation will have a large vapor flow because it will require a large reflux ratio. Also, refluxed material has to be vaporized and condensed, which directly affects the utility costs for operating the column. Therefore, it makes sen.se to try to minimize the total of the vapor flows for a system of columns. 

By definition, component 1 is less retained than component 2. The initial condition of the problem corresponds to a system of columns that are all initially empty of feed but contain only the liquid and the solid phases in equilibrium. The boundary conditions result from the SMB design, i.e., what flows into a column depends on the composition of the fluid phase that is eluting from the previous column. These conditions are... 

The application of these equations to speciflc cases is demonstrated in Example 3.2. The number and complexity of the equations for multistage, multi-component columns are much higher, which rules out the elimination and substitution methods. General systematic techniques for solving systems of column equations are discussed in Chapter 13. 

APPLICATION OF THE THETA METHOD OF CONVERGENCE TO COMPLEX COLUMNS AND TO SYSTEMS OF 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. 

For the system of columns shown in Fig. 3-8, there exists one independent component-material balance per column, namely... 

A GENERALIZED FORMULATION FOR A SYSTEM OF COLUMNS IN WHICH THE TOTAL-FLOW RATES OF THE EXTERNAL STREAMS ARE SPECIFIED... 

If each column in the system of columns shown in Fig. 3-8 is at total reflux of the type in which all of the total-flow rates of the terminal streams (F, Du D2, Bu B2) are nonzero, finite, and positive,... 

The 2N Newton-Raphson method may be applied to any type of distillation column or to any system of interconnected columns. Absorbers, strippers, reboiled absorbers, and distillation columns are treated in Sec. 4-1. Selected numerical methods for solving the 2N Newton-Raphson equations are presented in Sec. 4-2. In Sec. 4-3, two methods for solving problems involving systems of columns interconnected by recycle streams are presented. 

Recommended procedures for solving problems involving single columns as well as systems of columns interconnected by recycle streams are summarized in Table 4-1 for the case where mixtures to be separated form ideal or near solutions throughout the column. As shown there, the 2N Newton-Raphson method is recommended for solving problems involving absorber-type columns (any column which does not possess both an overhead condenser and a reboiler such as absorbers, strippers, and reboiled absorbers). 

Also, to demonstrate the effect of the number of plates on the computer time required to obtain a solution, the first column of the system of columns of Example 3-3 was solved as a complex column and referred to as Example 4-7. The recycle stream B2 (see Fig. 3-12) was taken to be an independent feed and its composition was taken to be the solution set of values given in Table 3-5. The solution sets of temperatures, total-flow rates, and composition were the same as those presented for Example 3-3 in Table 3-5. 

SYSTEMS OF COLUMNS IN THE SERVICE OF SEPARATING MIXTURES OF NONIDEAL SOLUTIONS... 

Figure 6-11 Sketch of system of columns used in Examples 6-1 and 6-2.

SOLUTION OF SYSTEMS OF COLUMNS WITH ENERGY EXCHANGE BETWEEN RECYCLE STREAMS... 

There follows a description of the use of the capital 0 method in the determination of a solution for the system of columns shown in Fig. 7-1 such that the reboiler duty of column 1 is equal to the condenser duty of column 2, that is, such that QRl = QC2. For the system of columns shown in Fig. 7-1, suppose that column 1 has a partial condenser and that the flow rate, composition, and thermal condition of the feed are fixed as well as the pressure, the number of stages, and the feed-plate location. Likewise, suppose that the number of plates, the feed-plate location, and the pressure of column 2 are fixed. Two additional specifications may be made on each column. Since the precise choice of these additional specifications determines the form of the calculational procedures for each column as well as the system, only these additional specifications are listed in the problems considered. In each instance it is supposed, however, that the usual specifications of the type listed above have been made. 

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