Turbulent flow in closed and free-surface unbaffled tanks stirred by radial impellers

Pergamon

Chemical Engineering Science, Vol. 51, No. 14, pp. 3557 3573 1996 Copyright f:) 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/96 $15.00 + 0.00

S0009-2509(96)00004-8

T U R B U L E N T F L O W IN C L O S E D A N D F R E E - S U R F A C E U N B A F F L E D T A N K S STIRRED BY R A D I A L I M P E L L E R S MICHELE CIOFALO* Dipartimento di Ingegneria Nucleare Universit~ di Palermo, Viale delle Scienze, 90128 Palermo, Italy and A L B E R T O BRUCATO, F R A N C O G R I S A F I and N I C O L A T O R R A C A Dipartimento di Ingegneria Chimica dei Processi e dei Materiali, Universit~t di Palermo, Viale delle Scienze, 90128 Palermo, Italy (First received 16 March 1995; revised manuscript received 14 September 1995; accepted 22 December 1995)

Abstract--The three-dimensional turbulent flow field in unbaffled tanks stirred by radial impellers was numerically simulated by a finite-volume method on body-fitted, co-located grids. Simulations were run, with no recourse to empirical input, in the rotating reference frame of the impeller. Free-surface problems were also simulated, in which the profile of the central vortex was computed as part of the solution by means of an iterative technique. Predicted velocity and turbulence fields in the whole vessel and power consumptions were assessed against available literature data; free-surface profiles were also compared with original experimental data obtained in a model tank. Both the eddy-viscosity k-e turbulence model and a second-order differential stress (DS) model were used and compared: satisfactory results were obtained only by using the latter model. The need for including source terms arising from fluctuating Coriolis forces in the Reynolds stress transport equations is highlighted. Copyright © 1996 Elsevier Science Ltd

1. I N T R O D U C T I O N : B A F F L E D VS U N B A F F L E D T A N K S

Stirred tanks are widely used in the process industries: mining, food, oil, chemicals, pharmaceuticals, paper and power industries, as well as municipal and industrial wastewater treatment, are major examples of fields where stirred tanks are currently employed. Depending on the purpose of the operation carried out in the mixer (blending of miscible liquids, dispersing of gases or immiscible liquids into a liquid phase, solids suspension, heat transfer promotion, chemical reaction, etc.), the best choice for the geometry of the tank and impeller type can vary widely. The geometry illustrated in Fig. l(a), often referred to as the "standard" geometry, is the one that has been most extensively investigated so far. The impeller utilized CRushton turbine") is shown in greater detail in Fig. 2(a), and belongs to the class of "radial" impellers, as the flow discharge of this turbine is mainly radial. An alternative impeller type (8-blade flat paddle) is shown in Fig. 2(b). If the baffles which characterize the standard geometry are not supplied, the liquid tends to move mainly along circular trajectories, resulting in small relative velocities between impeller and fluid and weak radial flows directed towards the tank walls. This results in a poor axial mixing and in the forma-

*Corresponding author. Tel.:(39)91-232 257. Fax:(39)91232 215.

tion of a pronounced vortex on the free surface of the liquid, whose depth depends on the stirrer rotational speed [Fig. l(b)]. Installation of the baffles effectively destroys the circular liquid patterns, inhibiting the vortex formation so that the liquid surface becomes almost fiat [Fig. l(a)]. Moreover, axial flows become much stronger, leading to an improved mixing rate. For these reasons, baffled tanks are the more widely used in industrial applications, and have received much more attention both by experimentalists and modellers. However, there are cases in which the use of unbaffled tanks may be desirable. First, baffles are usually omitted in the case of very viscous fluids (Re< 20), where they, giving rise to dead zones, may actually worsen the mixer performance, and where vortex formation is inhibited by the low rotational speed and by the high friction on the cylindrical wall (Nagata, 1975). Unbaffled tanks are also advisable in crystallizers, where the presence of baffles may promote the particle attrition phenomenon (Mazzarotta, 1993). Finally, unbaffled tanks give rise to higher fluid-particle mass transfer rates for a given power consumption, which may be desirable in a number of processes (Grisafi et al., 1994). In free-surface unbaffled tanks, the central vortex, usually regarded as a drawback, may be desirable in a n u m b e r of situations (Smit and During, 1991). For example, the central vortex is effective in drawing down floating solid particles or in removing gas bubbles from the liquid, thus reducing foam formation.

3557

3558

M. CIOFALOet al. 9

H

cI

H

DID m

i, T

T

(a)

(b)

Fig. 1. Shape and characteristic dimensions of stirred tanks. (a) Standard baffled vessel; (b) free-surface

unbaffled vessel.

q

D

•

~b D

•

a q

d

~'

(a)

(b)

Fig. 2. Shape and characteristic dimensions of the two radial impellers considered in this study. (a) Rushton turbine; (b) eight-blade paddle impeller.

Most published experimental and predictive work on mechanically stirred vessels has regarded baffled tanks. For example, detailed measurement of the flow field, obtained by Laser-Doppler velocimetry (LDV), have been presented, among others, by Costes and Couderc (1988), Wu and Patterson (1989), Stoots and Calabrese (1989) and Ranade and Joshi (1990a). In numerical simulations, the presence of baffles allows the top free surface (if present) to be treated as a fiat no-shear surface (symmetry plane), thus avoiding all computational difficulties connected with a non-planar free surface profile. Predictions of the flow field in baffled tanks stirred by radial (Rushton) impellers have been presented, for example, by Middleton et al. (1986), Gosman et al. (1989), Ranade and Joshi (1990b), and by some of the present authors (Brucato et al., 1989, 1990). All the above simulations used the k-~ turbulence model and treated the impel-

ler as a "black box", i.e. prescribed empirical values of mean velocities and turbulence quantities on its boundaries. However, some authors (e.g. Fokema et al., 1994) have pointed out the importance and the difficulty of describing the impeller correctly in this kind of simulations. A number of approaches have recently been proposed which allow the explicit simulation of the whole flow field without any recourse to empirical data; in this case, a major problem to be faced is, of course, the relative motion between rotating blades and stationary baffles. To this purpose, overlapping or sliding mesh techniques have been implemented in the latest versions of commercial codes like F L U E N T (Murthy et al., 1994), STAR-CD (Luo et al., 1993; Bode, 1994) or R F L O W (Takeda et al., 1993). An iterative "inner-outer" technique was proposed by some of the present authors (Brucato et al., 1994).

Turbulent flow in closed and free-surface unbaffted tanks stirred by radial impellers Comparatively less effort has been dedicated, both by modellers and by experimentalists, to unbaffled stirred tanks in turbulent flow. Early experimental results for the power number as a function of the Reynolds and Froude numbers for free-surface vessels stirred by a variety of impellers were reported by Rushton et al. (1950). More complete data, including the behaviour of the power and pumping numbers and relatively crude flow field measurements, obtained by Pitot-static probes, were collected by Nagata (1975); only paddle impellers were considered. Recently, LDV measurements of mean and fluctuating velocities in unbaffled tanks were presented by Dong et al. (1994) and by Armenante et al. (1994). Dong et al. used an eight-blade, flat-paddle impeller having D ~ 2.5 cm at rotational speeds--oL..100 and 150 rpm. Thus, the Reynolds number was ~ 1000 (transitional, rather than fully turbulent, flow), and no significant central vortex developed. Armenante et al. used a six-blade, 45 ° pitched turbine having D = 10.2cm at a rotational speed of 300rpm (Re ~ 50 000). The vessel was equipped with a lid, and thus no free surface existed. They also conducted numerical simulations using the F L U E N T code and alternative turbulence models (k-e or ASM); however, they did not simulate the impeller region explicitly but rather modelled it by imposing appropriate, empirically derived velocity conditions on its boundaries, as is customary in the above discussed baffled-tank simulations. The full numerical simulation of the turbulent flow in unbaffled vessels is relevant and challenging, at least for two reasons which will be stressed here. First, the nature of the problem, characterized by strong turbulence anisotropy, streamline curvature, and rotation, makes it a severe benchmark for turbulence models. At the same time, the relatively simple geometry allows one to conduct full simulations while still using standard numerical techniques (thus avoiding either the need for empirical data or an excessive computational complexity, which may both obscure the interpretation of the results). In free-surface unbarfled tanks, Fig. l(b), the geometry of the central vortex is controlled by the overall flow field which is established in the vessel at any given rotational speed. Thus, a correct prediction of the vortex shape (for which experimental data are easily obtained) is not trivial and can be regarded as a good validation of the overall flow field simulation. A second, and perhaps more important, reason to be interested in the present problem is that many of the advanced techniques used to predict the flow field in baffled tanks, like the above-mentioned "innerouter" approach of Brucato et al. (1994), include the numerical simulation of the inner vessel region surrounding the impeller, which is closely related to the case of an unbaffled tank and presents much the same computational difficulties. Thus, advances in the simulation of unbaffled tanks may well result in improvements in our capability to perform complete

3559

simulations of the, industrially more important, baffled vessels. 2. T H E C E N T R A L V O R T E X I N F R E E - S U R F A C E U N B A F F L E D VESSELS

A simplified, potential flow theory of the vortex geometry in unbaffled stirred vessels is given by Nagata (1975). It is summarized here taking into account also the slightly modified formulation due to Smit and Dfiring (1991). By assuming that the velocity in the vessel is purely tangential and depends only on the radial coordinate r, and by neglecting viscous forces, the momentum (Navier-Stokes) equations are reduced to: u0~ 0p p -- r Or -

(1)

Op ez

= =--.

P9

(2)

By imposing p(h) = constant (e.g. atmospheric pressure) one has: dh

u02 -

dr

(3)

r 9"

Thus, the profile h(r) of the free surface can be derived once the velocity uo(r) is known. To this purpose, Nagata assumes that the whole flow field can be subdivided into an inner region r ~< rc (forced vortex), which exhibits a rigid-body motion with the angular velocity c,J of the stirrer, and an outer region r > r,, (free vortex), in which the angular momentum (ruo) is a constant [Fig. 3(a)]. There follows the velocity profile: Uo = cor

(r <~ rc)

(4a)

Uo = o)rZ/r

(r > re).

(4b)

By introducing the dimensionless quantities: U~ =

UO - -

Utip

UO

~D/2

(5a)

r

= -D/2

(5b)

h h* = D

(5c)

(i.e. normalising the radius to the impeller radius D/2, the height to the impeller diameter D, and the velocity to the peripheral velocity of the impeller, Utip = o)D/2), and introducing the Froude number Fr = N2D/9, eqn (3) leads to the following "universal" freesurface profile: 7[2

h*=(H*-h*)+-fFr~Z h* = (/-/* + h~) + 5 - e r ~

}1 _ }~

(~<<~c)

(6a)

(~ > ~ ) (6b)

3560

M. CIOFALO et al.

I I

I I

[

(a)

(b)

Fig. 3. Schematic representation of circumferential velocity profile (a) and free-surface height (b) in unbaffled tanks according to the simplified theory of Nagata.

in which H* = H I D is the dimensionless average elevation of the fluid free surface, ~r and ~.care, respectively, equal to T / D (dimensionless diameter of the vessel) and to 2 r i D (dimensionless radius of the forced vortex), and h~', h~ are the dimensionless depth and height of the vortex, respectively below and above the mean elevation, which are given by simple volume balances as: h* = ~2 F r [ ~2 _ ~--~ g~I ~-ln

~]1

(7a)

h* = rcz Fr g~r

The free surface profile given by eqns (6) and (7) is sketched in Fig. 3(b). The only free parameter of the model is the dimensionless radius of the forced vortex, {~. This can be estimated from measurements of the vortex depth, h*; an alternative method, based on the comparison of the power consumption of any given impeller with that of a paddle having the same values of D and b, is described by Nagata (1975). The same authors reports typical values of ~ of about 0.600.65 for D / T = 0.3~).7 and Reynolds numbers (Re = p N D 2 / # ) above 4 x 104, and smaller values for lower Reynolds numbers or very low D / T ratios. The dependence of ~ upon the vessel and impeller geometry was investigated experimentally by Yamamoto [reported by Nagata (1975)] who, for the case of flat-blade paddle impellers, proposed the following correlation: ¢,.= 1.23(0.57 + 0.35 D ) ( b )

°°36

Smit and Diiring (1991) experimented with vessels equipped with multiple impellers having D / T = 0.8, and ranging in size from T = 0.25 m to 3.5 m (fullscale equipment). They found that, for Froude numbers < 0.4, h* and h* correlated with Fr as: h* = 2.20 Fr,

h* = 1.13 Ft.

(9a, b)

These values do not agree well with Nagata's model; from eqns (7a) (9a) a value of about 0.57 is obtained for ~.c,while eqns (7b)-(9b) yield ~c = 0.67. In order to clarify the reasons for this disagreement, Smit and Dfiring measured the tangential velocity Uo at various heights and suggested that experimental data were best reproduced if eqns (4a) and (4b) were modified as follows: Uo = 0.825cor

(r ~< re)

(10a)

Uo = 0.825~orc(r~/r) °'6

(r > re)

(10b)

i.e. by assuming a lower angular velocity than ~o in the forced vortex region and a weaker dependence of Uo on r than r 1 in the free vortex region. Best fit of experimental data with eqns (10a) and (10b) yielded gc = 0.675. Using eqns (10a) and (10b) as a basis for the recalculation of the vortex shape, a better agreement with experimental free-surface profile was obtained than using the Nagata model, although some further empirical adjustment was required in order to accurately reproduce the empirical vortex height (h* + h* = 0.33 Fr). Of course, these results can be extended only qualitatively to single-stage impellers having lower values of D / T , as those on which attention is focussed in the present work. 3. M O D E L A N D M E T H O D S

X n b0'116

Re

lO00 + 1.43 R e

.

(8)

Numerical simulations of turbulent fluid flow in unbaffled, free-surface tanks stirred by single-stage

Turbulent flow in closed and free-surface unbaffled tanks stirred by radial impellers radial impellers have been conducted by some of the present authors for some years (Brucato et al., 1989, 1990). The models and numerical methods used will be briefly described here, together with computational details and programming aspects.

the transport equations for k and e being: /u., \ div(puk) = d i v ~ g r a d k ) + G - p e

(

div(pue) = d i v / 4 f f grad e + (C1G - C2pe) ~ 3.1. Equations of motion and turbulence models All numerical simulations were conducted in the rotating (non-inertial) reference frame in which the impeller is stationary. The equations which were solved were the Reynolds continuity and momentum equations for an incompressible fluid in steady-state turbulent motion, which, in cylindrical coordinates r, 0, z, take the form: div(u) = 0 u0z div(puu~) - p r -

div(puuo) + p

(11)

~p ~00 ~?r + div(r,) - --r + F~

U.Uo r

(12a)

1 ~p Z~o + div(~0) + - - + Fo r 00 r

(12b) @ div(puu=) = - ~z + div(~=) + F~.

(12c)

For simplicity, overbars have been omitted, here and in the following, from mean (i.e. time- or ensembleaveraged) quantities. F,, Fo and F~ are body forces including both gravity and the centrifugal and Coriolis terms which arise in the non-inertial reference frame; they are given by: F, = p ( 0 3 2 r +

(13a)

203Uo)

Fo = p ( - 203u~)

(13b)

F= = - P9.

(13c)

z,, ~0, ~= are the row vectors making up the effective stress tensor zu, sum of the Reynolds stresses ~'ij,turb = - pu[u) and of the viscous terms zu,~i~ = - 2pSi). So is the mean strain rate tensor, whose six independent components are given, in explicit form and in cylindrical coordinates (Hinze, 1975) by:

~Ur

s. =-~r,

&°=2\r

1 dUo u. So0 ~ r~-O + r ' ~Uz

s= = 7 [ '

S,=

2\az

(16)

(17)

Here, #elf is the effective or total viscosity, given by p + Pr, and G is the dissipation function -"~ijSij, which, in eddy-viscosity models, can be computed as 2perfSoSi ;. The values of the model constants were the "consensus" ones C, = 0.09, C1 = 1.44, C2 = 1.92, ak = 1.0 and o~ = 1.3. The mean normal Reynolds stress, (2/3)pk, was added to the static pressure p to give a total pressure/~ replacing p in the momentum equations (12). When the differential stress (DS) model (Launder et al., 1975) was used, the turbulent stresses Zlj.turb were computed by solving six independent transport equations having the general form: div(pn'cij,turb) = G i j --

div(d o) + ~ij - eo + FU

(18)

in which the terms on the right-hand side represent generation, diffusion, pressure-strain correlation, dissipation and generation by fluctuating body forces, respectively. As it is well known, in the DS model a further transport equation for the dissipation e, similar to eqn (17), is also solved, while the turbulence energy k, where required, is computed as the half sum of the diagonal terms Zkk,t~rb. Details of the model and of the form taken by the stress transport equations in cylindrical coordinates are cumbersome and will not be given here. The implementation of the DS model in the CFDSFLOW3D code is discussed by Clarke and Wilkes (1989). In the present computations, "wall reflection" terms q)~,: J were not explicitly included in the pressure-strain correlation. It is important to observe that fluctuating Coriolis forcesfi (f, = 2p~ou'o, fo = - 2p03u;) give rise to source terms F~j =f~u) +J)ul in the Reynolds stress transport equations; in the present case, these take the explicit form: F , = 403 ~'r0,turb,

Fro = - 2 0 3 ( T r r , t u r b - - 27rO,turb)

Foo = - 4o9 Tr0,turb,

Frz = 2 ( 0 ~'0z,turb

F==0,

Foz = - - 2 6 0 Trz,tur b.

?0 + r ~ r r )

(14)

+ ar ]

1/10u=

OUo~

So= = ~ ~ 7 - ~ + ez

7

When the eddy-viscosity k-e model (Launder and Spalding, 1974) was used, the traceless part of the Reynolds stress tensor was computed as the viscous stress tensor, but with the laminar viscosity p replaced by a turbulent viscosity gr. This was expressed as: k2 Pr = C U P - -

3561

(15)

(19)

Equation (19) shows that these terms have an opposite effect on the radial and azimuthal normal Reynolds stresses and thus contribute to the turbulence anisotropy. As will be discussed latter, they play an important role in determining the form of the velocity field. It is worth noting that, since the sum of the diagonal components Fkk is zero, the Coriolis terms have no net effect on the turbulent kinetic energy k; therefore, they cannot be accounted for when the simpler k-e eddy-viscosity model is employed. It should also be observed that the centrifugal

3562

M. C1OFALOet al.

force does not possess fluctuating components; therefore, it does not contribute source terms to the Reynolds stress transport equations. 3.2. Boundary and initial conditions For symmetry reasons, only 1In of the tank was simulated, n being the number of impeller blades, and periodicity conditions were imposed on the planes 0 = 0 and 0 = 2gin. The tangential velocity - cor was imposed on the peripheral and bottom walls (which are still in the laboratory reference frame); on all solid walls, conventional linear logarithmic "wall functions" were used (Launder and Spalding, 1974). The free surface, if present, was treated as a noshear surface characterized by the conditions: u, = O,

Ou,/On = 0

FRAME

CONDITION A

CONDITION B

_~

=~ ,~

O

<'

(20a,b)

"n" and "t" being the directions normal and tangential to the surface, respectively. Zero-normal derivative conditions akin to (20b) were also imposed to k and e. When the differential stress model was used, the shear components of the Reynolds stress tensor were made to vanish on the free surface, while zero-normal derivative conditions were imposed on all the remaining components. Launder (1989) suggested that normal stresses vanish on free surfaces. However, he did not give further details. Since this assumption does not seem to be justifiable on physical grounds, it was not followed here. The method used to compute the freesurface profile is described in Section 3.4 below. In all the simulations described in the next section, the initial conditions for velocity were simply ur = u= = O, Uo = - ~or. Since velocities are relative to a reference frame rotating with the impeller, this is equivalent to assuming that the fluid is initially still in the laboratory frame throughout the vessel. Some runs were repeated by setting Uo = 0 for r <~ D/2; this is equivalent to assuming that the fluid in the inner cylinder r <<.D/2 initially rotates rigidly with the impeller and slips on the outer, still fluid. The two alternative initial profiles of the circumferential velocity Uo are schematically illustrated in Fig. 4 as seen in both reference frames. No significant difference was observed in the converged solution, but convergence was delayed by the latter choice of initial conditions, especially when the DS model was used; actually, the initial flow field was destroyed and replaced by practically still fluid before new velocity profiles started to grow. In all cases, simple uniform initial values were prescribed for the turbulence energy k and the dissipation 8 (k = 0.01 U2p, ~ = C3/4k3/Z/D). In DS simulations, an eddy viscosity model was used for the first 100 S I M P L E C iterations so that initial values of the Reynolds stresses need not be prescribed. 3.3. Numerical methods The governing equations were solved by using a finite-difference method (Burns and Wilkes, 1987),

Fig. 4. Schematic representation of different initial conditions adopted for Uo, shown in the impeller and in the laboratory reference frame. implemented in the computer code C F D S - F L O W 3 D (Release 2). It uses a co-located grid approach, based on the Rhie Chow (1983) algorithm to prevent "chequerboard" oscillations, and allows the use of non-orthogonal body-fitted grids (full use of this was made in free-surface computations). The S I M P L E C algorithm (VanDoormal and Raithby, 1984) was used to couple the continuity and Navier-Stokes equations. Underrelaxation factors between 0.6 and 0.7 were adopted in conjunction with the k-~ model, while smaller values (0.1-0.3) were found to be necessary when the DS model was adopted. Pressure was never underrelaxed, as required by the S I M P L E C algorithm. Reducing the underrelaxation factors further during the last stages of a simulation helped to reduce the final mass source residual. The resulting sets of linear equations were solved by using a three-dimensional version of the Strongly Implicit Procedure (Stone, 1968) for the m o m e n t u m and scalar-transport equations, and a conjugate gradient method with incomplete Cholesky preconditioning (Kightley, 1985) for the pressure-correction equation. Either the upwind or the Q U I C K discretization schemes were used for the convective terms; no significant difference was ever observed in the results. 3.4. Algorithm f o r the free-surface computation In free-surface problems, the free-surface profile was determined by applying the iterative procedure sketched in Fig. 5. It can be summarized as follows: Initialization: Impose a tentative free-surface profile hl °) (height of the generic "surface element" Ai, defined as the annulus r i < ~ r < r i + l ) , with a prescribed average H = ~,Aihi10)/X~ Ai"

Turbulent flow in closed and free-surface unbaffled tanks stirred by radial impellers

I

START

]

assumefiat free surface ~

generatecomputational grid compute flOWfield (SIMPLECiteration)

•

NO

~

YES

compute new free surface

3563

used for each run; convergence problems arising with the k-e model are discussed in the following section. In free-surface computations using the DS model, a satisfactory reduction of the mass source residual was generally attained at each external iteration after 1000-1500 internal iterations, according to problem conditions, initial conditions chosen and grid size. A larger number of S I M P L E C iterations (e.g. 2000) were used during the first external iteration. During the last stages of the simulation, the mass source residual decreased to about 10- 5 times the circulation flow rate, which is proportional to poJDHT. With the finest grids, having roughly 20000 volumes, computation times were about 6 s per iteration on an H P 9000-712 RISC workstation for the k-e model, and about 15 s with the DS model. Thus, the overall C P U time typically ranged from 15 to 40 h for a complete simulation. The storage requirements for completely in-core computations were within 20 Mbytes. 4. RESULTSAND DISCUSSION

NO

END

1

Fig. 5. Flow-chart of the iterative procedure used for the computation of the free-surface height. Generic external iteration k: (I) Build a body-fitted grid; Compute the flow field and the surface pressures plk) by a n u m b e r of internal S I M P L E C iterations, restarting from the results of the last external iteration; Compute the average surface pressure p(k)= Aiplk)/v~ A,; Compute 6pl k) = pl k) - pig). (II) Check convergence; exit if max (abs [6plk)]) ~< 6pmax. (III) Compute new heights hl"~w) = H + 6plk)/(pg); Underrelax h(ik+ 1) = (1 - fl)hl k) + flhl new). (IV) Let k = k + 1; go back to (I). The above method converges rapidly; 4 to 8 external iterations were usually sufficient, depending on the specific vessel geometry and impeller rotational speed, for the profile h(r) to exhibit no further appreciable changes. The adoption of an underrelaxation factor /3 of about 0.7 was found to be useful to reduce overshoots and oscillations of the computed free surface profiles. 3.5. Further computational details The size of the computational grids used ranged from 2 0 x 16× 30 to 26× 1 6 × 4 6 ( r ' O ' z ) control volumes. U p to 9000 S I M P L E C (internal) iterations were

4.1. Comparison of turbulence models for closed vessels First, simulations were conducted for a flat-bottomed vessel provided with a lid (i.e. not possessing a free surface) and stirred by an 8-blade paddle impeller, located at mid-height. This will be referred to as Geometry A in the following. The physical dimensions are summarized in Table 1. The rotational speed was 1.2 rps (72 rpm), giving a Reynolds n u m b e r of about 105 . The computational grid used is shown in Fig. 6. For this problem, only poor results were obtained when the k-e turbulence model was used. Starting from zero velocities everywhere in the laboratory frame (see Fig. 4, condition A), radial profiles of Uo similar to those in Fig. 3(a) developed after a moderate n u m b e r of S I M P L E C iterations, see Fig. 7(a), and a corresponding, fairly intense, double-circulation loop was established in constant-0 planes, as shown in Fig. 7(b). Here, and in the following discussion, results are averaged over the circumferential direction 0. Similar, or even more realistic results, were obtained after the same number of iterations when the initial condition B in Fig. 4 was used. Such velocity distributions are qualitatively acceptable as compared with the classic results of Nagata, and have actually been reported by some of the present authors in previous work on unbaffled tanks (Brucato et al., 1989, 1990). However, as the n u m b e r of iterations increased, radial profiles of Uo slowly drifted Table 1. Data for geometries A and B (Nagata, closed and free-surface) Impeller diameter, D (8-blade paddle) Liquid depth, H Tank diameter, T Height of impeller from bottom, C Blade length, a Blade width, b

0.3 m 0.6 m (2D) 0.6 m (2D) 0.3 m (1D) 0.14 m 0.06 m

M. CIOFALOet al.

3564 i l l l l l l l l l l i l l l i l l l l l i l l l i l i l l l l l l l l l l l i l l l n l l i l l l

lillillilllllilinlllilill

lUliilillinllilililllllli

lllllllllllllllllllllllll lllllllllllllllllllllllll Illllllllllllllllllllllll nllllllllllllllllllllllll lllllllllllllllllllllllll Illllllllllllllllllllllll Illllllllllllllllllllllll lllllllllllllllllllllllll lllllllllllllllllllllllll lllllllllllllllllllllllll Ullllllllllllllllllllllll Illllllllllllllllllllllll lllllllllllllllllllllllll lllllllllllllllllllllllll Illllllllllllllllllllllll ~ ) ~ m m m m m m l m m i m i m ~;~ilnnimmmmulum ~ ~ m n l l l n l n m m m m n

~ ~ m m m m m m m m m m m m n ~ ~ m u m m m m m m m m n m u

iuunuuuuuunnnnnuiumnnnunn ninuunnuuuuunuuunumnuunun iuuninununuunuuuinmnuunun nnuuuuuunJunuuuuunmnuuunn nnnnnnnnnnuununuunmuunnnu nlnlunuluiluniilllninuini lllllllllllllllllllllllll Illllllllllllllllllllllll Illllllllllllllllllllllll nllllllllllllllllllllllll lllllllllllllllllllllllll Illllllllllllllllllllllll lllllllllllllllllllllllll lllllllllllllllllllllllll Illllllllllllllllllllllll illllllllllllllllllllllll lllllllllllllllllllllllll Illllllllllllllllllllllll lllIIIIIIIIIHIIIlIlIlIII lllllllllllllllllllllllll

Fig. 6. Computational grid used for geometry A.

towards rigid-body rotation (RBR) profiles (uo = o3r), while the corresponding secondary motion (double circulation loop) decreased in intensity and shrank in size until it was confined to only a small region around the impeller. This is clearly shown in Fig. 7(c) and (d), which are relative to 9000 SIMPLEC iterations. At this point, the mass source residual was very low (about 10 -5 times the total mass flow in the tank) but the solution still continued to change slowly approaching the RBR solution. Much more satisfactory predictions were obtained by using the DS turbulence model. Figure 8(a) reports radial profiles of uo computed after 9000 SIMPLEC iterations; these exhibit a clear maximum and are in agreement with the simplified theory of Nagata based on the concept of "forced" and "free" vortices, see Section 2. The corresponding circumferentially-averaged flow field in an axial plane is shown in Fig. 8(b); it exhibits the expected double-loop circulation pattern. What is most important, the solution was fully converged in that it did not exhibit any further drift, although the mass source residual was comparable with that obtained after the same number of iterations by using the k-~ model. In order to assess the influence of the terms F,.j appearing in the Reynolds stress transport equations because of fluctuating Coriolis forces, see eqn (19), DS simulations were also repeated without these contributions. Only marginal differences were observed in the predicted profiles of u0, Fig. 8(c), as compared with those reported in Fig. 8(a). However, much more relevant effects were obtained on the secondary flow; this is shown in Fig. 8(d), which can be compared with Fig. 8(b). The radial jet is now broader and less intense, and induces significant secondary velocities

only in the peripheral region of the double recirculation loop. A residual lack of symmetry can be observed in the solution, which suggests the presence of instabilities and convergence difficulties. It can be concluded that the terms Fzj do have a significant impact on the predicted flow field, and should never be neglected. On the whole, these preliminary simulations relative to closed vessels show that, even without the additional complication of a free surface, only a second-order turbulence model can capture the complex interrelations between main and secondary flow, turbulent stresses, and inertial (rotation-induced) forces, and can thus describe correctly the actual flow field in unbaffled stirred tanks. In the following sections, where the case of free-surface vessels will be considered, only predictions based on the DS model (with the explicit inclusion of the terms F~j) will be discussed. 4.2. Free-surface problems: comparison with the classic results of Nagata The first free-surface case discussed here (referred to as geometry B in the following) is that of an unbaffled tank, stirred by an 8-flat blade paddle, mid-height located impeller, rotating at 1.2 rps (72 rpm), for which Nagata (1975) gives experimental flow data based on Pitot-meters. Apart from the free surface, the tank geometry is identical to that considered in Section 4.1 as geometry A and described in Table 1. The simulation started with a flat top surface and a grid identical to that used in the previous, closed vessel, simulations. Successive free-surface profiles and corresponding, progressively distorted, grids are shown in Fig. 9.

Turbulent flow in closed and free-surface unbaffled tanks stirred by radial impellers

....

till.'-"

3565

~l

' "'////'",tl , ttlllll/-~{| t tll/ZZl1-~.

}

fl

;.

~ ~ \ \\\\',.-/,

t, I

•

(a)

(b) U tip

,

,

r

/

.

t

.

! t

'

P p

o

~-~

'

'

"

-

Flit.

(c)

.

~

,

t

'

"

"

,

.

4

" .

"

"

-

-~',

.....

II~

I l~;'.,

t r ,

(d)

Fig. 7. k - e results for geometry A. (a) u o profiles, 1000iterations; (b) flow field, 1000iterations; (c) uo profiles, 9000 iterations; (d) flow field, 9000 iterations. It has to be observed that the use of the DS model in free-surface problems led to the development of "chequerboard" oscillations in the normal velocity component and in the Reynolds stresses near the top boundary. A similar difficulty arose when the closed (symmetric) tank discussed in the previous section was simulated by limiting the computational domain to one half of the vessel and treating the midplane as a plane of symmetry. The origin of these oscillations is not yet completely clear; presumably, it is related to the inadequacy of the Rhie-Chow algorithm to completely prevent even-odd cell de-coupling of velocity components when non-diffusive terms (gradients of Reynolds stresses) are present in the momentum equa-

tions. Here, the problem was simply "cured" by averaging over the last couple of cells below the free surface the velocity component normal to it. No oscillation was observed in other quantities. Predicted radial profiles of the tangential velocity Uo, circumferentially averaged and normalized with respect to U,iv = ~ D / 2 , are shown in Fig. 10 for different axial locations, and are compared with Nagata's experimental profiles. The computed free-surface profile is also reported and is compared with that predicted by eqns (6a) and (6b) using the value of ~c (0.75) given by the Yamamoto correlation, eqn (8). Computed profiles of Uo at different heights are very close to one another and in excellent agreement with

3566

M. CIOFALO et al.

iiiii! ...... T ooo°oo~ .°°°°o~

::!i:ii o°tttf

(a)

(b)

Utip

rm..~..o.d~ • ~G4J

"tl • it

,.&|~ddlll~ 9.~bbLb~,k~

~::::IIIlll

(c)

(d)

Fig. 8. DS results for geometry A (9000 iterations). (a) uo profiles; (b) flow field (terms F u included); (c) uo profiles; (d) flow field (terms F~j omitted).

Nagata's experimental results. Maxima of Uo occur at ~c ~ 0.78 (with the exception of the region near the impeller midplane, where they are slightly farther away from the axis); this value is very close to that predicted by the Yamamoto correlation. The shape and depth of the vortex are also satisfactorily reproduced. Axial profiles of the circumferentially-averaged radial velocity u, are shown in Fig. 11 for different radial locations. Corresponding profiles from Nagata (1975) are shown in an inset for comparison purposes. The overall flow pattern, characterized by a radial disc jet issuing from the impeller with inwardly-di-

rected flow away from it, is qualitatively reproduced. However, the lateral spreading of the radial jet is less marked than that reported by Nagata; also the predicted distribution of the inward-directed flow recirculating from the jet (which has its maximum intensity near the free surface) is different from that reported by this author (which corresponds to no radial motion on the free surface). Radial profiles of the circumferentially-averaged axial velocity uz are reported in Fig. 12; as above, Nagata's results are shown in an inset for comparison purposes. Again, some disagreement can be observed. In the central (forced,vortex) region r ~ re, the

Turbulent flow in closed and free-surface unbaffled tanks stirred by radial impellers llllllllllllllllllllllllll Illlllllllllllllllllllllll IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII

I|tll-i$l~ll|Illllll; .181111lllllllllllllllllI IIIII ill iliilIIllIIll immm~l ~ i I I ilimmmllil lnmullll|lllll Ill IIIIIIP itliIlalI nl I I illltlllltIllllllIIIlIlI iiIlIlllllllIIlIllllllll IllIlllllllllllllllIIIII IlmllllllllllllIIIliIIII

llllllllllllllllllllllllll IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII

::::::::::::::::::::::::: inlillplllllllllllIIIIII; iiliiiiilIlIIlIIIIl|llill iiiiiiiiiiiiiiiiiiiiiiiii iIiiiiiiIIIIIiiIIIIIIIIll iiiIiiIIIIIIIIIIIIIIIIIll iiiiiiIIIIIIIiiiiiIIIIIll w.-...........i.l.m.....,

IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII

I|||||ii|||||I||II|!||i|i iiiiii iiiiiiiiiiiiiiiii

i,... ....i!|H|||i!3|~ii~Eh:||| ::::::::::::::::::::::...

Iiiiil==lllllllllllllllll

Illllllllllllllllllllllll IIIIIIIIIIIIIIIIIIIIIIIII

llllmlmmllmmmilllllllmIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIII

IIIIIIIIIIIIIIlilllllllll IIIIIIIIIIIIIIIIIIIiiiiii IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIi IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIII

llllllllllllllllllllllllll IIIIIIIIIIIIIIIIIIIIIIIIlI

i i i i i i i[i!i i i i i (o)

H[ll][ll[lltll:llttlltltll

i[iiiiiii!!!ii!iiiiiiiiiill

(1)

LLL

Illlllllllilllll

:::::::::::::::::::::::::::

L|lllllllllllilllllllllllll ~ll[lllJIIIIl[lllllllllllll li[llllilllllllllllllllllll ~lllllllIIIIIIIIIIIIIIIIII [][l[lllllflllll[llllllllll [llllllllllllliIIIIIIIIIIII [[llllllllllllllllll

IIIIIIIIIIlilllllllllllllll [llllllllillllllilll[llllll IlIIIIIIIIIllll)lllllIl|lll illlllllllllllllll[lll[llll IIIIIIIIIIIIIIIIIIIIIIII

!llllllllllllllllllllIIIIII L~JIIIIIIIIIIIIIIIIIlilIIII Illlllllllllllllllllllllil IIIIIIIIJllllllIllllllllll lll[lllllIl]lllllllllIIIII llllllillIIIIIIIIIIIl[llll Illlliilllllllllllllllll ±~,,~,,,,,,J~,,,~,,,,~,tl Iliillllllllll[l[llillll[I IIIIIIit111111111111111111 IIIIIIIIIIIIIIIIIIIII1[11 ~lll[llIllIIIItlllllllllll

I[lllllllllltlllIIlllllll IltttltlllttllllttilltliilI IJJJJJJlIlJJJltJIJllllll#ll Illllllllllllllllllllll .~. l.l .[ [.l.l .l l.l .l l.l .[ l.l.l l.l.l l.l.l l.l. . . . . ]|llllllllllillllllllllll

L,,,,,,,,,,,,,,,,,,,IIIIIII

(z)

3567

~, I [ l l.l.l.l.l.l.[.l.l.l.l.l.l .l .l .l .E. i. l. l. l.i l l

III III]

ILlllllllllllllllllll[llll

Illlllllllllllllllllllllll llilllllillllllllllllllll IIIIIIIIIIIIIIl[l[llllllll 111[111111111111111111111 Illtlllllllllllllllllllill

(3)

Fig. 9. DS results for geometry B (open tank): evolution of the free-surfaceprofile and of the computational grid in successive "external" iterations.

upward-directed velocities reported by Nagata are not present in the simulated flow field. In the outer region, the predicted peaks of upward-directed velocity are more narrow and intense than the experimental ones. Finally, Fig. 13 reports a vector plot of the circumferentially-averaged flow field in a plane containing the axis. As shown also by the predicted velocity profiles reported in the previous figures, the whole forced-vortex region r ~< rc is characterized by a virtual absence of secondary flow. Experimentally-derived streamlines are reported by Nagata only for the top half of the tank, and are reproduced in an inset. The predicted location of the recirculation centre is farther from the midplane than reported by Nagata. In the forced-vortex region, this author indicates a weak counter-rotating circulation, coherent with the

upward-directed axial velocities visible in the inset of Fig. 12 but not reproduced by the present numerical simulations. An important engineering quantity for stirred vessels is the power consumption P, often expressed in dimensionless form as power number Np (see Notation). Nagata reports a number of empirical results and correlations expressing Np as a function of the Reynolds number and the geometry of vessel and impeller, both for unbaffled and baffled tanks. For the unbaffled, stirred vessel considered here, equipped with an 8-flat blade paddle, he reports Np = 0.95. In the present simulations, power consumption was computed as follows. First, the torque on the single blade included in the computational domain was computed from the pressure distributions on its upstream and downstream faces. This was multiplied by

3568

M. CIOFALO et al.

the number of blades to give the total torque acting on the impeller; the contribution from shear stresses on the shaft surface was neglected• Finally, the total torque was multiplied by the angular velocity o~ = N/(21r) to give the power P. For N = 1.2 rps (72 rpm), this gave Np ~ 1, in excellent agreement with the above figure from Nagata. A second important global quantity is the discharge flow rate Q, which can be made dimensionless as a pumping number NQ = Q/(ND3). For the geometry and rotational speed considered here, Nagata reports NQ = 0.34; the simulation yielded a very close value 0.37. The agreement of computed power and pumping numbers with the corresponding experimental values (within 5% and 9%, respectively) is quite satisfactory, considering that these are important engineering quantities and are also sensitive indicators of the overall accuracy of the flow and turbulence field predictions.

Fig. 10. DS results for geometry B. Computed radial profiles of uo (solid lines) are compared with experimental results from Nagata (1975)(dots). The computed free-surfaceprofile (solid line) is compared with that predicted by eqns (6) and (8) (dashed line).

U tip

....

[cmlsec]

blV~ / d~

V,

r.p.m.-2o o I , J

+20

,

J

Waler~

R"

so,,

,i

• l.::.'/i 4

8 121620 r

262925

[cm I

U tip

Fig. 11. DS results for geometry B: axial profiles of u,. The inset reports corresponding experimental results from Nagata (1975).

Turbulent flow in closed and free-surface unbaffled tanks stirred by radial impellers

~N=72r.p.m.

Z

20

" 15

3569

~~

.-"-'~/...___._~;~-~ 20 i, ~ ' ' [ 1 0,.,Vz I/ ' I-~,

| 10

~"

,'r

v

,

"'/I "Fo - 2 0

'

,,~J[,lllIlllE

10

-

20 29.25

r[em]

U tip

Fig. 12. DS results for geometry B: radial profiles of u=. The inset reports corresponding experimental results from Nagata (1975).

z

/" F;//7\

=

~

r

U tip

Fig. 13. DS results for geometry B: flow field. The inset reports qualitative streamlines from Nagata (1975).

M. CIOFALOet al.

3570

4.3. Comparison with laboratory results in the model tank Experiments were conducted in a model tank in order to obtain reliable data on the free-surface profile at various rotational speeds, to be used for the validation of the computational procedure described above. The geometry of the tank (geometry C) is summarized in Table 2. It was built of perspex and was stirred by a standard six-blade Rushton turbine, located at one third of the (average) height from the bottom wall. The working fluid was ambient-temperature water. A vertically-adjustable finger, transvers-

(a)

ing the tank diameter, was used for the assessment of the free-surface height (vortex shape). A high-speed video camera was also used for qualitative flow visualization by injecting dye at various locations. Table 2. Data for geometry C (model tank) Impeller diameter, D (Rushton turbine) Liquid depth, H Tank diameter, T Height of impeller from bottom, C Blade length, a Blade width, b

0.095 m 0.19 m (2D) 0.19 m (2D) 0.063 m (0.67D) 0.023 m 0.019 m

(b)

U tip

(c) Fig. 14. DS results for geometry C (model tank): flow fields. (a) 139 rpm; (b) 194 rpm; (c) 240 rpm.

Turbulent flow in closed and free-surface unbaffted tanks stirred by radial impellers 22

22

20

20-

18-

18-

3571

i 16-

"~16-

14-

14-

12 0

I

2S

I

I

.5 r (cm)

7.5

12 10

0

i

I

I

25

5 r (cm)

7.5

10

(b)

(a)

22 20-

18-

16-

. s S []

14-

12 o

I

I

2.5

5

715

1o

r (crn) (C)

Fig. 15. Free-surfaceprofilesfor geometry C (model tank). Symbols:experiments;solid line:DS predictions; dashed line: eqns (6)-(8). (a) 139 rpm; (b) 194 rpm; (c) 240 rpm.

The grid used for the numerical simulations of this geometry included 20 x 16 x 30 (r" 0" z) nodes. The predicted velocity fields on the plane containing the axis are shown in Fig. 14(a)-(c) for N = 2.31, 3.23 and 4 rps, respectively (139, 194 and 240 rpm). Computed and measured profiles of the free surface are reported in Fig. 15(a)-(c) for the same rotational speeds; profiles predicted by eqns (6)-(8) are also reported for comparison purposes. A satisfactory agreement with the experimental results can be observed for the profiles computed by the iterative technique described in Section 3; the only reservation concerns the underprediction of the vortex depth in the region close to the shaft, especially at the highest speed. The free-surface profiles computed by using Nagata's simplified theory and Yamamoto's correlation for ~c, eqns (6)-(8), are also in good agreement with the experimental data. Computed values of the power number, Np, are compared in Table 3 with classic experimental results reported by Rushton et al. (1950) for the closest geometry studied (H/D = 2.3, T/D = 2.33, C/D = 0.7) and for different rotational speeds. It should be

Table 3. Comparison of computed power numbers with Rushton's results rpm

Re

Fr

Nv(computed )

Np(Rushton)

139 194 240

23 200 32 300 40000

0.052 0.101 0.155

1.19 1.14 1.09

1.47 1.39 1.33

observed that, in the case of free-surface tanks, Rushton presented his data for Np as functions of both the Reynolds and the Froude numbers. Rushton's results are underpredicted by about 18% by the present numerical simulations. However, the dependence of Np upon the rotational speed is correctly reproduced. 5. CONCLUSIONS Numerical predictions of the complex fluid flow structure in mechanically stirred unbaffled vessels were performed by using a finite difference method in conjunction with alternative (k-~ and Differential Stress) turbulence models, implemented in the computer code Harwell-FLOW3D (Release 2). Simulations

M. CIOFALO et al.

3572

were performed in the rotating reference frame of the impeller, without any recourse to empirical data. The case of free-surface tanks was also investigated; the vortex formation at the free surface was dealt with by an iterative method which used the code's capability to treat non-orthogonal, body-fitted grids. Predictions were compared with classical results of Nagata and Rushton and with measurements of the free-surface height conducted in a model tank. When the eddy-viscosity k-e model was adopted, results were disappointing; after a large number of iterations, the flow field approached rigid-body rotation and secondary motions tended to disappear. On the contrary, a satisfactory agreement was obtained for the overall flow patterns, the height of the free surface (vortex shape), and global parameters such as the power and pumping numbers, by using the differential stress turbulence model and including the effects of fluctuating Coriolis forces. To the authors' knowledge, these are among the first simulations of flow fields in mixing tanks which make use of a second-order turbulence closure model and which explicitly compute the free-surface profile. The results show that this apparently simple geometry actually presents a very complex turbulence structure and provides a severe benchmark for turbulence models and computational methods. Acknowledgement--Harwell-FLOW3D, Release 2.3 is a research version of the commercial package CFDS-FLOW3D and is a proprietary code of AEA Technology, UK. NOTATION a,b,C,d,D, H,T,W A C1, C2, Cu, Ck, C~ Fr, Fo, F~ Fr g h k N Ne NQ n P P

Q Re r, O, Z Ur, blO, Uz

u'r, u'o, u" Utip

tank and impeller geometrical parameters, m; see Figs 1 and 2 tank cross sectional area, m 2 constants of the k-e model body forces, N / m 3 Froude number (N2D/g) acceleration due to gravity, m/s 2 local free-surface height, m turbulent kinetic energy, J / K g rotational speed of the impeller, rps, s power number [P/(pN3DS)] pumping number [Q/(ND3)] number of blades in the impeller true (static) pressure, N / m 2 modified pressure (p + (2/3)pk), N / m 2 power consumption, W discharge flow rate, m3/s Reynolds number (pND2/#) cylindrical coordinates mean velocity components, m/s fluctuating velocity components, m/s impeller tip speed (reference velocity) (o~D/2), m/s

1

Greek letters fl underrelaxation factor for free-surface computations

]A,~ T, l~eff

P zij (D

dissipation rate of k, W/kg laminar, turbulent and effective viscosities, Ns/m 2 density, kg/m 3 components of the traceless total stress tensor, N / m 2 angular velocity, rad/s REFERENCES

Armenante, P., Chou, C.-C. and Hemrajani, R. R., 1994, Comparison of experimental and numerical fluid velocity distribution profiles in an unbaffled mixing vessel provided with a pitched-blade turbine. Instn Chem. Engng Symp. Ser. 136, 349-356. Bode, J., 1994, Computational fluid dynamics applications in the chemical industry. Comput. chem. Engn 9 18, $247$251. Brucato, A., Ciofalo, M., Grisafi, F. and Rizzuti, L., 1989, Application of a numerical fluid dynamics software to stirred tanks modelling, In Supercomputin 9 Tools for Science and Engineerin 9 (Edited by Laforenza, D. and Perego, R.), pp. 413-419. Franco Angeli, Milano. Brucato, A., Ciofalo, M., Grisafi, F. and Rizzuti, L., 1990, Computer simulation of turbulent fluid flow in baffled and unbaffled tanks stirred by radial impellers. Proceedings of the International Conference on Computer Applications to Batch Processes (CATBP 90) (Edited by Dovi, V. G.), pp. 69-86. Cengio, Italy. Brucato, A., Ciofalo, M., Grisafi, F. and Micale, G., 1994, Complete numerical simulation of flow fields in baffled stirred vessels: the inner-outer approach, lnstn Chem. Engng Syrup. Ser. 136, 155-162. Burns, A. D. and Wilkes, N. S., 1987, A finite-difference method for the computation of fluid flows in complex three-dimensional geometries. UKAEA Report AERE-R 12342, Harwell, U.K. Clarke, D. S. and Wilkes, N. S., 1989, The calculation of turbulent flows in complex geometries using a differential stress model. UKAEA Report AERE-R 13428, Harwell, U.K. Costes, J. and Couderc, J. P., 1988, Study by laser Doppler anemometry of the turbulent flow induced by a Rushton turbine in a stirred tank: influence of the size of the units (Parts I&II). Chem. Engn 9 Sci. 43 (10), 2754-2772. Cresta, S. M. and Wood, P. E., 1991, Prediction of the three-dimensional turbulent flow in stirred tanks. A.1.Ch.E.J. 3, 448 460. Dong, L., Johansen, S. T. and Engh, T. A., 1994, Flow induced by an impeller in an unbaffled tank--I. Experimental. Chem. Engn9 Sci. 49 (4), 549-560. Fokema, M. D., Cresta, S. M. and Wood, P. E., 1994, Importance of using the correct impeller boundary conditions for CFD simulations of stirred tanks. Can. J. Chem. Engn9 72, 177 183. Gosman, A. D., Issa, R, I., Lekakou, C., Looney, M. K. and Politis, S., 1989, Multidimensional modelling of turbulent two-phase flow in stirred vessels. Proceedings of the A.I.Ch.E. Annual Meetin 9. San Francisco, CA. Grisafi, F., Brucato, A. and Rizzuti, L., 1994, Solid liquid mass transfer coefficients in mixing tanks: influence of side wall roughness. Instn Chem. Engn9 Symp. Ser. 136, 571-578. Hinze, J. O., 1975, Turbulence, 2nd Edition. McGraw-Hill, New York. Kightley, J. R., 1985, The conjugate gradient method applied to turbulent flow calculations. UKAEA Report CSS 184 (HL85/1584), Harwell, U.K. Launder, B.E., 1989, Second-moment closure and its use in modeling turbulent industrial flows. Int. J. Number. Meth. Fluids 9, 963-985. Launder, B. E. and Spalding, D. B., 1974, The numerical computation of turbulent flows. Comp. Meth. Appl. Mech. Engnq 3, 269-289.

Turbulent flow in closed and free-surface unbaffled tanks stirred by radial impellers Launder, B. E., Reece, G. J. and Rodi, W., 1975, Progress in the development of a Reynolds-stress turbulence closure. J. Fluid Mech. 68, 537-566. Lug, J. Y., Gosman, A. D., Issa, R. I., Middleton, J. C. and Fitzgerald, M. K., 1993, Full flow field computation of mixing in baffled stirred vessels. Trans. Instn Chem. Engrs 71, Part A, pp. 342-344. Mazzarotta, B., 1993, Comminution phenomena in stirred sugar suspensions. A.I.Ch.E. Symp. Ser. 89 (293), 112 117. Middleton, J. C., Pierce, F. and Lynch, P. M., 1986, Computation of flow fields and complex reaction yield in turbulent stirred reactors and comparison with experimental data. Chem. Engn9 Res. Des. 64, 18-22. Murthy, J. Y., Mathur, S. R. and Choudhury, D., 1994, CFD simulation of flows in stirred tank reactors using a sliding mesh technique, lnstn Chem. Engng Syrup. Ser. 136, 341-348. Nagata, S., 1975, Mixing: Principle and Applications. Wiley, New York. Ranade, V. V. and Joshi, J. B., 1990a, Flow generated by a disc turbine: Part I. Experimental. Trans. lnstn Chem. Engng 68(A), 19 33. Ranade, V. V. and Joshi, J. B., 1990b, Flow generated by a disc turbine: Part II. Mathematical modelling and comparison with experimental data. Trans. Instn Chem. Engrs 68(A), 34 50.

3573

Rhie, C. M. and Chow, W. L., 1983, A numerical study of the turbulent flow past an aerofoil with trailing edge separation. A.I.A.A.J. 21, 1525-1532. Rushton, J. H., Costich, E. W. and Everett, H. J., 1950, Power characteristics of mixing impellers (Parts I and If). Chem. Engng Prog. 46, 395 403 and 467 476. Smit, L. and D0ring, J., 1991, Vortex geometry in stirred vessels. Proceedings of the 7th European Congress on Mixin(], Vol. 2, pp. 633-639. Bruges, Belgium. Stoots, C. M. and Calabrese, R. V., 1989, The flow field relative to a stirred tank turbine blade. Proceedings of the A.I.Ch.E Annual Meeting. San Francisco, CA. Stone, H. L., 1968, Iterative solution of implicit approximations of multidimensional partial differential equations. SIAM J. Numer. Anal. fi, 530-558. Takeda, H., Narasaki, K., Kitajima, H., Sudoh, S., Onofusa, M. and Iguchi, S., 1993, Numerical simulation of mixing flows in agitated vessels with impellers and baffles. Comput. Fluids 22 (2/3), 223-228. Van Doormal, J. P. and Raithby, G. D., 1984, Enhancements of the SIMPLE method for predicting incompressible fluid flows. Numer. Heat Transfer 7, 147-163. Wu, H. and Patterson, G. K., 1989, Laser-Doppler measurements of turbulent-flow parameters in a stirred mixer. Chem. En,qng Sci. 44 (10), 2207-2221.