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Free jet expansion and gas entrainment characteristics of a plunging liquid jet
ELSEVIER
Free Jet Expansion and Gas Entrainment Characteristics of a Plunging Liquid Jet G. M. Evans G. J. Jameson Department of Chemical Engineering, University of Newcastle, Newcastle, Australia C. D. RieUy Department of Chemical Engineering, University of Cambridge, Cambridge, England
mThe change in effective jet diameter is measured as a function of free jet length for vertical liquid jets passing through air. The data are incorporated into a model to predict the rate of gas entrainment for a liquid jet plunging into a confined column of liquid. In the model it was assumed that the total gas entrainment rate included gas contained within (1) the effective diameter of the free jet at the plunge point and (2) an annular film adjacent to the surface of the jet, where the outer boundary of the film was defined to be the separating streamline between the entrained and unentrained components of the moving gas boundary layer. It was further assumed that the radial location of the separating streamline was independent of both liquid and gas flow rates and system geometry. Excellent agreement between model predictions and gas entrainment measurements were obtained once a number of experimental parameters were determined. Keywords: plunging liquid jets, gas entrainment, free jet expansion
INTRODUCTION A plunging liquid jet is defined as a moving column of liquid that passes through a gaseous headspace before impinging on the horizontal free surface of the receiving liquid. Examples of plunging liquid jets in nature include waterfalls and breaking waves. In industry, plunging liquid jets can be found in the steel teeming process, roll-coating systems, and plunging liquid jet bubble columns [I]. As shown in Fig. 1, a feature of plunging liquid jets is that a depression is formed in the receiving liquid free surface, referred to as the induction trumpet [2] due to its characteristic trumpetlike profile. Gas is drawn into the induction trumpet by the free jet before being entrained into the receiving liquid and broken up into fine bubbles. In some instances, such as roll coating and steel teeming, the presence of bubbles is detrimental to the process because they cause imperfections to the final product. However, in many processes the entrainment characteristics of a plunging liquid jet are beneficial, especially those involving mass transfer and aeration operations. In many industrial processes, such as those involving mineral floatation, gas absorption, and chemical reactions, confined plunging jets are applied. In these systems the free jet passes through an enclosed headspace before plunging into the receiving liquid contained within a vertical column, as shown in Fig. 2. Typically, the c o l u m n / j e t diameter ratio is of the order of 10 : 1. The action of the jet plunging into a confined volume generates intense recirculation and high-energy dissipation rates, resulting
in the manufacture of very fine bubbles, in the range of 100-500 ttm diameter, and large amounts of interfacial area. In general, the amount of interracial area produced is increased by increasing the gas/liquid flow ratio entering the column. However, an increase in this ratio can also lead to instabilities and in some cases to the complete collapse of the system. This is shown in Fig. 3. At low gas rates into the column, the level of the gas-liquid mixture (froth) is sustained just below the level of the nozzle, indicating that the plunging jet can effectively entrain all of the gas supplied to the headspace. If the gas flow rate is increased, a point is reached where the jet can no longer entrain all of the gas being introduced into the headspace, and the froth level in the column starts to drop. At this point a compensatory phenomenon is at work that has the effect that the rate of entrainment of air by the plunging jet increases up to a point as the length of free jet increases. The phenomenon is possibly linked to the increase in the effective jet diameter with increase in the free jet length. Whatever the reason, a new equilibrium height is reached inside the column, marking the length where the jet can effectively entrain all of the entering gas. In a sense the jet is "self-regulating" in that the length of the free jet will increase to accommodate the increase in the amount of gas added to the column (up to a point). At very high gas rates, bubble coalescence results in the formation of large gas slugs lower in the column that restrict the downward flow of gas. This leads to an
Address correspondence to Dr. Geoffrey Evans, Department of Chemical Engineering, The University of Newcastle, Newcastle, New South Wales 2308, Australia.
Experimental Thermaland Fluid Science 1996; 12:142-149 © Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010
0894-1777/96/$15.00 SSDI 0894-1777(95)00095-X
A Plunging Liquid Jet
143
IIncreasinggas feed rate I [Increasinggas recirculationI
free surface
-------t>~
receiving
induction trumpet
liquid
K
o'O ooO o
0
0
entrained
gas
!~iil
o
Figure 1. Plunging liquid jet. IDecreasingcontactareaI
- - ~I QG
Ioee~easin~stabitity I
I t2L fA
Figure 3. Column stability. quantity of gas, QT, that is trapped within the effective diameter of the jet (trapped gas component). This can be written as QE = QF + QT"
N,M
Figure 2. Confined plunging jet. increase in the pressure in the headspace, causing a drop in the froth level and, in the extreme case, the complete collapse of the system. To effectively operate confined plunging liquid jet reactors it is essential to know the free jet length as a function of gas and liquid flow rates, column diameter, and liquid physical properties. A number of entrainment models have been proposed for unconfined jets [1]. However, the weakness of all these models is that they do not contain the column diameter as a variable, whereas it has been found experimentally that column diameter has a significant effect on the rate of gas entrainment into industrial confined plunging jet reactors. The purpose of this paper is to present a gas entrainment model for a confined plunging liquid jet that takes into account column diameter and reentrainment of recirculated gas. The model is based on the effective jet diameter as a function of free jet length. THEORETICAL The proposed model is based on the assumption that the total gas entrainment rate, QE, for a plunging liquid jet is the sum of (1) the gas that is contained within a thin annular film of gas, QF, which is carried along adjacent to the jet free surface (filmwise component), and (2) the
(1)
A further assumption (which applies only when the effective jet diameter, defined as Dj in Fig. 4, is less than D*) is that the outer boundary of the entrained gas film is defined by a time-average streamline, with diameter D* as shown in Fig. 5, which separates the entrained and unentrained components of the moving gas boundary layer associated with the free jet. On the basis of the above-mentioned assumptions, three distinct regions of gas entrainment can be identified, depending on the length of the jet.
Region 1 Once the jet leaves the nozzle, the velocity profile changes from pipe to plug flow. Usually, this takes about 3 - 5 nozzle diameters, and any change in
jet length
L1
t envelope
Figure 4. Effective jet diameter.
144 G.M. Evans and G. J. Jameson jet diameter is due to the relaxation in the velocity profile. During this period there is no gas penetration into the outer boundary of jet, that is, no trapped gas, and only filmwise entrainment takes place. This can be written as "B"
QE = OF = ~-( D*2 -- DJe)VF,
un-entrained ~
1
separating streamline
(2)
where v F is the average velocity of the gas inside the film, bounded by D* and the outer boundary of the jet, Dj, that is, the effective jet diameter. entrained
R e g i o n 2 For jets longer than about 3 - 5 D N, the velocity profile has relaxed to approximately plug flow, although there is a slight reduction in the surface velocity due to the drag exerted by the surrounding gas phase. There is also an increase in the average je t velocity due to the acceleration of gravity. Both of these terms are small, however, compared with the average velocity of the jet, and the free jet velocity can be considered constant and independent of jet length. Waves on the jet surface result in gas being entrained within the jet envelope (trapped gas) as well as in the film adjacent to the jet free surface and within D*. The general expression for the rate of entrainment (i.e., filmwise plus trapped gas) is
increases in region 1. In region 2 it is assumed to remain approximately constant. Thus, in region 1 the average velocity of the gas inside the film may be taken to be proportional to the velocity of the jet leaving the nozzle and the length of the jet, that is,
QE = QF + QT = ~-( 0 . 2 -- D2)VF + ~"/'f( O j 2 -- O2)CN,
REGION 1
o°O oo°0 o
0
0
o
Figure 5. Entrained gas component.
(3) where D 1 is the effective diameter of the jet at the end of region 1 and CN is the average jet velocty at the nozzle. The first term on the right-hand side of Eq. (3) represents the filmwise entrainment, and the second term is the trapped gas component. In this last term, it is assumed that there is no slip between the liquid and entrained gas.
CF = K | ( L j / D N ) U N ,
(5)
where D N is the nozzle diameter, Lj is the free jet length, and K 1 is a constant. The length of region 1 is approximately 3D N. Thus, in regions 2 and 3, where the flow in the gas film is independent of length of jet, the average gas velocity is given by REGIONS 2 AND 3
3 It may be assumed that the radial position of the separating streamline remains fixed (i.e., the diameter D* in Fig. 5 remains constant), and so the thickness of the film decreases as the effective jet diameter increases. A point is reached where the gas film thickness reduces to zero, when Dj equals D*, and gas entrainment may be entirely attributed to the trapped gas component, that is, Region
7/"
QE = a T = --~( D 2 -- D 2 ) U N •
(4)
Equations (2)-(4) can be used to predict the total rate of gas entrainment for a plunging liquid jet, provided (1) the diameter and velocity relaxation characteristics of the free jet and (2) the velocity profile of the entrained gas film are known. The velocity profile of the entrained gas stream is not straightforward to measure. The flow in the gas film is generated by the interfacial drag, and so it might be expected that the average velocity in the film, v F, would be related to the velocity on the surface of the jet. Initially the jet surface velocity is zero leaving the nozzle and
UF = 3KlV N.
(6)
The entrainment model outlined above is useful for determining the total amount of gas a plunging liquid jet can entrain below a liquid surface. It is applicable to both confined and unconfined plunging jet systems. For confined systems, as shown in Fig. 2, it is more important to know the quantity of gas feed, QG, entering the headspace at the top of the column. This is equal to the total rate of gas entrainment minus the recirculated gas, QE, which disengages from the recirculating liquid at the top of the column and is reentrained by the jet; that is, Oo = QE - QR.
(7)
The total rate of gas entrainment is given by Eqs. (2)-(4). An estimate of the quantity of recirculated gas can be obtained by considering the way in which the gas (bubbles) is released from the receiving liquid. As shown in Fig. 2, a confined plunging liquid jet generates a recirculating bubbly flow region at the top of the columnJ I The recirculation regions is referred to as the mixingzone.
A Plunging Liquid Jet Superimposed on the bulk flow is the rise velocity of the bubbles, u,, which results in a number of bubbles disengaging from the flow and being recirculated back into the headspace at the top of the column. Mathematically, the interaction between bulk flow and bubble rise velocity is very complex. Qualitatively at least, one would expect the amount of recirculated gas, Qn, to increase with increasing bubble rise velocity and decreasing recirculation velocity, vn. A simple formulation that allows us to achieve this relationship is a dimensionless power law, Qn = KZ(v,/vn)K3,
VB =
QB/&
(9)
Barchilon and Curtet [3] have quantified culation as a function of the Crayer-Curtet Qn = Qr(O.37/C,
the bulk recirnumber, C,:
- 0.64),
&= &?&
(11)
Eqs. (91 and (101 into (8) gives K3
’
(12)
where vc is the superficial downward liquid velocity inside the column. The bubble rise velocity can be calculated using the formula given by Peebles and Garber [51,
v, = 1.35
fTg( p - PG)
(
0z5
=,Ki[(z)‘-
(%)‘I
-K, I
“r uc
(
+ [($)‘-
Cr 0.37 - 0.64C,
)I
Region 3
QG -= QL
’
EXPERIMENTAL The experimental apparatus [61 is shown in Fig. 6. It consisted of a perspex column mounted vertically above a constant-level bath. The top of the column was enclosed, and the opening at the base of the column was located 25 mm below the free surface of the bath. The column was effectively isolated from the atmosphere, and gas (air in this case) could enter the column only through the metered inlet at the top. The metered liquid feed entered the column via a jet delivery tube located at the top of the column. The delivery tube was aligned with the vertical axis of the column so that the liquid jet plunged into the constant-level bath below. The inside diameters, D,, of the jet nozzles used in this study were 2.38, 4.76, and 7.12 mm. Each nozzle consisted of a conical entrance section that fitted neatly inside the nozzle delivery tube, giving a smooth transition. The tapered entrance led to the throat of the nozzle, which
--i
P F
by-pass \
z=K~($)[(g)‘($1’1 K3
’
(16)
where DJD, is related empirically to L, by Eq. (171, which can also be used to calculate the jet diameter, D,, at the end of region 1. The unknowns in Eqs. (14)-(16) are K,, K,, K,, and D* and can be obtained from regression analysis applied to the experimental measurements.
i
Region 1
(15)
’
liqtlid feed
where u is the surface tension and p and pG are the liquid and gas densities, respectively. Finally, combining Eqs. (21-02) leads to the following expressions for the normalized gas feed rate for a confined plunging liquid jet:
(%)‘I
K3
(131
7 P2
2
(101
where Qr is the jet volumetric flow rate. The numerical constants in Eq. (10) are those proposed by Liu and Berkelew [4]. For confined plunging liquid jets, the Crayer-Curtet number is related to the nozzle diameter, D,, and column diameter, D,, by
Substituting
Region 2
(81
where K, and K, are constants. The velocity profile in the bulk recirculation region is very complex. However, an estimate of the average bulk circulating velocity can be obtained by dividing the bulk recirculation flow rate, Qn, by the flow cross-sectional area of the column, A,:
145
(14) Figure 6. Experimental
apparatus.
146
G . M . Evans and G. J. Jameson sion and a 150-mm telephoto lens were again used, with an aperture setting of f/5.6. The shutter speed was increased to 1/250s, and photographs were taken in a darkened room to eliminate any influence of ambient light. The 10-p.s flash was able to effectively "freeze" the motion of the jet, which allowed qualitative comments to be made about the characteristic of the jet surface.
consisted of a straight length of precision-bore brass tubing. The throat length for each nozzle was approximately 17D N. The liquids used were tap water, kerosene, and 16 and 28 wt % sucrose solutions. Ten parts per million of Teric 407 (Imperial Chemical Industries) was added to the water to prevent bubble coalescence, and the subsequent formation of large gas slugs, inside the column. The experimental procedure involved adjusting the valve .in the bypass line to give the desired superficial liquid velocity through the nozzle. The gas inlet valve at the top of the column was opened, and the entrained gas volumetric flow rate was recorded. Once the resultant two-phase mixture (froth) had reached a steady height inside the column, the free jet length, L j , w a s measured from the tip of the nozzle to the point where the jet penetrated the froth. While maintaining the same liquid flow rate, the gas inlet valve was used to vary the gas flow rate, and each corresponding equilibrium froth height was recorded. The above procedure was repeated using each nozzle for a range of liquid flow rates (jet velocity) and column diameters. The experimental conditions for each trial are summarized in Table 1. In addition to the gas entrainment measurements, the effective jet diameter was measured as a function of free jet length (see Fig. 4). For these measurements the column was removed from the apparatus to eliminate distortion created by photographing through a curved surface. The moving jet was photographed under natural lighting conditions using a Linhoff Polaroid camera (aperture setting f/5.6; shutter speed 1 / 6 0 s ) fitted with a 156-mm telephoto lens and extension bellows. The 1/60-s exposure produced a time-averaged image of the outline of the jet that corresponded to the effective jet diameter. The surface structure of the free jet was also examined. A black background was placed behind the jet, and a 10-/zs-duration flash was mounted to the side at 45 ° to the vertical focal plane of the Linhoff camera. A white reflecting screen was placed on the other side of the camera to produce an even lighting of the jet. The bellows exten-
RESULTS The uncertainty in the experimental results is due mainly to (1) the difficulty in defining the outer boundary of the jet surface, that is, the effective jet diameter, from the time-average photographs and (2) the uncertainty in the actual length of the free jet for the gas entrainment experiments. (The latter uncertainty is due to the difficulty in determining the distance below the horizontal free surface in the column where the free jet came into contact with the froth.) It is estimated that the uncertainty of the effective jet diameter measurements as a function of free jet length is on the order of 20% [6].
Jet Diameter Expansion A flash photograph of the jet is given in Fig. 7, which shows the presence of a large number of disturbances on the jet free surface. Moreover, the structure of the jet appears to change at around three to five jet lengths from what could be described as a random distribution of small amplitude disturbances to a much larger scale helical disturbance. This is consistent with observations from other studies [7-9], that found that the relative velocity between the gas phase and the moving jet resulted in the formation of both axisymmetric and nonaxisymmetric (helical) disturbances [10] being generated on the jet surface. Furthermore, according to the analysis of Mattingly and Chang [11], the axisymmetric disturbances have greatest amplification in the first three diameters of the jet, whereas the helical waves dominate further downstream.
Table 1. Experimental Conditions Liquid
D C (mm)
D N (mm)
10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 16 wt % Sucrose 28 wt % Sucrose Kerosene 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407
44 44 44 44 44 44 44 44 74 74 74 74 74 95 95 95 95 95 95
7.12 4.76 4.76 4.76 2.38 4.76 4.76 4.76 7.12 4.76 4.76 4.76 2.38 7.12 7.12 7.12 4.76 4.76 4.76
UN
(m / s) 11.5 15.0 11.5 7.8 11.5 11.5 11.5 11.5 11.5 15.0 11.5 7.8 11.5 15.0 11.5 7.8 15.0 11.5 7.8
o" (N / m)
Iz (Pa s)
p (kg / m 3)
0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.028 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063
0.0009 0.0009 0.0009 0.0009 0.0009 0.0017 0.0029 0.0021 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009
998 998 998 998 998 1061 1114 784 998 998 998 998 998 998 998 998 998 998 998
A Plunging Liquid Jet ....
0.8
I,
.,,I,,
,,I,
,,,
I,,,.
I,,.,I,,.,
147
I,,,,
/
+20%
0.7. 0.6. 0.5. 0.4"
LjIDN
t~ "d 0.3.
0.2: 0.1. A.
Figure 7. Jet photograph.
0
f.,..l,,,,l,.,,l...,l....l
0 For each jet arrangement, the time-average photographs were used to determine the effective jet diameter as a function of free jet length. The measurements were correlated by the dimensionless expression Oj
2
where Re L is the jet Reynolds number ( = P U N L j / ~ ) based on tla~e liquid physical properties and free jet length L j; and Oh is t h e ~ Ohnesorge or stability number [9], given by /~/~/pDNO'. The numerical parameters in Eq. (17) were correlated for the range 3 < L j / D N < 30 and apply to highly turbulent jets issuing from nozzles with throat lengths of approximately 16DN. For other jet systems the numerical parameters may be different. A graphical comparison of measured and predicted normalized jet diameter is given in Fig. 8. It can be seen from the results that the agreement with predictions from Eq. (17) is approximately +20%, which is of the same order as the uncertainty of the measurements.
1.6
DN
Rate of Gas Entrainment The normalized gas entrainment QG/QL and free jet l e n g t h L j / D N measurements for L j / D N _> 3 were used to obtain the numerical values for the model parameters. A comparison between experimental and predicted gas entrainment rates is shown in Fig. 9 (predictions based on parameter values obtained by curve fit of Eq. (15) to experimental data). The comparison covers the range of column and jet diameters, jet velocities, and liquid physical properties listed in Table 1. It can be seen from the results that the agreement with predictions is within _+20%, which is of the same order of uncertainty for the jet diameter versus jet length model. A general comparison in the model predictions for Q~/QL with those from previous studies [1] can be made by considering the results for the 4.76-mm-diameter jet with velocity 11.5 m / s for the 44-, 74-, and 95-mm-diameter columns. The model predicts recirculated gas ratios Qr~/QL of 0.64, 0.82, and 0.93 for the 44-, 74-, and
0.2
,...i....i.~..
0.3 0.4 0.5 0.6 Measured value
0.7
0.8
Figure 8. Measured vs. predicted [from Eq. (17)] normalized jet diameter.
(17)
- - - 1 = 0.0085 O h °'s3 R e °'63 Lj
0.1
I ....
hp,,l.,.il
....
I ....
~
h 'I fl ....
I ....
l....l
....
I
o -
1.4•
o
1.2-"
o
-20% -
**
.
0.80.6-
O 0.2-
A
,
f
0 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Measured value
2
Figure 9. Measured vs. predicted normalized gas feed rate for confined jet. 95-mm-diameter columns, respectively. It can be seen that the recirculated gas component increases with increasing column diameter, which is not accounted for in any other model. The recirculated gas component cannot be ignored in confined plunging jet systems, as the total entrainment rate of the jet, QE/QL, is approximately 0.6-1.5. Hence, gas recirculation significantly reduces the amount of fresh gas feed that can be introduced into the system. The numerical parameters used in conjunction with Eq. (15) to obtain the predictions are listed in Table 2. First, it can be seen that the normalized position of the separating
148 G.M. Evans and G. J. Jameson Table 2. Model Parameter Values Parameter
Value
D1/D N D*/D N
1.1 2.4 0.08 0.86 0.58
K1 K2 K3
streamline was found to be 2.4, which means that all data used in this study fell in region 2 of the gas entrainment model. In practice, most of the experimental data applicable to confined plunging jets used in industry lie within region 2. For example, region 2 for a 5-mm-diameter w a t e r jet traveling at 15 m / s covers the jet length range 15-1000 mm; typically, the desired operating length is 15-100 mm. The value of 0.08K1, the ratio of the entrained film to the jet velocity, indicates that the average velocity of the gas in the film in region 2 rises to about 24% of the liquid nozzle velocity. In the absence of experimental measurements it is not possible to verify this value, except to say that it is the right order of magnitude considering the geometry of the entrained gas flow field. Similarly, it was not possible to experimentally verify the numerical values for K 2 and K 3. It appears that Eq. (8), in conjunction with the numerical values listed for K 2 and K3, c a n be used to quantify the recirculated gas component. However, further work is required to verify the model assumptions and numerical parameters. PRACTICAL SIGNIFICANCE/USEFULNESS The practical significance of this study lies in the development of a theoretical model to determine the rate of gas entrainment of a confined plunging liquid jet for a given set of free jet characteristics. Using this model the gas entrainment rate has been directly related to the expansion of the jet. The model incorporates two mechanisms: (1) gas trapping in the free jet through surface instabilities and (2) entrainment of the gas film at the plunge point, both of which are related to the jet diameter. Confined plunging jet reactors are finding increasing use in a number of industrial applications, including mineral flotation and gas absorption operations. For each application, it is important to know how much gas will be entrained into the reactor for a given jet and column configuration, to ensure optimum performance. CONCLUSIONS The effective jet diameter has been measured as a function of free jet length for a vertical water jet passing through air. It was found that the jet diameter increased approximately linearly for about three jet lengths downstream of the nozzle. In this region axisymmetric disturbances were present on the jet free surface. Further downstream, helical disturbances dominated the surface characteristics of the jet and the normalized jet diameter increased with increasing jet length. In this region the jet expansion was correlated with the jet stability and Reynolds number. The jet diameter measurements were incorporated into a model to predict the rate of gas
entrainment for a confined plunging jet. In the model it was assumed that the total gas entrainment rate included gas contained within (1) the effective diameter of the free jet at the plunge point and (2) an annular film adjacent to the surface of the jet, where the outer boundary of the film was defined as a streamline separating the entrained and unentrained components of the moving gas boundary layer. It was further assumed that the radial location of the separating streamline was independent of system geometry and gas and liquid flow rates. Good agreement between model predictions and gas entrainment measurements was obtained, and a number of experimental parameters were determined, including the fraction of gas recirculated back to the headspace. The novelty of the model is that it includes both filmwise and trapped gas entrainment, based on the jet diameter at the plunge point, and also accounts for reentrainment of recirculated gas in a confined system. At this stage a number of parameters need to be determined experimentally, and current work is under way to predict the parameter values from first principles.
NOMENCLATURE A C cross-sectional area of column, m 2 C T Crayer-Curtet number, defined by Eq. (11), dimensionless D c column diameter, m Dj effective jet diameter, defined in Fig. 4, m D N nozzle diameter, m D 1 effective jet diameter at the end of region 1, m D* diameter of separating streamline, defined in Fig. 4, m KI, 2,3 empirical constants, dimensionless Lj free jet length, m Oh Ohnesorge or stability number ( = Ix/p~/-~Ntr), dimensionless QB total bulk recirculating flow rate, m3/s QE total volumetric gas entrainment rate, m3/s QF filmwise volumetric gas entrainment rate, m3/s QG volumetric flow rate of gas entering column headspace, m3/s QT trapped gas volumetric gas entrainment rate, m3/s QR liquid volumetric flow rate, m3/s QR volumetric flow rate of gas recirculated back to column headspace, m3/s ReLj jet Reynolds number ( = P V N L j / t x ) , dimensionless v B average bulk recirculation velocity inside column, m/s v c superficial liquid velocity inside colum, m / s v F average velocity of gas inside film bounded by Dj and D*, m / s v N superficial liquid velocity based on nozzle diameter, m / s Greek Symbols /x liquid absolute viscosity, Pa s p liquid density, k g / m 3 tr surface tension, N / m
A Plunging Liquid Jet REFERENCES 1. Bifi, A. K., Gas Entrainment by Plunging Liquid Jets, Chem. Eng. Sci. 48, 3585-3630, 1993. 2. McCarthy, M. J., Entrainment by Plunging Jets, Ph.D. Thesis, Univ. Newcastle, Newcastle, Australia, 1972. 3. Barchilon, M., and Curtet, R., Some Details of the Structure of an Axisymmetric Jet with Backflow, J. Basic Eng. 86, 777-787, 1964. 4. Liu, C., and Barkelew, C., Numerical Analysis of Jet-Stirred Reactors with Turbulent Flows and Homogeneous Reactions, AIChEJ. 32, 1813-1820, 1986. 5. Peebles, F., and Garber, H., Studies of the Motion of Gas Bubbles in Liquids, Chem. Eng. Prog. 49, 88-97, 1953. 6. Evans, G., A Study of a Plunging Jet Bubble Column, Ph.D. Thesis, Univ. Newcastle, Newcastle, Australia, 1990.
149
7. Hoyt, J. W., and Taylor, J. J., Waves on Water Jets, J. Fluid Mech. 83, 119-127, 1977. 8. Taylor, J. J., and Hoyt, J. W., Water Jet Photography--Techniques and Methods, Exp. Fluids 1, 113-120, 1983. 9. McCarthy, M. J., and Molloy, N. A., Review of Stability of Liquid Jets and the Influence of Nozzle Design, Chem. Eng. J. 7, 1-20, 1972. 10. Arai, A., and Hashimoto, H., Helical Surface Instability of Cylindrical Liquid Jet in Concurrent Gas Stream, Bull. JSME 247, 77-821, 1986. 11. Mattingly, G. E., and Chang, C. C., Unstable Waves on an Axisymmetric Jet Column, J. Fluid Mech. 65, 541, 1974.
Accepted August 24, 1995
Free Jet Expansion and Gas Entrainment Characteristics of a Plunging Liquid Jet G. M. Evans G. J. Jameson Department of Chemical Engineering, University of Newcastle, Newcastle, Australia C. D. RieUy Department of Chemical Engineering, University of Cambridge, Cambridge, England
mThe change in effective jet diameter is measured as a function of free jet length for vertical liquid jets passing through air. The data are incorporated into a model to predict the rate of gas entrainment for a liquid jet plunging into a confined column of liquid. In the model it was assumed that the total gas entrainment rate included gas contained within (1) the effective diameter of the free jet at the plunge point and (2) an annular film adjacent to the surface of the jet, where the outer boundary of the film was defined to be the separating streamline between the entrained and unentrained components of the moving gas boundary layer. It was further assumed that the radial location of the separating streamline was independent of both liquid and gas flow rates and system geometry. Excellent agreement between model predictions and gas entrainment measurements were obtained once a number of experimental parameters were determined. Keywords: plunging liquid jets, gas entrainment, free jet expansion
INTRODUCTION A plunging liquid jet is defined as a moving column of liquid that passes through a gaseous headspace before impinging on the horizontal free surface of the receiving liquid. Examples of plunging liquid jets in nature include waterfalls and breaking waves. In industry, plunging liquid jets can be found in the steel teeming process, roll-coating systems, and plunging liquid jet bubble columns [I]. As shown in Fig. 1, a feature of plunging liquid jets is that a depression is formed in the receiving liquid free surface, referred to as the induction trumpet [2] due to its characteristic trumpetlike profile. Gas is drawn into the induction trumpet by the free jet before being entrained into the receiving liquid and broken up into fine bubbles. In some instances, such as roll coating and steel teeming, the presence of bubbles is detrimental to the process because they cause imperfections to the final product. However, in many processes the entrainment characteristics of a plunging liquid jet are beneficial, especially those involving mass transfer and aeration operations. In many industrial processes, such as those involving mineral floatation, gas absorption, and chemical reactions, confined plunging jets are applied. In these systems the free jet passes through an enclosed headspace before plunging into the receiving liquid contained within a vertical column, as shown in Fig. 2. Typically, the c o l u m n / j e t diameter ratio is of the order of 10 : 1. The action of the jet plunging into a confined volume generates intense recirculation and high-energy dissipation rates, resulting
in the manufacture of very fine bubbles, in the range of 100-500 ttm diameter, and large amounts of interfacial area. In general, the amount of interracial area produced is increased by increasing the gas/liquid flow ratio entering the column. However, an increase in this ratio can also lead to instabilities and in some cases to the complete collapse of the system. This is shown in Fig. 3. At low gas rates into the column, the level of the gas-liquid mixture (froth) is sustained just below the level of the nozzle, indicating that the plunging jet can effectively entrain all of the gas supplied to the headspace. If the gas flow rate is increased, a point is reached where the jet can no longer entrain all of the gas being introduced into the headspace, and the froth level in the column starts to drop. At this point a compensatory phenomenon is at work that has the effect that the rate of entrainment of air by the plunging jet increases up to a point as the length of free jet increases. The phenomenon is possibly linked to the increase in the effective jet diameter with increase in the free jet length. Whatever the reason, a new equilibrium height is reached inside the column, marking the length where the jet can effectively entrain all of the entering gas. In a sense the jet is "self-regulating" in that the length of the free jet will increase to accommodate the increase in the amount of gas added to the column (up to a point). At very high gas rates, bubble coalescence results in the formation of large gas slugs lower in the column that restrict the downward flow of gas. This leads to an
Address correspondence to Dr. Geoffrey Evans, Department of Chemical Engineering, The University of Newcastle, Newcastle, New South Wales 2308, Australia.
Experimental Thermaland Fluid Science 1996; 12:142-149 © Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010
0894-1777/96/$15.00 SSDI 0894-1777(95)00095-X
A Plunging Liquid Jet
143
IIncreasinggas feed rate I [Increasinggas recirculationI
free surface
-------t>~
receiving
induction trumpet
liquid
K
o'O ooO o
0
0
entrained
gas
!~iil
o
Figure 1. Plunging liquid jet. IDecreasingcontactareaI
- - ~I QG
Ioee~easin~stabitity I
I t2L fA
Figure 3. Column stability. quantity of gas, QT, that is trapped within the effective diameter of the jet (trapped gas component). This can be written as QE = QF + QT"
N,M
Figure 2. Confined plunging jet. increase in the pressure in the headspace, causing a drop in the froth level and, in the extreme case, the complete collapse of the system. To effectively operate confined plunging liquid jet reactors it is essential to know the free jet length as a function of gas and liquid flow rates, column diameter, and liquid physical properties. A number of entrainment models have been proposed for unconfined jets [1]. However, the weakness of all these models is that they do not contain the column diameter as a variable, whereas it has been found experimentally that column diameter has a significant effect on the rate of gas entrainment into industrial confined plunging jet reactors. The purpose of this paper is to present a gas entrainment model for a confined plunging liquid jet that takes into account column diameter and reentrainment of recirculated gas. The model is based on the effective jet diameter as a function of free jet length. THEORETICAL The proposed model is based on the assumption that the total gas entrainment rate, QE, for a plunging liquid jet is the sum of (1) the gas that is contained within a thin annular film of gas, QF, which is carried along adjacent to the jet free surface (filmwise component), and (2) the
(1)
A further assumption (which applies only when the effective jet diameter, defined as Dj in Fig. 4, is less than D*) is that the outer boundary of the entrained gas film is defined by a time-average streamline, with diameter D* as shown in Fig. 5, which separates the entrained and unentrained components of the moving gas boundary layer associated with the free jet. On the basis of the above-mentioned assumptions, three distinct regions of gas entrainment can be identified, depending on the length of the jet.
Region 1 Once the jet leaves the nozzle, the velocity profile changes from pipe to plug flow. Usually, this takes about 3 - 5 nozzle diameters, and any change in
jet length
L1
t envelope
Figure 4. Effective jet diameter.
144 G.M. Evans and G. J. Jameson jet diameter is due to the relaxation in the velocity profile. During this period there is no gas penetration into the outer boundary of jet, that is, no trapped gas, and only filmwise entrainment takes place. This can be written as "B"
QE = OF = ~-( D*2 -- DJe)VF,
un-entrained ~
1
separating streamline
(2)
where v F is the average velocity of the gas inside the film, bounded by D* and the outer boundary of the jet, Dj, that is, the effective jet diameter. entrained
R e g i o n 2 For jets longer than about 3 - 5 D N, the velocity profile has relaxed to approximately plug flow, although there is a slight reduction in the surface velocity due to the drag exerted by the surrounding gas phase. There is also an increase in the average je t velocity due to the acceleration of gravity. Both of these terms are small, however, compared with the average velocity of the jet, and the free jet velocity can be considered constant and independent of jet length. Waves on the jet surface result in gas being entrained within the jet envelope (trapped gas) as well as in the film adjacent to the jet free surface and within D*. The general expression for the rate of entrainment (i.e., filmwise plus trapped gas) is
increases in region 1. In region 2 it is assumed to remain approximately constant. Thus, in region 1 the average velocity of the gas inside the film may be taken to be proportional to the velocity of the jet leaving the nozzle and the length of the jet, that is,
QE = QF + QT = ~-( 0 . 2 -- D2)VF + ~"/'f( O j 2 -- O2)CN,
REGION 1
o°O oo°0 o
0
0
o
Figure 5. Entrained gas component.
(3) where D 1 is the effective diameter of the jet at the end of region 1 and CN is the average jet velocty at the nozzle. The first term on the right-hand side of Eq. (3) represents the filmwise entrainment, and the second term is the trapped gas component. In this last term, it is assumed that there is no slip between the liquid and entrained gas.
CF = K | ( L j / D N ) U N ,
(5)
where D N is the nozzle diameter, Lj is the free jet length, and K 1 is a constant. The length of region 1 is approximately 3D N. Thus, in regions 2 and 3, where the flow in the gas film is independent of length of jet, the average gas velocity is given by REGIONS 2 AND 3
3 It may be assumed that the radial position of the separating streamline remains fixed (i.e., the diameter D* in Fig. 5 remains constant), and so the thickness of the film decreases as the effective jet diameter increases. A point is reached where the gas film thickness reduces to zero, when Dj equals D*, and gas entrainment may be entirely attributed to the trapped gas component, that is, Region
7/"
QE = a T = --~( D 2 -- D 2 ) U N •
(4)
Equations (2)-(4) can be used to predict the total rate of gas entrainment for a plunging liquid jet, provided (1) the diameter and velocity relaxation characteristics of the free jet and (2) the velocity profile of the entrained gas film are known. The velocity profile of the entrained gas stream is not straightforward to measure. The flow in the gas film is generated by the interfacial drag, and so it might be expected that the average velocity in the film, v F, would be related to the velocity on the surface of the jet. Initially the jet surface velocity is zero leaving the nozzle and
UF = 3KlV N.
(6)
The entrainment model outlined above is useful for determining the total amount of gas a plunging liquid jet can entrain below a liquid surface. It is applicable to both confined and unconfined plunging jet systems. For confined systems, as shown in Fig. 2, it is more important to know the quantity of gas feed, QG, entering the headspace at the top of the column. This is equal to the total rate of gas entrainment minus the recirculated gas, QE, which disengages from the recirculating liquid at the top of the column and is reentrained by the jet; that is, Oo = QE - QR.
(7)
The total rate of gas entrainment is given by Eqs. (2)-(4). An estimate of the quantity of recirculated gas can be obtained by considering the way in which the gas (bubbles) is released from the receiving liquid. As shown in Fig. 2, a confined plunging liquid jet generates a recirculating bubbly flow region at the top of the columnJ I The recirculation regions is referred to as the mixingzone.
A Plunging Liquid Jet Superimposed on the bulk flow is the rise velocity of the bubbles, u,, which results in a number of bubbles disengaging from the flow and being recirculated back into the headspace at the top of the column. Mathematically, the interaction between bulk flow and bubble rise velocity is very complex. Qualitatively at least, one would expect the amount of recirculated gas, Qn, to increase with increasing bubble rise velocity and decreasing recirculation velocity, vn. A simple formulation that allows us to achieve this relationship is a dimensionless power law, Qn = KZ(v,/vn)K3,
VB =
QB/&
(9)
Barchilon and Curtet [3] have quantified culation as a function of the Crayer-Curtet Qn = Qr(O.37/C,
the bulk recirnumber, C,:
- 0.64),
&= &?&
(11)
Eqs. (91 and (101 into (8) gives K3
’
(12)
where vc is the superficial downward liquid velocity inside the column. The bubble rise velocity can be calculated using the formula given by Peebles and Garber [51,
v, = 1.35
fTg( p - PG)
(
0z5
=,Ki[(z)‘-
(%)‘I
-K, I
“r uc
(
+ [($)‘-
Cr 0.37 - 0.64C,
)I
Region 3
QG -= QL
’
EXPERIMENTAL The experimental apparatus [61 is shown in Fig. 6. It consisted of a perspex column mounted vertically above a constant-level bath. The top of the column was enclosed, and the opening at the base of the column was located 25 mm below the free surface of the bath. The column was effectively isolated from the atmosphere, and gas (air in this case) could enter the column only through the metered inlet at the top. The metered liquid feed entered the column via a jet delivery tube located at the top of the column. The delivery tube was aligned with the vertical axis of the column so that the liquid jet plunged into the constant-level bath below. The inside diameters, D,, of the jet nozzles used in this study were 2.38, 4.76, and 7.12 mm. Each nozzle consisted of a conical entrance section that fitted neatly inside the nozzle delivery tube, giving a smooth transition. The tapered entrance led to the throat of the nozzle, which
--i
P F
by-pass \
z=K~($)[(g)‘($1’1 K3
’
(16)
where DJD, is related empirically to L, by Eq. (171, which can also be used to calculate the jet diameter, D,, at the end of region 1. The unknowns in Eqs. (14)-(16) are K,, K,, K,, and D* and can be obtained from regression analysis applied to the experimental measurements.
i
Region 1
(15)
’
liqtlid feed
where u is the surface tension and p and pG are the liquid and gas densities, respectively. Finally, combining Eqs. (21-02) leads to the following expressions for the normalized gas feed rate for a confined plunging liquid jet:
(%)‘I
K3
(131
7 P2
2
(101
where Qr is the jet volumetric flow rate. The numerical constants in Eq. (10) are those proposed by Liu and Berkelew [4]. For confined plunging liquid jets, the Crayer-Curtet number is related to the nozzle diameter, D,, and column diameter, D,, by
Substituting
Region 2
(81
where K, and K, are constants. The velocity profile in the bulk recirculation region is very complex. However, an estimate of the average bulk circulating velocity can be obtained by dividing the bulk recirculation flow rate, Qn, by the flow cross-sectional area of the column, A,:
145
(14) Figure 6. Experimental
apparatus.
146
G . M . Evans and G. J. Jameson sion and a 150-mm telephoto lens were again used, with an aperture setting of f/5.6. The shutter speed was increased to 1/250s, and photographs were taken in a darkened room to eliminate any influence of ambient light. The 10-p.s flash was able to effectively "freeze" the motion of the jet, which allowed qualitative comments to be made about the characteristic of the jet surface.
consisted of a straight length of precision-bore brass tubing. The throat length for each nozzle was approximately 17D N. The liquids used were tap water, kerosene, and 16 and 28 wt % sucrose solutions. Ten parts per million of Teric 407 (Imperial Chemical Industries) was added to the water to prevent bubble coalescence, and the subsequent formation of large gas slugs, inside the column. The experimental procedure involved adjusting the valve .in the bypass line to give the desired superficial liquid velocity through the nozzle. The gas inlet valve at the top of the column was opened, and the entrained gas volumetric flow rate was recorded. Once the resultant two-phase mixture (froth) had reached a steady height inside the column, the free jet length, L j , w a s measured from the tip of the nozzle to the point where the jet penetrated the froth. While maintaining the same liquid flow rate, the gas inlet valve was used to vary the gas flow rate, and each corresponding equilibrium froth height was recorded. The above procedure was repeated using each nozzle for a range of liquid flow rates (jet velocity) and column diameters. The experimental conditions for each trial are summarized in Table 1. In addition to the gas entrainment measurements, the effective jet diameter was measured as a function of free jet length (see Fig. 4). For these measurements the column was removed from the apparatus to eliminate distortion created by photographing through a curved surface. The moving jet was photographed under natural lighting conditions using a Linhoff Polaroid camera (aperture setting f/5.6; shutter speed 1 / 6 0 s ) fitted with a 156-mm telephoto lens and extension bellows. The 1/60-s exposure produced a time-averaged image of the outline of the jet that corresponded to the effective jet diameter. The surface structure of the free jet was also examined. A black background was placed behind the jet, and a 10-/zs-duration flash was mounted to the side at 45 ° to the vertical focal plane of the Linhoff camera. A white reflecting screen was placed on the other side of the camera to produce an even lighting of the jet. The bellows exten-
RESULTS The uncertainty in the experimental results is due mainly to (1) the difficulty in defining the outer boundary of the jet surface, that is, the effective jet diameter, from the time-average photographs and (2) the uncertainty in the actual length of the free jet for the gas entrainment experiments. (The latter uncertainty is due to the difficulty in determining the distance below the horizontal free surface in the column where the free jet came into contact with the froth.) It is estimated that the uncertainty of the effective jet diameter measurements as a function of free jet length is on the order of 20% [6].
Jet Diameter Expansion A flash photograph of the jet is given in Fig. 7, which shows the presence of a large number of disturbances on the jet free surface. Moreover, the structure of the jet appears to change at around three to five jet lengths from what could be described as a random distribution of small amplitude disturbances to a much larger scale helical disturbance. This is consistent with observations from other studies [7-9], that found that the relative velocity between the gas phase and the moving jet resulted in the formation of both axisymmetric and nonaxisymmetric (helical) disturbances [10] being generated on the jet surface. Furthermore, according to the analysis of Mattingly and Chang [11], the axisymmetric disturbances have greatest amplification in the first three diameters of the jet, whereas the helical waves dominate further downstream.
Table 1. Experimental Conditions Liquid
D C (mm)
D N (mm)
10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 16 wt % Sucrose 28 wt % Sucrose Kerosene 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407 10 ppm Teric 407
44 44 44 44 44 44 44 44 74 74 74 74 74 95 95 95 95 95 95
7.12 4.76 4.76 4.76 2.38 4.76 4.76 4.76 7.12 4.76 4.76 4.76 2.38 7.12 7.12 7.12 4.76 4.76 4.76
UN
(m / s) 11.5 15.0 11.5 7.8 11.5 11.5 11.5 11.5 11.5 15.0 11.5 7.8 11.5 15.0 11.5 7.8 15.0 11.5 7.8
o" (N / m)
Iz (Pa s)
p (kg / m 3)
0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.028 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063
0.0009 0.0009 0.0009 0.0009 0.0009 0.0017 0.0029 0.0021 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009
998 998 998 998 998 1061 1114 784 998 998 998 998 998 998 998 998 998 998 998
A Plunging Liquid Jet ....
0.8
I,
.,,I,,
,,I,
,,,
I,,,.
I,,.,I,,.,
147
I,,,,
/
+20%
0.7. 0.6. 0.5. 0.4"
LjIDN
t~ "d 0.3.
0.2: 0.1. A.
Figure 7. Jet photograph.
0
f.,..l,,,,l,.,,l...,l....l
0 For each jet arrangement, the time-average photographs were used to determine the effective jet diameter as a function of free jet length. The measurements were correlated by the dimensionless expression Oj
2
where Re L is the jet Reynolds number ( = P U N L j / ~ ) based on tla~e liquid physical properties and free jet length L j; and Oh is t h e ~ Ohnesorge or stability number [9], given by /~/~/pDNO'. The numerical parameters in Eq. (17) were correlated for the range 3 < L j / D N < 30 and apply to highly turbulent jets issuing from nozzles with throat lengths of approximately 16DN. For other jet systems the numerical parameters may be different. A graphical comparison of measured and predicted normalized jet diameter is given in Fig. 8. It can be seen from the results that the agreement with predictions from Eq. (17) is approximately +20%, which is of the same order as the uncertainty of the measurements.
1.6
DN
Rate of Gas Entrainment The normalized gas entrainment QG/QL and free jet l e n g t h L j / D N measurements for L j / D N _> 3 were used to obtain the numerical values for the model parameters. A comparison between experimental and predicted gas entrainment rates is shown in Fig. 9 (predictions based on parameter values obtained by curve fit of Eq. (15) to experimental data). The comparison covers the range of column and jet diameters, jet velocities, and liquid physical properties listed in Table 1. It can be seen from the results that the agreement with predictions is within _+20%, which is of the same order of uncertainty for the jet diameter versus jet length model. A general comparison in the model predictions for Q~/QL with those from previous studies [1] can be made by considering the results for the 4.76-mm-diameter jet with velocity 11.5 m / s for the 44-, 74-, and 95-mm-diameter columns. The model predicts recirculated gas ratios Qr~/QL of 0.64, 0.82, and 0.93 for the 44-, 74-, and
0.2
,...i....i.~..
0.3 0.4 0.5 0.6 Measured value
0.7
0.8
Figure 8. Measured vs. predicted [from Eq. (17)] normalized jet diameter.
(17)
- - - 1 = 0.0085 O h °'s3 R e °'63 Lj
0.1
I ....
hp,,l.,.il
....
I ....
~
h 'I fl ....
I ....
l....l
....
I
o -
1.4•
o
1.2-"
o
-20% -
**
.
0.80.6-
O 0.2-
A
,
f
0 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Measured value
2
Figure 9. Measured vs. predicted normalized gas feed rate for confined jet. 95-mm-diameter columns, respectively. It can be seen that the recirculated gas component increases with increasing column diameter, which is not accounted for in any other model. The recirculated gas component cannot be ignored in confined plunging jet systems, as the total entrainment rate of the jet, QE/QL, is approximately 0.6-1.5. Hence, gas recirculation significantly reduces the amount of fresh gas feed that can be introduced into the system. The numerical parameters used in conjunction with Eq. (15) to obtain the predictions are listed in Table 2. First, it can be seen that the normalized position of the separating
148 G.M. Evans and G. J. Jameson Table 2. Model Parameter Values Parameter
Value
D1/D N D*/D N
1.1 2.4 0.08 0.86 0.58
K1 K2 K3
streamline was found to be 2.4, which means that all data used in this study fell in region 2 of the gas entrainment model. In practice, most of the experimental data applicable to confined plunging jets used in industry lie within region 2. For example, region 2 for a 5-mm-diameter w a t e r jet traveling at 15 m / s covers the jet length range 15-1000 mm; typically, the desired operating length is 15-100 mm. The value of 0.08K1, the ratio of the entrained film to the jet velocity, indicates that the average velocity of the gas in the film in region 2 rises to about 24% of the liquid nozzle velocity. In the absence of experimental measurements it is not possible to verify this value, except to say that it is the right order of magnitude considering the geometry of the entrained gas flow field. Similarly, it was not possible to experimentally verify the numerical values for K 2 and K 3. It appears that Eq. (8), in conjunction with the numerical values listed for K 2 and K3, c a n be used to quantify the recirculated gas component. However, further work is required to verify the model assumptions and numerical parameters. PRACTICAL SIGNIFICANCE/USEFULNESS The practical significance of this study lies in the development of a theoretical model to determine the rate of gas entrainment of a confined plunging liquid jet for a given set of free jet characteristics. Using this model the gas entrainment rate has been directly related to the expansion of the jet. The model incorporates two mechanisms: (1) gas trapping in the free jet through surface instabilities and (2) entrainment of the gas film at the plunge point, both of which are related to the jet diameter. Confined plunging jet reactors are finding increasing use in a number of industrial applications, including mineral flotation and gas absorption operations. For each application, it is important to know how much gas will be entrained into the reactor for a given jet and column configuration, to ensure optimum performance. CONCLUSIONS The effective jet diameter has been measured as a function of free jet length for a vertical water jet passing through air. It was found that the jet diameter increased approximately linearly for about three jet lengths downstream of the nozzle. In this region axisymmetric disturbances were present on the jet free surface. Further downstream, helical disturbances dominated the surface characteristics of the jet and the normalized jet diameter increased with increasing jet length. In this region the jet expansion was correlated with the jet stability and Reynolds number. The jet diameter measurements were incorporated into a model to predict the rate of gas
entrainment for a confined plunging jet. In the model it was assumed that the total gas entrainment rate included gas contained within (1) the effective diameter of the free jet at the plunge point and (2) an annular film adjacent to the surface of the jet, where the outer boundary of the film was defined as a streamline separating the entrained and unentrained components of the moving gas boundary layer. It was further assumed that the radial location of the separating streamline was independent of system geometry and gas and liquid flow rates. Good agreement between model predictions and gas entrainment measurements was obtained, and a number of experimental parameters were determined, including the fraction of gas recirculated back to the headspace. The novelty of the model is that it includes both filmwise and trapped gas entrainment, based on the jet diameter at the plunge point, and also accounts for reentrainment of recirculated gas in a confined system. At this stage a number of parameters need to be determined experimentally, and current work is under way to predict the parameter values from first principles.
NOMENCLATURE A C cross-sectional area of column, m 2 C T Crayer-Curtet number, defined by Eq. (11), dimensionless D c column diameter, m Dj effective jet diameter, defined in Fig. 4, m D N nozzle diameter, m D 1 effective jet diameter at the end of region 1, m D* diameter of separating streamline, defined in Fig. 4, m KI, 2,3 empirical constants, dimensionless Lj free jet length, m Oh Ohnesorge or stability number ( = Ix/p~/-~Ntr), dimensionless QB total bulk recirculating flow rate, m3/s QE total volumetric gas entrainment rate, m3/s QF filmwise volumetric gas entrainment rate, m3/s QG volumetric flow rate of gas entering column headspace, m3/s QT trapped gas volumetric gas entrainment rate, m3/s QR liquid volumetric flow rate, m3/s QR volumetric flow rate of gas recirculated back to column headspace, m3/s ReLj jet Reynolds number ( = P V N L j / t x ) , dimensionless v B average bulk recirculation velocity inside column, m/s v c superficial liquid velocity inside colum, m / s v F average velocity of gas inside film bounded by Dj and D*, m / s v N superficial liquid velocity based on nozzle diameter, m / s Greek Symbols /x liquid absolute viscosity, Pa s p liquid density, k g / m 3 tr surface tension, N / m
A Plunging Liquid Jet REFERENCES 1. Bifi, A. K., Gas Entrainment by Plunging Liquid Jets, Chem. Eng. Sci. 48, 3585-3630, 1993. 2. McCarthy, M. J., Entrainment by Plunging Jets, Ph.D. Thesis, Univ. Newcastle, Newcastle, Australia, 1972. 3. Barchilon, M., and Curtet, R., Some Details of the Structure of an Axisymmetric Jet with Backflow, J. Basic Eng. 86, 777-787, 1964. 4. Liu, C., and Barkelew, C., Numerical Analysis of Jet-Stirred Reactors with Turbulent Flows and Homogeneous Reactions, AIChEJ. 32, 1813-1820, 1986. 5. Peebles, F., and Garber, H., Studies of the Motion of Gas Bubbles in Liquids, Chem. Eng. Prog. 49, 88-97, 1953. 6. Evans, G., A Study of a Plunging Jet Bubble Column, Ph.D. Thesis, Univ. Newcastle, Newcastle, Australia, 1990.
149
7. Hoyt, J. W., and Taylor, J. J., Waves on Water Jets, J. Fluid Mech. 83, 119-127, 1977. 8. Taylor, J. J., and Hoyt, J. W., Water Jet Photography--Techniques and Methods, Exp. Fluids 1, 113-120, 1983. 9. McCarthy, M. J., and Molloy, N. A., Review of Stability of Liquid Jets and the Influence of Nozzle Design, Chem. Eng. J. 7, 1-20, 1972. 10. Arai, A., and Hashimoto, H., Helical Surface Instability of Cylindrical Liquid Jet in Concurrent Gas Stream, Bull. JSME 247, 77-821, 1986. 11. Mattingly, G. E., and Chang, C. C., Unstable Waves on an Axisymmetric Jet Column, J. Fluid Mech. 65, 541, 1974.
Accepted August 24, 1995