190
Ind. Eng. Chem. Process Des. Dev. 1980, 79, 190-195
Gas Absorption by Newtonian and Non-Newtonian Liquids in a Bubble Column Morio Nakanoh and Fumitake Yoshida" Chemical Engineering Department, Kyoto University, Kyoto 606,Japan
On the basis of experimental data for oxygen absorption into two Newtonian and two nowNewtonian liquids in a bubble column, a dimensionless equation for k,a is proposed. In addition to common liquid properties and operating variables, the equation includes the Deborah number, which represents the viscoelastic behavior of non-Newtonian liquids.
Introduction The bubble column is widely used for gas-liquid reactions and aerobic fermentations because of its simplicity of construction and superior performance. Although polymer solutions and fermentation broths often behave as non-Newtonian liquids, validity of existing correlations for mass transfer rates in bubble columns is limited to Newtonian liquids. The present work was intended to obtain a correlation for the volumetric liquid-phase mass transfer coefficient hLa in non-Newtonian liquids in bubble columns analogous to the correlation of Yagi and Yoshida (1975) for kLa in non-Newtonian liquids in sparged mechanically agitated vessels. Data on oxygen absorption and gas holdups in non-Newtonian as well as Newtonian liquids in a bubble column were obtained. Experimental Section Newtonian liquids used were water and aqueous sucrose solutions. Aqueous solutions of sodium carboxylmethyl cellulose (CMC) and of sodium polyacrylate (PA) of various concentrations were used as non-Newtonian liquids. Figure 1 is a schematic diagram of the experimental setup. The bubble column, 14.55 cm in inside diameter and 190 cm in height, was made of transparent polyvinyl chloride resin. A single-orifice gas sparger, with a 0.4 cm hole diameter, was installed at the bottom of the column. The clear liquid height was 110 cm in all the runs. An oxygen electrode, connected to a Beckman-Toshiba Type 777 oxygen analyzer, was inserted horizontally through the column wall a t a height of 80 cm from the bottom. One thermometer and two pressure taps for gas holdup determination were also attached to the column wall. Temperature was controlled at 30 f 1 "C by recirculating water from a thermostatted reservoir through the polyvinyl chloride tubing wound around the column. The whole column was heat-insulated with glass wool. Bubble size distribution at a superficial gas velocity of 100 m / h (2.78 cm/s) was determined with water and 0.05% PA solution by means of a photographic technique similar to that used by Akita and Yoshida (1974). To prevent distortion of pictures, part of the column was enclosed in a water-tight transparent acrylic resin box with a square cross section, with water filled in the space between the column and the box. All the experiments were carried out batchwise with respect to liquids. Air from an oil-free compressor (Hitachi OBT-5T) was saturated with water by bubbling through water, temperature-controlled at 30 "C, and metered with
* Correspondence should be addressed to F. Yoshida, Seisan Kaihatsu Kagaku Kenkyusho, Shimogamo, Kyoto 606, Japan 0019-7882/80/1119-0190$01 .OO/O
a calibrated orifice, before it was supplied to the column. The fractional gas holdup, tG, defined here with respect to the clear liquid volume, i.e., the liquid volume excluding bubbles, was determined by measuring Az, the difference in the liquid heights in the two manometer tubes connected to the two pressure taps located a vertical distance z apart. Thus f G = Az/(z - AZ) (1) The volumetric coefficient of liquid-phase mass transfer kLa used in this work is also defined with respect to the clear liquid. Prior to an experiment for kLa determination, the liquid in the column was sparged with a sufficient amount of nitrogen from a cylinder until the concentration of dissolved oxygen in the liquid became negligible. Then, air supply to the column was started. The dissolved oxygen concentration C increased with time t . One run took 40 s to 8 min, depending on kLa values. From an oxygen balance dC/dt = kLa(C* - C j (2) where C* is the value of C in equilibrium with the average oxygen partial pressure in the air and can be obtained from the reading of the oxygen meter E , when it reaches a constant value after sufficient aeration. Upon integration of eq 2 one obtains In (C* - C) = -kLat + constant (3) Since C was proportional to E , kLa was determined from the slope of the In E vs. t plot for each run. This method of evaluating kLa was based on three assumptions: (1) The liquid phase was perfectly mixed. (2) The driving potential for mass transfer was uniform throughout the column. (3) Response of the oxygen electrode to a change of the dissolved oxygen concentration was sufficiently fast. Validity of these assumptions was confirmed by experiments or calculation. Values of the dissolved oxygen concentrations in 0.1 % PA solutions measured under operating conditions with electrodes at 60 and 100 cm from the column bottom agreed within experimental errors, indicating that assumption (1)was valid. Thus, an electrode was installed a t 80 cm from the column bottom in most runs. In case assumption (1)holds, the validity of assumption ( 2 ) should depend on the degree of back-mixing in the gas phase. However, in view of the small difference between the oxygen partial pressures in the gas entering and leaving the column, which could be estimated by calculation, we assumed that the driving potential for oxygen transfer was practically uniform throughout the column, regardless of the degree of back-mixing in the gas phase. C 1979 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980
191
Table I. Properties of Liquids Used (30 "C)
D~ x
I-
'4 5
105
x
pL,
u,
solutions
cm'is
g/cm3
pis'
cm2/s
water sucrose 10% 30% 50%
2.60
0.995
71
0.804
2.26 1.66 1.06
1.03 1.13 1.23
71 71 71
1.05 2.25 8.82
V L
1 0 2 ,
A. s
CMC 0.3% 0.5%
1.0%
2.6 2.6 2.5
1.00 1.00 1.01
71 71 68.4
0 0 0
2.6 2.6 2.6
0.998 0.998 0.999
71 64.5 64.4
0.16 1.5 4.6
PA
0.01% U
Figure 1. Experimental apparatus: 1,bubble column; 2, enclosure for photographing; 3, thermometer; 4, manometers; 5, oxygen electrode; 6, oxygen analyzer; 7, recorder; 8, nitrogen cylinder; 9, orifice; 10, heater; 11, saturator; 12, air compressor.
The time constant of the particular oxygen electrode used was determined by response experiments, i.e., first immersing the electrode in a liquid in a vessel sparged with nitrogen and then dipping it quickly in the liquid in the column which had been sparged with air for a sufficiently long time. The time constant varied slightly with liquid properties and the superficial gas velocity, indicating that the resistance of the liquid film on the surface of the electrode membrane was not negligible. Its values were 5 to 6 s for water and 9 to 10 s for 60% sucrose solution. The following theoretical equation (Nakanoh, 1976) gives the relationship between time t and the dissolved oxygen concentration C' observed by an electrode with a time constant T . In (C* - C') = In [e-kL,ot- kLaTe-"ir]+ In [C*/(l - k ~ a T ) l(4) Curves representing eq 4, with values of the time constant T such as given above, are practically parallel to the straight line representing eq 4 with T = 0, Le., eq 3, except for the initial portion, unless the kLavalue is large, say over 0.1 s-l. Since in all the runs in the present work kLa was less than 0.1 s-l, we were able t o evaluate kLa from the slope of the straight portion of the curve without worrying about the time constant. If kLa is greater than 0.1 SKI, eq 4 must be solved for kLa with known values of C', t , and
T. Properties of Liquids Relevant physical properties at 30 "C of the liquids used are listed in Tables I and 11. Surface tensions of water and sucrose solution were available in the literature, and those of CMC and PA solutions were measured with use of a du Nouy-type instrument. In using the instrument, the possible error due to elasticity of the sample was prevented by pausing for a while, say several seconds, each time the tension of the wire was increased very minutely. After such a pause, which was always longer than the relaxation time, elasticity of the sample should have disappeared. It took 10 to 15 min to determine surface tension of a sample. Diffusivities of oxygen in sucrose and CMC solutions were determined by Akita (1978). Those in PA solutions were regarded as equal to the oxygen diffusivity in water for the reason mentioned by Yagi and Yoshida (1975). Viscosities of sucrose solutions were measured with use of a capillary viscometer, and those of CMC and PA so-
0.05% 0.1%
Table 11. Apparent Viscosities of CMC a n d PA Solutions a t 30 "C (cP) shear rate, s - ' CMC 0.3% 0.5% 1.0% PA 0.01% 0.05% 0.1%
40
80
150
300
600
10.3 15.3 64
10.2 15.1 60
10.0 15.0 54
9.7 15.0 48
9.5 15.0 41
5.8 22 41
4.7 16 29
3.9 12 21
3.1 8.8 15
2.5 6.4 11
lutions were determined using a coaxial rotating cylinder viscometer. Both CMC and PA solutions are pseudoplastic, the former only slightly, and follow the power law model. The apparent viscosity ka,often adopted in correlating data with such power law model liquids, is defined by pa = K y - 1 (5) where 9 is the shear rate. Values of the apparent viscosities of CMC and PA solutions a t various shear rates are given in Table 11. The relaxation time (characteristic material time) X for the PA solutions used were determined by the method of Prest et al. (1970), as described by Yagi and Yoshida (19751, and are given in Table I. Elbirli and Shaw (1978) made an analysis of the possible error in the values of X obtained from shear-viscosity data. They conclude that data a t low shear rates are critical in obtaining accurate values of A. However, it is unlikely that possible errors in the values of X we obtained are so serious as to affect the type of the mass transfer correlation which will be proposed later. Values of X for CMC solutions were regarded as practically zero for the concentrations covered in the present work. It might be mentioned that rheological properties of solutions of polymers such as PA and CMC are not exactly reproducible, since they vary somewhat with degrees of polymerization and degradation. Bases for Correlations Akita and Yoshida (1973, 1974) proposed the following empirical dimensionless equations for the gas holdup 70, the volumetric liquid-phase mass transfer coefficient kLB, and the volume-surface mean bubble diameter d,, in bubble columns based on their data with various Newtonian liquids. ? G / (1 - ? G ) ~= ~ . ~ ~ N B ~ ~ / ~ N G (6)~ ~ / ~ ~ N
kLBD2/ DL = 0.60Ns,0'5N~,0'62N~80.31?~1'1 (7) d,, = 26NBo4.50N 4.lZN -0.12 ( 8) Ga Fr
192
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980
where
t water
NBo= Bond number = gD2pL/a
(9)
NGa = Galilei number = gD3/vL2
(10)
NFr= Froude number = UG/(gD)'l2 Nsc = Schmidt number = vL/DL
(11)
0
(12)
Equation 6 holds when UG is lower than 1000 m/h, and the constant 0.20 should be replaced by 0.25 for electrolyte solutions. In eq 6 and 7, ZG and kL6 are defined with respect to the aerated liquid volume, whereas EG and kLa used in the present work are defined with respect to the clear liquid volume. Distinction between the two different bases for defining the gas holdup and the volumetric mass transfer coefficient should be noted. Thus, by definition ?G = t G / ( 1 + ' G ) (13) and kLa = kL6(l
+ tG)
(14)
With the relation of eq 13 the left-hand side of eq 6 becomes ? G / ( 1 - ?G)4 = f G ( 1 + f G ) 3 (15) The specific gas-liquid interfacial area, d or a, depending on whether it is based on the aerated or clear liquid volume, could be estimated, using the following well-known relationship d = 6(Z~/d,,) (16a) a = 6(%/d,,)
(16b)
Perez and Sandal1 (1974) proposed an equation for kLa in non-Newtonian liquids in sparged agitated vessels using the apparent viscosity. Their study did not involve viscoelastic liquids. Yagi and Yoshida (1975) proposed an empirical dimensionless equation for kLa in Newtonian and non-Newtonian liquids in sparged agitated vessels, in which they introduced a term including the Deborah number ND, to take care of the behavior of viscoelastic liquids. They defined the Deborah number as the product of the relaxation time and the rotational impeller speed. Ranade and Ulbrecht (1978) proposed an empirical equation for k ~ a in non-Newtonian liquids in sparged agitated vessels in which they also adopted the Deborah number defined in the same way. Baykara and Ulbrecht (1978) observed that kLa in a bubble column decreased substantially with addition of a very small amount of certain polymers possibly with nonzero relaxation time, which fact could not be explained by viscosity change alone. Nishikawa et al. (1977) estimated the average shear rates in aerated nowNewtonian liquids in a bubble column, 15 cm in diameter, from their data for the heat transfer coefficients on the column wall and on the surface of a coil immersed in the core section of the column. First, they estimated apparent viscosities of non-Newtonian liquids by correlating non-Newtonian heat transfer data by an equation for the heat transfer coefficient in Newtonian liquids. Then, they calculated the average shear rate by the equation +a" = (pa/ml/(n-l) (17) which was obtained from eq 5 and defined the average shear rate. They correlated the average shear rate with the superficial gas velocity by the following dimensional equation qav= 5 0 U ~ (18) where qavis in s-l and UGin cm s-l. At
UG
30 % sucrose
50 Yo sucrose
higher than
,
004 ,'-1'-,;A
,
A
I 0 % CMC 001 % ,PA
0 05 PA 0 10 % PA 004
002002
006
010
020
1 0.30
E G pred
Figure 2. Gas holdup data compared with predictions by eq 6 and 15. Table 111. Bubble Size Distribution Data (U,= 2.78 cm/s) mean diam, range, d, mm mm
0-1 1-2 2- 3 3-4 4- 5 5-6 6-8 8-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45
0.5 1.5 2.5 3.5 4.5 5.5 7.0 9.0 12.5 17.5 22.5 27.5 32.5 37.5 42.5 total
d,= I:N d 3 / xN d 2 d, by eq 8
water
NixN
N 792 49 5 245 84 23 5 1 1
0.05% PA s o h
N
NIXN
0.481 86496 2185 0.301 0.149 716 0.0510 238 0.0140 88 3.04 x 10-3 31 6.07 x 10.' 27 6.07 X 18 18 7 8.5 1 1.5 3 0.5 2 1647 1.00 89839.5 6.33 m m 5.78 m m
0.963 0.0243 7.97 x 10-3 2.65 x 9.80 x 10-4
3.45 x 1 0 . ~ 3.01 x 2.00 X 2.00 x 1 0 . ~
1.00
9.75 m m 10.5 m m
4 cm/s, eq 18 holds for the shear rates near the column wall as well as in the core section. A t lower gas velocities, the local average shear rate in the core section was higher than that given by eq 18, and the average shear rate near the column wall was lower than by eq 18. However, even at superficial gas velocities lower than 4 cm/s, the overall average shear rate for the entire column section could be approximated by eq 18. Results a n d Discussions Gas Holdup. Figure 2 compares experimental data for the gas holdups in water, sucrose solutions, CMC solutions, and PA solutions with prediction by eq 6. For non-Newtonian liquids the apparent viscosities pawere used in place of p. Except for the data for CMC solutions, observed and predicted values of the gas holdup agree within 30%. All the data points for CMC solutions lie above the 45 "C line in Figure 2 for an unknown reason. Thus, eq 6 seems to hold roughly for non-Newtonian liquids, if pa is used in place of p . Bubble Size. Photographs revealed that bubble size distributions in viscoelastic solutions were quite different from those in inelastic liquids. Table I11 gives the results of bubble size measurements with water and 0.05% PA solution, both at 30 "C and for a superficial gas velocity of 2.74 cm/s (100 m/h). In water most bubbles were smaller than a few millimeters, whereas in the PA solution relatively large bubbles mingled with a great number of
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980 193 I0 '
0.I
t
water
0 IO 'lo sucrose 0
30 %sucrose 50 % sucrose
v
0.01
t
0 30 % sucrose
50% s u c r o s e
I0
3001,
UE
2
10'1 N;cs
Figure 3. kLa in water and sucrose solutions.
N;oz
I 0' No39
Go
very fine bubbles less than 1mm in diameter. A mechanistic explanation of the bubble size and its distribution in viscoelastic liquids was given by Ranade and Ulbrecht (1978). Table I11 also gives the volume-surface mean diameter d,, calculated from the experimental data and the d,, predicted by eq 8 with use of y, in place of p. It is seen that eq 8 based on data with Newtonian liquids can predict d,, in non-Newtonian liquids fairly well. k L a in Inelastic Liquids. Figure 3 shows the kLa values in water and sucrose solutions plotted on log-log coordinates against the superficial gas velocity UG. The slope of the straight lines is unity at least for the range of UGstudied, i.e., below 10 cm/s. The reason why the line for water is very slightly below the line for 10% sucrose solutions is unknown. These kLa data agree approximately with those predicted by eq 6, 7, 14,and 15. Predictions of kLa by these equations involves estimation of the gas holdup. It would be more practical if kLa could be correlated directly with the superficial gas velocity or the Froude number. It was difficult in the work of Akita and Yoshida (1973), since they defined kLti and ?G with respect to the aerated liquid volume, which is a function of UG. From eq 7,13, and 14,one obtains
kLaD2
--
I
I os
crn/sec
- 0.60Ns,0.5NBo0.62NGa0.31 CG 1 1(1 tG)4'.1 (19) DL For the range of t G < 0.15, it can be shown that tG1'l(l + 6G)-01 N 0 . 6 t ~ ( 1 + tG)3 (20) '
From eq 6, 15, 19, and 20 we get
The present data for kLa in water and solutions of sucrose and CMC are shown in Figure 4, in which k L a D 2 / D Lare plotted against N~0'5NBo0'75NGa039NF~1'0 in accordance with eq 21. The straight line in the figure represents the following equation with a slightly larger value of the constant (0.09) than in eq 21.
"
0
rr
Figure 4. kLa in water, sucrose solutions, and CMC solutions, correlation in accordance with eq 21. 1
106,
1'
0.01 % PA 0.05% PA 0.I O % PA
0
2
10'-
s t c
1
lo4.
I os
Ng:
'N ::
IO'
NZT NLP
Figure 5. kLa in PA solutions, comparison with prediction by eq 22.
tated vessel. Viscoelasticity of CMC solutions was negligible for the range of concentrations covered in the present work. After Yagi and Yoshida we introduced into eq 22 a term including the Deborah number to take care of the dependence of kLa on viscoelasticity. In correlating their data with a sparged agitated vessel, Yagi and Yoshida defined the Deborah number as the ratio of the relaxation (characteristic material) time X to the reciprocal of the rotational impeller speed N which could be regarded as the characteristic process time. Our problem was how to define the characteristic process time for an operating bubble column. We tentatively adopted the time required for a bubble of an average diameter to rise the vertical distance equal to the average diameter. The vertical rise velocity UB was calculated from the superficial gas velocity lJG and the gas holdup tG, assuming the following relationship.
UB = U G ( +~ ~ G ) / € G Then, the characteristic process time = dvs/UB
7
(23) could be given by (24)
and the Deborah number could be defined as
k L ain Viscoelastic Liquids. Figure 5 compares the kLa values for PA solutions with those predicted by eq 22 using pa estimated by eq 17 and 18. The kLa values for PA solutions are considerably lower than predicted by eq 22. Apparently this is due to the viscoelasticity of PA solutions, as with the data of Yagi and Yoshida (1975) for kLa in viscoelastic liquids in a sparged mechanically agi-
N,, = X / 7
= UBX/dv,
(25)
Such a definition of the Deborah number is arbitrary. Some people may call the group defined by eq 25 the Weissenberg number. Although the definitions of the Deborah and Weissenberg numbers are not identical, the distinction between the two does not seem to be very clear in some cases.
194
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980
Table IV. Data for PA Solutions concn,
UB,
UG,
cmls
%
cm/s
d,,, cm
kLarE/ kLavE
0.01
1.57 3.56 5.64 8.39
43.3 45.7 50.4 57.1
0.931 0.812 0.737 0.681
1.45 1.36 1.55 1.63
0.05
2.06 4.22 8.56 1.94 3.22 5.53 8.44
52.4 55.7 64.4 65.5 73.9 77.5 79.6
1.13 0.962 0.821 1.31 1.16 1.02 0.924
2.48 2.26 2.67 3.21 3.94 4.46 4.76
0.01
N,,
A 0
7.44 9.00 10.9 13.4
0
69.5 86.9 118 229 292 349 396
1 0 */e CMC 001% PA 0 0 5 % PA
I
j
+ water 0
10 %sucrose
0
30 %sucrose 50 %sucrose
i I
04 b 5
I I0 '
I06
Ng:
Ni075
$7 NFPi 1
i
0 13N:y)
'
Figure 7. Generalized correlation for kLa in Newtonian and nonNewtonian liquids.
- I
IO
IOz
IO'
N D.
Figure 6. kLa in PA solutions, effect of viscoelasticity.
Table IV lists the values of UB, d,,, and ND,, as well as the ratio of the kLa values predicted by eq 22 for inelastic liquids (designated as kLaIE) to the kLa values observed with viscoelastic PA solutions of various concentrations (kLuvE) a t various superficial gas velocities. The ratio of kLaIEto kLaWis seen to increase with increasing Deborah number. Noting that the kLa in inelastic liquids, for which the Deborah number is zero, can be correlated by eq 22, we assumed a relationship of the following type. kLaE/kLavE = f(NDe)= 1 + cNDem
(26)
If this relationship holds, a log-log plot of [f(ND,) - 11 against ND,should give a straight line with a slope of m, as shown in Figure 6. Thus, the following dimensionless equation for kLa in Newtonian and non-Newtonian liquids in bubble columns is proposed on the basis of our data with two Newtonian and two non-Newtonaian, including one viscoelastic, liquids.
~ L U D / D L= 0.09NSc0'5NBo0.75NG~3gNF~1'o( 1 + CNDe m)-l (27)
The values of c and m obtained from the present data are 0.13 and 0.55, respectively. For inelastic liquids, the term in the parentheses reduces to unity. Equation 27 is based on the data for Uc lower than 10 cm/s. Figure 7 shows that all of the kLa data with Newtonian and non-Newtonian liquids used in the present work agree within 40% with the prediction by eq 27 with the values of c and m given above. Exact values of c and m should be determined in the future, when more extensive data for various viscoelastic liquids become available. It is even conceivable that the value (-1) of the exponent on the term in the parentheses in eq 27 might be replaced by another value. If the gas holdup and the volume-surface mean bubble diameter in bubble columns are not affected by viscoelasticity of liquids as stated before, the relationship of eq 6 requires that the specific interfacial area a also be independent of viscoelasticity. Then decrease in kLa with
increasing viscoelasticity should mean decrease in kL rather than a. We have demonstrated that in viscoelastic liquids large bubbles mingle with very fine bubbles, whereas bubbles in inelastic liquids are relatively uniform in size. Since a kL value should be an average value over the whole interfacial area by definition, a decrease in k L values in a viscoelastic liquid could be explained by the relative ineffectiveness for mass transfer of the surfaces of very fine bubbles, which would move very slowly relative to the liquid. In other words, the effective interfacial areas in viscoelastic liquids seem to be smaller than those in inelastic liquids. Conclusions Equation 27 is tentatively proposed for the volumetric liquid phase mass transfer coefficient kLa in Newtonian and non-Newtonian liquids in bubble columns. In addition to common liquid properties, the equation includes a term which takes care of viscoelasticity of liquids. More extensive data with various viscoelastic liquids are desired to test general usefulness of eq 27.
Acknowledgment The authors are grateful to Drs. K. Akita and H. Yagi for helpful discussions and comments. Dr. Akita supplied us with some diffusivity data he obtained. Nomenclature a = specific gas-liquid interfacial area based on clear liquid volume, cm2/cm3,or cm-' ii = specific gas-liquid interfacial area based on aerated liquid volume, cm-' C = dissolved oxygen concentration, cm3 cm-3 C* = C in equilibrium with oxygen partial pressure c = constant D = column diameter, cm DL = liquid phase diffusivity, cm2 s-l d,, = volume-surface mean bubble diameter, cm f = function g = gravitational constant, cm s-' K = fluid consistency index, g cm-' sn-* m = exponent N = number of bubbles per unit clear liquid volume, cm-3 NBo= Bond number, = g D2 p ~ / c dimensionless , NDe= Deborah number = uBx/d,,, dimensionless NFr= Froude number = UG/(gD)"', dimensionless NGa = Galilei number = g D3/vL2,dimensionless n = flow behavior index, dimensionless T = time constant of oxygen electrode, s t = time, s U , = superficial gas velocity with respect to cross section of column, cm s-l UB = bubble rise velocity defined by eq 21, cm s-' z = height of liquid in manometer tube, cm
Ind. Eng. Chem, Process Des. D e v . 1980, 19, 195-197
Greek Letters shear rate, SS’ = average shear rate, eG = gas holdup, fraction of clear liquid volume, dimensionless G : = gas holdup, fraction of aerated liquid volume, dimensionless X = characteristic material time, s p = liquid viscosity, g c m - l s ~ ~ p a = apparent viscosity, g cm-’ s-l vL = kinematic viscosity of liquid = p / p L , cm2 s-l pL = liquid density, g u = surface tension, g T = characteristic process time = d v s / U B ,s $ =
Subscripts IE = inelastic
195
VE = viscoelastic
Literature Cited Akita, K., Yoshida, F., Ind. Eng. Chem. Process Des. Dev, 12, 76 (1973). Akita, K., Yoshida. F., Ind. Eng. Chem. Process Des. Dev, 13, 84 (1974). Akita, K., private communication, Tokushima University, 1978. Baykara, 2. S.,Ulbrecht, J., Biotech. Bioeng., 20, 287 (1978). Elbirli. B., Shaw, M. T., J . Rheo/ogy, 2 2 , 561 (1978). Nakanoh, M., M.Eng. Thesis, Kyoto University, 1976. Nishikawa, M., Kato, H., Hashimoto, K., Ind. Eng. Chem. Process Des. Dev., 16, 133 (1977). Perez, J. F., Sandall, 0. C..A I C h E J . . 20, 770 (1974). Prest, W. M., Porter, R. S.,O’Reilly, J. M., J . Appl. Po/ym. Sci., 14, 2697 (1970). Ranade, V. R., Ulbrecht, J. J., A I C h E J . , 24, 796 (1978). Yagi, H., Yoshida, F., Ind. Eng. Chem. Process Des. D e v . , 14, 488 (1975).
Recezued f o r reuiew August 25, 1978 Accepted October 10, 1979
COMMUNICATIONS Optimization of a Nonlinear System Consisting of Weakly Connected Subsystems Optimization of a very large complex system described by nonlinear equations is often beyond the capability of presently available procedures if the system is treated in its entirety. Many such systems are found to contain “weakly connected” subsystems which may be considered independently. A revised approach which makes use of this special structure to substantially reduce the size of overall system optimization is proposed. Under some circumstances, this method reduces a complex nonlinear integer or mixed integer problem into an equivalent small dimension continuous variable optimization problem.
Introduction Optimization of nonlinear systems in which some or all of the decision variables are integers is generally difficult. When the number of integer variables considered is relatively small there are several methods which can provide satisfactory results. For example, the authors have found computer programs based on the adaptive random search procedure of Heuckroth et al. (1976) useful in moderate size problems. However, there does not appear to be any fully satisfactory procedure for optimization of very complex nonlinear systems with many continuous and integer variables. There are a number of practical mixed integer and integer optimization problems encountered in the process industries which are too large to be successfully treated when the system is considered as a whole. One area in which difficulties have been encountered has been in the optimization of process systems subject to stochastic reliability or availability constraints. Burdick et al. (1977) have pointed out that in many such problems the system may be decomposed into stochastically independent subsystems. By using a Monte Carlo simulation technique they were able to determine bounds for the subsystem behavior and these bounds could then be used in a simplified overall system optimization. It is the purpose of this communication to present an improved procedure for the establishment of subsystem characteristics where the subsystems are “weakly connected” and to show that the general approach is applicable to a wider class of problems than originally considered. Problem Formulation In their paper, Burdick et al. (1977) optimized the design 0019-7882/80/1119-0195$01.00/0
of the shutdown heat removal system (SHRS) of a liquid metal fast breeder reactor (LMFBR) minimizing cost subject to SHRS unavailability. They used eight integer variables in their nonlinear optimization problem. They further showed that a cost-unavailability relationship for the SHRS could be obtained by generating a Monte Carlo plot of SHRS cost vs. SHRS unavailability for randomly selected designs. A lower bounding curve was obtained for the plot. The curve provided an approximate functional SHRS cost for a given SHRS unavailability. The advantage of such an approach is that it reduces a complex nonlinear integer or mixed integer system optimization problem into a simple continuous problem. Determination of the bounding curves via Monte Carlo simulation can be very time consuming since a large number of configurations must be examined if reasonably accurate bounding curves are to be obtained. However, a simpler and more exact approach can be used to obtain the functional relationship between unavailability and SHRS cost. Solution of the problem minimize C( Q) (1) subject to Q 5 h, (2) locates one point on the bounding curve. Resolving the problem for a number of different k, delineates the entire bounding curve. The revised approach has been found to be a significant improvement over that initially proposed. The cloud of points in Figure 1 was obtained by computing costs and unavailability for all possible configurations of the SHRS. The solid curve shown in Figure 1 has been drawn through the results obtained from a series of @ 1979 American Chemical Society
