2370
Ind. Eng. Chem. Res. 1992, 31, 2310-2319
Comparison of Axial Flow Impellers Using a Laser Doppler Anemometer V. V. Ranade,?V. P. Mishra, V. S. Saraph, G. B. Deshpande, and J. B. Joshi* Department of Chemical Technology, University of Bombay, Matunga, Bombay 400 019, India
Influence of shapes of eight axial flow impellers on flow in agitated vessels was studied using a laser Doppler anemometer. The tank diameter was 500 mm with a flat bottom and provided with four standard (width = T/10) baffles. In all cases the tank to impeller diameter ratio was 3 and the impellers were centrally located. The flow generated by different axial impellers has been compared in terms of mean velocities, turbulent kinetic energy, pumping effectiveness, and hydraulic efficiency. The measured flow data near the impeller have been presented in the form suitable for specifying the boundary conditions to the numercial model. The two-equation (k-e) turbulence model has been shown to be adequate for predicting the bulk flow in the case of all impellers. 1. Introduction 1.1. Background. Agitated reactors are widely used in chemical and allied industries. There are many different ways to provide agitation and mixing in a vessel. The present paper is primarily concerned with impeller-induced agitation. Numerous parameters affect flow in agitated tanks. The tank geometry and the type and arrangement of impellers are the two most important parameters which determine the flow field in the vessel and therefore the performance of a stirred vessel. In most of the cases, the basic shape of a agitated vessel is cylindrical with either a flat or dished bottom and a number of baffles. However, a wide variety of impellers with different shapes and sizes are being used in practice (Tatterson, 1991;Rewatkar and Joshi, 1991). There are fundamental reasons for choosing one impeller over another, which depend on a complete definition of the mixing requirement. It therefore appears that a calculation method which predicts accurately the flow field around an impeller of arbitrary shape would be of enormous benefit to the reactor designer. Unfortunately, such a general method does not exist, even for the most commonly used radial flow impellers. The flow field is usually so complex and highly turbulent that it defies analytical description. Traditional designs have therefore had to rely on experience and a set of well-defined geometrical relationships between impeller dimensions, its positioning, and tank size. Recent advances in experimental and numerical modeling techniques can enhance our understanding about the influence of impeller design on flow in agitated tanks. This paper describes an attempt to develop a coherent basis for impeller design by establishing defiite relationships between impeller shape and the generated flow characteristics. 1.2. Types of Impellers. Impellers are classified as two general types: axial flow and radial flow. An axial flow impeller discharges fluid along the axis of the impeller (parallel to the impeller shaft) while radial flow impeller discharges flow along the impeller radius. General flow patterns of these impellers are well-known and well documented (Joshi et al., 1982; Mann,1983; Fort, 1986, Oldshue, 1984; Nagata, 1975). However, knowledge of turbulence characteristics of the flow generated by these impellers is not readily available. No adequate information is available to make proper selection of impellers. A widely cited guide for impeller selection uses arbitrary parameters
* Author to whom correspondence should be addressed.
+ Present address: Chemical Engineering Division, National Chemical Laboratory, Pune 411 008, India.
0888-5885I92 12631-2370%03.00IO
like scale of agitation and considers only standard impellers. Recently many new, modified impellers have been proposed for a variety of process results. These impellers need to be investigated and placed at their appropriate place in a general perspective. 1.3. Present Contribution. The influence of various impellers on flow in agitated vessels is investigated in detail. A combined approach of experimental as well as numerical simulation is adopted to generate a coherent picture of flow characteristics of various impellers. Experimental data useful for specifying the impeller boundary conditions is reported for eight axial flow impellers. Hydraulic efficiencies and pumping effectiveness factors of various impellers are reported to aid the designer for making the proper selection of an impeller. 2. Tools of Investigation
Recent advances in measurement techniques such as the hot film anemometer/laser Doppler anemometer enable us to characterize complex, three-dimensional turbulent flows generated by impellers. However, detailed characterization of the flow field is very expensive and time consuming. Numercial simulations of the generated flow field can prove very economical if properly validated models are developed. Recently, Ranade and Jbshi (199Ob) and Ranade et al. (1989) have presented results of numerical simulations of flow generated by a disk turbine and pitched blade turbines. They have obtained good agreement between simulated and measured flow characteristics. Their models, however, require boundary conditions near the impeller. Flow near and inside the impeller can in principle be simulated using a suitable turbulence model. However, periodic movement of blades and coherent structure behind the blade raise severe and often intractable problems for flow simulation. Alternatively, approximate models for the flow inside the impeller can be constructed using drag and lift coefficienta for the impeller blades. Such models can give a reasonable estimate of mean flow characteristics near the impeller. This issue of flow near the impeller is discussed again in the subsection of impeller boundary conditions. Since the objective of the present study is to investigate the influence of impeller designs, uncertainty in the flow near the impeller should be minimized. Therefore, we have adopted a hybrid approach in the present work (1)flow near the impeller will be characterized accurately using measurements with the laser Doppler anemometer; (2) using this information, flow in the bulk of the vessel will be simulated using a turbulence model. Details of these two stages are given in the following subsections. 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 10,1992 2371 Table I. Impeller Design Details impeller name PTD PTD PTD FTD PTD MPTD MODPTD-1 PROPELLER CURPTD CONPTD a
figure n0.O
D A E C
B
i.d. 16 16 16 16 16 16 16 16 16 16
hub details, mm 0.d. height 25 28 25 28 25 27 25 28 25 28 32 31 32 28 25 51 25 50 25 50
blade width, WB,mm 0.30 0.30 0.30 0.20 0.40 41 21-29‘ 50 52
blade thickness 2 2 2 2 2 3 3 2 2 2
blade angle, deg 45 60 30 45 45 45 45
no. of blades 6 6 6 6 6 12 2 3 6 6
45 45
See Figure 2. *Near the hub and at the impeller tip, respectively.
2.1. Physical Modeling. The experimental system was composed of an agitator assembly (shown schematically in Figure lA), a cylindrical vessel (500-mm i.d.), a DANTEC 55 X modular series laser Doppler anemometer (LDA) and ita associated data processing instruments, and a PDP 11/03 Digital Equipment Corporation minicomputer. The LDA was operated in a forward scatter mode. Figure 1B shows the experimental arrangement of the LDA. The lasers (15-mW He-Ne laser and 5-W Ar ion laser, both manufactured by Spectra Physics, USA) and optics were mounted on a bench which has a one-dimensional traversing mechanism. To identify the flow reversals correctly, a frequency shift was given to one of the beams by means of a Bragg cell with electronic downmixing. Data validation and signal processing were carried out using the DISA 55L96a Doppler signal processor or “counter”. The PDP 11/03 minicomputer functioned as the central data acquisition and reduction controller via a DISA 57G20 buffer interface. Other details of setup, experimental methodology, and quantification of measurement error are described by Ranade and Joshi (1989). The flow generated by eight different impeller designs was measured. These included three six-bladed pitched downflow turbines (PTD) with blade angles of 30°,4 5 O , and 60°, a multiple-bladed pitched downflow turbine (MPTD), two modified designs of the PTD, a propeller, and a modified propeller. The schematic diagrams of these impellers (excluding standard pitched blade turbines with different blade angles) are given in Figure 2. In the first modified PTD design, a two-bladed pitched downflow turbine was selected. Near the blade end, the blade was cut and raised as shown in Figure 2A. The impeller will be called MODPTD-1. A similar design has been suggested by Ekata Company. In the second modified design, a six-bladed downflow turbine was selected. The blades of these impeller were curved along the blade width (Figure 2B). The curvature was given in such a way that the curved portion was an arc and the curvilinear length was equal to one-sixth of the perimeter of a circle having the same radius as that of the arc. This means that, if all six blades were kept side by side, it formed a cylindrical shape. The convex portion of this impeller leads during rotation. This impeller will be called a convex pitched bladed downflow turbine and denoted by CONPTD. In the third modified design the blades were curved lengthwise (Figure 2C). In this case also, the curved portion formed an arc of a circle where the arc length was one-sixth of the perimeter. This impeller will be called a curved pitched bladed turbine and will be denoted by CURF’TD. It was also throught desirable to inveatigate the design details of the standard PTD. For this purpose, three blade angles (30°, 45O, and 60°) were selected (Ranade and Joshi (1989) have studied the flow characteristics of the pitched blade turbines in a 300-mm vessel and have established that
Q4
9 I
I
4 b
I
DOPPLER S I G N A L
B
C
1
2372 Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992
A
/IMPERMEABLE
5
WALL
r!
PER IO D Ic BOUNDA R IES
‘A\
B
DIRECTION OF ROTATION
1 1 TOP SURFACE
r---l-l--T
::YETRY ALONG TANK
C
BAFFLE
-TANK
WALL
DIRECTION OF ROTATION
BOTTOM SURFACE
Figure 3. Solution domain for the mathematical model.
D
E
I
Figure 2. Various designs of axial flow impellers. (A) Modified pitched bladed d o d o w turbine (MODF’TD-1). (B)Convex pitched bladed downflow turbine (CONPTD). (C) Curved pitched bladed downflow turbine (CURPTD). (D) Multiple pitched bladed downflow turbine (12 blades, MPTD). (E) Propeller.
pitch equal to 0.3 times the impeller diameter (Figure 2E). The details of all the impellers are given in Table I. In all casea the impeller diameter was 167 mm, the liquid height was equal to the tank diameter, and the impellers were centrally located. 2.2. Numerical Modeling. 2.2.1. Model Equations. Recently Ranade and Joshi (1990a) have reviewed the previous attempta on the modeling of flow in stirred tanks and recommended the use of a standard k-c model. In the present work also, this model will be used to simulate the flow generated by various axial flow impellers. The system investigated consists of a cylindrical tank with four baffles (of width T/10)equally spaced around the perimeter. The impeller shaft is concentric with the axis of the vessel, and the impeller is located at H / 3 from the bottom. Under steady-state conditions, flow in fully baffled vessel can be divided into four symmetric parts (in the present case,each part has a baffle at ita center). The solution of governing flow equations for any one of these four parts is sufficient. The solution domain is shown in Figure 3. All the relevant equations can be written as a general transport equation:
where p, is the effective viscosity for the corresponding transport variable 4. S, is the source term. The corresponding 4 and S, terms are given in Table 11. 2.2.2. Boundary Conditions. Specification of boundary conditions for the flow in stirred tanks has been discussed recently by Ranade and Joehi (1990a)and Ranade et al. (1989) and therefore will not be repeated here. The
Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992 2373 Table I1
Q
S
P4
1
0
U
Peff
W
POff
0
V
Table 111. Boundary Conditions 1. symmetry axis, r = 0 u = v = asjar = o where Q # U (radial) and V (tangential) 2. impermeable walls normal component of velocity = 0 p UWC~’I‘k’I2 Tw = In (EY+) where y+ = wall Reynolds number; U, = absolute value of mean velocity parallel to the wall at the near wall node akpy = o
code FIATdescribed by b a d e et al. (1990b). The mean velocity, pressure, and turbulence variables need to be under-relaxed (with factors in the range of 0.148) to achieve the convergence. The iterations for flow computations were continued till the mass residue became less than lo4 times the mass flow rate through the impeller. Most of the flow computations were carried out using a grid structure of 15 X 31 X 5 (some of the computations were carried out using 30 X 46 X 5 grids; however, differences between the computed results of the two grids were not significant).
CD3/4k3/2
e=-
kY 3. parameters of the k-t model CD = 0.09 1.0 C1 = 1.43 as = 1.3 Cz = 1.92 4. parameters of the wall function k = 0.4 E = 9.0
boundary conditions used in the present work are listed in Table 111. Wall functions have been used to impose a neslip boundary condition at the impermeable wall. The treatment of impeller boundary conditions plays a crucial role in these simulations. The problem of relating the steady-state discharge in the rotating frame of reference of impeller to the stationary boundary conditions generated by the walle and baffles of the vessel needs to be solved. As mentioned in the Introduction, flow inside the impeller swept volume has not been modeled in detail in the present work. This requires the development of suitable boundary conditions on the impeller swept surface. The impeller blade fluid slip and the geometrical influences should be appropriately incorporated in these boundary conditions. An easy way out of this problem is to specify boundary conditions using the experimental data. Alternatively,one can develop a semiempirid submodel for the impeller (bladea)using the drag/lift coefficient framework. However, it is rather difficult to simulate the existence of trailing vortices using such a drag/lift coefficient format. However, since the major objective is to present the comparative flow characteristice of various axial impellers, we have wed experime3tal data for specifying the boundary conditions for the complex-shaped impellers. The data used for specifying the boundary conditions for various impellers are discussed in the next section. 26.3. Solution of Model Equations. Flow computations were carried out using the extended version of the
3. Results and Discussion 3.1. Formulation of Impeller Boundary Conditions. Turbulent flow inside the impeller swept volume is extremely difficult to simulate using the methodology described in this paper. The use of a rotating frame of reference can eliminate the problem of simulating periodic movement of bladea (Kaminoyama et al., 1990). However, even then the remaining problems of complex geometry, wall boundary conditions, fine grids,and slow convergence often make the complete solution of flow in the stirred vessel including impeller swept volume impractical. Therefore, in this paper (asdiscussed in section 2) we have opted for specifying the impeller boundary conditions which will account for the rotation of the impeller. For the axial impellers, specification of boundary conditions at the bottom surface of the impeller swept surface is sufficient to generate reasonable predictions. Analytical models for the flow in the impeller region can be developed to specify boundary conditions at the bottom surface of the axial impellers along the lines of the models of Placek and Tavlarides (1985) and Fort (1987). Alternatively, one can develop a semiempirical submodel for axial impellers using the drag/lift coefficient framework. Figure 4 shows a force and velocity diagram for a section of one blade. From this, the tangential and axial momentum contributions arising from resolved lift and drag components (Wallis, 1961; Duncan et al., 1970; Pericleous and Patel, 1987) can be written as 1
M, = -p[CD1 cos 6 + CL2sin 6][(wrcos 2 1
L‘P
- PIC,
(2)
M, = p[CLI cos 6 - CD2sin 6][(wrsin 6) - W]Cp{ (3)
2374 Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992
e 3
O,':
b
t
W
e
FOR DOWN FLOW
!i W
>
o'2
B
B
E
rI 0.3X
ye Figure 4. Force and velocity diagram on a blade section (shape of the blade may be an airfoil or a flat rectangular blade).
in in W
0.4-
i ii, Z
W
0
B
0.5-
r0 where CD1and CD2and CL1and CL2are the resolved lift and drag coefficients (Figure 4). The drag and lift coefficients need to be specified using the experimental data. It should be noted that the angle of incidence (angle between flow and blade) is not known in the beginning of the solution. If the lift and drag coefficients are varying significantly with the angle of incidence for the shape of blade under consideration,the convergence rates of the iterative procedure will be slower. Pericleous and Pate1 (1987) have used such formulations in the two dimensional flow simulations in stirred tanks. However, the drag and lift coefficient data for the blades of various shapes (and their variation with the angle of incidence) are not available. Since the flow in the agitated tank is convection dominated, uncertainty in the impeller boundary conditions would greatly jeopardize the confidence in the simulated flow characteristics in the bulky region of the tank. To avoid this, we have generated the data for impeller boundary conditions using a laser Doppler anemometer. The boundary conditions of the mean velocities can be specified directly using these experimental data. The experimental data of turbulence intensities can be used to calculata the profile of turbulent kinetic energy k at the bottom surface of the impeller. Since the local values of energy dissipation rates e can not be calculated using the experimental data, we have followed the method of b a d e et al. (1989) of using the data of the hydraulic efficiency (defined in the next section) of the impeller. They have specified the E boundary conditions using the shape of the profie obtained without specifying any boundary condition of e and then scaling it up using the value of hydraulic efficiency. This boundary condition was shown to yield a reasonable prediction. The data used for the specification of these boundary conditions for various impellers are discussed below. 3.2. Data for Impeller Boundary Conditions. As disqussed by Ranade et al. (1989), flow data at the bottom surface of the impeller swept volume are necessary for specifying boundary conditions. These data were measured at constant 2 plane 30 mm below the impeller center plane and are shown in Figure 5. Most of the impellers generate strongly accelerating flow in the impeller region. From Figure 5A it can be seen that the PTD impellers develop a high-speed jet flow in the downward direction below the impeller. Further, the blade angle of the pitched blade turbine has a strong influence on the generated flow. An increase in the blade angle increases the maximum axial velocity and the pumping capacity of the impeller. However, the blade width does not have a significant influence on the generated flow. Blade width equal to 0.3
m i
0
o.6
t
0.7
0
0
0.1
0.2
0.3
0.4
DIMENSIONLESS RADIAL COORDINATE, r / R
c
3
,
3
e
0.1-
a
c
s
0.2-
0 a
0
8
W
a
0.31
0
I
I
I
0.1
0.2
0.3
I
0.4
DIMENSIONLESS R A D I A L COORDINATE,r/R
Figure 5. (A) Axial mean velocity at 30 mm below the impeller center plane for the PTD impellers and propeller. (#) pitched blade turbine with blade angle = 30'; (m) pitched blade turbine with blade angle = 45'; ( 0 )pitched blade turbine with blade angle = 60'; (E) propeller. (B) Axial mean velocity at 30 mm below the impeller centre plane for various axial flow impellers. (0)CURPTD (a) CONPTD (0)MODPTD-1; (e) MPTD.
times the impeller diameter was found to give maximum pumping capacity ( M a d e and Joshi, 1989). The maximum axial velocity is lower for the propeller (0.4U than for PTD (O.515UGJ. The MODPTD shows a consi erably lower maximum axial velocity (0.3Uq). The axial velocity of the convex pitched blade turbine is also low (0.29U~p). The multiblade pitched blade turbine (PMT)generates a less intense downward jet, and the position of the maximum axial velocity for thisjet is shifted radially outward. The curved pitched blade turbine has a relatively higher maximum axial velocity (0.55Utip)as compared with the standard PTD (0.515Utip).
3)
Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992 2375 Table IV. Location and Magnitude of Maximum Resultant Velocity (Dimensionless) at z = -30 mm impeller point of max z(RI)-' max resultant vel 0.65 0.313 standad PTD (U a 46', WB/D = 0.3) 0.60 0.295 curved pitched blade turbine flat profile 0.200 convex pitched blade turbine 0.24 0.232 propeller 0.75 0.120 MODPTD-1 1.00 0.08 MPTD
A
~~
~
at z = +30 mm point of max x(R1)-' max resultant vel 0.78 0.61 0.86 0.63 flat profile 0.29 0.73 0.45 0.75 0.20 1.00 0.30
0.12
A
0
O.1°
t
-
O
"
4
I
0
0
E
0
m E
E
o
E
BI
E B
I
t
I-
B
ezE
m
0,3
v)
o
m
z
P
0
v)
0.6
fj
8
0
0
0
8
7
1
I
I
I
0.1
0.2
0.3
DIMENSIONLES RADIAL COORDINATE
0
0
W J
8
E
1
a
"
1
1
O
0.4
, r/R
-
e *
o
J
9
e
0
2
-0.05
e
0
0 0 0
0 0
0
m
;
ln W v) J
v)
W
2
p
0 W -0,l
Y
li
E
031
0
0
0
0
8
0 0
0
-0 15;
1
I
1
01
02
03
I
04
DIMENSIONLESS RADIAL COORDINATE, r/R
F b m 6. (A)Radial mean velocity a t 30 mm below the impeller center plane for PTD impellere and Propeller. symbols 88 in F b w e 5A. (B)Radial mean velocity at 30 mm below the impeller center plane for varioua axial flow impellers. Symbols 88 in Figure 5B.
At the bottom surface, dimensionless radial velocities were very low in magnitude (see Figure 6). The range of the velocities for all the impellers was in the range of 0.03-0.1 times the tip speed. The tangential velocities at the impeller top surface were more or less flat for all the impellers, but those at the bottom surface exhibited sharp
-0 4;
I
I
I
01
02
03
1
04
DIMENSIONLESS RADIAL COORDINATE, r/R
Figure. 7. (A)Mean tangential velocity at 30 mm below the impeller center plane for PTD impellera and propeller. Symbols 88 in F i e 5A. (B)Mean tangential velocity at 30 mm above the impeller center plane for various axial flow impellers. Symbols 88 in Figure 5B.
maxima (Figure 7). The maximum dimensionless tangential velocity was observed for the standard PTD and the curved pitched turbine, and was 0.34Uti No sharp maximum was observed for the propeller. f o r the multiblade pitched turbine, it was 0.18Uti, while for MODPTD, it was 0.13Uti,. The maximum &directional velocity for the propeller was 0.23Uti,. The position of the
.
2376 Ind. Eng. Chem. Res., Vol. 31, No. 10,1992 Table V. Maximum Resultant Velocity (Dimensionless) with Point of Maxima at z = -30 mm impeller point of max r / R max resultant vel standard PTD 0.216 0.313 curved pitched blade turbine 0.2 0.295 convex pitched blade turbine flat profile 0.200 propeller 0.08 0.232 MODPTD-1 0.25 0.120 MPTD 0.33 0.08
&directional maximum was shifted toward the wall for MODPTD and the convex pitched blade turbine. For all other impellers, the velocities were still lower and in the range of 0.02-0.07times the impeller tip speed. Table IV summarizes the data on maximum mean velocities. Table V gives the locations and values of resultant velocities for all the impellers. The turbulence kinetic energy values at the bottom impeller surfaces are shown in Figure 8. At the bottom impeller surface of MPTD, a sharp peak was observed near the impeller tip. The discharge streams of MODPTD-1 was characterized by low turbulent kinetic energy profiles, and the values ranged between 0.005 and 0.01. Unlike the bottom impeller surface, the profile was flat at the top impeller surface of MPTD, and the magnitudes were also very small (0.0024l.005). MODPTD-1 shows a flat profile at the top surface, but the values were lower as compared to the bottom surface. At the top surface, the curved pitched blade turbine shows increased maximum turbulent kinetic energy in the impeller stream. The k profiles for the propeller and convex pitched bladed turbine were almost flat at the top surface of the impeller. Profiles at the bottom were similar to that of the resultant velocity profiles except that the point of maxima was shifted outward in the former case. The k profile for the convex blade showed a peak on the vertical surface while for the other impellers the profiles were practically flat. 3.3. Mean Flow Characteristics. Using the impeller boundary conditions described above, flow generated by various impellers can be computed using the computer code FIAT ( h a d e et al., 198913; h a d e and Joshi, 1990). h a d e et al. (1991) have used an extended version of the FIATcode to predict the flow characteristics of the standard PTD in a 300-mm vessel. They have observed excellent agreement between the experimental data and predicted results. In the present work we have also observed the similar agreement for all eight axial flow impellers. Parta A, B, and C of Figure 9 show a typical comparison between the model predictions and the axial velocity in the bulk region of the tank for the propeller, CURPTD, and CONPTD. Though the data in the bulk region of the tank for the various axial flow impellers studied in the present work are not reported here, these can be obtained from the authors and are reported by h a d e (1989), Deshpande (19881, Saraph (19891, and Mishra (1992). Thus the specification of impeller boundary conditions using the data reported in this paper and the FIAT code is sufficient to describe the flow generated by the various impeller designs. Such predicted flow results can be used to estimate the process performance of these impellers. 3.4. Comparison of Impellers. Convective flow and turbulence generated by impellers in stirred vessel reactors determine the rates of various transport processes occurring in the reactor and its performance. However, since convection and turbulent dispersion affect various pro~88888like mixing, solid suspension, gas-liquid dispersion, and heat transfer differently, performance comparison of the impellers can be carried out with respect to a particular process. To avoid discussion of comparison of the axial
at z = +30 mm point of max r / R rnax raultant vel 0.26 0.61 0.286 0.63 flat profile 0.29 0.243 0.45 0.25 0.20 0.33 0.30
A N .@
2
0
'
1 0
4
0.120
-
0.1
0
0,08O
-
B
0,06
0,04-
0
'
0
DIMENSIONLESS
8
8
RADIAL COORDINATE, r / R
B 006 I
I
0
0
2
Q
0
: I-
0.04
Y
0
I-
z
W -1
3
t
0.03
m
a
0
0
0
t
3 I-
*
0
0
0.02
ln
e
z
0
0.01
e
0.1
I
I
0.2
0.3
I 0.4
DIMENSIONLESS RADIAL COORDINATE, r / R
(A) Turbulent kinetic energy at 30 mm below the impeller center plane for PTD impellers and propeller. Symbols a~ in Figure 5A. (B)Turbulent kinetic energy at 30 mm above the impeller center plane for various axial flow impellers. Symbols in Figure 5B. Figure 8.
impellers studied in this work with respect to all these transport proceeees, we have decided to use two parameters to characterize the impellers. The fmt parameter is hydraulic efficiency (BH)which is defined as the ratio of the rate of kinetic energy (mean and turbulent) flowing out of the impeller swept volume and the total energy input rate:
Ind. Eng. Chem. Res., Vol. 31, No. 10,1992 2377
A
VH
I
f
01-
s
/-
211.-0
-
I2
01-0 1-
KE, =
.’
z’R*O
ZIR.
02
03
05
04
DIMENSIONLESS
or
OS
08
09
10
R A D I A L COORDINATE, r / R
B -05
/
-025 IfR.0 0 25
/
05c
0 0
+ W + V + 3u2 + w2 + u2) dz
I
0
The impeller region was defined as z1 = -30 mm, z2 = 30 mm, rl = 0, and r2 = 1.5RI.This definition implies that, for applications where large turbulent energy dissipation near the impeller is desired (like for gas-liquid dispersion), an impeller with a lower hydraulic efficiency is desirable. However, for applications where uniform power dissipation rates are desired, an impeller with a higher hydraulic efficiency is desirable. The second parameter, called pumping effectiveness (qE), is a measure of convective flow generated by an impeller for a given power and is defined as
Z/R.O2
i
02
TE = N Q / N P
04
$
+ v2 + u2 + 3w2 + u2)rdr
(7)
m 01
i2*(27rU)p((F t
-00 2 1
0
(5)
(6)
A 03
+
1,rs(27rW)p(l.P W
+ 0 12
KE, =
O
+ KE,
where
01-
2
KE,
E D
-01-
3 0
D
(4)
where Em is the energy input rate. EDis the net kinetic (mean plus turbulent) energy imparted to the discharged stream:
- 00 2 1
O
=ED/kiP
/
/
where
-025 -
i i?
025-
YI
05
(8)
o
Y 0
0 025
?
OB 0
OB 0
01
02 0
01
02
03
04
05
08
07
08
09
10
DIMENSIONLESS RADIAL C W R D I N A T E , r/R
C
r
”
I z/1.-02
-
01
02
y!
I
O’t
0
01
02
03
04
OS
DIMENSIONLESS R A D I A L
06
07
08
09
10
COORDINATE, r / R
Figure 9. Mean axial velocity in the bulk comparison between model predictions and experimental data. (-) Model predictions; other symbols as in Figure 5. (A) Propeller. (B)CURPTD. (C) CONPTD.
and N p is the power number. Impellers with higher pumping effectiveness will generate more convective flow than one with lower pumping effectiveness and therefore will be desirable for the convection-controlled processes like mixing of miscible liquids. These two parameters are summarized in Table VI for the eight axial impellers studied in the present work. The propeller showed the maximum hydraulic efficiency (81%1. The energy dissipation rate was very low in the impeller region. This may be due to the contoured shape of the propeller blades (reduced form drag). The maximum part of the energy input by the propeller was utilized for pumping of the fluid. This also resulted in a high pumping effectiveness (89.08%). Therefore, the propeller is characterized as ycirculation” type impeller. From Table VI it can be seen that the hydraulic efficiency of all the PTD impellers is within a small range of 5446%. However, the power number increases with an increase in the projected area (either increase in blade angle or blade width). Therefore, the power dissipated in the impeller region increases with an increase in the projected area. The pumping effectiveness was found to increase with a reduction in the projected area. This means that the flow leaving the impeller is less turbulent when the projected area is less (refer to eqs 4-9). The hydraulic efficiency of the curved pitched blade turbine (66.9%) was slightly greater than that of the standard pitched blade turbine (65%). The pumping effectiveness was also not significantly different. Energy dissipation in the impeller region for the curved pitched blade turbine was only decreased by 4% in comparison with the standard PTD. The hydraulic efficiency of convex pitched blade turbine was 23.5%. It clearly showed that
2378 Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992 Table VI. Hydraulic and Mechanical Efficiencies of Different ImDellers NP NQ 1 standard PTD ( a 45O, WnlD = 0.3) 2.1 0.94 3.33 1.08 2 PTD (a = 60°, W,/D = 0.35' 0.65 0.87 PTD ((Y = 30°, WB/D = 0.3) 3 1.47 PTD ( a 45O, WB/D = 0.2) 0.85 4 2.47 5 PTD ( a = 45O, WB/D = 0.4) 0.82 2.41 6 curved pitched blade turbine 0.39 7 convex pitched blade turbine 2.29 0.39 0.89 0.79 8 propeller 0.64 0.41 MODPTD-1 9 1.16 10 MPTD 0.41
curvature along the width results in reduction of pumping capacities but increase in the turbulence level. Decreased pumping effectiveness of the convex pitched blade turbines also showed the greater form drag offered by the modified shape of the blades. An interesting result can be seen in Table VI for the cases of the modified pitched blade turbine (MODPTD-1). Both parameters, the hydraulic efficiency and pumping effectiveness, are higher than those for the standard pitched blade turbine. Thus change in the shape of impeller tip decreases the energy dissipation rate in the impeller region. The secondary flow number is defined by the following equation:
where Qs is the overall liquid circulation rate. Its value is calculated using eq 9; however, the upper integration limit is the radius at which flow reversal occurs. The ratio NB, to N p is given in the last column of Table VI. This number (which may be called pumping effectiveness-2) gives the relative comparison in terms of average liquid circulation velocity per unit power consumption. For almost all the impellers, it follows the same trend as that of the primary pumping effectiveness. The two parameters described above can help a design engineer to make a suitable choice of the impeller design, depending on the controlling transport process in the reactor. The boundary conditions described in this paper and the predicted flow in agitated reactors using these conditions can also be used to directly simulate a particular transport process in the reactor rather than using the two parameters described above for the design purpose. Ranade et al. (1991) have developed a mathematical model for the relationship between flow and mixing. The work of simulating the mixing performance of the impellers studied in this work is in progress along similar lines. 4. Conclusions
It has been shown that the different axial impellers develop markedly different flows in the impeller vicinity as well as in the bulk. Flow data for the specification of the impeller boundary conditions have been reported for eight axial flow impellers. The model described in the work can be used to simulate flow generated by these impellers. Two parameters were defined to characterize the flow characteristics of these impellers. The hydraulic efficiency hae been found to vary from 23 to 85% whereas the pumping effectiveness was found to vary from 19 to 89%. This difference in the flow will have a great influence on the values of design parameters such as the extent of mixing, the critical impeller speeds for gas dispersion and solid suspension, and the heat-transfer coefficient. A systematic investigation regarding the flow structure and mixing is in progress.
NQs
?H
?E
1.74 2.01 1.18 1.9 2.03 1.76 0.81 1.25 0.68 0.68
65 66 57 54 62 67 23 81 85 85
44.8 32.4 74.7 57.8 33.0 45.1 17.6 89.1 64.2 64.2
NQsINP
100
82.9 60.9 135.6 129.3 82.2 73.0 35.4 140.4 106.3 71.6
Acknowledgment V.V.R., V.P.M., V.S.S., and G.B.D. are grateful to the University Grants Commission, Government of India, for the award of fellowships. The research work was supported by a grant under the Indo-US Collaborative Materials Science Programme (CE 1). Nomenclature
C, = constant of k-e model Cz = constant of k-e model C D = drag coefficient C, = chord length C, = span of blade D = impeller diameter, m E D = energy dissipation rate in the bulk of the tank outside the impeller region, W EIMP= energy input rate, W H = height of liquid in the tank, m k = turbulent kinetic energy per unit mass, m2 M, = tangential contribution to momentum from lift and drag M, = axial contribution to momentum from lift and drag N = impeller speed, s-l N p = power number, P/pJPD5 N Q = flow number, Q/ND3 N,, = secondary flow number, Qs/ND3 P = power consumption, W p = pressure, N m-2 Q = impeller pumping capacity, m3 s-l Qs = overall volumetric liquid circulation rate, m3 s-l R = vessel radius, m RI = impeller radius, m r = radial coordinate, m T = tank diameter, m U = radial mean velocity, m s-l U, = absolute value of mean velocity parallel to the wall at the near wall node, m s-l Vtip= impeller tip velocity, m 8-l u = radial rms velocity, m s-l V = mean tangential velocity, m s-l u = tangential rms velocity, m s-l W = mean axial velocity, m s-l W, = blade width, m w = axial rms velocity, m s-l y = distance from the wall, m y+ = wall Reynolds number z = axial coordinate, m Greek Letters a=
blade angle
Ind. Eng. Chem. Res. 1992,31, 2379-2385
6 = angle made by flow direction with horizontal plane
6 = angle of incidence 4 = rate of turbulent energy dissipation per unit mas, m2 s3 t = solidity coefficient VH = hydraulic efficiency, (eq 4) = pumping effectiveness (eq 8) 0 = tangential coordinate cc = molecular viscosity, Pes cct = eddy viscosity, Pes cceff = total effective viscosity, Pes p4 = effective viscosity for the variable 4 p = liquid density, kg m-3 = Prandtl number for turbulent kinetic energy = Prandtl number for turbulent energy dissipation rate 4 = generalized notation for transport variable T, = wall shear stress, N m-2 Qk
u,
Literature Cited Deshpande, G. B. Fluid Mechanics of Agitated Reactors. M. Chem. Eng. Thesis, University of Bombay,, 1988. Duncan, W. J.; Thom, A. S.; Young, A. D. Mechanics of Fluids; Arnold: London, 1970. Fort, I. Mixing: Theory and Practice; Uhl, V. W., Gray, J. B., Eds.; Academic Press: New York, 1986;Vol. 111. Joshi, J. B.; Pandit, A. B.; Sharma, M. M. Mechanically Agitated Gas-Liquid Reactors. Chem. Eng. Sci. 1982,37,813-844. Kaminoyama, M.; Saito, F.; Kamiwano, M. Numerical Analysis of 3D Flow of Pseudo-Plastic Liquid in a Stirred Vessel with a Turbine Impeller. Znt. Chem. Eng. 1990,30,720-728. Mann, R. Gas-Liquid Contacting in Mixing Vessels; Inst. Chem. Eng.: Rugby, 1983. Mishra, V. P. Unpublished results, 1992. Nagata, S. Mixing: Principles and Applications; Kodanska Ltd.1 Wiley: Tokyo, New York, 1975.
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Oldshue, J. Y.Fluid Mixing Technology; McGraw-Hik New York, 1984. Pericleous, K. A.; Patel, M. K. The Modelling of Tangential and Axial Agitators in Chemical Reactors. PCH, Physicochem. Hydrodyn. 1987,8,105-123. Placek, J.; Tavlarides, L. L. Turbulent Flow in Stirred Tanks-I Turbulent Flow in the Turbine Impeller Region. AIChE J. 1985, 31, 1113-1120. Ranade, V. V. Design of Multiphase Reactors. Ph.D. Thesis, University of Bombay, India, 1988. Ranade, V. V.; Joshi, J. B. Flow Generated by Pitched Blade Turbines-I Experimental. Chem. Eng. Commun. 1989,81, 197-224. Ranade, V. V.;Joshi, J. B. Flow Generated by a Disc Turbine-I Experimental. Chem. Eng. Res. Des. 1990a,68,19-33. Ranade, V. V.; Joshi, J. B. Flow Generated by a Disc Turbine-11: Mathematical Model and Comparison with Experimental Data. Chem. Eng. Res. Des. 1990b,68, 34-50. Ranade, V. V.; Joshi, J. B.; Marathe, A. G. Flow Generated by Pitched Blade Turbines-11 Mathematical Model and Comparison with Experimental Data. Chem. Eng. Commun. 1989,81, 225-247. Ranade, V. V.; Bourne, J. R.; Joshi, J. B. Fluid Mechanics and Blending in Agitated Tanks. Chem. Erg. Sci. 1991,46,1883-1893. Rewatkar, V. B.; Joshi, J. B. Effect of Impeller Design on Liquid Phase Mixing in Mechanically Agitated Reactors. Chem. Eng. Commun. 1991,91,322-353. Saraph, V. S.Fluid Mechanics of Agitated Reactors. M. Chem. Eng. Thesis, University of Bombay, 1989. Tatterson, G. B. Fluid Mixing and Gas Dispersion in Agitated Tanks; McGraw-Hilk New York, 1991. Wallis, R. A. Axial Flow Fans: Design and Practice; Newnes: London, 1961.
Received for review January 9, 1992 Revised manuscript received June 10, 1992 Accepted June 29, 1992
Fluid Flow in Capillary Suction Apparatus D. J. Lee**+ and Y. H. Hsu Department of Chemical Engineering, Yuan-Ze Institute of Technology, Taoyuan, Taiwan, 32026, R.O.C.
The fluid flow through porous media in capillary suction apparatus (CSA)was investigated experimentally and theoretidy. Water, with and without surfactant additives, methanol, and ethylene glycol were used to study the effects of liquid on capillary suction time of apparatus with various column radii and sampling locations. A model based on the saturation-diffusion mechanism was developed. The results showed satisfactory agreement with the experimental data. A modified capillary suction time was proposed based on the model which incorporated only the liquid effect and was independent of the apparatus used. Experiments with kaolin slurry showed that the model could be used to describe the fluid flow behavior in CSA with cake formation, if the liquid saturation under the inner cylinder was taken as less than unity.
Introduction Capillary suction apparatus (CSA) has been widely used since first developed by Gale and Baskerville in 1967 (Wilcox et al., 1987; Dohanyos et al., 1988; Vesilind and Davis, 1988; Lu et al., 1989; King and Forster, 1990). A CSA is composed of two plastic plates, Whatman No. 17 chromatugraphy paper, a stainless steel cylindrical column, several electrodes serving as sensors, and a timer. Sludge is poured into the column and the time the filtrate needs to travel between two concentric circles is called the capillary suction time (CST). Since the filter paper is ma-
* T o whom correspondence should be addressed.
Present address: Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan, 10764,ROC.
chine-made, the filtrate would move faster along the grain than across. Gale and Baskerville (1967) found that the difference is about 20%. Baskerville and Gale (1968) obtained a correlation between the CST and the specific resistance under filtration of sludge. It was shown that a sludge with a long CST indicated a sludge that when filtrated would form a cake with high specific resistance. Karr and Keinath (1978) investigated the factors influencing the sludge dewaterability. In their work, a CSA with columns of two inner radii was used. It was clear that the CST was a strong function of the column radius. Nguyen (1980) developed a theory for modeling the CSA by assuming the liquid moving in the filter paper as a piston-like process. An equivalent radius defined as the square root of the product of the long and short principle
0888-5885f 92f 2631-2379$Q3.oO/Q 0 1992 American Chemical Society
