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Ind. Eng. Chem. Res. 1996, 35, 2777-2781
2777
Evolution of Pinch-Point Zones in Multicomponent Distillation Columns. 1. Rectifying Section Yang Zhicai,* Wu Di,† Yang Zhaoxia, and Zhang Ronglan† Chemical Engineering Research Center, Tainjin University, Tainjin 300072, People’s Republic of China
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 are presented in this work. Analytical formulas are obtained for calculating the pinchpoint zones as well as the distillate exactly. It has been found that the evolution of pinch-point zones in the stripping section is much similar to the evolution of pinch-point zones in the rectifying section. Based on these new findings, a new method for calculating the minimum reflux ratio of multicomponent continuous distillation, as well as a new shortcut method for the design calculation of a batch distillation process, can be developed. Introduction The method for calculating the minimum reflux ratio that is probably most widely used was introduced by Underwood (1946a,b). Necessary assumptions of the method are constant volatility and constant sectional flows between the pinch-point zones of the two column sections. One of the defects of Underwood’s method is that it cannot help to calculate pinch-point zones and distillate while calculating the minimum reflux ratio. Tavana and Hanson (1979) proposed a computational technique for the calculation of minimum flows and energy loads for the distillation column. The method works by calculating the pinch-point zones. However, it is too complex to be adopted in a shortcut method for distillation calculation. While inspecting the concentration profiles of nonholdup batch distillation, we found 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 and constant vapor stream entering the rectifying section. Based on these principles, a new method for calculating pinch-point zones and distillate is derived. The assumptions involved in our work are (1) gas phase is ideal gas, (2) liquid phase is ideal solution which obeys Raoult’s law, and (3) gas and liquid are with equal molar overflows. When the concentration profile in a multicomponent nonholdup batch distillation column with a finite number of theoretical stages is calculated with the abovementioned assumptions, and if the number of stages is high enough, there are certainly some stages that have almost the same composition and temperature. A pinchpoint zone is defined as a zone of constant composition 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. Stability of Pinch-Point Zones Several liquid composition profiles in a rectifying section with 100 theoretical stages are shown in Figure 1. The molar composition of the vapor entering the † Permanent address: Daqing Oilfield Construction Design and Research Institute, 163712 Daqing, People’s Republic of China.
Table 1. Antoine Coefficients for Benzene, Toluene, and Ethylbenzenea Antoine coefficients component
A
B
C
benzene toluene ethylbenzene
15.9008 16.0137 16.0195
2788.51 3096.52 3279.47
-52.36 -53.67 -59.95
a
ln P ) A - B/(T + C). P (mmHg); T (K).
Table 2. Molar Distillate Compositions with 100 Theoretical Stages as a Function of the Reflux Ratio R
benzene
toluene
ethylbenzene
1 1.449 2 4 4.420 8
0.4597 0.5405 0.6257 0.9350 0.9999 1.0000
0.4304 0.4595 0.3743 0.0650 0.0001 0.0000
0.1100 0.0000 0.0000 0.0000 0.0000 0.0000
rectifying section is 0.2799 for benzene, 0.3652 for toluene, and 0.3550 for ethylbenzene. The vapor is assumed to enter the rectifying section at its dew 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 batch distillation simulation method and the Thomas method to calculate the composition profile in the rectifying section. It can be seen in Figure 1 that the 100 theoretical-stage profile is relatively too many for the benzene-toluene-ethylbenzene system, so there appear zones of constant composition (pinch-point zones), with the reflux ratio varying from 1 to 8. When the reflux ratio R takes the value of 1, a pinch-point zone appears at the bottom of the rectifying section, which has a liquid molar composition of 0.1 for benzene, 0.3 for toluene, and 0.6 for ethylbenzene. The liquid phase composition is found in vapor-liquid equilibrium, with the vapor phase entering the rectifying section at an absolute pressure of 760 mmHg. When R takes the values of 2 and 4, pinch-point zones appear inside the rectifying section, and when R takes the value of 8, a pinch-point zone appears on the top of the rectifying section. It is most interesting that, when R changes from 2 to 4, the distillate composition (shown in Table 2) changes, but the pinch-point composition does not change, which suggests that with the assumptions in this work pinchpoint zone composition is likely to vary with the reflux ratio not continuously but in a step-by-step way. Calculation done on many different systems shows the
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2778 Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996
Figure 1. Liquid composition profiles in a rectifying section with 100 theoretical stages.
same phenomenon, which reflects an important principle of the evolution of pinch-point zones in the rectifying section. Result 1. The composition of pinch-point zones in a rectifying section with an infinite number of stages varies with the reflux ratio not continuously but in a step-by-step way provided that the vapor stream entering the rectifying section does not change.
Let stage i in Figure 2 be an arbitrary stage in a pinch-point zone in a rectifying section with an infinite number of stages. From the definition of a pinch-point zone, we have
Two States of the Pinch-Point Zones in a Rectifying Section
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 top of the rectifying section and through the pinchpoint zone gives
The balance of the components around the top of the rectifying section and through the pinch-point zone shows that a component must be absent in the distillate if it is absent in the pinch-point zone; a component must appear in the distillate if it appears in the pinch-point zone. Two states can be distinguished for the pinchpoint zones in the rectifying section in light of whether they have the same number of components as the distillate or not: (a) first stateshaving the same number of components as the distillate (e.g., R ) 1 in Figure 1); (b) second stateshaving more component(s) than the distillate (e.g., R ) 1.449 in Figure 1).
xij ) xi-1,j ) xpj
(j ) 1, 2, ..., nc)
ypj ) Wxpj + (1 - W)xDj
(1)
(2)
where ypj refers to the molar liquid composition of component j in the pinch-point zone, W represents the
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Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2779 0 ppj > WP
(j ) 1, 2, ..., h)
(6)
where h refers to the number of the heaviest component in the distillate and the first-state pinch-point zone (i.e., the number of components in the distillate and the firststate pinch-point zone). The combination of eq 5 and normalization conditions gives h
∑ j)1
h
∑ j)1
xpj )
(1 - W)xDj 0 ppj
)1
(7)
-W
P
Now define a function F so that h
Figure 2. Schematic diagram of a rectifying section.
liquid-vapor ratio, and xDj denotes the molar distillate composition of component j. According to the assumptions mentioned above, the relationship between the vapor compositions and liquid compositions can be expressed in terms of the saturated vapor pressures of pure components and the system pressure as follows:
ypj )
(
)
0 ppj - W xpj P
(1 - W)xDj 0 ppj -W P
h
dF
∑ j)1
(8)
-W
( ) 0 ppj
P
2
(9)
dTp
-W
Let T1 be the boiling point of component h under pressure WP, and T2 be the boiling point of component h under pressure P. From 0 < W < 1 and dp0pj/dTp > 0, we have T2 > T1. Inspection of eq 6 gives
Tp > T1
(10)
F(T1+) ) +∞
(11)
(4) From eq 8, we have
h
0 * WP) (ppj
-1
0 (1 - W)xDj dppj
)dTp
Provided that p0pj/P - W * 0, i.e., p0pj * WP, rearrangement gives
xpj )
0 ppj
and derive this function
(3)
where p0pj refers to the standard 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
(1 - W)xDj
P
P
p0pi x P pi
(1 - W)xDj )
∑ j)1
F ) F(Tp) )
F(T2) e
(5)
It can be proved that not more than two first-state pinch-point zones can be in the rectifying section at the same time. The derivation is given below: Assuming that the distillate composition and liquidvapor ratio are known and the pinch-point zone composition and liquid-vapor ratio are known, the pinchpoint 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 can 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 firststate pinch-point zone composition and temperature based on mass balance and vapor-liquid equilibrium conditions below. From xDj > 0, xpj > 0, 1 - W > 0 (since total reflux, i.e., W ) 1, is a special case in which the pinch-point zone is located at the top of the rectifying section and has only the lightest component, we restrict W < 1 in this paper) and from eq 4, we have
∑ j)1
(1 - W)xDj P P
-1
-W
while h
∑ j)1
(1 - W)xDj P P
-W
h
-1)
xDj - 1 ) 0 ∑ j)1
So
F(T2) e 0
(12)
Only when h ) 1 are the two sides in eq 12 equal. From eq 9, we have
dF/dTp < 0
(13)
It can be seen from eq 8 that function F is continuous in the open interval (T1, +∞); the combination of eqs 11-13 shows that there is one and only one root of equation F(Tp) ) 0 in the interval (T1, +∞). Equation 10 shows that the pinch-point temperature Tp lies in
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2780 Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996
the interval (T1, +∞). So, it can be concluded that the rectifying section has one and only one first-state pinchpoint zone under a certain reflux ratio. The combination of eqs 10-12 shows that Tp lies in a narrower interval (T1, T2 ). Further inspection shows that there cannot be more than one second-state pinch-point zone at the same time in the rectifying section. The derivation is as follows: Assume that component k appears in a second-state pinch-point zone and is absent in the distillate, i.e., xDk ) 0, xpk * 0. Inspection of eq 4 gives 0 ppk ) WP
(14)
The saturated vapor pressure of different components at the same temperature are different, so it can be concluded that there is only one component appearing in a second-state pinch-point zone that is absent in the distillate and that component is the heaviest component appearing in the second-state component, i.e., k ) h. So 0 ) WP pph
(15)
Assume two second-state pinch-point zones appear simultaneously in the rectifying section and component k is the heaviest component in the upper second pinchpoint zone. It is obvious that component k appears in the lower second-state pinch-point zone and is the heaviest component in the lower second pinch-point zone. Inspection of eq 15 shows that the temperatures of the two second-state pinch-point zones are the same. Inspection of eq 4 shows that the two second-state pinchpoint zones are actually the same. It can also be concluded from the above discussion that there must be a first-state pinch-point zone above the second-state pinch-point zone. Inspection of eq 15 shows that a pinch-point zone is in the second state only under a certain value of the liquid-vapor ratio. The above discussion can be summarized in Result 2 and Result 3. Result 2. The pinch-point zones in the rectifying section can exist in two states: (a) first stateshaving the same number of components as the distillate; (b) second stateshaving one more component (i.e., the heaviest component in the pinch-point zone) than the distillate. Result 3. There may be one or two pinch-point zones in the rectifying section at the same time. In the case of one pinch-point zone in the rectifying section, the pinch-point zone is a first-state pinch-point zone. In the case of two pinch-point zones in the rectifying section at the same time, the upper pinch-point zone is a firststate pinch-point zone, while the lower pinch-point zone is a second-state pinch-point zone. Figure 3 shows all the possible cases and locations of pinch-point zones in the rectifying section.
Figure 3. Locations of pinch-point zones in a rectifying section for (a) onesat the bottom; (b) twosbottom to middle; (c) onesin the middle; (d) twosmiddle to middle; (e) twosmiddle to top; and (f) onesat the top.
the reflux ratio) or a decrease by 1 (with an increase of the reflux ratio) of the number of components appearing in the distillate. The cases shown in parts b, d, and e of Figure 3 are transition states for the replacement of pinch-point zones. In Figure 1, R ) 1 corresponds to Figure 3a; R ) 2 and R ) 4 correspond to Figure 3c; R ) 8 corresponds to Figure 3f; and R ) 1.449 and R ) 4.420 correspond to parts b and e of Figure 3, respectively. With infinitesimal reflux, no separation takes place in the rectifying section. The rectifying section is dominated by a first-state pinch-point zone which has the same vapor composition as the vapor entering the rectifying section. Inspection of Result 1 leads to Result 5. Result 5. With the gradual increase of the reflux ratio from zero, the first pinch-point zone, which is located at the bottom of the rectifying section, has the same vapor composition as the vapor entering the rectifying section.
Replacement of Pinch-Point Zones
Calculation Procedure of the Pinch-Point Zone and the Distillate
The combination of the above three results leads to Result 4. Result 4. The replacement of pinch-point zones with variation of the reflux ratio in the rectifying section with a constant vapor stream entering the rectifying section is accomplished through the transition state under a certain reflux, in which two pinch-point zones appear simultaneously. The replacement of pinch-point zones is accompanied by an increase by 1 (with a decrease of
Based on the above five results, a procedure for calculating the pinch-point zone temperature and composition under various reflux ratios, as well as the distillate composition, is developed. The basic calculation procedure follows: (1) Calculate the temperature and liquid composition of the pinch-point zone at the bottom of the rectifying section following Result 5 and the vapor-liquid equilibrium condition.
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Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2781 Table 3. Molar Distillate Compositions with an Infinite Number of Stages as a Function of the Reflux Ratio R
benzene
toluene
ethylbenzene
1 1.449 2 4 4.420 8
0.4597 0.5405 0.6257 0.9351 1.0000 1.0000
0.4303 0.4595 0.3743 0.0649 0.0000 0.0000
0.1100 0.0000 0.0000 0.0000 0.0000 0.0000
(2) Calculate the critical liquid-vapor ratio, Wc, of the transition state for the replacement of pinch-point zones in Figure 3b. Rearrangement of eq 15 gives
Wc )
0 pph /P
(16)
(3) Check if the specified liquid-vapor ratio W is less than or equal to Wc. If W e Wc corresponds to this pinch-point zone, the distillate can be calculated. Rearrangement of eq 4 gives 0 ppj -W P x (j ) 1, 2, ..., h) xDj ) 1 - W pj
(17)
If W > Wc, calculate the distillate composition of the transition state with eq 17. (4) Calculate the temperature and composition of the upper 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 (T1, T2). Otherwise, false results will be obtained. (5) Calculate the Wc 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 lightest component is reached, the specified W corresponds to this pinch-point zone and only the lightest component is in the distillate. Example Use the data for Figure 1 to calculate the distillate compositions, pinch-point zones temperatures, and compositions when an infinite number of stages are involved. Results are summarized as follows: The three pinch-point zones solved are (1) Bottom pinch-point zone
Tp ) 390.84 K xp1 ) 0.1
xp2 ) 0.3
xp3 ) 0.6
The reflux ratio interval is (0, 1.449). (2) Intermediate pinch-point zone
Tp ) 376.75 K xp1 ) 0.1617
xp2 ) 0.8383
xp3 ) 0
The reflux ratio interval is (1.449, 4.420). (3) Top pinch-point zone
xp1 ) 1
xp2 ) 0
xp3 ) 0
The reflux ratio interval is (4.420, +∞).
The distillate compositions obtained in this way under various reflux 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 by a rigorous simulation method. The results can be regarded as a good corroboration of the correctness of the five results presented in this paper. Conclusion (1) Five results concerning the evolution of pinchpoint zones in the rectifying section of the multicomponent distillation column are obtained. (2) A simple calculation procedure is developed for calculating the distillate composition, pinch-point zone temperature, and composition. (3) It has been found that the evolution of pinch-point zones in the stripping section is similar to the evolution of pinch-point zones in the rectifying section. Based on these new findings, a new method for calculating the minimum reflux ratio of a multicomponent continuous distillation, as well as a new shortcut method for the design calculation of the batch distillation process, can be developed. Notation h ) number of the heaviest component in a pinch-point zone nc ) total number of components p0pj ) the saturated vapor pressure of component j at pinch-point zone temperature Tp P ) system pressure R ) reflux ratio xDj ) molar liquid composition of component j in the distillate xpj ) molar liquid composition of component j in the pinchpoint zone Tp ) pinch-point zone temperature W ) liquid-vapor ratio
Literature Cited Tavana, M.; Hanson, D. N. The Exact Calculation of Minimum Flows in Distillation Columns. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 154. Underwood, A. J. V. Fractional Distillation of Ternary Mixtures. J. Inst. Petrol. 1946a, 32, 598. Underwood, A. J. V. Fractional Distillation of Multicomponent Mixtures. J. Inst. Petrol. 1946b, 32, 614. Wu, D. 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 November 20, 1995 Revised manuscript received May 10, 1996 Accepted May 10, 1996X IE950697A
X Abstract published in Advance ACS Abstracts, July 15, 1996.
