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N(2c + 3) + 3

Since the number of variables exceeds the number of equations by three, it is necessary to fix three variables in order to obtain a solution to the N(2c + 3) equations. For example, the distillate rate D, reflux rate L1? and the column pressure may be specified.

When it is supposed that the vapor and liquid streams form ideal solutions, the enthalpy per mole of vapor and the enthalpy per mole of liquid leaving plate j are given by the following expressions (as shown in Chap. 14)

where the enthalpy of each pure component i in the vapor and liquid streams leaving plate j are represented by H^ and hji9 respectively. These enthalpies are of course evaluated at the temperature and pressure of plate j. The meaning of Hd depends upon the type of condenser employed. For a total condenser, D is withdrawn from the accumulator as a liquid at its bubble-point temperature 7\ at the column pressure, and y2i = xu = XDi. Thus

For a partial condenser, D is withdrawn from the accumulator as a vapor at its dew-point temperature 7\ at the column pressure, and yu = XDi. Thus c c

The enthalpy per mole of bottoms has double representation, hB and hN, that is, c c ha = £ hBixBi = £ hNixSi = hs (2-5)

The symbols Qc and QR are used to denote the condenser and reboiler duties, respectively. The condenser duty Qc is equal to the net amount of heat removed per unit time by the condenser, and the reboiler duty QR is equal to the net amount of heat introduced to the reboiler per unit time.

A wide variety of numerical methods have been proposed for solving the set of equations represented by Eq. (2-1). Two fundamentally different iterative procedures have been proposed for solving these equations; namely the Lewis and Matheson method10 and the Thiele and Geddes method.14 In the Lewis and Matheson method, the terminal compositions {Xand {xBi} are taken to be the independent variables, and in the Thiele and Geddes method, the temperatures (the temperature of each stage) are taken to be the independent variables. Up until about 1963, the Lewis and Matheson choice of independent variables was used almost exclusively, and since then, the Thiele and Geddes choice of the independent variables has become the most popular.

Merely the statement that the Thiele and Geddes choice of independent variables (or the Thiele and Geddes method) has been employed to solve a problem is not sufficient to describe the calculational procedure. In the solution of a set of nonlinear equations by iterative techniques, the convergence or divergence of a given calculational procedure depends not only upon the initial choice of the independent variables but also upon the precise arrangement and order in which each equation of the set is solved. Over a period of several years, the author has investigated a variety of arrangements and combinations of the expressions given by Eq. (2-1). Of these, the calculational procedure described below was found to converge for almost all problems involving distillation columns. To achieve this result, it was necessary to include the 0 method of convergence in the calculational procedure.

2-2 FORMULATION AND APPLICATION OF THE 6 METHOD OF CONVERGENCE, THE Kb METHOD, AND THE CONSTANT-COMPOSITION METHOD

The order of presentation of the topics in this section is the same order in which the combined set of methods listed above are applied in the calculational procedure. First, the component-material balances given by Eq. (2-1) are restated in terms of the component-flow rates. The component-flow rates for the liquid phase are eliminated from this set of equations by use of the equilibrium relationships given by Eq. (2-1). Then the 0 method is presented. The 0 method is used to compute an improved set of compositions on the basis of the most recent set of calculated values of the component-flow rates. The compositions so obtained are used to compute a new set of temperatures by use of the Kb method. The new sets of compositions and temperatures are then used to compute a new set of total flow rates by use of the constant-composition method for solving the enthalpy balances. Numerical examples are used to demonstrate the application of these methods.

Statement of the Component-Material Balances and Equilibrium Relationships as a Tridiagonal Matrix Equation

Although the equations utilized in this procedure differ in form from those presented by Eq. (2-1), they are an equivalent independent set. In the case of the component-material balances, a new set of variables—the component-flow rates in the vapor and liquid phases—are introduced, namely, vji = vjyji and /,,- = LjXj, (2-6)

Also, the flow rates of component i in the distillate and bottoms are represented by di = DX Di and &,= Bxm (2-7)

and the flow rates of component i in the vapor and liquid parts of the feed by vFi = vryti and lFi = LFxFi (2-8)

The equilibrium relationship y}i = K^x^ may be restated in an equivalent form in terms of the component-flow rates vn and lJt as follows. First, observe that through the use of Eq. (2-6), the expression yjt = K^x^ may be restated in the form

where the absorption factor An and the stripping factor are defined as follows

Instead of enclosing the ends of the column and the respective plates in each section of the column as demonstrated by Eq. (2-1) and Fig. 2-1, an equivalent set of component-material balances is obtained by enclosing each stage (j — 1, 2, ..., N — 1, N) by a component-material balance as demonstrated in Fig. 2-3. The corresponding set of material balances for each component i are as follows

Material balances

h-1. i - V» - !» + vj+ i = 0, (/ = 3, 4, ...,/- 2)

f/" — 2, / - «>/- 1, « - lf~ 1. « + Vfi = - ^ (2"12) //- I. « — ^/i — + 1. f = - J«

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