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Use of the results of field tests in the modeling of packed distillation columns and packed absorbers—III
Chemical Engineering Science, 1970, Vol. 25, pp. 633-65 1.
Pergamon Press.
Printed in Great Britain.
Use of the results of field tests in the modeling of packed distillation columns and packed absorbers-III RONALD MCDANIEL?, A. A. BASSYONI and C. D. HOLLAND Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843, U.S.A. (First received6
June 1969; accepted
24 September
1969)
Abstract-The use of the results of a series of field tests in the modeling of a packed distillation column and a packed absorber is demonstrated. The modeling techniques proposed make use of data which are commonly available for existing columns. The parameters determined from the field tests consisted of the number of plates or-mass transfer sections, the location of the feed plate of the distillation column, one component factor Et,,,, for each component for all plates and all runs, and two plate factors for all runs as described under modification (1) of the procedure for the determination of vaporization efficiencies.
A SERIES of several field tests were made on a packed distillation column and a packed absorber. These columns were parts of the Zoller gas plant (See Fig. 1) located near Refugio, Texas. The undesirable characteristics of both the Murphree-type efficiencies and the overall mass transfer coefficients as demonstrated previously[2,1 l] has provided further support for the use of vaporization efficiencies in the modeling of packed columns as proposed by McDaniel et af.[ 151 and Rubac et al.[ 191. The modeling technique proposed herein gave satisfactory results for both of the columns. Briefly, this procedure consisted of finding the number of perfect plates or perfect mass transfer sections which minimized the deviation of the absolute value of the logarithm of the ratio of the observed and calculated product distributions. A further improvement in the agreement of these product distributions was achieved through the use of vaporization efficiencies. EXPERIMENTAL
EQUIPMENT REDUCTION
AND
DATA
Brief descriptions of the packed distillation column, the packed absorber, analytical equipment, and reduction of the data follow. Packed distillation column Detailed descriptions of the Zoller gas plant have been given by others[3,13,14,18]. A
schematic diagram of the packed distillation column is shown in Fig. 2. Its mechanical detail has also been described by others[3,18]. As designed by Stearns-Roger Manufacturing Company, its major components consisted of a total condenser, two packed sections, a feed mixing section, and a reboiler. Both packed sections were 17 ft in length. The upper section contained 3340 lb of 2 in. metallic Pall Rings while the lower section contained 5920 lb of this packing. The upper packed section was three feet in diameter while the lower section was four feet in diameter. In addition to the temperatures of the external streams, the temperature profile for the packed sections was determined by means of 17 ironconstantan thermocouples (18 in. in length), which extended about 15 in. into the packing. The flow rates of the distillate and the bottoms and the reflux rate were measured by means of orifice meters. The feed rate was calculated by material balance. Also for each test run, samples were withdrawn from the feed, distillate, and bottoms lines. These samples were analyzed by means of a gas-liquid chromatograph (a Varian Aerograph Model 1520) and a digital integrator (Infotronics CRS-104). The adsorption columns used were a 30 ft and a 40 ft column, both packed with 25 per cent d.c., 200/500,30/60 Chromosorb P. To satisfy the necessary material balances, the
tPresent address: The Chemshare Corp., P.O. BOX23 10, Norman, Oklahoma 73069, U.S.A.
633
R. MCDANIEL,
A. A. BASSYONI
and C. D. HOLLAND *VAPOR
PACKING HOLD-DOWN
GRATING
TO CONDENSER
REFLUX
INLET
THERMOCOUPLE
r
PACKED RECTIFYING SECTION
PACKING SUPPORT GRATING -
LIQUID
DISTRIBUTOR
PLATE
PACKING HOLD-D
-PACKED
STRIPPING SECTION
PACKWG SLRPORT GRATING LEAN OIL FROM HEATER
-
+?----I INDlCATES LOCATION OF
THERMOWELLS
LEAN OIL TO HEATER
I
/
4 -w=
Fig. 2. The packed distillation column at the Zoller gas plant.
feed composition was computed on the basis of the analyses of the distillate and bottoms and the respective flow rates. The computed composition was in good agreement with that determined directly by analysis [3]. A summary of the operating conditions for the 17 field tests made on the distillation column
is given in Table 1. A complete set of experimental results for Run 10 1 is given in Table 3. Packed
absorber
A schematic diagram of the packed absorber is shown in Fig. 3. This absorber, which has been described in detail previously [13,14], had a 23 ft
634
Fig.
1.The Zoller gas plant.
(Facing page 634)
Modeling of packed distillation columns - 11I Table 1. Summary of the operating conditions for the field tests made on the packed distillation column Temperature, “F
Flow rate, lb-moleslhr Field Test No.
Feed
Reflux
Distillate
Feed
Reflux
001 002 003 101 102 103 104 105 201 202 203 204 205 301 303 304 305
408.9 411.3 408.3 405.4 368.3 373.3 420.7 405.6 407.6 390.8 372.6 425.8 439.3 438.1 385.9 370.9 352.1
130.3 184.6 238.9 183.0 180.3 180.2 184.0 183.4 221.0 221.6 223.5 227.1 229.5 149.1 147.9 147.3 144.2
161.6 161.8 161.3 153.8 146.0 146,7 155.2 153.7 155.8 152.6 151.6 157-o 157.4 156.3 147.9 144.4 138.0
358.3 356.6 360.0 360.0 356.0 360.3 364.5 356.0 351.8 353.0 342.5 362.5 364.5 363.0 355.0 352.3 375,8
82.0 97.0 107.0 93.0 99.0 100.0 109.0 98.0 90.0 90.0 90.0 88.0 88.0 88.0 89.0 88.0 88.0
packed section with a diameter of three feet. The packing consisted of 4900 lb of 2 in. metallic Pall Rings. This column was also designed by the Stearns-Roger Manufacturing Company. In addition to the temperatures of the terminal streams, the temperature profile in the packed section was determined. This was accomplished by means of 7 copper-constantan thermocouples (18 in. in length), which extended about 15 in. into the packing. The flow rates of the lean oil, lean gas, and rich oil were obtained by means of orifice meters. The flow rate of the rich gas was computed by means of a material balance about the packed section of the absorber. Samples were withdrawn from the lean gas, lean oil, and the plant inlet gas (a composite of the rich gas and the condensate streams). These samples were analyzed by the equipment mentioned previously. The composition of the rich gas (the gas actually entering the bottom of the packing) was computed by means of a simple flash calculation on the basis of the analysis of the plant inlet gas. These analyses, total flow rates, and calculated results were used to determine the composition of the rich oil by material balances. This procedure was
Overhead Reboiler vapor 493.0 498.0 499.0 494.0 490.0 490.0 490.0 495.0 494.0 500.0 494.0 502.0 502.0 494.0 500.0 500.0 500.0
176-O 181.0 180.0 175-o 174.0 174.0 175.0 175.0 170.0 170.0 170.0 171.0 171.0 180.0 179.0 178.0 180.0
Pressure PSIA 165 165 165 163 165 162 165 165 176 175 175 176 176 177 175 175 175
checked for several runs by analyzing the rich oil and condensate streams. The computed compositions were found to be in good agreement with the analyses. A summary of the operating conditions for the 18 field tests made on the absorber is given in Table 2. A complete set of experimental results for a typical test run (Run 303) is given in Table 4. In the analysis of the field tests on the distillation column, the reboiler and condenser duties were computed by use of the customary energy balances. This procedure permitted errors in the enthalpies to be absorbed into the calculated values of the reboiler and condenser duties. However, there were no heat sources or sinks that could absorb these errors for the absorber data. Therefore the absorber runs were placed in the best energy balance possible by adjusting the enthalpies. The errors in the vapor and liquid enthalpies for all components were regarded as being equally likely, and thus the enthalpies were adjusted using the following expressions
635
h,l,,
= hi + ECiL(T - TD)
R. MCDANIEL,
A. A. BASSYONI
and C. D. HOLLAND
f-b
HO1
LEAN
GAS
PACKING .D- DOWN GRATING
LEAN OIL
THERMOCOUPLE
PACKING SUPPORT
VAPOR CHIMNEYS
~LIQIJID
RICH GAS ----+v
\
GRATING
DRAW- OFF
TRAY
I
CONDENSATE AND ETHYLENE
*CONDENSATE
-
INDICATES LOCATION OF THCRMOWELLS
Fig. 3. The absorber at the Zoller gas plant.
H* lcorr = Hi+eCi”(T-To).
MODELING
The curve fits for hi and Hi are available elsewhere[ 141. These enthalpies were used to compute Ci” and CiL over the range of the curve fits. The value of E was computed by minimizing the sum of the squares of the heat losses over all runs. 636
OF
qACKED
COLUMNS
Concepts utilized in the transformation of the continuous mass transfer processes into equivalent stage processes follow as well as a description of the various calculational procedures employed. procedure The same general modeling procedures
General
calculational
were
Modeling Table 2. Summary Field Test No.
Lean oil
101 102 103 201 202 203 204 205 301 302 303 304 305 401 402 403 404 405
188.9 190.5 188.7 192-I 162.8 165.6 205.4 195.9 197.5 184.1 166.8 214.3 227.3 226.7 214.5 189.0 172.0 159.6
of packed
of the operating
distillation
conditions
- 111
for the field tests made on the packed Temperature,
Flow rates, lb-moles/hr
Table
columns
absorber
“F
Rich gas
Lean gas
Lean oil
Rich oil
Rich gas
Lean gas
Pressure PSIA
2994.1 2995.4 2971.4 2977.8 2902.0 2731.2 2787.5 3106.5 2769.0 2769.5 2792.4 2786.1 2785.4 2788.2 2799.3 2635.2 2665.0 2675.7
2721.1 2726.0 2701.2 2700.7 2644.4 2474.6 2494.9 2824.7 2486.2 2493.4 2528.9 2504.7 2492.9 2491.8 2496.7 2364.9 2411.2 2435.1
2.9 2.9 2.9 1.5 0.0 0.4 7.0 0.3 -2.0 -3.0 -2.5 1.0 3.0 3.0 2.5 - 1.0 - 1.0 -1.5
18.6 18.6 18.6 18.0 16.3 18.0 20.0 18.5 16.0 16.0 16.0 17.0 17.0 18.0 18.0 18.0 17-o 16.0
0.0 0.0 0.0 -1.0 - 1.5 -2.0 -2.0 2.2 -1.5 -2.1 -2.5 -1.0 0.0 0.0 0.0 -2.0 -2.5 -2.0
24.0 24.0 24.0 23.0 20.5 22.5 23.0 24.0 23.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0
807 807 793 780 784 778 792 786 794 794 792 790 790 781 780 780 782 784
3. Experimental
Component Ethane Propane i-Butane n-Butane i-Pentane n-Pentane Hexane Heptane Octane Nonane Decane tu = unknown
results
Feed rates FXi 0.8330 784683 31.1295 24-4780 8.1443 4.7172 6.9601 32.7723 90.5138 68.6985 58.7195 product
obtained for run 101 on the packed column
Product dist. (Wi)errr
ut 0.20/X 0.2370 x 0.3714 x 0.7040 x 0.7919 x 0.1135 x 0-3610x l4 U
lO-2 lo-* 10-l 10-I loo 102 103
distribution.
used for both packed distillation columns and packed absorbers. Thus, to avoid repetition, these procedures are described for only distillation columns.
Depth of packing (ft) O(Reflux) OWapor) 1.5 2.5 3.5 4.5 5.5 6.5 8.5 10.5 12.5 14.5 15.5 17XKl 18.5 20.0 21.5 25.5 29.5 32.5 34-o
distillation
Temp. (“F) To= 93.0 T, = 175.0 219.5 196.0 202.5 233.0 216.5 237.5 263.5 272.2 288.2 302.7 313.5 T,v = 360.0 399.8 429.0 418.2 427.8 435.5 451.5 TN+, = 494.0
The results from the field tests yielded values for those variables commonly specified in the design of a distillation column, namely; the distillate rate D, the external reflux rate L,,, the
637
R. MCDANIEL,
A. A. BASSYONI
and C. D. HOLLAND
Table 4. Exoerimental results obtained for run 303 on the uacked absorber
Component
Rich gas %+1.i
Carbon dioxide Nitrogen Methane Ethane Propane i-Butane n-Butane i-Pentane n-Pentane Hexane Heptane Octane Nonane Decane
19.24727 7.81712 2477.91602 174.88483 7668777 19.03183 11.84179 2.97810 140726 0.48697 0.07558 0.01194 0*00061 o+wOO7
Lean oil c
Product dist. J?JilVH 0.11786 o-09517 O-04485 0.48578 4.81826 31.65155 116.61901 193.30931 103.0743 68.82089 67.19069 76.46271 193.09148 509.09302
0.0 0.0 0.0 o-0
O-24520 0.05504 0.02168 o-07506 o-04504 O-16847 11*04901 69.82089 55.84227 33.90868
type of condenser (partial or total), the column pressure, and the complete definition of the feed F. This group of operating variables plus the number of mass transfer sections and the location of the feed section are hereafter referred to as the “usual specifications.” These variables are those that are necessary to fix the product distributions {bi/di} and the temperature profile. In addition to the usual specifications, values for the product distributions and the temperature profile were determined. This group is referred to as the “additional specifications.” Both columns were divided into elements as demonstrated in Figs. 4 and 5. The numbering notation used in the model of the distillation column follows immediately from Figs. 4 and 6. Likewise, the numbering of absorbers follows immediately from Fig. 5. This notation is seen to be the same as that employed in the analysis of columns with plates [9]. Figure 6 is a sketch of the feed mixing section of the distillation column. At relatively high feed flow rates, where the outlet of the feed-entrance pipe is submerged, all of the vapor from the lower packed bed tended to contact all of the liquid on the feed distributor plate. For this case, which is representative of these 17 field tests, the feed mixing section is perhaps best represented as a mass transfer section or plate. At relatively 10~ flow rates, where the holdup on the distributor
Depth of packing (ft)
Temp. (“F)
O(lean oil) O(lean gas) 2.0 6.0 10.0 14.0 18.0 21.5 23.0 23(rich oil) 23(rich gas)
-2.5 25.0 19.5 27.0 23.5 19.5 15.5 8.4 -2.5 16.0 -2.5
i
Feed
Mixinq
Fig. 4. Sketch of a packed distillation column.
plate is low, the feed mixing section is perhaps best represented by a mixing zone with0t.R ma88 transfer as demonstrated previously[lg]. The use of the concept of the mass transfer
638
Modeling of packed distillation columns - I II Yji =
Fig. 5. Sketch of a typical packed absorber.
_&edsez:ng
Fig. 6. Feed mixing section for a submerged-feed
section
to
represent
a
by a stage process previously[ 15,191. If it behavior of each element element) is represented relationship[5,9,10, Ill, process
continuous
entrance.
separation
has been demonstrated is supposed that the of packing (say thejth by the mass transfer
Ejx& Ki
Xjf
(1)
then each element of packing is in effect transformed into an equivalent plate or mass transfer section, and thus the packed column is transformed into an equivalent column with plates. In other words, the component-material, totalmaterial, and energy balances reduce to the same form as those for conventional columns. This transformation has been previously given in detail[3,13,15,18,19]. Given a set of values for {Eji}in addition to the usual specifications, the product distribution and the temperature profile may be determined by use of any one of the well known procedures which are available for columns with plates [ 1,6,7,9, lo]. Similarly, given a set of values for {bJ4] and {Tj> in addition to the usual specifications, the set {Eji}may be computed by the well known procedure given for columns with plates[9]. This procedure was first extended to packed columns by McDaniel ef al.[ 151 and Rubac et a/.[ 191. Because of the modifications to come later, it is necessary to discuss in detail a portion of this procedure. For a distillation column with a partial condenser, the set {Eji}contains at most c(N+ 2) elements. For a distillation with a total condenser, the set contains c(N+ 1) elements. Even though the number of product distributions is c, there are only (c - 1) degrees of freedom gained by their specification. For, given (c - 1) product distributions the remaining distribution may be computed by use of component-material balances and the knowledge of the distillate rate D. The column used to make the field tests was equipped with a total condenser; hence, the set { Tj} contains N + 1 elements. Therefore, a total of (c + N) degrees of freedom is available from the results of the field tests. TO calculate the set {Eji} of c(N + 1) elements from the (c + N) degrees of freedom, Harris [8] proposed the following model Eji= Eipj where Ei is the component
639
(2)
efficiency factor and
R. MCDANIEL,
A. A. BASSYONI
Modification (1). Additional specifications consist of all or part of the set {bi/di} and the set {T,,TN+,}; the set {Ei}, PI and PN+l are to be determined. Consider first the case where the complete set {b,/d,} and the temperatures T1 and TN+I are known. In this modification two section factors, p, and PN+I, which satisfy the specifications T, and TN+~ are computed. The factors for the intermediate sections are taken to be the geometric mean of PI and PN+l; that is,
& is the plate or element factor. Notice, however, there are c elements in the set {I!?,} and N+ 1 elements in the set {&} ; thus, there are c + N + 1 variables to be computed. To obtain unique sets {Ei} and {&}, an additional restriction was needed. This condition was taken by Taylor et a1.[21] to be the requirement that the product of the elements of the set {pj} be equal to unity; that is, N+l rl[
@j=
and C. D. HOLLAND
1.
j=1
(&)ca = In this normalization procedure, value flj is defined by
[(kh)ca(~N+l)ca11’2,
(2
the normalized
5
j
5
N). (6)
The normalization procedure using Eqs. (4) and (5) leads to the following sets of normalized values
(4)
PI = (h/PN+I);f The subscript “~a” is used to identify the most recent calculated sets {pj} and {I?,} after each iteration of the calculational procedure. To keep the set {&} unchanged, the normalized value of Ei must be defined by the formula
PNfl pj=
&
l/PI
1, (25jZ
(Ei)ca[(Pl)ca(PN+1)ca11’2.
(8)
N)
(9) (10)
Since this procedure assumes no knowledge of the temperature profile in the packing, this profile ( T2 5 Tj S TN) is computed by means of the Kb-method[9, 161. For the case where part of the set {b,/d,} is unknown, modifications of the general calculation procedure have been proposed previously [5,10,19,21]. The modification employed in this work was used for the case where certain of the components were in either of the product streams, B or D, only in trace amounts. For these components, the calculated value of I?, was assigned the value of unity.
This particular normalization procedure has the desirable characteristic that at total reflux, the ratio of the product distributions bJd, for any two components depends only on the ratio of their respective values of lY?iand Ki[5, lo]. Modifications of the calculational procedure for the determination of the vaporization eficiencies The following modifications were formulated for the purpose of making it possible to model existing columns on the basis of information which is either presently available or can be readily obtained. This information most often contains only a portion of the group of specifications referred to as the additional specifications in the previous section. For example, only part of the product distributions and the temperatures of the terminal streams may be known.
=
=
(7)
ModiJication (2). Additional speciJcations consist of all or part of the set {b,/d,} and the set {Tl,TN+I}; the set {&} and a single p are to be determined. This proposed modification consists of a minor variation of the first in that instead of determining two section factors such that the two terminal
640
Modeling of packed distillation columns - I I I
temperature specifications are satisfied, a single factor /3 is to be found such that a weighted sum of the squares of the temperature functions fi and&+, is minimized; where,
fl =
lil&mli X1*-
(11)
1
TL..,
xv+1=
g1EmN+l,*&i+1,i -
1.
(12)
r.+,.
em
(Note that for the case of a perfect mass transfer section (I& = l), fi and fN+1 reduce to the well known bubble-point functions.) Thus, a value of p is computed such that 0 (p) is minimized. 0 (P) = Wfi2 +
~2f~+1*
(13)
The quantities w1 and w2 are arbitrary weighting factors which satisfy the conditions: WI 2 0 w, e 0 w,+w,
= 1.
(14)
The equation to be used to compute p is found by differentiating Eq. (13) with respect to p and equating the result to zero and then solving for p. The result is
As in the first modification, the temperature for each element is found by use of the Z&method and by using the value for (/3)ca as the element factor for each element. The normalization procedure using Eqs. (4) and (5) yields the following normalized values of pi and Ei, p*= 1,
(1 Sj
5 N+l)
(16) 641
Cl3 Vol. 25 No. 4- F
(17) Thus, it is obvious that a new set of K-values for each component has been created by the multiplication of each K-value function by its corresponding Ei a If any of the product distributions are unknown, the calculational procedure is modified as described for the first modification. The function O(p), defined by Eq. (13), was examined and found to exhibit the anticipated behavior; that is, as wr(O 5 w, s 1) was increased the calculated value of T1 approached the experimental value as demonstrated by Bassyoni [3]. The enthalpy and K-data used in the calculation of the efficiencies for the distillation column and the absorber have been presented by Bassyoni [3] and McDaniel [ 133.
Formulation of objective functions Several conceivable methods for using the results of field tests to formulate suitable mathematical models for a packed distillation column exist. For example, consider the case where the additional specifications (consisting of the sets { Tj} and {bJdi}) have been determined experimentally as well. as the usual specifications or operating conditions. It is evident from the description of the calculational procedure described above that for each choice N 2 3, an exact representation of the packed distillation column exists. That is, for any given N 2 3, sets {I!?,} and (pJ} may be found such that a one-toone correspondence exists between all observed and calculated values of the variables. Thus, in general, the number of packed sections may be picked arbitrarily. Consequently, the number of packed sections may be picked as required to best satisfy a particular objective. For example, in the formulation of a model for the purposes of process control, the objective might well be to minimize the number of equations for describing the column, in which case the number of packed sections may be taken as two, one above and one below the feed mixing section.
R. MCDANIEL,
A. A. BASSYONI
and C. D. HOLLAND
{,?$}, {pJ} such that the deviation of each calculated product distribution from its corresponding experimental value is minimized. The objective is represented by the average of 0, over all runs R,
The primary objective of the work described herein was the formulation of a model, possessing relatively few parameters, which could be used to predict accurate product distributions and terminal temperatures over wide ranges of operating conditions. In an effort to reduce the total number of parameters, the two modifications of the basic procedure for the determination of the &‘s, wherein only one or two &‘s are determined, was developed as shown above. In the applications which follow, Modification (1) which involves the determination of P1 andPN+I or {Pj] = @lYPN+I) is employed. The problem then consists of finding the feeddistributor-plate location f, the number of mass transfer sections N, the sets {&} and (pj} such that the deviation of each calculated product distribution (b,/dJ,, from its corresponding experimental value (b*/di),,p is minimized. In addition it is desired that the deviation of the calculated values of the terminal temperatures from the experimental values be minimized. Although the deviations of these temperatures do not appear explicitly in the following objective function, they are included implicitly in the determination of the p’s. For any given run, the objective function to be minimized is given by
The quantity 13~ is defined by (19) where the symbol (b,/di),, refers to the calculated value of the product distribution for component i. Note that for any given set {f, N}, there exists a set Of Ei’s and a set of fij’s such that 0, = 0. Since there exists infinitely many sets {AN}, it follows that the function 0, has infinitely many zeroes. However, this statement is not necessarily true if it is required that the same values of the arguments of 0, be used over all runs R, which represents the problem of interest. That is, the objective is to find the sets (AN},
where the subscript r denotes the run number. This particular form of argument ( Ilog, BiTI) was selected because it assigns the same weight to deviations resulting from values of &, < 1 as it does to deviations resulting from &, > 1. Note again that if for each run a set of Ei’S and one set of pj’s are permitted for all runs for a given choice off and N, the problem of finding the sets {Ei}, {PJ1 and LW f or which a1 takes on its minimum value is recognized as a problem in optimization. Such a problem may be solved by use of multivariable-sequential search techniques [ 12,221. However, such a search was considered and regarded as impractical because of the large amount of computer time which would have been required to find the global minimum (the smallest of all possible minima) of 0,. Instead of searching 6, sequentially over all variables, the function 0, was searched over allf and N at perfect plates or perfect mass transfer sections (Eji = 1 for all i and j). Then on the basis of the solution set {AN) so obtained, the E,‘s and Pj’S were determined for each run. From these sets of &‘s and pJ’s, a single set of &‘s and one set of &‘s were computed by means of an appropriate averaging procedure developed below. Although the approach described may not produce the global minimum in &, this disadvantage is more than offset by the fact that both 0, and 0, were unimodal (possessed a single minimum) at perfect mass transfer sections. The desire to choose an objective function which always has one and only one minimum with respect to f and N stems from the fact that commercial packings are commonly rated with respect to their “equivalent number of theoretical plates.”
642
Modeling of packed distillation columns - 111
The purpose of the following development is to demonstrate the existance of functional relationships between the &‘s, and Bi’s, and Tj’S. The relationships so obtained suggest the possible use of other objective functions closely related to 0,. For a distillation column equipped with a total condenser at the operating condition of total reflux, the product distribution computed on the basis of perfect mass transfer sections is given by
bi
BID
0iTica=N+l n
0
=
Ea(Kji)exp
log 0, = -
(N + 1) lOgeE, + 10&Xcif&
(25)
Note that at total reflux Ri = 1. The relationship given by Eq. (25) may be used in the formulation of several objective functions. By taking the absolute values of each term in Eq. (25) followed by the averaging of each term over all components and all runs yield
BID
N+l n
where Kjb is an arbitrarily selected base component. To account for the effect of operating refluxes other than total, let the function log& be added to the right-hand side of Eq. (23) to give
where
BID exp
(24)
Kji
When vaporization efficiencies are determined such that the additional specifications are satisfied, the corresponding expression for the product distribution at total reflux is given by
h
Kji = ai ( Kjb)ci
(21)
-
j=l
z
To account for the variation of the Kji’s with temperature, the following relationship, suggested by Winn [23] for hydrocarbons, may be employed,
=EiN+’
I=1
E
(KjiIexp‘
j=1
(22) The final expression on the right-hand side of Eq. (22) follows as a result of the normalization procedure which requires that the product of the &‘s over allj be equal to unity [see Eqs. (3) and (4)]. The subscript “exp” on the Kji’S is used to denote the fact that the Kj(‘S are to be evaluated at the experimentally determined temperatures where a complete set of pj’s is computed. If only one or two p’s are determined, then the subscript “exp” on the Kji’S refers to the values of these variables as determined by the proposed calculational procedures [see Modifications (1) and (2)]. For any one run let the members of Eq. (22) be divided by the corresponding members of Eq. (21). The result so obtained may be stated in logarithmic form as follows: N+l
l”&Ni=-((N+l)
lO&Ei+lO&
n L J=l t
Let the order of operations used to obtain Eq. (26) be altered whereby the members of Eq. (25) are first averaged over all runs R, and then the absolute value of each term of the resulting expression is taken. After each term of the expression so obtained has been averaged over all components c, the following result is obtained.
+
de,,
i=l
r=l
I*
’
(23) This expression .
j$ f: 5 10&X~CiRir’ (27)
643
may be stated in terms of geo-
R. MCDANIEL,
A. A. BASSYONI
metric means as follows:
greater than the absolute value of the sum. Next, note that if for each component i, each eir < 1 over all runs R or 8i,. > 1 over all runs R, then Eq. (33) reduces to a1 = ii,, and if the &,.‘s satisfy analogous conditions, then 0, = 6,. If a1 = 0, and 0, = 04, then it follows from Eqs. (26) and (27) that
10g,e,,
+f
log,Xtnc”Oim (28)
i:
and C. D. HOLLAND
i=l
where the subscript “WZ”is used to denote the geometric mean of the variable over all runs R. For example, Eii, = +? (EilEiZE*s . . . &).
(29)
The left-hand side of Eq. (26) is recognized as the objective function ol, defined by Eq. (20), and 0, may be obtained by omitting the summation over all R used to obtain Eq. (26). The left-hand side of Eq. (27) suggests the definition of the following objective function
G=&
$i
hk&l.
(30)
The right-hand sides of Eqs. (26) and (27) suggest the definitions of two other objective functions; namely,
hi& a,,
6 4 =$jy
$ log&I i=l
(31)
(32)
r=l
respectively. As Eqs. (26) and (27) indicate, all of these objective functions are interrelated. Now observe that s, 2 0,
(33)
0, SZs,
(34)
since the sum of the absolute values is equal to or
ii,
5
a,+& i r=l
~llo&x~ciQrl.
(35)
i=l
Although this inequality does not necessarily hold unless the conditions upon which it is based are satisfied, it does serve to emphasize that these two objective functions are related. The objective function 6, was used in the modeling procedure proposed in the next section, while the averaging procedure (the geometric mean of the Ei’s) contained by the function b4 was used. However, the use of 0, and 0, as objective functions in the modeling procedure was ruled out because their use led to the subsequent multimodal behavior in the function 0, for the absorber tests [ 141. In addition to the functions 0, and a4, several other objective functions involving either the efficiencies or the temperatures were investigated. The functions involving the efficiencies were searched over all runs for the distillation column for thef and N for which the deviations of all sections from perfect sections was minimized. In particular, the functions investigated by Bassyoni [2], McDaniel[ 141, and Rubac[ 181 included: (1) the average over all components c, sections N + 1, and runs R of the squares of the deviations of the &‘s from unity; (2) the average over the same set of parameters listed in (1) of the absolute values of the deviations of the Eji’s from unity, and (3) the average over the same set of parameters listed in (1) of the squares of the deviations of the Eji’s from unity per element per component. Although these functions always exhibited a single minimum for each of the field tests made on the packed distillation column (3), they failed to exhibit unimodal behavior for the absorber runs (14), and consequently this class
644
Modeling of packed distillation columns-
of functions was discarded as potential objective functions for the general modeling of all types of packed columns. The possibility of employing objective functions based on the set of observed temperatures {T+,,} is suggested by the second function on the right-hand side of Eq. (26). Functions of this general type were introduced by Rubac et a/.[ 191. These functions were analogous to those described for the &‘s except that the argument consisted of the deviation of the ratio of the calculated to the experimental temperatures from unity. Although this class of objective functions exhibited unimodal behavior for all runs for both the packed distillation column and the packed absorber, they were discarded for general modeling purposes because temperature profiles are not generally available for commercial columns. Of the possible methods considered for modeling packed columns, the following one appeared the most desirable.
111
borhood containing the optimum solution set (f= 6, N = 11) for the packed distillation column are presented in Table 5. In this work, a simultaneous search as indicated in Table 5 was employed in order to determine the general behavior of the objective functions over the domain of the independent variables f and N. However, since the functions were observed to be unimodal, many reliable techniques[22] for searching O,(f,N,{E,, = 1)) exist. For example, the method of Hooke and Jeeves[l2] may be applied in a manner similar to that described by Srygley et af.[20].
Recommended modeling procedure for packed columns The first step in the proposed modeling procedure for packed distillation columns consists of a logical extension of the concept of the “Height Equivalent to a Theoretical Plate” (called an HETP), proposed by Peters[l7], to columns in the process of separating multicomponent mixtures. For such a column, a set {AN} of perfect mass transfer sections does not necessarily exist such that all calculated and observed product distributions may be placed in a one-to-one correspondence. For any given set f and N, the objective functions 0, and 0, are seen to give an adequate measure of the deviations of the calculated and experimental product distributions for all components. The first step of the modeling procedure is to find that pair of valuesfand N for which 0, (f,Ny{Eji = 1)) takes on its minimum value. Both of the functions 0, and 0, exhibited unimodal behavior with well defined minima as shown in Fig. 7 for distillation columns and in Fig. 8 for absorbers. Values of ol(f,N,{Ejl = 1)) in a small neigh-
TOTAL NUMBER OF MASS TRANSFER SECTION. N I l
F_ig.7. Variation of the function 0, and 0, with the total number of perfect mass transfer sections and at equal numbers of packed sections above and below the feed-distributor plate [f= (N- 1)/Z] of the distillation column.
TO.7 G &!%A _0.5L z ;;_
1 2
4
6
’ 8
’
’ IO
’
J 12
NUMBER OF MASS TRANSFER SECTIONS, N
F_ig.8. Variation of the functions 0, and 0, with the number of perfect mass transfer sections in the absorber.
645
R. MCDANIEL,
A. A. BASSYONI
Table 5. Values of the objective function 0, (f,N,{E,, = 1)) in the neighborhood of the solution set {f= 6, N = 11) for the distillation column Feed plate f
Number? of sections N
2 3 4 5 6 7 8 9 7 6 6 5
0,(&N,{&
3 5 7 9 11 13 15 17 12 10 12 10
= 1))
1.4544 1.0878 0.7675 0.4884 0*3459$ 0.4223 0.6212 0.8798 0.3749 0.4652 0.3832 0.3769
tN = number of packed sections plus the feed distributor-plate. ‘*Minimum value of the objective function.
The second step in the proposed modeling procedure consists of the determination of the vaporization efficiencies for each run at the optimum f and N for the distillation column and the optimum N for the absorber by use of any
and C. D. HOLLAND
one of the procedures described previously for the determination of the &‘s. Since the &‘s place the calculated and observed product distributions in a one-to-one correspondence, the second step has the effect of driving the objective function O1 (f,iV) to zero for each run. However, because of the large number of parameters {II&} resulting from the second step, a procedure was needed for averaging the efficiencies in some appropriate manner. The third step of the modeling procedure consists of the averaging of the El’s as well as the flj’s and checking for the dependency of the efficiencies upon the operating conditions. The geometric mean was used to compute the average of the l?i’s as indicated by Eq. (29). The sets {I&} of geometric mean values of the &‘s over all runs R for the packed distillation column and for the packed absorber are presented in Table 6. The geometric mean values of the @j’s are also presented in Table 6. When the p’s are averaged in this particular way, the mean values so obtained are consistent with the geometric average of the I?cs. TO check for the significance of the Eim’s
Table 6. Geometric mean values of the efficiencies over all runs made on the packed distillation column and packed absorber Distillation column f=6,N+l=
Component Carbon dioxi de Nitrogen Methane Ethane Propane i-Butane n-Butane i-Pentane n-Pentane Hexane Heptane Octane Nonane Decane
12 Modification (1) &n 1%l43 0.9869 1.0216 1.1693 1.1972 1.1908 1.2169 1*OS04 1.0754 1.0099 1+l448 8, = 1.1077 0,~ = 0.9028
Absorber
f=6,N+l= 12 Modification (2) &It
1wjO2 1.0023 I.0347 1.1084 1.0886 I.1256 I*0128 0.9834 1.0623 0.9899 1.0332 w1 = 0.5
646
N=8 Modification (I) &I
N=8 Modification (2) &I
2.2551 0.0435 14671 0.9256 0.7411 0.9323 1.0220 0.9016 0.9295 0.5302 0.4405 0.6342 0.5529 0.4046
2.1297 0.0528 1.3792 0.8875 0.7476 0.9534 I.0506 0.9585 1+lOso 0.6623 0.4858 0.7003 0.6108 04468
/3, = 1.0854 & = 0.9213
w, = 0.5
Modeling of packed distillation columns- III
(averaged over all runs R) in the reduction of & the product distributions for each run were recomputed by use of the &,,,‘s and at the optimum solution set {f,N} found in step one. These product distributions were then used in the re-evaluation of & and the result obtained was 0,(6,11,
{&n),
{&J)
= 0.1618
where p1 = l-1077, p N+1= 09028, and {Er,} is given in Table 6. The minimum value obtained for perfect mass transfer sections in step one was O,(6,11) {Ej, = 1)) = O-3459. Since the set of Ei,‘s gave approximately a 50 per cent reduction in bI, it is evident that the set of mean &,‘s and pjm’s over all runs R provided a significant improvement in the calculated values of the product distributions over those predicted on the basis of perfect plates. Similar results were obtained over all runs R for the absorber; namely, O,(N, &J,
I&m]) = O-3316
where N = 8 and &, PN, and the {Ei,} are given in Table 6. The minimum value obtained for is, in step one for the absorber was O,(N, {Eji = 1)) = O-7834 for N = 8. Thus, the set of l?im’s (over all runs R made on the absorber) gave a significant reduction in the error of the product distributions. Next, a test for the possible effect of operating conditions on the &‘s for the packed distillation column was made in the following manner. After the runs made on the distillation column had been divided into three subsets in accordance with the reflux ratios employed, sets of geometric mean values of the &‘s and pJ’s were computed for each subset of runs. Product distributions were re-determined for each subset of runs on the basis of the corresponding set of efficiencies (@*mL ~Plm~PN+l,mI)~ and the product distri647
butions so obtained were used to evaluate 0, for each subset of runs. The results given in Table 7 show that no significant improvement in the calculated product distributions was achieved by using a different set of efficiencies for each of the three sets of runs. This result implies that over the range of refluxes investigated, the efficiencies were independent of reflux ratio. This same conclusion was reached by Bassyoni[3] upon making the F-test on the Ei’s. A comparison of the calculated and observed product distributions for the packed distillation column and packed absorber are presented in Figs. 9 and 10. The calculated product distributions are based on the mean Els and pi’s given in Table 6 under “Modification (I).” A comparison of the average deviation of the calculated from the experimental values for the terminal temperatures for the distillation column and the absorber are presented in Table 8. The Pjm’s for the terminal sections (j = 1, N + 1 for the distillation column and j = 1, N for the absorber) together with the set of {Ei,} for each of the two columns reduced the deviations obtained for perfect sections by factors ranging from about 2 to 6. Although the proposed modeling procedure may not yield the global minimum in & this Table 7. Values obtained for the objective function 0, for each of four sets of runs by use of the corresponding sets of &‘s and/.$‘s [determined by modification
(I)1 No. runs used Reflux No. runs to compute ratio ’ 0% ,,,I and
h/D
s’:t
@A3N+l.m1
a
-
17 5 6 6 Averaget
17 5 6 6 -
0.1618 0.3155 O-0280 0.1675 0.1618
0.8 l-2 l-4 -
tin the computation of the average of 6, for the three subsets of runs (at LJD =0*8, 1.2 and l-4), the value of 8, for each subset was weighted according to the number of runs in the subset.
R. MCDANIEL,
A. A. BASSYONI
and C. D. HOLLAND
1000 f=6 J*I= 12
/
;?
:
./
loo
___
IO
.
.yI
I’
/
IQ 8
Y
s
0-I
o-01
i-
1
0001
/ r
OoO? 001
C 101
IO
C
100
1000
Fig. 9. Comparison of the product distributions computed on the basis of the mean values/?, ~ and Pl,m (listed under modification (1) Table 6) with the experimental product distributions for the packed &stillation column.
Table 8. Average deviation of the calculated from the experimental values for the terminal temperatures for the distillation column and the absorber
Temp.
Distillation column Average deviationt Perfect sections Modification (1)
T, (Overhead vapor) TN+I(Reboller)
0.02695 040785
T, (Lean gas) TN(Rich oil)
040792 040129
tAverage deviation of temperatureT,(j=
Temp.
Absorber Average deviationt Perfect sections Modification (1)
l,N,N+
648
1) = f $1
0.01756 0.02944 -=I,
040917 040606
where T,is in OR.
Modeling of packed distillation columns - III
,.
N =8
:.::I I ..i’._
.‘.
,/ ‘.v
.
.;
../ I..‘,,<
.:;: t.
/
Fig. 10. Comparison of the product distributions computed on the basis of the mean values .8?,;,,and a,,,, (listed under modification (1) Table 6) with the experimental distributions for the packed absorber.
deficiency is counterbalanced many times by the very definite unimodal behavior exhibited by 0, for all each run and by d, for all runs where perfect mass transfer sections are employed as described in step one. Brosilow et a1.[4] proposed a method somewhat similar to the one described in step one except that instead of perfect mass transfer sections, an approximation of the mass transfer rate expression was employed. CONCLUSIONS
The proposed modeling procedure makes use of an objective function oI(f,N,{lZji = 1)) for the distillation column and b,(N,{E,, = 1)) for the absorber which appear to always exhibit unimodal behavior. A suitable method for averaging the &‘s and flj’s over each subset or over all runs is provided by the proposed modeling procedure. Also, the proposed procedure provides a method for testing for the dependency of
the &‘s upon the operating conditions. Furthermore, the proposed procedure involves the use of relatively few parameters, and these parameters appear to remain approximately constant over reasonable ranges of operating conditions. Finally, the proposed modeling procedure is obviously directly applicable to columns with plates. Acknowledgments-The support provided by the National Science Foundation and Texaco, Inc. is gratefully acknowledged. The assistance given by W. E. Vaughn, J. W. Thompson, J. D. Dyal, and J. P. Smith of Hunt Oil Co., and J. H. Galloway, M. F. Clegg, Jr., and J. L. Shanks of Humble Oil and Refining Company; and R. V. Randall and W. E. Muzacz of Esso Production Research Company is appreciated.
649
NOTATION ai
bi
B
factor that appears in Eq. (24) molal flow rate of component i in the bottom product total molal flow rate of the bottoms
R. MCDANIEL,
A. A. BASSYONI
and C. D. HOLLAND
To datum temperature total number of components T, temperature of mass transfer section j factor that appears in Eq. (24) molal heat capacities for pure compositive constants lying between 0 Wl,WZ ponent i in the liquid and vapor and 1 and having a sum of unity phases; used in the expressions for xji mole fraction of component i in the the corrected liquid and vapor vapor leaving the jth mass transfer enthalpies section molal flow rate of component i in the yji mole fraction of component i in the distillate vector leaving the jth mass transfer total molal flow rate of the distillate section vaporization efficiency for componGreek symbols ent i and mass transfer section j component efficiency for component i Pj mass transfer factor for the jth mass transfer section number of the feed-distribution plate E factor appearing in the corrected of feed mixing section enthalpies mass transfer functions evaluated at activity coefficient for component i in the experimental values of T, and the liquid leaving the jth mass TN+l *definedbyEqs.(ll)and(l2), transfer section respectively multiplier defined by Eq. (19) enthalpy of pure component i in the function defined below Eq. (25) liquid, B.t.u./lb mole function of operating conditions haventhalpy of pure component i in the ing a value of unity at total reflux vapor, B.t.u./lb mole (or recycle) value of K for component i at the temperature of the liquid leaving Subscripts the jth mass transfer section b base component number of packed sections plus the ca calculated value feed distribution plate; N+ 1 = experimental value ew total number of mass transfer seci component number tions + feed-distribution plate + mass transfer section number j reboiler (for a distillation column m mean value with a total condenser). For the r run number absorber, N = total number of mass transfer sections Mathematical symbols objective function; defined by Eq. sum over all component from i = 1 (13) through i = c objective function for any number of absolute value of the argument x runs R; defined by Eq. (20). objective functions; defined by Eqs. set of all l?f’S; El. E-2. . . E, (30), (3 I), and (32), respectively product of all elements from x1 run number total number of runs through & 7
REFERENCES [l] AMUNDSON N. R. and PONTINEN A. J., Ind. Engng Chem. 1958 50 730. [2] BASSYONI A. A., MCDANIEL R. and HOLLAND C. D., Chem. Engng Sci. 1970 25 437. [3] BASSYONI A. A., Ph.D. Dissertation, Texas A&M University, College Station,Texas 1969.
650
Modeling of packed distillation columns - 111 [4] BROSILOW
C., TANNER R. and TUREFF H., Optimization of Staged Counter Current’Processes, presented at the 63rd Nat. Mtg. of the Am. Inst. Chem. Engrs, St. Louis, Missouri, February 1968. [5] DAVIS P., TAYLOR D. L. and HOLLAND C. D.,A.I.Ch.E.JI 1965 11678. [6] GREENSTADT J., BARD Y. and MORSE B., Ind. Engng Chem. 1968 50 1644. [7] HANSON D. N., DUFFIN J. H. and SOMERVILLE G. F., Computation of Multistage Separation Processes. Reinhold 1962. [8] HARRIS T. R., M.S. Thesis, Texas A&M University, College Station, Texas 1962. [9] HOLLAND C. D., Multicomponent Distillation. Prentice-Hall 1963. [lo] HOLLAND C. D., Unsteady State Processes with Applications in Multicomponent Distillation, Prentice-Hall, Englewood Cliffs, N.J. 1966. [l l] HOLLAND C. D., McMAHON K. S., Chem. Engng Sci. 1970 25 43 I. [12] HOOKE R. and JEEVES T. A., J. Ass. comput. Mach. 19618 2. [13] MCDANIEL R., M.S. Thesis, Texas A&M University, College.Station, Texas 1968. [14] MCDANIEL R., Ph.D. Dissertation, Texas A&M University, College Station, Texas 1969. [15] MCDANIEL R., RUBAC R. E. and HOLLAND C. D., Packed Distillation Columns and Absorbers at Steady Operation, presented at the 63rd Nat. Mtg. of the Am. Inst. Chem. Engrs., St. Louis, Missouri, February 1968. [16] NARTKERT. A., SRYGLEY J. M. and HOLLAND C. D., Can.J. them. Engng 196644217. [ 171 PETERS W. A., Jr., Ind. Engng Chem. 1922 14 476. [ 181 RUBAC R. E., Ph.D. Dissertation, Texas A&M University, College Station, Texas 1968. 1191 RUBAC R. E., MCDANIEL R. and HOLLAND C. D.,A.I.Ch.E.JI 1969 15 568. 120] SRYGLEY J. M. and HOLLAND C. D.,A.I.Ch.E.Jf 1965 11695. 1211 TAYLOR D. L.. DAVIS P. and HOLLAND C. D..A.I.Ch.E.JI 1964 10 864. i22j WILDE D. J., Optimum Seeking Methods. Prentice-Hall 1964. [23] WINN F. W., Petrol. Re$ner 1958 37 216. Resume- On demontre I’utilisation des resultats dune serie d’essais de service en vue de I’ttablissement d’un mod&Ii de colonne de distillation gamie et d’un absorbeur gami. Les mdthodes proposees utilisent lesdonntes communement disponibles pour les colonnes existantes. Les parambtres ddterminis d’apres les colonnes existantes. Les parametres determines d’apres les essais de service comprennent le nombre de plateaux ou de sections de transfert de masse, la position du plateau d’entree dans la colonne de distillation, un facteur consistuant Ei, pour chaque composant de tous les plateaux et de toutes les series de production et deux facteurs de plateaux pour toutes les series ainsi qu’il est dicrit dans la modification (1) de la procedure pour la determination des rendements de vaporisation. Zusammenfassung-Die Anwendung der Ergebnisse einer Reihe von Gelandeversuchen in der mathematischen Darstellung einer Fiillkorperkolonne und eines Filllkiirperabsorbers wird vorgefiihrt. Die vorgeschlagenen Methoden mathematischer Darstellung machen Gebrauch von Messwerten, die gewuhnlich fur bestende Kolonnen erhaltlich sind. Die in den Geliindeversuchen ermittelten Parameter bestanden aus der Zahl der Boden oder $toffaustauschabteilungen, der Lage des Aufgabebodens der Kolonne, einem Komponenten-faktor E,.,” fur jede Komponente fur alIe Bidden und alle Versuche, und zwei Bodenfaktoren fur alle Versuche, wie beschrieben unter Anderung (i) der Verfahrens fiir die Bestimmung von Verdampfungswirkungsgraden.
651
Pergamon Press.
Printed in Great Britain.
Use of the results of field tests in the modeling of packed distillation columns and packed absorbers-III RONALD MCDANIEL?, A. A. BASSYONI and C. D. HOLLAND Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843, U.S.A. (First received6
June 1969; accepted
24 September
1969)
Abstract-The use of the results of a series of field tests in the modeling of a packed distillation column and a packed absorber is demonstrated. The modeling techniques proposed make use of data which are commonly available for existing columns. The parameters determined from the field tests consisted of the number of plates or-mass transfer sections, the location of the feed plate of the distillation column, one component factor Et,,,, for each component for all plates and all runs, and two plate factors for all runs as described under modification (1) of the procedure for the determination of vaporization efficiencies.
A SERIES of several field tests were made on a packed distillation column and a packed absorber. These columns were parts of the Zoller gas plant (See Fig. 1) located near Refugio, Texas. The undesirable characteristics of both the Murphree-type efficiencies and the overall mass transfer coefficients as demonstrated previously[2,1 l] has provided further support for the use of vaporization efficiencies in the modeling of packed columns as proposed by McDaniel et af.[ 151 and Rubac et al.[ 191. The modeling technique proposed herein gave satisfactory results for both of the columns. Briefly, this procedure consisted of finding the number of perfect plates or perfect mass transfer sections which minimized the deviation of the absolute value of the logarithm of the ratio of the observed and calculated product distributions. A further improvement in the agreement of these product distributions was achieved through the use of vaporization efficiencies. EXPERIMENTAL
EQUIPMENT REDUCTION
AND
DATA
Brief descriptions of the packed distillation column, the packed absorber, analytical equipment, and reduction of the data follow. Packed distillation column Detailed descriptions of the Zoller gas plant have been given by others[3,13,14,18]. A
schematic diagram of the packed distillation column is shown in Fig. 2. Its mechanical detail has also been described by others[3,18]. As designed by Stearns-Roger Manufacturing Company, its major components consisted of a total condenser, two packed sections, a feed mixing section, and a reboiler. Both packed sections were 17 ft in length. The upper section contained 3340 lb of 2 in. metallic Pall Rings while the lower section contained 5920 lb of this packing. The upper packed section was three feet in diameter while the lower section was four feet in diameter. In addition to the temperatures of the external streams, the temperature profile for the packed sections was determined by means of 17 ironconstantan thermocouples (18 in. in length), which extended about 15 in. into the packing. The flow rates of the distillate and the bottoms and the reflux rate were measured by means of orifice meters. The feed rate was calculated by material balance. Also for each test run, samples were withdrawn from the feed, distillate, and bottoms lines. These samples were analyzed by means of a gas-liquid chromatograph (a Varian Aerograph Model 1520) and a digital integrator (Infotronics CRS-104). The adsorption columns used were a 30 ft and a 40 ft column, both packed with 25 per cent d.c., 200/500,30/60 Chromosorb P. To satisfy the necessary material balances, the
tPresent address: The Chemshare Corp., P.O. BOX23 10, Norman, Oklahoma 73069, U.S.A.
633
R. MCDANIEL,
A. A. BASSYONI
and C. D. HOLLAND *VAPOR
PACKING HOLD-DOWN
GRATING
TO CONDENSER
REFLUX
INLET
THERMOCOUPLE
r
PACKED RECTIFYING SECTION
PACKING SUPPORT GRATING -
LIQUID
DISTRIBUTOR
PLATE
PACKING HOLD-D
-PACKED
STRIPPING SECTION
PACKWG SLRPORT GRATING LEAN OIL FROM HEATER
-
+?----I INDlCATES LOCATION OF
THERMOWELLS
LEAN OIL TO HEATER
I
/
4 -w=
Fig. 2. The packed distillation column at the Zoller gas plant.
feed composition was computed on the basis of the analyses of the distillate and bottoms and the respective flow rates. The computed composition was in good agreement with that determined directly by analysis [3]. A summary of the operating conditions for the 17 field tests made on the distillation column
is given in Table 1. A complete set of experimental results for Run 10 1 is given in Table 3. Packed
absorber
A schematic diagram of the packed absorber is shown in Fig. 3. This absorber, which has been described in detail previously [13,14], had a 23 ft
634
Fig.
1.The Zoller gas plant.
(Facing page 634)
Modeling of packed distillation columns - 11I Table 1. Summary of the operating conditions for the field tests made on the packed distillation column Temperature, “F
Flow rate, lb-moleslhr Field Test No.
Feed
Reflux
Distillate
Feed
Reflux
001 002 003 101 102 103 104 105 201 202 203 204 205 301 303 304 305
408.9 411.3 408.3 405.4 368.3 373.3 420.7 405.6 407.6 390.8 372.6 425.8 439.3 438.1 385.9 370.9 352.1
130.3 184.6 238.9 183.0 180.3 180.2 184.0 183.4 221.0 221.6 223.5 227.1 229.5 149.1 147.9 147.3 144.2
161.6 161.8 161.3 153.8 146.0 146,7 155.2 153.7 155.8 152.6 151.6 157-o 157.4 156.3 147.9 144.4 138.0
358.3 356.6 360.0 360.0 356.0 360.3 364.5 356.0 351.8 353.0 342.5 362.5 364.5 363.0 355.0 352.3 375,8
82.0 97.0 107.0 93.0 99.0 100.0 109.0 98.0 90.0 90.0 90.0 88.0 88.0 88.0 89.0 88.0 88.0
packed section with a diameter of three feet. The packing consisted of 4900 lb of 2 in. metallic Pall Rings. This column was also designed by the Stearns-Roger Manufacturing Company. In addition to the temperatures of the terminal streams, the temperature profile in the packed section was determined. This was accomplished by means of 7 copper-constantan thermocouples (18 in. in length), which extended about 15 in. into the packing. The flow rates of the lean oil, lean gas, and rich oil were obtained by means of orifice meters. The flow rate of the rich gas was computed by means of a material balance about the packed section of the absorber. Samples were withdrawn from the lean gas, lean oil, and the plant inlet gas (a composite of the rich gas and the condensate streams). These samples were analyzed by the equipment mentioned previously. The composition of the rich gas (the gas actually entering the bottom of the packing) was computed by means of a simple flash calculation on the basis of the analysis of the plant inlet gas. These analyses, total flow rates, and calculated results were used to determine the composition of the rich oil by material balances. This procedure was
Overhead Reboiler vapor 493.0 498.0 499.0 494.0 490.0 490.0 490.0 495.0 494.0 500.0 494.0 502.0 502.0 494.0 500.0 500.0 500.0
176-O 181.0 180.0 175-o 174.0 174.0 175.0 175.0 170.0 170.0 170.0 171.0 171.0 180.0 179.0 178.0 180.0
Pressure PSIA 165 165 165 163 165 162 165 165 176 175 175 176 176 177 175 175 175
checked for several runs by analyzing the rich oil and condensate streams. The computed compositions were found to be in good agreement with the analyses. A summary of the operating conditions for the 18 field tests made on the absorber is given in Table 2. A complete set of experimental results for a typical test run (Run 303) is given in Table 4. In the analysis of the field tests on the distillation column, the reboiler and condenser duties were computed by use of the customary energy balances. This procedure permitted errors in the enthalpies to be absorbed into the calculated values of the reboiler and condenser duties. However, there were no heat sources or sinks that could absorb these errors for the absorber data. Therefore the absorber runs were placed in the best energy balance possible by adjusting the enthalpies. The errors in the vapor and liquid enthalpies for all components were regarded as being equally likely, and thus the enthalpies were adjusted using the following expressions
635
h,l,,
= hi + ECiL(T - TD)
R. MCDANIEL,
A. A. BASSYONI
and C. D. HOLLAND
f-b
HO1
LEAN
GAS
PACKING .D- DOWN GRATING
LEAN OIL
THERMOCOUPLE
PACKING SUPPORT
VAPOR CHIMNEYS
~LIQIJID
RICH GAS ----+v
\
GRATING
DRAW- OFF
TRAY
I
CONDENSATE AND ETHYLENE
*CONDENSATE
-
INDICATES LOCATION OF THCRMOWELLS
Fig. 3. The absorber at the Zoller gas plant.
H* lcorr = Hi+eCi”(T-To).
MODELING
The curve fits for hi and Hi are available elsewhere[ 141. These enthalpies were used to compute Ci” and CiL over the range of the curve fits. The value of E was computed by minimizing the sum of the squares of the heat losses over all runs. 636
OF
qACKED
COLUMNS
Concepts utilized in the transformation of the continuous mass transfer processes into equivalent stage processes follow as well as a description of the various calculational procedures employed. procedure The same general modeling procedures
General
calculational
were
Modeling Table 2. Summary Field Test No.
Lean oil
101 102 103 201 202 203 204 205 301 302 303 304 305 401 402 403 404 405
188.9 190.5 188.7 192-I 162.8 165.6 205.4 195.9 197.5 184.1 166.8 214.3 227.3 226.7 214.5 189.0 172.0 159.6
of packed
of the operating
distillation
conditions
- 111
for the field tests made on the packed Temperature,
Flow rates, lb-moles/hr
Table
columns
absorber
“F
Rich gas
Lean gas
Lean oil
Rich oil
Rich gas
Lean gas
Pressure PSIA
2994.1 2995.4 2971.4 2977.8 2902.0 2731.2 2787.5 3106.5 2769.0 2769.5 2792.4 2786.1 2785.4 2788.2 2799.3 2635.2 2665.0 2675.7
2721.1 2726.0 2701.2 2700.7 2644.4 2474.6 2494.9 2824.7 2486.2 2493.4 2528.9 2504.7 2492.9 2491.8 2496.7 2364.9 2411.2 2435.1
2.9 2.9 2.9 1.5 0.0 0.4 7.0 0.3 -2.0 -3.0 -2.5 1.0 3.0 3.0 2.5 - 1.0 - 1.0 -1.5
18.6 18.6 18.6 18.0 16.3 18.0 20.0 18.5 16.0 16.0 16.0 17.0 17.0 18.0 18.0 18.0 17-o 16.0
0.0 0.0 0.0 -1.0 - 1.5 -2.0 -2.0 2.2 -1.5 -2.1 -2.5 -1.0 0.0 0.0 0.0 -2.0 -2.5 -2.0
24.0 24.0 24.0 23.0 20.5 22.5 23.0 24.0 23.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0
807 807 793 780 784 778 792 786 794 794 792 790 790 781 780 780 782 784
3. Experimental
Component Ethane Propane i-Butane n-Butane i-Pentane n-Pentane Hexane Heptane Octane Nonane Decane tu = unknown
results
Feed rates FXi 0.8330 784683 31.1295 24-4780 8.1443 4.7172 6.9601 32.7723 90.5138 68.6985 58.7195 product
obtained for run 101 on the packed column
Product dist. (Wi)errr
ut 0.20/X 0.2370 x 0.3714 x 0.7040 x 0.7919 x 0.1135 x 0-3610x l4 U
lO-2 lo-* 10-l 10-I loo 102 103
distribution.
used for both packed distillation columns and packed absorbers. Thus, to avoid repetition, these procedures are described for only distillation columns.
Depth of packing (ft) O(Reflux) OWapor) 1.5 2.5 3.5 4.5 5.5 6.5 8.5 10.5 12.5 14.5 15.5 17XKl 18.5 20.0 21.5 25.5 29.5 32.5 34-o
distillation
Temp. (“F) To= 93.0 T, = 175.0 219.5 196.0 202.5 233.0 216.5 237.5 263.5 272.2 288.2 302.7 313.5 T,v = 360.0 399.8 429.0 418.2 427.8 435.5 451.5 TN+, = 494.0
The results from the field tests yielded values for those variables commonly specified in the design of a distillation column, namely; the distillate rate D, the external reflux rate L,,, the
637
R. MCDANIEL,
A. A. BASSYONI
and C. D. HOLLAND
Table 4. Exoerimental results obtained for run 303 on the uacked absorber
Component
Rich gas %+1.i
Carbon dioxide Nitrogen Methane Ethane Propane i-Butane n-Butane i-Pentane n-Pentane Hexane Heptane Octane Nonane Decane
19.24727 7.81712 2477.91602 174.88483 7668777 19.03183 11.84179 2.97810 140726 0.48697 0.07558 0.01194 0*00061 o+wOO7
Lean oil c
Product dist. J?JilVH 0.11786 o-09517 O-04485 0.48578 4.81826 31.65155 116.61901 193.30931 103.0743 68.82089 67.19069 76.46271 193.09148 509.09302
0.0 0.0 0.0 o-0
O-24520 0.05504 0.02168 o-07506 o-04504 O-16847 11*04901 69.82089 55.84227 33.90868
type of condenser (partial or total), the column pressure, and the complete definition of the feed F. This group of operating variables plus the number of mass transfer sections and the location of the feed section are hereafter referred to as the “usual specifications.” These variables are those that are necessary to fix the product distributions {bi/di} and the temperature profile. In addition to the usual specifications, values for the product distributions and the temperature profile were determined. This group is referred to as the “additional specifications.” Both columns were divided into elements as demonstrated in Figs. 4 and 5. The numbering notation used in the model of the distillation column follows immediately from Figs. 4 and 6. Likewise, the numbering of absorbers follows immediately from Fig. 5. This notation is seen to be the same as that employed in the analysis of columns with plates [9]. Figure 6 is a sketch of the feed mixing section of the distillation column. At relatively high feed flow rates, where the outlet of the feed-entrance pipe is submerged, all of the vapor from the lower packed bed tended to contact all of the liquid on the feed distributor plate. For this case, which is representative of these 17 field tests, the feed mixing section is perhaps best represented as a mass transfer section or plate. At relatively 10~ flow rates, where the holdup on the distributor
Depth of packing (ft)
Temp. (“F)
O(lean oil) O(lean gas) 2.0 6.0 10.0 14.0 18.0 21.5 23.0 23(rich oil) 23(rich gas)
-2.5 25.0 19.5 27.0 23.5 19.5 15.5 8.4 -2.5 16.0 -2.5
i
Feed
Mixinq
Fig. 4. Sketch of a packed distillation column.
plate is low, the feed mixing section is perhaps best represented by a mixing zone with0t.R ma88 transfer as demonstrated previously[lg]. The use of the concept of the mass transfer
638
Modeling of packed distillation columns - I II Yji =
Fig. 5. Sketch of a typical packed absorber.
_&edsez:ng
Fig. 6. Feed mixing section for a submerged-feed
section
to
represent
a
by a stage process previously[ 15,191. If it behavior of each element element) is represented relationship[5,9,10, Ill, process
continuous
entrance.
separation
has been demonstrated is supposed that the of packing (say thejth by the mass transfer
Ejx& Ki
Xjf
(1)
then each element of packing is in effect transformed into an equivalent plate or mass transfer section, and thus the packed column is transformed into an equivalent column with plates. In other words, the component-material, totalmaterial, and energy balances reduce to the same form as those for conventional columns. This transformation has been previously given in detail[3,13,15,18,19]. Given a set of values for {Eji}in addition to the usual specifications, the product distribution and the temperature profile may be determined by use of any one of the well known procedures which are available for columns with plates [ 1,6,7,9, lo]. Similarly, given a set of values for {bJ4] and {Tj> in addition to the usual specifications, the set {Eji}may be computed by the well known procedure given for columns with plates[9]. This procedure was first extended to packed columns by McDaniel ef al.[ 151 and Rubac et a/.[ 191. Because of the modifications to come later, it is necessary to discuss in detail a portion of this procedure. For a distillation column with a partial condenser, the set {Eji}contains at most c(N+ 2) elements. For a distillation with a total condenser, the set contains c(N+ 1) elements. Even though the number of product distributions is c, there are only (c - 1) degrees of freedom gained by their specification. For, given (c - 1) product distributions the remaining distribution may be computed by use of component-material balances and the knowledge of the distillate rate D. The column used to make the field tests was equipped with a total condenser; hence, the set { Tj} contains N + 1 elements. Therefore, a total of (c + N) degrees of freedom is available from the results of the field tests. TO calculate the set {Eji} of c(N + 1) elements from the (c + N) degrees of freedom, Harris [8] proposed the following model Eji= Eipj where Ei is the component
639
(2)
efficiency factor and
R. MCDANIEL,
A. A. BASSYONI
Modification (1). Additional specifications consist of all or part of the set {bi/di} and the set {T,,TN+,}; the set {Ei}, PI and PN+l are to be determined. Consider first the case where the complete set {b,/d,} and the temperatures T1 and TN+I are known. In this modification two section factors, p, and PN+I, which satisfy the specifications T, and TN+~ are computed. The factors for the intermediate sections are taken to be the geometric mean of PI and PN+l; that is,
& is the plate or element factor. Notice, however, there are c elements in the set {I!?,} and N+ 1 elements in the set {&} ; thus, there are c + N + 1 variables to be computed. To obtain unique sets {Ei} and {&}, an additional restriction was needed. This condition was taken by Taylor et a1.[21] to be the requirement that the product of the elements of the set {pj} be equal to unity; that is, N+l rl[
@j=
and C. D. HOLLAND
1.
j=1
(&)ca = In this normalization procedure, value flj is defined by
[(kh)ca(~N+l)ca11’2,
(2
the normalized
5
j
5
N). (6)
The normalization procedure using Eqs. (4) and (5) leads to the following sets of normalized values
(4)
PI = (h/PN+I);f The subscript “~a” is used to identify the most recent calculated sets {pj} and {I?,} after each iteration of the calculational procedure. To keep the set {&} unchanged, the normalized value of Ei must be defined by the formula
PNfl pj=
&
l/PI
1, (25jZ
(Ei)ca[(Pl)ca(PN+1)ca11’2.
(8)
N)
(9) (10)
Since this procedure assumes no knowledge of the temperature profile in the packing, this profile ( T2 5 Tj S TN) is computed by means of the Kb-method[9, 161. For the case where part of the set {b,/d,} is unknown, modifications of the general calculation procedure have been proposed previously [5,10,19,21]. The modification employed in this work was used for the case where certain of the components were in either of the product streams, B or D, only in trace amounts. For these components, the calculated value of I?, was assigned the value of unity.
This particular normalization procedure has the desirable characteristic that at total reflux, the ratio of the product distributions bJd, for any two components depends only on the ratio of their respective values of lY?iand Ki[5, lo]. Modifications of the calculational procedure for the determination of the vaporization eficiencies The following modifications were formulated for the purpose of making it possible to model existing columns on the basis of information which is either presently available or can be readily obtained. This information most often contains only a portion of the group of specifications referred to as the additional specifications in the previous section. For example, only part of the product distributions and the temperatures of the terminal streams may be known.
=
=
(7)
ModiJication (2). Additional speciJcations consist of all or part of the set {b,/d,} and the set {Tl,TN+I}; the set {&} and a single p are to be determined. This proposed modification consists of a minor variation of the first in that instead of determining two section factors such that the two terminal
640
Modeling of packed distillation columns - I I I
temperature specifications are satisfied, a single factor /3 is to be found such that a weighted sum of the squares of the temperature functions fi and&+, is minimized; where,
fl =
lil&mli X1*-
(11)
1
TL..,
xv+1=
g1EmN+l,*&i+1,i -
1.
(12)
r.+,.
em
(Note that for the case of a perfect mass transfer section (I& = l), fi and fN+1 reduce to the well known bubble-point functions.) Thus, a value of p is computed such that 0 (p) is minimized. 0 (P) = Wfi2 +
~2f~+1*
(13)
The quantities w1 and w2 are arbitrary weighting factors which satisfy the conditions: WI 2 0 w, e 0 w,+w,
= 1.
(14)
The equation to be used to compute p is found by differentiating Eq. (13) with respect to p and equating the result to zero and then solving for p. The result is
As in the first modification, the temperature for each element is found by use of the Z&method and by using the value for (/3)ca as the element factor for each element. The normalization procedure using Eqs. (4) and (5) yields the following normalized values of pi and Ei, p*= 1,
(1 Sj
5 N+l)
(16) 641
Cl3 Vol. 25 No. 4- F
(17) Thus, it is obvious that a new set of K-values for each component has been created by the multiplication of each K-value function by its corresponding Ei a If any of the product distributions are unknown, the calculational procedure is modified as described for the first modification. The function O(p), defined by Eq. (13), was examined and found to exhibit the anticipated behavior; that is, as wr(O 5 w, s 1) was increased the calculated value of T1 approached the experimental value as demonstrated by Bassyoni [3]. The enthalpy and K-data used in the calculation of the efficiencies for the distillation column and the absorber have been presented by Bassyoni [3] and McDaniel [ 133.
Formulation of objective functions Several conceivable methods for using the results of field tests to formulate suitable mathematical models for a packed distillation column exist. For example, consider the case where the additional specifications (consisting of the sets { Tj} and {bJdi}) have been determined experimentally as well. as the usual specifications or operating conditions. It is evident from the description of the calculational procedure described above that for each choice N 2 3, an exact representation of the packed distillation column exists. That is, for any given N 2 3, sets {I!?,} and (pJ} may be found such that a one-toone correspondence exists between all observed and calculated values of the variables. Thus, in general, the number of packed sections may be picked arbitrarily. Consequently, the number of packed sections may be picked as required to best satisfy a particular objective. For example, in the formulation of a model for the purposes of process control, the objective might well be to minimize the number of equations for describing the column, in which case the number of packed sections may be taken as two, one above and one below the feed mixing section.
R. MCDANIEL,
A. A. BASSYONI
and C. D. HOLLAND
{,?$}, {pJ} such that the deviation of each calculated product distribution from its corresponding experimental value is minimized. The objective is represented by the average of 0, over all runs R,
The primary objective of the work described herein was the formulation of a model, possessing relatively few parameters, which could be used to predict accurate product distributions and terminal temperatures over wide ranges of operating conditions. In an effort to reduce the total number of parameters, the two modifications of the basic procedure for the determination of the &‘s, wherein only one or two &‘s are determined, was developed as shown above. In the applications which follow, Modification (1) which involves the determination of P1 andPN+I or {Pj] = @lYPN+I) is employed. The problem then consists of finding the feeddistributor-plate location f, the number of mass transfer sections N, the sets {&} and (pj} such that the deviation of each calculated product distribution (b,/dJ,, from its corresponding experimental value (b*/di),,p is minimized. In addition it is desired that the deviation of the calculated values of the terminal temperatures from the experimental values be minimized. Although the deviations of these temperatures do not appear explicitly in the following objective function, they are included implicitly in the determination of the p’s. For any given run, the objective function to be minimized is given by
The quantity 13~ is defined by (19) where the symbol (b,/di),, refers to the calculated value of the product distribution for component i. Note that for any given set {f, N}, there exists a set Of Ei’s and a set of fij’s such that 0, = 0. Since there exists infinitely many sets {AN}, it follows that the function 0, has infinitely many zeroes. However, this statement is not necessarily true if it is required that the same values of the arguments of 0, be used over all runs R, which represents the problem of interest. That is, the objective is to find the sets (AN},
where the subscript r denotes the run number. This particular form of argument ( Ilog, BiTI) was selected because it assigns the same weight to deviations resulting from values of &, < 1 as it does to deviations resulting from &, > 1. Note again that if for each run a set of Ei’S and one set of pj’s are permitted for all runs for a given choice off and N, the problem of finding the sets {Ei}, {PJ1 and LW f or which a1 takes on its minimum value is recognized as a problem in optimization. Such a problem may be solved by use of multivariable-sequential search techniques [ 12,221. However, such a search was considered and regarded as impractical because of the large amount of computer time which would have been required to find the global minimum (the smallest of all possible minima) of 0,. Instead of searching 6, sequentially over all variables, the function 0, was searched over allf and N at perfect plates or perfect mass transfer sections (Eji = 1 for all i and j). Then on the basis of the solution set {AN) so obtained, the E,‘s and Pj’S were determined for each run. From these sets of &‘s and pJ’s, a single set of &‘s and one set of &‘s were computed by means of an appropriate averaging procedure developed below. Although the approach described may not produce the global minimum in &, this disadvantage is more than offset by the fact that both 0, and 0, were unimodal (possessed a single minimum) at perfect mass transfer sections. The desire to choose an objective function which always has one and only one minimum with respect to f and N stems from the fact that commercial packings are commonly rated with respect to their “equivalent number of theoretical plates.”
642
Modeling of packed distillation columns - 111
The purpose of the following development is to demonstrate the existance of functional relationships between the &‘s, and Bi’s, and Tj’S. The relationships so obtained suggest the possible use of other objective functions closely related to 0,. For a distillation column equipped with a total condenser at the operating condition of total reflux, the product distribution computed on the basis of perfect mass transfer sections is given by
bi
BID
0iTica=N+l n
0
=
Ea(Kji)exp
log 0, = -
(N + 1) lOgeE, + 10&Xcif&
(25)
Note that at total reflux Ri = 1. The relationship given by Eq. (25) may be used in the formulation of several objective functions. By taking the absolute values of each term in Eq. (25) followed by the averaging of each term over all components and all runs yield
BID
N+l n
where Kjb is an arbitrarily selected base component. To account for the effect of operating refluxes other than total, let the function log& be added to the right-hand side of Eq. (23) to give
where
BID exp
(24)
Kji
When vaporization efficiencies are determined such that the additional specifications are satisfied, the corresponding expression for the product distribution at total reflux is given by
h
Kji = ai ( Kjb)ci
(21)
-
j=l
z
To account for the variation of the Kji’s with temperature, the following relationship, suggested by Winn [23] for hydrocarbons, may be employed,
=EiN+’
I=1
E
(KjiIexp‘
j=1
(22) The final expression on the right-hand side of Eq. (22) follows as a result of the normalization procedure which requires that the product of the &‘s over allj be equal to unity [see Eqs. (3) and (4)]. The subscript “exp” on the Kji’S is used to denote the fact that the Kj(‘S are to be evaluated at the experimentally determined temperatures where a complete set of pj’s is computed. If only one or two p’s are determined, then the subscript “exp” on the Kji’S refers to the values of these variables as determined by the proposed calculational procedures [see Modifications (1) and (2)]. For any one run let the members of Eq. (22) be divided by the corresponding members of Eq. (21). The result so obtained may be stated in logarithmic form as follows: N+l
l”&Ni=-((N+l)
lO&Ei+lO&
n L J=l t
Let the order of operations used to obtain Eq. (26) be altered whereby the members of Eq. (25) are first averaged over all runs R, and then the absolute value of each term of the resulting expression is taken. After each term of the expression so obtained has been averaged over all components c, the following result is obtained.
+
de,,
i=l
r=l
I*
’
(23) This expression .
j$ f: 5 10&X~CiRir’ (27)
643
may be stated in terms of geo-
R. MCDANIEL,
A. A. BASSYONI
metric means as follows:
greater than the absolute value of the sum. Next, note that if for each component i, each eir < 1 over all runs R or 8i,. > 1 over all runs R, then Eq. (33) reduces to a1 = ii,, and if the &,.‘s satisfy analogous conditions, then 0, = 6,. If a1 = 0, and 0, = 04, then it follows from Eqs. (26) and (27) that
10g,e,,
+f
log,Xtnc”Oim (28)
i:
and C. D. HOLLAND
i=l
where the subscript “WZ”is used to denote the geometric mean of the variable over all runs R. For example, Eii, = +? (EilEiZE*s . . . &).
(29)
The left-hand side of Eq. (26) is recognized as the objective function ol, defined by Eq. (20), and 0, may be obtained by omitting the summation over all R used to obtain Eq. (26). The left-hand side of Eq. (27) suggests the definition of the following objective function
G=&
$i
hk&l.
(30)
The right-hand sides of Eqs. (26) and (27) suggest the definitions of two other objective functions; namely,
hi& a,,
6 4 =$jy
$ log&I i=l
(31)
(32)
r=l
respectively. As Eqs. (26) and (27) indicate, all of these objective functions are interrelated. Now observe that s, 2 0,
(33)
0, SZs,
(34)
since the sum of the absolute values is equal to or
ii,
5
a,+& i r=l
~llo&x~ciQrl.
(35)
i=l
Although this inequality does not necessarily hold unless the conditions upon which it is based are satisfied, it does serve to emphasize that these two objective functions are related. The objective function 6, was used in the modeling procedure proposed in the next section, while the averaging procedure (the geometric mean of the Ei’s) contained by the function b4 was used. However, the use of 0, and 0, as objective functions in the modeling procedure was ruled out because their use led to the subsequent multimodal behavior in the function 0, for the absorber tests [ 141. In addition to the functions 0, and a4, several other objective functions involving either the efficiencies or the temperatures were investigated. The functions involving the efficiencies were searched over all runs for the distillation column for thef and N for which the deviations of all sections from perfect sections was minimized. In particular, the functions investigated by Bassyoni [2], McDaniel[ 141, and Rubac[ 181 included: (1) the average over all components c, sections N + 1, and runs R of the squares of the deviations of the &‘s from unity; (2) the average over the same set of parameters listed in (1) of the absolute values of the deviations of the Eji’s from unity, and (3) the average over the same set of parameters listed in (1) of the squares of the deviations of the Eji’s from unity per element per component. Although these functions always exhibited a single minimum for each of the field tests made on the packed distillation column (3), they failed to exhibit unimodal behavior for the absorber runs (14), and consequently this class
644
Modeling of packed distillation columns-
of functions was discarded as potential objective functions for the general modeling of all types of packed columns. The possibility of employing objective functions based on the set of observed temperatures {T+,,} is suggested by the second function on the right-hand side of Eq. (26). Functions of this general type were introduced by Rubac et a/.[ 191. These functions were analogous to those described for the &‘s except that the argument consisted of the deviation of the ratio of the calculated to the experimental temperatures from unity. Although this class of objective functions exhibited unimodal behavior for all runs for both the packed distillation column and the packed absorber, they were discarded for general modeling purposes because temperature profiles are not generally available for commercial columns. Of the possible methods considered for modeling packed columns, the following one appeared the most desirable.
111
borhood containing the optimum solution set (f= 6, N = 11) for the packed distillation column are presented in Table 5. In this work, a simultaneous search as indicated in Table 5 was employed in order to determine the general behavior of the objective functions over the domain of the independent variables f and N. However, since the functions were observed to be unimodal, many reliable techniques[22] for searching O,(f,N,{E,, = 1)) exist. For example, the method of Hooke and Jeeves[l2] may be applied in a manner similar to that described by Srygley et af.[20].
Recommended modeling procedure for packed columns The first step in the proposed modeling procedure for packed distillation columns consists of a logical extension of the concept of the “Height Equivalent to a Theoretical Plate” (called an HETP), proposed by Peters[l7], to columns in the process of separating multicomponent mixtures. For such a column, a set {AN} of perfect mass transfer sections does not necessarily exist such that all calculated and observed product distributions may be placed in a one-to-one correspondence. For any given set f and N, the objective functions 0, and 0, are seen to give an adequate measure of the deviations of the calculated and experimental product distributions for all components. The first step of the modeling procedure is to find that pair of valuesfand N for which 0, (f,Ny{Eji = 1)) takes on its minimum value. Both of the functions 0, and 0, exhibited unimodal behavior with well defined minima as shown in Fig. 7 for distillation columns and in Fig. 8 for absorbers. Values of ol(f,N,{Ejl = 1)) in a small neigh-
TOTAL NUMBER OF MASS TRANSFER SECTION. N I l
F_ig.7. Variation of the function 0, and 0, with the total number of perfect mass transfer sections and at equal numbers of packed sections above and below the feed-distributor plate [f= (N- 1)/Z] of the distillation column.
TO.7 G &!%A _0.5L z ;;_
1 2
4
6
’ 8
’
’ IO
’
J 12
NUMBER OF MASS TRANSFER SECTIONS, N
F_ig.8. Variation of the functions 0, and 0, with the number of perfect mass transfer sections in the absorber.
645
R. MCDANIEL,
A. A. BASSYONI
Table 5. Values of the objective function 0, (f,N,{E,, = 1)) in the neighborhood of the solution set {f= 6, N = 11) for the distillation column Feed plate f
Number? of sections N
2 3 4 5 6 7 8 9 7 6 6 5
0,(&N,{&
3 5 7 9 11 13 15 17 12 10 12 10
= 1))
1.4544 1.0878 0.7675 0.4884 0*3459$ 0.4223 0.6212 0.8798 0.3749 0.4652 0.3832 0.3769
tN = number of packed sections plus the feed distributor-plate. ‘*Minimum value of the objective function.
The second step in the proposed modeling procedure consists of the determination of the vaporization efficiencies for each run at the optimum f and N for the distillation column and the optimum N for the absorber by use of any
and C. D. HOLLAND
one of the procedures described previously for the determination of the &‘s. Since the &‘s place the calculated and observed product distributions in a one-to-one correspondence, the second step has the effect of driving the objective function O1 (f,iV) to zero for each run. However, because of the large number of parameters {II&} resulting from the second step, a procedure was needed for averaging the efficiencies in some appropriate manner. The third step of the modeling procedure consists of the averaging of the El’s as well as the flj’s and checking for the dependency of the efficiencies upon the operating conditions. The geometric mean was used to compute the average of the l?i’s as indicated by Eq. (29). The sets {I&} of geometric mean values of the &‘s over all runs R for the packed distillation column and for the packed absorber are presented in Table 6. The geometric mean values of the @j’s are also presented in Table 6. When the p’s are averaged in this particular way, the mean values so obtained are consistent with the geometric average of the I?cs. TO check for the significance of the Eim’s
Table 6. Geometric mean values of the efficiencies over all runs made on the packed distillation column and packed absorber Distillation column f=6,N+l=
Component Carbon dioxi de Nitrogen Methane Ethane Propane i-Butane n-Butane i-Pentane n-Pentane Hexane Heptane Octane Nonane Decane
12 Modification (1) &n 1%l43 0.9869 1.0216 1.1693 1.1972 1.1908 1.2169 1*OS04 1.0754 1.0099 1+l448 8, = 1.1077 0,~ = 0.9028
Absorber
f=6,N+l= 12 Modification (2) &It
1wjO2 1.0023 I.0347 1.1084 1.0886 I.1256 I*0128 0.9834 1.0623 0.9899 1.0332 w1 = 0.5
646
N=8 Modification (I) &I
N=8 Modification (2) &I
2.2551 0.0435 14671 0.9256 0.7411 0.9323 1.0220 0.9016 0.9295 0.5302 0.4405 0.6342 0.5529 0.4046
2.1297 0.0528 1.3792 0.8875 0.7476 0.9534 I.0506 0.9585 1+lOso 0.6623 0.4858 0.7003 0.6108 04468
/3, = 1.0854 & = 0.9213
w, = 0.5
Modeling of packed distillation columns- III
(averaged over all runs R) in the reduction of & the product distributions for each run were recomputed by use of the &,,,‘s and at the optimum solution set {f,N} found in step one. These product distributions were then used in the re-evaluation of & and the result obtained was 0,(6,11,
{&n),
{&J)
= 0.1618
where p1 = l-1077, p N+1= 09028, and {Er,} is given in Table 6. The minimum value obtained for perfect mass transfer sections in step one was O,(6,11) {Ej, = 1)) = O-3459. Since the set of Ei,‘s gave approximately a 50 per cent reduction in bI, it is evident that the set of mean &,‘s and pjm’s over all runs R provided a significant improvement in the calculated values of the product distributions over those predicted on the basis of perfect plates. Similar results were obtained over all runs R for the absorber; namely, O,(N, &J,
I&m]) = O-3316
where N = 8 and &, PN, and the {Ei,} are given in Table 6. The minimum value obtained for is, in step one for the absorber was O,(N, {Eji = 1)) = O-7834 for N = 8. Thus, the set of l?im’s (over all runs R made on the absorber) gave a significant reduction in the error of the product distributions. Next, a test for the possible effect of operating conditions on the &‘s for the packed distillation column was made in the following manner. After the runs made on the distillation column had been divided into three subsets in accordance with the reflux ratios employed, sets of geometric mean values of the &‘s and pJ’s were computed for each subset of runs. Product distributions were re-determined for each subset of runs on the basis of the corresponding set of efficiencies (@*mL ~Plm~PN+l,mI)~ and the product distri647
butions so obtained were used to evaluate 0, for each subset of runs. The results given in Table 7 show that no significant improvement in the calculated product distributions was achieved by using a different set of efficiencies for each of the three sets of runs. This result implies that over the range of refluxes investigated, the efficiencies were independent of reflux ratio. This same conclusion was reached by Bassyoni[3] upon making the F-test on the Ei’s. A comparison of the calculated and observed product distributions for the packed distillation column and packed absorber are presented in Figs. 9 and 10. The calculated product distributions are based on the mean Els and pi’s given in Table 6 under “Modification (I).” A comparison of the average deviation of the calculated from the experimental values for the terminal temperatures for the distillation column and the absorber are presented in Table 8. The Pjm’s for the terminal sections (j = 1, N + 1 for the distillation column and j = 1, N for the absorber) together with the set of {Ei,} for each of the two columns reduced the deviations obtained for perfect sections by factors ranging from about 2 to 6. Although the proposed modeling procedure may not yield the global minimum in & this Table 7. Values obtained for the objective function 0, for each of four sets of runs by use of the corresponding sets of &‘s and/.$‘s [determined by modification
(I)1 No. runs used Reflux No. runs to compute ratio ’ 0% ,,,I and
h/D
s’:t
@A3N+l.m1
a
-
17 5 6 6 Averaget
17 5 6 6 -
0.1618 0.3155 O-0280 0.1675 0.1618
0.8 l-2 l-4 -
tin the computation of the average of 6, for the three subsets of runs (at LJD =0*8, 1.2 and l-4), the value of 8, for each subset was weighted according to the number of runs in the subset.
R. MCDANIEL,
A. A. BASSYONI
and C. D. HOLLAND
1000 f=6 J*I= 12
/
;?
:
./
loo
___
IO
.
.yI
I’
/
IQ 8
Y
s
0-I
o-01
i-
1
0001
/ r
OoO? 001
C 101
IO
C
100
1000
Fig. 9. Comparison of the product distributions computed on the basis of the mean values/?, ~ and Pl,m (listed under modification (1) Table 6) with the experimental product distributions for the packed &stillation column.
Table 8. Average deviation of the calculated from the experimental values for the terminal temperatures for the distillation column and the absorber
Temp.
Distillation column Average deviationt Perfect sections Modification (1)
T, (Overhead vapor) TN+I(Reboller)
0.02695 040785
T, (Lean gas) TN(Rich oil)
040792 040129
tAverage deviation of temperatureT,(j=
Temp.
Absorber Average deviationt Perfect sections Modification (1)
l,N,N+
648
1) = f $1
0.01756 0.02944 -=I,
040917 040606
where T,is in OR.
Modeling of packed distillation columns - III
,.
N =8
:.::I I ..i’._
.‘.
,/ ‘.v
.
.;
../ I..‘,,<
.:;: t.
/
Fig. 10. Comparison of the product distributions computed on the basis of the mean values .8?,;,,and a,,,, (listed under modification (1) Table 6) with the experimental distributions for the packed absorber.
deficiency is counterbalanced many times by the very definite unimodal behavior exhibited by 0, for all each run and by d, for all runs where perfect mass transfer sections are employed as described in step one. Brosilow et a1.[4] proposed a method somewhat similar to the one described in step one except that instead of perfect mass transfer sections, an approximation of the mass transfer rate expression was employed. CONCLUSIONS
The proposed modeling procedure makes use of an objective function oI(f,N,{lZji = 1)) for the distillation column and b,(N,{E,, = 1)) for the absorber which appear to always exhibit unimodal behavior. A suitable method for averaging the &‘s and flj’s over each subset or over all runs is provided by the proposed modeling procedure. Also, the proposed procedure provides a method for testing for the dependency of
the &‘s upon the operating conditions. Furthermore, the proposed procedure involves the use of relatively few parameters, and these parameters appear to remain approximately constant over reasonable ranges of operating conditions. Finally, the proposed modeling procedure is obviously directly applicable to columns with plates. Acknowledgments-The support provided by the National Science Foundation and Texaco, Inc. is gratefully acknowledged. The assistance given by W. E. Vaughn, J. W. Thompson, J. D. Dyal, and J. P. Smith of Hunt Oil Co., and J. H. Galloway, M. F. Clegg, Jr., and J. L. Shanks of Humble Oil and Refining Company; and R. V. Randall and W. E. Muzacz of Esso Production Research Company is appreciated.
649
NOTATION ai
bi
B
factor that appears in Eq. (24) molal flow rate of component i in the bottom product total molal flow rate of the bottoms
R. MCDANIEL,
A. A. BASSYONI
and C. D. HOLLAND
To datum temperature total number of components T, temperature of mass transfer section j factor that appears in Eq. (24) molal heat capacities for pure compositive constants lying between 0 Wl,WZ ponent i in the liquid and vapor and 1 and having a sum of unity phases; used in the expressions for xji mole fraction of component i in the the corrected liquid and vapor vapor leaving the jth mass transfer enthalpies section molal flow rate of component i in the yji mole fraction of component i in the distillate vector leaving the jth mass transfer total molal flow rate of the distillate section vaporization efficiency for componGreek symbols ent i and mass transfer section j component efficiency for component i Pj mass transfer factor for the jth mass transfer section number of the feed-distribution plate E factor appearing in the corrected of feed mixing section enthalpies mass transfer functions evaluated at activity coefficient for component i in the experimental values of T, and the liquid leaving the jth mass TN+l *definedbyEqs.(ll)and(l2), transfer section respectively multiplier defined by Eq. (19) enthalpy of pure component i in the function defined below Eq. (25) liquid, B.t.u./lb mole function of operating conditions haventhalpy of pure component i in the ing a value of unity at total reflux vapor, B.t.u./lb mole (or recycle) value of K for component i at the temperature of the liquid leaving Subscripts the jth mass transfer section b base component number of packed sections plus the ca calculated value feed distribution plate; N+ 1 = experimental value ew total number of mass transfer seci component number tions + feed-distribution plate + mass transfer section number j reboiler (for a distillation column m mean value with a total condenser). For the r run number absorber, N = total number of mass transfer sections Mathematical symbols objective function; defined by Eq. sum over all component from i = 1 (13) through i = c objective function for any number of absolute value of the argument x runs R; defined by Eq. (20). objective functions; defined by Eqs. set of all l?f’S; El. E-2. . . E, (30), (3 I), and (32), respectively product of all elements from x1 run number total number of runs through & 7
REFERENCES [l] AMUNDSON N. R. and PONTINEN A. J., Ind. Engng Chem. 1958 50 730. [2] BASSYONI A. A., MCDANIEL R. and HOLLAND C. D., Chem. Engng Sci. 1970 25 437. [3] BASSYONI A. A., Ph.D. Dissertation, Texas A&M University, College Station,Texas 1969.
650
Modeling of packed distillation columns - 111 [4] BROSILOW
C., TANNER R. and TUREFF H., Optimization of Staged Counter Current’Processes, presented at the 63rd Nat. Mtg. of the Am. Inst. Chem. Engrs, St. Louis, Missouri, February 1968. [5] DAVIS P., TAYLOR D. L. and HOLLAND C. D.,A.I.Ch.E.JI 1965 11678. [6] GREENSTADT J., BARD Y. and MORSE B., Ind. Engng Chem. 1968 50 1644. [7] HANSON D. N., DUFFIN J. H. and SOMERVILLE G. F., Computation of Multistage Separation Processes. Reinhold 1962. [8] HARRIS T. R., M.S. Thesis, Texas A&M University, College Station, Texas 1962. [9] HOLLAND C. D., Multicomponent Distillation. Prentice-Hall 1963. [lo] HOLLAND C. D., Unsteady State Processes with Applications in Multicomponent Distillation, Prentice-Hall, Englewood Cliffs, N.J. 1966. [l l] HOLLAND C. D., McMAHON K. S., Chem. Engng Sci. 1970 25 43 I. [12] HOOKE R. and JEEVES T. A., J. Ass. comput. Mach. 19618 2. [13] MCDANIEL R., M.S. Thesis, Texas A&M University, College.Station, Texas 1968. [14] MCDANIEL R., Ph.D. Dissertation, Texas A&M University, College Station, Texas 1969. [15] MCDANIEL R., RUBAC R. E. and HOLLAND C. D., Packed Distillation Columns and Absorbers at Steady Operation, presented at the 63rd Nat. Mtg. of the Am. Inst. Chem. Engrs., St. Louis, Missouri, February 1968. [16] NARTKERT. A., SRYGLEY J. M. and HOLLAND C. D., Can.J. them. Engng 196644217. [ 171 PETERS W. A., Jr., Ind. Engng Chem. 1922 14 476. [ 181 RUBAC R. E., Ph.D. Dissertation, Texas A&M University, College Station, Texas 1968. 1191 RUBAC R. E., MCDANIEL R. and HOLLAND C. D.,A.I.Ch.E.JI 1969 15 568. 120] SRYGLEY J. M. and HOLLAND C. D.,A.I.Ch.E.Jf 1965 11695. 1211 TAYLOR D. L.. DAVIS P. and HOLLAND C. D..A.I.Ch.E.JI 1964 10 864. i22j WILDE D. J., Optimum Seeking Methods. Prentice-Hall 1964. [23] WINN F. W., Petrol. Re$ner 1958 37 216. Resume- On demontre I’utilisation des resultats dune serie d’essais de service en vue de I’ttablissement d’un mod&Ii de colonne de distillation gamie et d’un absorbeur gami. Les mdthodes proposees utilisent lesdonntes communement disponibles pour les colonnes existantes. Les parambtres ddterminis d’apres les colonnes existantes. Les parametres determines d’apres les essais de service comprennent le nombre de plateaux ou de sections de transfert de masse, la position du plateau d’entree dans la colonne de distillation, un facteur consistuant Ei, pour chaque composant de tous les plateaux et de toutes les series de production et deux facteurs de plateaux pour toutes les series ainsi qu’il est dicrit dans la modification (1) de la procedure pour la determination des rendements de vaporisation. Zusammenfassung-Die Anwendung der Ergebnisse einer Reihe von Gelandeversuchen in der mathematischen Darstellung einer Fiillkorperkolonne und eines Filllkiirperabsorbers wird vorgefiihrt. Die vorgeschlagenen Methoden mathematischer Darstellung machen Gebrauch von Messwerten, die gewuhnlich fur bestende Kolonnen erhaltlich sind. Die in den Geliindeversuchen ermittelten Parameter bestanden aus der Zahl der Boden oder $toffaustauschabteilungen, der Lage des Aufgabebodens der Kolonne, einem Komponenten-faktor E,.,” fur jede Komponente fur alIe Bidden und alle Versuche, und zwei Bodenfaktoren fur alle Versuche, wie beschrieben unter Anderung (i) der Verfahrens fiir die Bestimmung von Verdampfungswirkungsgraden.
651