Ind. Eng. Chem. Res. 1991, 30, 1784-1791
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Wiedmann, J. A.; Steiff, A.; Weinspach, P. M. Investigation of “Power Consumption, Suspension and Flooding Characteristics of Stirred, Aerated Slurry Reactors”. Chem. Eng. Commun. 1980,
Rewatkar, V. B.; Raghava Rao, K. S. M. S.; Joshi, J. B. Some Aspects of Solid Suspension in Mechanically Agitated Reactors. AIChE
J. 1989, 35, 1577.
Rewatkar, V. B.; Raghava Rao, K. S. M. S.; Joshi, J. B. Dependence of Power Number on Impeller Design. Chem. Eng. Commun.
6, 245.
Wong, C. W.; Wang, J. P.; Huang, S. T. Investigations of Fluid Dynamics in Mechanically Stirred Aerated Slurry Reactors. Can. J. Chem. Eng. 1987, 65, 412. Zwietering, Th. N. Suspending of Solid Particles in Liquid by Agitators. Chem. Eng. Sci. 1958, 8, 244.
1990, 88, 69. Sai to, F.; Kamiwano, M. Power Consumption, Gas Dispersion and Solid Suspension in Three Phase Mixing Vessels. Proceedings of the 6th European Conference on Mixing; BHRA: Cranfield, Bedford, U.K., 1988; p 407. Subbarao, D.; Taneja, V. K. Three Phase Suspension Agitated Vessels. Proc. 3rd Eur. Conf. Mixing 1979, 1, 229.
Received for review August 14, 1990 Accepted February 25, 1991
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Critical Impeller Speed for Solid Suspension in Mechanically Agitated Three-Phase Reactors. 2. Mathematical Model Vilas B. Rewatkar and Jyeshtharaj B. Joshi* Department of Chemical Technology, University of Bombay, Matunga, Bombay-400 019, India
A mathematical model has been developed for describing the solid suspension on the basis of a sedimentation-dispersion model. All the parameters such as vessel diameter, impeller diameter, sparger design, particle size, and loading have been included in the model. The flow pattern in mechanically agitated reactors (measured by laser-doppler anemometer together with mixing time) has been used for the estimation of model parameters. All the experimental data collected in part 1 have been analyzed on the basis of the proposed mathematical model.
Introduction
small vessel diameter used by Narayanan et al. (1969). Similar is the case with the values of Baldi et al. (1978) (lower by 50%) and for similar reasons. The critical speeds of Kneule and Weinspach (1967) deviate from Zwietering’s values in the range -15% to +11%. The discrepancy is perhaps due to the dished tank bottom used by these investigators as against the flat bottom used by Zwietering
The importance of solid-phase suspension in mechanically agitated two- and three-phase reactors has attracted the attention of many investigators. The focus has been on the development of criterion for the critical impeller speed for solid suspension. Joshi et al. (1982) and Gray and Oldshue (1986) have critically reviewed the literature on mechanically agitated contactors. The aim of present paper is to develop rational correlations for two- and three-phase systems.
(1958).
The measured Ns value also depends upon the suspension criterion. Einenkel (1980) used the particle concentration variance (with respect to height) as the suspension criterion, whereas in the studies mentioned above the criterion was “the speed at which particles do not remain on the tank bottom for more than 1 or 2 s” (Zwietering, 1958). Due to the difference in the suspension criteria, the Einenkel values deviate by about 70% from those of Zwietering. Bohnet and Niesmak (1980) have used “relative standard deviation of concentration”, and these values deviated by about 43% from Zwietering’s values. Kolar (1961) has used absorbance at two locations (one above the impeller and the other below the impeller) as a criterion of suspension. With this criterion for lowdensity particles his values are lower by 24% and for high-density particles they are higher by 30%. Nienow (1968) used the same criterion as that of Zwietering (1958), and his values agree very well with those calculated by Zwietering’s correlation (7%). Recently, Chudacek (1986) systematically investigated the solid suspension in mechanically agitated reactors. He showed the strong dependence of suspension speed on the suspension criterion. Further, the scaleup was also shown to be dependent on the suspension criterion in addition to the impeller and vessel geometry.
Review of Experimental Work The suspension of solid particles in a two-phase (solidliquid) system has been extensively studied in the past.
The range of variables covered by the various investigators is summarized in Table I. The relative comparison of these studies is shown in Figure 1. Zwietering (1958) carried out a systematic study of solid suspension over a wide range of variables. He proposed the following empirical equation:
Ns
-
sdp°-2r-01(gAp/#)L)0-46X013/D0·86
(1)
where the exponents of y, d„, gAp/pL, and X were found to be independent of impeller diameter, vessel size, impeller clearance, and impeller to vessel diameter ratio. All the variations in the system were accounted by the dimensional constant s. The Ns values of Narayanan et al. (1969) as seen in Figure 1 are lower than those predicted by Zwietering’s correlation (22%). This could be attributed to the very *
Thus from Figure 1, it can be seen that the critical impeller speeds observed by different investigators deviate in the range -25 to +70% from Zwietering’s values.
To whom correspondence should be addressed.
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©
1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 8,1991
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Table I. Literature Survey of Suspension of Solid Particles in Mechanically Agitated Contactors (Solid-Liquid System) X, 10% ref T, m D, m 10% m Pl. kg/m3 impeller type W/D w/w % kg/ms dp, Mm Zwietering (1958)
2, paddles
6,
Narayanan et al.
DT
propellers vaned disk 8, paddle
510-1810
125-850
0.5-20
0.32-9.1
0.036, 0.046, 0.057
140-1600
106-600
2.5-20
1.0
0.14, 0.238, 0.289
0.051, 0.1
1600-
45-140
10-50
1-2
0.165, 0.218, 0.345 0.14
0.052, 0.07, 0.11
150
570-1640
3-20
1.0
530-1660
153-9000
0.1-1.0
1.0
1640
220-1120
0-6.1
1.0
1414-
50-545
0.2-2
0.0645-
63-3070
30
3.17 1.0
14.5-45.6 0.7-5.4
1.0,10
12.55-
1.0
0.154, 0.192, 0.24, 0.29 0.45, 0.6
0.06, 0.08, 0.112, 0.16 0.224
0.114, 0.141
0.5, 0.25 0.2 0.1
(1969)
Weismann and
6, paddle
Efferding
8700
(1960)
Kolar (1961)
propeller 6, PTD
Nienow (1968)
6,
DT
0.0364, 0.049, 0.073 0.08
0.2
0.122, 0.19, 0.229 0.164
0.032, 0.04, 0.048
0.25
0.051
1666-
PTD propeller
0.3, 1.0 0.29
0.05, 0.1, 0.333 0.1
490-3470 50-1480
19-5000
propeller
0.365, 0.79
0.12, 0.24
1870, 1460
200-630
50-1900
80-2800
154
700
1650
157-2000
7.6, 32
5 1.0
259, 1600
85-4000
2.56-12
1.0
Bourne and propeller Sharma (1974) Baldi et al. (1978) 8, DT Subbarao and Taneja (1979)
Herringe (1979) Bohnet and Niesmark
3,
propeller
0.172
1800 2010
6,
1050 2480
1.0
(1980)
Einenkel (1980) Chapman et al. (1983)
Musil et al. (1984)
Buurman et al.
6, DT 4, PTU 4, 6, PTD 3, PTD 24°
PTD PTD
0.14, 0.28 0.457, 0.9 0.79 0.15, 0.195 0.245, 0.315
0.2, 0.3
0.48, 4.25
0.2, 1.72
0.25
0.15, 0.2 0.3, 0.4 0.5
0.10-0.54 0.165
0.2
2.0
1650
110-290
6.1-24.4
1.0
0.5, 1.0
0.167, 0.33
0.2
2.9
1650
77-290
14.7-46.0
1.0
0.292
0.064
350-2200
1000, 3200
1-10
1.0
1520
100-2000
0.7-50
1.0
(1985)
Ditl and Rieger (1985)
Chudacek (1985) Chudacek (1986)
Gray (1987)
3, 6, PTD 6, FBT 3, propeller 6, 4, ADT 3, propeller 6, PTD 6, FBT
3, PTD DT Raghav Rao et al. 6, DT (1988) 6, PTU 6, PTD
0-3.0
0.29, 0.3 0.56, 0.91 1.83 0.6
33°, 45° 6, 4,
41.8
0.3, 0.4 0.57, 1.0 1.5
0.0976 0.102 0.19, 0.285 0.19 0.10, 0.142, 0.19, 0.25, 0.33, 0.365, 0.5
However, it will be interesting to note that the exponents on various parameters in the studies cited above are almost similar (Table II). Further, the proportionality constant s and its dependence on T/D ratio are different for different types of impeller. The value of s has been found to be low for the axial impellers (propellers) and high for the radial impellers, placing pitched turbines in the range inbetween. Bohnet and Niesmak (1980) calculated critical impeller speeds using nine correlations developed by various investigators (Gray and Oldshue, 1986). The calculated N$ values were in the range -56 to 250% of the experimental results of Bohnet and Niesmak. These large deviations of the calculated suspension speeds were probably due to the test parameters outside the range of those used in the tests on which the nine correlations were based. In addition, the suspension criteria were different for different studies. Further, design parameters such as blade width and blade thickness have an influence on Ns. Usually these details are not reported, and the differences in these parameters are also partially responsible for the discrepancies in the observed values of different investigators.
0.2 0.25 0.3 0.35
3.2, 9.5
0.5, 4.0, 15.0
2.3, 0.8
1.0
1.0
4.3, 6.4
Review of Mathematical Models for Solid Suspension Table III summarizes the mathematical models developed in the past. The assumptions made in these models have been listed in Table IV. These models can be classified into two categories. In the first category, either force or energy balance was established in the vicinity of the vessel bottom. In the second category the force or energy balance was established in the main body of the vessel. The first category criteria were developed to represent the lifting of solid particles from the bottom. On the other hand the force and energy balance in the bulk region represents a condition of homogeneous solid suspension. Both these criteria suffer from the following
limitations: (1) The force and energy balances have been rather incomplete. This is because the momentum and the energy transport are complex in stirred vessels. Therefore, with the present status of knowledge, the quantification of various force and energy terms is very difficult. For instance, (i) the energy dissipation rate is inhomogeneous,
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Table II. Correlations for the Critical Impeller Speed for Solid Suspension values of expts on governing pararas (eq X ref 7 Ap/pl Baldi et al. (1978) Bohnet and Niesmak (1980) Chudacek (1986) Chapman et al. (1983)
0.125 0.3 0.08 0.12
Einenkel (1980)
0.3
Gates et al. (1976) Herringe (1979) Kneule and Weinspach (1967) Narayanan et al. (1959) Nienow (1968) Raghav Rao et al. (1988)
Rewatkar et al. (1989) Wiesman and Efferding (1960) Zwietering (1958) 0
Exponent
on
0.17 0.08
0.4 0.3
0.1
0.4
0.5 0.1
0.18 0.25 0.22 0.12 0.1 (PTD, PTU) 0.125 (DT)
0-0.1
0.13-0.2 0.4 0.5 0.5 0.43
0.14 0.52 0.15
0.17-0.67 0.13-0.2 0.3 0 to 0.2 0.5 0.21 0.11 (PTD, PTU) 0.15 (DT)
terminal settling velocity terminal settling velocity
0.17 0.13
0.1
0.45
1)
0.2
D (T = const) -1.9 (DT) -0.8 (propeller) -0.58“ (PTD) -1.67 (PTD) -2.45 (DT) -1.15 (propeller) -1.7 (propeller) -2.4 -0.7“ (PTD) -1.7 (propeller) -2.0 (paddle) -2.25 (DT) -1.15 (PTD) -1.15 (PTD) -1.67 (paddle) -2.35 (DT) -1.6 (propeller)
D at constant D/T.
Table III. Review of Mathematical Models for Solid Suspension ref (type of impeller flow) basis/criterion for solid suspension
assumption net force acting on the suspended particles equated to zero. In the W force balance equation, the downward force included the gravity and the force exerted by the slurry head above the particle. The upward force consisted of buoyancy and the drag exerted by the rising fluid. Subbarao and Taneja (1979) (axial flow impeller) suspension assumed to be achieved when the settling velocity of a 5,6 particle equals the rise velocity in the bulk region 7-11 Baldi et al. (1978) (radial flow impeller) turbulent eddies of the order of particle size assumed responsible for the suspension of solid particles. The energy balance was established between the kinetic energy imparted by the eddies to the particles and the potential energy gained by the particle (pLV'2
Narayanan et al. (1969) (axial flow impeller)
“
dpApg)
Musil and Vlk (1978) (axial flow impeller)
at the condition of solid suspension, energy dissipated at the solid-liquid interface assumed equal to some fraction of the energy supplied by the impeller. The fraction was estimated using a decay
Voit and Mersmann (1986)
two criteria developed depending upon the dt/T ratio (i) for dJT < 5, 9, 10, 18-26 10"4, the mean axial velocity (Vc) assumed equal to the hindered settling velocity of a particle (Vg). In case of higher concentration of solids, the ratio of V, to Vg„ was assumed to be constant. This resulted into a constant Froude number criterion: Fr* = = const, (ii) For dp/T > 1CT4, Froude number was jVg2D2/oL/dpApg defined on the basis of vessel diameter: FrT* + N2D2pi/TApg = const suspension criterion depends upon the particle size of the Archimedes 27-32 number (Ar). The suspension of fine-grained particles (Ar < 40) immersed in viscous sublayer occurs due to the shear stress in the wall boundary layer of the vessel bottom. Equating the shear force and the buoyancy force the relationship between Reynolds number and Archimedes number (fieT2 = 2/3Ar) has been derived. In the case of coarse-grained particles (Ar > 40), the particle sedimentation velocity was assumed to be insignificant as compared with the fluid velocity. The liquid velocity was estimated using the impeller pumping capacity 28, 33 particle suspension occurs due to a characteristics velocity of liquid (VB) acting on the particle. This characteristic velocity was calculated by measuring actual values of shear rate (rB) in the boundary layer using electrodiffusion method [Wichterle (1988)]. The value of minimum shear rate acting at the bottom has been considered for the model. At the condition of suspension, the ratio of characteristic velocity (yBdp) to the particle settling velocity (V,) was assumed to be a constant (Bj,). On this basis the following relationship was obtained between the normalized critical impeller speed (jVg*) and the normalized particles diameter (dp*): Nig* =
12-17
function
Molerus and Latzel (1987) (axial flow impeller)
Wichterle (1988) (axial flow impeller)
[(Bjs/AmJ(dp*/(18 + 0.6dp*3/2)]2/3
and complete descripation of the energy dissipation pattern in stirred vessels is not available, (ii) The drag coefficient for a particle in a turbulent field (as in the stirred vessel) is markedly different from the single-particle drag coefficient. The estimation of drag coefficient in stirred vessels is very difficult, (iii) The solid-phase concentration profiles
in the radial and axial directions have been incompletely understood, (iv) Information on velocity and turbulence structure as a function of impeller design, geometry, solid loading, and size is still lacking. (2) The force or energy balance in the bulk is expressed in different ways: (i) The particle settling velocity equals
Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991
1787
Table IV. List of Assumptions 1. fractional holdup of solids is low 2. solids uniformly distributed throughout the vessel so that there are no radial and axial gradients in solid-phase concentration 3. solids completely wetted by liquid, and there is no slip between the particle and fluid _
drag coefficient for the particle can be considered to have a unique value and equal to 1 average liquid velocity can be calculated by using pumping capacity of the impeller and the cross-sectional area for the liquid flow in the upward direction [tCT2 D8)/4] 6. for estimation of hindered settling velocity of a particle under turbulent conditions, the correlation for solid-liquid fluidized bed can
4. 5.
-
be used
turbulent eddies having sizes of the order of particle size responsible for lifting the particle ratio of energy dissipated at the tank based (Eb) to the average energy dissipation in the vessel (E¡) remains constant power number (JVp) remains constant in fully baffled and turbulent flow energy uniformly distributed in the vessel, and the average dissipation of power can be calculated from the impeller power number mean velocity of liquid is proportion to ND2/T condition of solid suspension in agitated vessels can be viewed as a state of complete fluidization of particles with a constant-concentration profile 13. relative movement of the solid particles in the bottom layer roughly obeys the laws of fully developed sedimentation 14. energy used (per unit time) by the fluidized particles for remaining in suspension to the power supplied by the impeller 15. physical properties of the mixed slurry can be used to calculate the power number 16. local velocity (u) and the local turbulence intensity (ur) are linear functions of the impeller tip speed 17. rate of energy dissipation can be calculated using E = pLu*//', where l' is the turbulence scale 18. wall for a turbulent flow around a spherical particle the drag coefficient is independent of the impeller Reynolds number 19. total power consumption of impeller equal to the product of the volumetric flow of liquid and the pressure drop across the impeller = const x pL(w,2) 20. impeller discharge velocity and the pressure drop ( ) related as 21. additional pressure drop occurs due to the circulation of solid particles that is proportional to the volumetric holdup of solid and the square of the swarm settling velocity of particles 22. ratio of swarm settling velocity to the average velocity of fluid remains constant (for higher concentration of solids) 23. drag coefficient of a single particle in a turbulent flow equal to 0.4 and does not change with the degree of turbulence in the liquid 24. solid particles deposited on the tank bottom have a constant bed porosity 25. particles lifted from the bottom with a certain velocity. The particle velocity decays as the particle elevation increases. The decrease in kinetic energy of a particle is compensated for by the increase in pot(mtial energy 26. constant tip speed criterion can be used for scaleup 27. for fine-grained particles, the mean circulation velocity of a fluid exceeds the particle settling velocity by several orders of
7. 8. 9. 10. 11. 12.
magnitude for fine particles, At < 40, the wall shear stress is responsible for solid suspension from the boundary layer wall shear stress (rw) at the point of suspension can be calculated from the assumption of plane turbulent boundary layer flow along a flat plate at a distance ’/2 from the leading edge of the plate 30. frictional forces between the solid particles and the wall are insignificant 31. particle settling velocity (Vs) insignificant as compared with the fluid velocity, because of which two-phase (solid-liquid) flow occurs with high velocity from top to bottom 32. mean upward fluid velocity can be calculated by using pumping capacity of liquid after the correction for solid concentration 33. shear force acting on the particle (in the boundary layer) at the bottom of a vessel responsible for solid suspension in both types (radial and axial flow) of the impeller 28. 29.
Prfdicttd
values of N|
suspension condition in bulk, they do not necessarily ensure the suspension from the vessel bottom. (3) The suspension criteria need to take into account the geometrical design details. Minor details such as blade thickness and hub diameter also influence the particle suspension. It has been difficult to include the effect of the shape of vessel bottom in the mathematical model. Chudacek (1985,1986) has discussed this problem in detail. From the foregoing discussion, it is clear that the principal hurdle in developing a good model has been the lack of knowledge of fluid mechanics in stirred vessels. In the present work the flow pattern measured by laser-Doppler aneometer [Ranade and Joshi (1989a, 1989b, 1990a, 1990b)] will be used for describing the suspension condi-
tion.
Figure
1.
correlation.
Comparison of various investigations with Zwietering’s
the upward velocity of fluid, (ii) The net force acting on the particle is zero. Or (iii) The energy dissipation rate at the solid-liquid interface is some fraction of the impeller input energy. Though these criteria perhaps describe the
At this stage we would like to bring out interesting point. Values of Ns for disk and PTD impeller using different particle sizes and in 0.57-, 1.0-, and 1.5-m-i.d. vessles are shown in Table V. The suspension speeds for disk turbine were obtained by Zwitering’s correlation, and in some cases our own measurements [Raghav Rao (1987)] have been used (both agree within 12%). It can be seen from Table V that the suspension speeds for disk turbine and for PTD are practically the same. To explain this observation, it was thought desirable to compare the flow pattern generated by these impellers at the points of solid suspension. The process of solid susension was described in part 1. In the case of DT, the suspension of particles in the central region below the impeller occurs at the end. In contrast, in the case of a PTD impeller, the last suspension occurs
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Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991
Table V. 773, X =
JV8 1
Calculated from Correlation (D
wt/wt
=
773, C
(Vs
=
%)
m
0.3 0.3 0.3 0.3 0.57 0.57 0.57 0.57 1.0 1.0 1.0 1.0 1.5 1.5 1.5 1.5
dp, µ 120 180 460
revolutions/s PTD DT“
Vs., m/s
6.56
0.02 0.034 0.08 0.20 0.02 0.034 0.08 0.20 0.02 0.034 0.08 0.20 0.02 0.034 0.08 0.20
2000 120 180 460 2000 120 180 460 2000 120 180 460 2000
“Calculated from Zwietering correlation (eq
1
Ds
7.49 8.12 9.80 13.15 4.34 4.71 5.68 7.62 2.715 2.944 3.55 4.77
7.61 9.67 12.49 3.80 4.41 5.60 7.24 2.38 2.76 3.5 4.53 1.67 1.94 2.46 3.18
in part 1).
The motion of solid particles in a gas-solid, solid-liquid, three-phase fluidized beds is usually described by the dispersion model. A similar description will be used for the stirred vessels. The dispersion model is given by following equation: (2)
The particles move in the downward direction because of the settling velocity (Vs), and the downward flux is given by the right-hand side of eq 2. The dispersion or distribution of solid particles occurs because of the turbulent eddies. The dispersion mechanism tries to make the concentration homogeneous. The flux of solid particles due to dispersion is given by the left-hand side of eq 2. Equation 2 describes the mass balance at any cross section. There is one more parameter that moves the solid particles in the upward direction. In a batch stirred vessel, the net liquid flow across any cross section is zero. However, the solid-phase concentration is expected to be more in the region of liquid upflow as compared to the region of liquid downflow. Because of this, the liquid circulation helps in solid suspension. This effect can be included in eq 2 as follows: =
(Vg
-
a Vc)
