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The trailing vortex system produced by Rushton turbine agitators
Chemical Engineering Science, 1975. Vol. 30, pp. 109~1105.
Pergamon Press.
Printed in Great Britain
THE TRAILING VORTEX SYSTEM PRODUCED BY RUSHTON TURBINE AGITATORS K. VAN ‘t RIET and JOHN M. SMITH Physical Technology Laboratory, Technical University of Delft, The Netherlands (Received 21 November 1974; accepted 4 March 1975)
AbstractFurther studies of the discharge from a six blade Rushton turbine are reported. Photographic velocity measurements have been used to examine the trailing vortex system in water and aqueous glycerol. An analysis of the vortex field in terms of a solution to the simplifiedNavier-Stokes equations shows reasonable agreement with the measured velocity profiles. Pressure field determinations at two different scales prove that Reynolds number alone is a sufficient scale-up parameter. The centrifugal accelerations and shear rates associated with the tmiling vortices are shown to be much greater than previous literature has implied. Although restricted to single phase flow the results presented are of importance to most applications of disc turbines to dispersion operations. INTRODUCTION
Recent literature (Biesecker [ l]), Van ‘t Riet et al. 121and Bruijn et al. [3] shows qualitatively that in a turbine stirred vessel the flow in the stirrer blade region controls the power consumption, coalescence and dispersion in gas liquid mixtures to a large extent. A particularly important feature of the flow field is a trailing vortex pair that exists behind each stirrer blade (see Fig. IA-B). This trailing
vortex pair originates as a combined effect of the vortex motion in the dead water zone at the rear of the vertical inside edge of the blade and the wrapping of the vortex sheets behind both horizontal edges. The smooth outlIow at the vertical outer edge of the blade that is a necessary hydrodynamic condition is indeed present. In the present paper we present measurements of the velocity and pressure distributions within the trailing vortex. The
Fii. 1A. Schematic three dimensional view of the &ailingvortex pair.
position of the vortex axis is also determined. To describe the vortex motion analytically simplified Navier-Stokes (N.S.) equations for the vortex motion will be solved. men the velocity and pressure are known it is in principle possible to determine the influence of the vortex motion on dispersion processes. It is also possible to assess the influence of the &ailing vortex motion on turbulence measurements with stationary probes as made by Cutter[4], Rao[51, etc.
Fig. 1B. Schematic two dimensional view of the flow field in the stirrer blade region. S, stagnation point.
APPARATUS Measurements were obtained in two fully baffled vessels of 44 cm and 120cm dia. provided with six bladed disc turbine agitators of 17.6 cm and 48 cm dia. respectively. The proportions of the stirrer are given by the
1094
K. VAN‘t
&ET
ratios stirrer diameter to blade length to blade height being 20:5:4. The velocity measurements were obtained using the smaller tank and the turntable equipment described by Van ‘t Riet [2]. Photographs were made with stroboscopic illumination by a single lens reflex camera that rotated synchronously with the agitator. The stirrer speed could be as high as 5rev/s. Normal tap water and water/glycerine solutions were used with added polystyrene particles of 0.5 mm dia. as tracers Pressure was measured with an impact Pitot tube (O-5mm dia.) rotating together with the stirrer, thus stationary relative to the blade. For the 17.6 cm turbine the pressure on the impact tube was brought out via a hollow stirrer shaft and gland to an external manometer. With the 48 cm impeller a closed limb manometer, was mounted on and rotated with the stirrer shaft.
and J. M. Surt’n
Fig. 3A. Sinusoidalprojectionof a particlepath alongthe trailing vortex.
MEASURING METHOD AD
example of a photograph obtained with the rotating camera is shown in Fig. 2. In this photograph two tracer particle paths in the vortex pair are seen. From such a photograph it cannot be concluded whether these tracers are situated in the upper or lower part of the vortex pair, but there is ample additional evidence to establish the sense of rotation in the trailing vortices. The path of a tracer particle travelling out along a trailing vortex is seen on the photographs as a sinusoidal wave represented schematically in Fig. 3A. The circular cylindrical coordinate system used to describe this vortex motion is shown in Fig. 3B. The circumferential and axial velocity components v and w respectively are determined from measurements of the axial displacement s, the radius r and the corresponding time for one sine cycle. The radial velocity component u was too small to be measured.
Fig. 3B. Circular cylindrical coiirdmate system with corresponding velocity components.
Some aspects of non ideality of the vortex motion which influence the accuracy of the measurements must be considered however. The linear axis of the coordinate system is only an approximation to the curved line of the vortex axis. The fit is particularly poor in the section near the tip of the blade. Most of the measurements were done further from the blade where the vortex axis is more nearly linear. The effect of a curved vortex axis would be to change the circumferential velocity for two successive half sine cycles and indeed no such effect could be found in this region. Strictly speaking the vortex is asymmetric
Fig. 2. Stroboscopic photograph of the flow field in the stirrer blade region.
The trailing vortex system produced by Rushton turbine agitators
in that the centres of rotation for different radii are not coincident. This is particularly important in the region just behind the blade where the differences of position of the centres of rotation between small and large radii amounted to some millimeters. Besides that the streamlines in a plane perpendicular to the axis are not exactly circular, but somewhat elliptical. Due to some erratic movement of the vortex the position of the vortex axis is not fixed. Its position had to be estimated separately for each of the measurements from the sine shaped curve in question. The axial velocity at a defined vortex radius shows local values that deviate from the average value (the distance from the stirrer axis is important for example). However, with our technique, local values could not be measured accurately so an average axial velocity over one sine cycle is presented. The effects described above, and in particular the need to estimate the position of the vortex axis for each measurement, can result in an estimated experimental inaccuracy’ for individual measurements of 20% in the axial distance, an error of 30% in the radius from the vortex axis and an error of 20% in the time needed for one sine cycle. It has been established that the error due to tracer particle dimensions, the position of the camera relative to the vortex and the density difference between these tracer particles and fluid can be neglected. To determine the static pressure, the total pressure as read from the manometer has to be corrected for two effects. Firstly the influence of the centrifugal forces on the fluid in the Pitot tube that rotates with the stirrer. This effect is given by the pressure difference in a forced vortex having a radius the same as the distance from the axis to the tip of the Pitot tube. Secondly there is a dynamic pressure due to local fluid velocities directed into the Pitot tube. A correction for this can be estimated from the velocity measurements. When seen from the stirrer axis at the outer side of the vortex in the region where the vortex has left the blade the correction may become large (even larger than the value of the downpressure itself) as well as inaccurate. Near the centre of the vortex the correction becomes relatively small as well as accurately known.
COORDINATESYSTEMAND RELATEDN.S. EQUATIONS
To describe the vortex motion analytically a circular cylindrical coordinate system is used. The z-axis is located in a horizontal plane and its origin is at the outer edge of the blade, 3 mm below the upper tip at an angle of 82 degrees to the cutting line of the horizontal plane and the stirrer blade (see Fig. 4). This z-axis fits the vortex axis reasonably once the vortex has left the blade. A variation of the axis angle to the blade and its distance from the upper tip with various NRC is permitted because the expressions for the velocity which will be derived later are independent of these variables. The constant rotation of this coordinate system with the blade requires the inclusion of the fictitious coriolis force F, and centrifugal force Fcf given by (vector notation) F,, = -pa
x (a x RP)
(I)
1095
Stirrer
z-axis
Fig. 4. The coordinate system used to describe the vortex motion.
and F,, = -p2a x v,
(2)
in which R = stirrer angular velocity, R0 = distance between a given point and the stirrer axis (see Fig. 4) and u, = velocity relative to the rotating coordinate system. The N.S. equation in the rotating coordinate system thus becomes: pa=pf-2pnxv,-pnx(fixR,)-;Vp+vAv,
(3)
in which a = acceleration, f = body force per unit mass, p = pressure and TJ’=dynamic viscosity. The centrifugal and the gravitational forces, being dependent on the coordinates only, can be written as a pressure. If the gravitational. force is the only body force this leads to: (4) in which p*=ptpgrsin4_fp(nxR,)‘tp,
(9
p* is the modified pressure, the remaining part of the pressure arising from the fluid motion. p1 is a constant. The extent to which the coriolis force effects the fluid motion can be expressed in a Rossby number NRo (see Batchelor[61, Chap. 7). Nn, can be evaluated in this particular case as NRo= R/r
(6)
in which R = stirrer radius. Batchelor states that the coriolis force can be neglected if N,s, 9 1. With the 17.6 cm stirrer this condition is satisfied when r < 10 mm. Even at, the maximum vorte% radius measured Nn. does not fall below a value of about 3.5 so only a minor influence may be expected. The coriolis force arising from the circumferential velocity component is shown qualitatively in Fig. 5. From this figure it can be deduced that the effect of this component of the coriolis force on the average axial velocity will be smoothed out because of its
K. VAN‘t Rrw and J. M. Sf,fm+
lo!m
the angular momentum is the transferable quantity. At
some radii the circulation distribution will deviate from eqn (9). For very small radii a solid body rotation must exist (I 0: r*). At increasing radii eqn (9) predicts an ever increasing circulation, which is impossible from a physical viewpoint. The circulation presented by eqn (9) can only occur somewhere between these two extremes.
Enveloping cylinder around particle path
(b)
Fi. 5. Coriolis force due to the circumferential velocity component:(a) Plan; (b) Section.0, Vector directed out of the plane; X,Vectordirectedin to the plane. alternating character. The coriolis force due to the axial velocity component is always directed inwards perpendicular to the vortex axis. This component does not have a real effect on the average circumferential velocity, again because of the alternating character. The curvature of the vortex axis is produced by this component. Thus it can be concluded that the coriolis forces at the radii of interest will be small in any case and their effect on the average velocity components will be smoothed out. They can therefore be left out of the N.S. equation which becomes (7)
This equation will be completed with the equation of continuity for an incompressible medium
v . v, = 0.
htpow theory Writing out eqn (7) in its components and leaving out the derivatives with respect to t and 4 shows that neither the equation for the circumferential component nor the equation of continuity irtcludes the pressure. These two equations can be regarded as the differential equations for the velocity field. After these equations are solved, the pressure can be derived from the remaining two equations. The derivation of eqns (7) and (8) starts from a laminar flow. If present, the influence of turbulence can be taken into account by assuming a constant kinematic eddy viscosity. The two equations for the velocity field contain three variables. Therefore we choose here on the basis of the experiments, w = 2bz + k,
so that from the equation of continuity it can be calculated that u = -br.
(11)
Now, when assuming avlaz = 0, it can be shown that v
=
&
(1 _
e-(br21*vQ
in which b and k are constants, Y = kinematic eddy or laminar viscosity and I, = circulation at infinity. The solution corresponds approximately to the velocity distribution presented by Rott [9]. Important properties of eqn (12) are as
r -*m
then
v --fI,/2rr
(13)
(8)
Equations (7) and (8) describe the velocity and pressure distribution within the vortex.
-at large radii the vortex tends to a free vortex with circulation r, as
THEORETICAL DESCRlFllONS OPTHE VORTEX VELOCITYFDZLD
r+O
then
o+(Lb/hv)r
(14)
-at small radii the vortex becomes a forced vortex with an angular velocity of I’mb/4w.
Logarithmic circulation distribution theory
Hoffmann [71and Saffman181 have studied the velocity distribution within a turbulent trailing vortex. For a degree of turbulence such that the velocity field is completely independent of the laminar viscosity they both derived (although in different ways): I/r, = b In (r/r,) + 1
(10)
(9)
in which I = circulation at a radius r, r, = circulation at a particular radius rl and c = constant. Both assumed that
and
v = urn at
rm = 1.12
J2v b
(19
-there is one particular radius r,,, at which the circumferential velocity has a maximum vn. Conclusions
For the turbulent case the velocity distribution presented by eqn (9) can be applied only in a region of intermediate radii. Due to its qualitative character the
The trailing vortex system producedby Rushtonturbine agitators velocity cannot be calculated from the given parameters
of the system. The inflow theory [eqns (lO-12)l can describe the flow field at any radius. If the flow is laminar the velocity distribution can be calculated from the system parameters. However in a turbulent system an eddy viscosity must be estimated in order that its value can be introduced in eqn (12). The circumferential velocity predicted by the inflow theory is independent of the distance z. This means that the damping out of the vortex motion that would normally be expected does not occur. This can indeed happen because of the transport of momentum by the radial velocity component. EXPERIMENTAL The position of the vortex axis
The position of the vortex axis has been determined from the photographs for the 17.6 cm stirrer. In the region immediately behind the blade the centre of rotation for smaller and larger radii differed by some millimeters. In the region where the vortices have left the blade, the variation in position for different radii was within experimental error. In particular far from the blade the erratic movement of the vortex and/or measurements behind the different blades, causes a variation in position of some 5-10mm. The results at the different Reynolds number NRI are presented in Fig. 6. This shows that the position of the vortex axis is independent of NRe at NRI >5 x 10’. No measurements were available in the region 500 < NRI < 5 X lo’. The vortices were first formed in the range 150< NR~< 250. Direction of
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dividing o by the stirrer angular speed 27rN. At fixed conditions the angular velocity appears to be independent of the distance along the vortex axis. Measurements showed the dimensionless angular velocity to be dependent only on NRI.However at 1.5 X 10’< NR~< 9 X la’ the differences in angular velocity were so small that they were within the experimental accuracy. The results are shown in Fig. 7. In this figure for 1.5 X 10’< Nn, < 9 X 10’ a number of points are shown. Each of these represents between 10 and 30 measured values (individual standard deviation less than 20%) taken over a range of radii centred on that used for the point. The measurements for the other curves presented in Fig. 7 were not numerous enough to allow determination of such averages. For Nn, = 300 a number of measuring points, whose distribution around the curve is typical for all the curves, are added. From Fig. 7 it can be seen that the vortex tends to a forced vortex (given by o = constant) at smaller radii. The rigid core is larger at lower NR~.At larger radii the inertia forces will become more important. The vortex in which the inertia forces are dominant is the free vortex given by slope = - 2, (or’ = constant). There is indeed a tendency for the slope to approach this value at increasing radii. A limitation can be expected to the maximum radius of rotation in a trailing vortex both because of the finite size of its object of generation and the influence of coriolis forces at large radii. Though it is difficult to determine the maximum radius is thought to be between 0.14 and 0.2 r/D. The circumferential velocity (N.B. not tangential
Blade
Fig. 6. The vortex axis position at different iVRc
The angular and circumferential velocities The angular velocity o within the vortex was measured by determining the time needed for one sine cycle at a particular radius. Measurements were only made in the region between the point where the vortices have just left the blade and imaginary extension of the following blade. NRI was varied between 300 and 9 x 10’ by variation of stirrer speed and fluid viscosity. For each N, s 5 X ld at least two different speeds that differed by a factor 4 were used. The results are expressed dimensionlessly by
velocity) has been calculated directly from the curves that present the dimensionless angular velocity (Fig. 7). The results shown in Fig. 8 are presented dimensionlessly in terms of v,, the circumferential velocity of tbe stirrer tip. At all Nm the dimensionless circumferential velocity shows a rather flat maximum beyond which the velocity decreases slowly. For l-5 x 10’< Nfi < 9 x lo’ this maximum velocity is approximately the same as the tip speed. At smaller NRI this maximum velocity is less while the radius at which the maximum occurs is larger.
K.
1098
VAN ‘t Rtm
and J. M. SMITH
01 0
1
3
8
Dimensionless 20 Dimensiontess
rodius
from
vortex
OXIS, f/DxlO’
distance
1 8 b a 0 05 along Ihe vortex axis, z/D
Fig. 9. The dimensionless axial velocity. 0, NE. = 300; x, J&.=500; t, Nn.=Sxlo’; 0, 1~5xlti
Fig.7. The dimensionless angular velocity distribution. 0, for 1.5x 10 < N, < 9 X l(r; 0, Measuringpoints for Nti = 300;D = 17.6cm.
Averages
I
I
I
1.5x104
@&z
:
I 5 Dimensionless
I
I radius
from vortex
I
I IO axis,
I
,
I
1 15
r/DxlO’
Fig. 8. The dimensionlesscircumferentialvelocitydistribution.D = 17.6cm. The axial velocity
The axial velocity is determined from the axial displacement needed for one sine cycle. A dependence between the axial velocity and the radius from the vortex axis appeared to be, if present, within experimental error. The axial velocity did show a clear dependence on zC,the distance along the vortex axis. For different NR~ the distance along the vortex axis is measured along the locus corresponding to the NRI in question. The measurements are shown in Fig. 9. The points are the average values each of about ten measuring points (individual standard deviation less than 30%) in the zC region in question. Figure 9 shows that the dimensionless axial velocity has a linear relation to the distance along the axis for each NR~. The pressure distribution The static pressure is calculated from the total pressure measured along horizontal lines that lie in a plane nearly
perpendicular to the vortex axis. For the 17.6 cm stirrer the vertical position is determined so that it includes the point of minimum pressure of the vertical cross section in question. This can be expected to coincide with the vortex axis. For the 48cm stirrer, for each horizontal line, measurements were done at six different vertical positions. In both vessels stirrer speed was limited by air entrainment from the surface. Stirrer speed of the 17.6 cm stirrer could be varied from 1 to 2 rev/s (NRI = 3 x 10’ and 6 x 10’resp.). The 48 cm stirrer,usuaIly operated with a stirrer speed of 1.35 rev/s (NRI = 3 x lo’). A few measurements were also done at a stirrer speed of 0*5rev/s (NRI = 1.1 x Id). All these pressure measurements were done with normal tap water. The downpressure AP (defined as the difference between the derived static pressure under operating conditions and the static pressure under rest conditions at the same point, given as a positive value), is presented
1099
The trailing vortex system produced by Rushton turbine agitators
I
1
I
I
I
I
1 J
3-
Vortex
axis /
Blade Disc J%
,‘%Ltian on the axis where measured
./+ /
G2 t .; 5 g ” ft t r a.
I
I IO'
IO'
105
IO'
Reynolds
Number.
I 106
IV&
Fig. 10. The pressure coefficient at the N,. region covered. t, Experimentalvalues D = 48 cm; X, experimental values D = 17.6cm; 0, calculated values D = 17.6cm; ~1regionwherevortextirst formed.
dimensionlessly as a pressure coefficient C, defined as C, = AP/;pv:;
(16)
in which ipu: is the stagnation pressure due to the stirrer tip velocity. As stated earlier the experimental inaccuracy varies strongly in different regions. Therefore in the presentation of the results two regions of experimental inaccuracy will be distinguished. The pressure appeared to show short time variations of about 20% of the average value. These arise from the erratic movement of the vortex axis and from fluctuations in velocity. These effects are excluded from the measurements by damping. Measurements were done in both vessels at different speeds to establish the relation to NR~.Four C, values found at one particular position are presented in Fig. 10 as the cross points. A clear N, dependence exists, and there is a significant indication that it is sullicient to use NRI to describe all the vortex effects associated with the scale of the apparatus. For the 48 cm stirrer the isobars in a vertical flat plane, (at the position indicated in Fig. 12A by A) are shown in Fig. 11. From similar graphs measured at five other positions the C, values along a horizontal line that includes the point of minimum pressure can be determined. The results presented as isobars in a plane through the horizontal lines in question are shown in Fig. 12A (NR, = 3 X 105).For the 17.6 cm stirrer the pressure along these lines is measured directly. The results for the stirrer speed of 2 rev/s (NRC= 6 x 10’)are shown in Fig. 12B. It is seen from Fig. 12 that the position of minimum pressure is at about one fourth blade length from the inner side of the blade. Experiments confirmed that the tirst cavitation originates there. The vertical part of the trailing vortex is immediately behind the inner edge of the blade. Measurements in this section showed that the value of the maximum downpressure there is comparable to that in the horizontal part.
v-
Disc plane
Fig. 11. The C, = 0.5 step 0.5 till 4.0 isobars in a vertical flat plane at the position indicated by A in Fig. 12A. -, Isobars with accuracy tO.4; ---, isobars with accuracy 20.8; D = 48 cm; Na =3x1@. DI!JCUssION OF THE RESULTS Threedimensional oorkx shape
Some impression of the three dimensional shape can be obtained from the pressure distribution in the vertical plane shown in Fig. 11. No quantitative conclusions can be drawn because other pressure effects besides that due only to the vortex motion exist. This means that the fluid streamlines are not simply given by the isobars, and the 0 or O-5 isobar is not sufficient to define the limit of circulation. However some information can be obtained from the isobars near the centre of rotation. Here the other downpressure effects can be expected to be of similar magnitude along _an isobar. From Fig. 11 and similar graphs it could be deduced that further from the
1100
K. VAN 't Ri~r and J. M. SMITH 2"0
".qnCgi
'
Experiments xlO 4
//
/
i/
//
,,,,,:;;
e..iD=5.6xlO
1"5
/
~
/// ! ,,, i,/
,
,,,=,o.,,~
#/-
TM
eqn(12)
¢ L, ~
ii/
....
I'0
~ 0"5
i
.E e~
0
I O'l Dimensionless
Fig. 12A.The Ca = 0.5 step 0.5 till 5.5 isobars in the "horizontal" plane through the vortex axis. --, Isobars with accuracy +0.4; - - - , isobars with accuracy -+0.8;D = 48 cm; N~, = 3 x 105.
¢
=3'5
Fig. 12B. The Ca = 0.5 step 0.5 till 3.5 isobars in the "horizontal" plane through the vortex axis. --, Isobars with accuracy +0.4; - - - , isobars with accuracy ±0.8; D = 17.6cm; N~, = 6 x 10". blade the differences in local radius of curvature along one streamline are of the order of some tens of percents. In this region the difference in position between the centre of rotation and the vortex axis can be up to twenty percent of the radius.
05
5
radius f r o m v o r t e x axis,
r/r m
Fig. 13. The circulation distribution related to the circulation at the radius at which the velocity reaches its maximum. D = 17.6cm. flattening suggested by eqn (12). Although eqn (9) is therefore valid at NR, > 1.5 x 10' the inflow theory [eqn (12)] must also be considered. Reasons are threefold. Firstly eqn (9) has only a qualitative character: the velocity distribution cannot be calculated from the known parameters of the system, Further eqn (9) also does not explain the constancy of circumferential velocity at increasing zc values. Lastly no conclusion can be drawn for Nst < 1.5 x 104because of the lack of data at r > r,. In eqn (12) the turbulent character of the flow can be taken into account by replacing the laminar viscosity by an eddy viscosity. The eddy viscosity at defined conditions is assumed to be a constant. This is not necessarily valid since the effective viscosity may vary with radius, as will be shown later. In order to check the theory it is not necessary to take this effect into account. The factors needed to calculate the velocity distribution as presented in eqn (12) are F®, b and v. These factors proved to be linearly dependent on the tip speed and therefore will be presented as velocity independent terms by division by yr. At NR~ > 1.5 × 104 the value of F®can be derived from measurements at larger radii. Then it must be considered that in the turbulent case an "overshoot" of circulation occurs as Hoffmann [7] shows. The amount of this effect is assumed to be 20~. Alternatively the value of F® can be calculated from the equation derived from eqn (12).
Theoretical description of the velocity distribution To what extent the results of the velocity measurements agree with the theories can now be discussed. Firstly the theory describing a logarithmic circulation distribution [eqn (9)] will be considered. The measured circulation distribution at 1.5 × 104< NR, < 9 × 104 related to the circulation Fm at the radius rm at which the circumferential velocity reaches its maximum is shown in Fig. 13. The experimental points fit the circulation predicted by eqn (9) and do not have the characteristic
r~
--= Fm
1.40.
(17)
The values calculated in these two ways are presented in Table 1 as F®, and F.~ respectively and an average value of 0.1 for all Ns, is seen to be reasonable. The factor b is given by the half slope of the axial velocity versus z~ graph (Fig. 9). It is assumed then that where the vortices have left the blade, the vortex axis corresponds to the z
1101
The trailing vortex systemproducedby Rushtonturbine agitators axis of the coordinate system. The value of b is 4-0x u,, independent of N, The eddy viscosity can be determined from either eqn (14) or eqn (15). The results are presented in Table 1 as v,+, and v,-,_ respectively. The eddy viscosity shows a larger value at r = r,,, than r +O. This difference, which is huger than the admitted inaccuracy that arises from extrapolating the curves of Figs. 7 or 8 to r = 0, can be explained by the fact that at r = r,,, the turbulent stresses are more intense than those in the region r +O. To calculate the velocity distribution the average value ti, also presented in Table 1, is used. The linear dependence of the parameters on tip speed
region with two dimensional vortices in the vortex street behind an obstacle. It can be concluded that if turbulence were completely absent eqn (12) would describe the circumferential velocity distribution in the trailing vortex. The axial velocity proved to be described by eqn (10). This means that the inflow theory describes the vortex motion, and explains why the circumferential velocity is independent of the ze values. Some problems arise because of the turbulent character of the flow. Equation (12) can still be used by introducing a kinematic eddy viscosity. For the higher NRI the vortex is described qualitatively by eqn (9).
Table 1. Factors used to calculate the velocity distribution by means of eqn (12) ;,”
t
(do+)
4.0
160
4.0
230
130 180
4.0
420
340
4.0
610
520
results in a calculated dimensionless circumferential velocity only dependent on NR~and radius. The results for I.5 x la’ C Nm < 9 x 10’ and Nat = 300 are presented in Fig. 14. It shows that the velocity distribution calculated from eqn (12) fits the experimental curve better at the lower NR*.This is explained by the decrease of the influence of the turbulent stresses. This decrease becomes clear from the ratio between the eddy viscosity and the corresponding laminar viscosity of the fluid shown in Table 2. At NRI > 1.5 x 10’the ratio has such a magnitude that only a very minor dependence between the two viscosities can be expected. This confirms the results derived from eqn (9). At Nm = 300 the ratio is near the value of 2.4. This means that there will be at least some relation between laminar and eddy viscosities. This relation explains the dependence of the circumferential velocity on Nn. (which is the subject of the following paragraph). The ratio of 2.4 is about the same as Timme [lo] found in his experiments done in the same NRI
Table 2. The ratio between the eddy and laminar viscosity at different NR,
NR~and scale e#ects
The relation to N, will be discussed by means of the equations found from the inflow theory which proved to describe the velocity distributions satisfactorily. Equation (12) can be developed to:
Fig. 14. The theoretical and experimental velocity distribution. -, Experimental; ---, theoretical (eqn 12); -.-.-, theoretical (eqn 9, c = 0.42); D = 17.6cm.
1102
K. VAN‘t Rra’rand J. M. SMITH
The factor TJv,D can be expected to be constant as a result of the definition of the circulation. The factor bD/v, should also be constant because the axial velocity can be expected to show a geometric similarity. The only factor that remains, besides r/D, is vtD/v in the exponent. This factor will be discussed for the different conditions of turbulence intensity. For the fully turbulent case the Prandtl mixing length theory predicts that the eddy viscosity will be linearly dependent on a representative velocity and diameter (Hinze[ll], Chap. 5). The results concerning the eddy viscosity of the previous paragraph point in this direction and Squire[ 121 and Saffman[8] in work on wing tip vortices confirm this relation. In this way the dirnensionless angular velocity for the fully turbulent case proves to be a function of r/D only. For the simple viscous case the factor vrD/v is NR~.This means that the velocity is a function of NRI and r/D. In the intermediate region a weakened N, dependence can be expected. The velocity measurements have already shown this dependence with variation of laminar viscosity and tip speed. The influence of diameter has not been investigated separately, though it is clear from the pressure measurements done at the two scales that it shows the same Reynolds number relation. One important question yet to be answered is the velocity at which the flow field becomes independent of NRI. The circumferential velocity, when derived from pressure measurements presented in Fig. 10 can be seen to increase by some 25% with an NRI increasing from 3 x 10’ to 3 x 105.This seems to be in contradiction with the experimental velocity measurements where no increase was found at 1.5 x 10”< NRI < 9 x 104.This however might be explained by the experimental inaccuracy in the Nn, region covered. It is clear from the pressure measurements that even at very large NRI the flow in the vortex does not yet have a fully turbulent character. Some, though relatively small, influence of NRCon the circumferential velocity will occur even at large NR~.This means that the change’of the flow from laminar to fully turbulent will not be reached. However the influence of a variation of NRCbecomes relatively smaller at larger NR~. This can also be concluded from the ratio between laminar and eddy viscosity presented in Table 2, when it is remembered that the eddy viscosity used is the average value of an eddy viscosity varying with radius. The fact that the value of r, will continue to decrease at increasing NRI is also related to this. It has already been pointed out that the influence of that effect also becomes relatively smaller at increasing NR~. The final conclusion is that the angular velocity graph as shown in Fig. 7 validly represents the velocity as a function of NRCand r/D. At very large values of NR. the velocity can be expected to increase somewhat beyond the curve of 1.5 x 10’< Nn, < 9 X 104. The position of the vortex axis The vortex axis position, as determined from the velocity measurements, agrees in its main lines with that found from the pressure measurements. Just behind the blade the position of the point of minimum pressure, C,,, is very asymmetric relative to the isobars. This confirms
the conclusion already drawn from the photographs that in this region the centre of rotation of each of the streamlines does not coincide with the vortex axis. The pressure measurements show that the position of the vortex axis behind the upper horizontal edge tends to move downwards slightly as the outwards flow takes place. From Fig. 6 it can be seen that the position of the vortex axis is dependent on NRI. This variation in position has important consequences for the pumping capacity (that is the fluid voltime discharged by the impeller per unit time) and the related vessel circulation time t,. Measurements of the circulation time are presented by Voncken[l3]. The measurements show that the dimensionless circulation time Nt, decreases up to NRI = 100. At 100< NR<< 250, Nt, increases. This is just the range where the vortex originates. The measurements of Nt, showed a large scatter in this area of Nn,. This is explained by the more or less random origin of the vortex and the influence of the starting conditions. At 300
The pressure can be calculated from eqn (7) (see also Rott[9]). By introducing eqns (lo)-(12) the pressure becomes: p*=pa-g(b2r2+4b
(19)
in which p t is the pressure at r = 0 and z = 0. In the relation between p * and r the integral term in eqn (19) is the most important factor. By means of this integral the pressure distribution in the vortex can be calculated from the measured velocity distribution. It is assumed that the pressure differences due to the gravitational forces and the centrifugal forces caused by the rotation of the coordinate system being the difference between p* and the real pressure p can be neglected. The pressure distribution calculated in this way compared with the measured one is presented in Fig. 15. The difference between the two curves to a large extent will be due to the deviations from circular geometry of the vortex. Local values for v2 and r-’ ought to have been taken, a difference that can easily cause the difference between the two curves. On the right hand side of the vortex axis rotation with the blade occurs. This causes the difference between the measured C, values on the right and left hand sides of Fig. 15. Measured C,, values at different Nn. are shown in Fig. 10. When assuming a constant relative difference between measured and derived C,, values, the C,, values at NRC= 5 x lo’, 500 and 300 can be calculated. Though this method of calculation is not exact it gives a reasonable estimate for C,,, for the low Nn, where measurements are inaccurate or impossible. This calculation starts from the
The trailing vortex system produced by Rushton turbine agitators
1103
centrifugal acceleration is 7.x 10’ms-* or 7OOg,a remarkably high value. It should be pointed out that the data presented here go much further than merely substantiating the conclusions drawn from our first measurements which were reported in our earlier paper[2]. From eqn (20) it is clear that also a,_,/a, = f(NRc and r/D) only.
The shear rate
The rotational vortex motion results in an additional shear field. The most important component is that due to the differences in angular velocity. For the velocity distribution presented in the eqns (lo)-(12) this component becomes:
5
Dimensionless
0
5
IO
-
radius from vortex
(21)
axis, r/Dw102
in which r = shear stress and Jo= shear rate. By means of eqn (21) the shear rate can be calculated from the angular velocity distribution. Some remarks about the accuracy of this method have to be made. Firstly, as has already been discussed, the vortex does not statement that C,,, at NRI = 4 x 10”is a constant times the calculated C, value for 1.5 x 104
Fig. 1.5.The pressure distribution along a horizontal line at the positionindicated by B in Fig. IZB.D = 174 cm; NRI = 6 X IO’.
K. VAN‘t Rnz’rand J. M. Sm
1104
5
Oimensionless
radius
from
vortex
axis,
r/DxlO*
Fig. 16. The shearrate calculatedfromthe angularvelocitycurves. CONCLUSIONS The velocity and pressure distributions within the
vortex are measured. It is shown that velocity and pressure can be scaled up in terms of NRI only. The flow is turbulent even at N, = 300. Despite the turbulence the flow field can be seen as a solution of the N.S. equations by introducing an eddy viscosity. This solution shows that a radial intlow exists which allows the circumferential velocity to be independent of the axial coordinate despite dissipation. The pressure field and related centrifugal acceleration mean that gas bubbles will be sucked into the vortex. The large centrifugal acceleration will also be significant for liquid-liquid or solid-liquid dispersions when density differences between the two phases are present. The relatively low speed and the unexpected position at which cavitation first originates is also explained by the pressure measurements. The shear rate in the vortex at large N, is at least 10 times that expected from literature. This high shear rate is significant for dispersions, in particular in non-coalescing systems where a relatively small volume of high shear can inlhrence the whole dispersion. The vortex motion has important influences on the turbulence measurements done with a stagnant probe. The vortex sweeping over the probe adds a signal to it that can be interpreted as a turbulent signal while not necessarily being so.
constant stirrer diameter, m d vortex dimeter, m Fct centrifugal force, Nm-’ FC. coriolis force, Nm-” f body force, ms-’ gravitational acceleration, ms-* constant, ms-’ N stirrer speed, s-’ AP downpressure, Nm-* pressure, Nm-* P,P*,Pt,PI R stirrer radius, m RP radius to stirrer shaft, m NR. Reynolds number (pND*v-‘) NRO Rossby number (Rr-‘) I vortex radius, m rm vortex radius at maximum velocity, m s length, m t time, s tc circulation time, s U radial velocity, ms-’ V circumferential velocity, ms-’ velocity, ms-’ lb velocity at r = r,, ms-’ V, stirrer tip velocity, ms-’ VI W axial velocity, ms-’ z coordinate, m distance along the vortex axis, m t ;
k”
Acknowledgements-Wewould like to thank Prof. J. 0. Hinzefor valuable discussions and Ir. L. P. H. R. M. Mulders, D. F. van Berckelaer and J. K. Bliek for their contributions to the experimental programme. NOTATION
acceleration, ms-* centrifugal acceleration, ms-* centrifugal acceleration at the stirrer tip, ms-’ constant, s-l pressure coefficient maximum pressure coefficient
Greek symbols V circulation, m*s-’
circulation at r = rm, m*s+ circulation at infinity, m*s-’ shear rate, s-l dynamic viscosity, Nsm-* kinematic viscosity, m*s-’ density, kgm-” shear stress, Nm-* coordinate stirrer angular velocity, s-’ angular velocity, s-’
The trailing vortex system produced by Rushton turbine agitators ItEmRENcEs
[l] Biesecker B., V.D.I. Forschungsheft 1972584. [2] Van ‘t Riet K. and Smith John M., Chem. Engng Sci. 197328 1031. [3] Bruijn W., Van ‘t Riet K. and Smith John M., Trans. Inst. Chem. Engrs 197452 88. [4] Cutter L. A., A.1.Ch.E. .R 196612 35. [5] Rao M. A. and Brodkey R. S., Chem. Engng Sci. 197227 137. [6] Batchelor G. K., An Introduction to Fluid Dynamics. Cambridge University Press 1967.
1105
[7] Hoffmann E. R. and Joubert P. N., J. jIuidMech. 1%3 16395. [RI SaRman P. G., Phys. Fluids 197316 1181. [9] Rott N., Z.A.M.P. 1958 9b 543. [lo] Timme A., Ing. Arch. 1957 25 205. [ll] Hinxe J. O., Turbulence. McGraw-Hill, New York 1959. (121 Squire H. B., The Aer. Quart. 1%5 16 302. [13] Voncken R. M., Circukztion and mixing in stirred vessels,
Ph.D. Thesis, Delft (Dutch) 1966. [14] Metzner A. 8. and Otto R. E., A.LCh.E. .R 19573 3.
Pergamon Press.
Printed in Great Britain
THE TRAILING VORTEX SYSTEM PRODUCED BY RUSHTON TURBINE AGITATORS K. VAN ‘t RIET and JOHN M. SMITH Physical Technology Laboratory, Technical University of Delft, The Netherlands (Received 21 November 1974; accepted 4 March 1975)
AbstractFurther studies of the discharge from a six blade Rushton turbine are reported. Photographic velocity measurements have been used to examine the trailing vortex system in water and aqueous glycerol. An analysis of the vortex field in terms of a solution to the simplifiedNavier-Stokes equations shows reasonable agreement with the measured velocity profiles. Pressure field determinations at two different scales prove that Reynolds number alone is a sufficient scale-up parameter. The centrifugal accelerations and shear rates associated with the tmiling vortices are shown to be much greater than previous literature has implied. Although restricted to single phase flow the results presented are of importance to most applications of disc turbines to dispersion operations. INTRODUCTION
Recent literature (Biesecker [ l]), Van ‘t Riet et al. 121and Bruijn et al. [3] shows qualitatively that in a turbine stirred vessel the flow in the stirrer blade region controls the power consumption, coalescence and dispersion in gas liquid mixtures to a large extent. A particularly important feature of the flow field is a trailing vortex pair that exists behind each stirrer blade (see Fig. IA-B). This trailing
vortex pair originates as a combined effect of the vortex motion in the dead water zone at the rear of the vertical inside edge of the blade and the wrapping of the vortex sheets behind both horizontal edges. The smooth outlIow at the vertical outer edge of the blade that is a necessary hydrodynamic condition is indeed present. In the present paper we present measurements of the velocity and pressure distributions within the trailing vortex. The
Fii. 1A. Schematic three dimensional view of the &ailingvortex pair.
position of the vortex axis is also determined. To describe the vortex motion analytically simplified Navier-Stokes (N.S.) equations for the vortex motion will be solved. men the velocity and pressure are known it is in principle possible to determine the influence of the vortex motion on dispersion processes. It is also possible to assess the influence of the &ailing vortex motion on turbulence measurements with stationary probes as made by Cutter[4], Rao[51, etc.
Fig. 1B. Schematic two dimensional view of the flow field in the stirrer blade region. S, stagnation point.
APPARATUS Measurements were obtained in two fully baffled vessels of 44 cm and 120cm dia. provided with six bladed disc turbine agitators of 17.6 cm and 48 cm dia. respectively. The proportions of the stirrer are given by the
1094
K. VAN‘t
&ET
ratios stirrer diameter to blade length to blade height being 20:5:4. The velocity measurements were obtained using the smaller tank and the turntable equipment described by Van ‘t Riet [2]. Photographs were made with stroboscopic illumination by a single lens reflex camera that rotated synchronously with the agitator. The stirrer speed could be as high as 5rev/s. Normal tap water and water/glycerine solutions were used with added polystyrene particles of 0.5 mm dia. as tracers Pressure was measured with an impact Pitot tube (O-5mm dia.) rotating together with the stirrer, thus stationary relative to the blade. For the 17.6 cm turbine the pressure on the impact tube was brought out via a hollow stirrer shaft and gland to an external manometer. With the 48 cm impeller a closed limb manometer, was mounted on and rotated with the stirrer shaft.
and J. M. Surt’n
Fig. 3A. Sinusoidalprojectionof a particlepath alongthe trailing vortex.
MEASURING METHOD AD
example of a photograph obtained with the rotating camera is shown in Fig. 2. In this photograph two tracer particle paths in the vortex pair are seen. From such a photograph it cannot be concluded whether these tracers are situated in the upper or lower part of the vortex pair, but there is ample additional evidence to establish the sense of rotation in the trailing vortices. The path of a tracer particle travelling out along a trailing vortex is seen on the photographs as a sinusoidal wave represented schematically in Fig. 3A. The circular cylindrical coordinate system used to describe this vortex motion is shown in Fig. 3B. The circumferential and axial velocity components v and w respectively are determined from measurements of the axial displacement s, the radius r and the corresponding time for one sine cycle. The radial velocity component u was too small to be measured.
Fig. 3B. Circular cylindrical coiirdmate system with corresponding velocity components.
Some aspects of non ideality of the vortex motion which influence the accuracy of the measurements must be considered however. The linear axis of the coordinate system is only an approximation to the curved line of the vortex axis. The fit is particularly poor in the section near the tip of the blade. Most of the measurements were done further from the blade where the vortex axis is more nearly linear. The effect of a curved vortex axis would be to change the circumferential velocity for two successive half sine cycles and indeed no such effect could be found in this region. Strictly speaking the vortex is asymmetric
Fig. 2. Stroboscopic photograph of the flow field in the stirrer blade region.
The trailing vortex system produced by Rushton turbine agitators
in that the centres of rotation for different radii are not coincident. This is particularly important in the region just behind the blade where the differences of position of the centres of rotation between small and large radii amounted to some millimeters. Besides that the streamlines in a plane perpendicular to the axis are not exactly circular, but somewhat elliptical. Due to some erratic movement of the vortex the position of the vortex axis is not fixed. Its position had to be estimated separately for each of the measurements from the sine shaped curve in question. The axial velocity at a defined vortex radius shows local values that deviate from the average value (the distance from the stirrer axis is important for example). However, with our technique, local values could not be measured accurately so an average axial velocity over one sine cycle is presented. The effects described above, and in particular the need to estimate the position of the vortex axis for each measurement, can result in an estimated experimental inaccuracy’ for individual measurements of 20% in the axial distance, an error of 30% in the radius from the vortex axis and an error of 20% in the time needed for one sine cycle. It has been established that the error due to tracer particle dimensions, the position of the camera relative to the vortex and the density difference between these tracer particles and fluid can be neglected. To determine the static pressure, the total pressure as read from the manometer has to be corrected for two effects. Firstly the influence of the centrifugal forces on the fluid in the Pitot tube that rotates with the stirrer. This effect is given by the pressure difference in a forced vortex having a radius the same as the distance from the axis to the tip of the Pitot tube. Secondly there is a dynamic pressure due to local fluid velocities directed into the Pitot tube. A correction for this can be estimated from the velocity measurements. When seen from the stirrer axis at the outer side of the vortex in the region where the vortex has left the blade the correction may become large (even larger than the value of the downpressure itself) as well as inaccurate. Near the centre of the vortex the correction becomes relatively small as well as accurately known.
COORDINATESYSTEMAND RELATEDN.S. EQUATIONS
To describe the vortex motion analytically a circular cylindrical coordinate system is used. The z-axis is located in a horizontal plane and its origin is at the outer edge of the blade, 3 mm below the upper tip at an angle of 82 degrees to the cutting line of the horizontal plane and the stirrer blade (see Fig. 4). This z-axis fits the vortex axis reasonably once the vortex has left the blade. A variation of the axis angle to the blade and its distance from the upper tip with various NRC is permitted because the expressions for the velocity which will be derived later are independent of these variables. The constant rotation of this coordinate system with the blade requires the inclusion of the fictitious coriolis force F, and centrifugal force Fcf given by (vector notation) F,, = -pa
x (a x RP)
(I)
1095
Stirrer
z-axis
Fig. 4. The coordinate system used to describe the vortex motion.
and F,, = -p2a x v,
(2)
in which R = stirrer angular velocity, R0 = distance between a given point and the stirrer axis (see Fig. 4) and u, = velocity relative to the rotating coordinate system. The N.S. equation in the rotating coordinate system thus becomes: pa=pf-2pnxv,-pnx(fixR,)-;Vp+vAv,
(3)
in which a = acceleration, f = body force per unit mass, p = pressure and TJ’=dynamic viscosity. The centrifugal and the gravitational forces, being dependent on the coordinates only, can be written as a pressure. If the gravitational. force is the only body force this leads to: (4) in which p*=ptpgrsin4_fp(nxR,)‘tp,
(9
p* is the modified pressure, the remaining part of the pressure arising from the fluid motion. p1 is a constant. The extent to which the coriolis force effects the fluid motion can be expressed in a Rossby number NRo (see Batchelor[61, Chap. 7). Nn, can be evaluated in this particular case as NRo= R/r
(6)
in which R = stirrer radius. Batchelor states that the coriolis force can be neglected if N,s, 9 1. With the 17.6 cm stirrer this condition is satisfied when r < 10 mm. Even at, the maximum vorte% radius measured Nn. does not fall below a value of about 3.5 so only a minor influence may be expected. The coriolis force arising from the circumferential velocity component is shown qualitatively in Fig. 5. From this figure it can be deduced that the effect of this component of the coriolis force on the average axial velocity will be smoothed out because of its
K. VAN‘t Rrw and J. M. Sf,fm+
lo!m
the angular momentum is the transferable quantity. At
some radii the circulation distribution will deviate from eqn (9). For very small radii a solid body rotation must exist (I 0: r*). At increasing radii eqn (9) predicts an ever increasing circulation, which is impossible from a physical viewpoint. The circulation presented by eqn (9) can only occur somewhere between these two extremes.
Enveloping cylinder around particle path
(b)
Fi. 5. Coriolis force due to the circumferential velocity component:(a) Plan; (b) Section.0, Vector directed out of the plane; X,Vectordirectedin to the plane. alternating character. The coriolis force due to the axial velocity component is always directed inwards perpendicular to the vortex axis. This component does not have a real effect on the average circumferential velocity, again because of the alternating character. The curvature of the vortex axis is produced by this component. Thus it can be concluded that the coriolis forces at the radii of interest will be small in any case and their effect on the average velocity components will be smoothed out. They can therefore be left out of the N.S. equation which becomes (7)
This equation will be completed with the equation of continuity for an incompressible medium
v . v, = 0.
htpow theory Writing out eqn (7) in its components and leaving out the derivatives with respect to t and 4 shows that neither the equation for the circumferential component nor the equation of continuity irtcludes the pressure. These two equations can be regarded as the differential equations for the velocity field. After these equations are solved, the pressure can be derived from the remaining two equations. The derivation of eqns (7) and (8) starts from a laminar flow. If present, the influence of turbulence can be taken into account by assuming a constant kinematic eddy viscosity. The two equations for the velocity field contain three variables. Therefore we choose here on the basis of the experiments, w = 2bz + k,
so that from the equation of continuity it can be calculated that u = -br.
(11)
Now, when assuming avlaz = 0, it can be shown that v
=
&
(1 _
e-(br21*vQ
in which b and k are constants, Y = kinematic eddy or laminar viscosity and I, = circulation at infinity. The solution corresponds approximately to the velocity distribution presented by Rott [9]. Important properties of eqn (12) are as
r -*m
then
v --fI,/2rr
(13)
(8)
Equations (7) and (8) describe the velocity and pressure distribution within the vortex.
-at large radii the vortex tends to a free vortex with circulation r, as
THEORETICAL DESCRlFllONS OPTHE VORTEX VELOCITYFDZLD
r+O
then
o+(Lb/hv)r
(14)
-at small radii the vortex becomes a forced vortex with an angular velocity of I’mb/4w.
Logarithmic circulation distribution theory
Hoffmann [71and Saffman181 have studied the velocity distribution within a turbulent trailing vortex. For a degree of turbulence such that the velocity field is completely independent of the laminar viscosity they both derived (although in different ways): I/r, = b In (r/r,) + 1
(10)
(9)
in which I = circulation at a radius r, r, = circulation at a particular radius rl and c = constant. Both assumed that
and
v = urn at
rm = 1.12
J2v b
(19
-there is one particular radius r,,, at which the circumferential velocity has a maximum vn. Conclusions
For the turbulent case the velocity distribution presented by eqn (9) can be applied only in a region of intermediate radii. Due to its qualitative character the
The trailing vortex system producedby Rushtonturbine agitators velocity cannot be calculated from the given parameters
of the system. The inflow theory [eqns (lO-12)l can describe the flow field at any radius. If the flow is laminar the velocity distribution can be calculated from the system parameters. However in a turbulent system an eddy viscosity must be estimated in order that its value can be introduced in eqn (12). The circumferential velocity predicted by the inflow theory is independent of the distance z. This means that the damping out of the vortex motion that would normally be expected does not occur. This can indeed happen because of the transport of momentum by the radial velocity component. EXPERIMENTAL The position of the vortex axis
The position of the vortex axis has been determined from the photographs for the 17.6 cm stirrer. In the region immediately behind the blade the centre of rotation for smaller and larger radii differed by some millimeters. In the region where the vortices have left the blade, the variation in position for different radii was within experimental error. In particular far from the blade the erratic movement of the vortex and/or measurements behind the different blades, causes a variation in position of some 5-10mm. The results at the different Reynolds number NRI are presented in Fig. 6. This shows that the position of the vortex axis is independent of NRe at NRI >5 x 10’. No measurements were available in the region 500 < NRI < 5 X lo’. The vortices were first formed in the range 150< NR~< 250. Direction of
1097
dividing o by the stirrer angular speed 27rN. At fixed conditions the angular velocity appears to be independent of the distance along the vortex axis. Measurements showed the dimensionless angular velocity to be dependent only on NRI.However at 1.5 X 10’< NR~< 9 X la’ the differences in angular velocity were so small that they were within the experimental accuracy. The results are shown in Fig. 7. In this figure for 1.5 X 10’< Nn, < 9 X 10’ a number of points are shown. Each of these represents between 10 and 30 measured values (individual standard deviation less than 20%) taken over a range of radii centred on that used for the point. The measurements for the other curves presented in Fig. 7 were not numerous enough to allow determination of such averages. For Nn, = 300 a number of measuring points, whose distribution around the curve is typical for all the curves, are added. From Fig. 7 it can be seen that the vortex tends to a forced vortex (given by o = constant) at smaller radii. The rigid core is larger at lower NR~.At larger radii the inertia forces will become more important. The vortex in which the inertia forces are dominant is the free vortex given by slope = - 2, (or’ = constant). There is indeed a tendency for the slope to approach this value at increasing radii. A limitation can be expected to the maximum radius of rotation in a trailing vortex both because of the finite size of its object of generation and the influence of coriolis forces at large radii. Though it is difficult to determine the maximum radius is thought to be between 0.14 and 0.2 r/D. The circumferential velocity (N.B. not tangential
Blade
Fig. 6. The vortex axis position at different iVRc
The angular and circumferential velocities The angular velocity o within the vortex was measured by determining the time needed for one sine cycle at a particular radius. Measurements were only made in the region between the point where the vortices have just left the blade and imaginary extension of the following blade. NRI was varied between 300 and 9 x 10’ by variation of stirrer speed and fluid viscosity. For each N, s 5 X ld at least two different speeds that differed by a factor 4 were used. The results are expressed dimensionlessly by
velocity) has been calculated directly from the curves that present the dimensionless angular velocity (Fig. 7). The results shown in Fig. 8 are presented dimensionlessly in terms of v,, the circumferential velocity of tbe stirrer tip. At all Nm the dimensionless circumferential velocity shows a rather flat maximum beyond which the velocity decreases slowly. For l-5 x 10’< Nfi < 9 x lo’ this maximum velocity is approximately the same as the tip speed. At smaller NRI this maximum velocity is less while the radius at which the maximum occurs is larger.
K.
1098
VAN ‘t Rtm
and J. M. SMITH
01 0
1
3
8
Dimensionless 20 Dimensiontess
rodius
from
vortex
OXIS, f/DxlO’
distance
1 8 b a 0 05 along Ihe vortex axis, z/D
Fig. 9. The dimensionless axial velocity. 0, NE. = 300; x, J&.=500; t, Nn.=Sxlo’; 0, 1~5xlti
Fig.7. The dimensionless angular velocity distribution. 0, for 1.5x 10 < N, < 9 X l(r; 0, Measuringpoints for Nti = 300;D = 17.6cm.
Averages
I
I
I
1.5x104
@&z
:
I 5 Dimensionless
I
I radius
from vortex
I
I IO axis,
I
,
I
1 15
r/DxlO’
Fig. 8. The dimensionlesscircumferentialvelocitydistribution.D = 17.6cm. The axial velocity
The axial velocity is determined from the axial displacement needed for one sine cycle. A dependence between the axial velocity and the radius from the vortex axis appeared to be, if present, within experimental error. The axial velocity did show a clear dependence on zC,the distance along the vortex axis. For different NR~ the distance along the vortex axis is measured along the locus corresponding to the NRI in question. The measurements are shown in Fig. 9. The points are the average values each of about ten measuring points (individual standard deviation less than 30%) in the zC region in question. Figure 9 shows that the dimensionless axial velocity has a linear relation to the distance along the axis for each NR~. The pressure distribution The static pressure is calculated from the total pressure measured along horizontal lines that lie in a plane nearly
perpendicular to the vortex axis. For the 17.6 cm stirrer the vertical position is determined so that it includes the point of minimum pressure of the vertical cross section in question. This can be expected to coincide with the vortex axis. For the 48cm stirrer, for each horizontal line, measurements were done at six different vertical positions. In both vessels stirrer speed was limited by air entrainment from the surface. Stirrer speed of the 17.6 cm stirrer could be varied from 1 to 2 rev/s (NRI = 3 x 10’ and 6 x 10’resp.). The 48 cm stirrer,usuaIly operated with a stirrer speed of 1.35 rev/s (NRI = 3 x lo’). A few measurements were also done at a stirrer speed of 0*5rev/s (NRI = 1.1 x Id). All these pressure measurements were done with normal tap water. The downpressure AP (defined as the difference between the derived static pressure under operating conditions and the static pressure under rest conditions at the same point, given as a positive value), is presented
1099
The trailing vortex system produced by Rushton turbine agitators
I
1
I
I
I
I
1 J
3-
Vortex
axis /
Blade Disc J%
,‘%Ltian on the axis where measured
./+ /
G2 t .; 5 g ” ft t r a.
I
I IO'
IO'
105
IO'
Reynolds
Number.
I 106
IV&
Fig. 10. The pressure coefficient at the N,. region covered. t, Experimentalvalues D = 48 cm; X, experimental values D = 17.6cm; 0, calculated values D = 17.6cm; ~1regionwherevortextirst formed.
dimensionlessly as a pressure coefficient C, defined as C, = AP/;pv:;
(16)
in which ipu: is the stagnation pressure due to the stirrer tip velocity. As stated earlier the experimental inaccuracy varies strongly in different regions. Therefore in the presentation of the results two regions of experimental inaccuracy will be distinguished. The pressure appeared to show short time variations of about 20% of the average value. These arise from the erratic movement of the vortex axis and from fluctuations in velocity. These effects are excluded from the measurements by damping. Measurements were done in both vessels at different speeds to establish the relation to NR~.Four C, values found at one particular position are presented in Fig. 10 as the cross points. A clear N, dependence exists, and there is a significant indication that it is sullicient to use NRI to describe all the vortex effects associated with the scale of the apparatus. For the 48 cm stirrer the isobars in a vertical flat plane, (at the position indicated in Fig. 12A by A) are shown in Fig. 11. From similar graphs measured at five other positions the C, values along a horizontal line that includes the point of minimum pressure can be determined. The results presented as isobars in a plane through the horizontal lines in question are shown in Fig. 12A (NR, = 3 X 105).For the 17.6 cm stirrer the pressure along these lines is measured directly. The results for the stirrer speed of 2 rev/s (NRC= 6 x 10’)are shown in Fig. 12B. It is seen from Fig. 12 that the position of minimum pressure is at about one fourth blade length from the inner side of the blade. Experiments confirmed that the tirst cavitation originates there. The vertical part of the trailing vortex is immediately behind the inner edge of the blade. Measurements in this section showed that the value of the maximum downpressure there is comparable to that in the horizontal part.
v-
Disc plane
Fig. 11. The C, = 0.5 step 0.5 till 4.0 isobars in a vertical flat plane at the position indicated by A in Fig. 12A. -, Isobars with accuracy tO.4; ---, isobars with accuracy 20.8; D = 48 cm; Na =3x1@. DI!JCUssION OF THE RESULTS Threedimensional oorkx shape
Some impression of the three dimensional shape can be obtained from the pressure distribution in the vertical plane shown in Fig. 11. No quantitative conclusions can be drawn because other pressure effects besides that due only to the vortex motion exist. This means that the fluid streamlines are not simply given by the isobars, and the 0 or O-5 isobar is not sufficient to define the limit of circulation. However some information can be obtained from the isobars near the centre of rotation. Here the other downpressure effects can be expected to be of similar magnitude along _an isobar. From Fig. 11 and similar graphs it could be deduced that further from the
1100
K. VAN 't Ri~r and J. M. SMITH 2"0
".qnCgi
'
Experiments xlO 4
//
/
i/
//
,,,,,:;;
e..iD=5.6xlO
1"5
/
~
/// ! ,,, i,/
,
,,,=,o.,,~
#/-
TM
eqn(12)
¢ L, ~
ii/
....
I'0
~ 0"5
i
.E e~
0
I O'l Dimensionless
Fig. 12A.The Ca = 0.5 step 0.5 till 5.5 isobars in the "horizontal" plane through the vortex axis. --, Isobars with accuracy +0.4; - - - , isobars with accuracy -+0.8;D = 48 cm; N~, = 3 x 105.
¢
=3'5
Fig. 12B. The Ca = 0.5 step 0.5 till 3.5 isobars in the "horizontal" plane through the vortex axis. --, Isobars with accuracy +0.4; - - - , isobars with accuracy ±0.8; D = 17.6cm; N~, = 6 x 10". blade the differences in local radius of curvature along one streamline are of the order of some tens of percents. In this region the difference in position between the centre of rotation and the vortex axis can be up to twenty percent of the radius.
05
5
radius f r o m v o r t e x axis,
r/r m
Fig. 13. The circulation distribution related to the circulation at the radius at which the velocity reaches its maximum. D = 17.6cm. flattening suggested by eqn (12). Although eqn (9) is therefore valid at NR, > 1.5 x 10' the inflow theory [eqn (12)] must also be considered. Reasons are threefold. Firstly eqn (9) has only a qualitative character: the velocity distribution cannot be calculated from the known parameters of the system, Further eqn (9) also does not explain the constancy of circumferential velocity at increasing zc values. Lastly no conclusion can be drawn for Nst < 1.5 x 104because of the lack of data at r > r,. In eqn (12) the turbulent character of the flow can be taken into account by replacing the laminar viscosity by an eddy viscosity. The eddy viscosity at defined conditions is assumed to be a constant. This is not necessarily valid since the effective viscosity may vary with radius, as will be shown later. In order to check the theory it is not necessary to take this effect into account. The factors needed to calculate the velocity distribution as presented in eqn (12) are F®, b and v. These factors proved to be linearly dependent on the tip speed and therefore will be presented as velocity independent terms by division by yr. At NR~ > 1.5 × 104 the value of F®can be derived from measurements at larger radii. Then it must be considered that in the turbulent case an "overshoot" of circulation occurs as Hoffmann [7] shows. The amount of this effect is assumed to be 20~. Alternatively the value of F® can be calculated from the equation derived from eqn (12).
Theoretical description of the velocity distribution To what extent the results of the velocity measurements agree with the theories can now be discussed. Firstly the theory describing a logarithmic circulation distribution [eqn (9)] will be considered. The measured circulation distribution at 1.5 × 104< NR, < 9 × 104 related to the circulation Fm at the radius rm at which the circumferential velocity reaches its maximum is shown in Fig. 13. The experimental points fit the circulation predicted by eqn (9) and do not have the characteristic
r~
--= Fm
1.40.
(17)
The values calculated in these two ways are presented in Table 1 as F®, and F.~ respectively and an average value of 0.1 for all Ns, is seen to be reasonable. The factor b is given by the half slope of the axial velocity versus z~ graph (Fig. 9). It is assumed then that where the vortices have left the blade, the vortex axis corresponds to the z
1101
The trailing vortex systemproducedby Rushtonturbine agitators axis of the coordinate system. The value of b is 4-0x u,, independent of N, The eddy viscosity can be determined from either eqn (14) or eqn (15). The results are presented in Table 1 as v,+, and v,-,_ respectively. The eddy viscosity shows a larger value at r = r,,, than r +O. This difference, which is huger than the admitted inaccuracy that arises from extrapolating the curves of Figs. 7 or 8 to r = 0, can be explained by the fact that at r = r,,, the turbulent stresses are more intense than those in the region r +O. To calculate the velocity distribution the average value ti, also presented in Table 1, is used. The linear dependence of the parameters on tip speed
region with two dimensional vortices in the vortex street behind an obstacle. It can be concluded that if turbulence were completely absent eqn (12) would describe the circumferential velocity distribution in the trailing vortex. The axial velocity proved to be described by eqn (10). This means that the inflow theory describes the vortex motion, and explains why the circumferential velocity is independent of the ze values. Some problems arise because of the turbulent character of the flow. Equation (12) can still be used by introducing a kinematic eddy viscosity. For the higher NRI the vortex is described qualitatively by eqn (9).
Table 1. Factors used to calculate the velocity distribution by means of eqn (12) ;,”
t
(do+)
4.0
160
4.0
230
130 180
4.0
420
340
4.0
610
520
results in a calculated dimensionless circumferential velocity only dependent on NR~and radius. The results for I.5 x la’ C Nm < 9 x 10’ and Nat = 300 are presented in Fig. 14. It shows that the velocity distribution calculated from eqn (12) fits the experimental curve better at the lower NR*.This is explained by the decrease of the influence of the turbulent stresses. This decrease becomes clear from the ratio between the eddy viscosity and the corresponding laminar viscosity of the fluid shown in Table 2. At NRI > 1.5 x 10’the ratio has such a magnitude that only a very minor dependence between the two viscosities can be expected. This confirms the results derived from eqn (9). At Nm = 300 the ratio is near the value of 2.4. This means that there will be at least some relation between laminar and eddy viscosities. This relation explains the dependence of the circumferential velocity on Nn. (which is the subject of the following paragraph). The ratio of 2.4 is about the same as Timme [lo] found in his experiments done in the same NRI
Table 2. The ratio between the eddy and laminar viscosity at different NR,
NR~and scale e#ects
The relation to N, will be discussed by means of the equations found from the inflow theory which proved to describe the velocity distributions satisfactorily. Equation (12) can be developed to:
Fig. 14. The theoretical and experimental velocity distribution. -, Experimental; ---, theoretical (eqn 12); -.-.-, theoretical (eqn 9, c = 0.42); D = 17.6cm.
1102
K. VAN‘t Rra’rand J. M. SMITH
The factor TJv,D can be expected to be constant as a result of the definition of the circulation. The factor bD/v, should also be constant because the axial velocity can be expected to show a geometric similarity. The only factor that remains, besides r/D, is vtD/v in the exponent. This factor will be discussed for the different conditions of turbulence intensity. For the fully turbulent case the Prandtl mixing length theory predicts that the eddy viscosity will be linearly dependent on a representative velocity and diameter (Hinze[ll], Chap. 5). The results concerning the eddy viscosity of the previous paragraph point in this direction and Squire[ 121 and Saffman[8] in work on wing tip vortices confirm this relation. In this way the dirnensionless angular velocity for the fully turbulent case proves to be a function of r/D only. For the simple viscous case the factor vrD/v is NR~.This means that the velocity is a function of NRI and r/D. In the intermediate region a weakened N, dependence can be expected. The velocity measurements have already shown this dependence with variation of laminar viscosity and tip speed. The influence of diameter has not been investigated separately, though it is clear from the pressure measurements done at the two scales that it shows the same Reynolds number relation. One important question yet to be answered is the velocity at which the flow field becomes independent of NRI. The circumferential velocity, when derived from pressure measurements presented in Fig. 10 can be seen to increase by some 25% with an NRI increasing from 3 x 10’ to 3 x 105.This seems to be in contradiction with the experimental velocity measurements where no increase was found at 1.5 x 10”< NRI < 9 x 104.This however might be explained by the experimental inaccuracy in the Nn, region covered. It is clear from the pressure measurements that even at very large NRI the flow in the vortex does not yet have a fully turbulent character. Some, though relatively small, influence of NRCon the circumferential velocity will occur even at large NR~.This means that the change’of the flow from laminar to fully turbulent will not be reached. However the influence of a variation of NRCbecomes relatively smaller at larger NR~. This can also be concluded from the ratio between laminar and eddy viscosity presented in Table 2, when it is remembered that the eddy viscosity used is the average value of an eddy viscosity varying with radius. The fact that the value of r, will continue to decrease at increasing NRI is also related to this. It has already been pointed out that the influence of that effect also becomes relatively smaller at increasing NR~. The final conclusion is that the angular velocity graph as shown in Fig. 7 validly represents the velocity as a function of NRCand r/D. At very large values of NR. the velocity can be expected to increase somewhat beyond the curve of 1.5 x 10’< Nn, < 9 X 104. The position of the vortex axis The vortex axis position, as determined from the velocity measurements, agrees in its main lines with that found from the pressure measurements. Just behind the blade the position of the point of minimum pressure, C,,, is very asymmetric relative to the isobars. This confirms
the conclusion already drawn from the photographs that in this region the centre of rotation of each of the streamlines does not coincide with the vortex axis. The pressure measurements show that the position of the vortex axis behind the upper horizontal edge tends to move downwards slightly as the outwards flow takes place. From Fig. 6 it can be seen that the position of the vortex axis is dependent on NRI. This variation in position has important consequences for the pumping capacity (that is the fluid voltime discharged by the impeller per unit time) and the related vessel circulation time t,. Measurements of the circulation time are presented by Voncken[l3]. The measurements show that the dimensionless circulation time Nt, decreases up to NRI = 100. At 100< NR<< 250, Nt, increases. This is just the range where the vortex originates. The measurements of Nt, showed a large scatter in this area of Nn,. This is explained by the more or less random origin of the vortex and the influence of the starting conditions. At 300
The pressure can be calculated from eqn (7) (see also Rott[9]). By introducing eqns (lo)-(12) the pressure becomes: p*=pa-g(b2r2+4b
(19)
in which p t is the pressure at r = 0 and z = 0. In the relation between p * and r the integral term in eqn (19) is the most important factor. By means of this integral the pressure distribution in the vortex can be calculated from the measured velocity distribution. It is assumed that the pressure differences due to the gravitational forces and the centrifugal forces caused by the rotation of the coordinate system being the difference between p* and the real pressure p can be neglected. The pressure distribution calculated in this way compared with the measured one is presented in Fig. 15. The difference between the two curves to a large extent will be due to the deviations from circular geometry of the vortex. Local values for v2 and r-’ ought to have been taken, a difference that can easily cause the difference between the two curves. On the right hand side of the vortex axis rotation with the blade occurs. This causes the difference between the measured C, values on the right and left hand sides of Fig. 15. Measured C,, values at different Nn. are shown in Fig. 10. When assuming a constant relative difference between measured and derived C,, values, the C,, values at NRC= 5 x lo’, 500 and 300 can be calculated. Though this method of calculation is not exact it gives a reasonable estimate for C,,, for the low Nn, where measurements are inaccurate or impossible. This calculation starts from the
The trailing vortex system produced by Rushton turbine agitators
1103
centrifugal acceleration is 7.x 10’ms-* or 7OOg,a remarkably high value. It should be pointed out that the data presented here go much further than merely substantiating the conclusions drawn from our first measurements which were reported in our earlier paper[2]. From eqn (20) it is clear that also a,_,/a, = f(NRc and r/D) only.
The shear rate
The rotational vortex motion results in an additional shear field. The most important component is that due to the differences in angular velocity. For the velocity distribution presented in the eqns (lo)-(12) this component becomes:
5
Dimensionless
0
5
IO
-
radius from vortex
(21)
axis, r/Dw102
in which r = shear stress and Jo= shear rate. By means of eqn (21) the shear rate can be calculated from the angular velocity distribution. Some remarks about the accuracy of this method have to be made. Firstly, as has already been discussed, the vortex does not statement that C,,, at NRI = 4 x 10”is a constant times the calculated C, value for 1.5 x 104
Fig. 1.5.The pressure distribution along a horizontal line at the positionindicated by B in Fig. IZB.D = 174 cm; NRI = 6 X IO’.
K. VAN‘t Rnz’rand J. M. Sm
1104
5
Oimensionless
radius
from
vortex
axis,
r/DxlO*
Fig. 16. The shearrate calculatedfromthe angularvelocitycurves. CONCLUSIONS The velocity and pressure distributions within the
vortex are measured. It is shown that velocity and pressure can be scaled up in terms of NRI only. The flow is turbulent even at N, = 300. Despite the turbulence the flow field can be seen as a solution of the N.S. equations by introducing an eddy viscosity. This solution shows that a radial intlow exists which allows the circumferential velocity to be independent of the axial coordinate despite dissipation. The pressure field and related centrifugal acceleration mean that gas bubbles will be sucked into the vortex. The large centrifugal acceleration will also be significant for liquid-liquid or solid-liquid dispersions when density differences between the two phases are present. The relatively low speed and the unexpected position at which cavitation first originates is also explained by the pressure measurements. The shear rate in the vortex at large N, is at least 10 times that expected from literature. This high shear rate is significant for dispersions, in particular in non-coalescing systems where a relatively small volume of high shear can inlhrence the whole dispersion. The vortex motion has important influences on the turbulence measurements done with a stagnant probe. The vortex sweeping over the probe adds a signal to it that can be interpreted as a turbulent signal while not necessarily being so.
constant stirrer diameter, m d vortex dimeter, m Fct centrifugal force, Nm-’ FC. coriolis force, Nm-” f body force, ms-’ gravitational acceleration, ms-* constant, ms-’ N stirrer speed, s-’ AP downpressure, Nm-* pressure, Nm-* P,P*,Pt,PI R stirrer radius, m RP radius to stirrer shaft, m NR. Reynolds number (pND*v-‘) NRO Rossby number (Rr-‘) I vortex radius, m rm vortex radius at maximum velocity, m s length, m t time, s tc circulation time, s U radial velocity, ms-’ V circumferential velocity, ms-’ velocity, ms-’ lb velocity at r = r,, ms-’ V, stirrer tip velocity, ms-’ VI W axial velocity, ms-’ z coordinate, m distance along the vortex axis, m t ;
k”
Acknowledgements-Wewould like to thank Prof. J. 0. Hinzefor valuable discussions and Ir. L. P. H. R. M. Mulders, D. F. van Berckelaer and J. K. Bliek for their contributions to the experimental programme. NOTATION
acceleration, ms-* centrifugal acceleration, ms-* centrifugal acceleration at the stirrer tip, ms-’ constant, s-l pressure coefficient maximum pressure coefficient
Greek symbols V circulation, m*s-’
circulation at r = rm, m*s+ circulation at infinity, m*s-’ shear rate, s-l dynamic viscosity, Nsm-* kinematic viscosity, m*s-’ density, kgm-” shear stress, Nm-* coordinate stirrer angular velocity, s-’ angular velocity, s-’
The trailing vortex system produced by Rushton turbine agitators ItEmRENcEs
[l] Biesecker B., V.D.I. Forschungsheft 1972584. [2] Van ‘t Riet K. and Smith John M., Chem. Engng Sci. 197328 1031. [3] Bruijn W., Van ‘t Riet K. and Smith John M., Trans. Inst. Chem. Engrs 197452 88. [4] Cutter L. A., A.1.Ch.E. .R 196612 35. [5] Rao M. A. and Brodkey R. S., Chem. Engng Sci. 197227 137. [6] Batchelor G. K., An Introduction to Fluid Dynamics. Cambridge University Press 1967.
1105
[7] Hoffmann E. R. and Joubert P. N., J. jIuidMech. 1%3 16395. [RI SaRman P. G., Phys. Fluids 197316 1181. [9] Rott N., Z.A.M.P. 1958 9b 543. [lo] Timme A., Ing. Arch. 1957 25 205. [ll] Hinxe J. O., Turbulence. McGraw-Hill, New York 1959. (121 Squire H. B., The Aer. Quart. 1%5 16 302. [13] Voncken R. M., Circukztion and mixing in stirred vessels,
Ph.D. Thesis, Delft (Dutch) 1966. [14] Metzner A. 8. and Otto R. E., A.LCh.E. .R 19573 3.