2226
Ind. Eng. Chem. Res. 1994,33,2226-2241
Gas Inducing Type Mechanically Agitated Contactors Kaliannagounder Saravanan, Vishwas D. Mundale, and Jyeshtharaj B. Joshi' Department of Chemical Technology, University of Bombay, Matunga, Bombay 400 019, India
The rate of gas induction was measured in 0.57,1.0, and 1.5 m i.d. gas inducing types of mechanically agitated contactors (GIMAC). The ratio of impeller diameter to vessel diameter was varied in the range 0.13 < D/T< 0.59. The effect of impeller submergence from the top and impeller clearance from the bottom was investigated in detail. T h e effect of impeller speed was studied in the range 0.13 < N < 13.5 r/s. A model has been developed for the critical speed for gas induction, and for the rate of gas induction with the description of forced vortex structure in the impeller region. A unified theme has been proposed for the understanding of phenomena such as the critical speed for gas induction and the rate of gas induction. The physical significance of model parameters has been interpreted on the basis of flow information in the impeller region. 1. Introduction In gas-liquid reacting systems, recirculation of gas from the head space back into the liquid is required when the conversion of gas per pass is very low or pure gases are being used. In the latter case, the systems are known as dead-end systems. Hydrogenation and oxidation using pure hydrogen and oxygen are typical examples. In case of low conversions of gas per pass, the gas recycle is necessary for conserving it. In case of reactions with pure gases, the gas recycle is required to ensure an optimum gas-liquid mass-transfer coefficient. Many designs are commercially available to tackle such situations. They include (a) the sparged loop reactor (SLR), where the gas is recompressed and sparged back into the contactor; (b) the jet loop reactor (JLR), where the gas is recycled by eductor action of a jet of liquid pumped from the reactor; (c) the surface aerator (SARI, where the gas is sucked back into the surface of the mechanically agitated liquid due to the phenomenon of surface aeration; and (d) the gas inducing mechanically agitated contactor (GIMAC),where the gas is sucked from the head space-standpipe by the impeller. The mechanism and phenomena that underlie the phenomenon of gas induction are not adequately known. In high-pressure reactions the principal advantage of the gas inducing mechanicallyagitated contactor (GIMAC) and the surface aerating reactor (SAR) over the sparged loop reactor (SLR) and the jet loop reactor (JLR) lies in avoidance of equipment external to the reactor. Thus the gas cooler and compressor in the SLR and the liquid cooler and pump in the JLR are avoided with consequent reduction is capital and operating costs. Between the SAR and the GIMAC, the latter offers greater latitude in design. Thus the stator and standpipe can be used to manipulate the gas induction and dispersion functions conveniently, while isolating the impeller from deleterious effects of the liquid ingress into the impeller zone from the vessel. In the SAR the entire surface of the liquid acts in the gas induction function. In the GIMAC the standpipe acts as an inlet channel for the gas. This offers a way of controlling the gas induction process. It also allows the use of suction generated by the impeller to suck contaminated gases to clean them by absorbing their contaminants in the liquid and releasing back the gas to process or ambient. Thus the GIMAC has all the advantages of the SLR, the JLR, and the SAR. Reports on the utility of the GIMAC in a wide range of reactions employing pure gases (the dead-end systems)
* Author to whom correspondence should be addressed.
are available. Examples are hydrogenation (Perry, 19841, fermentation (Matsumura, 1982),oxidation, ammonolysis, alkylation, hydrohalogenation, addition halogenation, and sulfoxidation (Joshi and Sharma, 1977;Bollenrath, 1986). The GIMAC is also useful in many gas-liquid-solid noncatalytic operations. Examples are flue gas desulfurization (Babcock and Hitachi, 19801, oxyhydrolysis in ore leaching (Litz, 19851,and oxidative leaching of metals (Halwacha, 1986). The GIMAC design can also be an efficient liquid-liquid contactor, and many patent references are available for this kind of reactor. However, the widest application of this type of equipment has been in the unit operation of froth flotation, which is widely practiced for beneficiation of ores (Yarar, 1986; Harris, 1986) and water and waste water treatment (Burkhardt, et al., 1978;Churchill and Tacchi, 1978).The GIMAC has been subjected to extensive studies by froth flotation researchers. However, the focus of the majority of these studies has been on metallurgical performance rather than hydrodynamic aspects of a gas-liquid reactor. As a result very little data of relevance to the design of a gas-liquid or gas-liquid-solid reactor are available. More importantly, studies of the mechanism of gas induction and other related phenomena are a subject of few investigations so far. Due to such a lack of insight it is not yet possible to design or scale up a GIMAC as a gas recirculating reactor with total confidence. The present work aims at developing (i) a systematic understanding of the gas induction performance of an impeller rotating within astator (Figure lA), (ii) a rational procedure of GIMAC design and scale-up, with respect to the rate of gas induction, based on suitable semiempirical models, and experimental verification of it for a particular design of impeller. 2. Literature Survey
The process of gas induction is characterized by the critical speed of the impeller for gas induction and by the gas induction rate at all impeller speeds higher than the critical speed. A survey of the literature shows that these phenomena are qualitatively studied by a number of investigators, but quantitative studies are few. The available literature will be briefly reviewed here, by describing the effect of various operational variables on the process of gas induction. Various correlations available in the published literature for the rate of gas induction are shown in Table 1, and comparison of various gas inducing type impellers, taken from Joshi e t al. (1983), is shown in Figure 2.
0000-5005/94/2633-2226$04.50/00 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994 2227
A 1
I . STAND PIPE 2 . IMPELLER HOOD 3. STATOR L . STATOR SHROUD 5, STATOR AIR RING 6. STATOR VANE
Table 1. Correlation for Rates of Gas Induction for GIMAC of Stator Type investigator correlation
+
Degner and Treweek (1976) Qs a D2.4 ln(N)
Wemco 1 1flotation cell aeration mechanism over a scale ratio of 1 : l O
White and Devilliers (1977)
Slotted pipe stator, peristaltic pump rotor air-water air-aqueous solution
7. I M P E L L E R
8 11
7
8. I M P E L L E R SHROUD 9. I M P E L L E R AIR RING
10. I M P E L L E R SLAOE 1 I . SHAFT 1 2 . HUB
2 L
NQ,(DIS)O.5= 0.2310(Fr - Frc)l.@ NQ,(D/S)o.5 = O.O977(Fr - Frc)l.@ Fr = WD2/gS Fr, = WJ12/gS S=S-APJlpg
12
5
equipment, system details
3
Sawant et al. (1980)
9
IO
Qg = 51.2(Fr - Frc)0.m(D/S)0.6 Fr = WD2/gS Fr, = W@/gS
I
6-
Sawant et 01. (1981) Qg = 0.0021 (W - Wc)0,"6D3 Raidoo et al. (1987) Qg = 2.68 X 10"(APD2)'.@ A =f(Sv) 0
Wemco flotation cell of 300 mm i.d. air-water system Denver cells of 100,140,172, 380 mm sizes, SIT = 1.0 air-water system air-water system
PTD impeller inside the stator
AP = pressure rise across air inlet manifold; AP = 0 for natural
aeration. 5 -
B
3. FLATTENED CYLIDRICAL 4. MODIFIED PIPE IMPELLER
-4-
m
5
thick 12 no
-1
c 6
z3-
0 b-
V
3
-z 0
m2-
a
0 LL
0
W
s
= 1 -
I
Figure 1. (A, B)Various components of the stator-rotor system for gas inducing impellers.
2.1. Description of the Process of Gas Induction. The reader is referred to Figure 1 parts A and B which shows a typical arrangement of the impeller within a stator. When this assembly is placed inside a vessel containing liquid and the impeller is static, the liquid level is the standpipe equals the level in the vessel outside. This height when measured from the center plane of the impeller is defined as the submergence of the impeller. As the impeller begins to rotate, the liquid level in the standpipe is found to decrease. The decrease is dependent on speed, and ultimately a speed is reached at which bubbles of gas begin to appear in the radially outgoing impeller stream. The bubbles are best detected at the free surface of the liquid. The minimum impeller speed
I
I
I
I
I
I
POWER CONSUMPTION PER UNIT LIQUID VOLUME, Pg / V ( k W / m J
Figure 2. Gas induction rate vs power density relation for various impeller designs.
at which the gas bubbles appear is called the critical speed of gas induction. A simple analysis of the phenomenon of the critical speed of gas induction is provided by Braginsky (1964). The local hydrostatic head at the impeller tip is reduced by the velocity head imparted to the fluid. The energy balance is expressed as
As the fluid velocity increases, the local pressure decreases, and when the reduction equals the hydrostatic
2228 Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994
head of the fluid, the gas from the head space is sucked into the impeller. This is represented by a constant Froude number based on submergence.
Fr, = KB
(2)
This overall analysis is able to explain the data on the critical speed of gas induction for various impeller types in a unified semiempirical correlation (Sawant and Joshi, 1980),within f15% accuracy. The influence of viscosity has been accounted for empirically by Sawant and Joshi (1980) by extending Braginsky's correlation
When the speed is increased beyond the critical speed, gas induction is observed to increase progressively. An important feature of all the equations (Table 1) is that they are essentially empirical. They do not offer any insight into the process of gas induction. Attempts at quantitative modeling of the gas induction process at speeds greater than the critical speed have been very few. Many investigators, notably Degner and Treweek (1976), recommended that the objective of design and scale-up is best met by a systematic empirical approach rather than an analytical approach. 2.2. Models of Gas Induction. Arbiter et al. (1969) reviewed the previous efforts of modeling for power consumption in gas inducing flotation cells and developed a new model. This is not reviewed further as it is out of the scope of this paper. Zundelevich (1979) attempted the simulation of gas induction by drawing an analogy to aliquid jet gas ejector, with the following assumptions. (i) Gas induction was assumed to be due to the action of liquid flow educting gas at the periphery of the impeller. (ii) The dimensionless liquid flow number (7)was assumed to be a constant for the impeller at all impeller speeds. (iii) The proportion of the induced gas flow rate to the liquid flow rate was assumed to be the same as that for a geometrically similar gas eductor. For this purpose a geometrical similitude was assumed between the impeller-stator assembly and the water jet ejector. (iv) An empirical equation for the water jet ejector (correlating the ratio of the gas rate to the liquid rate with pressure differential and mixing chamber pressure differential) was assumed to be valid for the gas inducing impeller. (v) A certain design of stator and impeller was assumed (Zundelevich, 1978). With these assumptions, Zundelevich (1978)showed that a unique relationship should exist between two dimensionless parameters, viz., the impeller head coefficient CH and the gas Euler number Eu,. The relation is
where
CH = gS/(NDI2
Eug = gS/(QJD2)2
assumptions that underlie the analysis. However, experiments by Raidoo et al. (1987) on turboaerator design at 0.15, 0.20, and 0.25 m diameter scale did not produce a unique relationship between the impeller head number CHand the gas-phase Euler number,Eu,. The discrepancy could be understood in terms of the effects of the resistance at the outlet of gas, employed in the case of Zundelevich's experiment to simulate liquid height. In the case of experiments by Raidoo et al. (19871, an increase in submergence was achieved by an actual increase in liquid height. The pressure drop caused by this resistance is expected to raise the liquid level in the standpipe so as to flood the impeller adequately, as soon as the gas induction begins. The turboaerator design is open at the bottom in the central portion where impeller blades do not exist. Therefore, the impeller is always assured of being filled with the liquid over the entire radial width leading to a constant liquid flow number of the impeller. For the experiments of Raidoo et al. (1987) no such impeller zone flooding can occur and the hydrodynamics is expected to be different in the two cases. The analogy relating a gas eductor to a gas inducing impeller also needs re-examination. A number of other qualitative observations is also available describing the process of gas induction. We propose that the effects described in the literature can be explained in terms of the phenomenon of forced vortex formation in the stator by the rotating impeller (Table 2). 2.3. Proposed Description of the Gas Induction Process Based on Vortex Formation. According to Arbiter and Harris (1962)a vortex is formed by the impeller rotating within a stator. White and devilliers (1977) and Degner (1985)have also identified the existence of a vortex in fluid adjacent to the rotor. Degner (1985) describes it as running along the interior walls of the standpipe down through the central region of the rotor. Simultaneously, suction is created in the standpipe and two-phase flow occurs with the vortex motion. Such a gas induction is expected to affect the overall flow pattern as well as vortex motion. Thus, the vortex causes gas induction and is in turn affected by it. We propose that this forced vortex formation in the impeller zone plays a key role in the gas induction process. Vortex formation can occur in the stator-impeller assembly due to the protection offered by the stator and standpipe to the rotating impeller from the influence of liquid flow. Although the stator is open for radial flow at the plane of the impeller, the radially outward flow generated by the impeller could generate adequate velocity head to protect itself from the radially inward flow that can be caused by the baffles. The situation of an impeller rotating within a stator is then essentially similar to an impeller rotating in a vessel at a high DIT ratio. In understanding this situation, observations of Brennan (1976) should be a useful guide. They showed experimentally that, for a given impeller vessel combination, the total depth of the vortex with respect to the highest liquid level in the vessel is given by
(4)
Such a unique relationship was demonstrated by Zundelevich for two designs of gas inducing impeller-stator pairs and for three different impeller sizes between 0.08 and 0.12 m diameter. The success obtained by Zundelevich in correlating his own data is remarkable. This implies the validity of the
The constant kD depends on the vessel and impeller combination. Experimentally, when the value of h approaches the impeller submergence S, gas bubbles do get trapped in the impeller stream. This is essentially the same as the phenomenon of the critical speed of gas
Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994 2229 induction described earlier. In terms of eq 5, substituting
S for h and simplifying, we obtain Fr, = l/KD
(6)
The similarity of this equation to that reported by Sawant and Joshi (1979) is obvious. It is, therefore, apparent that vortex formation is a key phenomenon in the beginning of the gas induction process. We present a hypothesis of a sequence of events leading to gas induction based on the phenomenon of vortex formation. This hypothesis is extendable to a semiempirical model as shown later. 1. At speeds less than the critical speed, the rotating liquid would form a vortex within the stator as described above. The forced vortex liquid surface would be paraboloid in nature, and the vortex of the paraboloid would be above the impeller at speeds less than the critical speed. At the critical speed the vortex tip would reach the impeller and gas induction would begin. 2. The meeting of the tip of the vortex with the impeller is followed by the entrapment of bubbles in the liquid from the gas-liquid interface. This can occur in the following ways: (a) shear caused by the rotating impeller and (b) small vortices generated at the interface which entrap a bubble each, in a mechanism similar to that for surface aeration. (3) Having formed a bubble, the next step consists of conveying it hydraulically by the radially outward flow generated by the impeller to the tip of the impeller and dispersing it radially. The process is similar to radial dispersion of gas by impellers. However, it is also necessary for the dispersion to have adequate momentum, to cross the impeller-stator gap and the radial width of the stator. Gasinduction can be said to have begun only when a bubble enters the bulk. The momentum required to cross over to the bulk would depend on physical properties of the gas-liquid dispersion and the stator-impeller gap and resistance characteristics of the stator width. Gas bubbles that cannot cross over from the stator inside to outside would be floating up within the impeller zone, causing an internal circulation and no net gas induction. 4. At speeds greater than critical, the vortex paraboloid would occupy a wider area at the impeller plane exposing a greater area of the gas-liquid interface to the shearing action of the impeller plane. 5. As more gas bubbles are ingested and effectively carried away by the radially outward flow into the bulk of the liquid, a suction would be created at the impeller center and gas induction would continue. 6. For sustained and stable gas induction, both the liquid and the gas have to be coming into the impeller in an appropriate proportion. This is because the liquid is a conveying medium for the gas bubbles and has to go out with the gas bubbles. For sustained action, a material balance of liquid has to be set in. When an appropriate quantity of liquid is not available for the impeller to flow radially outward, gas induction could be adversely affected and instabilities could occur. 7. A larger vortex at the impeller center would leave a smaller radial blade length for imparting momentum to the liquid and generating a vortex as well as a radially outward flow. If more liquid comes to the impeller than that necessary for the gas carriage, a greater blade length may be required to pump the extra liquid and cause the vortex to shrink. This will reduce the gas induction rate. Ultimately an equilibrium will be reached. 8. The process of reaching equilibrium with respect to gas induction rate and liquid circulation rate and ap-
portioning the impeller blade lengths between the central gas ingesting zone and the outward gas-liquid dispersion zone would be helpful in explaining various oscillating behaviors of the gas inducing impeller. 9. When the central zone is practically filled with gas, there will be virtually no power consumption in that zone. There will be no liquid to form any vortex, and the gas induction will be reduced due to a breakdown of the mechanism explained in 1-8. With this broad outline, we propose to understand the reported behavior of the gas inducing systems in subsequent sections. 2.4. Effect of Design and Operational Variables. The rate of gas induction depends upon the design and operating variables such as impeller diameter, impeller design, stator design, impeller submergence, impeller clearance, vessel diameter, and physical properties. This subject has been actively researched during the past 50 years. The summary of this research work is given in Table 2. 3. Gas Induction Models
3.1. Role of the Stator-Impeller Gap in the Gas Induction Process. The hypothesis of the gas induction process described in section 2.3 requires that a stable vortex appear above the impeller. As the impeller can pump out the liquid continuously, the liquid level inside the stator is lower than the liquid submergence in the vessel. However, the liquid surface inside the stator takes the vortex shape. The height to which the liquid could be retained in the standpipe can be estimated from application of Bernoulli's theorem to this channel between location 1, the topmost height of liquid in the standpipe, and location 2, the free liquid level (see Figure 3B). Now at points 1 and 2, P1 = P2 = Patm
u1 = u2 = 0
(7)
With these simplifications, a simplified mechanical energy form of Bernoulli's equation can be written as zlg-
Wf
+ wi = zag
(8)
Note that all heights are with respect to the impeller center plane and are increasing upward. Expressing eq 8 in terms of a depression in the standpipe with respect to the level of liquid in the vessel, Az,
az = z2 - z1
(9)
g(az) = wi - Wf
(10)
then
The values of wi and wf can be written in terms of the impeller tip velocity
Note that ki is a head gain coefficient and kf is a head loss coefficient, based on the tangential velocity of the fluid at the impeller tip which forms one corner of the channel. The constants also take into account the variation of liquid level inside the stator. It may be pointed out that the lowering of the liquid level and the formation and maintenance of a vortex are very important for the process of gas induction. Thus it is seen that encroachment up along the standpipe has to
2230 Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994
v)
E
2
E
2
3
m 0
2
u'
.-* e c 0
P
-.-IP e,
0 W
e
%
.-E! m
W
t-
W
Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994 2231 ( 6 )
(A)
SI M P L l Fl ED PHYSICAL ANALOGUE
CONVENTIONAL ASSEMBLY
I
I
J!L
9LIQUID FLOW
t+ A
~
k IRwfm)'
29 # B
BOUNDARY CONDITIONS
r- R, P=
P,,,Z=#A
Figure 3. Simplifiedrepresentations of the impellemtator assembly.
253
0 d i
be optimal. Excessive encroachment would flood the standpipe destroying the vortex, and inadequate encroachment would render it devoid of liquid, preventing stable vortex formation. This explains the optimality of the stator-rotor gap experimentally reported by Zaitsev and Guseva (1983) and Klassen and Mokrusov (1963). Obviously, an impeller which pumps liquid from outside to inside will never induce gas. The points discussed above lead to an explanation as to how a rotating mass of liquid can emerge above the impeller in the stator standpipe assembly, essentially caused by the pitched turbine downflow (PTD) impeller in the stator and the standpipe, and be maintained by the controlled liquid access through the stator-impeller gap along the impeller hood and standpipe. Such a rotating liquid mass is known to take a vortex shape (Brennan, 1976) as discussed in earlier, qualitatively, leading to an interaction of the gas phase with the impeller and the beginning of gas induction. 3.2. Model for the Critical Speed for Gas Induction. For a mathematical expression representing the critical speed of gas induction in terms of vortex formation, we proceed by making a set of assumptions: (1) The entire gas induction process is assumed to occur at the center plane of the impeller. Submergence of the impeller and clearance from the bottom are both assumed to be measured with reference to the impeller center plane. Any effect caused by the impeller width and other factors is assumed to be incorporated implicitly in a submergence reduction parameter, a. The effective submergence is then taken as S - a. An impeller that is more capable of gas induction will be represented by a positive value of a. A design detrimental to gas induction will be represented by a < 0. (2) The pressure of the gas above the liquid level as well as in the standpipe is assumed to be constant at all times, both before and after gas induction. This implies negligible resistance in the gas line connected to the standpipe as well as at the outlet. (3) The area of the standpipe is assumed to be negligibly small as compared to area of the vessel. A correction factor is incorporated
2232 Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994
later to account for the finite standpipe area. (4) The complex geometry of the stator-impeller hood and standpipe is replaced with its hydraulic equivalent as in Figure 3. This simplification is essentially for understanding the vortex and other related phenomena. (5) The fluctuating nature of the flows and any oscillations caused by the hydraulic/pneumatic phenomena are disregarded for the current analysis. Steady states and steady flows are assumed for the impeller at all times. (6) The impeller hood and standpipe are assumed to be impermeable to liquid anywhere in the vessel. Gas will enter only at the top of the standpipe with negligible pressure drop as shown in Figure 3. (7) The axially upward flow at the impeller eye is assumed to be limited to a central core of radius I. At 0 Ir IZ, the impeller is assumed to be flooded with liquid that is not conforming in its motion to that of the impeller. So far as the tangential motion of liquid in the impeller and standpipe zone is concerned, this central zone of the impeller is assumed to be entirely ineffective. The effect of the impeller on the tangential motion is assumed to be restricted between I Ir IR. Recycling of gas bubbles via this recycling fluid is assumed to be negligible. Any gas induction observed is then gross gas induction. Any internal recycling of gas bubbles within the stator-impeller zone is assumed to be represented by the model parameters. (8) The radially outward and axially downward flow at the impeller periphery and axially upward flow at the impeller eye are assumed to be in steady state with each other. The magnitude of this flow is assumed to be adequate to entrap a bubble at the gas-liquid interface and convey it radially through the stator. However, the magnitude of these flows is considered negligible as compared to the tangential flows. For the purposes of simulation, only the tangential flow is, therefore, considered in modeling the vortex. Any effect of the neglected flows is assumed to be incorporated in appropriate parameters of the model, notably f m , described below. (9) The cylinder of liquid of radius R , at and above the impeller plane, is assumed to be rotating at a constant angular velocity of fmw where f m is called a conformity factor. Effects of the unsteady flow behavior of the liquid in the impeller zone, or the neglected radial and axial flows, would influence this conformity factor. (10) The influence of the impeller is assumed to decline in the stator-impeller gap. The liquid height in this gap, expressed as a depression relative to the static liquid height outside the stator and it, is expressed by an equation
where k is the net head coefficient of the impeller. 3.2. Equation for the Critical Speed of Gas Induction. Published literature indicates that vortex shape in an unbaffaled vessel obeys a very complex equation (Riger et al., 1979). However, the following simplified model is adequate for our consideration. Adopting a cylindrical coordinate system with Z increasing in the vertical downward direction and the origin at the intersection of the shaft axis and the static liquid level (Figure 3B), it can be shown that the differential equation governing the tangential motion of a liquid is as follows (Bird et al., 1960) d p = pfm2wrdr + pg dz One boundary condition is that at r=O
(13)
Integrating to any other r, z , p value
The shape of free surface of a liquid corresponds to a locus of all points where p = Pat. Substituting and simplifying with V = Rw
This is the equation of a paraboloid, and the z coordinate essentially defines a depression seen at various radial positions in the standpipe. The maximum depression is obtained at r = 0. However, as per assumptions of section 3.1 a central portion of the impeller between 0 5 r II is not effective. The vortex is destroyed in the central portion. Therefore, the maximum depression with reference to the initial unagitated height will be at r = I and
The vortex would not exist at 0 Ir 5 I (see Figure 4A). When this depression equals the effective submergence, S - a, the gas phase would get connected to the impeller and gas induction could potentially begin. This is mathematically shown as
-1
S = a (vf + 12[ z (1+ k ) - 12 (16b) 2g R2 We can simplify the above equation with the following substitutions, for convenience,where the subscript c refers to the critical speed of gas induction.
+
(1 k,)f,; = 4,
a = a,
I:/(l
+ k,) = Z,*2
(17)
Substitution of eq 17 in eq 16b modified eq 16b to
s = a, + $,(
$( - $) 1
This is the equation for the critical speed of gas induction. The constant 4, is interpreted as a lumped parameter defining the ability of the impeller to form a vortex and hence called the vortexing constant at critical speed. For all systems, where gas induction is feasible, & > 0. The gas induction is more facile for larger values of 4,. This value of & increases with an increase in the value in jmc. This means a greater conformity between the impeller and the fluid rotated by it. An increase in k, means a greater head developed by the impeller in excess of the head lost due to fictional head loss of liquid in flowing across the stator. The value of 9, tending to zero means that an impeller which rotates a t high speed will rotate the liquid and cause a vortex, with a consequent beginning of gas induction. Values of 4, < 0 imply a value of kf >> ki resulting in a negative value of the k, and 1+ It, terms. This implies that all the radial flow head generated by the fluid is lost due to friction at the stator and the gas cannot be carried across the stator and be effectively induced. Gas induction can never occur for such systems, as the
Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994 2233 ( B1
(A) AT V < V c
(C 1
AT V s Vc
AT V
> Vc
I
#A
=
JJI
zs
k (Rwfrn)2 29 GAS P H A K INGRESSING LIQUID ZONE (INEFFECTIYE )
#C
#G LIQUID FILLED ZONE
VORTEX INTERSECTION WITH IMPELEER PLANE
kg (Rwfrng)?
=
0 c
aep
29
GAS INDUCING ZONE LIQUID FLOW G A S LIQUID DISPERSION
Figure 4. Status of forced vortex in the standpipe before, at, and after the critical speed of gas induction.
critical speed of gas induction is an imaginary number. Likewise a, is the reduction in submergence afforded by the ability of the impeller-stator assembly to induce gas prior to the vortex reaching the impeller. Whereas I, is the actual radius of the ineffective portion at the impeller eye at the critical speed, I,* is the value scaled with the head reduction coefficient at the periphery of the impeller. 3.3. Correction Due to Finite Standpipe Area. One of the assumptions in deriving eq 18 is that the area of the standpipe is negligible as compared to the vessel area, whereby the clear height of liquid remains a constant at any state. However, this is not true in actual practice as the liquid in the standpipe gets displaced into the vessel with an increase in the submergence of liquid. The following correction needs to be incorporated
For large setups fsp = 1. However, for small laboratory setups fsp > 1and the correction may become significant. 3.4. Equation for Vortex Shape at Speeds Greater than Critical. The basic equation for the paraboloid surface of a vortex is given in eq 15. It is also seen that the maximum height of the depression, Zm,, occurs at r = I and, as one traverses up along the vortex surface, the radius of the vortex shape is given implicitly by
have just been reached, at the impeller tip speed V. We would write for this situation
Note that we have replaced a,,f m c , k,, and I, by %, fW, k,, and I, to account for the significant differences between the impeller situation at the critical speed and while inducing gas. At the critical speed the liquid in the standpipe and the impeller zone is practically free of gas bubbles. However, during gas induction, the liquid phase is rich in gas bubbles. Under such conditions all coefficients are expected to be different. Now according to eq 20 the vertical distance S, - S defines the radius r at the impeller plane which is exposed to the gas phase. Substituting from eq 21 and comparing s, - S to Z m , - Z of eq 20
Analogous to 4,, we can define a vortexing constant 4, + k,)fW2 and for simplicity, we define the radius r which is playing arole in gas induction as X for convenience of notation. So the modified eq 22 gives the radius of the intersection of the paraboloid of the vortex with the impeller plane. = (1
_ x 2-- (1 +k,)
Now consider any operational impeller tip speed in excess of the critical speed or V > V,. Let S, be the hypothetical submergence at which the critical speed would
[ "ziag']
1(23) R2 The vortex shape for this case and other details are shown in Figure 4C. 3.5. Assumptions made in Modeling the Gas Induction Rate. The model developed here essentially quantifies the description of the hypothesis proposed in
2234 Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994
section 2.3. Some of the aspects of quantification are amplified along the assumption. (1)The gas induction rate is assumed to be proportional to the interfacial area of gas-liquid contact at the impeller plane and the impeller's speed of rotation. The proportionality constant, which has dimensions of length, is identified as the gas induction modulus, X. It has a significance of being the thickness of the gas film successfully imbibed and induced into the vessel. The dependence of the impeller speed arises from the process of gas bubble inclusion as well as gas bubble carriage. The gas bubble inclusion is due to shear caused by the impeller, and gas bubble carriage is due to liquid flow caused by the impeller. Both these process are proportional to the impeller speed. (2)The interfacial area of gas-liquid contact at the interface is assumed to be proportional to the area of intersection of the vortex with the impeller plane. The actual shape of the interface is expected to be complex. However, if we assume it to be flat for simplicity, the proportionality constant, if any, will merge with the gas induction modulus. (3) The central core of the impeller, influenced by the ingress of liquid entering into the eye of the impeller, will continue to be ineffective as far as the vortexing action of the impeller is concerned. The net gas-liquid interfacial I:). Any curvature area is, therefore, proportional to (Pfactor discussed above will also merge with the value of Ig and A. (4)The impeller is assumed to be radially similar and continuous, for hydrodynamic considerations. This is a justification for assuming a constant value of the gas induction modulus and the vortex parameters for any size of the vortex for a given impeller. Following these assumptions,
Q, = ANr(X2- Ip")
Substituting eq 25 in to eq 24
We now define scaled parameters and adapt eq 26 to form eq 27 by substituting = I t / ( l + k,)
(27)
This equation can be easily adapted for applying regression analysis to data of gas induction rates for impellers of various diameters operated at varying submergences and impeller speeds. For regression, this can be adapted to Qg
= X*NR2 - X*122N- (A*/$,)
(NR")+
LEVEL INDICATOR
1
-
nC1'1
I
I Figure 5. Schematic of the experimental setup.
Evaluation of each regression coefficient allows us to r$,, and ag. The evaluate all four constants, A*, conformity factor fw cannot be extracted from this analysis due to a limited number of constants available. 3.6. Correction for Finite Standpipe Area. For reasons similar to those discussed in section 3.2 the submergence of the impeller is required to be corrected. After correction for finite standpipe area the equation for gas induction is modified to
(24)
where X is to be called the modulus of gas induction and has a significance of being the thickness of the gas film imbibed into liquid for every revolution of the impeller.
A* = ?r(l + k,)X
CONTROL PANEL
4. Experimental Section 4.1. Experimental Setup. The schematic diagram of the experimental setup is shown in Figure 5. The prime mover was a DC motor, driven through a thyristor controlled power source. The speed of rotation of the shaft was controlled by regulating the voltage applied to the armature of the motor by a 10 turn potentiometer. The entire device was able to control the speed and maintain it within i O . 1 7 6 accuracy. The entire assembly was mounted on a rigid frame of channels concentric to a torque table on the ground. The torque table was mounted on a pair of taper rollers and ball bearings and was practically frictionless. The table was mounted on a screw jack, which can be raised or lowered adequately, to maintain the desired clearance between the vessel bottom and the impeller. Five pairs of impellers and stators of different sizes were fabricated from carbon steel, the details of which are listed in Table 3 and schematically shown in Figure 1B. The gas was conducted to the standpipe from the ambient through a pulsation dampener with a volume of 200 L all connected by a 50 mm i.d. rigid PVC hose. The pulsation dampener and the flanged hose connections were periodically tested for leakage under a pressure of air at 1.0 m of water column. 4.2. Instruments and Measuring Techniques. The air flow rate was measured by a fractory calibrated turbine type anemometer. All measurements were carried out
Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994 2235 Table 3. Design Details of the Impeller-Stator Assemblies and Vessel.
Table 4. Summary of Condition for Runs during Gas Induction ~~
unit impeller nominal diam, m Db, mm W, mm stator DE, mm
We ha
19 0.19 186.5 40.5
2 22 0.22 224.5 43.2
3 33 0.33 327.0 65.5
4 40 0.40 390.0 81.5
260 34 44
315 40 52
466 60 77
560 72 92
1
5 50 0.550 495.0 104.0
mnemonic 5719 5719 1019 1022 1033 1519 1522 1533 1540 1550
700
90 115
aCommon parameters: (1) hub diameter, Hb = 50 mm, (2) extension of hub below impeller = 15 mm, (3) shaft diameter below impeller = 50 mm, (4) impeller blade angle = 43-46', (5) blade length = 0.5(D - Hb)+ 12 mm, (6) blade thickness of all impellers = 2.8 mm, (7) plate thickness for all stators = 3.5 mm, (8) number of vanes in stator = 12, (9) number of blades in impeller = 6, (10) angle of vanes = +30° to radius vector, (11)direction of rotation = clockwise when viewed from top, (12) vertical height of impeller hood = 35 mm. b Measured between center and center of opposite blades, and the average of three values is reported.
using atmospheric air and tap water. Ambient temperature for all runs at 30 f 3 "C. The main source of error in the flow measurement was intrinsic fluctuations in the process of gas induction. This source of error was minimized by a 200-Lpulsation dampener. Virtually no fluctuations were observed in the case of runs with large impellers. Some fluctuations were observed for runs with small diameter impellers especially at low submergences in large vessels. To counter the inaccuracies of fluctuations, each flow reading was taken 10 times and the average value was used for calculating air flow rates. The observed gas flow rates were corrected to standard temperature and pressure, for reporting. Air line pressures were also constantly monitored at the flowmeter, the pulsation dampener, and the standpipe using water manometers and were always practically equal to atmospheric pressure. The critical speed of gas induction was measured by noticing the smallest speed at which air bubbles first appeared on the free surface of the liquid. This was very easily detected for most runs. However, for small diameter impellers and at submergences in excess of 1m, the critical speed varied widely. The gas was induced in spurts. The critical speed was identified to be the lowest speed at which about 10 spurts were issued every 100 s, more or less uniformly. The definition was empirical and based on the observation that this was generally the largest stable frequency of gas issued in spurts. Above the critical speed the gas rate tended to be continuous. Below this frequency the gas issue was very irregular.
~~~
minlmax values
+ (L
*a
;
C,m
N , rls 0.09510.285 0.150l0.600 4.8112.8 0.09510.285 0.15010.600 3.418.60 0.160/0.500 0.15010.600 4.517.30 0.16010.500 0.15010.600 3.517.80 0.09010.500 0.150/0.600 2.217.80 0.50010.750 0.15010.600 3.219.80 0.50010.750 0.150/0.600 2.719.10 0.50010.750 0.15010.600 2.217.80 0.500/0.750 0.150/0.600 1.714.40 0.50010.750 0.15010.600 1.413.30 C: 0.600m C: 0.313m
C:
S,m
1
Qs, nL/s 0.517.800 0.96113.6 1.117.300 1.0116.30 1.0133.30 0.817.900 0.9112.30 1.0110.30 1.27119.4 1.42120.3
1IMPELLER?E;""
0.260m
C = o 160m
-
e
3.5 0.75
I
0.3
0.35
I
I
I
4
0.6 0.65 0.5 SUBMERGENCE S,lrnl
0.55
I
0.6
(
55
Figure 6. Critical speed of gas induction aaa function of submergence and clearance for run 1033. Table 5. Data on &''and a, Values for Various Combinations of Tank and Impeller Sizes
T 57 57 10 10 10
D 19 22 19 22 33
&'I
0.7500 0.8595 0.7000 0.7991 0.9848
a, 0.0895 0.0944 0.1068 0.0946 0.0739
T 15 15 15 15 15
D 19 22 33 40 50
&'I
0.8965 0.9552 0.8518 1.0261 1.0009
a, 0.1076 0.1038 0.1219 0.1219 0.1300
5.1. Critical Impeller Speed for Gas Induction. Typical data on the critical impeller speed for gas induction versus impeller submergence, at different clearances from bottom for the 1033 run are presented in Figure 6. The monotonously increasing critical speeds lie on the same curve for C 1 0.16 m. The model equation is
5. Results and Discussion
A number of runs were performed for measuring the critical speed of gas induction and rates of gas induction. These measurements were made for various combinations of vessel diameter, impeller diameter, impeller clearances, and liquid height, as well as different impeller speeds. Even the submergence of the impeller was independently varied for some runs. The range of variables covered for each impeller-vessel combination is summarized in Table 4. For the sake of brevity, the runs are identified with a four digit number where the first two digits signify vessel size and the next two digits signify impeller size. The run identification numbers also appear in Table 4. Other abbreviations used are summarized in the Nomenclature section.
+
= Ic2/(l k,). where 4, = (1 + kc)fmc2and For a given impeller-vessel combination this reduces to
(32) where &" = (&/fsp)(l - Ic*2/R2). For 1033 runs, &" = 0.9848 and a, = 0.07390 m with a correlation coefficient (CC)of 0.995 at 16 DF. In view of the strong evidence for correlation in terms of eq 31 and no evidence of any relation to D or T,a regression on the entire data set was performed. The values of &" and a, for different vessel-impeller combinations are given in Table 5.
2236 Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994 Regression of the entire data set of 107 points showed that the following coefficients (significant at >99% ) represent it with a CC of 0.989.
4, = 1.065
I,*2 = 0.00342 m2, I,* = 58.50 mm
a, = 0.0394 m (33)
In conclusion, the following equation is recommended for prediction of the critical speed of gas induction, within &8% for S > 0.20 and C > 0.25 m.
for 0.19 ID 50.50 m, 0.57 IT I 1.5 m, and 0.12 C DIT C 0.59. Values of constants are in eq 33. A parity line is shown in Figure 7 parts A-C. Note that for all practical purposes N, is independent of T. 5.2. Gas Induction Rate. The most important variables influencing the rate are the impeller diameter, its speed, and its submergence. Figures 8-10 show gas induction rate versus impeller speed data for various impellers, submergence, and vessel combinations. Figure 8 shows data for all impellers studied in a vessel with a diameter of 1.5m for a submergence of 0.3 m and clearances in excess of 0.25 m. The effect of submergence is shown in Figure 9 for 1033 runs. Figure 10 shows the effect of induction rate versus impeller speed for the same impeller in different vessels. Figure 11 shows the effect of submergence on the critical speed of gas induction for various impellers in a vessel with a diameter of 1.5 m. From our previous discussion, the model for the process of gas induction is represented by the following equation.
C O R R E L A T E D VALUES O F C R l T l C A L IMPELLER N, (YlsI
SPEED FOR G A S INDUCTION
1
I
1
I
I
I
I
I
7 3 5 1 CORRELAlEO VALUES OF CRITICAL IMPELLER SPEED FOR GAS INOUCTION N ITIS I
C
9 v
A* = a(l
1
7
U
+/
UJ
where
+ k,)A,
Ig*2 = I;/(l+ k,),
For a given impeller-vessel combination, the equation is adapted to
aA ms
*/g-
I
I
where
UJ
A*(R2-
(37a)
Pg = fspA*R2/+,
(37b)
YQ = A*R2a,/4,
(37c)
Lug =
From the values of a,0, and y the following relations are obtained.
X*(R2- Ig*’) = ‘YQ (38) The adequacy of eqs 35 and 36 to represent the effect of submergence was gauged by studying the error between the experimental value and the predicted value obtained
I
I
I
7 3 5 CORRELATED VALUES OF CRITICAL IMPELLER SPEED MR GAS INDUCllOh N, I T 1 5 1 1
Figure7. Parity line for the critical speed of gas induction correlation for the (A) 15XX,(B)IOXX,and (C)57XX runs.
by the regression equation. The error did not vary systematically with the submergence of impeller except at S < 0.20. This shows that the influence of submergence is adequately represented in the Froude number terms of these equations for S > 0.20. 5.2.1. Effect of Clearance from Vessel Bottom. Various values of ~ Q / @ Q~, Q / @ Qand , CYQfor 1033 runs at different clearances from the bottom are shown in Figure 12 for clearances in excess of 0.25 m. The values of aQand f f Q / @ Q which are proportional to A* and $ , respectively, are constant for all practical purposes. The value of YQ/ PQ which is proportional to ag is also constant. A t low clearance, both the gas induction modulus A* and the vortexing constant 4, are seen to drop significantly.
Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994 2237 f D =0.500m
*
D
=
0.400m
x D = 0.330m
+ D = 0.225m A D
0
1.5
3
4.5
IMPELLER
6
7.5
= 0.190
9
m
10.5
Figure 8. Gas induction rate as a function of impeller speed and impeller diameter for run 15XX.
8
6
SPEED, N(rls)
IMPELLER
SPEED, N ( r / s )
Figure 10. Gas induction rate as a function of impeller speed for the same impeller diameter in different vessels (D = 0.225 m).
+
S-0.60m
0 1550
t 15-40
;C 1533
E 1522
X 1033
i.
S=0.50m
4 1022
A 1019
X 5722
0
5719
0 1519
0
S=0.40m
X
S=0.30m S=0*15m
A
S=0.70m
0.09 IMPELLER
SPEED, N(r/s)
0.19
0.29
0.39
049
0.59
SUBMERGENCE, S ( m )
Figure 9. Gas induction rate as a function of impeller speed and submergence for run 1033.
Figure 11. Critical speed of gas induction for various impeller and submergence values.
Both are attributable to resistance caused by the proximity of the vessel bottom. The vortexting constant q5g is a lumped parameter consisting of (1 + k,)fmg2.This could
drop due to the resistance caused by the vessel bottom. The gas induction modulus could drop due to inability of the impeller to generate adequate radial velocities to carry
2238 Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994 4
Table 6. Coefficients of Gas Induction Equation for Various Runs run S range Q range a P Y alP YIP C
a
3.5
'1
1550 0.25-0.60 1.40-20.2
8.5084 7.9916 0.3828 1.0647 0.0479 0.25-0.75
1540 0.25-0.60 1.20-19.4
5.4307 5.4690 0.1206 0.9929 0.0221 0.50-0.75
1533 0.50.75 1.00-10.3
2.7896 2.9117 0.0919 0.9580 0.0315 0.50-0.75
]
1522 0.50.75 0.90-7.90
1.8559 1.7870 0.0793 1.0385 0.1473 0.5-0.75
Figure 12. Coefficients of gas induction correlation as a function of clearance for run 1033.
1519 0.50-0.75 0.80-4.50
0.3097 0.3006 0
away the gas bubbles. The reduction in flow generated by the impeller is attributable to starvation of the impeller of liquid due to bottom resistance to flow for return of liquid to impeller zone from the vessel. Gas that is not carried away as fast as it is induced can be lost back to gas space reducing the effective rate of gas induction. The average values of a, alp, and yl/3 shown below were obtained by performing regression on all 1033 data for C 1 0.25 m. An error analysis of the data showed that most of the data was predicted by the equation within *lo%. Neglecting the data at clearances I0.167 m and Qg 5 0.96 us,
1033 4.1609 5.0844 0.0191 0.8184 0.033 0.50-0.75 1.00-33.27
2.5
% d
I
I
1.5
0.5
0
1
r/B '
0 .
0.05
J '
0.15
0.25 0.35 IMPELLER C L E A R A N C E , [ m l
I
0.15
I
The significance of the coefficients increases beyond that obtained while considering the low gas rate data. The low gas rate data is scattered around the predictions of the proposed equation. This could be due to the effect of the phenomenon of gas induction even before the vortex physically reaches the impeller, which is not included in the model. At low rates of gas induction the effect of downflow generated by the impeller may not be completely simulated by the submergence correction term ag alone, causing a systematic error. In further data processing, the data of low gas induction rates (-0.96 LIS) and low clearances was eliminated if it showed very large deviations. 5.2.2. Effect of Impeller and Vessel Diameter. Equation 36 was fitted to all impeller-vessel combination data over a wide range of submergences. The data are summarized in Table 6. Equation 36 also helps to correlate the values of "9, PQ, and YQ with impeller radius. 5.2.3. Overall Correlation for Gas Induction Data. Equation 35 was fitted and a CC of 0.942 was obtained at 587 DF and a standard error of estimate of 0.45 nLIs. All coefficients of regression were significant at the >99 % level. Values of the parameters obtained were A* = 172.37 mm
Ig = 61.7
X*Ig*' = 0.65 m3 4g = 0.929 X*/4, = 160.2 mm
However, error analysis showed that for some points in the 1519 run and all points in the 5722 and 1022 runs, the correlation showed a systematic underestimate. Although
1.0305 0
0.50-0.75
0.16-0.50
1022 0.25-0.60 1.00-19.3
1.8385 2.6305 0.0720 0.6989 0.0273 0.16-0.50
1019 0.25-0.60 1.00-7.30
0.7993 1.0564 0.0178 0.8277 0.0168 0.16-0.50
5722 0.15-0.60 0.96-13.6
1.5960 3.3745 0.2203 0.4729 0.0652 0.095-0.285
5719 0.15-0.60
0.9569 1.6594 0.0702 0.5767 0.0423 0.095-0.285
this error was within 15% limits, it was considered worthwhile to isolate this data and process the two lots separately. Thus, for the 1550,1540,1533,1522,1519 (for S < 0.75 m), 1033,1019, and 5719 runs the CC was 0.974 at 478 DF and a standard error flow rate estimation of 0.26 nLIs, with the following regression coefficients, A* = 169.37 mm
Ig* = 58.28 mm X*IC2 = 0.574 4g = 1.101 X*/r#Jg = 153.78 mm (39)
for 0.20 < S < 0.80 m, C > 0.145 m for D 50.22 m, and CID > 0.5 for D 10.33 m. Parity lines are shown in Figure 13 parts A-C. With these constants the gas induction rate is predicted within f10% for Q < 1.5 nL/s, For design purposes eq 39 is considered adequate and is valid within the following range of variables. 0.20 < S < 0.75 m, C > 0.145 m for D 5 0.22 m, 0.01 < 2 g S l F < 1.16, 0.124 < DIT < 0.394, 0.19 < D < 0.50 m, 0.57 < T < 1.5 m, Q > 1.0 nL/s The evaluated parameters are summarized in Table 7. I t is seen that there is a significant difference in the vortexing constant and submergence correction factor between the data of set 1 and set 2. The hydrodynamic reasons for this behavior need further study. However, the coefficients of set 1produce an overestimate of the gas flow rate, therefore only the coefficients of set 2 are recommended for the purpose of design. Comparingthe coefficientsabove the critical speed and at the critical speed, the value of the central ineffective
Ind. Eng. Chem. Res., Vol. 33, No. 9,1994 2239
A-
gas induction process is occurring. The plane appears to move downward as gas induction begins. The vortexing constant is also reduced from 1.065 to 0.929. This decrease implies that the flow resistance coefficient of the stator to flow of the gas-liquid suspension (kf)increases or the coefficient of head generation by the impeller (ki)is reduced, simultaneously with a reduction in rotational speed of the liquid vis a vis the impeller. These effects are lumped up in 4,, and further work is necessary to resolve them.
. el
g ul I-
IO
-
9 -
0 -
6. Conclusions 4
50
5
5 -
I 0
, , , , , , , , , 1
1
1
1
(
,
1
The published literature on typical gas induction systems consisting of a rotor with a stator has been reviewed. An analysis of the gas induction phenomena and the published models has been presented. Various aspects of the process of gas induction in the system with a stator can be interpreted in terms of the phenomenon of vortex formation. Semiempirical models have been developed for the critical speed of gas induction and the gas induction rate. The model parameters have physical significancein terms of vortex related phenomena. The critical impeller speed of gas induction is represented by three parameters, viz., 4, (the vortexting constant), a, (the submergence reduction), and I,* (the scaled ineffective core radius) through eq 31. The gas induction rate is represented by four parameters, viz., A* (the scaled gas induction modulus), & (the vortexing constant in the gassed state), andI,* (the scaled central ineffective core radius in the gassed state),and a, through eq 35 or 36. The models developed in the previous section are satisfactory for design and scale-up for the critical speed of gas induction (eq 34) and the gas induction rate (eq 351, for 0.19 < D < 0.50 and 0.57 < T < 1.5 m.
, , , , , ,
3 5 7 9 11 I3 15 17 19 CORRELATED RATE OF GAS INDUCTION,OgIn.LIs I
71
B -
r V
o
r
m 0 Q Y
0
f a
0 & 8 I? 16 20 24 28 CORRELATED RATE OF GAS I N D U C T I O N , O q ( n . L / s I
32
C
Acknowledgment
K.S.and V.D.M. are greatful to the University Grants Commission and the Government of India for the award of fellowships. The research work was sponsored by a Grant under the Council of Scientific and Industrial Research (CSIR), New Delhi, India [4(104)/EMR-II of 10-4-893. Nomenclature ILI
a ul
o
l 0
Y
l
~
l
~
l
l
l
l
~
l
7 4 6 8 10 12 CORRELATED RATE O F GAS INDUCTION Oq ( n . L / S 1
l
l
Figure 13. Parity line for eq 36 for the (A)15XX,(B)LOXX,and (C)57XX runs. Table 7. Summary of Parameters Representating Equation for Critical Speed and Gas Induction Rate at during gas induction parameters speed set '1 set2' vortexing constant & 1.065 4, = 0.929 Qg = 1.101 a, = 0.0394 a, = 0.128 e, = 0.0 submergence correction a,m ineffective core radius Z,mm Zc= 58.5 Zg= 61.7 Zg= 58.28 gas induction modulus A*, mm A* = 169.3 A* = 169.3 0 Set 1 : authors' data. Set 2: others (5719,1019,1519,1033,1522, 1533,1540,1550).
core is practically the same. The submergence correction is reduced, and the reduction is marginally significant. This is interpretable as a shift in the plane a t which the
a
la
= interfacial area (m2, m-3)
l =l submergence correction at impeller periphery (m)
C = impeller clearance from bottom (m) CH = impeller head coefficient in eq 4 (dimensionless) c = constant depending on impeller design CC = correlation coefficient D = impeller diameter (m) d = constant depending on impeller design d~~~ = maximum gas bubble size in gas liquid dispersion (m) DF = degrees of freedom ED = density of energy input (W, ma) Eu, = gas-phase Euler number in eq 4 (dimensionless) f m = conformity factor F f = Froude number based on corrected submergenceWD2/ gS (dimensionless) Fr = Froude number based on uncorrected submergence W D 2 / g S , (dimensionless) Fr,= Froude number at critical speedWD21gs(dimensionless) g = gravitational acceleration (m s - ~ ) GIMAC = gas inducing mechnically agitated contactor h = depth of vortex (m) H = liquid height (m)
2240 Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994
I = ineffective radius at impeller eye for gas induction process (m) I*= scale ineffective radius at impeller eye for gas induction process (m) JLR = jet loop reactor ke = Brenner defined constant in eq 3 k D = Sawant defined constant in equation 6 kl = head reduction coefficient,see eqs 1and 5 (dimensionless) N = impeller rotational speed (r/s) NQ= gas flow number, Qg/ND3 (dimensionless) p = pressure (Pa) PTD = pitched turbine downflow Q = volumetric flow rate (m3 s-l) Qg = gas induction rate (m3 8-1) R = impeller radius (m) SAR = surface aerating reactor SLR = sparged loop reactor S = submergence of impeller in clear liquid (m) S' = corrected submergence, see Table 3 (m) T = tank diameter (m) V , = superficial gas velocity (m 9-1) V = impeller tip velocity (m s-l) VL = volume of liquid in dispersion w = specific work (Wkgl) Wp = vertical projected width (m) Z = term related to concentration of gas in impeller zone Greek Letters a = aspect ratio of impeller, W/D (dimensionless) A = drag coefficient of impeller eg = gas hold up, (dimensionless) q = liquid-phase flow number (dimensionless) p = density of liquid (kg m-9 P A = density of dispersed gas (kg m-3) pa = density of gas-liquid dispersion in impeller zone (kg m-3) p = viscosity (Pa s) u = surface tension of liquid (N m-l) 4 = vortexting constant of PTD design (dimensionless) &" = vortexting constant of particular impeller size (dimensionless) X = gas induction modulus (mm) w = angular velocity (rad-s-1) Subscripts 1 = liquid g = gas, gassed state c = critical state for gas induction sp = stand pipe atm = atmospheric i = impeller Superscript * = scaled parameter Literature Cited Arbiter, N.; Harris, C. C. Flotation Machines. In Froth Flotation; Fuerstenau, D. W., Ed.; AIME New York, 1962;Chapter 14,pp 347-364. Arbiter, N.;Harris, C. C.; Yap, R. J. Hydrodynamics of Flotation Cells. Trans. SOC.Min. Eng. AZME 1969,244,134-148. Arbiter,N.;Harris,C.C.;Yap,R.J.TheAirFlowNumberinF1otation Machine Scale-up. Znt. J. Miner. Process. 1976,3,257-280. Babcock-Hitachi, K. K. Gas Liquid Contactor. Japanese Patent Tokkyo Koho 80,106,532, August 15,1980;Chem. Abstr. 1981,94, P158800m. Bollenrath, F. M. Stirring System and Methods for Gassing Liquids. German Patent DE 3,516,027November 06,1986;Chem. Abstr. 1987,106,P52295b. Brennan, D. J. Vortex Geometry in Unbaffled Vessels with Impeller Agitation. Trans. Znst. Chem. Eng. 1976,54,209-217.
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Zaitaev, V. A.; Guseva, T. P. Study of Hydrodynamics in Apparatus with Aerating Units. Khim. Neft. Mashinostr. 1983,10, 13-14; Chem. Abstr. 1984,100,53592j. Zlokarnik, M.; Judat, H. Tube and Disk Stirrers-Two Efhcient Stirrers for the Gassing of Liquids. Chem. Zng. Tech. 1967,39 (20),1163-1168. Zundelevich, Y. Power Consumption and Gas Capacity of Self Inducing Turbo Aerations. AZChE J. 1979,25 (5),763-773. Received for review February 23, 1994 Accepted May 16,1994@ @
Abstract published in Advance ACS Abstracts, July 1,1994.
