Ind. Eng. Chem. Process Des. Dev. 1084, 23, 566-573
566
Thermodynamics of Petroleum Mixtures Containing Heavy Hydrocarbons. 2. Flash and PVTCalculations with the SRK Equation of State Karen Schou Pedersen,+ Per Thomassen,$ and Aage Fredenslund'' Insfifuftef for Kemkeknik, Danmarks Tekniske H Ojskole, DK-2800 Lyngby, Denmark, and STA TOIL. Den Norske Sfafs O~eselskapa s , N-400 1 Sfavanger, Norway
The oil and gas mixtures described in part 1 of this series are subjected to three PVT experiments commonly performed on oil and gas mixtures: constant mass expansion, differential liberatbn, and constant volume depletion. In addition, P I T flashes at separator condltlons are determined experlmentally. These experimental results are simulated with the SRK equation of state plus various correlations for the liquld-phase molar volumes. It is found that the hydrocarbon characterization methods presented in part 1 give satisfactory simulations of the P,Tflash and PVT data. The most suitable liquidghase molar volume correlations are the Standing-Katz and Alani-Kennedy
correlations.
1. Introduction The first article in our series on phase equilibrium calculations on petroleum mixtures using the SRK equation of state (Pedersen et al., 1984) dealt with phase envelope calculations. It also presented the methods used for characterization of the heavy hydrocarbon fractions. The present part of the series reports the results of P,T flash calculations and simulations of three PVT experimenta commonly performed on oil and gas mixtures. The flash calculations are based on a computer program developed by Michelsen (1982).
2. PVT Experiments 2.1. Constant Mass Expansion. The principle is sketched in Figure 1. The reservoir fluid is kept in a cell a t reservoir conditions. The pressure is reduced in steps a t constant temperature and the change in volume is measured. The saturation point volume, Vsat,is used as standard value and the results are presented as relative volumes, i.e., total volume at a given pressure divided by Vmv For oil systems above the saturation point pressure, Pmt,this enables determination of the oil compressibility, co
where V is the volume at a pressure P > Peat. Below the saturation point, the results are expressed by means of the "Y factor"
where V , is the total gas and liquid volume. For gas condensate systems the compressibility factor, 2, of the gas phase is determined. In the two-phase region the relative liquid volume, Le., volume percent liquid of Vsat,is determined in addition. 2.2. Differential Liberation. This experiment is only carried out on heavy oils. The reservoir fluid is kept in a cell at saturation point pressure and reservoir temperature. The pressure is reduced in steps and all the liberated Institutted for Kemiteknik.
* STATOIL.
0196-4305/84/1123-0566$01.50/0
gas is displaced and flashed to standard conditions, i.e., 1 atm and 15 "C. This procedure, which is shown schematically in Figure 2, is repeated 6-10 times. The final pressure is 1atm. At the end the temperature of the cell is lowered to 15 "C. The resulting volume of the remaining oil, V,, is used as a normalizing factor, and the relative oil volumes or oil formation factors, Bo Bo = V / V , (3) are reported at the various pressure levels. V is the oil volume at the given pressure and cell temperature. Another interesting results is the so-called gas oil ratio, GOR, which is the volume measured at standard conditions of gas "dissolved" in the oil divided by V,. 2.3. Constant Volume Depletion. This test is used for gas condensates and volatile oils. The principle is shown in Figure 3. The reservoir fluid is kept in a cell at reservoir temperature and saturation point pressure. The pressure is reduced in steps, and at each level so much gas is removed that the volume of the remaining gas and oil mixture equals the saturation point volume. The released gas is eventually flashed to standard conditions. The mole fraction removed and the liquid volume are recorded at each pressure. If the last test point is at standard conditions it enables determination of oil formation factors and GORs. 2.4. Flash Separations. The flash separation data result from either offshore test separations or laboratory separations. In offshore test separations, high-pressure samples are collected of the gas and liquid phases in equilibrium. The gas composition is measured directly as explained by Pedersen et al. (1984). The liquid composition, however, cannot be determined before the highpressure liquid is flashed to standard conditions. The composition of the original liquid phase is calculated from the measured compositions and amounts of evolved gas and liquid phases. 2.5. Accuracy of the Measurements. It is difficult to ascertain the absolute accuracy of the PVT experiments. The analyses are complex, and many additional factors can influence the result. The discussion is therefore based on our experience of reproducibility of the results. 2.5.1. Constant Mass Expansion. In this experiment the total volume of the hydrocarbon system at constant temperature is determined as a function of pressure. This 0 1984 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984
_I n c_i p i e n t
f o r m a t i o n of g a s 1
T I I
I 5
I ~
Pi”Psat
9’Psat
“3=Psat
P4 0 100
200
\ ...... .....;\
......... S K
1.3-
1.2
-
1.1
-
\
300 Pressure. a t m
Figure 5. Graphical representation of constant mass expansion results for sample no. 14 at 103.5 'C.
I8
culated and experimental results, even at pressures near the saturation point. The Alani-Kennedy equation is seen to yield slightly better results than the other liquid density correlations. A constant mass expansion experiment has been performed on sample no. 13 at 93.3 "C. This is a heavy oil and a t the stated conditions the total volume is mainly governed by a dense liquid phase. The process has been simulated and the results, presented as total volumes divided by saturation point volumes, are given in the supplementary material. All the tested density correlations give reasonable results, but quite surprisingly the best results were obtained with the SRK equation. It should be made clear that satisfactory prediction of the liquid compressibility does not necessarily imply good predictions of the liquid density as the saturation point volume is used as normalizing factor. 4.3. Differential Liberation. A seven-stage differential liberation experiment has been carried out on sample no. 13 at 93.3 "C. This experiment has been simulated, and the results, comprising molar fractions removed at each stage, Compositions, and 2 factors of the liberated gas, relative volumes, oil densities, and gas oil ratios are given in the supplementary material. A summary of the results is presented in Table I11 and in Figures 6 and 7. Table I11 shows measured and calculated gas phase compositions after the lst, 5th, and 7th stages; Figure 6 measured and calculated relative volumes and Figure 7 oil densities.
I
400
300
200
0
100
Pressure, a t m
Figure 6. Differential liberation calculation for sample no. 13 at 93.3 "C. h
-
Experiment
---_ Peneloux _ .
1.0
...........
SK SRK AK I
/
> 0.9
/
Il 5
I 400
300
200
100
0
Pressure, a t m
Figure 7. Differential liberation calculation for sample no. 13 at 93.3 "C.
570
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984
Table IV. Constant Volume Depletion a t 121.0 "C of Sample No. 11. Composition (mol %) of Removed Gas After lst, 5th, and 7th Pressure Stages P = 313.1 atm P = 131.1 atm
=
54.4 atm
comp
exptl
calcd
% deva
exptl
calcd
% dev
exptl
calcd
% dev
N, CO,
0.78 9.33 71.33 8.39 4.81 0.72 1.24 0.41 0.41 0.45 0.56 0.53 0.28 0.76
0.65 9.29 71.01 8.49 4.90 0.73 1.27 0.42 0.44 0.47 0.59 0.58 0.30 0.86
-16.7 -0.4 -0.4 1.2 1.9 1.4 2.4
0.18 9.39 72.03 8.49 4.89 0.13 1.28 0.42 0.42 0.44 0.46 0.38 0.21 0.07
0.66 9.39 71.91 8.55 4.88 0.12 1.25 0.40 0.41 0.43 0.50 0.45 0.20 0.24
-15.4 0.0 -0.2 0 .I -0.2 -1.4 -2.3 -
0.16 9.28 10.75 8.67 5.28 0.84 0.52 0.55 0.57 0.59 0.58 0.37 0.13 0.12
0.66 9.40 71.68 8.63 4.99 0.74 1.29 0.42 0.43 0.45 0.52 0.45 0.18 0.02
-13.2 1.3 1.3 -0.5 -5.5 -11.9 148.1 -23.6 -24.6 -23.7 -10.3
c, C"
cl
i-C
n-C, i-C n-C, C6
c, c* c9
Clot a %
P
-
5.4 9.4
-
13.2
-
-
-
-
dev defined in Table I (only determined if the experimental mol % is above 0.5).
E
5 1 4.0
_ _ _ EP ea np eelroi mu xe n t ...... ..
7.0
S R K ( g a s ) + SK(liquid)
Ea 6.0
I
3
-
5.0
0
x 4.0
E 3.0
-30
0
20
D 2
1.0
100
20 0
300
~-
Pressure. a t m
I
100
200
300 Pressure, atm
Figure 8. Graphical representation of constant volume depletion results for sample no. 11 at 121 O C .
Comparison of the results of Figures 6 and 7 reveals that though the best results for the relative volumes are obtained with the SRK equation, this is not reflected in the prediction of the liquid densities for which better results are obtained with the Alani-Kennedy equation and the Standing-Katz method. The Peneloux and Lee-Kesler (not shown) liquid densities are too large and deviate more from the experimental results than do the SRK densities. 4.4. Constant Volume Depletion. Experimental results from a seven-stage constant volume depletion experiment are available for sample no. 11 of Pedersen et al. (1984) (a gas condensate). Measured and calculated results comprising amounts of gas removed at each stage, molar compositions, Z factors of the gas phase removed at each pressure level, and liquid volumes are given in the supplementary material. In Table IV are shown the molar compositions after the lst, 5th, and 7th pressure stages. Measured and predicted liquid volumes using the SRK equation, the method of Peneloux et al. (1982), and the Alani-Kennedy equation are shown for sample no. 11 in Figure 8 and for sample no. 14 in Figure 9. For sample no. 11, discrepancies between measured and calculated volumes are found close to saturation just as was the case with the constant mass expansion of sample no. 1. Again, the predicted results for sample no. 14 are in excellent agreement with the mea-
Figure 9. Graphical representation of constant volume depletion results for sample no. 14 at 103.5 "C.
sured ones. The best results are obtained with the Alani-Kennedy equation, the correlation of Standing and Katz (not shown), and the method of Peneloux et al. (1982). The SRK volumes are too large, and those of Lee and Kesler (not shown) are too small. 5. Discussion The results show overall satisfactory agreement between experimental and calculated values. The largest discrepancies arise for the liquid precipitation from the gas condensates no. 1and 11for which graphical representations are given in Figures 4 and 8. Though the liquid fraction constitutes only a few volume percent of the total mixture, the discrepancy between measured and calculated results may be somewhat large. The predicted liquid volumes increase too much with decreasing pressure just below the saturation point. Since no compositional data are available for these liquid phases it is at present not obvious which of the components cause these troubles. However, results from other calculations give some indication in this regard. The flash results for sample no. 1presented in Tables I and I1 showed that the calculations tend to overestimate the content of lighter components in the liquid phase. Another indication of this may be obtained from the composition of the gas removed during the constant volume depletion test (see Table IV). The measured methane content is larger than the calculated value, and hence the liquid phase contains less methane than predicted. Although the deviation for the gas phase is small, this may have a pronounced effect on
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984 571
Table V. Constants of the Alani-Kennedy Equation for Pure Hydrocarbons K n component 61.893223 91 60.6413 C , (70-300 O F ) 3247.4533 147.47333 C. 1301-460 O F ) -404.4884 4 46703.573 C i (100-249 'Fj 34.1635 5 1 17495.343 C , (250-460 O F ) 190.24420 20247.7 5 1 c 3 131.63171 32204.420 i-C, n-C, 33016.212 146.15445 299.62630 37 046.234 n-C n-C, 254.56097 5209 3.006 a
m x
lo4
3.3162472 -14.072637 5.15 209 81 2.8201736 2.1586448 3.3862284 2.9021257 2.1954185 3.6961858
C
0.50874303 1.8326695 0.52239654 0.6 23 09871 0.908 32 5 19 1.1013834 1.1168144 1.4 364289 1.5929406
Units: P in psia; Tin O R ; Vin ft3/lb-mol;R = 10.7335 1b-ft3/in., "R lb-mol.
Table VI. Standing-Katz Pure Component Densities at 1 atm and 15 "C component
density, g/cm3
H,S
0.7970 0.5072 0.5625 0.5836 0.6241 0.6305 0.6850
c 3
i-C, n-C, i-C , n-C, n-C,
the liquid phase because methane constitutes more than 70 mol ?% of the total mixture. The mixing rules recommended for the SRK equition ihvolve a binary interaction pararfieter. This parameter is in our work set to zero for interactions between two hydrocarbons. One way of controlling the calculated distribution of lighter components between the two phases might be to b e binary interaction parameters different from zero. No problems are encountered with the prediction of the liquid precipitation from sample no. 14. This is a gas condensate which has a lower content of aromatic compounds than samples no.'l and 11. Another difference is the composition of the +- fraction, for which estimated results are presented in the supplementthy material of this paper and in Pedersen et al. (1984). The heaviest carbon number fractions present in sample no. 14 with a mole fraction above 5-104 is C3@ The same criterion used on samples no. 1and 11leads to CW and Cd5as the heaviest carbon number fractions. This may be an artifact stemming from deficiencies in the characterization of the heaviest fractions. On the other hand, the good agreement between measured and calculated dew points found by Pedersen et al. using the same characterization procedure is an indication of a reasonable characterization of the heaviest components because the dew points are almost exclusively determined by these. If there are severe shortcomings in the characterization methods, deviations between experimental and calculated results must be expected to build up during the series of flashes involved in the differential liberation and constant volume depletion tests. The results of Tables I11 and N show that after the 5th step, where about half of the initial mixtures has been removed, the deviation between the measured and calculated compositions of the gas phase is still at the same low level as after the 1st step. Hereafter the deviations increase, but apparently not in any systematic manner. For the gas-phase compressibility factors, the best results are obtained with the Lee-Kesler and Peneloux methods while those determined from the SRK equation are somewhat too high (data are presented in the supplementary material). Five different methods for determination of liquid volumes have been tried. For gas condensates the best results are obtained with the method of Peneloux et al., the Alani-Kennedy equation, and the Standing-Katz
procedure. The SRK equation generally predicts too large liquid volumes and the essence of Peneloux's method is to determine how much the SRK volumes should be reduced. Apparently this works well for gas condensates, whereas for heavy oils the errors resulting from the SRK equation are highly overestimated. Figure 7 shows that for heavy oils the best determination of the liquid phase density is obtained with the Alani-Kennedy equation and the Standing-Katz method but both these methods have the drawback that they can only be used for the liquid phase. In those cases where it is essential to use the same procedure for both phases of a heavy oil, it is better to use the unmodified SRK equation than the method of Peneloux or that of Lee and Kesler. Finally, it is worth noting that the best prediction of the liquid compressibilitiesand the gas oil ratia has been achieved with the SRK equation. Sim and Daubert (1980) have evaluated procedures for predicting flash volumes of petroleum mixtures with heavy residua. They also obtained satisfactory flash volumes using the unmodified Soave procedure. However, the discrepancies between predicted and experimental values are, especially at low liquid flash volumes, somewhat larger than those reported in this work. This is most likely due to differences in the procedures used for characterizing the heavy residue. 6. Conclusion The hydrocarbon computational procedures presented by Pedersen et al. (1984), suitable for determination of dew and bubble points, have been extended to simulate P,T flashes and laboratory PVT experiments on gas condensates and heavy oils. Generally it is possible to predict the phase compositions with an accuracy which is close to what can be achieved experimentally, but there seems to be a tendency to somewhat overestimate the content of lighter components in the liquid phase. The most suitable methods for determination of liquid phase densities seem to be the Alani-Kennedy equation and the Standing-Katz method. If a consistent method is needed, the method of Peneloux et al. is recommended for gas condensates, while the unmodified SRK equation gives better results for heavy oils. Acknowledgment The authors are thankful to Arne Hole of STATOIL and Michael L. Michelsen of Instituttet for Kemiteknik for their continued interest in this work. Appendix The Alani-Kennedy equation (1960) has the following form
where R = gas constant, T = absolute temperature, P = pressure, and V = molar volume. For pure substances a = K@fT (A2)
572
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984
b=mT+C (A31 where K , n, m, and C are constants. Values for the lighter hydrocarbons are given in Table V. It is suggested to use the methane values also for N2, COP,and HzS. The values of a and b of a C7+ fraction are found from the following expressions In ac,+ = 3.8405985 X 10-3MW - 9.5638281 X 10-4MW/p + 2.6180818 X 102/T + 7.3104464 X 104MW + 10.753517 (A4)
+
b q + = 3.4992740 X lO-’MW - 7.2725403~ 2.2323950 X 10-4T - 1.6322572 X 10-2MW/p 6.2256545 (A5)
+
where MW is the modecular weight of the C,+ fraction, p is the density in g/cm3, and Tis the temperature in OR. a and b of mixtures are found as follows amix= E a & (A61 i bmix = Cbixi (-47) 1 where ai and bi are the pure component values and xi are molar fractions. Equation A1 is a cubic equation which may have one or three real roots. In case there are three real roots, the lowest root expresses the liquid volume. The Standing-Katz procedure involves the following steps (the densities are expressed in g/cm3). 1. (HzS C3+) Density. The density at standard conditions of the (HzS C3+) fraction is calculated as follows
+
+
The index i refers to H2S, C3and heavier components. The pure component densities can be taken from Table VI, whereas the C7+ density must be measured. 2. Correction for C2. The weight fraction Wz of Cz of the (HzS + C2+) fraction is determined. The density at standard conditions of the (H2S+ Cz+) fraction is then calculated from the equation p(H2S + Cz+) = p(H2S + C3+) - A0 - Aiai - AZa2 (A9) where A , = 0.3158Wz A1 = -0.2583w2 A2 = 0.01457 W2 a1 = 3.3 - 5.0p(HzS + C3+) U Z = 1 + 15(p(H,S + C3+) - 0.46) X (2.5p(H2S + C,+) - 2.15) 3. Correction for C02. The density of the (COz+ HzS + Cz+) fraction at standard conditions is calculated on additive volume basis using p(HzS + Cz+) and a COz density of 0.8215 g/cm3. 4. Correction for C1 and N2. The weight fraction Wl of (C, + NJ of the total mixture is calculated. The density po of the total mixture at standard conditions is then calculated from the equation PO = p(CO2 + H2S + C2) - Bo - Bib1 (A10) where Bo = 0.088255 - 0.095509b2 + 0.007403b3 - O.OO603b4 B1 = 0.142079 - 0.150175b2 + 0.006679b3 + O.OO1163b4 b, = p(C02 + H2S
+ Cz+)
- 0.65
bz = 1 -
low,
+ 3ow1(5w1 - 1) b4 = 1 - 60W1 + 75Owl2- 250OWl3 =1
b3
5. Pressure Correction. The density pp at the actual pressure P (in psia) and 60 “C is found from the equation pp = PO - Co
- ClCi - C2~2- C3~3
(All)
where Co = -0.034674
+ 0.026806~4+ 0.003705~5+ 0.000465C6 c1 = -0.022712 + 0.015148~4+ 0.004263~5+ O . O O O ~ ~ ~ C ~ cz = -0.007692 + 0.003521~,+ 0.002482~5+ 0.000397C6 C3 = -0.001261 - 0.000294~4+ 0.000941~5+ 0.000313~, c,=l-2-
P - 500 10000
P - 5 0 0 P - 500 ~2=1+6---10000 10000
(
c3 = 1 - 12p-500 10000
-y
+ 30( P - 500 Cq
C5
=1
1)
- 20(
-)P - 500
10000 = 3.4 - 5po
10000
+ 15(p0- 0.48)(2.5po - 2.2)
cg = 1 - 3o(p0- 0.48) + 187.5(po - 0.48)2- 312.5(po - 0.48)3
6. Temperature Correction. Finally, the density p of the total mixture at the actual temperature T (in O F ) and pressure can be calculated p = pp - Eo- Elel - E2e2- E3e3 (-412)
where Eo = 0.055846 - 0.060601e4+ 0.005275e5- 0.000750e6
El = 0.037809 - 0.049262e4+ 0.012043e5+ 0.000455e6 E2 = 0.021769 - 0.032396e4 + 0.011015e5+ 0.000247e6 E3 = 0.009675 - 0.015500e4 + 0.006520e5- 0.000653e6 e l = l - 2 - T - 520 200
e3 = 1 - 12-
T - 520 + 30( T - 520)’ - 20( T - 520)3 200 200 200 e4 = 3.6 - 5pp
e5 = 1 + 15(pp- 0.52)(2.5pp- 2.3) e6 = 1 - 30(pp - 0.52)
+ 187.5(pp- 0.52)2- 312.5(pp - 0.52)3
Nomenclature Bo = oil formation factor defined in eq 3 c, = isothermal compressibility defined in eq 1 GOR = volume measured at standard conditions of gas “dissolved”in the oil at a given pressure per unit volume residual oil Psat= saturation point pressure V = volume V , = volume of residual oil at 1 atm and 15 “C (defined in section 2.2) V,,, = saturation point volume V , = volume of total mixture
Ind. Eng. Chem. Process ms. mv. 1984, 23,573-576
Y = defined in eq 2 2 = compressibility factor Registry No. N2,7727-37-9;C02, 124-38-9; methane, 74-82-8.
Literature Cited Alani, H. G.; Kennedy, H. T. Pet. Trans. AIM€ 1980, 219, 288. Lee, E. I.; Kesler, M. 0. A I C M J . 1975. 2 1 , 510. Michelsen, M. L. NuM Phase Equl/lb. 1982, 9 , 1. Pedersen, K. S.; Thomassen, P.; Fredenslund, Aa. Ind. Eng. Chem. Process Des. D e v . 1984. 2 3 , 163. PBneloux, A.; FrBze, P. FIuM Phese Equlllb. 1982, 8, 7. Sim, W. J.; Daubert, T. E. Ind. Eng. Chem. Process Des. Dev. 1980, 19,
386. Standing, M. E.; Katz, D. L. Trans. Am. Inst. Mln. Metall. Eng. 1942, 146, 159.
Received for review September 20, 1982 Accepted September 27, 1983
573
Supplementary Material Available: (A) Analytical data for the samples 13 and 14 used in this work, comprising molar compositions, specific gravities, molecular weights of all fractions. (B) The results of the characterization procedure for samples 13 and 14 including all the data which enter the SRK equation, Le., mole fractions, critical temperatures, and pressures and acentric factors. (C)Measured and calculated results of flashes, constant mass expansions, differential liberations, and constant volume depletion studies (15 tables). The data include molar compositions, gas phase compressibility factors, liquid densities, and amounts of gas removed during the differential liberation and constant volume depletion processes (23 pages). Ordering information is given on any current masthead page.
Measurement of Bubble Size in Fluidized Beds K. Vlswanathan’ and D. Subba Rao Department of Chemical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi- 1 100 16, India
A simple method is developed to estimate bubble size variation with height in fluidized beds from axial pressure measurements. Experiments are performed and results are presented to indicate the procedure of using the method developed. Bubble sizes thus obtained compare reasonably well with available bubble growth correlations. The present method is expected to be useful for bubble size measurements at high temperatures and pressures and under complex reacting condltions.
Introduction Bubble size is an important parameter for bubbling bed models to predict the performance of a fluidized bed reactor (Viswanathan, 1982). Existing fluidized bed reactor models require information on axial variation of average bubble size at any cross section. Equations based on experimental measurements are available in the literature to estimate bubble size at the distributor (Miwa et al., 1972) and the subsequent increase in its size with height (Kato and Wen, 1969; Mori and Wen, 1975). However, the measurements on which the above bubble growth correlations are based are a t ambient conditions and their applicability is uncertain at high temperatures, pressures, under reacting conditions and in the presence of internals. Hence bubble size measurements for these different conditions are in great need. Bubble measurement techniques have been reviewed by Rowe (1971). Photographic techniques are employed in two dimensional (2-D) beds, but 2-D beds are not practically useful. For 3-D beds X-ray and y-ray absorption techniques are used. These methods are not very practical for large diameter beds in presence of swarms of bubbles. The other technique generally used measures electrical properties such as capacitance, impedance, conductivity, etc. These techniques lead to measurement of local bubble size, and to get an average bubble size at any level requires considerable effort. Further, it may not be possible to use such probes under reacting conditions. Bubble size, bubble rise velocity, bubble holdup, and exchange between various phases are interrelated. Measurement of any one of these is sufficient to predict the other parameters. Bubble holdup variation can be obtained from pressure measurements along the length of the 0196-4305f84f 1123-0573$01.50f0
bed as it is related to the pressure gradient through the equation
where emf is the assumed porosity in the emulsion phase and 6 is the bubble fraction at any cross section. Pressure measurements have been used to obtain bubble fraction and average bubble size (Matsen, 1973; Davidson et al., 1977). Pressure fluctuation signals have been used by Lirag and Littman (1970) to obtain bubble size and by Swinehart (1966) to obtain bubble rise velocity. An attempt is made in this paper to develop a method for obtaining bubble size variation with height using eq 1 and experimental pressure measurements. Relevant Equations for Back-Calculating Bubble Size from Bubble Fraction The treatment is essentially based on the assumptions of the Kunii and Levenspiel (1969) model. The equations are modified incorporating the fact that solids move up in wakes and down in clouds, whereas gas moves up both in clouds and wakes. The various interrelated equations (Viswanathan and Rao, 1980, 1983) are
0 1984 American Chemical Society
