When the column pressure P, the rich gas rates {vN+lti} and the temperature Tn+ 1? the lean gas rates {/0l} and the temperature T0, and the number of plates N are fixed (or specified), the set of equations represented by Eq. (4-1) contains N(2c + 3) unknowns: {i-}, {/„}, {V;}, {!,•}, and {7}}.

The N equilibrium functions are formulated by first restating the second and third expressions of Eq. (4-1) as follows

Elimination of the s by use of the first expression of Eq. (4-1) and restatement of the result so obtained in functional form yields

The N enthalpy functions are obtained by dividing the sum of the input terms of the last expression of Eq. (4-1) by the sum of the output terms. When the result so obtained is restated in functional form, one obtains

The functions F; and G; contain the dependent variables {t^}, {/;i}, and {Vj}. For any choice of values of the independent variables {7}} and {Lj/Vj}, expressions are needed for computing the corresponding values of the dependent variables.

First, an equation for computing the for any set of assumed Lj/V/s is developed. Summation of the component-material balances over all components followed by the elimination of the summations through the use of the second and third expressions of Eq. (4-1) yields

Vj+l + LJ.l - Vj - Lj = 0 (j = 2, 3, ..., N - 1) (4-5)

For any given set of Lj/V/s it is desired to solve the total-material balances for the corresponding set of vapor rates {Vj}. In the restatement of the total-material balances, it is convenient to define the new variable 6j as follows

where (Lj/Vj)a is any arbitrary value of Lj/Vj. Taking this assumed ratio equal to the most recently assumed value of Lj/Vj serves to normalize the 0/s so that at convergence 9j approaches unity for all j. Let Eq. (4-6) be restated as follows

where Rj is defined by

Equation (4-7) may be used to restate the total material balances in terms of either the vapor or the liquid rates. For any interior plate j (j = 2, 3,..., N — 1), the total material balance may be restated in terms of the vapor rates as follows

The complete set of total-material balances may be represented by the matrix equation

where

y=[Vl V2 ••• Vn]t 3r = [L0 0 ••• 0 VN+l]T

For a given set of values of the independent variables {6j} and {7}}, the corresponding sets of values of the component-flow rates {t?^.} and {/,-,} are needed in order to evaluate the functions {F,} and {G,-}. These rates may be computed through the use of the first and fourth expressions of Eq. (4-1) which may be rearranged to the form hi = AjiVji (4-11)

where

" KjiVj KjAVjh

After the lj-s 0=1, 2, N) have been eliminated from the component-material balances by use of Eq. (4-11), the resulting set of equations may be restated in the matrix form of Eq. (2-18), A. v, = -/i, where each absorption factor appearing in Af is given by Eq. (4-11) and v, and / have the following elements v, = K- v2i •••

Now observe that for any given set of 6/s and T/s (and some arbitrary set of (Lj/Vj)a*s), sets of numerical values may be found for the V/s and the v^s by solving Eqs. (4-10) and (2-18) respectively. After the VJs have been found, the Lj s may be computed by use of Eq. (4-7). Similarly, after the t^./s have been computed, the corresponding /,,'s may be computed by use of the equilibrium relationship, Eq. (4-11). In summary, it is desired to find the set of IN independent variables

X = [01 02 0N Tx T2 ••• Tn]t which satisfy the IN independent functions f=[F1 F2 ••• Fn G, G2 - Gn]t simultaneously.

The Newton-Raphson equations for solving the IN functions {Fj, Gj} for the IN independent variables {0j, 7}} may be represented by the matrix equation

where the jacobian J has the representation

-dFl |
ÔF, |
dFi |
ÔF,- |

38 i |
deN |
ar, |
dTN |

d'FN |
dFs |
dFN |
dFs |

eel |
deN |
ÔT, |
stn |

dGl |
3Gi |
301 |
dGt |

00! |
dh, |
ÔT, |
dTN |

dGN |
dGN |
3Gk |
sgn |

. Mi |
d0N |
8Tt |
ôtn_ |

Next, the 2N Newton-Raphson method is applied to reboiled absorbers, conventional distillation columns, and complex distillation columns, and then a procedure which makes use of the calculus of matrices for solving these equations is presented.

In the formulation of the Newton-Raphson equations, each of the functions to be employed may be obtained by any combination of the independent equations which produces an independent function. One of the most important steps in the application of the Newton-Raphson method is the formulation of the functions because the precise form of the functions determines the region of convergence. To illustrate this concept, consider the formulation of the isothermal flash function. Although a variety of flash functions may be developed by different combinations of the 2c + 2 equilibrium and component-material balances, many of these functions could prove unsatisfactory for solving the adiaba-tic flash problem by use of a formulation of the Newton-Raphson method which involves two independent variables and two independent functions. In general it is desirable to construct functions which are monotonie in the independent variables throughout the region of convergence. For example, the flash function given by Eq. (1-30) is not monotonie in the independent variable throughout the solution domain 0 < < 1. Although the function P(*¥) may be used satisfactorily to find the solution to the isothermal flash problem by starting at = 1, its use could lead to difficulties in the solution of the adiabatic flash problem by a Newton-Raphson formulation in terms of and T. Examination of P(*F) shown in Fig. 1-8 shows that when any value of > 0 to the left of the minimum is used, Newton's method will predict a negative The customary procedure used in n-dimensional space consists of the successive reduction of the corrections [A*F for the function P(*F)] by factors of 1/2 until positive values of the variables are obtained. For the function P(*F), this procedure fails because all values of to the left of the minimum are outside of the region of convergence for the positive nonzero root. In this case, trials would be made at successively smaller values of and the trivial solution = 0 would be approached as this procedure is applied indefinitely. To obtain the desired solution (the > 0 which makes P(*F) = 0), a new starting value which is to the right of the minimum value P(lF) must be selected. Selection of an initial set of the variables which are in the region of convergence can prove difficult for «-dimensional problems unless the functions are very carefully formulated.

Another type of serious difficulty arises when /(x) exhibits the following type of behavior. Suppose that after having passed through the x axis at some x > 0, it then passes through a minimum (or a maximum) and then approaches zero asymptotically. Thus, for larger and larger values of x to the right of the minimum (or maximum), the function f(x) becomes smaller and smaller. The function in Prob. 4-5(b) behaves in this manner. Although it is difficult to deduce the behavior of a function in w-dimensional space, the traces of the function can be examined in two-dimensional space and an attempt should be made to formulate functions whose traces are monotonie.

The sketch for a typical reboiled absorber is shown in Fig. 4-2. To demonstrate the formulation of the IN Newton-Raphson method for reboiled absorbers, two different sets of specifications are considered.

Specification set 1 P, F, {ATJ, thermal condition of F, L0, {x0i}, T0,/, N and QR In order to solve a problem of this type by the IN Newton-Raphson method,

the following sets of IN independent functions are selected

X = [0i 02 • • 0s Tx T2 - Tn]t f=[F, F2 ••• Fn Gx G2 • • G„]7

The matrix A, of the component-material balances are of the same general form as the A, given by Eq. (2-18) for distillation columns. When plate 1 is assumed to behave according to model 1 (see Fig. 1-13) and plate /is assumed to have the behavior characterized by model 2 (see Fig. 2-2), the component-material balances may be represented by Eq. (2-18); provided that the elements of /i are taken to be the following set

where vFi lies in row / — 1 and lhi lies in row /

The second set of specifications differs from the first in that the boilup ratio Vv/B is specified instead of the reboiler duty.

Specification set 2 P, F, {X,}, thermal condition of F, L0, T0, j\ N, Vs/B

For this set of specifications, the IN independent variables are given by

X = [0l 02 " fl.v-i TXT2'" TV^ Tn QR]r (4-15)

and the 2N independent functions f are the same set listed for Specification Set 1. The N vapor-liquid equilibrium functions are given by Eq. (4-3), and the enthalpy balance functions are given by Eq. (4-4) for all stages except j =/— 1,/, N. For stage /— 1 and / the functions x and Gf contain the additional term c c

in their denominators, respectively. For stage N, the normalized form of Gv is given by c

Conventional Distillation Columns

From the sketches of a conventional distillation column (Fig. 2-1) and a reboiled absorber (Fig. 4-2), it is seen that the geometrical configuration of a conventional distillation column is obtained by replacing plate 1 of the reboiled absorber by a condenser-accumulator section (stage 1) and by eliminating the feed L0. The condenser-accumulator section is assigned the stage number 1, and when the condenser duty Qc is specified the independent variables corresponding to this stage are 0X (where 0X = /D) and 7\.

The matrix equation representing the component-material balance is again given by Eq. (2-18), and the elements of the matrices A,, vf, and / have the meanings stated below Eq. (2-18).

For a column having a partial condenser, the dew-point functions are given by Eq. (4-3) for j = 2, 3, ..., N. For j = 1, D and </, play the same role as V} and vj{ in Eq. (4-3) and the dew-point function Ft is given by

For a column having a total condenser, the bubble-point function for the distillate is used for Fu namely,

u ¡=i the enthalpy balance functions are given by Eqs. (4-4) and = 1, the normalized form of the enthalpy balance function is i [dtHu + tuhu] + Qc

Hv2iH2i 1=1

where H^ = Hu for a partial condenser and HK = hu for a total condenser. For stages 7 = 2, 3, ...,/- 2, /+ 1, /+ 2, ..., and N - 1, the enthalpy balance functions G, are given by Eq. (4-4). The functions Gf _ x and Gf are formulated as described below Eq. (4-15) and GN is given by Eq. (4-16).

The independent variables for different sets of specifications are listed in Table 4-2.

Complex columns were defined in Chap. 3 and illustrated by Figs. 3-1 and 3-4. To illustrate the application of the 2N Newton-Raphson method to the solution of problems involving complex columns, consider the simple case where the sidestream Wx is withdrawn in the liquid phase from some interior plate p. The withdrawal of the sidestream Wt gives rise to one specification in addition to those stated for conventional columns, in items 1 through 4 of Table 4-2. When this additional specification is taken to be either the total-flow rate Wl or the ratio Wi/Lp, the sets of specifications, independent variables, and functions for this complex column are the same as those stated in Table 4-2 except that either W1 or Wi /Lp should be added to each set of specifications.

When Wx /Lp is specified, the total-material balance for plate p, Vp+ i + Lp_ i - Vp - Lp - Wt = 0 may be restated as follows

Rp_ t Vp„ 1 - (l 4■ *F + ^RPJ Vp + Vp+1 = 0 (4-20)

Except for stage 1, (4-16). For stage j ■ given by

Table 4-2 Specifications, independent variables, and functions for conventional distillation columns

1. Specifications: P, F, {X,}, thermal condition of F,/, N, Qc> QR, a partial condenser, and the model of the feed plate. Independent variables: 0,, 02, .0S_,,0N, T,, T2, ,, TN.

Functions: F„ F2, FN, G„ G2 GN.V Gs. F, is given by Eq. (4-17).

2. Specifications: P, F, {XJ, thermal condition of the feed F, / N, LJD, VN/B, and a partial condenser.

Independent variables: Q0 02, 03, ..., T,, T2, ..., TN_,, Tv.

Functions: Same as item 1.

3. Specifications: Same as item 1 above except that a total condenser instead of a partial condenser is to be used. Independent variables: Same as item I. Functions: Same as item 1 except F, is given by Eq. (4-18).

4. Specifications: Same as item 2 above except that a total condenser instead of a partial condenser is to be used. Independent variables: Same as item 2. Functions: Same as item 3.

Thus, the matrix equation RV = [Eq. (4-10)] applies, provided that the element lying on the central diagonal of row p of R is changed from (1 + Rp) to

When Wt is specified, the total-material balance for plate p takes the form

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