Gas-inducing impeller design and performance characteristics

Pergamon PII:

Chemical Enflineering Science, Vol. 53, No. 4, pp. 603-615, 1998 ,t~, 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0009-2509/98 $19.00 + 0.00

S0009-2509(97)00352-7

Gas-inducing impeller design and performance characteristics S. E. Forrester,* C. D. Rielly ** and K. J. Carpenter* * Department of Chemical Engineering, Cambridge University, Pembroke Street, Cambridge CB2 3RA, U.K.; tZeneca Fine Chemicals Manufacturing Organisation, PO Box A38, Leeds Road, Huddersfield HD2 I FF, U.K. Received 24 June 1997; accepted 6 October 1997

Abstract--A theoretical and experimental study on the design and performance characteristics of gas-inducing impellers is presented. In particular, the model developed by Evans et al. (1991, A.I.Ch.E. Spring National Meeting, Houston, TX, Paper 33e) is critically reviewed and, as a result, improvements to the kinetic energy pressure loss analysis and to the initial conditions are proposed. In addition, the model is successfully extended to account for multiple gas outlet orifice on each blade. Experimental measurements of the power consumption, rate of gas induction, mass transfer coefficient and detached bubble size for a partially optimised, 0.154 m diameter, six-bladed concave gas-inducing impeller are presented. A significant increase in the induced gas rate is observed by adding more outlet orifices to each blade. The principal advantage of using multiple orifices is that similar size bubbles are produced, compared to a single orifice, but larger interfacial areas are generated; the aerated power input is only slightly reduced from its ungassed value. Mass transfer coefficients, kLa, of the order of 0.02 s- 1 are attainable for a single outlet orifice on each blade; kLa is significantly increased by using multiple orifices. The dimensionless bubble size distributions, d/dgm are independent of the impeller speed over the range 4-8 rps, and can be successfully represented by a log-normal distribution. © 1998 Elsevier Science Ltd. All rights reserved

Keywords: Gas-inducing impeller; gas-liquid mixing; concave-blade impeller.

INTRODUCTION Gas-inducing impellers may be used as a specialised method of gas-liquid contacting in stirred tank reactors. As the impeller rotates, the liquid phase is accelerated over the surface of the contoured impeller blades, resulting in the formation of a reduced pressure region (see Fig. 1). This reduced pressure region on the blade surface is connected to the reactor headspace, via a gas inlet on the shaft above the liquid level, hollow shaft and blades, and an outlet orifice on each blade. The pressure difference between the blade surface and the headspace produces a gas induction effect. The magnitude of this driving force depends on the impeller speed and the radial position of the orifice; gas induction commences when the pressure at the orifice falls to the headspace pressure, i.e. when the static head of liquid above the orifice has been overcome. The speed at which this occurs is known as the critical impeller speed. These reactors may be operated in 'dead-end mode', so that almost all of the feed gas may be

*Corresponding author. Tel.: 01223 334777; fax: 01223 334796.

absorbed in the liquid. Thus gas-inducing devices display an advantage over conventional sparger plus recycle systems for a number of industrially important gas-liquid unit operations. In addition to being widely used for minerals separation using froth flotation (Joshi et al., 1982), potential applications are in the fine chemicals industry for reactions where external recycling of the headspace gas would be hazardous, e.g. for hydrogenations or chlorinations. Often surface aeration mass transfer devices are used to perform these gas-liquid reactions, but their performance is highly dependent on the static head of liquid above the impeller (Albal et al., 1983); small changes in batch volume can result in very significant reductions in the entrained gas flow and mass transfer rate. In contrast, Evans et al. (1991) have shown that the gas flow rate in gas-inducing systems is much less dependent on the liquid fill level. There are two principal objectives to this research. The first is to gain a better understanding of the gas induction mechanism such that an improved and general design method for gas-inducing impellers can be developed. To this end, the importance of understanding how the bubbles form and detach from the gas outlet orifices has previously been highlighted

603

S. E. Forrester et al.

604

Gas induction rate

N headspace baffle

hollow shaft T lO gas outlet orifice

T

Fig. I. Geometry of the gas-inducing system.

(Forrester and Rielly, 1993); a more detailed study of the bubble growth is presented below. A critical review of an existing model for gas-inducing impeller design has been conducted, in addition to experimental measurements of the bubble sizes generated within the gas-inducing system. The second objective is to measure the performance characteristics of a six-bladed concave gas-inducing impeller, for comparison with other self-inducing and gas-sparged devices. In particular, measurements of the gassed and ungassed power consumption, mass transfer coefficient, and the effect of using multiple gas outlet orifices per blade on the rate of gas induction rate have been conducted. The concave impeller was selected for this purpose based on gas induction rate measurements by Evans et al. (1991) and Rielly et al. (1992) on a number of different blade designs; the concave shape gave good gas-inducing characteristics and liquid pumping, without producing a large trailing gas cavity behind the blade. As a result, the impeller power consumption is not significantly affected by the presence of the gas over a range of aeration numbers, and high rates of mass transfer can be maintained.

PREVIOUS WORK

This section presents a critical review of the gasinducing impeller literature relevant to this research. The first stage in the design of gas-inducing devices is to determine the critical impeller speed which marks the onset of gas induction. This has proved to be a relatively simple problem and good agreement between the theory (summarised below in the Theory section) and the experimental results have been reported by Evans et al. (1991) for a range of blade designs. The next stage is to predict the gas induction rate for conditions above the critical impeller speed. This is a significantly more complex problem and a discussion of the various approaches employed is given below. Finally, literature on the mass transfer performance of gas-inducing impellers is considered.

The ability to make reliable predictions of the induced gas flow rate at a given impeller speed is of prime importance in the design of gas-inducing devices. The empirical correlations proposed by Raidoo et al. (1987) and Sawant et al. (1981) consider the gas flow rate to be a function of the impeller speed, N, diameter, D, and submersion depth, h. However, such correlations take no account of the physical processes involved and, therefore, are unlikely to give acceptable predictions of the gas induction rate outwith the range of experimental conditions for which they were derived. This is illustrated by the lack of quantitative agreement between these workers on the exact dependence of induced gas flow rate on each of the three parameters. More reliable design methods require a better understanding of the fluid dynamics involved. The semi-theoretical model described by Martin (1972) is based on the assumption that the dominant pressure drop along the gas pathway is across the gas outlet orifice on the impeller blade. As a result, the following modified form of the orifice equation is proposed for the induced gas flow rate through a single orifice:

(

Qo = CoAoK~_PL(c'pu '2 - 29 h) \Po

0.5

- KIK2

(1) where Co is the loss coefficient for the orifice, Ao is the open area of the orifice, K1 is a dimensionless empirical constant to correct for the pressure losses not included in the analysis, C~ is a dimensionless pressure coefficient, U' is the absolute velocity of the orifice, h is the impeller submersion depth, and Kz is an empirical constant with units of m 3 s - 1. The Kx K2 term on the right-hand side of eq. (1) is a correction such that the condition of zero gas flow rate at the critical impeller speed is satisfied. Evans et al. (1991) have shown that the pressure drop across the orifice typically accounts for only around 30% of the total pressure driving force available, and that there are other significant pressure losses in the system which cannot be modelled satisfactorily through the empirical form of eq. (1). Evans et al. (1991) extended the work of Martin (1972) by using a simplified representation of the induction process in the development of a theoretical model for the gas induction rate. Their model is based on an energy balance equating the total pressure driving force for gas induction to the sum of the pressure losses along the gas pathway (see Theory section for more detail). This approach produced good agreement with experimental induced gas flow rate data for a cylindrical impeller blade, but gave poorer predictions for both concave and flat-blade impellers (Forrester, 1992). Although the model has a sound physical basis, it is possible to identify areas where the simplifications employed fail to adequately describe the real situation; principally, the proposed mechanisms of bubble growth and detachment from

Gas-inducing impeller design and performance characteristics the orifices on the moving impeller blades suffer from a current poor understanding of the processes involved (Forrester and Rielly, 1993). As a result, the literature on bubble formation in liquid cross-flows has been studied and is summarised briefly below. Bubble formation in a liquid cross-flow Bubble formation from the moving blades of a gasinducing impeller is essentially the same process as bubbles forming from a stationary orifice in a liquid cross-flow. Modes of bubble formation in a liquid cross-flow can be divided into three regimes (Marshall, 1990): (i) single bubbling; (ii) pulse bubbling; and (iii) jetting. For a given gas-liquid system, the principal parameters which determine the bubble formation regime are the liquid cross-flow velocity, the gas velocity through the orifice and the orifice diameter. Single bubbling is typically observed at very low gas flow rates through the orifice and is characterised by the regular production of nearly spherical bubbles of approximately uniform size, which detach close to the orifice. The equivalent spherical diameter of the detached bubble, d, can be predicted from a horizontal force balance equating the drag force produced by the liquid stream to the surface tension force (Johnson et al., 1982): ( 1 6 r o 7 ,~o.5 d = \CaU2pLj (2)

where Ce is a drag coefficient due to the liquid crossflow, which typically has a value of around 0.5 (dependent on the Reynolds number for the cross-flow, defined as pLUod/#L, and the bubble shape at detachment), Uo is the liquid velocity over the orifice, ro is the orifice radius, and 7 is the surface tension. Equation (2) is limited to very low gas flow rates where the drag due to the liquid cross-flow provides the dominant force for bubble detachment, and fails as Uo is reduced towards zero in which case buoyancy effects become significant. As the gas flow rate is increased a transition to pulse bubbling is observed. The bubbles become increasingly non-spherical and detachment no longer occurs close to the orifice, but by the severing of an elongated neck joining the bubble to the orifice. Detailed observation of the bubbles generated indicate that they consist of two or three pulses resulting from successive bubbles running into each other to produce agglomerated double or triple bubbles (Marshall, 1990). A number of bubble formation models have been proposed to describe pulse bubbling (Ghosh and Ulbrecht, 1989; Marshall 1990), however these models are limited to either: (i) very low liquid cross-flow velocities (typically less than 0.5 m s-1, whereas the small bubbles introduced via a gas-inducing device form on the surface of blades moving at around 4 m s-1); or (ii) spherical bubble growth, which is not a valid assumption at high cross-flow velocities. At very high gas flow rates through the orifice a gas jet forms. Quantitative definitions of the jetting

605

regime are somewhat vague, qualitatively the gas stream emerging from the orifice takes on the appearance of a continuous jet which subsequently breaks up at a small distance from the orifice, resulting in the production of irregularly sized bubbles. The mean bubble size formed by this process can be predicted by the jet break-up model originally developed by Rayleigh (1892), and subsequently confirmed by the experimental work of Silberman (1957) giving: d = 2.4(Qo/Uo) °5.

(3)

Generally it is difficult to predict which regime of bubble formation will predominate in a given system without experimental evidence. It is not possible to make detailed visual observations of the bubbles forming from the blades on a gas-inducing impeller within the stirred tank set-up. As a result, currently the major limitation in the modelling of gas-inducing impellers is a lack of knowledge on how the bubbles form and detach from the orifices on the moving blades. Mass transfer measurements The gas-liquid volumetric mass transfer coefficient, kLa, for gas-inducing impeller design is an important performance characteristic that may be readily compared with other gas-inducing and gas-sparged systems. Work in this area is limited: Sawant et al. (1981) measured kLa values for a Denver froth flotation cell; while Joshi and Sharma (1977) studied a cylindrical bladed open-ended gas-inducing impeller. In contrast, mass transfer characteristics in sparged systems have received much more attention. Often the volumetric mass transfer coefficient is determined from unsteady-state oxygen absorption experiments and an assumed model for the gas absorption driving force; the oxygen concentration can be measured with relative ease using dissolved oxygen probes. The development of equations to describe the absorption driving force require, for example, an assumption of ideal mixing of the two phases. However, due to uncertainties in the oxygen concentration profiles in the two phases (particularly in the gas phase) a number of different models have been proposed, and the resulting value of kLa can be highly dependent on which model is applied to the experimental data (Linek et al., 1982). In sparged reactors, mass transfer results are generally fitted to an empirical correlation of the form: kLa = x(Pg/VL) y V;

(4)

where Po is the gassed power input, Vt is the total liquid volume in the tank, Vg is the gas superficial velocity, and x, y and z are empirical constants. It is likely that a similar type of expression could be applied to gas-inducing impellers, the difference being that the specific power input and superficial gas velocity are no longer independent parameters (both are related to the impeller speed). Indeed, both Joshi and Sharma (1977) and Sawant et al. (1981) produced

606

S. E. Forrester et al.

correlations similar to eq. (4) which provide a useful comparison with the experimental results presented later in this paper, and are respectively (in S.I. units): kLa = O.O068(Pg/Vp,) T M V°o "5

(5)

kLa = O.O00617(Po/VL) ° 5 .

(6)

Independent estimates of the exponents in eq. (5) were determined from experiments in which the diameter of the gas outlet orifice was varied. Unfortunately, the calculation of the mass transfer coefficient from eq. (5) requires a further relationship to predict the induced superficial gas velocity for given gas-inducing impeller operating conditions. THEORY

The theory developed by Evans et al. (1991) is based on an energy balance equating the total pressure driving force for gas induction to the sum of the pressure losses along the gas pathway. This section presents an overview of their model, followed by a more detailed discussion of the areas which have been identified for improvement. Finally, their model is extended to allow for multiple orifices on each blade. M o d e l overview Pressure distribution over the blade surface. The

driving force for gas induction is the difference in pressure between the orifice on the blade surface, Po, and the headspace, Ph. An expression for this driving force can be found from the application of Bernoulli's equation; in the frame of reference where liquid flows over a stationary blade, the liquid velocity upstream of the blade is U, and the local liquid velocity over the orifice is Uo, giving: Po - Ph = p z g h - l2 PL (Uo2 -- U2).

(7)

Referring to an orifice positioned at a radial distance Ro from the vertical impeller axis on the blade of

a gas-inducing impeller, eq. (7) can be written as Po - Ph = PLgh -- ½PL(2~ZNRo( 1 - K))2(Cp - 1)

(8) where K is a blade slip factor and Cp is an orifice pressure coefficient. The blade slip factor is the ratio of the liquid velocity upstream of orifice to the orifice velocity, as given by U = 2~rNRo(1 - K ) .

(9)

The orifice pressure coefficient is defined as in aerofoil theory, and is a measure of the acceleration of the liquid phase over the blade surface (Rielly et al., 1992). These model constants, which characterise the performance of a gas-inducing impeller design, are almost independent of the vessel geometry and the impeller Reynolds number in the turbulent flow regime (Rielly et al., 1992). Significant changes in the values of K and Cp are only found to occur when the impeller is very close to either the base of the vessel or the free surface.

Their values can be determined for any blade design, by independent measurements of the total pressure driving force ( A P o = Po - P h ) against impeller speed prior to the onset of gas induction [see eq. (8) and Evans et al. (1990)]. Gas induction commences when the absolute pressure at the gas outlet orifice falls below the headspace pressure, corresponding to the total pressure driving force becoming negative. Therefore, an equation for the critical impeller speed, No, can be obtained by solving eq. (8) for APt, = 0 at N = N,., giving Nc =

2(~zRo(1 - K))2(Cp - 1)

"

(10)

Equation (10) has been found to give reliable predictions of the critical impeller speed over a wide range of blade designs and geometries (Evans et al., 1991; Rielly et al., 1992). For a given blade shape and orifice position, a single pair of model constants (K and Cp) determined from preliminary independent experiments can be used in eq. (10) to predict the critical impeller speed. Energy balance on the gas flow. The rate of gas induction is found from the following energy balance which equates the total pressure driving force, generated by the flow of liquid over the blade, to the sum of the pressure losses associated with the gas flow through the system: - APD = A P e + APo + AP~ + A P r E

(11)

where APD is the total pressure driving force, A P e is the frictional pressure loss along the gas pathway through the shaft and blade, APo is the pressure drop across the gas orifice on the blade surface, AP.~ is the pressure loss associated with the work done against surface tension at the bubble surface, and APrE is the pressure loss due to kinetic energy imparted to the liquid during the bubble formation process. The total pressure driving force, APD = Po -- Ph, is found from eq. (8), while Evans et al. (1991) propose simple correlations to estimate each of the pressure loss contributions (as summarised in Table 1). It is in the evaluation of these terms, and in particular APt,, APKE and AP~, that the limitations of this model become apparent. M o d e l improvements (i) Gas f l o w rate predictions close to the critical impeller speed: Close inspection of the gas induction model proposed by Evans et al. (1991) reveals the

wrong asymptotic behaviour as the impeller speed is reduced towards its critical value. In their model, the gas flow rate through a single orifice, Qo, is given by 4~zr3 1 Qo - - 3 td

4~r3d U C °'5 3

(rd+ro)

(12)

where td is the bubble detachment time, and r d is the bubble radius at detachment. Thus as the impeller speed decreases and approaches No, the gas flow rate

607

Gas-inducing impeller design and performance characteristics Table 1. Summary of the model equations for both single and multiple orifices Single orifice equations

Multiple orifice equations

- A P o = APKe + APp + APo + AP~ APD = prgh

-- IpLU2(Cp

-

1)

APp = c'Qg + c"Q2g

--APD(i)= APrdi) + APp + APo(i) + AP./(i) APD(i) = prgh

2 2 2CoAo

2~ APy = - rd APrE

-

1)

APp = c'Qg + c"Q2

pgQ2 APo =

-- ½pL U(i)2(Ce(i)

pgQo(i)2 APo(i)

2Co(i)2Ao(i) 2

27 AP~(i) = - rd(i)

3flpLQ 2 32rc2rd(r3_ r3)

Qg = n~Qo

3flprOo(i) 2 APKe(i) = 32~2ra(i)(rd(i)3 _ ro(i)a) Qo =nb ~ Qo(i) /=1

Qo

4rc(r3 -- roa) 3tn

ra + ro td = UCO.5

falls to zero and eq. (12) suggests that the detached bubble radius should also tend to zero giving an unrealistically large pressure difference due to surface tension, AP v (see Table 1), This behaviour causes the model to fail at impeller speeds close to the critical value. In practice the bubble diameter formed at low gas flow rates cannot be less than the orifice diameter, and therefore this problem can be overcome by assuming the bubble growth to start from a sphere of radius equal to that of the orifice. The gas flow rate through a single orifice now becomes

Qo =

4x(r 3 -- ro3) 1 3 td

Qo(i) =

4n(r~(i)3 - ro(i)3) 3ta(i)

rd(i) + ro(i) tn(i) = U(i)C.(i)o.5

stagnant liquid

~

growing bubble

~,,\\\\\\\\\\\\\"~ \\\\\\\\\\\\\\\\'q (a)

gas in ao

relative liquid flow 2 ~/Ro(1 - K)

ro + rd

4rt(r3 -- r3o) UC °'5 3 (rd + ro)" (13)

As a result of this alteration, the detached bubble radius approaches the orifice radius and the total pressure driving force approaches a finite value as the impeller speed is reduced towards No. The only other equation which is affected by this change is the pressure drop due to kinetic energy imparted to the liquid, which is discussed below. This modification to the initial bubble size only has a significant effect on the gas flow rate predictions close to the critical impeller speed. (ii) Pressure loss due to kinetic energy imparted to the liquid: Evans et al. (1991) showed that the pressure drop due to kinetic energy imparted to the liquid by the growing bubble often accounts for the largest fraction of the total pressure driving force. The original model described by these workers estimates this term from the simple analysis of Davidson and Schiller (1960) which assumes spherical bubble growth into an infinite stagnant liquid [see Fig. 2(a)], and neglects the effect of the impeller blades and the relative liquid cross-flow on the kinetic energy trans-

(b)

ro

liquid velocity profile ~U°b=f(rb~

(C)

K\\\\\\\\\\\\\'~"

~\\\\\\\\\\\\\\\~

Fig. 2. Bubble growth assumptions. (a) Bubble formation mechanism; (b) bubble detachment mechanism; (c) timeaveraging the total pressure driving force.

fer. For a stationary orifice in a stagnant liquid the detached bubble radius is relatively large, and as a result the kinetic energy pressure drop is negligible. However, in gas-inducing systems the bubbles form in

6O8

S. E. Forrester et al.

a strong liquid cross-flow, resulting in the production of a very large number of smaller bubbles and the APt
(14)

where fl is a constant determined from numerical integration of the velocity field generated by the expanding bubble to be approximately 1.5. Thus the total kinetic energy imparted to the liquid by a spherical bubble growing in point contact with a solid surface is ~ 1.5 times that of a spherical bubble growing into an infinite stagnant liquid. (iii) Bubble detachment criterion: Equation (14) indicates that to solve the gas induction model proposed by Evans et al. (1991) an estimate of the detached bubble size is required. The analysis employed by these workers assumes spherical bubble growth and therefore a detachment criterion needs to be specified. The original model assumes that the bubble detaches from the orifice when its base has travelled a distance equal to the orifice radius plus the detached bubble radius [see Fig. 2(b)]. The velocity of the bubble base is assumed equal to the liquid velocity at this point, and thus the detachment time, td, can be found from: ta .

ro + re Uo.

4rc(r3 - r 3) . . 3Qo ;

known, e.g. by assuming inviscid flow for a cylindrical impeller blade (Forrester, 1992), but in general the flow field is not known and cannot easily be estimated. Time-averaging of the total pressure driving force can have a significant effect on the model predictions as was illustrated by Forrester (1992) based on the cylindrical impeller blade. Extendiny the model to more than one orifice per blade The original model proposed by Evans et al. (1991) only considered the case where there is a single gas orifice on each blade, however their analysis can easily be extended to multiple orifices. Consider a gas-inducing impeller with no outlet orifices on each blade positioned at radial distances Ro(i) for i = 1,2 . . . . , no from the impeller axis of rotation. Individual energy balances can be written for each of the no orifices in a similar manner to the single orifice situation given by eq. (ll), i.e.: --APD(i) = APj, + APo(i) + AP.,.(i) + APrE(i)

i = 1,2. . . . . no.

(16)

The derivation of eq. (16) is based on the assumption that the frictional and inertial loss term, APe, is the same for each orifice along the blade, i.e. the frictional loss along the hollow blade between the orifices is negligible compared to the losses as the gas enters and flows down the hollow shaft, into the blade channel. The remaining pressure loss terms have different values for each orifice. The full set of multiple orifice model equations are given in Table 1, and can be combined into no equations for no unknown values of the detached bubble radii, rd(i), giving 1

z

pLgh -- ~pzU(i) (Ce(i) - 1) Uo

UCp0.5 . (15)

4

The physical basis for the selection of this detachment distance is weak, and furthermore, in both in this analysis, and that in (i) and (ii) above a number of assumptions are necessary, namely: spherical bubble growth; constant volumetric gas flow rate through the orifice; bubble growth into an initially stagnant liquid; no slip between the bubble and liquid; and, the relevant velocity for detachment is that at the base of the bubble. (iv) Time-averagin 9 the total pressure drivinyforce: The final important aspect of the model to consider concerns the total pressure driving force, APD (the pressure difference between the reactor headspace and the top of the growing bubble). The liquid velocity profile above the orifice is unlikely to be uniform and thus as the bubble grows away from the orifice the local liquid velocity at its top surface will vary [see Fig. 2(c)]. Therefore the pressure at the top of the bubble will also be time dependent. Evans et al. (1991) propose a time-averaged total pressure driving force which requires an estimate of the mean square liquid velocity over each bubble. This velocity can only be determined if the liquid flow field over the blade is

+'~rcc nb ~ U ( j ) C p ( j ) °'5 r 16=2.,,,2/ ~ . . . . . . . . .

o.sfrd(j) 3 - r o ( j ) 3 ~ 2

+ 9 Co(i)2Ao(i) 2 k. +

1,

,oi,,3

k. r d [ t ) - + r ~ ) i l l

{ . . . . . . . . o5 (rd(i) 3 - r o ( i } 3) 52

2? + r-~ = 0

i = 1 , 2 . . . . . no

(17)

where U(i) = 2~ZNRo(i)(1 - K(i))

i = 1,2. . . . . no (18)

nb is the number of impeller blades, and c' and c" are the empirical constants in the frictional pressure loss correlation. The resulting system of non-linear algebraic equations can be solved using the multi-dimensional Newton-Raphson method (Press et al., 1986) to give the detached bubble size for each orifice. An important point to note from this analysis is that since

609

Gas-inducing impeller design and performance characteristics the frictional pressure loss term is common to all the orifices, the rate of gas induction for multiple orifices is not simply the sum of the induced gas flow rates through the individual orifices on their own. To summarise, from the model of Evans et al. (1991) it is possible to predict the critical impeller speed marking the onset of gas induction with confidence for a wide range of impeller designs and experimental conditions. These predictions are based on two dimensionless model constants, K and Cp, which can be determined from independent experiments and have been found to be relatively insensitive to tank geometry and impeller clearance; in the turbulent flow regime, they are also relatively independent of the impeller Reynolds number. The applicability of this model is demonstrated by the good agreement between the experimental and theoretical gas induction rates for a cylindrical impeller blade (where time-averaging can be properly taken into account). It is too insufficient knowledge of the bubble formation and detachment mechanisms that prevent good predictions of the gas induction rate for more complex blade designs. The model has been extended such that it is now possible to examine the effect of increasing the number of orifices per blade on the gas-inducing characteristics. EXPERIMENTAL

The experimental apparatus employed in this work has previously been described by Evans et al. (1991). In summary, the power input to the impeller is measured using shaft-mounted strain gauges and an F M telemetry system, the local pressure at the orifice on the blade surface is determined via a pressure tapping and micro-manometer connected to a sealed bearing

gas outlet orifice relative liquid flo~

2ff.NRo(1- K)

)

Fig. 4. Concave blade profile.

assembly fitted to the shaft above the liquid level, and a calibrated wet gas meter allows the induced gas flow rate to be monitored. A six-bladed concave gas-inducing impeller has been used for this study, the design of which is shown in Figs 3 and 4 and Table 2. Bubble size measurements

The bubbles were recorded photographically through the wall of the transparent vessel and the bubble size distributions were measured by image analysis using the O P T O M A X V Particle Size Analyser software (Version 6.05). A ruler fixed to the inside of the tank was used as a scale and bubbles close to the tank wall, and in approximately the same horizontal plane as the impeller, were focussed upon and photographed. At low impeller speeds the photographs were taken using ordinary lighting, however at higher impeller speeds the bubbles moved too quickly to obtain a clear image by this method and electronic flash lighting was used. Three photographs were taken at each impeller speed and approximately 150 bubbles were analysed on each photograph. Mass transfer measurements

~ection ofrotation

(a)

(b)

i i

Fig. 3. Six-bladed concave impeller design. (a) Plan vicw; (b) section through X X.

Experimental values of the mass transfer coefficient were determined by the dynamic method (Linek et al., 1982). The water was initially de-oxygenated by distributing nitrogen from a ring sparger positioned beneath the rotating impeller until steady state was reached (with the air inlet on the hollow shaft closed). The impeller motion was stopped for a few seconds to allow the nitrogen bubbles to disengage from the free surface. Agitation was then restarted with gas induction through an open air inlet and the change in oxygen concentration with time was measured using two Russell CD400 dissolved oxygen probes, one positioned near the liquid surface and the other close to the base of the vessel. The probe response times were measured experimentally to be 2.8 + 0.2 s, suggesting that neglecting this parameter in the calculation of the mass transfer coefficient, kLa, will have a negligible effect on the resulting values (van't Riet, 1979). To evaluate the mass transfer coefficient from the unsteady dissolved oxygen concentration response a number of simplifying assumptions are made,

610

S. E. Forrester et al. Table 2. Basis for the experimental and modelling work (a) Tank geometry T (m)

H (m)

h (m)

0.45

0.45

0.225

(b) Impeller geometry Design

nb

D (m)

de (m)

dw (m)

0

Ne

10- 6c' (kg m-4 s- 1)

10-12c, (kg m- 7)

concave

6

0.154

0.02

0.004

60°

3.2

0.233

0.00585

(c) Orifice geometry Ro (m)

ro (mm)

Co

Cp

K

0.068 0.061 0.054 0.046

0.5 0.5 0.5 0.5

0.7 0.7 0.7 0.7

2.4 3.1 3.4 3.3

0 0 0 0

principally: (i) a well mixed liquid phase; and (ii) negligible gas-phase mass transfer resistance. As a resuit, the mass balance on the oxygen in the liquid phase is given by

V dCL L --d-i" = VLkLa (C* -- eL)

1.1

(19) Pg 1.01

where VL is the liquid volume in the vessel, CL is the oxygen concentration in the bulk liquid, and c~ is the equilibrium concentration of oxygen in the liquid. The concentration driving force, (c* - cD, is dependent on one further assumption concerning the behaviour of the gas phase, i.e. well-mixed or plug flow. Initial calculations showed the kLa values to be insensitive to the gas-phase model used, indicating negligible gas-phase oxygen depletion. Thus, from the integration of eq. (19) the mass transfer coefficient can be determined from the gradient of In {1 - (CL/C*)} plotted against time. Mass transfer coefficients have been measured for three different situations: (i) normal operation of the gas-inducing system; (ii) with the gas inlet on the shaft closed, so that the contribution of surface aeration could be estimated; and, (iii) with the gas inlet on the shaft closed and a gas flow rate provided through a ring sparger positioned below the impeller (with 17 x 1 mm diameter holes, and overall diameter equal to that of the impeller). For situation (iii) the gas flow rate from the ring sparger was set to be equal to the flow rate that would have been induced had the gasinducing inlet been open (this had previously been determined in separate experiments). RESULTS AND DISCUSSION

Power measurements Experimental measurements of the power input for the six-bladed concave impeller described in

eu

0.9 0.00

0.01

0.02 Qg/(ND 3)

0.03

Fig. 5. Effect of number of open orifices per blade on the ratio of gassed to ungassed power consumption for the sixbladed concave impeller. Ro (m): 0.068 [--©--]; 0.068, 0.061 [-4=]--]; 0.068, 0.061, 0.054 [--~--]; 0.068, 0.061, 0.054, 0.046 [ ~ - ] .

Figs 3-4 and Table 2 produced an ungassed power number of 3.2. The corresponding gassed power characteristics are shown in Fig. 5, in the form of the ratio of gassed to ungassed power input as a function of the non-dimensionalised gas induction rate, Qg/(ND3), for between one and four open orifices on each blade. This figure confirms that the drop in power consumption under gassed conditions for the concave blade design is small, i.e. less than 10% for values of Qo/(ND 3) of up to 0.020. Furthermore, this power reduction is approximately independent of the number of orifices open on each blade.

Gas-inducing impeller design and performance characteristics 0.04

800

•

,

.

6tl ,

,

•• s~

600

0.03

• t& 0 • ",#~ 0 /" ," " o f [] ,ss A O//D

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G

DD

%

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•

//

./y- . 0.01

200

Y / / / / /~J / "/ / ,

4

5

,

6 7 N (rps)

,

,

8

9

10

Fig. 6. Effect of number of outlet orifices on the gas induction performance of the six-bladed concave gas-inducing impeller. Ro (m):0.068 [©, experimental, theory]; 0.068, 0.061 [[2, - - - - ] ; 0.068, 0.061, 0.054 [~, - - ]; 0.068, 0.061, 0.054, 0.046 [• . . . . . . ].

-e xx x

x

x

x x

x

560 ' 1000' 15'00' 2dO0' 2500 Pg/VL (W / m 3) Fig. 7. Mass transfer coefficientas a function of the specific power input for /ihe six-bladed concave impeller with Ro = 0.061 m. Gas-inducing [ , ] ; Ring sparger [[Z]; Surface aeration [x]; Smith et al. (1977) [eq. (20) ]; Sawant et al. (1981) [eq. (6)- --]; Joshi and Sharma (19771 [eq. (5) ---],

Gas induction rate results

Figure 6 shows the total rate of gas induction plotted against impeller speed for the six-bladed concave impeller with between one and four gas outlet orifices on each blade• Two sets of data are presented: (i) experimental results; and (ii) predictions based on the modified version of Evans et al. (1991) model, as described in the Theory section. The impeller speed at which gas induction commences when using more than a single outlet orifice on each blade, is equal to the lowest value from those of the individual orifices (in this case for the orifice positioned at Ro = 0.061 m). Indeed, it is from this orifice that bubbles can first be seen to emerge. The experimental results suggest that increasing the number of orifices on each blade produces a significant increases in the rate of gas induction. Increasing from one to two outlet orifices almost doubles the total induced gas flow rate at a given impeller speed, however further increases in the number of orifices shows a continually diminishing increase in the induced gas flow rate. These experimental results correspond to V V M s of up to 0.6, and superficial gas velocities of up to 4.4 mms 1, suggesting that gas throughputs typical of those used in industrial and laboratory reactors can be obtained using the concave gas-inducing impeller design (Nienow et al., 1977). The model predictions (based on the input variables given in Table 2) show good agreement with the experimental data for a single orifice on each blade. However, as the number of orifices is increased the model starts to overpredict the rate of gas induction; for two outlet orifices the flow rates are overpredicted by around 10%, while for both three and four orifices this increases to approximately 20%. Nevertheless, even with all four orifices active on each blade, the model continues to give acceptable predictions of the rate of gas induction, indicating that the analysis of

Evans et al. (1991) can successfully be extended to account for multiple orifices by the method described in the Theory section. Mass transfer measurements

Preliminary mass transfer experiments showed the two dissolved oxygen probes (one positioned near the base of the vessel, and the other close to the free surface) to give very similar responses confirming the validity of the well-mixed liquid assumption• The mass transfer coefficient for the vessel is therefore calculated from the average rate of change of the dissolved oxygen concentration measured by the two probes. Figure 7 shows the experimentally determined mass transfer coefficient plotted against the gassed power input per unit volume of liquid for the six-bladed concave impeller using a single orifice on each blade (positioned at Ro = 0.061 m). The range of specific power inputs correspond to impeller speeds of between 5 and 10 rps, and gas induction rates of between 30 and 340 cm 3 s-1. In addition to the experimental gas-inducing results, this figure also shows: (a) the contribution to kLa due to aeration from the free surface (obtained by closing off the air inlets to the hollow shaft and to the sparge ring, leaving the free surface as the only method by which air can enter the watert; (b) the values of kLa obtained by closing the air inlet on the shaft, and running the impeller as a gas-sparged system (in which the gas flow rate through the ring sparger was set equal to the experimentally determined gas flow rate that would have been produced from the gasinducing system, at the same impeller speed); and

S. E. Forrester et al.

612

(c) the empirical correlations proposed by: (i) Joshi and Sharma (1977) for a two-bladed open-ended cylindrical gas-inducing impeller [given by eq. (5)]; (ii) Sawant et al. (1981) for a Denver type froth flotation cell [given by eq. (6)]; and (iii) Smith et al. (1977) representing a typical correlation used in sparged coalescing gas-liquid systems, i.e. in S.I. units: kLa = O.OI(Pg/VL) °475 V °'4

(20)

e~

where Vg is the superficial gas velocity, determined from the experimental gas induction rate measurements as given in Fig. 6. The experimental results indicate that mass transfer coefficients of the order of 0.02 s-~ are attainable using the six-bladed concave design operating in the gas-inducing mode (but not a fully optimised design). Furthermore these values compare well with those obtained with the impeller operating in the gas-sparging mode, and with the literature correlations for alternative gas-inducing and gas-sparged devices operating at similar specific power inputs. The data appears to be in reasonable agreement with both the empirical correlations of both Sawant et al. (1981) and Smith et al. (1977). An empirical correlation, of the form given by eq. (4), but with the superficial gas velocity term, V~, and the constant term, x, lumped together can be fitted to the experimental gas-inducing data presented in Fig. 7, giving (in S.I. units) kLa = (76 _+ 21) x 106(pg/VL) 0"80 + 0.06

(21)

with 95% confidence. The contribution of surface aeration to the overall mass transfer coefficient is always small, typically representing less than 10% of the total. The trend in the experimental gas-inducing and gas-sparged results is worthy of closer examination. For gas-inducing systems the bubble size increases slightly with impeller speed, whereas for sparged systems the bubble size decreases slightly with impeller speed (see bubble size measurements). As a result it might be expected that gas-sparged systems would show increasingly improved mass transfer characteristics with increasing impeller speed compared to gasinducing systems, However this is not observed in Fig. 7; the gas-inducing mode of operation gives the higher values of kza as the impeller speed is increased. This improved mass transfer performance may be the result of gas-inducing devices providing a more effective method of bubble formation, since the bubbles are generated in the region of the reactor where the mass transfer coefficient, kL, is largest. Bubble size measurements

A plot of the detached bubble diameter against impeller speed is shown in Fig. 8. In addition to the experimental results, the plot also shows the bubble sizes as predicted by: (i) the model of Evans et al. (1991) using the model constants given in Table 2; (ii) the theoretical correlation for single bubbles forming

10 N (rps) Fig. 8. Detached bubble diameter against impeller speed for the six-bladed concave impeller with Ro = 0.061 m. Experimental [O]; Evans et al. (1991) [eq. ( 1 5 ) - - ] ; Calderbank (1958) [eq. (22) - - - - ] ; Single bubbling [eq. (2) - - -].

in a liquid cross-flow [given by eq. (2)]; and (iii) the empirical correlation proposed by Calderbank (1958) for the Sauter mean bubble size in sparged coalescing systems, This final correlation, which was developed from experiments on a Rushton turbine, is given by f

~0.6

\

d32 = 4 . 1 5 ~ i p g / v - - ~ , p O . 2 ) c ~ ° s

~ = \--~-j

+0.000216\.

+ 0.0009 (m)

~

(22)

j\~-~j (23)

where ~ is the gas volume fraction, V~ is the terminal rise velocity of the bubble calculated from the Wallis correlation for solid spheres (Kay and Nedderman, 1985a), and V~is the superficial gas velocity evaluated from the experimental gas induction rate measurements. The model of Evans et al. (1991) and the single bubbling analysis both assume uniform spherical bubbles. The experimental results and the empirical correlation of Calderbank (1958) are based on the Sauter mean bubble diameter, d32. Furthermore the correlation given by Calderbank (1958) represents the mean bubble size over the whole tank, whereas the experimental results represent local bubble size measurements taken close to the wall and in the same horizontal plane as the impeller. Figure 8 illustrates that for the 0.154 m diameter six-bladed concave impeller geometry (see Figs 3 and 4 and Table 2) bubbles of approximately 2 - 2.5 mm in diameter are formed from the 1 mm diameter gas outlet orifices. The overall trend in the experimental data is of a slightly increasing bubble size with impeller speed, as also predicted by the model of Evans etal. (1991). However, the agreement between the actual values is poor due to the simple bubble growth model and arbitrary detachment criterion employed by these workers (see Model improvements in the

Gas-inducing impeller design and performance characteristics

613

10.0 1[

Jln(d/dgm)]q

pn:l°°x~Ll+er~/ ~-~ )J

--1.0 d

~ m m l

a~m

~ •

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.01

.1

i

I

4.03-1.69

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2.04 1.95 2.20 1.81 2.26

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2.31 2.20 2.41 2.47 2.37 I

I

; 1'0 2o30 50 08'0

--2.21

99

99.9 99.99

Percent undersize, Pn Fig. 9. Log-normal probability plot of the bubble size distributions for the six-bladed concave impeller with Ro = 0.061 m.

Theory section). The results from the gas-sparging correlation proposed by Calderbank (1958) show an initial rapid increase in bubble size to approximately 2.5 mm at 5 rps, followed by a very gradual decrease in bubble size with impeller speed. The bubble diameters calculated from the single bubbling correlation, given by eq. (2), provide a lower limit on the bubble sizes produced from the orifice. These are approximately a factor of five less than the experimental measurements suggesting that the gasinducing impeller does not operate in the single bubbling regime. The bubble sizes predicted by the jetting correlation, given by eq. (3), are not shown on Fig. 8. Equation (3) provides an upper limit to the bubble size generated and predicts bubble diameters an order of magnitude greater than those measured experimentally. If a jet exists at the orifice, then the bubbles formed would undergo further break-up by the turbulence created by the blades, in a manner similar to sparged systems. Bubble size distributions The bubble size distributions corresponding to bubble size data presented in Fig. 8 are shown in Fig. 9. The dimensionless size distributions, d/do,,, were found to be best represented by a log-normal distribution with cumulative distribution function: F(d/d.m) =

Jo

(2n)°~ a(d/do,,) exp

x d(d/dom )

20-2

] (24)

where do,. is the geometric mean bubble diameter [-related to the Sauter mean bubble diameter by dom= d32exp(-5a2/2); Kay and Nedderman, (1985b)], and the parameter a has a value of 0.31 _+ 0.03 at the 95% confidence level. Figure 9

illustrates that the bubble size distributions are independent of impeller speed over the range 4-8 rps suggesting that the mechanism of bubble formation does not change over this range of speeds.

CONCLUSIONS The model for gas-inducing impeller design proposed by Evans et al. (1991) has been critically reviewed. Their analysis allows the critical impeller speed to be reliably predicted for any blade design, however, predictions of the gas induction rate for shapes other than cylindrical have met with only limited success (based on model constants from single phase experiments). A number of improvements to the model are proposed; the correct behaviour is now observed close to the critical impeller speed, and the pressure loss due to the kinetic energy imparted to the liquid by the expanding bubble has been modified to account for the presence of the blade. It has been shown that the theory of Evans et al. (1991) suffers from a lack of understanding on how the gas bubbles grow and detach from the outlet orifices on the impeller blades. It is difficult to make direct visual observations of the bubble formation and detachment mechanisms within the gas-inducing system. Experimental measurements of the detached bubble diameter produced values of the order of 2 2.5 mm from the 1 mm orifice diameter. The corresponding bubble size distributions, d/do,, , were best represented by a log-normal distribution with a standard deviation, a, of 0.31 _+ 0.03. Furthermore, the distributions were independent of impeller speed over the range 4 8 rps indicating that the mechanism of bubble formation does not change over this range of speeds. It is clear from these studies that to develop further the design methods for gas-inducing impellers

614

S. E. Forrester et al.

it is necessary to have an improved understanding of the mechanisms of bubble formation, growth and detachment in a strong liquid cross-flow. An experimental study on the performance of a 0.154m diameter six-bladed concave gas-inducing impeller in a 72 1 vessel has been conducted. The ungassed power number for this impeller configuration was measured at 3.2, and the reduction in power under gassed conditions was confirmed to be small, i.e. less than 10%, for dimensionless gas flow rates, Qg/(ND3), of up to 0.020. The rate of gas induction was significantly increased by adding more outlet orifices to each blade (at the same angle of attack but different radial positions), with VVMs of up to 0.6 attainable using four open orifices on each of the six blades. The model for gas-inducing impeller design proposed by Evans et al. (1991) has been successfully extended to account for the effect of multiple gas outlet orifices, predicting the gas induction rate to within 20% of the experimental data for four outlet orifices on each blade. Mass transfer measurements produced kLa values of the order of 0.02 s- ~ based on a single outlet orifice on each blade of the six-bladed concave impeller. These values compared well with the mass transfer data reported in the literature for other gas-inducing and gas-sparged devices.

K1 K2 nb no N Nc Np p, P Pg

Ph Po Ps P, Qg

Qo rb

rd ro Ro t

Acknowledgements This work was carried out with support from the EPSRC and Zeneca Fine Chemicals Manufacturing Organisation, which SEF gratefully acknowledges.

ta T U'

Uo Uob

Zo CI

CI' CL

c~ Cd Co Cp

c'. d d~

d.m dw d32 D 9 h H kL

kLa K

NOTATION open area of orifice, m 2 empirical constant in frictional pressure drop equation, kg m -4 s- ~ empirical constant in frictional pressure drop equation, kg m - 7 oxygen concentration in the bulk liquid, kmolm -3 equilibrium oxygen concentration in the bulk liquid, kmol m - 3 drag coefficient, dimensionless orifice coefficient, dimensionless orifice pressure coefficient { = 2 ( P s - Po)/ (pLU~)}, dimensionless orifice pressure coefficient { = 2 ( P - Po)/ (pL U' 2)} , dimensionless equivalent spherical bubble diameter, m blade diameter (see Fig. 4), m geometric mean bubble diameter, m blade thickness (see Fig. 4), m Sauter mean bubble diameter, m impeller diameter, m gravitational acceleration, m s- 2 impeller submersion depth, m total liquid height, m liquid film mass transfer coefficient, m s-~ mass transfer coefficient, s-x blade slip factor, dimensionless

U

VVM Vo VL V~

empirical constant in eq. (1), dimensionless dimensional empirical constant in eq. (1), m3s-1 number of blades on impeller, dimensionless number of orifices on each blade, dimensionless impeller speed, rps critical impeller speed, rps impeller power number, dimensionless percent undersize, dimensionless liquid pressure upstream of the orifice on the blade surface, N m - 2 gassed impeller power consumption, W headspace pressure, N m - z pressure at the outlet orifice on the blade surface, N m - 2 pressure at the front stagnation point on the blade surface, N m - 2 ungassed impeller power consumption, W total induced gas flow rate, m 3 s-1 gas flow rate through a single orifice, m 3 s bubble radius, m bubble radius at detachment, m outlet orifice radius, m orifice radial distance from the impeller axis, m time, s total bubble formation time, s vessel width, m absolute velocity of the orifice, m sliquid velocity over the orifice relative to the orifice velocity, m s- 1 liquid velocity over the growing bubble relative to the orifice velocity, m sliquid velocity upstream of the orifice relative to the orifice velocity, m s- 1 volume of gas per unit volume of liquid per minute, m i n superficial gas velocity, m s- 1 total liquid volume in tank, m 3 terminal rise velocity of the bubble, m s-

Greek letters fl

APD APKE APo APp AP~ 7 0 #L pg

pt, a ~b

gas voidage fraction, dimensionless constant in eq. (14), dimensionless total pressure driving force, N m - 2 pressure losses due to kinetic energy imparted to the liquid, N m - 2 pressure losses across the orifice, N m - 2 pressure losses due to friction along the gas pathway, N m - 2 pressure losses due to surface tension at the gas-liquid interface, N m - 2 surface tension, N m - 1 orifice angular position on blade, dimensionless liquid viscosity, kg m - 1s- 1 gas density, kg m - 3 liquid density, kg m - 3 standard deviation, dimensionless blade angle of attack, dimensionless

Gas-inducing impeller design and performance characteristics REFERENCES Albal, R. S., Shah, Y. T. and Schumpe, A. (1983) Mass transfer in multiphase agitated contactors. The Chem. Engny J. 27, 61-80. Calderbank, P. F. (1958) Physical rate processes in industrial fermentation, Part I. The interracial area in gas-liquid contacting with mechanical agitation. Trans. lnstn Chem. Engrs 36, 443-463. Davidson, J. F. and Schiiler, B. O. G. (1960) Bubble formation at an orifice in a viscous liquid. Trans. lnstn Chem. Engrs 38, 144-154. Evans, G. M., Rielly, C. D., Davidson, J. F. and Carpenter, K. J. (1990) A fundamental study of gas-inducing impeller design, In Fluid Mixing IV, Bradford. I. Chem. Engng Syrup. Ser. No. 121, pp. 137 152. Evans, G. M., Rielly, C. D., Davidson, J. F. and Carpenter, K. J. (1991) Modelling and design of a gas-inducing reactor. A.I.Ch.E. Spring National Meeting, Houston, TX, Paper 33e. Forrester, S. E. (1992) An Investigation of Gas-inducing Impeller Design. CPGS Report, University of Cambridge, Cambridge. Forrester, S. E. and Rielly, C. D. (1993) Mass transfer and mixing studies with a concave-bladed gasinducing impeller. I.Chem.E. Res. Event, Birmingham, 2, 666- 668. Ghosh, A. K. and Ulbrecht, J. J., (1989) Bubble formation from a sparger in polymer solutions--II. Moving liquids. Chem. Engng Sci. 44, 969-977. Johnson, B. D., Gershey, R. M., Cooke, R. C. and Sutcliffe, W. H., Jr. (1982) A theoretical model for bubble formation at a frit surface in a shear field. Sep. Sci. Technol. 17, 1027 1039. Joshi, J. B., Pandit, A. B. and Sharma, M. M. (1982) Mechanically agitated gas-liquid reactors. Chem. Engng Sci. 37, 813-844. Joshi, J. B. and Sharma, M. M. (1977) Mass transfer and hydrodynamic characteristics of gas-inducing type of agitated contactors. Can. J. Chem. Engng 55, 683-695. Kay, J. M. and Nedderman, R. M. (1985a) Fluid Mechanics and Transfer Processes, pp. 504-508. Cambridge University Press, Cambridge. Kay, J. M. and Nedderman, R. M. (1985b) Fluid Mechanics and Transfer Processes, pp. 540. Cambridge University Press, Cambridge.

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Linek, V., Benes, P., Vacek, V. and Hovorka, F. (1982) Analysis of differences in kLa values determined by steady-state and dynamic methods in stirred tanks. Chem. Engng J. 25, 77-88. Martin, G. O. (1972) Gas-inducing agitator. Ind. Eng. Chem. Process Des. Dev. 11, 397-404. Marshall, S. H. (1990) Air bubble formation from an orifice with liquid cross-flow, Ph.D thesis, University of Sydney, Australia. Nienow, A. W., Wisdom, D. J. and Middleton, J. C. (1977) The effect of scale and geometry on flooding, recirculation, and power in gassed stirred vessels. In Proc. 2nd European Conj'. on Mixing, Cambridge, U.K., Paper F1. Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T. (1986) Numerical Recipes, pp. 269 273. Cambridge University Press, New York, U.S.A. Raidoo, A. D., Raghav Rao, K. S. M. S., Sawant, S. B. and Joshi, J. B. (1987) Improvements in gas-inducing impeller design. Chem. Engng Commun. 54, 241-264. Rayleigh Lord (1892) On the stability of cylindrical fluid surfaces. Phil. Mag. 34, 177 180. Rielly, C. D., Evans, G. M., Davidson, J. F. and Carpenter, K. J. (1992) Effect of vessel scaleup on the hydrodynamics of a self-aerating concave blade impeller. Chem. Engng Sci. 47, 3395-3402. Sawant, S. B., Joshi, J. B., Pangarkar, V. G. and Mhaskar, R. D. (1981) Mass transfer and hydrodynamic characteristics of the Denver type of flotation cell. The Chem. Engng J. 21, 11 19. Silberman, E. (1957) Production of bubbles by the disintegration of gas jets in liquids. In Proc. 5th Midwestern Conf. Fluid Mech., pp. 263-284. Smith, J. M., Van't Riet, K. and Middleton, J. C. (1977) Scale-up of agitated gas liquid reactors for mass transfer. In Proc 2nd European Conf. on Mixing, Cambridge, U.K., Paper F4. van't Riet, K. (1979) Review of measuring methods and results in nonviscous gas-liquid mass transfer in stirred vessels. Ind. Engng Chem. Process Des. Dev. 18, 357 364. Witze, C. P., Schrock, V. E. and ChambrO, P. L. (19681 Flow about a growing sphere in contact with a plane wall. Int. J. Heat Mass Tran.~fer. 11, 1637 1652.