Hald, A., “Statistical Decision Theory With Engineering Applications,” p 117, Wiley, Sew York, NY, 1952. Hooke, R., Jeeves, T. A., J . Assoc. Comput. Xachin., 8 , 119 11961). Peiry, R. H., Singer, E., Chem. Eng., 75, 163 (1968). Takamatsu, L., Hashimoto, I., Ohno, H., Ind. Eng. Chem. Process Des. Develop., 9 , 368 (1970). Tong, L. S., Currin, H. B., Thorp, A. G., 11, LVuc/eonics,21 (j), 43 (1963). Weibull, W., J . A p p l . Mechan., 18, 293 (1951).
Weisman, J., “Engineering Design Optimization Under Risk,” PhD thesis, Cniversity of Pittsburgh, 1968. Weisman, J., Holzman, A. G., Trans. Amer. ,\-ucl. Soc., 12, 141 11969). \
- I
Weisman, J., Wood, C. F., Rivlin, L., Chem. Eng. Progr. Symp. Ser., 61 (Z),50 (1965). Zangwill, R. J., Xanag. Sci., 1 3 , 344 (1967). RECEIVED for review June 15, 1971 ACCEPTED February 10, 1972
Gas-Inducing Agitator Godfrey Q. Martin Shell Development Co., Emeryville, CA 94608
A technique has been developed for recycling gas from a gas cap for gas-liquid contacting in stirred tanks without the use c f an external loop and recycle compressor. This could be particularly useful at high pressures or with toxic gases. This gas-inducing agitator consists of a hollow shaft connected to a hollow impeller. An orifice in the shaft above the liquid level allows the gas to enter and another in the impeller allows the gas to leave. As the impeller i s rotated through the fluid, the liquid around the impeller i s accelerated, leading to a decrease in pressure. When this pressure i s low enough to overcome the liquid head over the orifice inside the shaft, gas i s recirculated through the liquid from the gas cap. Experimental results have been correlated and used to develop a procedure to design the agitator to give a specified mass transfer rate.
M o s t gas-liquid reactions require extensive and intimate contact between the gas and liquid phases. Stirred tank reactors conveniently provide long residence times for the liquid phase. However, once-through gas sparging of the gas phase would give only limited gas contact times and so the gas phase is often recycled into the reactor through an external loop using a compressor. FTe have devised a self-inducing sparger which will recycle the gas from the gas cap back through the liquid phase without the use of the external loop or compressor. This will reduce capital cost. -21~0,the above device is particularly useful for reactions involving toxic gases and/or high pressures since a leakproof system could be maintained much more easily than a system involving a n external loop. The sparger consists essentially of a hollow shaft connected to a hollow impeller, for example, a hollow tube. An orifice is drilled into the shaft above the liquid level to allow the gas to enter and another orifice (or orifices) is drilled into the hollow impeller to allow the gas to leave. As the impeller is rotated through the liquid, the liquid is accelerated around it, leading to a decrease in pressure. At a critical velocity, the pressure becomes low enough to overcome the liquid head over the orifice inside the shaft and impeller, and gas st’arts to sparge into the liquid. This report presents a theoretical analysis of the gas sparging rate, laboratory studies on a small model, and a design method. Reactors of capacity 300 and 1350 gallons based on this method have been operated successfully in commercial plants. Subsequent to t’his work, the rates of sparging and mass transfer coefficient for several hollow stirrers of various designs were reported by Zlokarnik (1966, 1967). Because of the drastically different geometries and lack of data on power
consumption to provide a basis for comparison, no effort has been made to compare his results with this investigation. Experimental Equipment
Experiments m-ere made with a n air-water system in a n 11-in. diameter flat-bottomed glass vessel fitted with baffles (Figure 1). The shaft of the impeller was a hollow tube fitted a t its lower end with a tee into which the impeller tubes could be screwed. An orifice was drilled into the shaft above t’heliquid to allow the gas to enter into the shaft a d impeller. The diameter of this orifice was substantially larger than the diameter of the sparging orifice to ensure that the resistance controlling the flow of the gas n-ould be a t the sparging orifice. The orifice on the rotating shaft was covered with a stationary sleeve fitted with a n inlet tube so that measurements of the gas flow rate could be made. The gas flow rate was measured by a rotameter which had been calibrated against a wet test meter. The pressure inside the impeller was measured by a manometer. The surface of the liquid above the sparging orifice was normally exposed t o the atmosphere. Two impeller profiles were tested in the laboratory. Aicylindrical profile was first chosen because of ease of construction and the availability of theoretical and experimental work on the pressure distributions around this profile. Improvements on the efficiency of the rate of gas induct,ion were then obtained by flattening the cylinder and mounting it a t different angles of attack, in analogy to a n airfoil (Figure 2 ) . The impeller was rotated a t different speeds, the speed of rotation being measured with a tachometer. The gas flow rates were measured a t each speed, as was the pressure inside the impeller with the gas flowing and also with the gas flow choked off. The pressure measured wit’h the gas choked off Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
397
Table 1. Gas Flow Rate and Pressure Profiles as Function of Impeller Speed-Cylinder RotaAngular tianal loca- speed, tion, n, eo rprn
70
80
90
105
120
200 245 283 316 346 360 332 300 264 223 200 245 283 316 346 360 332 300 264 223 200 245 283 316 346 360 332 300 264 228 200 245 283 316 346 360 332 300 264 223 200 245 283 316 346 360 332 200 264 223
Gas flow rate.
Static pressure, h a ft
-0.19 -0.73 -1.34 -1.95 -2.56 -2.98 -2.26 -1.64 -1.05 -0.51 -0.23 -0.80 -1.49 -2.06 -2.79 -3.14 -2.42 -1.76 -1.12 -0.51 -0.17 -0.69 -1.30 -1.76 -2.56 -2.88 -2.23 -1.61 -0.94 -0.40 -0.11 -0.52 -1.00 -1.54 -2.09 -2.35 -1.83 -1.28 -0.79 -0.33 -0.00 -0.33 -0.60 -0.85 -1.08 -1.19 -0.96 -0.72 -0.48 -0.15
ox cfm
ft"2
Orifice speed, ft/sec
0.00 1.10 2.45 3.95 4.90 5.45 4.40 3.30 1.95 0.25 0.00 1.30 3.40 4.35 5.35 6.15 4.90 3.60 1.95 0.60 0.00 1.30 3.45 5.45 7.00 7.80 6.40 4.35 2.05 0.30 0.00 1.75 4.25 6.30 7.95 8.30 7.00 5.30 3.15 0.65 0.00 0.80 2.90 5.05 6.95 7.65 6.05 4.10 2.15 0.00
0.44 0.85 1.16 1.40 1.60 1.73 1.50 1.28 1.02 0.71 0.48 0.89 1.22 1.43 1.67 1.77 1.56 1.33 1.06 0.70 0.41 0.83 1.14 1.33 1.60 1.70 1.49 1.27 0.97 0.63 0.33 0.72 1.00 1.24 1.45 1.53 1.35 1.13 0.89 0.57 0.00 0.57 0.77 0.92 1.04 1.09 0.98 0.85 0.69 0.39
6.98 8.55 9.85 11.00 12.10 12.55 11.60 10.47 9.20 7.76 6.98 8.55 9.85 11.00 12.10 12.55 11.60 10.47 9.20 7.76 6.98 8.55 9.85 11 .oo 12.10 12.55 11.60 10.47 9.20 7.76 6.98 8.55 9.85 11.00 12.10 12.55 11.60 10.47 9.20 7.76 6.98 8.55 9.85 11.00 12.10 12.55 11.60 10.47 9.20 7.76
102,
(-IlJl'*,
p' = (ha - hd/ (V2/2 9 )
-1.14 -1.23 -1.33 -1.38 -1.49 -1.49 -1.41 -1.36 -1.30 -1.24 -1.18 -1.29 -1.43 -1.43 -1.53 -1.55 -1.49 -1.43 -1.35 -1.24 -1.10 -1.19 -1.31 -1.28 -1.42 -1.45 -1.40 -1.34 -1.21 -1.12 -1.03 -1.04 -1.11 -1.17 -1.22 -1.23 -1.20 -1.14 -1.10 -1.06 -0.89 -0.88 -0.84 -0.81 -0.77 -0.75 -0.78 -0.82 -0.87 -0.86
corresponded to a static pressure measurement (Tables I and 11)* The orifice coefficients were determined by applying a vacuum to the surface of the liquid and therefore drawing air through a wet test meter, a rotameter and finally through the orifice and liquid above it. The pressure drop across the orifice and the liquid above i t was measured by a manometer. The orifice coefficient mas determined from the slope of the curve of flow rate against the square root of the pressure drop 398
Ind. Eng. Chern. Process Der. Develop., Vol. 1 1 ,
No. 3, 1972
,031"
Figure 1. Vessel internals
Figure 2. Sketch of impeller made from '/*-in. 0.d. brass tubing flattened to above dimensions
through the orifice. The value of 0.63 for this submerged orifice agrees within experimental error Ivith coefficients for nonsubmerged orifices. Hydrodynamics of Flow Past Immersed Bodies
The rate of flow of a fluid through an orifice is normally given by a n equation of the form
Q
=
C 0 A 0 V 2g(h1 - h?)
(1)
For the case of the gas sparger, hl is the fluid head inside the impeller equal to the pressure head in the gas cap and hg is the fluid head just outside of the orifice. Since the rate of sparging is critically dependent on the pressure outside of the orifice h2, a brief discussion of the effects of the hydrodynamics of flow on this pressure will help in an understanding of the correlations used and their limitations. Consider a cylinder immersed in a flowing fluid. As the fluid flows by the cylinder, i t is accelerated leading to a decrease in the pressure of the fluid. If the fluid were a n ideal frictionless fluid, the pressure outside the cylinder a t any angular location from the stagnation point e would be given by the equation (Bird et al., 1960)
shown graphically in Figure 3. Actually, most fluids are not ideal. The flow of real fluids past a cylinder may nevertheless be treated by dividing the region into a boundary layer region where the viscous effects are important and the remaining region where the viscous effects are negligible. The boundary layer formed initially is laminar. At low Reynolds number, we have subcritical flow where the boundary layer remain3 laminar until i t separates from the cylinder a t about 80' from the stagnation point, giving a pressure distribution of the form shown in Figure 3. Above a certain critical Reynolds number, we have supercritical flow where the laminar layer becomes turbulent before separating, giving another pressure distribution also
0,
I
SUB-CRITICAL
Table II. Gas Flow Rate and Pressure Profiles as Function of Impeller Speed-Flattened Cylinder
w
. "-1
/
I
Angle of attack, Pa
0
-3.5
3 80 100 120 ANGLE FROM STAGNATION POINT, e
60
Figure 3. Pressure profiles around a cylinder for different flow regimes
-
P' = (h, h d / ( V 2 / 2 g) Angle from stagnation point, 9
shown in the above figure. The critical Re (based on d i ) is 300,000 for no turbulence but is much lower for the highly turbulent conditions in an agitated vessel. It can be seen that the transition from supercritical to subcritical flow could cause a threefold decrease in the dimensionless pressure with a n even more drastic effect on the rate of sparging of the gas into the liquid. Care should therefore be used in eytending the yesults of this investigation, where it is felt the flow regime was supercritical, to the case of very viscous liquids, where the flow regime could conceivably be subcritical. Khatever the flow regime, however, it would be logical to correlate the pressure, using the dimensionless pressure
as a function of angular location with the velocity as a parameter. There are other complicating factors which further affect the pressure profiles around a n impeller in a stirred tank. As the impeller is moved through the fluid, the fluid also moves. While i t is easy to measure the velocity of a point on the impeller relative t o an observer outside the mixer system, it is difficult to estimate the velocity of this point relative to the fluid around it and it is presently impossible to predict the effects of reactor internals as, for example, baffles on this relative velocity. I n addition, the impeller must always move in either its own wake or the wake of another impeller. Because of all these complicating factors, experimentally determined empirical factors must necessarily be used in correlating data and caution must be used in extrapolating to other geometries.
5
10
15
20
Discussion of Results
Cylindrical Profile. Pressure Variation Tt-ithout Gas Flow. Some experimental points showing the variation of diniensionless pressure with angular location a t impeller Reynolds number of 35,700 and 52,000 are plotted in Figure 4. The velocities were calculated relative to a stationary observer outside the vessel rather than relative to the fluid outside the impeller. Also shown is a curve (Galloway and Sage, 1967) shoa ing the variation of dimensionless pressure with angular location for a cylinder in a wind tunnel with a well-controlled uniform velocity field whose turbulence characteristics had also been measured by hot wire measure-
30
Rotational speed,
n, rpm
200 245 283 316 346 360 332 300 264 223 200 245 283 316 346 360 332 300 264 223 200 245 283 316 346 360 332 300 264 223 200 245 283 316 346 360 332 300 264 223 200 245 283 316 346 360 332 300 264 223 200 245 283 316 346 360 332 300 264 223
Gas flow rate, Static pressure, ha, ft
QX
-0.01 -0.32 -0.70 -1.05 -1.45 -1.62 -1.22 -0.87 -0.53 -0.08 -0.00 -0.68 -1.24 -1.68 -2.16 -2.42 -1.98 -1.42 -0.91 -0.46 -0.34 -0.97 -1.57 -2.13 -2.87 -3.12 -2.54 -1.83 -1.24 -0.66 -0.36 -1.06 -1.91 -2.71 -3.26 .,. -3.06 -2.31 -1.56 -0.76 -0.41 -1.07 -2.01 -3.10 -3.96 -4.50 -3.56 -2.48 -1.53 -0.74 -0.26 -0.83 -1.54 -2.17 -2.84 -3.25 -2.54 -1.88 -1.25 -0.57
0.0 0.0 1.6 3.4 5.3 6.0 4.1 2.6 0.8 0.0 0.0 1.7 4.3 6.0 7.3 8.5 6.8 4.8 2.5 0.0 0.0 3.0 5.5 7.3 9.0 9.6 8.4 6.7 4.3 1.6 ... 3.8 6.2 7.9 9.0 10.1 8.8 7.1 5.2 1.9 15 4.6 6.7 8.6 9.5 10.4 9.0 7.6 5.8 3.0 0.8 5.1 6.8 8.5 9.7 10.7 9.1 7.7 6.2 3.6
102, cfm
Orifice speed, ft/sec
p' = (h, - h i ) / (V2/2 9)
6.98 8.55 9.85 11.oo 12.10 12.55 11.60 10.47 9.20 7.76 6.98 8.55 9.85 11.00 12.10 12.55 11.60 10.47 9.20 7.76 6.98 8.55 9.85 11 .oo 12.10 12.55 11.60 10.47 9.20 7.76 6.98 8.55 9.85 11.00 12.10 12.55 11.60 10.47 9.20 7.76 6.98 8.55 9.85 11.oo 12.10 12.55 11.60 10.47 9.20 7.76 6.98 8.55 9.85 11.00 12.10 12.55 11.60 10.47 9.20 7.76
-0.90 -0.87 -0.91 -0.91 -0.93 -0.93 -0.91 -0.91 -0.91 -0.79 -0.88 -1.19 -1.26 -1.24 -1.25 -1.26 -1.27 -1.23 -1.19 -1.19 -1.34 -1.45 -1.48 -1.48 -1.56 -1.54 -1.54 -1.47 -1.44 ... -1.36 -1.48 -1.70 -1.80 -1.73 ... -1.79 -1.75 -1.69 -1.51 -1.43 -1.53 -1.77 -1.99 -2.04 -2.10 -2.03 -1.85 -1.66 -1.49 -1.23 -1.32 -1.46 -1.50 -1.55 -1.59 -1.54 -1.50 -1.45 -1.31
drn, ft"2
0.10 0.56 0.83 1.02 1.20 1.27 1.10 0.93 0.73 ... ... 0.82 1.11 1.30 1.47 1.56 1.41 1.19 0.95 0.68 0.58 0.98 1.25 1.46 1.69 1.77 1.59 1.35 1.11 0.81 0.60 1.03 1.38 1.65 1.80 ... 1.75 1.52 1.25 0.87 0.64 1.03 1.42 1.76 1 99 2 12 1.89 1.57 1.24 0.86 0.51 0.91 1.24 1.47 1.69 1.80 1.59 1.37 1.12 0.75
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
399
'r
0
AUTHOR Np.
=
_=I
35,700
ANGULAR LOCATlON 70'
WIND TUNNEL TEST (GALLOWAY AND SAGE, 1967)
> -1.0 >
A
v
r L
80' 90' 105' 120'
0.10 -2,o-
( N R o= 7 3 . M : INTENSITY =
I
-2.5.
I
I
I
I%%)
I
I
I
,
Figure 6. Effect of driving force on flow rate for a cylinder Gas flow rate, ft3/min Pressure driving force, ( -hJ'/2
ft'/2
ANGULAR LOCATION 700 800
2v
: :1 1209
7
8
IO
9
11
12
13
ORIFICE SPEED, V FT/SEC
Figure 5. Effect of orifice speed on dimensionless liquid head for a cylinder P' = (ha
-
ht)/(V2/2 9 ) Orifice speed, V ft/sec
ments. The high turbulence level of 13.50/, was chosen from the two turbulence levels (1.3 and 13.5%) reported because it represents more nearly the turbulence of (approximately 10%) existing in the wake of a cylinder (Townsend, 1949; Czkan and Reynolds, 1967). Since in the stirred tank reactor used in this investigation, the cylinder is continually moving in the wake of the cylinder comprising the other arm of the impeller, there is surprisingly close agreement between the results obtained in the complicated hydrodynamics of a baffled stirred tank mixer and the wind tunnel tests mentioned above. These results indicate that the flow regime is supercritical because of the high level of free stream turbulence in which the impeller always moves. This hypothesis was tested further by introducing a Prandtl tripping wire to induce the laminar boundary layer, if it existed, to turn turbulent and thus give supercritical flow. No change in the pressure profiles was noted, indicating that the flow regime was already supercritical. The results also indicate that the velocity of the impeller, relative to the fluid around it, is very close to the velocity measured by a stationary observer outside the system. Reference to the relative locations of the impeller tip and the vessel baffles in Figure 1 shows that this would not be an unreasonable assumption. Care should be taken in 400 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1 9 7 2
applying the results of this work to systems where the relative velocities of the impeller with respect to the fluid and to a stationary observer are not the same. The effect of orifice speed on the dimensionless pressure with the angular location of the orifice as a parameter is shown in Figure 5. The dimensionless pressure decreases with orifice speed except a t an angular location of 120' where a reverse trend is noted. This anomaly may be explained by the fact that separation and consequently a reversal of flow takes place between 105' and 120'. Quite possibly, the orifice measures some component of an impact pressure of the reversed flow in addition to the negative pressure due to the acceleration of the fluid giving rise to higher measured pressures. Gas Sparging Rates. If the logarithm of gas sparging rate were plotted against the logarithm of a pressure driving force, (hl - h z ) ,a straight line with slope 2 would be obtained when Equation 1 was applicable. If the pressure in the gas cap is chosen as the reference pressure, then hi = 0. The pressure head h2 was chosen to be the static pressure head h, measured with no gas flow. When this driving force was actually plotted against the gas flow rate, a straight line with slope 1.8 was obtained. Previous work (Martin, 1956) on gas pressure drops through perforated plates with liquids flon-ing over them also gave a slope of 1.S.The slope of 2.0 implied that the fluid is frictionless so that the Bernouilli equation may be used. If one assumes that the coefficient C = V z / 2 gh is actually a weak function of the velocity of the form C 0: VO 2, then the experimental relation between the sparging rate and the driving force can be explained. For convenience of design and because of the assumptions already implicit in using a single phase equation for sparging into a liquid, i t was decided to correlate the results by plotting the flow rate against the square root of the applied driving force h,. Figure 6 shows that reasonably good correlations are obtained.
ANGLE OF ATTACK
e p
A
50
Io'
T 15' 200
*
30-
0 70
I
l
80
l I I I Po IW I IO ANGULAR LOCATION OF ORIFICE
I
Figure 7. Dependence of flow correction factor angular location K1
120
ORIFICE SPEED, fpi
Kl
on
Figure 8. Effect of orifice speed on dimensionless pressure for flattened cylinder
Angular location of orifice
P' = (hs = hIj/(V2/2 gj
However, the gas only starts to flow when d m exceeds 0.5 or h , exceeds 3.0 in. water. Part of this pressure may be attributed to the pressure required to overcome the surface tension forces (Ah = 2 u / r = 1.5 in. water). h similar effect was observed by Davidson and Schuler (1960) in their study of bubble formation at a horizontal, stationary orifice submerged below water. They speculate t h a t the difference between theoretical and experimental values could b e due partly to the upward current of water caused by the stream of rising bubbles and partly due to bubble deformation. Examination of Figure 6 also shows that the slopes of the curves are dependent 011 the angular location of the orifice. This effect may be due to the distortion of the pressure profiles owing to the gas flow from the cylinder or i t could be the effect of the relative velocity of the liquid past the orifice on the effective area of the "vena contracta." Whatever the reasons for the above effects, the data in Figure 6 may be correlated by an equation of the form
0"410.12
0.10
$
ANGLE OF ATTACK
e
-
-
o* 5'
/
A IO' V 15"
-
20' 30'
/
-
.
2 0.08
5 2 4"
-
0.06
-
0.04
-
where h, is given by .
(4)
0 4
I 0 8
i
I 1 2
PRESSURE DRIVING FORCE,
l
I
I
1 6
a,
0
ft"'
and
V
= 2
mr
(5)
K1 may be regarded as a n experimental constant required to account for the effects discussed above and is given in Figure 7 . It should be iioted that while K1 is dimensionless, the constant 0.00085 has the dimensions of ft3/sec in the above equation. Optimum Angular Location at Orijce. While the maximum vacuum a t a particular orifice velocity was drawn a t a n angle of 80" from the forward stagnation point, the maximum sparging rate was actually obtained a t a n angular location of 105" because of the effect of gas flow on the pressure driving force as discussed in the previous section. At this angular location, a t the higher sparging rates which would normally be of most interest, one can therefore assign values of 1.1 and -1.2 to the constants K 1and P', respectively. Flattened Cylindrical Impeller. It was felt t h a t lower pressures and higher $parging rates might be obtained a t
Figure 9. Effect of driving force on flow rote for flattened cylinder Gas flow rate, ft3/min Pressure driving force, ( - h a ) ' / ! , ft1/2
the same orifice speeds if the cylindrical impeller were flattened into a shape resembling a n airfoil and held a t different angles of attack. Because of ease of fabrication, a cylinder was rolled into the shape shown in Figure 2 , and an orifice was drilled as shown. Paralleling the previous discussion for a cylindrical impeller, the change in dimensionless pressure with orifice speed a t different angles of attack is shown in Figure 8, and the effect of the driving force on the rate of sparging is shown in Figure 9. At angles of attack between 0" and l5", the pressure drop required to start the flow of gas was the same as for the cylinder. This pressure drop decreased drastically for the Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , NO.3, 1 9 7 2
401
0.10
/r
0.06
10 I1 ORIFICE SPEED, V FT/SEC
9
I2
13
Figure 10. Comparison of sparging rates of the cylinder and the flattened cylinder Gas fiow rate, 4 cfm Orifice speed, V ft/sec
runs with angle of attack a t 20' and 30', perhaps because of the higher gas flow rates encountered. The minimum pressure was pulled a t an angle of attack of 20' but the maximum sparging rate was obtained when the angle of attack was 30'. The values of K1 and P' to be used in the design equation a t this maximum sparging rate are 0.90 and - 1.50, respectively. The rates of sparging a t the optimum orientation of the orifices for the cylindrical and flattened cylindrical impeller are compared in Figure 10; this shows clearly the superiority of the flattened cylinder over the ordinary cylinder on the basis of the sparging efficiency through single orifice. Design of the Agitator
No experimental work has been done to evaluate the mass transfer rates from the bubbles created by the sparger or the power required to turn the sparger arms. Nevertheless, work reported in the literature can be combined with the information in this report to design an agitator-sparger to perform a specified job. The design method is illustrated by the following hypothetical problem. A pure gas stream, A , is to be contacted with a liquid stream containing a component, B, in a stirred tank reactor. The component, -4,diffuses into the liquid where it reacts with B a t a known rate. We wish to design a cylindrical sparger-impeller to ensure that the rate of mass transfer of A into the liquid will be a t least large enough to ensure that the mass transfer rate will not be controlling. Assume that the following quantities are also known: Maximum allowable length of the impeller D , Maximum liquid level above impeller h 1 Power available to turn sparger-impeller Diameter of the impeller d , Step 1. The power required to turn the impeller can be estimated from the relationship Power
=
N,pLnaDZ6 Qc
where N , is the power number characteristic of such items as the impeller size and shape or reactor baffling. For a 402 Ind. Eng. Chem. Process Des. Develop., Vol. 11, No. 3, 1972
cylindrical impeller in a tank with three baffles, we have estimated the power number to be approximately 1.0 (see Appendix). When we know the length of the impeller and the power available, the speed of rotation may be estimated. Step 2. To minimize the departure from the cylindrical profile, assume that the diameter of the orifices will be one twelfth of the impeller diameter d f . Further, maintain a t least one impeller diameter between the outermost orifices and the ends of the impeller. To maximize the sparging rate, place the orifices so that the angular location of the center of the orifices are 105' from the forward stagnation point. To minimize interaction between the streams from the orifices, place them a t least two diameters apart, center to center, along the axis of the impeller. Step 3. Assume that the distance of the orifice nearest the shaft is controlled by the fact that no liquid should be drawn through it-that is, its velocity through the liquid should be high enough so that the vacuum drawn a t the orifice is just sufficient to balance the liquid head over it. When we know the liquid level above the impeller and the speed of rotation, this minimum distance can be calculated from Equations 4 and 5 and Figure 5. Because of the spacing defined in Step 2, the number of orifices and their distance from the axis of rotation have now been defined. Step 4. The volumetric rate of flow through each of the orifices and therefore the total volumetric rate of flow of gas can be estimated from Equations 3-5 and Figures 5 and 7. If we assume this flow of gas takes place through an area bounded by the impeller tips, the superficial velocity of the gas can be estimated. Step 5. If we use this superficial velocity, the gas holdup H (= volume of gas per unit volume of the liquid) and the liquid phase mass transfer coefficient can be estimated from work done by Towell et al. (1965). The latter data were taken for stationary spider spargers with l/l&. holes but should nevertheless give reasonable values. Step 6. The diameter of the bubbles may be estimated from a correlation by Calderbank (1948,1949).
where db and the term in the brackets are in cms. The power per unit volume ( P / v ) can be estimated conservatively, as recommended in the above paper (Towel1 et al., 1965) from the equation
where V Q ,pQ, and g are given in cgs units. A lower boundary on the bubble diameter is obtained when the impeller power per unit volume of reactor is used in the above equation. Since the area per unit volume for each bubble and therefore for all of the gas is equal t o 6/db, the total area available for mass transfer a is given by
a
=
6 vH/db.
Step 7. The concentration driving force for the transfer of A into solution may be assumed equal to the equilibrium concentration of A a t the liquid gas interface (CA*) if the concentration of A in the liquid is assumed to be zero because of the reaction between A and B. Step 8. When we know the mass transfer coefficient, the area, and the concentration driving force, the rate of mass transfer can be estimated from the equation
Step 9. If the mass transfer rate obtained is less than the reaction rate, some constraint will have to be lifted. Assuming that more power can be made available, the speed of rotation or the length of the sparger impeller will have to be increased to give larger recirculation rates of the gas through the liquid which, in turn, will give higher mass transfer rates. Scale-up Considerations
The design of this sparger on a plant scale may be deduced from the following arguments. Suppose that the laboratory model was to be scaled up to a plant size one thousand times the volume of the laboratory model. If we assume geometrical similarity, this would imply an increase of any characteristic dimension by a factor of (1000)1/3 = 10. Assume that the criterion of similarity to be used is when gas just begins to sparge into the liquid. Examination of Equations 3-5 reveals that the Froude number
V2 2ghz
The energy dissipated per second by differential section =
(dF)(V)
2ghz
nP2rp2
nnnt2rm2
(hJ,
(hJm
-
($=)a
= loll2 =
But V = 2 rnr where n is the speed of rotation. Therefore, energy dissipated per second by differential section
The total energy dissipated per second by impeller, assuming that drag coefficient is independent of r ,
The increase in power per unit volume equals
n 3r 2
T
- (2 rnr)2
would be the appropriate parameter for similarity between the model and its prototype. Therefore,
pa p2 n m rm
1
di +-
3.16
This increase in power per unit volume would have to be balanced against the power required to run a n external compressor now working a t increased pressure (10 times) because of the higher head involved in the vessel. Incidentally, scale-up will cause the Reynolds number based on the impeller diameter and speed to increase so that there is little danger of the flow regime around the impeller becoming subcritical on scale-up.
If di/Di = x, then the total energy dissipated per second by the impeller
APPENDIX
Determination of Power Number for Cylindrical Impeller. The power number for a cylindrical impeller does not appear to be available in the literature. We have, therefore, developed the theory to calculate the power number for any impeller geometry. We have used it to calculate the power number for a flat blade paddle and compared the calculated value to an experimental value reported in the literature. The correction factor so obtained has then been applied to the power number calculated for the cylindrical impeller. Consider a n impeller with tip-to-tip length D t and width dt projected in a plane perpendicular to its movement through the fluid. The force, d F , on a differential length, dr, a t distance r from the axis of rotation is given by
where N,
ir3
=
( C D ) ( Z)8
For a flat blade paddle with two blades in a baffled vessel, with di/Dt = 0.25, the power number equals approximately 2.0 (G. G. Brown, “Unit Operations,” p 207). The ratio of height to length for each blade is therefore 0.5. The drag coefficient for a plate with this ratio obtained presumably from wind tunnel tests = 1.18 (“Engineering Hydraulics” by H. Rouse, p 126). Therefore, N p=
where Cn
drag coefficient for the particular impeller profile density of the liquid V = speed of the differential section dr through the liquid A , = projected area on a plane perpendicular to its movement
1.18 X 0.5 X
8
=
2.23
=
p =
Such close agreement between the calculated value and that reported in the literature is probably fortuitous but does indicate that the theory does apply. The drag coefficient for a circular cylinder with diameter to length ratio = 0.10 is approximately 0.90. Ind. Eng. Chern. Process Des. Develop., Vol. 1 1 , No. 3, 1972
403
n = speed of rotation of shaft, revolutions per second N , = power number, dimensionless
N , for a circular cylinder =
(0.90)(0.1)
=
0.34
($
Because of the large effect of d i / D i and the disturbance of the flow patterns due to the flow of gas, i t would be safer to use a value for the power number of at least 1.0.
v= v,
GREEKLETTERS
Nomenclature
a = total area available for mass transfer, ft2 a,= area of bubbles or gas per unit volume, ft-1 A , = area of orifice, ft2
6 c
P = power p’ = dimensionless pressure = (h, - h l ) / ( V 2 / 2g ) & = gas flow rate through orifice, ft3/sec r = distance of orifice from axis of rotation, in. speed of orifice relative to stationary observer, ft/sec = superficial velocity of gas, ft/sec v = volume of gas-liquid mixture, ft3
width of baffle, in. vertical clearance between impeller and baffles, in. CA* equilibrium concentration of A at the gas-liquid interface, mole A/ft3 C, = conventional orifice coefficient, dimensionless db = bubble diameter, ft d , = impeller diameter, ft do = orifice diameter, ft D i = length of impeller, ft DT = diameter of vessel, ft g = acceleration due to gravity, ft hb = height of orifice above vessel bottom, in. h, = pressure in gas cap, ft h i = liquid head above orifice, ft h, = liquid head outside orifice when no gas is allowed to flow, ft hm = liquid head in the streamline forming the boundary of the cylinder a t a n infinite distance from it, ft H = gas holdup or volume of gas per total volume of gas and liquid dimensionless K 1 = experimentally determined const’ant indicating deviation between pressure driving force across orifice with gas flow from driving force without gas flow = = =
e
= angular location of orifice from the forward stagnation
point on cylinder in degrees angle of attack of flattened cylinder in degrees = density of gas, lb m/ft3 = density of liquid, lb m/ft3 = interfacial tension
p = pg
p~ u
literature Cited
Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” p 136, 1960. Calderbank, P. H., Trans. Znst. Chem. Eng.,26,443 (1948). Calderbank. P. H.. ibid.. 37. 173- (1949). - ~ Davidson, J. F., Schuler: B. 0. G., ibid.: 38,335 (1960). Galloway, T. R., Sage, B. H., AIChE J . , 12 (3), 563 (1967). Martin, G. Q., ?*IS thesis, Columbia Univ., New York, N.Y., 1956. Towell, G. D., Strand, C. P., Ackerman, G. H., “Mixing and Mass Transfer in Large Diameter Bubble Columns,” Proc., AIChE-I. Chem. E. meeting in London, June 1965, Symposium No. 10. Townsend. A. A.. Aust. J . Sci. Res.. 2A. 451 119491. Uzkan, T.; Reynolds, W. C., J . F l u i d hfech., 28, 803 (1967). Zlokarnik, >Chem. I., Ingr. Tech., 38, 357-66 (1966). Zlokarnik, AI., ibid.,pp 717-23. Zlokarnik, M., ibid.,39, 1163-8 (1967). I
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RECEIVED for review July 6, 1971 ACCEPTED January 31, 1972
Piston Bed and Its Design Herbert F. Wiegandt, Robert 1. Von Berg,’ and Jean P. Leinroth’ School of Chemical Engineering, Cornell University, Ithaca, h’Y 1.4850
Recovery of washed crystals from a slurry of crystals in mother liquor is a purification operation frequently practiced in the process Industries. One method to achieve this separation is with a hydraulic piston bed (Wiegandt, 1960) in whlch liquid flou s upward cocurrently with the suspended crystals, the crystals add to the bottom of an upwardly moving crystal bed, a displacement head of wash liquid flows from the top with a very small downward component, and both streams exit through a drainage zone flush to the moving crystal bed and intermediate to the top and bottom. The investigations of desalination by freezing processes or hydrate processes serve as the principal examples wherein piston bed operations have been applied. The design methods To whom correspondence should be addressed. Present address, Department of Chemical Engineering, LMFSsachusetts Institute of Technology, Cambridge, RIA 02139. 404
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
in this paper are developed with reference to one specific, real system: water, brine, and ice. However, the solutions are perfectly general and indeed have been applied to other systems (Wiegandt and Lafay, 1967; Landis and Wiegandt, 1971). The element of engineering importance for the hydraulic piston bed is the rate a t which washed crystals can be produced. Figure 1 schematically represents a piston bed and a buoyancy bed for ice. For most slurry systems the buoyancy option is excluded; for ice beds of closely packed small crystals, movement by buoyancy alone is too slow. An early analysis of ice bed hydraulics made by Bosworth et al. (1960) presented the integrated equation for pressure drop for a washer which used a conical section for upflow followed by a cylindrical section with a horizontal annular screen a t its bottom to carry out the brine and the net downflow of wash water. At the same time, Wiegandt (1960) made
