Three-dimensional turbulent flow in agitated vessels with a nonisotropic viscosity turbulence model

Three-Dimensional Turbulent Flow in Agitated Vessels with a Nonisotropic Viscosity Turbulence Model

S. Y. JU, T. M . MULVAHILL* and R. W. PIKE**

Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803

Solutions of the time-averaged equations of motion with a nonisotropic k--E model were developed for the three- dimensional turbulent flow field in turbine stirred tanks. These results were validated with the measurements of three velocity components with a hot wire anemometer and literature data. The nonisotropic turbulence model considered the rotation and curvature effect of the turbulence with a turbulent Richardson number term and accounted for the important three-dimensional effects through the nonisotropy of the viscosity. Also, it was found that a frequently used isotropic k-E turbulence model did not describe this three-dimensional turbulent flow field.

. .- ~~~~~~~

On a resolu les equations de mouvement comprenant un modkle non isotrope k-6 et moyennCes dans le temps pour etudier I’tcoulement tridimensionnel dans des reservoirs agites par une turbine. Les resultats ont ete validks par des mesures des trois composantes de vitesse 2 I’aide d’un anemornetre a fil chaud et par des donnCes publiees. Le modkle de turbulence non isotrope considere I’effet de rotation et de courbure de la turbulence avec un terme turbulent du nombre de Richardson et tient compte des effets tridimensionnels importants par la non isotropie de la viscosite. De meme, on a trouvC que le modele de turbulence isotrope k - E frkquemment utilisk ne convenait pas pour decrire cet ecoulement turbulent tridimensionnel.

Keywords: mixing, turbulent flow, nonisotropic viscosity, turbulence model.

gitated vessels are widely used in the chemical, A pharmaceutical and petroleum refinery industries for mixing and chemical reactions. An impeller is used to generate the turbulent flow field which minimizes temperature and concentration gradients, suspends solid particles, maintains emulsions, disperses a gas phase into a liquid, and combines chemical reactants.

The description of the turbulent flow field is required for use with the other transport equations to predict energy and mass transfer rates in agitated vessels. Also, a solution of the fluid dynamics can be used in place of experimental measurements for flow patterns, shear rates, and shear and normal stresses for design purpose.

To solve the time-averaged equations of motion for the turbulent flow field in an agitated vessel, a turbulence model is required for the Reynolds stress terms, e.g., the standard k-E model (Launder and Spalding, 1972). However, these turbulence models only provide approximations for the Rey- nolds stresses, and no universal turbulence model exists. Consequently, it is necessary to develop a turbulence model to accurately describe a particular flow field, in this case the one in agitated vessels.

In agitated vessels, the rotating turbine generates an impeller stream in the radial and tangential directions. This tangential jet flow divides at the wall, and the flow then cir- culates back into the impeller region. Thus, the flow field may be approximated by two regions. One is the region of the impeller stream, and the other is the recirculating bulk region. The tangential component of the flow resulting from the rotating impeller is converted to axial flow by the baffles which make the flow field in the region away from the impeller three-dimensional. Consequently, the turbulence model must account for the rotational effect on turbulence and the three-dimensional flow in the vessel.

To date, this three-dimensional turbulent flow has been simulated mostly by using the two-dimensional axisymmetric time-averaged equations of motion with different isotropic viscosity turbulence models. The only three-dimensional

*Arnoco Chemicals Corporation, Texas City, Texas. **Author to whom correspondence should be addressed

solution was by Middleton et al. (1986) who reported a solution with the standard k-E turbulence model and used the con- version of a simple competitive-consecutive reaction system to validate the performance of his solutions in three different sizes of turbine stirred tanks. For two-dimensional solutions, Platzer (1981) used the standard k-E model, Harvey and Greaves (1982) used the standard k-E model with different model coefficients, and Placek et al. (1986) used an isotropic three-equation model. Smith (1985) applied the two- dimensional flow field solution by Placek et al. (1986) to describe a dispersed phase system in the vessel. The comparisons of these two-dimensional solutions with experimental data have shown only qualitative agreement. In summary, there has been no solution that gave a satisfac- tory description of the turbulent flow field in these baffled tanks. Consequently, the purpose of this research was to describe accurately this three-dimensional flow field, and this required a nonisotropic viscosity turbulence model.

The previous solutions had not considered the effect of rotation on turbulence and the three-dimensional flow induced by the baffles except that a “drag term” was purposely added in the angular momentum equation in the two-dimensional solution by Harvey and Greaves (1982) to reduce the tangential velocity and increase the axial velocity, Accordingly, a turbulence model which can account for the three- dimensional characteristics of this flow field was needed. The nonisotropic viscosity model developed in this study should be beneficial to studies of the turbulent flow field in other types of mixing vessels such as tubular reactors, Berker and Whitaker (1978); gas stirred reactors, Salcudean et al. (1985); and jet-stirred reactors, Liu and Barkelew (1986).

Governing equations

The Reynolds stress tensor in the time-averaged equations of motion can be described by Reynolds stress transport models or turbulent eddy viscosity models. The Reynolds stress transport equations are a set of six partial differential equations containing correlations among fluctuating quanti- ties that have to be approximated in terms of mean quanti- ties. second-moment correlations and characteristic time

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 68, FEBRUARY, 1990 3

scales to give a closed set of seven partial differential equations for the Reynolds stress tensor and the kinetic energy dissipation rate, as detailed by Launder (1985). The turbulent eddy viscosity models relate the individual Rey- nolds stresses to the mean flow gradients, and the k-c model is the most general and widely used of these models. The eddy viscosity models ensure the Reynolds stresses follow the mean strain rate and give qualitatively plausible solutions whereas Reynolds stress models may exhibit unsuspected sin- gular behavior according to Bradshaw et al. (1981). Approx- imations to the Reynolds stress models which convert the partial differential equations to algebraic equations (algebraic stress models) are usually made for computational simplicity, and numerical solutions are required for these algebraic equations and two partial differential equations for the turbulent kinetic energy and dissipation rate. According to Launder (1985) convergence is reported to be more difficult with algebraic stress models than with k-E models. Consequently, the k - E model was used here, since there has not been any generally demonstrated superiority of the Reynolds stress models over the k-c models reported in the literature.

Nonisotropic k-E models have six different components for the turbulent eddy viscosity. To evaluate these components Lilley and Chigier (1971a, 1971b, 1972) and Lilley (1973, 1974) developed nonisotropic exchange coefficients to describe a two-dimensional turbulent swirling jet. Based on these works, Schetz (1980), Gupta et al. (1984), and Gupta and Lilley ( 1985) suggested using one of the six components of the effective viscosity (sum of the molecular and turbulent viscosities) as a reference, e.g., p r z . Then the other components were obtained from the following equation:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . /L,, = przlo,, (1)

where a,, is a viscosity ratio, and the subscript ij represents rr, u, 00, r0 and ez. If the five viscosity ratios are equal to one, the nonisotropic model reduces to an isotropic model. In this research, a set of viscosity ratios were determined for the three-dimensional turbulent flow in turbine agitated vessels.

TIME-AVERAGED EQUATIONS OF MOTION

Based on the concept of the nonisotropic viscosity components, the three-dimensional time-averaged equations of motion in cylindrical coordinates are as follows: Continuity equation:

I a(rV 1 aw au r ar r ae az . . . . . . . . . . . . . . + - - + - = o (2) - -

Radial momentum equation:

a ( p r z F) 2 P r z aw 2 - - + - p r z V "1 ae r2 0 0 0 ae aefi r2 az

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

Angular momentum equation:

. . . . . . . . . . . . . . . . . . . . . . . . . . [2 3 . . . . (4)

Axial momentum equation:

a ] ::r i a + --(WU) + -((u2) r a0 az

= --

The effective viscosity prz in the above equations is defined as the sum of the molecular and turbulent viscosity, i .e. ,

(6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . prz = p + pr

where pr is determined by a turbulence model; see Equa- tion (9).

NONISOTROPIC VISCOSITY TURBULENCE MODEL

The turbulent viscosity is determined from the solution of the transport equations for the turbulence kinetic energy k

4 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 68, FEBRUARY, 1990

and its dissipation rate E . The nonisotropic k-c model in cylindrical coordinates used for the turbulent flow in agitated vessels is expressed as follows: Equation for turbulence kinetic energy:

p [ t i ( r l % ) + --(Wk) i a + r a0

-1 ak + ; [ (i - 1) 2 3 + G,,, - Pc

ae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

Equation for dissipation rate of the turbulence kinetic energy:

a I - : , " , - - ( ~ V E ) + - - ( W E ) + - ( U E ) - - - i a l a

' [ r a r r a0 az

[C~G,,,, - C, (1 - C , R i , ) p ~ l . . . . . . . . . . . . . . (8)

The turbulent viscosity p, is defined as:

(9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 p, = C,pk / c

Other turbulence model parameters are given as follows:

c, = 0.09(1 + 20.5/R,) [ 1 - exp(-0.0165Rk)]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

C1 = 1.44 [ l + (0.05/(C,,/0.09))3] . . . . . . . . . . . (1 1)

C, = 1.92 11.0 - exp(-R:)] . . . . . . . . . . . . . . . (12)

R, = pk2/pE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Rk = pk"'y,,,/p (14)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c, = 0.2 (15)

a k = 0.87 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a, = 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

+ 1 (F+ y)2 + (F + E)2] (19) 0 8 z

According to Launder et al. (1977) and Sharma (1979), the rotational effect on turbulence is incorporated in the term with the turbulent Richardson number, Ri, in Equation (8). When Ri, is negative, the angular momentum of the mean flow decreases with radius; and the term with Ri, in Equa- tion (8) will decrease the energy dissipation rate thus increasing the turbulence kinetic energy. Also, the turbulent viscosity will become larger according to Equation (9). Like- wise, a positive value of Ri, will diminish the turbulent viscosity. When the turbulent Richardson number is equal to zero, Equation (8) reduces to the €-equation in the standard k-E model.

The expressions for C,, C, and C,, i.e., Equations (10-12), were suggested by Lam and Bremhorst (1981) for flow throughout the fully turbulent, transitional and laminar regions. This k-E model with coefficients given by Equations (10-12) does not require the use of the wall function method or the introduction of the additional term in the k and E equations for the low flow region. Thus, they appear to be par- ticularly suitable for the flow in agitated vessels where the flow is circulating from the higher velocity impeller stream through the lower velocity region in the rest of the tank.

The nonisotropic viscosity ratios appearing in Equations (3-5) and (7-8) were obtained using the procedure of Lilley (1973, 1974) who expressed the viscosity ratios for a swirl jet as a function of the local swirl number. However, those expressions could not be used in this study without modifi- cation and adjustment. The swirl jet studied by Lilley (1973, 1974) moved in the axial direction, but the impeller stream in this study was discharged in the radial and tangential directions of the cylindrical coordinate system of the stirred tank. Also, the impeller stream was affected by the wall, baffles, and the recirculating flow in the bulk region. As discussed below, Ju (1987) used the concepts of viscosity ratios and swirl number to represent the nonisotropic viscosity components and showed that the following values for the viscosity ratios were the ones to be used for the turbulent flow in turbine-agitated vessels:

u , ~ = uez = urr = uo8 = uzzz = 1.0 in bulk region . (20)

ar8 = a,, = 008 = uzz = 0.7 in impeller stream . . . (21)

agz = I + 2 ~ ; ' ~ in impeller stream . . . . . . . . . . . . (22)

The impeller stream in Equations (21-22) is defined as the region bounded by planes through the top and bottom of the turbine blades, i.e.,

1/2(Hl - Hb) I x,. I 1/2(H, + Hh) . . . . . . . . . (23)


where x , is the axial distance measured from the tank bottom, H I is the height of the fluid in the tank, and Hb is the width of the blades of the impeller. The remaining region in the tank is the bulk or recirculation region.

The values of 1 .O for the viscosity ratios in Equation (20) mean that isotropic viscosity is assumed for the turbulent flow in the bulk region of agitated vessels. The value of 0.7 for the viscosity ratios in Equation (21) was recommended by Lilley (1974) for the free swirling jet and is adopted here for the impeller stream.

In Equation (22) S, is the swirl number for the tangential jet from the rotating turbine and was defined by Ju (1987) as:

S, = ( I O / D , ) ( G ~ / G , ) ( 1 / 2 ~ / r ) ’ . . . . . . . . . . . . . . (24)

In the above equation, G, is the radial flux of radial momentum, Go is the radial flux of angular momentum, and D, is the diameter of the impeller. G, and Go are defined by:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . G, = J,e,pV2rd~ (25)

Go = {Jnp V2 Wr2dz . . . . . . . . . . . . . . . . . . . . . . . . . (26)

where the integral limit ‘7et” means that the integrals are evaluated over the impeller stream, defined by Equation (23). In Equation (24) S, is a function of l l r3 which leads to lower tangential velocities than those obtained from the corresponding isotropic viscosity turbulence model as will be shown later. The other two velocity components are affected also through the viscosity ratios and the swirl number. This description of the nonisotropic viscosity is incorporated in the conservation equations for an accurate representation of the three-dimensional flow field.

GENERAL FORM

The conservation equations for the flow field in a baffled vessel include the continuity and momentum equations and the nonisotropic k-E turbulence model. As conservation equations, they can all be put in the following general form:

where 4 represents the dependent variable or property, I’4 is the diffusive transport coefficient for 4 and S, is the source term for the generation of the property #. #, rQ and S, are listed in Table 1 for the three-dimensional conservation equations and nonisotropic model.

BOUNDARY CONDITIONS

The boundary conditions required for the solutions of the governing equations for the turbulent flow in agitated vessels are shown in Figure 1 for the r-z plane. These include no slip at the tank wall, the tank bottom and the baffles and no shear at the air-liquid interface. k and c are assumed to be

zero at the tank wall and tank bottom, and their gradients in the z-direction at the air-liquid interface are zero. At the shaft the axial and radial velocity components are zero, and the angular velocity component is equal to the velocity of the surface of the shaft. Also, there is no flux across the shaft and gradients of other variables are zero. At the centerline of the tank below the impeller, the radial and angular velocity components are zero, and all of the gradients of other variables in the r-direction are zero because of symmetry. At the impeller, the flow from the blades is described as a tangential jet (DeSouza and Pike, 1972). The three velocity components at the impeller tip are given as follows:

V = V,, [ l - tanh2(v/2)] . . . . . . . . . . . . . . . . . (28)

2r2 - a2 tanh(7) - 9 [ 1 - tanh2(v/2)1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (29)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W = V tan0, (30)

where

V,, = 1/2A(b/r3)”2(r2 - a2)1’4 . . . . . . . . . . . . . (31)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 = bzlr (32)

. . . . . . . . . . . . . . . . . 0, = tan-‘ [a/(r2 - (33)

The three parameters in this explicit model, i.e., a, b and A, were obtained from velocity measurements in the impeller stream and they are:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b = 12.621 (34)

a = 0.06924 (Dr - DI) I DT . . . . . . . . . . . . . . . . . (35)

A = 1.1436 [NDjl(R2 - a2)1/410.8337 . . . . . . . . . (36)

The gradients of k and E with respect to r at the blade tip were taken as zero. This was based on the experiments of Mujumdar et al. (1970) and Gunkel and Weber (1975) who showed slight changes in the turbulence parameters in the vicinity of the impeller. Also, the theoretical study of Placek and Tavlarides (1985) obtained the same results. Harvey and Greaves (1983) used this boundary condition in their axisymmetric solution, and Placek et al. (1986) used an equivalent boundary condition. The values of k and E at the impeller tip were thus determined from the solution of the k and E equations.

For the flow entering the top and bottom of the impeller, a material balance was used to determine the axial velocity as shown in the Appendix. The tangential velocity was assumed to be equal to the velocity of the surface of the rotating impeller. The gradients with respect to z of radial velocity, k and E were assumed to be zero at the top and bottom of the impeller. As will be demonstrated, the solutions of the governing equations with the above-mentioned boundary conditions provide a reasonable representation of the experimental data.


TABLE 1 Variables 4, and S, of the Three-Dimensional Conservation Equations with the Nonisotropic k - E Model.

+J- r;+- 1 au --)'. (f.3'1 0 0 2 r a8

Finite domain solution of the governing equations

The finite domain method has been effective in obtaining solutions for turbulent flow fields (Gosman et al., 1969; Patankar, 1980; Spalding 1981). Using the general form of the transport equations for turbulent flow in a baffled stirred tank, Equation (27), discretization equations were developed for the control volume shown in Figure 2. In this method, the calculation domain was divided into a number of six-sided finite subdomains surrounding each centrally located grid point. The discretization equations were derived by integrating Equation (27) over these subdomains. Then the convection and diffusion terms in the transport equations were approximated using the power law scheme of Patankar (1980), and the source term, S, in Table 1, was linearized, Ju (1987).

Because of symmetry associated with the four uniformly positioned vertical baffles, a quarter of the mixing tank was selected as the calculation domain. As shown in Figure 3, variables and their derivatives on boundary K = 2 are the same as those on K = KPT- 1. In the numerical solution rpk= = 4K=KPT-2 and r # ~ ~ = ~ = 4K=KPT were used to satisfy the symmetry conditions, where 4 ~ ~ 3 and r # J K = K P T - 2 were obtained from the latest computation.

The set of discretization equations was solved using the SIMPLE algorithm of Patankar and Spalding (1972). The solutions were obtained from FORTRAN programs run on the FPS-264 vector computer with an IBM 3084 as its front end processor. Results from two-dimensional solutions were used as the initial estimates for the iterative procedure for the three-dimensional computations.

Solutions were obtained for the three tank systems given in Table 2. The three tank systems had the standard tank configuration with the same tank diameter, two different impeller diameters, and three different speeds. The impellers were centrally located in the tank. Two non-uniform grids 26 x 30 x 25 and 23 X 30 x 25 shown in Table 2 were used where the grid points were distributed nonunifonnly to have a finer grid near the tank wall and the impeller. Also, they were arranged to coincide with the hot wire anemometer measurements which were used for validation of the theoretical results. These grids gave two sets of 92,736 and 8 1,144 nonlinear algebraic equations obtained from the discretization of the governing equations which were solved iter- atively using the SIMPLE algorithm. Typically, a solution required about 650 iterations to converge from a good starting point provided by a two-dimensional solution. The CPU time

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING. VOLUME 68, FEBRUARY, 1990 I

9 = 0

Shaft

a9 -=O except U = O al.

a@ dr

- = O e x c e p t U = O v =o W = 2 7 r r N

v = o I I I I I

w = 0

a9 -I 0 dr

except U,V,W from tangent i a I je t model

C Tank Wal l

b = 0

Figure 1 - Boundary conditions for flow in an agitated vessel.

U i * j * k

Figure 2 - Three-dimensional finite domain for a main grid point.

listed in Table 2 showed that the 26 X 30 X 25 grid consumed more CPU time because of the higher impeller speed and it had more equations than the 23 x 30 x 25 grid. Comparing solutions from the two grids at the same conditions gave an

K = 3 K= 2

K = I

U = 1

K= 2 (e = 0)

Figure 3 - Calculation domain in the r-0 plane.

TABLE 2 Three Stirred Tank Systems

Impeller Impeller Tank Grid CPU time on Tank speed diameter diameter size FPS-264

System (rpm) (min) (min) (r-z-0) (minutes)

1 250 76 292 2 6 x 3 0 ~ 2 5 223 2 150 102 292 23X30X25 153 3 200 102 292 2 3 x 3 0 ~ 2 5 170

estimate of accuracy which was at least three significant figures, and this was comparable to other comparisons made with two-dimensional solutions. Ju (1987).

The SIMPLE algorithm required the residual of the continuity equation to be zero as the criterion for convergence. A value of the residual of was used for entire calculation domain, and was used for each subdomain. Additional criteria of convergence required lop6 for the residuals of the momentum equations and lop5 for the residuals for the k and E equations with any subdomain. The residuals of the transport equations were determined after normalizing the other terms in the discretization equations with the central grid point term. Also, the relative change of lov4 or smaller was required from iteration to iteration for the variables at any grid point.

Experimental measurements

The impeller stream region has been studied extensively, but there have been limited measurements on the three- dimensional flow in the bulk region of the tank. Therefore, precise velocity measurements were required for this region over the range of conditions shown in Table 2. The three velocity components of the flow in the bulk region were measured with a Thermo-Systems Model 1053B constant temperature anemometer using a Model 1294 probe with three sensors. The sensor wires were tungsten and were mounted on the end of the 18-inch probe shaft. They were mutually perpendicular and each individual sensor was at a 54.7" angle from the probe shaft. Colorless dimethyl silo- cone fluid of 5.0 centistokes viscosity was used as the liquid in the three tank systems shown in Table 2. This noncon- ducting fluid served as an excellent medium for the hot wire


Results and discussion

/ / / I I t

\ \ \ i \ I

Figure 4 - The normalized velocity vectors on the 45" r-2 plane in Tank System 1.

probe since no electrolysis occurred from the heated wire. There were occasions when tiny, lint type particles would suddenly collect on a wire changing the reading dramatically. These were removed by dipping the wire in a chlorinated solvent.

The probe entered from the top of the tank and was mounted on a slotted arm which rotated. Thirteen axial measurements were taken at eight radial positions in each of the three tank systems on the 45" plane which was centered between two baffles. Also, measurements were taken along the tangential direction at two locations of r and z in each of the three tank systems.

The probe sensors were individually calibrated in a rotating vessel where the velocity was known, and King's Law calibration curves were obtained. The three velocity components in cylindrical coordinates were obtained from the three cooling velocities with yaw angle transformations. The error in velocity measurements was found to be 5 % based on the standard deviation, Mulvahill (1979).

First, velocity profiles from the solution of the conservation equations with the nonisotropic model are presented in the r-z plane and r-8 plane. This illustrates the magnitude and direction of the flow in the bulk region and in the impeller stream. Then the solutions of the conservation equations with the nonisotropic and isotropic models are compared with experimental data from this research and other investigators. It is found that the nonisotropic model is required to accurately describe the velocity field. The results are presented in terms of the following nondimensionalized variables based on impeller speed Nand impeller diameter D I , as has been done by other investigators.

r' = r / ( D , / 2 ) z ' = z / ( D 1 / 2 )

k' = kl(N2D;) E ' = c / (N3D;)

The axial distance z is measured from the impeller plane with positive values above the plane and negative values below it. The three velocity components were nondimensionalized by the tip velocity of the impeller (VnP = r N D I j .

TURBULENT FLOW FIELD RESULTS

Figure 4 shows the velocity vectors from the nonisotropic model solution for the vertical plane centered between two baffles in Tank System 1 . The magnitude and direction of each velocity vector in the plane are (V2 + U2)li2 and tan-' (UIV) respectively. These velocity vectors shown were also normalized with respect to the impeller tip velocity Vlip = r N D I . As shown in the figure, the radial velocity components dominate in the impeller stream and have a maximum magnitude of about 0.6Vlip near the surface of the impeller. The radial motion of the flow turns to the axial direction in the region near the wall. In this wall region the maximum magnitude of the velocity is about 0.15Vlip . In the regions near the top and bottom of the vessel, radial motion is relatively dominant and a maximum magnitude of the resultant velocity is about 0.1 Vtb. The flow near the centerline of the tank is mainly in the axial direction pointing back to the impeller region with a maximum magnitude of about 0.2Vlip. Also, the velocity vectors adjacent to the impeller blades are pointing toward the impeller stream, showing that the fluid is entrained by the impeller stream. In addition, the flow in the tank is asymmetric about the impeller plane because there is a rotating impeller shaft and free surface in the upper half of the tank and only a no-slip tank bottom in the lower half. However, there is symmetry in the jet zone close to the impeller. The flow from the rotating impeller dominates this region and overrides the effects of the flow from the bulk regions.

Figures 5 and 6 show the velocity vectors in two cross sec- tions of the tank, one in the impeller stream and one in the bulk region, respectively. In this case the ma nitude and direction of the velocity vectors are (W2 + V ) and tan-' ( W / V respectively. These velocity vectors were also nondimensionalized with respect to Vtb. As shown in Figure 5, the fluid in the impeller stream is discharged from the impeller and moves radially and tangentially toward the tank wall. The flow is basically two-dimensional in the region near the impeller because of the dominance of the rotating impeller

2 ,?*

THE CANADJAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 68, FEBRUARY, 1990 9

-p 2.0 Figure 5 - The normalized velocity vectors on the impeller plane (z ' = 0) in Tank System 1.

.' \

- - - - 1 4

-* 0 .5 Figure 6 - The normalized velocity vectors on the r-8 plane in the bulk region (z' = 1.5) of Tank System 1 .

over the effects from baffles and bulk flow. In the region close to the tank wall, there is recirculating flow behind the baffle. In Figure 6 which shows the velocity vectors in the bulk region of the tank, we see that the flow is moving radially inward near the centerline of the vessel and is mainly tangential near the periphery except in the regions behind the baffles where it recirculates. The velocities near the baffles are significantly lower than those in the wall region away from the baffles.

Typical results for the turbulence kinetic energy and its dissipation rates are given in Figures 7 and 8, respectively. Figure 7 presents a distribution of the nondimensional turbulence kinetic energy on the 45 O plane in Tank System 1. The turbulence kinetic energy close to the impeller tip is over ten times that in the regions away from the impeller. This means that the turbulence energy is consumed quickly in the

-3.50 -3.83

0.00 1.00 2.00 3.00 3.83 r I

Figure 7 - Block diagram of the turbulence kinetic energy on the 45" r-z plane in Tank System 1.

- 1 . 0 3

0 r '

Figure 8 - Contours of energy dissipation rates on the 45" r-z plane in Tank System 1.

impeller stream. This phenomenon can also be seen from the contours for the dissipation rates of turbulence kinetic energy in Figure 8. Most of the turbulent energy dissipation occurs in the impeller stream. In the rest of the agitated vessel, the dissipation rates of turbulence kinetic energy are relatively small, i.e., from 1 .O to 0.1 as compared to more than 32.0 to 4.0 in the impeller stream. Basically, the distribution of energy dissipation rates follows the same


10-

O B -

06

- NONISOTROPIC CHEN CUTTER I SOT ROPlC MUJUMDAR VAN OER MOLEN

DESOUZA AND VAN MAANEN

0

. -

1 IMPELLER BLADE

I I I I I I I I 1

I

I 0 I 5 2 0 2 5 3 0

r ’

Figure 9 -- Radial profiles of the radial velocity components on the impeller plane.

10 0 CHEN - NONISOTROPIC - ISOTROPIC

[ “,::::A 1, 0 8

IMPELLER BLADE

0 0 1 I I I I

0 0 5 10 1 5 2 0 2 5 3 0 r ‘

Figure 10 - Radial profiles of the tangential velocity components on the impeller plane.

pattern as the turbulence energy distribution. Further results were presented by Ju (1987).

COMPARISON WITH EXPERIMENTAL DATA

In fluid mechanics the validity of the equations governing the flow is usually established by comparing the predicted velocities with experimental data for the all velocity components. In the bulk region, comparisons will be made with the hot wire anemometry data from this research; and in the impeller stream region, comparisons will be made with the data obtained by hot wire anemometry , three-dimensional pitot tube, photographic methods, and laser velocimetry in other studies.

IMPELLER STREAM

The flow field adjacent to the impeller is complicated and three-dimensional with two ring vortices trailing behind the impeller blades. Van? Riet and Smith (1975), Placek et al. (1985, 1986) and Yianneskis (1987) have reported key experiments and analyses to describe this flow. However, the net result of the vortex flow and the flow between the vortices is two-dimensional which is described a tangential jet as illustrated by Placek et al. (1986), and the tangential jet equations were used as boundary conditions to describe the flow from the impeller.

Comparisons of the radial and tangential velocity components in the impeller stream predicted using the two models

DATA NONISOTROPIC

--- I SOTROPIC

2 ’ = 2 . 0

-0.10

- 0.15 0 10 20 30 40 5 0 60 70 00 90

------ V ‘ - 0 . 0 5

V ’ - 0 05

-0.10

m * -0. 15 I I 1 I I I I I I

0 10 20 30 40 50 60 70 00 90 e l

Figure 11 - Tangential profiles of the radial velocity components.

are made with literature data in Figures 9 and 10 respectively. Figure 9 shows the normalized radial velocities on the impeller plane as a function of radial position. The experimental data were measured by a photographic method Cutter (1966), a three-dimensional pitot tube DeSouza (1969), a hot wire anemometer Mujumdar et al. (1970), and a laser velocimeter Van der Molen and Van Maanen (1978) and Chen (1986). The radial velocity profile from the nonisotropic model fits the data in the impeller stream while the isotropic model underpredicts this data.

In Figure 10, the normalized tangential velocities in the impeller stream predicted by these two models are compared with the experimental data taken by Chen (1986), Cutter (1966), and DeSouza (1969). Again, the solution with the nonisotropic model describes the data, and the solution with the isotropic model overpredicts the tangential velocity components. This is comparable to the results for the bulk flow region as will be shown subsequently.

BULK REGION

In Figures 1 1 to 13 comparisons are given for the three velocity components as functions of tangential position for three values of r and z in the three tank systems. Predicted values were obtained using both the nonisotropic and isotropic models. In all three figures the baffles are located at 8’ = 0 and 8’ = 90°, and the impeller is rotating counter- clockwise. Figure 1 1 shows that the dimensionless time- averaged radial velocity V ’ is always negative which means that the flow is away from the tank wall and toward to the centerline of the vessels, and there is a significant variation of radial velocity along the &direction. From the shape of the curves, we find the largest changes in velocity occur


rn D A T A N O N I SOTROPIC ISOTROPIC ---

0 10 r l = 3.46

I I I 1 I 1 I I 1

Z I = 2.0

W ' 0.06

0.04

0.02

0.00 I 1 I I I I I I I J

0 10 2 0 3 0 4 0 5 0 60 70 80 90 ",:," 1 System 2

- -._ r l = 2.6 2 ' ' 1 . 5 Y-

- 0.05 I I I I I I I I I I 0 10 20 30 40 50 60 70 80 90

System 3 O Z o r r ' z 1 . 5

0. 15

w' 0.10 e m ern--

0.00 0.05 0 3 10 2 0 30 4 0 50 60 70 00 90

8l

Figure 12 - Tangential profiles of the tangential velocity Components.

0.20

0. I 5

u ' 0.10

0.05

DATA System I NONISOTROPIC r 1 = 3.46 - - - ISOTROPIC

'.lor System 3

0 ~ 5 1 ~ 1 = r l = I 5

000 U'

-0 .05 -0- I

1- 1- \.

- - . = = - / - -0 lo

-0 .15 1 I I I I I I I I I 0 10 20 30 4 0 50 60 70 80 90

8l

Figure 13 - Tangential profiles of the axial velocity components.

DATA N O N ISOTROPIC ISOTROPIC -_- 2 1 . 2 . 5

- 0.10 1 I

- 0 I S e A - - 0 0 0 5 I 0 1 5 2 0 2 5 3 0 3 5 4 0

O l o System 2 0.05 I' 1.875

8': 45

- 0 151 I

1 0 1 5 2 0 2 5 3 0 00 0 5

System 3 2 ' . - 0 7 5 e l = 4 5

0 10

I I I

10 1 5 2.0 2 5 3 0 0 101 0 0 0.5

r 1

Figure 14 - Radial profiles of the radial velocity components.

in the regions near the baffles. The radial velocities obtained from the solution of the nonisotropic model generally tit the experimental data within the accuracy of the measurements. Also, as can be seen from the comparison of the two models in this figure, the magnitude of I/' using the isotropic model was less than the corresponding values from the nonisotropic model by as much as 50% in each of the three tank systems.

Figure 12 shows similar comparisons for the nondimensional tangential velocities W ' . The solution of the nonisotropic model predicted tangential velocities within the accuracy of the data. However, the solution of the isotropic model overpredicted the tangential velocity components by as much as 100% as a consequence of excessive rotation. The largest deviation between nonisotropic results and data occurred in the region close to the backside of baffles (r' = 2.6 and z ' = 1.5 in Tank System 2). This is probably due more to interference by the hot wire anemometer probe in the velocity measurement rather than to a model deficiency. The negative values for W' in the range of 0°-15" from both predictions show that there is recirculation behind the baffles in that particular position. This phenomenon is reasonable and also was found experimentally by Yianneskis et al. (1987) using laser velocimetry.

As can be seen from Figure 13, the axial velocity components obtained from the nonisotropic model once again fit the data to within their accuracy, and the velocities predicted using the isotropic model are lower than the data by 10% to 100% in the three tank systems. The variation of the axial velocity in the &direction illustrates the three-dimensional character of the flow. In the region close to the tank wall, e.g., r' = 3.46 in Tank System 1 and r' = 2.6 in Tank System 2, the effect of baffles in turning the flow is seen. The axial velocity near the front face of the baffle is higher than the axial velocity on the back side.


W '

W '

W '

O lo

0 0 8 -

0 06

0 04

0 0 2 -

NONISOTROP~C I SOT ROPl C -.- - System I

z ' = 2 5 8'; 4 5

- /'*-.

6' 0001 I I I I I I

0 15 r System 2 I

I I 1 0 0 0 5 I 0 1 5 2 0 2 5 3 0 3 5 4 0

-0 l o -

I z I = 1.875 8'. 45

\--------- I I I I I I 0 001 I I I I I I 1

0 0 0 5 10 15 2 0 2 5 3 0

- 0 05 I I I I I 0 0 0 5 10 1 5 2 . 0 2 5 3 0

r l

Figure 15 - Radial profiles of the tangential velocity components.

In Figures 14 to 16 comparisons are made for the three velocity components as a function of radial position for three values of z and 19 in the three tank systems. In Figure 14, the negative measured radial velocities V' show that the flow is recirculating back from the tank wall to the centerline. The radial velocities predicted using both models demonstrate the same behavior. In Tank Systems 2 and 3, the nonisotropic results match the experimental data better than the isotropic solution does. In Tank System 1, the discrepancy between the experimental data and the computed radial velocities is the largest observed in all of the comparisons with experimental data. This could be due to a problem with probe interference, or it could be associated with the accuracy of the numerical solution or a model parameter.

In Figure 15, the radial profiles for tangential velocity W' from the two models are compared with experimental data. It is clear that the nonisotropic model describes the data while the solution with the isotropic model overpredicts the tangential velocity.

For the radial profiles of axial velocity U' shown in Figure 16, the nonisotropic model provides a better description of the measured values while the isotropic model underpredicts the data. For the position above the impeller plane, i.e., z' > 0, the flow is moving upward near the tank wall and moving downward in the region away from the wall. For z ' < 0, the flow is moving downward near the tank wall and moving upward in central region. There is a deviation between solution and data in the regions of r' < 1.5 in Tank System 2 where the calculated axial velocities are negative but the measured data are positive. The model predicts the axial flow is moving down toward the impeller stream and the experimental data show the axial flow is still directed up. It may be

DATA System I - NONISOTRDPIC I ' = 2 . 5 ISOTROPIC ---

0 .

U I 0.00 p;+1 -/----- - 0. lo

- 0 . 2 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

System 2

- 0 .

u' 0.00

- 0.201 L I I I 0.0 0.5 1 . 0 1 . 5 2 .0 2.5 3.0

r l

Figure 16 - Radial profiles of the axial velocity components.

possible to reduce this deviation by fine-tuning one of the turbulence model parameters.

Figure 11 to 16 show the typical comparisons of experimental data and the solutions with the two turbulence models. For a total of 1,014 measurements the average difference between the measurements and the solution with the nonisotropic model was 16% with a sample standard deviation of 6% while the average difference between the measurements and the solution with the isotropic model was 39% with a sample standard deviation of 8%. This demonstrates that the nonisotropic model represents the data better than the isotropic model.

In summary, the experimental data have confirmed that the nonisotropic model accurately described the velocity components in various tank systems. Also, comparisons of the calculated turbulence parameters were made with approxi- mate values reported in the literature, and qualitative agreement was shown. These details are reported by Ju (1987). One additional point should be made here is that the { p d V over the tank volume only predicts part of the power consumption obtained from power-number Reynolds-number correlations, because the region swept by the impeller is not included in the calculation, Harvey and Greaves (1982) and Ju (1987). Also, further information on power consumption in agitated vessels is reported by King et al. (1988).

Conclusions

The three-dimensional solution of the governing equations with a nonisotropic viscosity turbulence model was developed and validated with experimentally measured velocity profiles in agitated vessels with a standard tank configuration. Also, it was shown that the frequently used isotropic model did

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING. VOLUME 68, FEBRUARY, 1990 13

not adequately describe the turbulent stirred-tank flow field. In addition, the computed results show that there was symmetry in the region of the je t zone close to the impeller. The flow from the rotating impeller dominated this region and overrode effects from the baffles, tank wall and the flow in bulk region. Moreover, the solution showed that the impeller stream contained and consumed most of the turbulence kinetic energy in agitated vessels.

The nonisotropic model has the flexibility of having different viscosity components a t the same point. In the present nonisotropic model only one of five viscosity ratios was varied locally and the other four were held constant. It would be worthwhile to examine the sensitivity of the solution to different expressions for the viscosity ratios. Also, the solution can be used to obtain the shear rate and shear stress which are important in the design of agitated vessels. In addition, this solution for the velocity field provides the basis for the next step in obtaining the solutions of the energy equation and the species continuity equations to predict the performance of nonisothermal reacting flows in agitated vessels.

APPENDIX

A material balance over the blade-swept region shown in Figure 1 is performed here to obtain the equation used to compute the axial velocity on the top and bottom of the impeller from the flow out of the impeller, The inflow to this region is given in terms of the discretized axial velocities U ( I r , J ) and U ( I B , J ) which are on the horizontal faces of the impeller.

JR

Inflow = c p [ IU(I~ ,J ) lA(IpJ) -t J = 2

The outflow from this region is given in terms of the discretized radial velocity V(I,JR) on the plane circling the impeller tip.

In these two equations A is the area of the control surface, p is the liquid density, V(I,JR) is the radial velocity on the impeller tip which is obtained from the tangential jet, and U(Ir,J) and U(lB, J ) are the axial velocity components on the top and bottom faces of the impeller blades which are unknown.

From a material balance for the steady flow of an incompres- sible fluid, the flow into the top and bottom of the impeller is equal to the flow out from the impeller tip, and the following equation is obtained:

+ l U ( f B , J ) l A ( I B , J ) ] . . . . . . . . . . . . . . . . . . . . . . . . . . (Al)

The above equation involves two sets of unknowns, U(Ir, J) and U ( I B , J ) , both occurring on the right hand side.

One procedure is to have these two sets of unknown axial velocities be equal to the axial velocities on the planes one step above and below the impeller. This is called the zero-gradient assumption for the axial velocity. However, this would neglect the radial flow between these planes and the planes at the top and bottom of the impeller; and a material balance on the impeller would not be satisfied.

A convenient procedure to satisfy the material balance was developed to obtain the boundary conditions for the axial velocity components on the top and bottom side of the impeller by using a proportional constant a as shown below.

U(Ij-, J) = a U(IT + l , J ) for J = 2,3,. . . . J R . . . . . (A2)

. . . . . . . . . U ( I B , J ) = a U ( l B - l , J ) for J = 2 , 3 , JR (A31

U(IT + 1,J) and U(IB + 1,J) are adjacent axial velocities to the U(IT, J ) and U(IB, J) and are on the plane one step above and below the impeller. Values used for U(1, + 1,J) and U(ZB - 1,J) are from the latest iteration. If the proportional constant ty is known, then the axial velocities on the top and bottom faces can be computed from Equations (A2) and (A3) for use as the boundary conditions.

To obtain a, Equations (A2) and (A3) are substituted into Equa- tion (Al). The rearrangement will give the following equation for a:

IT C

I = IB v(f, JR) A (1, JR)

a = JR

C J = 2

[IU(I, + 1,J)1A(Ir9J) f IU(IB - ~ , J ) I A ( ~ B , J ) I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A41

Values for a were calculated using this procedure during the convergence of the numerical solution for both two- and three- dimensional solutions using both nonisotropic and isotropic models, Ju (1987). The values for a were 0.850 + 0.005 for Tank System 1, and 0.931 f 0.005 for Tank Systems 2 and 3. This means that the material balance for the blade-swept region would not be satisfied with the commonly used zero-gradient assumption (a = 1) for the axial velocities on the top and bottom faces of the impeller. The a’s for Tank System 2 and Tank System 3 are essentially the same which is probably due to the same impeller diameter and slightly different impeller speeds being used in these two tank systems.

Acknowledgement

This research was funded through the Louisiana Mining and Mineral Research Institute under Bureau of Mines Grant Number G1164112.

Nomenclature

A = tangential jet parameter in Equation (36) A(I, J ) = area of control surface in Equation (Al)

= tangential jet parameter in Equation (35) = tangential jet parameter in Equation (34) = turbulence model coefficient in Equation (1 1) = turbulence model coefficient in Equation (12) = turbulence model coefficient in Equation (15) = turbulence model coefficient in Equation (10) = tank diameter = impeller diameter = generation term in the turbulence energy equation,

Equation (19) = radial flux of radial momentum = radial flux of angular momentum = gravitational constant = width of impeller blade = impeller height from tank bottom = liquid height = turbulence kinetic energy = nondimensionalized turbulence kinetic energy = impeller rotation speed = time-averaged pressure = centrally located grid point in finite subdomain = impeller radius = turbulent Reynolds number, Equation (13) = turbulent Reynolds number, Equation (14) = turbulent Richardson number, Equation (1 8) = radial coordinate


nondimensionalized radial coordinate swirl number for the impeller stream, Equation (24) source term time-averaged axial velocity nondimensionalized time-averaged axial velocity time-averaged radial velocity nondimensionalized time-averaged radial velocity impeller tip velocity time-averaged tangential velocity nondimensionalized time-averaged tangential velocity axial distance measured from tank bottom radial distance measured from the centerline of the tank radial distance measured from the tank wall axial coordinate nondimensionalized axial coordinate

Creek leiters

proportional constant turbulence energy dissipation rate nondimensionalized turbulence energy dissipation rate tangential jet parameter exchange coefficient fluid viscosity turbulent viscosity, Equation (9) general variable in Equation (27) fluid density viscosity ratios in Equation (1) tangential coordinate in radians tangential coordinate in degrees angle velocity vectors emerge from impeller tip

Subscripts

B b

e

efs = effective value I , J , K i , j , k ij KPT ma = maximum value n

non = nonisotropic value R = related to the impeller tip r , 0, 7 = related to cylindrical coordinates S = related to the control face on the south side of

T = related to the grid point on top side of subdomain t = related to the control face on the top side of

W = related to the control face on the west side of

= related to the grid point on bottom of subdomain = related to the control face on the bottom of

= related to the control face on the east side of subdomain

subdomain

= related to coordinates in finite difference form = related to coordinates in finite difference form = related to rr, d, u, Oz, and 00 = number of grid point in the &direction

= related to the control face on the north side of subdomain

subdomain

subdomain

subdomain € = turbulence energy dissipation rate 9 = variable

References

Berker, A. and S. Whitaker, “A Two-Dimensional Model of Second Order Rapid Reactions in Turbulent Tubular Reactors”, Chem. Eng. Sci. 33, 889-895 (1978).

Bradshaw, P. T. Cebeci and J. H. Whitelaw, “Engineering Cal- culation Methods for Turbulent Flow”, Academic Press, New York (1981).

Chen, K. Y., “Experimental Measurements of Laser-Doppler Anemometry in a Baffled Mixing Tank”, Ph. D. Dissertation, University of Missouri-Rolla (1986).

Cutter, L. A., “Flow and Turbulence in a Stirred Tank”, AIChE

DeSouza, A., “Fluid Dynamics and Flow Patterns in Stirred Tanks with a Turbine Impeller”, Ph. D. Dissertation, Louisiana State University (1969).

DeSouza, A. and R. W. Pike, “Fluid Dynamics and Flow Patterns in Stirred Tanks with a Turbine Impeller”, Can. J. Chem. Eng. 50, 15-23 (1972).

Gosman, A. D., W. M. Pun, A. K. Runchal, D. B. Spalding, and M. Wolfshtein, “Heat and Mass Transfer in Recirculating Flows”, Academic Press, London (1969).

Gunkel, A. A. and M. E. Weber, “Flow Phenomena in Stirred Tanks”, AIChE J. 21, 931-939 (1975).

Gupta, A. K. and D. G. Lilley, “Flow Field Modeling and Diag- nostics”, Abacus Press, Kent (1985).

Gupta, A. K., D. G. Lilley and N. Syred, ‘‘Swirl Flows”, Abacus Press, Kent (1984).

Harvey, P. S. and M. Greaves, “Turbulent Flow in an Agitated Vessel”, Trans. I. Chem. E. 60, 195-210 (1982).

Harvey, P. S. and M. Greaves, “Turbulent Flow in an Agitated Vessel-Reply”, Chem. Eng. Res. Des. 61, 136-137 (1983).

Ju, S. Y., “Three-Dimensional Turbulent Flow Field in a Turbine Stirred Tank”, Ph. D. Dissertation, Louisiana State University (1987).

King, R. L., R. A. Hiller and G. B. Tatterson, “Power Consump- tion in a Mixer”, AIChE J. 34, 506-509 (1988).

Lam, C . K. G. and K. Bremhorst, “A Modified Form of the k-e Model for Predicting Wall Turbulence”, J. Fluids Eng. 103,

Launder, B. E., in “Theoretical Approaches to Turbulence”, D. L. Dwoyer, M. Y. Hussaini and R. G. Voigt, Eds. Springer Verlag, New York (1985), p. 155.

Launder, B. E., C. H. Priddin and B. I. Sharma, “The Calcula- tion of Turbulent Boundary Layers on Spinning and Curved Sur- faces”, J . Fluids Eng. 991, 231-239 (1977).

Launder, B. E. and D. B. Spalding, “Mathematical Models of Tur- bulence”, Academic Press, London (1972).

Lilley, D. G., “Prediction of Inert Turbulent Swirl Flows”,

Lilley, D. G., “Turbulent Swirling Flame Prediction”, AIAA J.

Lilley, D. G. and N. A. Chigier, “Nonisotropic Exchange Coeffi- cients in Turbulent Swirling Flames from Mean Value Distribu- tions”, Combustion and Flame 16, 177-189 (1971a).

Lilley, D. G. and N. A. Chigier, “Nonisotropic Turbulent Stress Distribution in Swirling Flows from Mean Value Distributions”, Int. J. Heat Mass Transfer 14, 573-585 (1971b).

Lilley, D. G. and N. A. Chigier, “An Inverse Solution Procedure for Turbulent Swirling Boundary Layer Combustion Flow”, J. Comp. Phys. 9, 237 (1972).

Liu, C. H. and C. H. Barkelew, “Numerical Analysis of Jet-Stirred Reactors with Turbulent Flows and Homogeneous Reactions”,

Middleton, J. C., F. Pierce and P. M. Lynch, “Computations of Flow Fields and Complex Reaction Yield in Turbulent Stirred Reactors and Comparison with Experimental Data”, Chem. Eng. Res. Des. 64, 18-22 (1986).

Mujumdar, A. S. , B. Huang, D. Wolf, M. E. Weber, and W. J. M. Douglas, “Turbulence Parameters in a Stirred Tank”, Can. J . Chem. Eng. 48, 475-483 (1970).

Mulvahill, T. M., “Velocity Profiles in A Stirred Tank Via Three- Dimensional Hot Wire Anemometry”, M. S. Thesis, Louisiana State University (1979).

Patankar, S. V., “Numerical Heat Transfer and Fluid Flow”, McGraw-Hill, New York (1980).

Patankar, S. V. and D. B. Spalding, “A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional Para- bolic Flows”, Int. J. Heat Mass Transfer 15, 1787-1806 (1972).

J . 12, 35-45 (1966).

456-460 (1981).

AIAA J. 11(7), 955-960 (1973).

12(2), 219-223 (1974).

AIChE 1. 32, 1813-1820 (1986).

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING. VOLUME 68, FEBRUARY, 1990 15

Placek, J., and L. L. Tavlarides, “Turbulent Flow in Stirred Tanks - Part I: Turbulent Flow in the Turbine Impeller Region”,

Placek, J., L. L. Tavlarides, G. W. Smith and I. Fort, “Turbulent Flow in Stirred Tanks - Part 11: A Two-Scale Model of Turbu- lence”, AIChE J. 32, 1771-1786 (1986).

Platzer, B., “A Contribution to the Evaluation of Turbulent Flow in Baffled Tanks with Radial Outflow Impellers”, Chem. Technik. 33, 16-19 (1981).

Salcudean, M., K. Y. M. Lai and R. I. L. Guthrie, “Multidimen- sional Heat, Mass and Flow Phenomena in Gas Stirred Reactors,” Can. J. Chem. Eng. 63, 51-61 (1985).

Schetz, J . A., “Injection and Mixing in Turbulent Flow”, Ameri- can Institute of Aeronautics and Astronautics, New York

Sharma, B. I. in “Turbulent Boundary Layers - Forced, Incom- pressible, Non-Reacting”, H. E. Weber, Ed., American Society of Mechanical Engineers, New York (1979), p. 15.

AIChE J. 31, 1113-20 (1985).

(1980).

Smith, G. W., “Simulation Modeling of Hydrodynamic Effects in Dispersed Phase Systems”, Ph. D. Dissertation, Illinois Insti- tute of Technology (1985).

Spalding, D. B., “A General Purpose Computer Program for Multi- Dimensional One- and Two-Phase Flow,” Mathematics and Computers in Simulation 23, 267-276 (1981).

Van Der Molen, K. and H. R. E. Van Maanen, “Laser-Doppler Measurements of the Turbulent Flow in Stirred Vessels to Estab- lish Scaling Rules”, Chem. Eng. Sci. 33, 1161-1168 (1978).

Van’t Riet, K. and J. M. Smith, “The Trailing Vortex System Produced by Rushton Turbine Agitators”, Chem. Eng. Sci. 30,

Yianneskis, M., Z. Popiolek and J. H. Whitelaw, “An Experimental Study of the Steady and Unsteady Flow Characteristics of Stirred Reactors”, J. Fluid Mech. 175, 537-555 (1987).

1093- 1 105 ( 1975).

Manuscript received June 30, 1988; revised manuscript received June 8, 1989; accepted for publication June 9, 1989.


Documents

Three-dimensional turbulent flow in agitated vessels with a nonisotropic viscosity turbulence model