Modelling of the local distributions of velocity components and turbulence parameters in agitated vessels—method and results

13

Modelling of the Local Distributions of Velocity Components and Turbulence Parameters in Agitated Vessels-Method and Results Modellierung der artlichen Verteilungen der Geschwindigkeitskomponenten Turbulenzparameter in Riihrbehgltern-Methode und Ergebnisse BERND

PLATZER

Technical

University

(Received

November

and GUNTER

und

NOLL

“Carl Schorlemmer”

Leuna-Merseburg,

18, 1986; in final form September

4200 Merseburg

(G.D.R.)

16, 1987)

Abstract Three correlations for the description of the velocity and turbulence parameter distributions are defined: (i) rotational, (ii) circulating, and (iii) jet stream flow. The two latter are based upon the classification of the stirred tank in regions of the cellular vortex and the jet stream from the impeller. These models are taken from equations of well-known flow situations (wall jet, rotating disc, free jet, etc.) and from generalizations obtained from many measurements. The resulting equations are simple in form. The model parameters can be calculated from geometrical data and power number. The comparison of the model with measured values shows good agreement for tanks with and without baffles, impellers with radial and axial flow, and for a wide range of Reynolds numbers. The accuracy obtained is comparable but less work is involved compared with the results obtained through the numerical solution of the Navier-Stokes equations. Thus the engineer has a simple method for calculation of the local distribution of velocity and turbulence parameters in stirred tanks. A methodical way for obtaining this model is demonstrated.

Kurzfassung Fiir die Beschreibung der Verteilungen der Geschwindigkeit und der Turbulenzeigenschaften werden drei charakteristische Strijmungsformen benutzt: (i) Rotations-, (ii) Zirkulationsund (iii) Strahlstromung. Die beiden letz-ten basieren auf der Unterteilung des Riihrbehalters in Zellularwirbel und Strahlbereiche. Die Modelle bauen auf bekannten Stromungssituationen (Wandstrahl, rotierende Scheibe, Freistrahl usw.) und auf Verallgemeinerungen experimenteller Ergebnisse auf. Die resultierenden Gleichungen haben eine einfache Form. Die Modellparameter kbnnen aus geometrischen Daten und dem Leistungsbeiwert berechnet werden. Fur Behglter mit und ohne Stromstorer, fur Ruhrer mit radialer und axialer Forderrichtung und fur einen grol3en Reynoldszahlbereich zeigen die Modellrechnungen eine gute Ubereinstimmung mit MeBwerten. Die Gleichungen sind sehr gut handhabbar und erreichen eine Genauigkeit, die vergleichbar mit der Giite der numerischen LSsung der Navier-Stokes-Gleichungen ist. Damit hat der Ingenieur eine einfache Methode zur Berechnung der Srtlichen Verteilungen der Geschwindigkeit und der Turbulenzparameter in Riihrbehhltern in den Handen. Die methodische Herangehensweise fur das Erstellen des Modells wird aufgezeigt.

Synopse Nachdem in der 2. H@fte unseres Jahrhunderts die Zahl der experimentellen Arbeiten zur Untersuchung der Prozesse in Riihrbehiiltern stark angestiegen ist, verstiirkten sich in den Ietzten Jahren die AnstrenModelle fur die darin ablaufenden Strogungen, mungsvorgiinge aufzustellen. Dabei beschritt man zwei 02%2701/88/$3.50

Chem. Eng. Procm.,

Wege: numerische Liisung der Navier-Stokes-Gleichungen und Erstellen von empirischen Gleichungen. Mittels physikalisch begriindeter VorstelIungen gelang es, eine Niiherungsliisung zu schaffen, die sich gegeniiber den von anderen Autoren vorgestellten durch ihre wesentlich breitere Anwendbarkeit und ihre einfache Handhabung auszeichnet. Im jblgenden wird eine systematische Zusammenstellung der beniitigten

23 ( 1988) 13-3 1

0 Elsevier Sequoia/Printed in The Netherlands

14

Gleichungen und in Tabelle 4 ein Algorithmus zu ihrer Abarbeitung gegeben, so da8 es schon mittels programmierbarer Taschenrechner miighch wird, in kiirzester Zeit die &lichen Verteilungen der Geschwindigkeit und wichtiger Turbulenzparameter anzugeben. Ausgangspunkt der Modellierung sind (i) die detaillierte Analyse der MeJwerte, (ii) die Bezugnahme auf eine Reihe gut bekannter Stromungssituationen und deren obertragung auf die Verhiiltnisse im Riihrbehdlter, (iii) die Anwendung vereinfachter Bilanzen, (iv) die Vorgabe von physikalischen Bedingungen zur Bestimmung der Modellparameter. Dadurch gelang eine Verallgemeinerung bisher vorhandenen Datenmaterials und die Kopplung mit den globalen Strijmungsparametern. Als giinstig erwies sich die Aufteilung in Rotations-, Zirkulationsund Strahlstriimung (Bild I). Bei der Rotationsstriimung wird van charakteristischen Verteilungen (GI. (I), (2), (6)) ausgegangen, deren Modellparameter in Abhiingigkeit von der Geometrie, Bewehrung und Reynolds-Zahl angegeben werden. Dadurch ist es gleichzeitig miiglich, Trombenverliiufe such in teilbewehrten Systemen nachzurechnen (Gl. (8), (9), (9a), Bild 3). Bei der Zirkulationsstriimung wird zwar von Potenzansiitzen (Cl. (10)-(12)) ausgegangen, durch die zur Koefizientenbestimmung benutzten physikalisch sinnvollen Zusammenhiinge konnte jedoch eine breite Anwendbarkeit gesichert werden. Die radiale und axiale Geschwindigkeitskomponente sind fiber die Kontinuitatsgleichung (GI. (13)) gekoppelt. Auch hier sind die Parameter wieder in Abhiingigkeit von den Abmessungen und Betriebsbedingungen vorausberechenbar (GI.’ (12), 08)_(30)). In Riihrernahe wird mit Strahlmodellen gearbeitet. Ausgangspunkt fur radialfordernde Riihrer ist das TangentialstrahlmodeN (GI. (32)-(35)), dessen Parameter mit den GI. (36)-(40) oorausberechnet werden ko’nnen. Fiir axialfordernde Riihrer wird das Model1 vom runden Freistrahl mit Drall modtjiziert (GI. (41), (42)), die Parameter sind nach Gl. (43), (44) zu berechnen. Zst es notwendig, gewiilbte Bidden zu beriicksichtigen, kann auf die Gl. (46)-(48) zuriickgegrtflen werden. Mit den Gl. (49)-(52) werden Hinweise fur die Beriicksichtigung von Leitrohren gegeben. iiber die Vorgabe des Turbulenzgrades (GI. (53)(5.5) Tab. I, 2) und des MakromaJstabs der Turbulenz (GI. (57)) gelingt auJerdem die Vorausberechnung wichtiger Turbulenzeigenschaften (GI. (56), (58)-(62)). Die Verwendung der Vorstellungen iiber die turbulente Wirbelkaskade (Gl. (63)(66j sowie Tab. 3) erlaubt die direkte Untersuchung der Wechselwirkung der Partikeln mit den turbulenten Wirbeln. Dadurch werden bessere Zugiinge zu Problemen des Mischens, Dispergierens und Suspendierens miiglich. Die gilder 2 bis 14 veranschaulichen durch den Vergleich mit MeJwerten und numerischen Liisungen die Anwendbarkeit und die erreichte Giite des Modells. Das ausfiihrhch dargestellte methodische Vorgehen ist such auf andere Riihrsysteme und sogar auf die Erstellung von Naherungsliisungen fur die Geschwindigkeitsverteilungen in viilhg anderen Apparaten iibertragbar.

1. Introduction and model basis The engineer often has to solve problems that are very difficult. These difficulties can be caused by: (i) interaction of single processes (e.g. simultaneous reaction, mixing and heat transfer in an agitated vessel); (ii) insufficient theoretical research (e.g. turbulence processes and multiphase flow in technical apparatus); (iii) geometrical and/or operational conditions of an apparatus differing from earlier experimental works; (iv) new principles of action. Flow patterns are often described after use of an apparatus. Prior to this, various simplified assumptions (e.g. hydrodynamic ideal reactors in reaction engineering) are made to calculate some of the effects. With the need for better control and more intensive processes, the actual processes (and so the flow patterns in the apparatus too) must be described more exactly. Agitated vessels are often used in industry, but it was only in the second half of this century that systematic research was carried out on processes in agiof tated vessels [I]. Nowadays the local distribution flow patterns is of increasing interest because it allows a deeper understanding and better control of processes in stirred tanks. Although a large number of experimental results have been published (summarized in refs. 2-5), there is increasing interest in describing these flow patterns with mathematical models. We can divide these models into two groups: (1) models based on numerical solutions of the Navier-Stokes equations [Cl 13; (2) models that divide the tank into characteristic regions, for which velocity profiles are defined, linked together to form an overall solution (linking of characteristic velocity profiles [ 12-141: jet flow, potential flow, etc.; linking of characteristic flow situations [ 151: ideal mixing, piston flow, bypass, back-flow, etc.). The first models have the disadvantage that their numerical calculations are expensive. Moreover, the impeller and the baffles must be substituted in the model by suitable boundary conditions for twodimensional problems. The second group has a great number of subregions (e.g. 6 [ 121, 7 [ 131, or 8 [ 141); this makes a scale-up for other situations difficult. Although today the importance of computer techniques is increasing, in the future the task of the engineer will not be to solve the balance equations numerically in all cases, because simpler models describing the flow in accordance with experimental data are often sufficient. The following modelling levels exist for the flow in vessels: (i) ideal mixed vessel (without consideration of velocity fields in the tank); (ii) characterization of the flow with global parameters (c,, cz); (iii) approximated solutions of velocity distribution in vessels (empirical equations or analytical

15 solutions of very simplified balance equations) to estimate the whole flow or characteristic flow situations; (iv) numerical solution of the more complete balance equations. One aim of this paper is to describe the way to obtain an empirical model of level (iii) which has parameters that can be calculated from the geometry and power number. For such a model sufficient experimen tal data or calculated results from level (iv) must be available. In this article much of the experimental data published in the last few years provide the basis. The other aim of this paper is the systematic description of results of earlier papers. In this way a treatment of other cases (uninvestigated impeller systems or even other apparatus) is possible for the user. In the first instance for the development of the model it was advantageous to consider level (ii). Because knowledge of cF and cz is insufficient, the

Jet

flow from

the

equations of these parameters are developed further here. From numerical solutions and experimental data only a variant catalogue of the investigated cases results. The advantage of our model is the possibility of a direct calculation of the local distribution of flow patterns for many technically interesting cases. The proposed model is valid for impellers with radial or axial flow, differing Reynolds number (laminar Re < 10-100, transition range l&100 < Re < 10 000, turbulent Re 3 lo4 [2]) and differing numbers of baffles ( 2 0). The model contains the essential characteristics of streaming. These considerations are also the basis of the models in the literature involving regions. But models with 6-8 regions reach the boundary of practical usage because of the great number of parameters (boundaries of regions, parameters of model equations). Therefore it was considered necessary to minimize the number of regions.

impeller

(W

(4 Fig. 1. Regions of the circulation flow and the impeller jet stream in stirred vessels: (a) radial flow impeller; (b) axial flow impeller; (c) impeller near tank bottom; (d) draft tube with impeller.

16

This first analysis shows that a division of flow into three characteristic model parts is useful: (1) rotational flow because of the impeller rotation; (2) cell structure of flow in agitated vessels as circulating flow (see Fig. 1), which is a desired secondary flow effect; (3) distinct jet flow (especially near the impellersee Fig. 1). If only a rough description of the velocity field in the impeller stream is necessary, two model parts are sufficient. In modelling methods, basic types of flow situations are useful. The model parameters are taken from physical conditions, so the model parameters may be predicted if the geometry and power consumption are known. For the modelling one must know that the character of the rotational tlow induces the circulating flow: a weak/strong rotational flow (e.g. a baffled/unbaffled vessel) induces a strong/weak circulating flow. Parameters such as cF and cz already show that. If it is possible to reflect this relation in the model constants, both model elements are separable in this way and the local velocity results from a suitable overlay of the results for any model part. The jet flow from the impeller, induced by the impeller rotation, is the driving force of the circulating flow. Its character differs from that of the circulating flow. Therefore, it is often impossible to calculate the jet flow region with the circulating flow model. Separate equations are needed for this region. In some cases other information is needed, such as time-averaged velocity fields. So special flow conditions must be known if processes in the characteristic regions of agitated vessels (near the impeller for dispersion conditions; near the vessel wall for heat transfer: etc.) and turbulence processes are to be described. The same methods of modelling can be used succesfully for this. Because this paper will demonstrate the way to obtain these models and the parameters, each section is divided into four parts: ( 1) analysis of experimental data; (2) model formulation and background; (3) model equations; (4) examples and application boundaries.

2. Rotational 2.1. Analysis

flow of experimental

data

Rotational flow characterizes the tangential velocity field in agitated tanks (taking into consideration a second part from the jet model). The following facts can be deduced from the experimental data. (i) The tangential velocity is dominant in the tank without baffles. The profiles are characterized by a linear slope for r < r, and a hyperbolic fall for r > r,, [ 17-191 (Fig. 2). In addition, there is a very steep fall in tangential velocity close to the wall, but this fact does not always need to be included in the model. (ii) The tangential velocity does not depend

0.6 0.4 0.2

i

0

r

Fig. 2. Comparison between experimental data (-) of the tangential velocity [ 161 and calculation (---, eqn. (2) or (6)) over a wide range of Reynolds numbers in a vessel without baffles.

greatly on the axial coordinate outside the impeller jet stream in the case of turbulence and unbaffled vessels [ l&20,21] (Fig. 2). (iii) The tangential velocity decreases with the use of baffles in the turbulent range. It can be said that where BW !: 0.2, then the tangential velocity is small outside the impeller stream and almost independent of location. The influence of baffles on the tangential velocity decreases in the case of non-turbulent flow with decrease in the value of the Reynolds number. 2.2. Model formulation

and background

For calculation of the rotational flow in unbaffled vessels and for Re > 105, several equations exist in the literature with almost equally good approximations and with no consideration of axial dependence (review in ref. 19). However, further development of these equations was necessary because: (1) the model parameters cannot be estimated adequately beforehand; (2) for greater heights of the liquid in the vessel, Re < IO3 and vessels with partial baffling the dependence of the velocity in the axial direction must be considered; (3) for Re < 40 the flow changes its character (combination of a Couette-flow situation with a rotating disc; see Fig. 2). Important parameters of the radial dependence of the rotational velocity are the velocity maximum v,,, and the corresponding radius ro. Often r. was estimated from the profiles of the vortex surface [2,4]. A dependence of the kind r, = f( Re) [4] or r,, = f(d,JD) [2] was obtained. This dependence was greatly influenced by the vortex equation used and by the

17

accuracy of the experimental data. But a direct analysis of the experimental u,(r) profiles shows that r0 = constant (eqn. (3)) is valid for different mixing equipment independent of the Re number and the axial distance from the impeller level (see also Fig. 2). The velocity uti can be calculated by an energy balance around the impeller (similar to the pump theory). Thereby, the profiles are substituted by characteristic mean values [ 191. Equation (4) resulting from this, is extended with an empirical function (eqn. (5)) for partial baffling and Re < 105. For the laminar flow the radial decrease is steeper. This can be approxi-

mated better with the law of Couette flow (eqn. (6)). Inside the impeller jet region the resulting tangential velocity must be described by the principle of superposition. Here the resulting kinematic energies of the impeller jet and rotational flow are the basis (eqn. (7)). In vessels that are not fully baffled the vortex shape (eqn. (8)) and the vortex depth (eqn. (9)) (dimensionless) are interesting. They can usually be calculated by using v,,~ (eqn. (2)). A graphical representation of eqn. (9) shows that the dependence of the geometrical parameters can also be approximated with a linear function (eqn. (9a)).

2.3. Model equations Fully bafled, turbulent [22, 231 l?*,J = 0.05 u

Re >, IO3

(I)

Not fully hafled or transition range [19, 241 c

uror f’R= rJO.5 + 0.5(r/ro)4]0.s

(2)

with r, = 0.77 rR

(3) Re > 10’

(4)

0.325~;($)&-0.9)] CP

* = cp for Re = lo5 and unbaffled

vessel

for Re = 10’ and unbaffled

vessel

cF * =

cF

z’ =

z - (hR/2)

z

Re < IO5

(5)

impeller with small blade width impeller with larger blade width

Laminar [24] r < rR

(6)

O,,R =

r > rR for an impeller, for a rotating

oIO/u from eqn. (5) disc, v,,,/u = exp(a,z’/z&)

Inside the impeller jet region [24] VI,,,, = (u,,s2 + v,,R2)o.5

(7)

Vortex shape

(8)

18

Vortex depth 119, 241

(9)

(94 with or without

baffles, Re > 40.

2.4. Examples and application boundaries Figure 2 shows the good agreement between data and equations for all Reynolds numbers. Figure 3 illustrates that for mixing systems with different degrees of baffling eqn. (9) can be effectively applied for calculation of the vortex depth. Thus, uCO(z = H - hE) was used in eqn. (9).

(ii) use of the local points where the velocities are zero; (iii) consideration of the condition of symmetry; (iv) use of the mass balance (upstream = downstream) in each plane. Analysis of the experimental data showed [ 19,261 that the velocity along the vessel axis is proportional to the velocity in the vicinity of the vessel wall (the opposing directions must be taken into consideration). This factor is an essential function of baffling. Further locations and the number of positions where the axial velocity is zero (in baffled vessels there is only one) depend to a large extent on the degree of baffling and to a lesser extent on the impeller type. Thus border conditions are: o,(r = R) = 2+&z) u,(r = r,) = 0

Fig. 3. Measured [t] and calculated values (-. eqn. 9) of the cT number as a function of the baffling: V, C&/D = 0.35; 0, d,JD = 0.3; 0, dR/D = 0.25.

3. Circulating

u,(r = 0) = k,u,(r = R) &,/arl,=, R 2n

flow

J

3.1. Analysis of experimental

data

The basis is the cell structure (see Fig. 1). It can be seen from experimental axial velocity profiles that -the u;(r) have similar profiles for different z values; ~ the radial dependence has a simple form. Therefore, it is possible to separate the flow into an axial. and a radial function (eqn. (10)). The axial dependence can be determined accurately near the vessel wall; polynomial equations can be used for this. 3.2. Model formulation

and background

The flow near the wall is characterized by an inducement of the wall jet flow through the stagnation point flow. It is possible to describe the local heat transfer coefficient through these flow types [25]. Here, the steep fall in velocity near the wall must be considered. For other problems it is not as important. If the steep fall is neglected, then the radial function of the axial velocity can be simplified. The following physical considerations are relevant to the estimation of constants: (i) use of velocities at characteristic points (vicinity of wall, vessel axis);

r

= 0

v,r dr =0

0

The modified conditions for the case of a rotating cylinder are given in ref. 24. So it is possible to estimate five model constants (see eqn. (12)). In eqn. (12) the steep fall in velocity close to the wall is not taken into account because the boundary layer thickness 6,, is very small compared with the tank diameter [25]. The equations from refs. 27731 give better results for the boundary layer of the wall jet, but the border condition u,(r = R - 6,,) = vws must then be used for estimating the parameters. The term v&z) represents the axial function in equation ( 10). The function for vws (eqn. (11)) used in ref. 32 is more suitable than the equation for v,, in ref. 26 because of its greater simplicity. For the estimation of the two model constants in this equation the circulating volume in the vessel (eqn. (19)) and the location of the circulating point (eqns. (23) and (26)--(28)) can be used and give eqns. (17)-(20). Equation (19) considers the discrepancy between the circulating volume in the whole vessel and the volume in the cellular vortex (the basis of this model). The radial velocity component results directly from the mass balance (eqns. ( 13)( 16)). The problem is then to estimate the parameters kq, rz, zz and vz. The first three parameters can be calculated

19

from experimental data [22,33-351 (eqns. (12) and (23)-(28)). Here, one must take into account that for turbulent flow the circulation point fluctuates irregularly up to O.lD [ 11. For axial flow impellers only the case of Re 3 IO4 and fully baffled vessels is considered (eqns. (27) and (28)) because the application of this mixing device is not useful for other conditions. Radial flow occurs increasingly for axial flow impellers for Re < Re,. Measured values from ref. 36 for axial flow impellers and fully baffled vessels can be represented using eqn. (3 1). In tanks without baffles the axial flow is not fully developed [ 371. With deep fixed impellers of the radial flow type one must consider that for a distance of the impeller from the bottom h,/H < 0.2 [38] the lower cellular vortex is not present (see Fig. 1 and eqn. (24)). For the circulating flow some correlations exist in the literature (for a review see ref. 34). However,

the range of validity of these correlations is narrow and there are different dependences on geometrical parameters. Analogous to the treatment in $2, a balance around the impeller was the basis for the calculation. The method for the calculation of the circulating flow (eqns. (21) and (22)) [34] and the pump capacity of the impeller (eqns. (40) and (44)) [ 191 results from the relation between the impulse generated by the impeller and the impulse of the impeller discharging stream, and the introduction of suitable mean values. These equations provide a good representation of the data of different authors. In comparison with other equations (see listing of equations in ref. 34) we find that eqns. (21) and (22) are often simpler and generally cover a wide range. When comparing data from different authors and differing experimental methods, one must consider that the data of cz can vary considerably [ 391 (similarly for cr [40]).

3.3. Model equations Axial velocity [24, 261 v, = F(z) G(r)

(10)

V ws=F(z)=B

(11)

G(r)=k,

(i>“+k,(i)?+k,(i)1+k,

(12)

where [ 19, 261 k, = -8.95

- 20.9k,

k, = 19.9 f 39.8k4 k 3 = -9.95

- 19.9k4 baffled [ 19,261; i.e. k, = 1.5, k2 = k, = 0 unbaffled [ 193

Radial velocity %v,) -+7= ar

WvJ

0

(13)

gives U, = - F’(z) G*(r)

(14)

with F’(z) =

avws

7

(1%

(16)

20 The two model constants B and m are obtained v,(z

=

zZ)

=

0

as,,

or

from [24]

= 0

-

i3Z

(17)

.z=rz

i.e.

m2!L1 zz

(18)

and knowing pz’,,

=

that the flow in the cellular vortex is

pz/2

radial flow impeller

3,

axial flow impeller

(19)

i.e. - vz’,, 271(1 - zz/zw) “(~zhvl.m

B=

(20)

with

Circulating capacity [34]

Location

radial flow impeller

(21)

axial flow impeller

(22)

of the points of circulation [19, 34, 351

- radial flow impellers 40 < Re < 730 Re > 730

fully baffled

Re > 1000 unbaffled,

(23) not fully baffled

with zw =

mm. (ha, H - hR) H

BW =

cw%,(WDXhdD)

h,lH h,lH

2 0.2 5 0.2

(24) (25)

with the drag coefficient 1 cw = 1 0.45

for for

of the baffle [2]

blade baffle tube

BW, = 0.2 r-2 -= R

0.42 Re0.0964 0.76

- axial flow impellers

40 464

(26)

[22, 331

zz = h, fully baffled, turbulent rz = 0.76 R

any baffling condition

(27) (28)

21

With the equations for the location of the point of circulation (eqns. (23), (26), (27) and (28)), eqn. (20) can be simplified for special cases. For example, eqns. (29) and (30) are obtained for radial flow impellers and Re > 730: B =

Vzw/(O. 1054 P) i tizw/(0.0526

Ah -_= h, i.e.

R*)

0.0144 Re’.‘”

fully baffled

(29)

unbafled

(30)

Re < 2000

Re > 2000, Ah = h, (fully developed Re-+O,

(31)

Re 2 2000

(1

Ah = 0

(equivalent

Ah is the axial distance 3.4. Examples

axial flow)

to radial flow)

where the middle streamline

reaches the tank wall, measured

from z = hE.

and application boundaries

Figure 4 shows for two examples that eqn. (11) represents the profiles near the wall very well. Figure 5 illustrates the usefulness of the equation for cz. Further comparisons for other data are contained in ref. 34. Recalculations of published data on measured velocity fields are in very good agreement (radial flow impellers, tanks with baffles, turbulent [26]; axial flow impellers, [ 331; radial flow impellers, unbaffled, turbulent [ 193; laminar and transition range [24]). Examples of these are given in Figs. 6,7 and 8. These are as

4SQmm

4bO

i0

b

i0

400

mm 450

r

I-

Fig. 6. The numerical (-- - [8]) in comparison (14)); 0, experimental

solution of the Navier-Stokes equations with the proposed model (eqns. (10) and data near the wall [B, 421.

Fig. 4. Comparison between experimental and calculated velocity values of the wall jet profile: 0, measured values of an 8-flat-blade paddle in a vessel with baffles [41] (cz = 2.28); x , measured values of a lbflat-blade paddle in a vessel without baffles [20] (cz = 0.58); -, calculated with eqn. (11).

I

mm

Fig. 5. cz and cF values of a flat-blade turbine in a vessel with baffles. Experimental data [2] for c,: 0, dR/D = 0.3,0.4,0.5; for cz: x , d,/D = 0.5; W, d,{D = 0.4; 0, d,JD = 0.3. Calculated: ---, eqn. (40); -, eqn. (21).

I

600

400

-r

200

0

200

400

60vmm

r-c

Fig. 7. Axial and radial velocity produced by an axial flow impeller in a vessel with baffles: -, experimental [43]; ---, calculated, eqns. (41). (42). (10) and (14).

22

1 4200 mm 4000

800

600

409

200 0 0

200

400mm

585

I-

Fig. 8. Axial velocities produced by a radial flow impeller in a vessel without bafflles: 0, experimental [37]; -, calculated, eqn. (IO).

accurate as results obtained by the numerical solution of the Navier-Stokes equations [8,26]. It must be noted for the analytical solution that, for axial flow impellers and H > D, secondary vortex streams occur [39] for which other values for r’,, zw, rz and zz are needed.

where turbulence parameters are included in the model [56]. In ref. 57 the radial jet with spin is also used in tanks without baffles. Apart from the usual three parameters of the tangential jet flow, a fourth parameter is used in ref. 52 to enable the axial drift to be described. This is brought into connection with hE [ 521. It also seems to be influenced by the shape of the tank bottom [43] and by the number of baffles [37]. The velocity equations can be derived by considering boundary layer conditions [SS, 591 (eqns. (32)+35)). A listing of the jet parameters for turbulent flow is given in ref. 55. A method which allows the estimation of the parameters for a wide range of conditions is presented in ref. 24. The physical boundary conditions are again the basis. It is known [51] that the rotational impulse is constant in the impeller jet streaming and that it is proportional to the power consumption. Further, one can demand that the discharge volume reaches the impeller discharge volume for r = rR and - hR/2 < z < hR/2. Parameter correlations that have a local maximum for r > rR and z = constant must be eliminated. The following equations take account of these demands [24,26]: m 2~

i --xi

r2v+ v,,s dz = const.

haI2

4. Jet flow from the impeller 4.1. Analysis of experimental

data

Trailing vortices dominate very close to the impeller blades [44-49]. There are several correlations in the literature for the trailing vortices, but they cannot be used to calculate the conditions near the impeller blades because the connection of the parameters with the mixing system is generally unknown. The vortices break up by r = 0.75d,-d, for radial flow impellers [46, 501. The jet stream is then rotationally quite symmetrical [ 5 1, 521, but for many purposes an extrapolation of the rotational symmetry is also possible for r < 0.75d,-d,. 4.2. Model formulation

and background

Trailing vortex Through utilization of experimental data and other flow types (trailing vortices after flat blades) it is possible to estimate the parameters from geometrical and operating conditions. The dependence of the parameters on operating and geometrical conditions for velocity equations of the vortex is described in ref. 53. A representation of the Sauter mean diameter in non-coalescent media is thereby possible [53]. Radial flow impellers Models of jet streaming exist in the literature, but there is again the problem of estimating the parameters. The tangential jet stream model is used for the Rushton turbine impeller [8, 12, 13, 24, 52-551. The radial jet model is suggested for tanks without baffles,

471rR

v,(r = rK) dz = Vr I cl

‘(‘,,s = vr,S (max)) < rR A correlation for estimating the jet parameter Q results from a utilization of experimental data (eqn. (36)). The other jet parameters A and a are then calculable with the above demands (eqns. (37) and (38)). The parameters depend only on geometry and on the pump capacity. The impeller discharge stream pr can be calculated with a simple impulse balance (eqn. (40)), similar to that in 9;3 [ 19, 341. For laminar flow the impeller operates as a rotating disc. Equation (39) takes account of this fact and unites eqn. (40) (impeller and higher Re numbers) with the dependence of the rotating disc. It must also be taken into account that the tangential velocity in the jet stream is overlayed with rotational flow (eqn. (7)). Axial flow impellers References 60 and 61 use a linear increment for r < r, with a following hyperbolic fall as a model for the impeller jet. The basis of our reflections was the fact that the circular jet with spin is a known characteristic flow situation [33]. Of course the equations cannot be used for agitated vessels without some modifications. As with the tangential jet model it is advantageous to use a drift parameter. With this parameter and taking into account that the up-stream is equal to the off-stream for each z-plane and that the total discharge volume is realized, eqns. (41) and (42) are obtained for the region of the impeller flow.

23

Outside this region the circulating flow model is valid. The model parameter k (eqn. (43)) results from consideration of the impeller and the total discharge vol-

ume [ 331. The pump capacity (eqn. (44)) [ 191 and the circulating capacity (eqn. (22)) are obtained from the above-mentioned impulse balance.

4.3. Model equations Radial flow impellers with or without bafles - tangential V r.S

V

jet stream [58, 591

ACT05 =~(rz-a2)o~25[l

r.s =

-tanh2e)]

_;2)o.5 0,s

(rZ

,4@2 _ a2)0.25

V&S=

(32)

zo0.5,,

(33)

2,.2 _ a2

Ttanh(z)-F[l-tanhi(z r -a2

1.5

(34)

Ap( 2r2 - a2)

ks = 2a

l.5r0.5(r2

_

with the parameters u = 19 exp( -0.108 a/rR =

(35)

a2)0.75

[24] cP)

(36)

[B*/( 1 + B*)]0.5 0.816

B* < 2 B*32

(37)

with 1.772 tanh[ah,/(2d,)] cr(a/c&.)“.’

4

0.282 (c,,rJa)‘,’ A /nd,’ =

(38)

0.21 c&.5

tanhbMWR)l ~ coefficient -3.94( cF =

B*<2 B*>2

of impeller discharge

[ 19, 341

1 + Re/40) Ree0,5f(j)

(c,cphR/dR)o.5

Re<40

(39

Re>40

(40)

with 0.91 Cl = 1 0.70

fully baffled vessel unbaffled vessel

j = hR(nRe/2)0.5/d, f(j)

= a + exp(bj’)

(analogous

to rotating

disc [34])

j-range

a

b

C

O
-1.0 -0.886

-0.3237 -0.4765

1.790 1.285

24 Axial flow impellers [33] - circular jet (valid for r -C rz and z -C hE) (1 + krz2)‘r2 rz2( 1 + kr2)”

v, = -0.667F(z)

v, = O.l67F’(z) with k

4~)

=

(41)

( 1 + kr,2)2r3 rz2( I + kr2)2

(42)

eqn. ( 11) and F’(z) E eqn. (15) and with the parameter

Wc.d0~5(rdd2- 1 rz2[1 - (cdc4 o.51

=

- coefficient

(43) [ 191

of impeller discharge

CF = 0.784 c&.0.5 4.4. Examples and application boundaries Figures 5 and 9 demonstrate equation of the pump capacity.

1.0 -

\ \

0.8 -

CF 0.6

the usefulness of the Figure 10 shows the

-

.

x

l

)’\ \” *

l

ZJ

0.4 -

=--___.

0.2 0

,

loo

101

10'

10'

10‘

10s

Re

Fig. 9. Effect of the Reynolds number on the discharge coefficient +. 0, x , experimental data of an &flat-blade paddle and an 8-flatblade turbine [ 161; calculated: - .- ., eqn. (39) - - -, eqn. 40; -, rotating disc.

calculated and experimental profiles of the velocity in the jet of a radial flow impeller for different radial distances and a wide range of Reynolds numbers. The usefulness of eqns. (41) and (42) for the velocity field of an axial flow impeller is demonstrated in Fig. 7. According to the facts already discussed the equations of an axial flow impeller are only useful in baffled vessels and for Re 2 2000. The tangential jet model is used instead of the circulating stream model so long as the radial velocity in the jet is greater than the velocity resulting from the circulating stream model (eqn. (14)). Without application of the jet models a partially acceptable accordance with the experimental data is obtained, except in the immediate vicinity of the impeller [26]. 5. Inclusion of the tank bottom shape 5.1. Analysis of the experimental

r-7.5

cm

Agitated vessels often have curved bottoms. The shape of the bottom influences the velocity field in this region, directs the stream and fixes the crossed plane (and in this way the mean velocity). 5.2. Model formulation

0

0.3

0.6

JL u Re - 44

0

0.6

a3

0

LL u r-40

03

0.6

v, u

Re =39

data

Re -89

0

0.3

a6 0 IL u

Re - 830

0.3 0.6 -%_ I/ Re-40’

cm

0 0.3 a6 0 03 a6 0 0.3 a6 0 0.3 0.6 0 0.3 a6 Ilr !L 3 2 3 u u u u u Fig. 10. Comparison between experimental ([ 14) and calculated (- -, eqn. (32)) values of the radial velocity in the jet from a radial flow impeller in a vessel without baffles.

and background

According to the requirements of the model the following possibilities exist to include the shape of the tank bottom [22]: (1) replacement of the tank with a curved bottom by a tank with a flat bottom; (2) use of a tangential jet model with drift parameter [52] for impellers located close to the tank bottom; (3) direct inclusion of the tank bottom shape when using the upper structure. The first possibility is very simple and all equations derived are valid without restrictions. The second, the modified jet model of ref. 52, could be useful, but more experimental data are necessary to estimate the model parameters for this case. The last possibility is demonstrated for the example of a

25 Kloepper bottom. The derivation of u, is the same as is demonstrated in the previous section. However, the radius R is substituted by R,(z) (eqn. (45)). The radial

velocity U, resulting from the mass balance equation ( 13) now has a slightly different structure (eqns. (46)(48)).

5.3. Model equations (220 - z*)O.~ R,(z)

=

z c 0.134 D zaO.134D

R 1

Through

substitution

(45)

of R by R,(z) in eqn. (12) it follows from eqn. (14) that

z) - F(z) G**(r, z)

G*(r,

v, = -F’(z)

Kloepper bottom cylindrical part

(46)

with I G**

=

1

r aG(r,

r

OR,

z>a&(z)

dr

(47)

Bz

0

For z -C0.134 D it follows that

(484 and for z 20.134

D

G**=O

(48b)

5.4. Examples

and application

boundaries

Generally, only a qualitative agreement with experimental data is obtainable with these three possibilities in the region of the vessel bottom. A better quantitative agreement requires a numerical solution of the balance equations, but there the problem of proper consideration of the curved wall exists too. 6. Considering 6.1. Analysis

draft tubes

of experimental

6.3. Model equations data

Draft tubes hinder the radial transport of mass between the inside and outside regions of the vessel. Experimental data show that rZ is nearly identical with the above correlations and zZ lies at half the height of the draft tube (eqns. (49) and (50)). Inside the draft tube the volume stream is constant over the height of the draft tube. Generally, the radius of the draft tube, r-r, is smaller than the radius r,. Therefore, for r,_ < r < rZ the stream has the same direction as in the draft tube. For r > rZ the stream is in the opposite direction (eqn. (52)). 6.2. Model formulation

about the height of the draft tube. A parabolic approximation for the axial dependence of vws gives a radial velocity profile for hEL -C z -C hEL + hL and r > r,_. A circulating flow exists here, but it is lower than that without draft tubes. For baffled vessels eqn. (1) is sufficient to calculate the tangential velocity. Some more details were discussed in ref. 22.

and background

Various possibilities exist by which to take into consideration draft tubes in the models [22]. The simplest is the assumption that vws is independent of the axial distance in the region of the draft tube (eqn. (5 l)), for example, the velocity profiles are constant

r, = 0.76 R

(49)

zz = hEL + hJ2

(50)

vws = const.

for

h,, < z < hE,_ + h,

(51)

that is, and

8, #f(z)

v, = 0

&lZ = col%t_ = riz(r > rZ) = PZ’,,LR + tiZI(r c rZ)

(52)

with PZ. LR

# f(z)

6.4. Examples

for

h,,Cz
and application

boundaries

Figure 11 shows the good agreement with experimental data in an agitated vessel with a draft tube

26

400

200

must be used. In an analysis of the experimental data of turbulent velocities it was possible to find some relations [22,23]. These offer the engineer a sufficiently exact description. The degree of turbulence Tu and the macroscale of turbulence A can be described in a simple manner (eqns. (53)-(57)). It can be shown [22] that the degree of baffling has a greater influence than the impeller equipment and the existence of draft tubes. The data of the macroscale of the turbulence vary considerably: A/dR = 0.0154.3

0 0

200 mm 400

$45

Fig. 11 Tangential and axial velocities produced by an axial flow impeller in a vessel with a draft tube and vessels with a curved bottom: 0, experimental [62]; -, numerical solution (71; -, calculated with this model [22].

when using the simple eqn. (51). The model has an agreement with the experimental data comparable with the numerical solution [ 71, but the mathematical expense is very much greater for the numerical solution. The very good validity of eqn. (1) over the entire range is surprising.

(utilization of literature data and measurements from ref. 51). According to ref. 51 the causes of this are the very different methods of measurement. A reasonable size is given by eqn. (57) [51]. With these assumptions it is possible to calculate further turbulence parameters (eqns. (58)-(62)). Equations (54) and (55) are valid only for fully developed turbulence. The values in Tables 1 and 2 (data of ref. 5 1) can be used as reference points of Tu for Re < 104. Up to the present day there are very few experimental data for velocities and turbulence parameters for the laminar/turbulent transition range in agitated vessels. Trailing

7. Local distribution of turbulence parameters 7.1. Analysis

of experimental

data

For many processes in agitated vessels it is advantageous to realize a turbulent velocity field. Therefore, knowledge of the turbulence parameters is often important. In an analysis of the turbulence in agitated vessels we must distinguish between stochastic and non-stochastic (e.g. trailing vortices [48, 501) flow situations. The trailing vortices near the impeller tips of radial and axial flow impellers produce a pseudo-turbulence measured with a fixed probe, but this is of another character. Likewise, one must distinguish whether the processes are connected with the local turbulence level or the direct interactions between the turbulence swirl and the dispersed particles (e.g., for a dispersion of particles in the vicinity of the impeller [53]). For the first process it is not useful to distinguish between turbulence and pseudoturbulence. For the second, more accurate knowledge may be necessary. This distinction is taken into account. 7.2. Model formulation Parameters

of the

and background turbulent

Turbulent

vortex cascade

The turbulent flow with its stochastic nature can be interpreted, for example, by using the turbulent vortex cascade mode1 [63]. Working on the assumption that there is a strong interaction between particles and vortices of the same magnitude, it is possible to obtain equations for this interaction (e.g. particle rotation [64]). The limits of the vortex cascade are the macro- and microscale of turbulence (eqns. (57) and (63)). The assumption that each vortex induces eight other vortices [63] allows assertions to be made regarding the structure of the cascade (eqns. (64) and (65)). Further equations exist for the turbulent viscosity of the turbulent vortex (see refs. 65 and 66, and eqn. (66)). TABLE 1. Effect of Reynolds number on the rate of turbulence Tu/Tu,, (per cent) in the discharge jet of a radial flow impeller in a vessel with baffles [2] (Tu, = Tu for Re = IO4 and r/ra = 1.05)

stream

In the previous sections, models representing the time-averaged velocity profiles were developed. Generally, the knowledge at present is insufficient to calculate the root-mean-squared velocities, dissipation rate, scale of turbulence and other details. Empirical relations or mathematically very complicated models

vortex

Some details of the trailing vortex in the vicinity of the impeller have already been discussed in $4. The model parameters can be found in ref. 53. An equation for calculating the turbulent vortex viscosity is also given there (eqn. (66)).

rlrR Re

1.05

1.2

1.5

1.7

104 185 56

100.0 40.9 22.1

100.0 40.9 22.7

109.1 27.3 9.1

109.1 13.6 4.5

27 TABLE 2. Effect of Reynolds number and the impeller height on the rate of turbulence with a radial flow impeller [2] (Tu, = Tu for Re = IO4 and r/rR = 1.05; see Table I)

Tu/T, (per cent) at the liquid surface in a baffled vessel

Re 111

3400

2300

1100

660

0.6

104.5

100

50.0

68.2

9.1

4.5

_

0.5 0.4 0.2

_ 95.5 90.9

_ 45.5 40.9

68.2 45.5 27.3

18.2 4.5

4.5 4.5 9.1

4.5 9.1 18.2

81.8 81.8

450

185

MD

56 _ 13.7 18.2 36.4

7.3. Model equations Degree of turbulence (22,231 Tu = v’jv = const.

(53)

with 21= (v,2 + v,2 + u,y Uf = (v:’ + v:‘+

,;)0.5

For a tank with baffles Tu = 0.35

(54)

For tanks without

baffles outside impeller jet

0.05 Tu=

-0.1

in impeller jet: (z( < 1.5/z,

H.o.2 h,z

Parameters

(55)

of turbulence [22,23]

v’ = Tu(q2 + vz2 + q2)0.’

(56)

AZ 0.4h,

or

(57)

v, = KAV’

with

A x O.O%d, K = 0.4 [67]

(58)

1, = pcpv,/Pr,

(59)

D, = v,/Sc,

(60)

E = lq2

(61)

E = 1.65 v’~/A

(62)

Turbulent vortex cascade [63] rw,o z 51, = 5(v3/e)“~” Re,“/Re,”

+ ’= 4.75,

(63) Rew = vm.wrw,ol~

(64)

rw,o “Irw,ont’ = 3.6

(65)

with the limits for the vortex radius A > rw 1 51, and with [65,66] VWIV= Re,/Re,,, Re,., = 36

(66) laminar

vortex, smallest element of turbulence

vortex cascade

z=ZOcm

0

10

20 cm

30

r-

0

Fig. 13. Comparison between experimental (- - - [ 671) and calculated (-, eqn. (60)) turbulent diffusion coefficients in an unbaflkd vessel with a radial Row impeller.

2OOmm 400 r

(4

in

900

9~~ v’

4.0

1

700

z

-1 _ 600 ms

0

_ 500 0.016 JWJIPJ

O.O?,

10.rrJx

IO.‘$J

x

I

1

0.013

0.6

400

0.04 10.011J1

0.0s 0.1‘ O.OS3JI0,OISJlO.t~ I

0.0, x

300

0.5 -

200

0.13 10.012;

0.014

0.031

400

0.1

I

-

0. I, .

lo;zlJ

(0.

0 0

,ioo

200

300 r

450

0.3

mm

0.015 IO.161

Fig. 12. Root-mean-squared velocity fluctuations. (a) Radial flow impeller in an unbarned vessel. Experimental data: x , I = 0.25 m [37]; n, r = 0.3 m [43]; l , z = 0.1 m [371. Calculated: -, eqn. (56). (b) Axial flow impeller in a vessel with draft tube and baffles: 0, experimental [62]. -, calculated, eqn. (56).

The curves in Fig. 12, calculated with eqns. (54)) (56), approximate well the turbulent velocity in different mixing equipment. Figure 13 shows values calculated for D, in a tank without baffles. The relatively strong dependence of D, on the location of the baffles [67] can only be estimated as an average [23]. This dependence does not exist in tanks without baffles. Figure 14 shows calculated values for E. The reproduction of the experimental data is good, although it is based upon very rough assumptions. Further calculated profiles are to be found in ref. 23. Care is to be taken here because the occurrence of local isotropic turbulence is only a rough estimation. More detailed facts about the validity of isotropy and about further components of the turbulence stress tensor are obtained in refs. 42 and

0. I, 10. Oat!

0.2-

(b)

7.4. Examples and application boundaries

0.09 0.061 lO.
IO.l~J

X ok

ISJ

0. A 10. 571

0. I? 0.n I O.f6SJlO.~3J I x 0.21 0.3‘ lO.llJfO.37J

1O:rSJ

0. f 6 SR

0

0

0.4

0.2

0.3

0.4

0.5

0.6

Cl? 08

0.9

f

Fig. 14. Comparison between experimental [51] and calculated (eqn. (62)) values of E. Experimental data (H = D): 0, D = 0.4 m; x , D = 1 .O. Calculated values in parentheses.

TABLE 3. Vortex cascade (eqns. (64)-(66)) Level

Rew”

n

N

N-l N-2 N-3 N-4 N-S N-6 ‘Re,,,;

36” 171 812 3858 18326 87050 413500

[64]

hv,o’

%.Wn

QJW”

(mm)

(m s-l)

(s-l)

0.189 0.248 0.328 0.433 0.572 0.754 0.995

2100 772 281 103 37.9 13.86 5.08

0.W

0.324 1.166 4.2 15.1 54.4 195.9

%ith 51, as characteristic value (31, < r,.,,, < l&12.51,).

29 TABLE

4. Steps for the application

of the presented

model Axial flow impeller

Radial flow impeller Enter: geometry,

impeller type. baffling, n, p. Y, cP

1. Regions of circulating and impeller flow =w zz

rz 2. Integral

Fig. l(a)

Fig. l(b)

Min. from (hF, H -/I,) Eqn. (23) Eqn. (26)

H

Eqn. (21) Eqns. (39), (40)

Eqn. (22) Eqn. (44)

Eqn. (27) Eqn. (28)

flow parameters

cz =i= vzw

Eqn. (19)

3. Circulating flow 3.1. Velocity parameters Eqn. (12) Eqn. (18) Eqn. (20)

k,, k,> k,> k, M B

3.2. Velocity components Eqn. (10) Eqn. (14)

rz 11, 4. Impeller flow 4.1. Velocity parameters 0 UIrR A k

Eqn. (36) Eqn. (37) Eqn. (38) Eqn. (43)

4.2. Velocity components Eqn. (42) Eqn. (41)

Eqn. (32) Eqn. (34)

0,. s oz. s 5. Rotational flow 5.1. v, inside impeller flow 5.2. v, outside impeller flow 6. Tank bottom

Eqn. (33) Eqns. (l), (2). (6)

shape Eqn. (45) Eqn. (10) Eqn. (46)

R,(z) “Z r, 7. Turbulence

parameters Eqns. Eqn. Eqn. Eqn. Eqn. Eqn. Eqn. Eqn.

Tu v’ A “I A, D, E E

68-70. Table 3 gives an example

of the conditions

for

the vortex cascade. 8. Summary With the models presented here it is possible, in a simple way, to estimate local distributions of velocity components and technically important parameters. Only geometrical data and power number are neces-

(54), (55) (56) (57) (58) (59) (60) (61) (62)

sary. Power number is usually easily obtainable for many impeller types [2-4, 711. Table 4 shows systematically the steps for application of the models presented. By comparison with experimental data the accuracy of these models was demonstrated. The model is applicable for radial and axial flow impellers, tanks with and without baffles and for a wide range of Reynolds numbers. Reasonable values are also obtained for rotating cylinders [24]. Compared with the numerical solution of the

30 Navier-Stokes equation the model gives results of the same accuracy at a much smaller expense. Through the use of the cellular structure a simple scale-up on other systems is possible, for example, on multistage impellers. The model described has a greater range of application and fewer parameters than other models. This results from the model strategy which is distinguished by: _ extensive analysis of the data; ~ application of well-known flow situations (e.g. jet models) and its transfer to the conditions in the agitated vessels; _ application of simple balance equations; - consideration of physically useful conditions to estimate the model parameters. With this strategy it was possible to generalize existing data and to connect the velocity profiles with global parameters of flow. Further, the presented model can be the basis for the formulation of boundary and initial conditions for more detailed calculations based on the numerical solution of the Navier-Stokes equation. The method is also useful for the generation of flow models of other mixing devices and apparatus.

al

B BW, BW,

6, Cl CF

CP cT

cw cz D, 4 D,

&,,

G(r) F’(z), G*(r) f g

G**(r,

H ::

hO,

hE h EL

h_ h,

z)

r

r. rL

rw rw,o rz SC, Tu u

Nomenclature

A,a

Ret Rew

tangential jet stream model parameters parameter, eqn. (6) wall jet velocity parameter, eqn. ( 11) baffling number, baffling number when fully baffled, eqn. (25) baffle width constant in eqn. (40) = V&d,‘, coefficient of discharge power number = Pipn’d,‘, coefficient of vortex depth in eqn. (9) drag coefficient = Vz/ndR3, coefficient of total circulation vessel and impeller diameter turbulence diffusivity specific turbulence energy functions in eqn. (10) functions in eqn. (14) function in eqn. (39) gravitational constant function in eqn. (46) fluid depth axial distance from vessel bottom = H - h,, axial distance to vortex baffle height axial distance from centre of impeller to vessel bottom height of draft tube from vessel bottom draft tube height impeller blade height

u

0’

u,t VI, v, vrs v,,svz,s v,o vt0.w u,,iZ

vws Z

=w ZZ

6ws

V

Vt P a

vortex depth parameters in eqn. ( 12) coefficient in eqns. (41) and (42) microscale of turbulence, eqn. (63) wall jet velocity parameter, eqn. (11) baffle number impeller rotational speed power consumption of impeller turbulent Prandtl number vessel and impeller radius vessel bottom radius, eqn. (45) = ndR2/v, Reynolds number of impeller Reynolds number of turbulence Reynolds number of trailing vortex, eqn. (64) radial coordinate radius of maximum tangential velocity draft tube radius vortex radius radius of maximum tangential velocity in vortex coordinate of circulation point (Fig. 1) turbulent Schmidt number rate of turbulence, eqn. (53) = md,, peripheral impeller velocity impeller discharge total circulation in vessel, in draft tube and in circulation swirl fluid velocity, eqn. (53) root-mean-squared velocity fluctuations radial, tangential and axial component of fluid velocity radial, tangential and axial component of tangential jet velocity maximum tangential velocity maximum tangential velocity in vortex tangential velocity of rotational flow maximum velocity of wall jet profile axial coordinate height of circulation swirl (Fig. 1) coordinate of circulation point (Fig. 1) wall jet scale, distance of vws from wall specific energy dissipation macroscale of turbulence thermal conductivity of turbulence dynamic viscosity of tangential jet stream model kinematic viscosity turbulence viscosity density tangential jet stream model parameter rotational velocity of vortex

31

References 1 R. M. Voncken, Br. Chem. Eng., 10 (1965) 12. 2 F. Liepe and H.-O. Mockel. Stoffvereinigen infiuiden Phasen, Vol. 1, VEB Deutscher Verlag fdr Grundstofindustrie, Leipzig, 1978; F. Liepe and H. WeiBglrber, Stojiiereinigen in fluiden Phasen, Vol. 2, VEB Deutscher Verlag fur Grundstoffindustrie, Leipzig, 1979. 3 V. W. Uhl and J. B. Gray, Mixing-Theory and Practice, Academic Press, New York, 1966. 4 S. Nagata, Mixing, Halsted, Tokyo, 1975. New 5 J. Y. Oldshue, Fluid Mixing Technology, McGraw-Hill, York, 1983. 6 H. Thiele, Dissertation, TU Berlin, 1972. 7 R. Kreher, Dissertation, TU Miinchen, 1976. 8 B. Platzer, Chem. Tech. (Leipzig), 33 (1981) 16. 9 J. Placek, Dissertation, VSCHT. Praha, 1980. 10 P. S. Harvey and M. Greaves, Trans. Inst. Chem. Eng., 60 (1982) 195 (part l), 201 (part 2). 11 M. Kuriyama, H. Inomata, K. Arai and S. Saito, AIChE J., 28 (1982) 385. 12 A. De Souza and R. W. Pike, Can. J. Chem Eng., 50 ( 1972) 15. 13 A. S. Ambegaonkar, A. S. Dhruv and L. L. Tavlarides, Can. J. Chem. Eng., 55 (1977) 414. 14 I. Foit, A. Obeid and V. Biezina, CON. Czech. Chem. Commun., 47 (1982) 226. 15 J. H. Olson and L. E. Stout, in V. W. Uhl and J. B. Gray (eds.), Mixing, Vol. 2, Academic Press, New York, 1967, pp, 115-167. 16 S. Nagata, K. Yamamoto, K. Hashimoto and Y. Naruse, Mem. Far. Eng., Kyoto Univ., 22 (1960) 68. 17 F. Liepe, Dissertarion, TU Dresden, 1972. 18 M. Hoffmeister, Maschinenbautechnik, 23 (1974) 387. 19 B. Platzer and G. Noll, Chem. Tech. (Leipzig), 35( 1983) 235. 20 S. Nagata, K. Yamamoto and M. Ujihara, Mem. Fat. Eng., Kyoto Univ., 20 (1958) 336. 21 D. J. Brennan, Trans. Inst. Chem. Eng., 54 (1976) 204. 22 B. Platzer and G. Nell, Chem. Tech. (Leipzig), 37 (1985) 243. 23 B. Platzer and G. Noll, Maschinenbautechnik, 33 (1984) 221. 35 (1986) 418. 24 B. Platzer and G. Nell, Maschinenbautechnik, 25 B. Platzer and G. Noll, Chem. Tech. (Leipzig), 35 (1983) 140. 26 B. Platzer and G. Nell, Chem. Tech. (Leipzig), 33 (1981) 648. 27 N. V. Swamy Chandrasekhara, B. H. Gowda Lakshmana and N. Chandrasekhar, Ind. J. Tech., 13 (1975) 103. 28 K. Weinhold, Dissertation, TU Dresden, 1964. 29 W. H. Schwarz and W. P. Cosart, .l. Fluid Mrch., 10( 1961) 481. 30 M. B. Glauert, J. Fluid Mech., 1 (1956) 625. 31 R. A. Seban and L. H. Back, Int. J. Heat Mass Tran.$er, 3 (1961) 255. 32 Y. Nagase and M. Kikuchi, Chem. Eng. J., 26 (1983) 13. 33 B. Platzer and G. Nell, Chem. Tech. (Leipzig), 34 (1982) 642. 34 B. Platzer and G. Noll, Maschinenbautechnik, 34 (1985) 512. 35 V. A. Orlov, J. V. Cepura and Ju. V. Tumanov, Tear. osnovy chim. Tech., lO(1976) 261. 36 H.-O. MGckel and Zastrow, data from ref. 2. 37 H. Thomae, Dissertation, TH Miinchen, 1969.

38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69

70 71

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