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Ind. Eng. Chem. Res. 2001, 40, 5810-5814
SEPARATIONS Determination of the Polynomial Defining Underwood’s Equations in Short-Cut Distillation Design Rosendo Monroy-Loperena*,† and Felipe D. Vargas-Villamil‡ Simulacio´ n Molecular and Matema´ ticas Aplicadas y Computacio´ n, Instituto Mexicano del Petro´ leo, Eje Central La´ zaro Ca´ rdenas 152, 07730 Distrito Federal, Me´ xico
The design of distillation columns requires the calculation of the minimum reflux. For ideal mixtures, the well-known Underwood equations, which can also be applied to complex columns, are used to calculate it (e.g., several feeds and side products and side-stream strippers and enrichers). In this work, it is shown that Underwood’s equations of an N-component mixture can be represented as a polynomial of degree N and general relations for its coefficients are derived. In addition, the advantage of solving Underwood’s equations in a polynomial form is shown. Any conventional numeric method may be used to find the roots of this polynomial because it is continuous and smooth (the discontinuities are eliminated). Introduction
is based on the transformation of Underwood’s equation to a polynomial.
Most of the research about the design of distillation columns and short-cut methods is limited to sharp splits in simple columns. However, distributed systems are important from an economic point of view because the optimum fractional recovery for a stream leaving a column is a tradeoff among the number of trays in the stripping and/or fractionation sections, the loss of material in the streams, and the recycling cost, except for those cases where the stream is constrained by the product specifications. The short-cut procedure most commonly used to obtain a quick estimate of the number of theoretical trays is based on the Fenske1-Underwood2-Gilliland3 procedure, and one of the most important methods to calculate the minimum reflux ratio, for simple distillation (i.e., one overhead and one bottom stream), was proposed by Underwood in 19324 and in 1945-1948.2,5-7 In these papers, Underwood developed a completely general solution to this problem, assuming constant relative volatility and constant molar overflow. The results obtained using this method are often qualitatively valid for real mixtures. Underwood4 also investigated the multicomponent distillation with side draws. Using two key components and the equations for binary mixtures, he extended his method to multicomponent distillation. These methods have had several modifications to deal with multiple feeds and/or side draws.8-18 In this paper, an efficient, robust, and reliable method to find the roots of Underwood’s equations, which are necessary to design a distillation column where distributed components are present, is developed. This method * Corresponding author. Tel: +52-5333-8105. Fax: +525333-6239. E-mail:
[email protected]. † Simulacio ´ n Molecular, Instituto Mexicano del Petro´leo. ‡ Matema ´ticas Aplicadas y Computacio´n, Instituto Mexicano del Petro´leo.
2. Underwood’s Equations Underwood7 developed an ingenious algebraic procedure to calculate the minimum reflux. For the rectifying section, he defined a quantity φ N
RiD(xi)D
∑ i)1 R
i
-φ
) Vmin
(1)
where N is the number of components; Ri, D, (xi)D, and Vmin are the relative volatility of component i (i ) 1, ..., N), the distillate flow rate, the distillate composition of component i (i ) 1, ..., N), and the minimum vapor flow rate, respectively. Similarly, for the stripping section, he defined a quantity φ′ N
RiB(xi)B
∑ i)1 R
i
- φ′
) -V′min
(2)
where B and (xi)B are the bottom flow rate and the bottom composition of component i (i ) 1, ..., N), respectively. Usually, the components in a mixture are numbered according to their boiling points. Equations 1 and 2 have, for a given product composition and reflux ratio, as many roots as there are components in the mixture. The roots of these equations may be ordered according to their volatilities as follows:
0 < φ1 < RN < φ2 < RN-1 < ... < R2 < φN < R1 (3) RN < φ′1 < RN-1 < ... < R2 < φ′N-1 < R1 < φ′N
(4)
As shown by Underwood,4,5 some roots of eq 1, φ, and eq 2, φ′, are identical when an infinite number of plates are assumed in both sections. Because these roots are
10.1021/ie010091o CCC: $20.00 © 2001 American Chemical Society Published on Web 11/01/2001
Ind. Eng. Chem. Res., Vol. 40, No. 24, 2001 5811
a solution of both equations, they are a solution of the sum N
RiD(xi)D
∑ i)1 R
i - φ
N
+
RiB(xi)B
∑ i)1 R
i - φ′
N
1
RiF(zi)F
∑ i)1 R
i - φ
N
2
1
) V - V′ ) (∆V)F
N-2+1
N-1
i1*k i2*k
Ri1Ri2...RiN-1 )
iN-1*k
N
(5) N
N
N
∑ ∑ ... )i∑ i )1i )i +1 i
a1,k ) Rkβk
1
) (∆V)F
N
Rk∑βj ∏ j)1 k)1
and because F(zi)F ) D(xi)D + B(xi)B, then N
N
∑ ∑ ... )i∑ i )1i )i +1 i
a0,k ) Rkβk
2
1
i1*k i2*k
(6)
N-3+1
N-2
Ri1Ri2...RiN-2 (11)
iN-2*k
l
If any nonkey component is distributed, the estimated values of D(xi)D cannot be used directly in eq 1. This is particularly true when a nonkey component has a volatility that is between the volatility of two key components. In this case, eq 6 is solved for m roots, φ, where m is equal to the number of distributed components minus 1. Furthermore, each root, φ, lies between an adjacent pair of relative volatilities of distributed components. With the m roots, eq 1 is written m times and solved simultaneously to obtain Vmin and the unknown values of D(xi)D. Another application of the Underwood equation is the calculation of the number of theoretical trays for multicomponent distillation.2
N
aN-2,k ) Rkβk
∑ Ri
1
i1)1 i1*k
aN-1,k ) Rkβk aN,k ) 0 with N
Rk ∏ k)1
b0 )
N
Ri -1 ∑ i )1
b1 ) b0
3. Polynomial Representation of Underwood’s Equations Underwood’s equations can be represented in a general form as N
R iβi
∑i R
i
-φ
N
(Ri Ri )-1 ∑ ∑ i )1i )i +1
∑ i)1R
i
N
N
(Rk - φ) ) 0 ∏(Rk - φ) - ψ∏ - φk)1 k)1
(8)
and performing some manipulations. The resulting polynomial of degree N is
qNφN + qN-1φN-1 + ... + q2φ2 + q1φ + q0 ) 0 (9) General relations for the coefficients qi are obtained by mathematical induction and presented in section 4. It is important to notice that all of the roots of this polynomial may be found using any conventional method19 and that there exist close-form solutions for systems up to four components. 4. Polynomial Coefficients The coefficients of the general polynomial (9), which represents the general Underwood equation (7), are N
qi ) (-1)i
where
ai,k + (-1)i+1ψbi ∑ k)1
for i ) 0, ..., N (10)
(12)
2
1
l N
N
N
... ∑ (Ri Ri ...Ri )-1 ) 1 ∑ ∑ i )1i )i +1 i )i +1
bN ) b0
1
1
Riβi
1
2
(7)
A polynomial in φ can be obtained by rearranging and equating this equation to zero N
N
b2 ) b0
1
)ψ
1
1
2
1
N
2
N
N-1
5. Numerical Examples Three examples are presented where the polynomial (9) and the nonpolynomial (6) forms of Underwood’s equation are compared. First, consider a vapor-liquid stream of six components, at a temperature of 395.52 K and a pressure of 2068 kPa, with the characteristics shown in Table 1. Under these conditions, the stream has a vaporization ratio V/F ) 0.50. The Ki values are assumed to be independent of the composition and are calculated using the polynomial expressions given by Holland.20 Because we are interested in the relation between φ and Underwood’s equation (6), which is used to design distillation columns, the following considerations are made: (1) The heaviest component is the reference to obtain the relative volatilities. Table 1. Hydrocarbon Mixture Characteristics at a Pressure of 2068 kPa and a Temperature of 395.52 K component
i
F(zi)F
Ki
Ri
C2 C3 iC4 nC4 iC5 nC5
1 2 3 4 5 6
0.05 0.10 0.15 0.30 0.30 0.10
5.9571 2.2719 1.3198 1.0775 0.6020 0.5171
11.5198 4.3934 2.5522 2.0836 1.1641 1.0000
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Ind. Eng. Chem. Res., Vol. 40, No. 24, 2001
Table 2. Intermediate Results for Example 1 i
ai,1
ai,2
ai,3
ai,4
ai,5
ai,6
bi
0 1 2 3 4 5 6
15.6652 46.3439 51.5270 26.7195 6.4472 0.5760 0.0000
31.3303 88.2763 91.0071 41.6704 8.0486 0.4393 0.0000
46.9955 124.6975 117.7912 47.4245 7.7182 0.3828 0.0000
93.9910 241.1126 217.5808 82.7292 12.8951 0.6251 0.0000
93.9910 205.4811 156.7850 52.4712 7.5256 0.3492 0.0000
31.3303 64.0772 47.0228 15.3620 2.1713 0.1000 0.0000
313.3033 954.0749 1111.0000 623.8488 175.3335 22.7131 1.0000
(2) The relative volatilities of the feed are equal to the average relative volatilities that are used to design the distillation column. From eq 6 and Table 1, the following expression is obtained:
0.5760 0.4393 0.3828 + + + 11.5198 - φ 4.3934 - φ 2.5522 - φ 0.6251 0.3492 0.1 + + - 0.5 ) 0 (13) 2.0836 - φ 1.1641 - φ 1 - φ Table 2 shows the intermediate results of applying eqs 10-12, which are used in eq 9 to obtain the following polynomial:
156.6517 - 292.9511φ + 126.2139φ2 + 45.5475φ3 42.8608φ4 + 8.8841φ5 - 0.5φ6 ) 0 (14)
Figure 1. Roots of the polynomial and Underwood’s equation for a hydrocarbon mixture.
Equations 13 and 14 are plotted in Figure 1. Notice that, although the roots of eqs 13 and 14 are the same, the shapes are very different; the former is discontinuous for singular points when φ f Ri, while the latter is continuous and smooth. So, any deflation algorithm may be used to find all of the roots of this polynomial. The roots of eqs 13 and 14 are
φ1 ) -1.9368 φ2 ) 1.0316 φ3 ) 1.5195 φ4 ) 2.4044
Figure 2. Roots of the polynomial and Underwood’s equation for a wide-boiling mixture.
φ5 ) 4.0169
the following polynomial:
φ6 ) 10.7366
13.5 - 6.5φ - 2.1667φ2 + 0.5φ3 ) 0
In this example, it is important to notice that the roots φ2 and φ4 are close to their adjacent relative volatilities. Therefore, the slope of eq 13 in these intervals is close to infinity. Thus, some numerical method will have difficulties converging. To show the usefulness of the polynomial representation, let us take two equimolar mixtures with a vaporization ratio V/F ) 0.5: one wide-boiling mixture, with relative volatilities R ) {9, 3, 1}, and another closeboiling mixture, with volatilities R ) {1.10, 1.01, 1}. Underwood’s equation (eq 6) for the former mixture is
1 0.3333 3 + + - 0.5 ) 0 9-φ 3-φ 1-φ
Figure 2 shows the shapes of eqs 15 and 16, whose roots are
φ1 ) -3.0000 φ2 ) 1.5585 φ3 ) 5.7749 Similarly, for the close-boiling mixture, we obtain
0.3366 0.3333 0.3666 + + - 0.5 ) 0 1.10 - φ 1.01 - φ 1-φ
(15)
When eqs 9-12 are applied, the former mixture yields
(16)
and
(17)
Ind. Eng. Chem. Res., Vol. 40, No. 24, 2001 5813
0.5555 - 0.5368φ - 0.5183φ2 + 0.5φ3 ) 0 (18) Figure 3 shows the shapes of eqs 17 and 18, with the following roots:
φ1 ) -1.0357 φ2 ) 1.0049 φ3 ) 1.0675 Figure 3 shows that for a very close-boiling mixture the polynomial representation offers a great advantage over Underwood’s equation, because it is smooth, and will be easily solved by any numerical algorithm. 6. Conclusions In this work, it has been shown that Underwood’s equations can be represented as a polynomial of degree N, where N is the number of components in the mixture. In addition, general relations to calculate the coefficients of this polynomial have been developed. The resulting polynomial is numerically efficient, robust, and reliable. By efficient, we mean that the polynomial coefficients can be obtained easily using eqs 9-12 and can be implemented easily using any computational language. It is robust because the procedure to obtain the polynomial form can be applied to problems with any number of components, and it is reliable because the resulting polynomial is continuous and smooth, which are the main characteristics sought in any deflation method.
Figure 3. Roots of the polynomial and Underwood’s equation for a close-boiling mixture.
qnφn + qn-1φn-1 + ... + q2φ2 + q1φ + q0 ) 0 where n
qi ) (-1)i a0,k )
ai,k + (-1)i+1ψbi ∑ k)1
Rkβk
n
n
for i ) 0, ..., n (A.5)
n
∑ ∑ ... ∑ Ri Ri ...Ri (n-1)!i )1i )1 i )1 1
1
2
2
f(i1,i2,...,in-1,k) )
n-1
n-1
n
q2φ2 + q1φ + q0 ) 0
a1,k )
Rkβk
n
n
n
∑ ∑ ... ∑ Ri Ri ...Ri (n-2)!i )1i )1 i )1 1
1
2
2
l n
∑ Ri f(i1,k)
an-2,k ) Rkβk
1
i1)1
q0 ) R1R2(x1 + x2) - ψ(R1R2)
an-1,k ) Rkβk
q1) -(R1x1 + R2x2) + ψ(R1 + R2)
an,k ) 0 Rk ∏ k)1
b0 )
For N ) 3, we obtain
n
q3φ + q2φ + q1φ + q0 ) 0
(A.6)
n
q2) -ψ
2
1
∑ i )1R
b1 ) b 0
(A.2)
i1
1
where
b2 )
q0 ) R1R2R3(x1 + x2 + x3) - ψ(R1R2R3) q1 ) -(R1R3 + R1R2)x1 - (R2R3 + R1R2)x2 -(R2R3 + R1R3)x3 + ψ(R1R3 + R1R2 + R2R3)
Now, solving for N ) n, we obtain
n
1
n
2!
f(i1,i2)
∑∑ i )1i )1 R
b0
1
i1Ri2
2
l bn )
q2 ) R1x1 + R2x2 + R3x3 - ψ(R1 + R2 + R3) q3 ) -ψ
f(i1,i2,...,in-2,k)
n-2
n-2
(A.1)
where
3
n
βj ∑ Rk∑ k)1 j)1
Appendix: Proof of Equations 10-12 Equations 10-12 can be obtained by mathematical induction by solving eq 8 for N ) 2, 3, ..., n. Solving eq 8 for N ) 2 and expressing it in polynomial form
(A.4)
n
1
n
n
n!
f(i1,i2,...,in)
∑ ∑ ... ∑ i )1i )1 i )1 R
b0
1
2
n
i1Ri2...Rin
)1
(A.7)
(A.3) and f(j1,j2,...,jn) is a boolean operator defined by the following rule:
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Ind. Eng. Chem. Res., Vol. 40, No. 24, 2001
f(j1,j2,...,jn) )
{
0 if any two elements of the set F ) {j1, j2, ..., jn} are equal 1 otherwise
}
(A.8)
Equations A.5-A.8 may be simplified by eliminating the redundant terms. Then eqs 10-12 are obtained. Literature Cited (1) Fenske, M. R. Fractionation of Straight-Run Pennsylvania Gasoline. Ind. Eng. Chem. 1932, 24, 482. (2) Underwood, A. J. V. Fractional Distillation of Multicomponent Mixtures. Chem. Eng. Prog. 1948, 44, 603. (3) Gilliland, E. R. Multicomponent Rectification. Ind. Eng. Chem. 1940, 32, 1220. (4) Underwood, A. J. V. The theory and practice of testing stills. Trans. AIChE 1932, 10, 112. (5) Underwood, A. J. V. Fractional Distillation of Ternary MixturessPart I. J. Inst. Pet. 1945, 31, 111. (6) Underwood, A. J. V. Fractional Distillation of Ternary MixturessPart II. J. Inst. Pet. 1946, 32, 598. (7) Underwood, A. J. V. Fractional Distillation of Multicomponent MixturessCalculation of Minimum Reflux Ratio. J. Inst. Pet. 1946, 32, 614. (8) Short, T. E.; Erbar, J. H. Minimum Reflux for Complex Fractionators. Petro/Chem. Eng. 1963, 11, 180. (9) Sugie, H.; Benjamin, C. Y. L. On the Determination of Minimum Reflux Ratio for a Multicomponent Distillation Column with any Number of Side-cut Streams. Chem. Eng. Sci. 1970, 25, 1837.
(10) Barnes, F. J.; Hanson, D. N.; King, C. J. Calculation of Minimum Reflux for Distillation Columns with Multiple Feeds. Ind. Eng. Chem. Process Des. Dev. 1972, 11, 136. (11) Solari, R. B.; Carballo, M.; Gabaldon, R.; Estevez, J. A short-cut method to evaluate minimum reflux in complex fractionators. Rev. Tec. INTEVEP 1981, 1, 153. (12) Glinos, K.; Malone, M. F. Design of Sidestream Distillation Columns. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 822. (13) Glinos, K.; Nikolaides, I. P.; Malone, M. F. New Complex Column Arrangements for Ideal Distillation. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 694. (14) Tsuo, F.-M.; Yaws, C. L.; Cheng, J.-S. Minimum Reflux for Sidestream Columns. Chem. Eng. 1986, 49. (15) Nikolaides, I. P.; Malone, M. F. Approximate Design of Multiple-Feed/Sidestream Distillation Systems. Ind. Eng. Chem. Res. 1987, 26, 1839. (16) Chou, S. M.; Yaws, C. L. Minimum Reflux for Complex Distillation. Chem. Eng. 1988, 79. (17) Wachter, J. A.; Ko, T. K. T.; Andres, R. P. Minimum Reflux Behavior of Complex Distillation Columns. AIChE J. 1988, 34, 1164. (18) Youssef, S.; Domenech, S.; Pibouleau, L. Calcul d’une Colonne de Rectification a` Multiples Alimentations et Soutirages. Chem. Eng. J. 1984, 42, 153. (19) Chapra, S. C.; Canale, R. P. Numerical methods for engineering: with programming and software applications, 3rd ed.; McGraw-Hill: New York, 1998. (20) Holland, C. D. Fundamentals of multicomponent distillation; McGraw-Hill: New York, 1981.
Received for review January 29, 2001 Revised manuscript received June 4, 2001 Accepted September 5, 2001 IE010091O
