Ind. Eng. Chem. Res. 1999, 38, 2151-2155
2151
Evolution of Pinch-Point Zones in Multicomponent Distillation Columns. 2. Stripping Section Wu Di,† Yang Zhicai,* Zhang Ronglan,† and Lu Shen Chemical Engineering Research Center, Tianjin University, Tianjin 300072, People’s Republic of China
The principles concerning the evolution of pinch-point zones with variation of the vapor-liquid ratio in the stripping section of a multicomponent distillation column with an infinite number of stages, which are similar to those of the pinch-point zones in the rectifying section published in a preceding paper, are given in this work. Based on these new findings, a new method for calculating the minimum reflux ratio of multicomponent continuous distillation can be developed. 1. Introduction The principles concerning the evolution of pinch-point zones with variation of the reflux ratio in the rectifying section of a multicomponent distillation column with an infinite number of stages have been presented in a preceding paper1 of this work. It has been found that the evolution of the pinch-point zones in the stripping section is much similar to the evolution of pinch-point zones in the rectifying section. The assumptions involved in this paper are identical with those in the preceding paper, i.e., (1) the gas phase is ideal gas, (2) the liquid phase is an ideal solution which obeys Raoult’s law, and (3) gas and liquid are with equal overflows. Similar to the case with the rectifying section, there exist in a stripping section certainly some stages wherein the variations of compositions and temperatures are tiny enough to be ignored provided that the number of stages is high enough. A pinch-point zone is usually defined as a zone of constant composition and temperature in a distillation column with an infinite number of stages. Here this definition is relaxed to include zones where there is almost no change in the composition of the liquid or vapor from stage to stage in a distillation column with a finite number of stages. However, analytical results are derived only when an infinite number of stages are present.
relatively too many for the toluene-ethylbenzenepropylbenzene-butylbenzene system, so there appear zones of constant composition (pinch-point zones), with the molar vapor-liquid ratio varying from 0.1 to 0.95. When the molar vapor-liquid ratio S takes the value of 0.1, a pinch-point zone appears on the top of the stripping section, which has the same liquid composition as the liquid entering the stripping section, i.e., 0.4 for toluene, 0.3 for ethylbenzene, 0.2 for propylbenzene, and 0.1 for butylbenzene. When S takes the values of 0.63 and 0.87, pinch-point zones appear inside the stripping section, and when S takes the value of 0.95, a pinchpoint zone appears at the bottom of the stripping section. It is most interesting that when S changes from 0.63 to 0.73, the residue composition (shown in Table 2) changes but the pinch-point zone composition does not change, which suggests that with the assumptions in this work pinch-point zone composition is likely to vary with the molar vapor-liquid ratio S not continuously but in a step-by-step way. Calculation done on many different systems shows the same phenomenon, which reflects an important principle of the evolution of pinch-point zones in the stripping section. Result 1. The composition of pinch-point zones in a stripping section with an infinite number of stages varies with the molar vapor-liquid ratio not continuously but in a step-by-step way provided that the liquid stream entering the stripping section does not change.
2. Stability of Pinch-Point Zones Several liquid composition profiles in a stripping section with 100 theoretical stages are shown in Figure 1. The molar composition of the liquid entering the stripping section is 0.4 for toluene, 0.3 for ethylbenzene, 0.2 for propylbenzene, and 0.1 for butylbenzene. The liquid is supposed to enter the stripping section at its bubble point under an absolute pressure of 760 mmHg. The thermodynamic data used in the calculation of the composition profiles are listed in Table 1. To ensure convergence, we use a combination of a rigorous simulation method and the Thomas method in the calculation of the composition profiles in the stripping section. It can be seen in Figure 1 that 100 theoretical stages are * To whom correspondence is addressed. Tel.: 0086-2227404493. Fax: 0086-22-27406197. E-mail: zcyang@ public1.tpttj.cn. † Permanent address: Daqing Oilfield Construction Design and Research Institute, 163712 Daqing, People’s Republic of China. Tel: 0086-459-5902523. Fax: 0086-459-5902772. Email:
[email protected].
3. Two States of the Pinch-Point Zones in a Stripping Section The balance of the components around the bottom of the stripping section and through the pinch-point zone shows that a component must be absent in the residue if it is absent in the pinch-point zone; a component does not necessarily appear in the residue if it appears in the pinch-point zone. Two states can be distinguished for the pinch-point zones in the stripping section in light of whether they have the same number of components as the residue or not: (a) first state, having the same number of components as the residue (e.g., S ) 0.73 in Figure 1); (b) second state, having more component(s) than the residue (e.g., S ) 0.6154 in Figure 1). Let stage i in Figure 2 be an arbitrary stage in a pinch-point zone in a stripping section with an infinite number of stages. From the definition of a pinch-point zone in a stripping section, we have
xij ) xi-1,j ) xpj (j ) 1, 2, ..., nc)
10.1021/ie980530s CCC: $18.00 © 1999 American Chemical Society Published on Web 04/15/1999
(1)
2152 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999
Figure 1. Liquid composition profiles in a stripping section with 100 theoretical stages: (a) toluene, (b) ethylbenzene, (c) propylbenzene, (d) butylbenzene. Table 1. Antoine Coefficients for Toluene, Ethylbenzene, Propylbenzene, and Butylbenzenea Antoine coefficients component
A
B
C
toluene ethylbenzene propylbenzene butylbenzene
16.0137 16.0195 16.0062 16.0793
3096.52 3279.47 3433.84 3633.4
-53.67 -59.95 -66.01 -71.77
a
ln P ) A - B/(T + C); P (mmHg), T (K).
Table 2. Molar Residue Compositions with 100 Theoretical Stages as a Function of the Vapor-Liquid Ratio S
toluene
ethylbenzene
propylbenzene
butylbenzene
0.1 0.6154 0.63 0.73 0.8048 0.87 0.9310 0.95
0.3722 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.3062 0.3898 0.3739 0.2192 0.0000 0.0000 0.0000 0.0000
0.2128 0.3837 0.3921 0.4743 0.5908 0.4282 0.0000 0.0000
0.1088 0.2265 0.2339 0.3064 0.4092 0.5718 1.0000 1.0000
xpj ) Sypj + (1 - S)xwj (j ) 1, 2, ..., nc)
(2)
where ypj refers to the molar vapor composition of component j in the pinch-point zone and xwj represents the molar residue composition of component j. According to the assumptions mentioned above, the relationship between the vapor compositions and the liquid compositions can be expressed in terms of the saturated vapor pressures of pure components and the system pressure as follows:
ypj )
p°pj x (j ) 1, 2, ..., nc) P pj
(3)
where p°pj refers to the saturated vapor pressure of component j at pinch-point zone temperature Tp and P denotes the system pressure. By substitution of eq 3, eq 2 can be rewritten as
(
p°pj x P pj
(1 - S)xwj ) 1 - S
)
(4)
Provided that 1 - Sp°pj/P * 0, i.e., p°pj * P/S, rearrangement gives
xpj )
Figure 2. Schematic diagram of a pinch-point zone.
where xij refers to the molar liquid composition of component j on stage i, xpj represents the molar liquid composition of component j in the pinch-point zone, and nc denotes the total number of components. Numbers of the components are counted in the order of volatility from strong to weak. A balance of component j around the bottom of the stripping section and through the pinch-point zone gives
(1 - S)xwj (p°pj * P/S) p°pj 1-S P
(5)
It can be proved that there can exist in the stripping section only one first-state pinch-point zone at the same time. The derivation is given below: Assuming that the residue composition and vaporliquid ratio are known, the pinch-point zone composition and temperature are to be determined. It is obvious that if we solve for the pinch-point zone composition and temperature by stage-by-stage calculation, we reach a first-state pinch-point zone when the stage number is high enough. Now, we will develop a straightforward method for the calculation of first-state
Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 2153
pinch-point zone composition and temperature based on mass balance and vapor-liquid equilibrium conditions below. From xwj > 0, xpj > 0, and 1 - S > 0 (since S ) 1 is a special case in which the pinch-point zone is located at the bottom of the stripping section and has only the heaviest component, we restrict S < 1 in this paper) and from eq 4, we have
p°pj < P/S (j ) k, k + 1, ..., nc)
(6)
where k refers to the number of the lightest component in the residue and the first-state pinch-point zone. The combination of eq 5 and normalization conditions gives nc
nc
xpj ) ∑ ∑ j)k j)k
(1 - S)xwj p°pj
1-S
)1
(7)
P
Now define a function F so that nc
F ) F(Tp) )
∑ j)k
(1 - S)xwj p°pj
1-S
-1
(8)
dTp
(1 - S)xwj dp°pj
nc
)-
∑ j)k
(
)
p°pj
P 1-S
P
2
dTp
(9)
Let T1 be the boiling point of component k under pressure P/S, and T2 be the boiling point of component k under pressure P. From 0 < S Sc, calculate the residue composition of the transition state with eq 17. (4) Calculate the temperature and composition of the lower pinch-point zone with eqs 8 and 5. When Newton’s method is used for the solution of Tp, the value of Tp should be limited in the interval (T2, T1). Otherwise, false results will be obtained. (5) Calculate the Sc of the transition state for the replacement of pinch-point zones in Figure 3d. (6) Repeat steps 3-5 until the solution is obtained. If a pinch-point zone which only has the heaviest component is reached, the specified S corresponds to this pinch-point zone and only the heaviest component is in the residue. 6. Example Use the data for Figure 1 to calculate the residue compositions, pinch-point zone temperatures, and compositions when an infinite number of stages are involved. Results are summarized as follows:
(1) Top pinch-point zone Tp ) 401.80 K xp1 ) 0.4
xp2 ) 0.3
xp3 ) 0.2
xp4 ) 0.1
The molar vapor-liquid ratio interval is [0, 0.6154].
(2) Intermediate pinch-point zone I Tp ) 417.62 K xp1 ) 0
xp2 ) 0.6369
xp3 ) 0.2524
xp4 ) 0.1107
The molar vapor-liquid ratio interval is [0.6154, 0.8048].
(3) Intermediate pinch-point zone II
5. Calculation Procedure of the Pinch-Point Zone and the Residue Based on the above five results, a procedure for calculating the pinch-point zone temperature and composition under various molar vapor-liquid ratios, as well as the residue composition, is developed. The basic calculation procedure follows: (1) Calculate the temperature and liquid composition of the pinch-point zone on the top of the stripping section following Result 5 and the vapor-liquid equilibrium condition. (2) Calculate the critical vapor-liquid ratio, Sc, of the transition state for the replacement of pinch-point zones in Figure 3b. Rearrangement of eq 15 gives
(17)
Tp ) 435.18 K xp1 ) 0
xp2 ) 0
xp3 ) 0.8512
xp4 ) 0.1488
The molar vapor-liquid ratio interval is [0.8048, 0.9310].
(4) Bottom pinch-point zone Tp ) 456.42 K xp1 ) 0
xp2 ) 0
xp3 ) 0
xp4 ) 1
The molar vapor-liquid ratio interval is [0.9310, 1].
Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 2155 Table 3. Molar Residue Compositions with an Infinite Number of Stages as a Function of the Vapor-Liquid Ratio S
toluene
ethylbenzene
propylbenzene
butylbenzene
0.1 0.6154 0.63 0.73 0.8048 0.87 0.9310 0.95
0.3722 0 0 0 0 0 0 0
0.3062 0.3898 0.3740 0.2194 0 0 0 0
0.2128 0.3837 0.3921 0.4743 0.5908 0.4282 0 0
0.1088 0.2265 0.2339 0.3064 0.4091 0.5718 1 1
The residue compositions obtained in this way under various vapor-liquid ratios are listed in Table 3. Comparison of the data in Tables 2 and 3 shows that the results of 100 stages and infinite stages are very close. It can also be seen that the pinch-point zone compositions calculated by the above procedure are very close to the pinch-point zone compositions obtained with 100 stages shown in Figure 1. The results can be regarded as a good corroboration of the correctness of the five results presented in this paper. 7. Conclusion (1) Five results concerning the evolution of pinchpoint zones in the stripping section of the multicomponent distillation column are obtained. (2) A simple calculation procedure is developed for calculating the residue composition, pinch-point zone temperature, and composition.
(3) Based on these new findings, a new method for calculating the minimum reflux ratio of multicomponent continuous distillation can be developed. Nomenclature k ) the number of the lightest components in a pinch-point zone nc ) total number of components p°pj ) saturated vapor pressure of component j at pinchpoint zone temperature Tp P ) system pressure S ) molar vapor-liquid ratio Tp ) pinch-point zone temperature xpj ) molar liquid composition of component j in the pinchpoint zone xwj ) molar composition of component j in the residue
Literature Cited (1) Zhicai, Y.; Di, W.; Zhaoxia, Y.; Ronglan, Z. Evolution of Pinch-Point Zones in Multicomponent Distillation Columns. 1. Rectifying Section. Ind. Eng. Chem. Res. 1996, 35, 2777. (2) Di, W. Study on Simulation and Design Calculation of Batch Distillation Processes. Academic Paper for Master’s Degree, Tianjin University, Tianjin, People’s Republic of China, 1992.
Received for review August 10, 1998 Revised manuscript received November 10, 1998 Accepted November 11, 1998 IE980530S
