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Gas holdup versus gas rate in the bubbly regime
International Journal of Mineral Processing, 29 (1990) 141-146
141
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
Technical Note
Gas holdup versus gas rate in the bubbly regime J.B. Yianatos and J.A. Finch Chemical Engineering, University of Santa Maria, Valparaiso, I 1O-V (Chile) Mining and Metallurgical Engineering, McGill University, Montreal, P.Q. H3A 2A 7 (Canada) (Received August 26, 1988; accepted after revision November 8, 1989 )
INTRODUCTION
Flotation columns can be characterized by two main zones: (a) the collection zone, where the feed slurry (entering near the top ) is contacted countercurrent with a bubble swarm (entering at the bottom) in order to selectively recover the hydrophobic particles; and (b) the cleaning (froth) zone, where the mineralized bubbles are washed countercurrently by means of a water stream added near the lip level. Normal practice is to operate the collection zone under bubbly flow conditions. This means keeping gas holdup, gas rate and bubble size within certain limits. Gas holdups are typically 20% and less, superficial gas rates 1-3 cm/s, and bubble sizes 0.5-2 mm (Dobby et al., 1985). In a recent article Shah et al. ( 1982 ) reviewed published correlations for predicting gas holdup in bubble columns. The conditions corresponded to churn-turbulent flow with gas rates larger than 4 cm/s. No general model to predict gas holdup is known. In this communication a model is developed to predict gas holdup from measurements of gas rate, liquid rate and bubble size for the bubbly flow conditions relevant to column flotation. MATHEMATICAL MODELING
Energy balance Joshi and Sharma (1978, 1979) and Joshi (1980) have described an energy balance model based on the work of Whalley and Davidson (1974). The method considers, for any multiphase contactor, that the supply of energy is from the introduction of the dispersed phase. When all the bubbles rise at their terminal velocity (U~ob) the fractional gas holdup is given by the following equation for countercurrent flow: 0301-7516/90/$03.50
© 1990 Elsevier Science Publishers B.V.
142
=
J.B. YIANATOS AND J.A. FINCH
-
Jg
[JL/(
1
(1)
]
with the symbols explained in the Notation. Under these conditions, all the energy introduced by the gas is dissipated at the gas-liquid interface and, as Joshi (1980) has shown, no liquid recirculation occurs. According to Joshi these conditions prevail in small diameter columns (less than 0.1 m) at relatively low superficial gas rates (less than 5 cm/s). In large diameter columns and at high superficial gas velocities, the experimental values of gas holdup are known to be less than those predicted by eq. 1. Joshi suggests this is due to liquid recirculation. Consequently the average residence time of the gas phase is reduced compared to that in the absence of liquid recirculation, and lower gas holdup results. No discussion was given on the impact which the change in bubble size, due to changes in gas rate, may have on the resulting gas holdup. Slip (or relative gas~liquid) velocity in a bubble swarm The hindered settling equations of Masliyah (1979) have been adapted (Yianatos et al., 1988 ) to correlate mean bubble size, gas holdup and gas velocity in liquid-bubble swarms. This approach has proven adequate to predict mean bubble size from gas holdup measurements (Yianatos et al., 1988 ). The basic relationships for a two phase (air-water) system are the adaptaNOTATION
C db g JL rn n
constant in eq. 4 bubble diameter (cm) gravity acceleration (cm/s 2) superficial gas rate (cm/s) superficial liquid rate (cm/s) parameter in eq. 2 constant in eq. 4
Reb
Reynolds number of bubbles,
Jg
USb U~ob
gL slip velocity between bubbles and liquid (cm/s) terminal velocity of single bubbles in the swarm (cm/s)
Greek symbols ~s PL /~L
db USbPL(1--~g)
fractional gas holdup liquid density (g/cm 3) liquid viscosity ( g c m - 1 s- 1 )
GAS HOLDUPVERSUSGAS RATEIN THE BUBBLYREGIME
143
tion of Masliyah's equation for slip velocity to the case of a bubble, namely:
USb-- -gd~, ( 1 --~g)m-lflL
(2)
18 ~L( I -4-0.15 Re °-687) and the bubble slip velocity definition (countercurrent system ) =Jg~
USb
~g
JL
(1 --Eg)
(3)
Comparison of eqs. 1 and 3 shows that the bubble slip velocity USbcorresponds to the terminal velocity of the bubbles in the swarm U~b (without liquid recirculation ). In order to develop a general correlation between gas holdup and gas rate, the variation of the average bubble size with gas rate must be considered. Dobby and Finch ( 1986 ) have shown that for porous gas spargers (as often used in flotation columns) the mean bubble diameter can be expressed as a function of gas rate:
db=C (Jg)"
(4)
where C and n are constants. At the gas velocities (1-3 c m / s ) used in flotation columns, n = 0 . 2 to 0.3. The constant C depends on the chemical environment, e.g. frother concentration and the relative sparger to column size (Manqiu and Finch, 1989); C corresponds to the mean bubble diameter at J~= 1 cm/s. Thus, combining eqs. 2-4, the gas holdup in a countercurrent bubble swarm can be predicted in terms of the liquid and gas rates. EXPERIMENTAL RESULTS AND DISCUSSION
Fig. 1 shows gas holdup versus gas rate data measured at different frother (Dowfroth 25°C) concentrations. The experimental set-up has been described elsewhere (Yianatos et al., 1985 ). The continuous lines represent the slip velocity model (eqns. 2-4). For the model the constant C (the mean bubble diameter at Jg = 1 c m / s ) was estimated using the technique of Yianatos et al. (1988), and n was taken as 0.25 (Finch and Dobby, 1986). The model adequately predicts the gas holdup for bubble diameters less than 2 mm; above this bubble size bubble motion becomes oscillatory and eq. 2 no longer applies (Yianatos et al., 1988 ). Fig. 2 is included as further evidence that n = 0.25 is appropriate; the trend of bubble diameter with gas rate reasonably follows the 0.25 power. Fig. 3 summarizes the agreement between measured and calculated gas holdup. Slight deviations at low gas holdups (% < 5%; Jg < 1 cm/s) can be attributed to the inaccuracy of measurement. Increase in scatter at larger gas holdups (%> 25%; Jg> 3 c m / s ) is probably associated with incipient liquid
144
J.B. YIANATOSAND J.A. FINCH
30
o o
J
-20 ~3 .J 0 "I" (5
....
10
DATA
M o d e l EQS. (2), (3), (4)
FRO~HER ppm
mCm
15
1.435 1.193 0.979
0 [] Z& I 1
0
I 2
I 3 GAS RATE, Jg (cm/s)
SUPERFICIAL
Fig. 1. Gas holdup as a function o f gas rate.
Model: db= C ' J~
db/C
i
0.5
EQS. FROTHER C 2-3 ppm mm 0 5 1.435 0 10 1.193 Z& 15 0.979
~
I
I
I
I
I
1
2
3
4
5
SUPERFICIAL GAS RATE, cm/s
Fig. 2. Effect of gas rate on bubble diameter.
I 4
J
j J
J
GAS HOLDUP VERSUS GAS RATE IN THE BUBBLY REGIME
145
/." / . ° / 30
/
-
/
///
,,,:/S /,-\,o, IlIAI / / f / I I II/I/
v
o.
, I I / U iI I~
r", .-I
ii/~,, U/11 / J //
o I o') .,,( (5
/// ~
20
I I I/.~D//II ///~Q//////
I.-Z LU
///~ /i
rr
LU O. X LU
/ "/~. ~ /
DATA
~ 10
o
,,N~ ,'+"I
J
FROTHER
- 5 . . . . . ;;~, ....
i
io
I
I
I
10
20
30
To
PREDICTED GAS HOLDUP, (%)
Fig. 3. Experimental vs. predicted gas holdup.
recirculation because of the increase in both average bubble size and variance in bubble size (larger than average bubbles rise at velocities higher than the average promoting recirculation). Under these conditions estimation of an average terminal velocity for bubbles in the swarm becomes unrealistic.
CONCLUSIONS For conditions typical of flotation column operation (db < 2 mm; Jg< 3 c m / s ) and assuming liquid recirculation is not significant, the gas holdup versus gas rate can be adequately described by the proposed model. The single parameter to be measured or estimated is the mean bubble diameter at Jg= 1 cm/s.
146
J.B. Y1ANATOS AND J.A. FINCH
REFERENCES Dobby, G.S., Amelunxen, R. and Finch, J.A., 1985. Column Flotation - Some Plant Experience and Model Development. IFAC Symp., Brisbane, pp. 259-263. Dobby, G.S. and Finch, J.A., 1986. Particle collection in columns - gas rate and bubble size effects. Can. Metall. Q., 25(1 ): 9-13. Joshi, J.B., 1980. Axial mixing in multiphase contactors - a unified correlation. Trans. Inst. Chem. Eng., 58:155. Joshi, J.B. and Sharma, M.M., 1978. Liquid phase back mixing in sparged contactors. Can. J. Chem. Eng., 56:116. Joshi, J.B. and Sharma, M.M., 1979. A circulation cell model for bubble column. Trans. Inst. Chem. Eng., 57: 244. Masliyah, J., 1979. Hindered settling in a multi-species particle system. Chem. Eng. Sci., 34: 1166-1168.
Manqiu Xu and Finch, J.A., 1989. Effect of sparger surface area on bubble diameter in flotation columns. Can. Metall. Q., 28(1 ): 1-6. Shah, Y.T., Kelkar, B.G., Godbole, S.P. and Deckwer, W.D., 1982. Design parameters estimation for bubble column reactors. AIChE J., 28 (3): 353-379. Whalley, P.B. and Davidson, J.F., 1974. Liquid circulation in bubble columns. Proc. Symp. Multiphase Flow Systems, Inst. Chem. Eng. Symp. Ser., No. 38, J5. Yianatos, J.B., Laplante, A.R. and Finch, J.A., 1985. Estimation of local holdup in the bubbling and froth zones of a gas-liquid column. Chem. Eng. Sci., 40 ( 10): 1965-1968. Yianatos, J.B., Finch, J.A., Dobby, G.S. and Manqiu Xu, 1988. Bubble size estimation in a bubble swarm. J. Colloid Interface Sci., 126 ( 1 ): 37-44.
141
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
Technical Note
Gas holdup versus gas rate in the bubbly regime J.B. Yianatos and J.A. Finch Chemical Engineering, University of Santa Maria, Valparaiso, I 1O-V (Chile) Mining and Metallurgical Engineering, McGill University, Montreal, P.Q. H3A 2A 7 (Canada) (Received August 26, 1988; accepted after revision November 8, 1989 )
INTRODUCTION
Flotation columns can be characterized by two main zones: (a) the collection zone, where the feed slurry (entering near the top ) is contacted countercurrent with a bubble swarm (entering at the bottom) in order to selectively recover the hydrophobic particles; and (b) the cleaning (froth) zone, where the mineralized bubbles are washed countercurrently by means of a water stream added near the lip level. Normal practice is to operate the collection zone under bubbly flow conditions. This means keeping gas holdup, gas rate and bubble size within certain limits. Gas holdups are typically 20% and less, superficial gas rates 1-3 cm/s, and bubble sizes 0.5-2 mm (Dobby et al., 1985). In a recent article Shah et al. ( 1982 ) reviewed published correlations for predicting gas holdup in bubble columns. The conditions corresponded to churn-turbulent flow with gas rates larger than 4 cm/s. No general model to predict gas holdup is known. In this communication a model is developed to predict gas holdup from measurements of gas rate, liquid rate and bubble size for the bubbly flow conditions relevant to column flotation. MATHEMATICAL MODELING
Energy balance Joshi and Sharma (1978, 1979) and Joshi (1980) have described an energy balance model based on the work of Whalley and Davidson (1974). The method considers, for any multiphase contactor, that the supply of energy is from the introduction of the dispersed phase. When all the bubbles rise at their terminal velocity (U~ob) the fractional gas holdup is given by the following equation for countercurrent flow: 0301-7516/90/$03.50
© 1990 Elsevier Science Publishers B.V.
142
=
J.B. YIANATOS AND J.A. FINCH
-
Jg
[JL/(
1
(1)
]
with the symbols explained in the Notation. Under these conditions, all the energy introduced by the gas is dissipated at the gas-liquid interface and, as Joshi (1980) has shown, no liquid recirculation occurs. According to Joshi these conditions prevail in small diameter columns (less than 0.1 m) at relatively low superficial gas rates (less than 5 cm/s). In large diameter columns and at high superficial gas velocities, the experimental values of gas holdup are known to be less than those predicted by eq. 1. Joshi suggests this is due to liquid recirculation. Consequently the average residence time of the gas phase is reduced compared to that in the absence of liquid recirculation, and lower gas holdup results. No discussion was given on the impact which the change in bubble size, due to changes in gas rate, may have on the resulting gas holdup. Slip (or relative gas~liquid) velocity in a bubble swarm The hindered settling equations of Masliyah (1979) have been adapted (Yianatos et al., 1988 ) to correlate mean bubble size, gas holdup and gas velocity in liquid-bubble swarms. This approach has proven adequate to predict mean bubble size from gas holdup measurements (Yianatos et al., 1988 ). The basic relationships for a two phase (air-water) system are the adaptaNOTATION
C db g JL rn n
constant in eq. 4 bubble diameter (cm) gravity acceleration (cm/s 2) superficial gas rate (cm/s) superficial liquid rate (cm/s) parameter in eq. 2 constant in eq. 4
Reb
Reynolds number of bubbles,
Jg
USb U~ob
gL slip velocity between bubbles and liquid (cm/s) terminal velocity of single bubbles in the swarm (cm/s)
Greek symbols ~s PL /~L
db USbPL(1--~g)
fractional gas holdup liquid density (g/cm 3) liquid viscosity ( g c m - 1 s- 1 )
GAS HOLDUPVERSUSGAS RATEIN THE BUBBLYREGIME
143
tion of Masliyah's equation for slip velocity to the case of a bubble, namely:
USb-- -gd~, ( 1 --~g)m-lflL
(2)
18 ~L( I -4-0.15 Re °-687) and the bubble slip velocity definition (countercurrent system ) =Jg~
USb
~g
JL
(1 --Eg)
(3)
Comparison of eqs. 1 and 3 shows that the bubble slip velocity USbcorresponds to the terminal velocity of the bubbles in the swarm U~b (without liquid recirculation ). In order to develop a general correlation between gas holdup and gas rate, the variation of the average bubble size with gas rate must be considered. Dobby and Finch ( 1986 ) have shown that for porous gas spargers (as often used in flotation columns) the mean bubble diameter can be expressed as a function of gas rate:
db=C (Jg)"
(4)
where C and n are constants. At the gas velocities (1-3 c m / s ) used in flotation columns, n = 0 . 2 to 0.3. The constant C depends on the chemical environment, e.g. frother concentration and the relative sparger to column size (Manqiu and Finch, 1989); C corresponds to the mean bubble diameter at J~= 1 cm/s. Thus, combining eqs. 2-4, the gas holdup in a countercurrent bubble swarm can be predicted in terms of the liquid and gas rates. EXPERIMENTAL RESULTS AND DISCUSSION
Fig. 1 shows gas holdup versus gas rate data measured at different frother (Dowfroth 25°C) concentrations. The experimental set-up has been described elsewhere (Yianatos et al., 1985 ). The continuous lines represent the slip velocity model (eqns. 2-4). For the model the constant C (the mean bubble diameter at Jg = 1 c m / s ) was estimated using the technique of Yianatos et al. (1988), and n was taken as 0.25 (Finch and Dobby, 1986). The model adequately predicts the gas holdup for bubble diameters less than 2 mm; above this bubble size bubble motion becomes oscillatory and eq. 2 no longer applies (Yianatos et al., 1988 ). Fig. 2 is included as further evidence that n = 0.25 is appropriate; the trend of bubble diameter with gas rate reasonably follows the 0.25 power. Fig. 3 summarizes the agreement between measured and calculated gas holdup. Slight deviations at low gas holdups (% < 5%; Jg < 1 cm/s) can be attributed to the inaccuracy of measurement. Increase in scatter at larger gas holdups (%> 25%; Jg> 3 c m / s ) is probably associated with incipient liquid
144
J.B. YIANATOSAND J.A. FINCH
30
o o
J
-20 ~3 .J 0 "I" (5
....
10
DATA
M o d e l EQS. (2), (3), (4)
FRO~HER ppm
mCm
15
1.435 1.193 0.979
0 [] Z& I 1
0
I 2
I 3 GAS RATE, Jg (cm/s)
SUPERFICIAL
Fig. 1. Gas holdup as a function o f gas rate.
Model: db= C ' J~
db/C
i
0.5
EQS. FROTHER C 2-3 ppm mm 0 5 1.435 0 10 1.193 Z& 15 0.979
~
I
I
I
I
I
1
2
3
4
5
SUPERFICIAL GAS RATE, cm/s
Fig. 2. Effect of gas rate on bubble diameter.
I 4
J
j J
J
GAS HOLDUP VERSUS GAS RATE IN THE BUBBLY REGIME
145
/." / . ° / 30
/
-
/
///
,,,:/S /,-\,o, IlIAI / / f / I I II/I/
v
o.
, I I / U iI I~
r", .-I
ii/~,, U/11 / J //
o I o') .,,( (5
/// ~
20
I I I/.~D//II ///~Q//////
I.-Z LU
///~ /i
rr
LU O. X LU
/ "/~. ~ /
DATA
~ 10
o
,,N~ ,'+"I
J
FROTHER
- 5 . . . . . ;;~, ....
i
io
I
I
I
10
20
30
To
PREDICTED GAS HOLDUP, (%)
Fig. 3. Experimental vs. predicted gas holdup.
recirculation because of the increase in both average bubble size and variance in bubble size (larger than average bubbles rise at velocities higher than the average promoting recirculation). Under these conditions estimation of an average terminal velocity for bubbles in the swarm becomes unrealistic.
CONCLUSIONS For conditions typical of flotation column operation (db < 2 mm; Jg< 3 c m / s ) and assuming liquid recirculation is not significant, the gas holdup versus gas rate can be adequately described by the proposed model. The single parameter to be measured or estimated is the mean bubble diameter at Jg= 1 cm/s.
146
J.B. Y1ANATOS AND J.A. FINCH
REFERENCES Dobby, G.S., Amelunxen, R. and Finch, J.A., 1985. Column Flotation - Some Plant Experience and Model Development. IFAC Symp., Brisbane, pp. 259-263. Dobby, G.S. and Finch, J.A., 1986. Particle collection in columns - gas rate and bubble size effects. Can. Metall. Q., 25(1 ): 9-13. Joshi, J.B., 1980. Axial mixing in multiphase contactors - a unified correlation. Trans. Inst. Chem. Eng., 58:155. Joshi, J.B. and Sharma, M.M., 1978. Liquid phase back mixing in sparged contactors. Can. J. Chem. Eng., 56:116. Joshi, J.B. and Sharma, M.M., 1979. A circulation cell model for bubble column. Trans. Inst. Chem. Eng., 57: 244. Masliyah, J., 1979. Hindered settling in a multi-species particle system. Chem. Eng. Sci., 34: 1166-1168.
Manqiu Xu and Finch, J.A., 1989. Effect of sparger surface area on bubble diameter in flotation columns. Can. Metall. Q., 28(1 ): 1-6. Shah, Y.T., Kelkar, B.G., Godbole, S.P. and Deckwer, W.D., 1982. Design parameters estimation for bubble column reactors. AIChE J., 28 (3): 353-379. Whalley, P.B. and Davidson, J.F., 1974. Liquid circulation in bubble columns. Proc. Symp. Multiphase Flow Systems, Inst. Chem. Eng. Symp. Ser., No. 38, J5. Yianatos, J.B., Laplante, A.R. and Finch, J.A., 1985. Estimation of local holdup in the bubbling and froth zones of a gas-liquid column. Chem. Eng. Sci., 40 ( 10): 1965-1968. Yianatos, J.B., Finch, J.A., Dobby, G.S. and Manqiu Xu, 1988. Bubble size estimation in a bubble swarm. J. Colloid Interface Sci., 126 ( 1 ): 37-44.