Power consumption and mixing times for liquid mixing in a ribbon mixer

The Chemical Engineering Journal~ 48 (1992) 135-140

135

Power consumption and mixing times for liquid mixing in a ribbon mixer S. Masiuk, H. ~:acki and F. Strek Department o f Chemical Engineering, Technical University of Szczecin~ A1. PiastSw 42, 71-055 Szczecin (Poland)

(Received May 7, 1991)

Abstract Measurements were taken of power consumption and mixing times for helical ribbon agitators of different pitch and ribbon width under laminar flow conditions in newtonian liquid. Empirical correlations for prediction of the power consumption, the mixing time and the energy required to achieve a prescribed degree of homogeneity are proposed. The results are compared with those of other works.

1. I n t r o d u c t i o n Different types o f rotating agitators are used for the acceleration o f the chemical process. One of the best ty p es o f low-speed rotary agitator for homogenization o f high viscosity liquid is a helical ribbon. The design configuration of this type agitator is shown in Fig. 1. A comprehensive outline of the literature available on this subject is given by Masiuk

[ 1 ]. He studied the usefulness of horizontal system agitators as mixers for solid materials. The purpose of this p a p e r and experimental work it describes is an investigation of the interaction between p o w e r consumption, mixing time and number of revolutions and the geometric parameters of helical ribbon agitators (Fig. 1) working in the vertical position, this being a case not previously investigated.

2. B a s i c c o n c e p t The dimensionless power n u m b e r [2l Po or the dimensionless mixing time [3] n t and, therefore, the mixing e n e r g y P t might be e x p e c t e d to be related to the dimensionless Reynolds n u m b e r Re and the dimensionless groups, representing the geometrical mixer parameters, by a functional equation of the form (see Appendix A) (Po, or n t , or Pt ) =fn(Re, dimensionless groups)

(1)

1__

Fig. 1. Schematic diagram of the ribbon agitator.

Practically, the function fn(o) is expressed by a p r o d u c t of the constant C~, and Reynolds n u m b e r with e x p o n e n t A~, and dimensionless moduli vr~with e x p o n e n t s a~ representing the effect of the geometrical mixer parameters. The subscripts i - - 1 , 2, 3 stand for the pow er number, mixing time and mixing energy respectively. The additional subscripts j = 1, 2, ... concerns the investigated mixer parameters. The values of C~, A~, and a~ were determined by a least-squares method.

Elsevier Sequoia

136 3. E x p e r i m e n t a l

S. M a s i u k et al. / L i q u i d m i x i n g i n a ribbon m i x e r

details

The measurements of power consumption and mixing time were made using the apparatus sketched in Fig. 2. T h e m a i n p a r t of the a p p a r a t u s w a s a f l a t - b o t t o m e d m i x e r with D = 0.2 m , c o o l e d b y w a t e r o n t h e cylindrical p a r t , e q u i p p e d with helical r i b b o n a g i t a t o r w i t h d = 0 . 1 8 m. T h e a g i t a t o r w a s driven b y a d.c. e l e c t r i c m o t o r with v a r i a b l e - s p e e d drive. In this p a p e r , in all c a s e s , t h e r o t a t i o n w a s s u c h t h a t t h e liquid c o u l d flow u p w a r d a n d d o w n w a r d f r o m the c e n t r e of t h e agitator. Glycerol w a s u s e d as t h e m i x e d liquid. T h e e l e c t r i c a l m e t h o d w a s u s e d to m e a s u r e the t o r q u e p r o d u c t b y t h e r o t a t i n g agitator. E q u a t i o n s f o r c a l c u l a t i o n o f t h e p o w e r c o n s u m p t i o n are g i v e n in ref. 1. T h e s o - c a l l e d c o n v e c t i v e m i x i n g t i m e w a s determ i n e d u s i n g t h e t h e r m a l - r e s p o n s e t e c h n i q u e prop o s e d b y H o o g e n d o o r n a n d H a r t o g [4 ]. This m e t h o d is b a s e d o n t h e p r i n c i p l e o f m o n i t o r i n g the c h a n g e in t e m p e r a t u r e in the m i x e d liquid a n d m e a s u r i n g t h e b u l k t e m p e r a t u r e in t h e v e s s e l at t w o locations. One t h e r m o c o u p l e is fLxed in t h e vicinity o f the b a t c h m e t e r b o t t o m ( d i s t u r b a n c e p o i n t ) at the liquid s u r f a c e a n d at a d i s t a n c e f r o m t h e wall o f the m i x i n g v e s s e l o f 0 . 0 5 m, a n d a n o t h e r t h e r m o c o u p l e is fixed o p p o s i t e t h e s h a f t w h e r e t h e cylindrical p a r t o f the m i x e r is c o n n e c t e d with t h e b o t t o m . T h e s e therm o c o u p l e s a r e c o n n e c t e d in a differential circuit. The mixing time experiments were conducted a c c o r d i n g to the following s c h e m e . 2 .

.

.

.

(1) B e f o r e a n y e x p e r i m e n t a l m e a s u r e m e n t s w e r e taken, the b u l k glycerol w a s c o o l e d to an a v e r a g e t e m p e r a t u r e o f a b o u t 20 °C b y a cold s t r e a m o f w a t e r flowed t h r o u g h the j a c k e t o f the mixer. N e x t the flow o f c o o l i n g w a t e r w a s s t o p p e d . (2) 0 . 0 0 1 5 m 3 glycerol w a s l o a d e d into t h e b a t c h m e t e r a n d h e a t e d to 90 °C. N e x t the h e a t e d glycerol w a s a d d e d to t h e b u l k glycerol with the a g i t a t o r at rest. The addition t i m e w a s a b o u t 1 s. (3) After l o a d i n g the s a m p l e the a g i t a t o r a n d the t e m p e r a t u r e r e c o r d e r w e r e s e t in m o t i o n simultaneously. Only o n e d i a g r a m (Fig. 3) of the t e m p e r a t u r e difference b e t w e e n the t w o m e a s u r e m e n t p o i n t s 1 a n d 2 (Fig. 2) is shown, d e r i v e d for a r o t a t i o n a l s p e e d o f 1.08 r e v s - 1 . T h e m i x i n g p r o c e s s is r e g a r d e d as c o m p l e t e w h e n the t e m p e r a t u r e difference AT1_2 is s m a l l e r t h a n 0.01 t i m e s the m a x i m u m t e m p e r a t u r e increase. The t i m e r e q u i r e d to a c h i e v e this is called the c o n v e c t i v e m i x i n g time.

4. R e s u l t s a n d d i s c u s s i o n

T h e effects o n p o w e r c o n s u m p t i o n a n d m i x i n g t i m e o f t h r e e m a j o r v a r i a b l e s w e r e i n v e s t i g a t e d in a total o f 160 r u n s carried o u t in t h e l a m i n a r region. T h e s e v a r i a b l e s included t h e r o t a t i o n a l s p e e d n, r a n g i n g f r o m 0 . 6 7 to 6.38 r e v s - l , the helix p i t c h s r a n g i n g f r o m 0.09 to 0.36 m, a n d t h e width o f the r i b b o n w r a n g i n g f r o m 0 . 0 0 8 to 0 . 0 6 m. F o l l o ~ n g the m e t h o d o f R u s h t o n e t a l . [2], t h e p o w e r n u m b e r Po w a s p l o t t e d o n a l o g - l o g g r a p h a g a i n s t the R e y n o l d s n u m b e r Re, f o r all data. The v a l u e s of Po, a n d Re w e r e c a l c u l a t e d a c c o r d i n g t o

.

z~T [ K]

'

3

n = 1.08 P=4.5

£

/vv Fig. 2. Experimental set-up: 1, d.c. electric motor; 2, batch meter; 3, measuring system; 4, recorder; 5, mixer; 6, helical ribbon.

o

r

L

I

I

I

4

8

12

16

I

'I' LJ~S~

Fig. 3. Measurement principle of determination of mixing time.

S. M a s i u k et aL / L i q u i d m i x i n g i n a ribbon m i x e r

the formulas given in Appendix A. All fluid properties in the dimensionless numbers were evaluated at the mean fluid temperature. The data for different values of dimensionless quantities s/d and w/d are shown in Figs. 4 and 5 respectively. For simplicity, the slope of the straight lines, as the average values

137

of the exponent of the Reynolds number, were taken to be - 1. This value of exponent is characteristic for laminar flow of liquids and is in good agreement with other investigators. Hence eqn. (1) for the dimensionless power number may, therefore, be written as W 0.38 S

s / d : 0.5

s/d=1 s/d =2

40

s/d

Point

0.5 1 2

[] 0 ~7

60

100

140

200

Re

Fig. 4. Power characteristic for various ribbon pitches.

The area terms in this equation indicate the effect of ribbon design on power consumption. Equation (2) is valid for the following ranges of geometrical parameters: 0.5<_s/d<_2; 0 . 0 4 4 < w ! d < 0.33; 38 _
nt=4OSRe-'l-~)

40

60

~0

100

140

200

Re

Fig. 5. Power characteristic for v a r i o u s ribbon widths.

--0,2

\--0.53/ \ 0 . 5

I~S )

(3)

Equation (3) is valid for the region 0.5
S. M a s i u k et al. / L i q u i d m i x i n g i n a r i b b o n m i x e r

138

I

!

I

,

~

I

~

I

(a)

I

I

I

PoRe= 421 ( w / d ) ° 4 ~

(b) PoRe= 32.8 [5.3.

/

(c) P°Re=726(w/d}"s

/

~

/

:

o-;./

c/d : 0.055 h/d 1.94 s/d : 1 =

I

I

0.02

0.06

I

0.10

I

I

I

I

0.14

0.18

0.22

0.26

I,

w/d

030

Fig. 6. C o m p a r i s o n o f o u r r e s u l t s w i t h t h o s e of o t h e r a u t h o r s : curve a, ref. 8; c u r v e b, ref. 7; curve c, ref. 6.

T

r (a)

600

500

T

1

: 32.815.3.6.9(s/e)

(b)

PoRe = 328(s/d) -o'G~

(c)

PoRe= 297 ( s/d )~o.~3

n.t 300

s/d

200

05 1 2

27

400

Point [] o 27

27

100

300 50

d/D :zoo c/d = 0.055 h/d = 1,94 w/d 0.167 100

0.5

I

\This ~ork

1.0

i

l

I

I

60

|

100

200

Re

PoRe = 240 (s/d) ~°'~

Fig. 8. Influence o f r i b b o n pitch o n the m i x i n g time characteristic.

s/d

power and mixing time together, as a product, is t h e so-called mixing energy. The effects of helix pitch and width of ribbon on the product P t are shown in Figs. 11 and 12. By fitting straight lines to the results this alternate correlation may be expressed by relationship (1). The equation for calculation of the mixing energy can be written in the following form

1.5

Fig. 7. C o m p a r i s o n o f o u r r e s u l t s w i t h t h o s e of o t h e r a u t h o r s : c u r v e a, ref. 7; c u r v e b, ref. 8; c u r v e c, ref. 6.

of this work and those of several other investigators is shown in Fig. 10 for values of Reynolds number ranging from 40 to 200. The agreement is seen to be not good. The different results may be explained by another type of helical agitator and by the different methods for measuring mixing time that were used. To assess the suitability of a given agitator for liquid homogenization, it is not sufficient to take into consideration the p o w e r consumption and mixing time separately. The term which connects mixing

/

\--0.15 /

Pt = 14Re°s/d )

\0,3

(d)

(4)

The generalized eqn. (4) is valid for the ranges of parameters, given for eqn. (3), with an average derivation of about _+10%.

S. M a s i u k et aL / L i q u i d m i x i n g i n a r i b b o n m i x e r

n,~

1

I

I

I

I

i

"~•~ 200

~O

i

w/d

Point

0.044

•

0.0833 0.167 0.333

0 A

300

139

Pt s/d 0.5

D

400

Point D o v

1

2

300

v,,,~

140

200 IO0 140

60

100 I

60 I

I

I

60

I

I

I

B0 100

200

I

I

I

I

B0 100

140

I

200

Re

Fig. 11. Product of power c o n s u m p t i o n and mixing time as a function of Reynolds n u m b e r for various ribbon pitches.

I

140

I

Re

Fig. 9. Influence of ribbon width on the mixing time characteristic. P-t

f I w/d | I I IPoint I

400 200

001

n.t

• 0.044 0.0833 0.167 0.333

I

• D

O A

o,r

•

200 [

1100 80 60

140[

This work n.t = 1053 Re"°'s ~

~" (a)

100L

1

60

40

1

I

I

60

i

I

80

I

I

Re

(0

5. C o n c l u s i o n s

I

100

200

(b)

16

40

14o

Fig. 12. Product of power c o n s u m p t i o n a n d mixing time as a function of Reynolds n u m b e r for various ribbon widths.

~

20

8o 1oo

200

I

Kr~e

Fig. 10. Comparison of o u r results with those of o t h e r authors: curve a, ref. 4; curve b, ref. 11; curve c, ref. 13; (equations not given). Work

aiD

s/d

w/d

Present 4 11 13

0.9 0.96 0.92 0.95

1 0.61 1 1

0.167 0.091 0.2 0.059

Several significant conclusions may be drawn from the material presented in this report. (1) It was shown that the rotational speed, helix pitch and width of helical ribbon of an agitator have a perceptible influence on power consumption and mixing time for the mixing of liquid in the laminar region. (2) Power consumption (2), and mixing time (3) correlations may be used to predict these values, depending on the Reynolds number for mixing and two-dimensionless moduli representing the effect of agitator geometry.

140

S. M a s i u k et al. / L i q u i d m i x i n g in a ribbon m i x e r

(3) On the basis of the investigation formula, eqn. (4) is p r o p o s e d for calculation of the mixing energy for mixing liquid in a ribbon agitator. (4) The experimental results have shown a continuous decrease in mixing number and continuous increase in mixing energy with an increase in the Reynolds number.

11 F. Rieger, V. Novak and D. Havelkova, Chem. Eng. J., 33 (1986) 143. 12 S.Hagata, T. Yokogama and M. Yaragimoto, Chem. Eng. Jpn., 21 (1957) 278. 13 J. B. Gray, Chem. Eng. Progr., 59 (1963) 55.

Appendix A: N o m e n c l a t u r e

References 1 S. Masiuk, P o w d e r Technol., 51 (1987) 217. 2 J. H. Rushton, E. W. Costich and H. J. Everett, Chem. Eng. Progr., 46 (1950) 395, 467. 3 H. Kramers, G. M. Baars and K. H. Knoll, Chem. Eng. Sci., 2 (1953) 35. 4 C. J. H o o g e n d o o r n and A. P. den Hartog, Chem. Eng. Sci., 22 ( 1 9 6 7 ) 1869. 5 V. N o v a k a n d F . Rieger, Trans. Inst. Chem. E n g . , 4 7 ( 1 9 6 9 ) T335. 6 L. R. Hall and J. C. Godfrey, Trans. Inst. Chem. Eng., 48 ( 1 9 7 0 ) T201. 7 H. Brauer and H. Schmidt-Traub, Chem. Ing. TechnoL, 44 ( 1 9 7 2 ) 1237. 8 H. Blasiftski and E. Rzyski, Chem. Eng. J., 19 (1980) 157., 9 P. Ayazi Shamlou and M. F. Edwards, Chem. Eng. Sci., 40 ( 1 9 8 5 ) 1773. 10 K. Takahasi, Y. Takahata, T. Yokota a n d H. Konno, J. Chem. Eng. Jpn., 18 (1985) 159.

d D h H n P Po

Re S t W

clearance between the agitator and the wall of the vessel (m) agitator diameter (m) inside diameter of the vessel (m) height of agitator (m) height of liquid level in the vessel (m) speed of agitator (s-1) power consumption (W) power number, Po = P / p n 3 d 5 Reynolds number, Re = n d ep/v pitch of the agitator (m) time of homogenization (mixing time) (s) ribbon width (m)

Greek s y m b o l s

V p r

viscosity (N s m -z) density (kg m -3) time (s)