Power Correlations for Close-Clearance Helical Impellers in Non-Newtonian Liquids Virendra V. Chavan* and Jaromir Ulbrecht Department of Chemical Engineering, University of Salford, Salford M.5 4 V T , Lancashire, England
Generalized power correlations have been presented for a variety of close-clearance helical impellers, i.e., helical screw impellers with a draught tube, helical ribbon impellers, and combined ribbon-screw impellers. The validity of the relations has been tested over a wide range of geometries using data on two classes of non-Newtonian liquids (inelastic shear thinning, elastic shear thinning). The experimental data from this work as well as those from literature are used to check the results. For inelastic shear-thinning liquids agitated by the screws with a draught tube the correlation is Po Re' = (n/2)(a)(d,/d)(4n/n(X2'n - 1 ) )"X2(1 ~ / c ) O . ~ ~ [ ( t - d,)/l,]-o.04G(c,/d)-0~03G.For ribbon and combined ribbon-screw impellers it is Po Re' = 2.5n(a)(d,/d)X2. { 4n/n(X?'n - 1 )
+
.
1".
T h e close-clearance lielical impellers (Figures 1 and 2 ) are sliowii to he the most efficieiit impellers for mixing of highcoiisisteiicy liquids by several workers (Gray, 1963; Hoogeridoni mid Den-Hartog, 1966). Most of the literature on the aspects of pon-er coiisuinptioii has been reviewed in our earlier publications (Chnvan, et al., 1972; C'havan and Ulbrecht, 1972). The most important single geometrical variable is the clearaiice betv-eeii the impeller arid the nearest solid boundary. 111the case of R helical screw with the draught tube it is the gap 1ietn.eeii the >crewaiid the draught tube and for the ribbon ini1)eller it is the gap between the inipeller and the vessel. For the relatioriships to be geiierally acceptable, it is necessary to extend their validity over a wider range of clearances and it i h also necessary to verify whether the relations will be applicable for viscoelastic liquids. For the combined ribbon-screw impellers the literature provides some data on Sewtonian liquids. Some results were obtained in this work for noli-Seivtonian liquids. The geometrical arrangements considered in this publication are listed iii Tables I-IT.
The viscobity ( p ) of the corn syrup n a s 28 P. The flow indices of Satrosol. CMC,PAA1, and P.-ih2 were 0.59, 0.47, 0.39, and 0.35>respectively, and the consistency indices ( k , dyn/ cm* sec-") of these materials Rere 108, 121, 48.5, and 104, respectively. All the e\periments were carried out a t 20 f 1'. Analysis of the Results
Screw Impellers with Draught Tube. d model based upoii the comparison of the highly sheared gap between the screw aiid the draught tube, and t h r flow between two coiiceiiti I C cylinders, was previously suggested by tlie authors (Chavaii, et al., 1972). The relation could be nritteii as
Po Re
=
2
=
Z'
(1)
for Newtonian liquids arid
P o Re'
for inelastic iioii-Newto~iianliquids, where
(3)
Experimental Procedure
The impellers were driven by a dc motor of shunt type capable of giving a maximum power of 1 hp. The electric current \vas supplied through a rectifier and a variac. Thus the speed of the motor could be adjusted by varying the voltage input. The shuiit motor gave practically constant speeds. -4 dynamometer based upoii the principle of having a weak joiiit oil the impeller shaft arid measuriiig the twist due to the weak joint was employed. The dynamometer was calibrated dyiiuiiii~allyusing two coi~eritriccylinders. The details 011 tlie calibratioii aiid tlie torque and speed (0.1-1.2 rps) nieasuremeiits are given elsewhere (Chavaii, 1972). .iqueous solutions of corn syrupl Satrosol, CMC, and PAL4 were used as test, liquids and the rheological properties (Chnvun, 1972) were measured on a rheogoniometer in the shear rate range 1-100 sec-1, which corresponds with the range of average shear rates in the vessel estimated on the basis of the hypothesis suggested by lletzner and Ott,o (1957). The properties of liquids were measured before and after each esperimerit and no substantial difference was found. 472
Ind. Eng. Chem. Process Des. Develop., Vol. 12,
NO. 4, 1973
2'
=
4x
.(.i($)X(
"
n(X2'" - 1)
A
c d(s
d)2+
(s d)'
=
15)
d, d,
71
+
(1 - [ l - 2 ( w ' d ) ] * ) n [ l - 2(w d)I(l,d)
17)
Further, if oiie considers the remaining geometrical variables arid assumes that effect of non-Sevvtonian characterlstics is solely represented by the function n(X2'"
-
1)
~~~
Table 1. Geometrical Variables for Helical Screw Impellers with Draught Tube No.
d, cm
fld
hld
1ld
sld
wld
cld
G.l G.2 G.3 G.4 G.5 G.6
30.5 20.35 20.35 20.35 19.05 14.00 12.70
1.50 2.25 2.25 2.25 2.40 3.28 3.60
1.94 2.70 2.70 2.70 3.10 4.24 4.65
1.50 2.25 2.25 2.25 2.40 3.24 2.96
0.96 0.50 1 .00 1.00 0.80 0.93 0.79
0.42 0.39 0.39 0.39 0.42 0.39 0.38
0.104 0.156 0.156 0.156 0.167 0.228 0.250
G.7
d,/d
1,ld
1.16 1.05 1.05 1.74 1.12 1.53 1.14
1.83 2.54 2.54 2.54 2.93 4.01 4.40
cJd
0 0 0 0 0 0 0
104 lt56 156 156 167 228 250
Table II. Geometrical Variables from Literature for Helical Screw Impellers with Draught Tube
G. 1 G.2 G.3 G.4 G.5 G.6 G.7 G.8 G.9
d, cm
Ref
No.
Chavan, e t d . Chavan, et $1. Chavan, et 21. Chavan, et $1. Chavan, et al. Chavan, et al. Xagata, e t d . S a g a t a , et ( X I . Xagata, et cal.
20 20 19 20 20 19 4 6 6
(1972) (1972) (1972) (1972) (1972) (1972) (1957) (1957) (1957)
tld
35 35 05 35 35 05 50 50 28
hid
2 25 2 25 2 40 1 50 1 50 1 60 2 22 1 54 1 59
2 2 2 1 1 1 2 1 1
l/d
63 63 80 75
sld
2 31 2 31 2 47 1 56 1 56 1 67 2 0 1 30 1 42
75 83 22 54 59
0 1 0 0 1 0 0 1 0
w/d
5 0 8 5 0 8 67 38 72
0 0 0 0 0 0
dr/d
cld
39 39 42 39 39 42
0 0 0 0 0 0 0 0 0
31 31 33 19 19 20 11 07 075
1 1 1 1
I 1 1 1 1
13 13 2 13 13 2 15 11 08
Ir/d
c,ld
2 2 2 1 1 I
0 0 0 0 0 0 0 0 0
25 25 4 50 50 60 1 55 1 23 1 26
19 19 2 13 13 14 25 16 16
Table 111. Geometrical Variables from literature for Helical Ribbon Impellers No.
Ref
G. 1
Kagar,a, et al. (1972) Xagal,a, et al. (1972) S a g a t a , et al. (1972) Sagai,a, et al. (1970) Nagala, et al. (1970) Naga’a, et al. (1970) Faga.a, et al. (1970) Kaga ta, et al. (1970) Kagata, et al. (1970) Kaga ta, et al. (1970) Kagata, et al. (1970) Muller (1971) Kappel and Seibring (19iO) Kappel and Seibring (1970) Kappel and Seibring (1970) 1;or's k (1970)
G.2 G.3 G.4 G.5 G.6 G.7 G.8 G.9 G.10 G.ll G.12 G.13
G.14 G.15 G.16
sld
Wld
NR
1 8 1 25
1 1 1 1 1 1 1 1 1 1 1 1 0
0 0 0 0 0 0 0 0 0 0 0 2 98
1 1 0 5 0 75 1 0 1 25 1 25 1 25 1 25 1 25 1 25 1 24 0 39
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.09 0.1
2 1 1 2 2 2 2 2 2 2 2 1 2
1 05
1 25
0 98
0 62
0.1
2
6
1 05
1 25
0 98
1 05
0.1
2
9 6
1 05
1 0
1 0
0.1
2
d, cm
+Id
hld
1 1
6
1 05 1 05 1 05 1 068 1 068 1 068 1 05 1 079 1 105 1 158 1 579 1 08 1 05
6
19 19 19 19 19 19 19 19
Ild
1
Table IV. Geometrical Variables for a Combined Ribbon-Screw Impeller No.
Ref
d i , cm
tldi
hldi
I/di
sddi
~ildi
G. 1 G.2 G.3 G.4 G.5
This wcrk Kagata et al. (1972) Xagata et al. (1972) S a g a t a et al. (1972) S a g a t a et al. (1972) Xagata, et al. (1972) Uurqbarher (1969)
45
103 105 105 105 105 I05 1 05
130 105 105 105 105 105 0 76
115 1 0 1 0 1 0 1 0 1 0 0 36
057 1 0 1 0 1 0 1 0 0 5 0 72
012 0 1 0 1 0 1 0 1 0 1 0 14
G.6 G.7
then one can write
Po Re Po Re’
R
= =
=
I?’
Zfl
=
2’~’~
(8) (9)
where fl=
jf(
cid,
t - d, 1, d
dild?
wrld.
sild-
NR
0286 042 042 0525 0525 035 0 72
0 4
1 0 2 2 1 5
1 2 1 2 2 1 I
1
0 33
1 5 0 72
influelice on poiver can be iieglected. The fuiiction SI accounts for all the rest of the geornetrical variables. Fuilctioiis such as (1 c i d ) , (t aud (c, d ) could he considered t o take into account such effects. Since the esperime~italvalues of lyere kxiowi for all the geometries in Table I and 11, it as pos.;ihIe t o relate I?,.,,, ‘Z to the variables (1 d c), ( t - d , lr), ant1 (c, d ) u3iiig niultilde regression technique. The result’ is given by
+
+
C,
2,;)
p )
(10,
Wheii the clearance of the draught tube from t h e level of the liquid is not small enough to be restrictive t o the flowv:its
(e, d ) -0.036 (1 1) Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 4, 1973
473
Table V. Power Correlation for Helical Screw Impellers with Draught l u b e in Newtonian Fluids" Geometry from Table I1
1 OO[(iexpt
G.l G.2 G.3 G.4 G.5 G.6 G.7 G.8 G. 9 Data taken from literature.
- Epredieted)/iexpt] -2 -3 +21 -6 - 17 +2 12 +2 12
+ +
~~
Table VI. Power Correlation for Helical Screw Impellers
' 7 ' with Draught Tube in Non-Newtonian Fluids A
Figure 1. Helical screw impeller with a draught tube
Liquid
Corn syrup
k
I
fl Natrosol
i
Figure 2. A single-bladed helical ribbon impeller and combined ribbon-screw impeller
Le.
For inelastic liquid from eq 9 one can write
Table V gives the comparison of the correlation viith the esperimental data on Xewtonian liquids from literature. (For geometrical arrangements refer to Table 11.) K'ovak (1970) recommended the use of theoretically derived espressions for screw extruders to describe the power consumption. Using this approach the values of Po Re ( L e . , I?) were calculated for the systems in Table I and compared v ith the esperimental values. The calculated values were much higher than those obtained from the experiments. K h e n the ratio of the draught tube diameter to the impeller diameter was more than 1.54, the calculated values of R were 10 times higher than the esperimental value,.. The reason for thls failure lies in the basic dissimilarities in the geometrieq of the screw extruder and the scretv miser. The gapq between the 474 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 4, 1973
Geometry from Table I
G. 1 G.2 G.3 G.4 G.5 G.6 G. 1 G.2 G.3 G.4 G.5 G .6 G.2 G. 1 G.2 G.3 G.4 G.5 G.6 G. 1 G.2 G.3 G.4 G.5 G.6
- OO[(Kexpt
-
Kpredieted)/Kexpt]
+-210 -4 +7 +I
-9 $2 $3 0 +20 +6 +5 -5 12 +5 -4 - 12 -6 +2 +16 -2 0 s 7 +6 $1
+
stationary wall (the barrel or the draught tube) are much higher in the case of screw misers. The ratio between the blade width and the channel width are very small for a n extruder. The theories developed for an estruder make use of certain assumpt,ions which are not valid for a screw mixer geometry. Further the assumlhon of a constant helix angle is violated in the case of a miser where there is fivefold variation in the angle along the width of the blade. The data obtained in this work for various liquids and for various geometries (from Table I) are compared with the above relations in Table VI. Two of the test liquids (PAA solutions) were viscoelastic and showed substantial normal forces on the Weissenberg rheogoniometer (Chavaii, 1972). HoiTever, the pon-er consumptions for these liquids could also be interpreted by the equations obtained for inelastic liquids. Thus the elasticity does not seem to have any effect on the pon-er consumption, a t least when the liquids are also showing viscous anomalies. From the correlations it can be inferred that the influence of geometrical variables outside t,he draught tube, Le., (t d,; I,) and (c,ld), is relatively very small and most of the polver
is consumed inside the draught tube and in the clearance betneeri the impeller and the bottom of the vessel. For (t - d, i,) > 0.11 aiid (cr/d) > 0.17, the effect of each of these on pon er ~ o u l be d less than 5%. Ribbon Impellers. In our previous publication (Chavan a n d Ulbrecht, 1972) a correlation based upon similar considerations as those for helical screw impellers in a d r a u g h t tube has been proposed. T h e correlation could be rewritten as follows Po R e
=
?I
4
2 . 5 ~ ~
=
n(X2'n
- 1)
where
d, t - = d d
- 2(w
a) /[ill I
{(t
d) - 1 - 2(W'd) (t d ) - 1
and a,tlie dimelisionless surface area, is given by
~~
Table VII. Power Correlation for Helical Ribbon Impellers in Newtonian Fluids" Geometry from Table 111
G. 1 G.2 G.3 G.4 G.5 G.6 G.7 G.8 G.9 G. 10 G.ll G.12 G.13 G.14 G.15 G.16 Data taken from literature.
1~
-
~ [ ( K e x p t Kpreciieteci)/Kexpt]
- 18
0 -3 - 14 -9 -6 -8 -1 -11 -3
+-2813 -41 - 37
- 27 -25
Table VIII. Power Correlation for Compared RibbonScrew Impellers in Newtonian Fluids Geometry from Table IV
approximately
u
=
r ( Z / d )( W j d ) / ' ( s ; ' d )
(18)
The corresponding Newtonian relat'ioiisliip could simply be obtained by substituting n = 1 and k = p
For Kewtonian this correlation was tested for 17 different geometrical arrniigeniente. It has also been verified for nonNewtonian liquids with varying degree of pseudoplasticity and five geometrical set-ups. The literature provides with data for 16 other geometrical set-ups. The values of Z? were calculated using eq 19 and comliared with the experimeiital values (Table VU). I n these calculations, the area has been obtained from eq 17. The areas given by eq 18 are 10-207G lower than those obtained from eq 17. For ribbons with more tliaii oiie > 1) the dimerisioiiless area should be obtained by blade (SR multiplying t'he area given by the above equations by the number of blades (.I-R). The correlation seeme to be satisfactory for first 11 geometries. Thus for 28 different geometrical set-ups 11.05 < t d < 1.58) it was possible to predict power coilsumption by the above relationships. However, tlie magnitudes of errors when the correlation was compared with tlie data from lluller (1971), Kappel and Seibriiig (19iO), and S o v a k (1970) are quite high. The reason for this discrepaiicy ma>-be in the fact, that eq 19 does not ccsnsider the probable influence of the clearaiice of the impellers from the bottom on the power conwmptiori. Unfortunately, magnitudes of this variable (c d)are not mentioned. by niaiiy authors arid thus a general relatioilship is not possible. It appeare, however, that the correlations (eq 14 aiid 19) predict the power sufficiently accurately as long as c / d is ea. 0.1. Combined Ribbon-Screw Impellers. E v e n oil the aspects of power requirements there is very litt,le previous information on these impellers. The geometries studies by S a p a t a , et aZ. (1972), and Burgbacher (1969) are detailed in Table ITalong with the geometry studied in this work. Since in this case the geometry acquires more complicat'ioiis. one may pose
Dimensionless area, am (ribbon)
1OO[(iiexpt Dimensionless area, a' (screw)
-
Kprzdioted)/ Kexptl
G.l* 0.98 0.176 - 16 G.2 0.47 0.26 -4 G.3 0.47 0.26 +2 G.4 0.47 0,44 -4 G.5 0.47 0.51 -4 G.6 0.81 0.20 i l l G.7 0.29 1.31 f81 The dimensionless area is with re3pect to the outer diameter of the ribhon. For CMC, Satrosol, PAA1, and PAA2 the errors were + 2 , -10, fl, and +14, respectively. more problems in relatiiig bhe power to the geometry and rlieologS-. Kheii the values of R (Po Re) were calculated (eq 19) considering only the geometry of the ribbon and compared with the experimeiital values of I? (Table VIII), it is observed that for the first' six geometrical set-ups R thus calculated are very near t,he experimental values of Z?. The order of magnitude of the differences in these cases is inucli tlie same as the p0'5ibk experimental errors. Hoxever. for the geometrical set-up of Burgbacher (1969) the experimental values of I? are 817, higher than the values obtained by using the rilibon correlation. 11ispectiiig the surface areas of both these elements one c m i conclude that as long as the surface area of the screw is smaller than or equal to the surface area of the ribbon the po\ver consumption for the combined ribbon-screw impeller (mi be safely predicted on the basis of ribbon geometry only. K h e n the screw area is much larger than the ribbon area (e.g., in the case of Burgbacher the screw area is more than four times the ribbon area), then the ribbon correlatioiis will underestimate the power coilsumption. The data obtained for four different non-Te\vtoiiiaii liquids are compared with the ribbon formulae (eq 14) and shon-ti in TableVIII. Conclusions
The arialpis presented in this work leads to some generalized power correlations for helical misers. Before using these Ind. Eng. Chem. Process Des. Develop., Vol. 12, No.
4, 1973 475
correlations it will be useful to specify the ranges of different geometrical variables for which these correlations have been verified. Such information could be obtained from the appropriate tables giving the geometrical variables (Tables IIV and the relevant tables in Charan, et d . , 1972; Chavan and Ulbrecht, 1972). I t may be worth stating that the correlations can be espected to be valid for impellers rotating in either direction of rotation. There is some previous evidence to confirm this statement. Moreover, the data collected from literature to support the correlations have been obtained in either direction of rotation. The liquids under consideration were of Newtonian, inelastic shear-thinning character. The correlations seem to be valid for liquids having flow indices between 0.35 to 1. Two of the liquids (PXA solutions) were viscoelastic. The extent of elasticity was det’ected by measuring the normal forces on the Weissenberg rheogoniometer (Chavan 1972). (Normal force per unit area in a cone and plate experiment was of the order of l o 3dynjcm2for the shear rate of 10 sec-l.) It appears (Table VI) that even though these liquids show elastic characteristics, the power consumption can be described by considering only the shear-thinning properties. It should also be pointed out that these correlations are valid for the low Reynolds number region. It seems that the critical values of these Reynolds numbers are dependent on the geometry of the system. Because of the lack of esperimental data it is not possible to find a relationship for the critical Reynolds numbers. From the available experimental evidence (Chavan, 1971; Chavan, 1972) i t appears that one may safely take the limiting Reynolds number (either Re or Re’) to be 10 in all the cases considered. The power consumption in a continuous system may also be predicted using the correlations obtained in this work as long as there are no major changes in the flow patterns, especially in the vicinity of t,he impeller. For example, in a downward flow system, if the inlet is a t the top of the draught tube near the impeller and the outlet is near the vessel walls a t the same level the general flox patterns should not differ much from the batch system. Under such circumstances the relevant equations may be used to estimate the power consumption even if the feed rates are smaller than the impeller discharge rates. Some data on torques n-ere collected by Chavan (1972) for helical-screw impeller in a draught tube under no-discharge-flow conditions and no measurable difference was observed between these values and the values obtained under the normal impeller-discharge-flox conditions. Nomenclature
a c
dimensionless surface area clearance b e b e e n the impeller and the bottom of the vessel = =
476 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 4, 1973
cr
= clearance between the draught tube and the bottom of the vessel d = impeller diameter de = equivalent diameter given by eq 6 and 16 d, = draught tube diameter h = height of the liquid column k- = consistency index in power law K = constant in power correlation for Newtonian liquids &’ = constant in power correlation for non-Newtonian liquids 1 = length of the impeller flight I , = length of the draught tube n = flow indes N = rotational speed of the impeller P = power consumed s = pitch of the impeller t = vessel diameter w = width of the impeller blade
GREEKSYMBOLS p p
= = =
density parameter in power correlations viscosity
DIMCSSIOSLESS NUMBERS Po = power number (P/pAV3d’) Re Re
= =
Reynolds number ( d 2 N p / p ) Reynolds number (d*X2--np/ k)
The above quantities may be expressed in any set of consistent units. literature Cited
Burgbacher, G., University of Stuttgart, Department of Xechanical Process Techniques, Study KO. 27 (1969). Chavan, V. V., 3 l . S ~ .Thesis, University of Salford, Salford, England, 1971. Chavan. J.. V.. Ph.D. Thesis, University of Salford, Salford, England, 1978. Chavan, V, V., Jhaveri, A. S., Ulbrecht, J., Trans. Znst. Chem. Eng., 50, No. 2, 147 (1972). Chavan, V, Ulbrecht, J., Chem. Eng. J . , 3, 308 (1972). Chavan, V. \ .,Ulbrecht, J., Chem. Eng. J . , submitted for publication. Gl~iz,11.D., Pavluschenko, I. S., Zh. Plikl. Khim. (Leningrad), 39, 2474 (1966). Gray, J. B., Chem. Eng. Progr., 59, 55 (1963). Hoogendorn, C. J., Den-Hartog, A. P., Chenz. Eng. Sci., 22, 1689 (1966). Kappel, M., Seibring, H., Verfahrenstechnik, 4, 470 (1970). bletzner, A. B., Otto, R. E., AZChE J., 3, 3 (1957). bliiller, \V., DECHEXA (Deut. Thes. Chevn. Apparatewesen) Xonogr., 66, 247 (1971). Sagata, S., Nishikawa, M., Katsube, T., Takaish, K., Znt. Chcm. Eng., 12, 172 (1972). Sagata, S., KishikaTa, AI,, Tada, H., Hivabaryashi, H., Gotoh, P., J . Chem. Eng. Jap., 3, 237 (1970). Nagata, S., Yanagimato, T., Yokoyama, T., Zcagaku Kogoku, 21, 278 (19*!7). Novak, J ., C.Sc. Thesis, I)ist,ributed by Statni Technicka Knihovna CSR, Narodni Xnihovna, Praha 1, Klementinum, Czechoslovakia, 1970. RECEIVED for review February 5 , 1973 -4CCEPTED May 14, 1973
\I,
