Almost band algorithms of the newtonraphson method

The Newton-Raphson formulations, called the Almost Band Algorithms are recommended for solving problems involving columns in the service of separating highly nonideal mixtures. In the Almost Band Algorithms, the independent variables in the Newton-Raphson method are taken to be either one or both sets of the component-flow rates {/Jt} and {i;,,}, the temperatures {7}}, and in some of the formulations one or more of the total flow rates {L,}. In Sec. 5-1 the independent variables are taken to be the component-flow rates, {/,,}, {t^-}, and the temperatures {7}). The formulation is presented in Sec. 5-1 for absorbers and strippers, and in Sec. 5-2 the formulation for conventional and complex columns is presented. Two modifications of Broyden's method are presented in Sec. 5-2. The modifications of Broyden's method preserve the sparsity of the initial jacob-ian matrix, whereas the original method as proposed by Broyden does not. The treatment of systems of columns in the service of separating highly nonideal solutions is presented in the next chapter.

Although the Almost Band Algorithms use a large number of independent variables, far less computer time is required to obtain a solution to a given distillation problem than might be expected. The computational speed results from the use of selected techniques of sparse matrices and the characteristics of homogeneous functions.

Sparsity of the jacobian matrix is achieved by a suitable ordering of the i variables and functions. The particular choice of variables and functions and their ordering (discussed below) leads to the unique form of the jacobian matrix • shown in Fig. 5-1. The well-known method of gaussian elimination may be ; applied in a stepwise fashion in the transformation of the matrix shown in ; Fig. 5-1 to the one shown in Fig. 5-2. At any one time, only four of the (c + 2) I square matrices along the diagonal and the two corresponding column matrices j are considered in the gaussian elimination process instead of the complete & N(c -I- 2) square matrix. No arithmetic is ever performed on any of the zero elements lying outside of the squares in Figs. 5-1 and 5-2. ■

Plate

Note:

I All elements outside of the shaded area are zero.

2. Each of the shaded squares contains one or more nonzero elements.

Figure 5-1 Structure of the jacobian matrix for an absorber.

Figure 5-2 Upper triangular matrix for an absorber.

Highly nonideal solutions are characterized by the fact that the activity coefficients and the partial molar enthalpies are strongly dependent upon composition. In order to compute the partial derivatives of these quantities which are needed in the application of the Newton-Raphson method, it is convenient to choose compositions or component-flow rates as members of the set of independent variables. Numerous choices of the independent variables have been made.6,7*8-13,15,17'19*20 To demonstrate the formulation of the Newton-Raphson method, the choice of independent variables proposed by Naphtali and Sandholm17 is used. The Almost Band Algorithm may be formulated for other choices of independent variables as shown by Gallun and Holland.7-8-9

5-1 ALMOST BAND ALGORITHMS FOR ABSORBERS AND STRIPPERS, INDEPENDENT VARIABLES:

As shown in Fig. 4-1, the plates of the absorber are numbered down from the top of the column, the top plate is assigned the number 1 and the bottom plate the number N. The variables regarded as fixed (or specified) in the developments which follow are:

1. {/0l}, liquid at T0 and at the inlet pressure P0 2- lvN+i,i}, vapor at Ty+1 and at the inlet pressure PN+l 3. the column pressure or the pressure on each stage

The N(2c + 3) equations required to describe the column may be stated in the following form:

Component- | vjL, + /, _! , - v}i - Iji = 0 material balances I

Energy balance

• X ['> i, 1.. + 0-1. ¡V i.f ~ vjiÛ» ~ 'ÂJ = 0 j, = i

where //,, and h}i denote the virtual values of the partial molar enthalpies; see Chap. 14. In the above statement of the equations, the component-material balances and the energy balances enclose each stage j.

Use of the second and third expressions of Eq. (5-1) to eliminate the total-flow rates from the equilibrium relationships yields a total of N(2c + 1) equations for the description of an absorber. When the independent variables are chosen as shown above, it is convenient to state each component-material balance and each energy balance for the enclosure of a single stage. Thus, after the total-flow rates have been eliminated from Eq. (5-l)'as described above, the resulting set of N(2c + 1) independent equations required to describe the column may be stated in functional form to give f _ yjiKjiht yjivji 0= i, 2,..., N)

ÎlVjiHji+Ijihy]

In the expressions for the activity coefficients yj;.} and the virtual values of the partial molar enthalpies {if;,, the mole fractions must have the sum of unity. This condition is satisfied by use of the following expressions for these mole fractions

In order to obtain a jacobian matrix having the form shown in Fig. 5-1 for an absorber, both the functions and the variables must be appropriately ordered. The functions must be ordered as follows f = [(fj, 1 fj, 2 '"he i "ij, 2 ' ' ' c Gj)j= i, jv]t (5'6)

where the subscript j = 1, N means that the argument is to be repeated for j = 1, 2,..., N. The variables must be ordered as follows x = Wj. i h.2 h.c vi i vj.2 Hc Ti)j-i.*]T (5_7)

By the ordering of the variables is meant the order in which the differentiation of each function is carried out in the Newton-Raphson method which is the same as the order in which the variables appear in the vector given by Eq. (5-7).

For example, the Newton-Raphson equation for any one function, say fjk (the equilibrium function for plate j and component k, where k denotes a particular one of the c components), is

(dfjk/dh. i) A/,. i + (dfjk/dlU2) AlU2 + ••• + (dfjk/dlUc) AlUe

+ • • • + (dfjk/dvNt c) Avn c + (d/ydT,) AT„

The complete set of Newton-Raphson equations may be stated in the following matrix form

J Ax = f

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