14
m = =− 10 2 121 40 0 0582 10 . log , . N Fr m = = 0 14 1 1212 0 0582 . . . Therefore, power P N NDN P A Fr m = ρ 3 5 = 20 950 15 0 61 1 1212 3 5 3 3 3 5 . . . . sec × × × × kg m rev m = 607.24 W = 0.61 kW (0.81 hp) Studies on various turbine agitators have shown that geometric ratios that vary from the standard design can cause different effects on the Power number N P in the turbulent regions [24]. For the flat, six-blade open turbine, N P (W/D A ) 1.0 . For the flat, six-blade open turbine, varying D A /D T from 0.25 to 0.5 has no effect on N P . When two six-blade open turbines are installed on the same shaft and the spacing between the two impellers (vertical distance between the bottom edges of the two turbines) is at least equal to D A , the total power is 1.9 times a single flat-blade impeller. For two six-blade pitched-blade (45°) turbines, the power is about 1.9 times that of a single pitched-blade impeller. A baffled, vertical square tank or a horizontal cylindrical tank has the same Power number as a vertical cylindrical tank. SCALE-UP OF MIXING SYSTEMS The calculation of power requirements for agitation is only a part of the mixer design. In any mixing problem, there are several defined objectives such as the time required for blending two immiscible liquids, rates of heat transfer from a heated jacket per unit volume of the agitated liquid, and mass transfer rate from gas bubbles dispersed by agitation in a liquid. For all these objectives, the process results are to achieve the optimum mixing and uniform blending. mixing of fluids 32

SCALE-UP OF MIXING SYSTEMS

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Page 1: SCALE-UP OF MIXING SYSTEMS

m = − = −1 0 2 12140

0 058210. log ,.

NFrm = =−0 14 1 12120 0582. ..

Therefore, power P N N D NP A Frm= ρ 3 5

= 2 0 950 1 5 0 61 1 12123 53

3

35. . . . •

sec•× × × ×

kgm

revm

= 607.24 W

= 0.61 kW (0.81 hp)Studies on various turbine agitators have shown that geometric

ratios that vary from the standard design can cause different effectson the Power number NP in the turbulent regions [24].

• For the flat, six-blade open turbine, NP ∝ (W/DA)1.0.• For the flat, six-blade open turbine, varying DA/DT from 0.25 to

0.5 has no effect on NP.• When two six-blade open turbines are installed on the same shaft

and the spacing between the two impellers (vertical distancebetween the bottom edges of the two turbines) is at least equalto DA, the total power is 1.9 times a single flat-blade impeller.For two six-blade pitched-blade (45°) turbines, the power is about1.9 times that of a single pitched-blade impeller.

• A baffled, vertical square tank or a horizontal cylindrical tank hasthe same Power number as a vertical cylindrical tank.

SCALE-UP OF MIXING SYSTEMS

The calculation of power requirements for agitation is only a partof the mixer design. In any mixing problem, there are several definedobjectives such as the time required for blending two immiscibleliquids, rates of heat transfer from a heated jacket per unit volume ofthe agitated liquid, and mass transfer rate from gas bubbles dispersedby agitation in a liquid. For all these objectives, the process resultsare to achieve the optimum mixing and uniform blending.

mixing of fluids

32

Page 2: SCALE-UP OF MIXING SYSTEMS

The process results are related to variables characterizing mixing,namely geometric dimensions, stirrer speed (rpm), agitator power, andphysical properties of the fluid (e.g., density, viscosity, and surfacetension) or their dimensionsless combinations (e.g., the Reynoldsnumber, Froude number, and Weber number, ρN2D3

A/σ). Sometimes,empirical relationships are established to relate process results andagitation parameters. Often, however, such relationships are non-existent. Laboratory scales of equipment using the same materials ason a large scale are then experimented with, and the desired processresult is obtained. The laboratory system can then be scaled-up topredict the conditions on the larger system.

For some scale-up problems, generalized correlations as shown inFigures 8-11, 8-12, 8-13, and 8-14 are available for scale-up. However,there is much diversity in the process to be scaled-up, and as such nosingle method can successfully handle all types of scale-up problems.

Various methods of scale-up have been proposed; all based ongeometric similarity between the laboratory equipment and the full-scale plant. It is not always possible to have the large and small vesselsgeometrically similar, although it is perhaps the simplest to attain. Ifgeometric similarity is achievable, dynamic and kinematic similaritycannot often be predicted at the same time. For these reasons, experi-ence and judgment are relied on with aspects to scale-up.

The main objectives in a fluid agitation process are [25]:

• Equivalent liquid motion (e.g., liquid blending where the liquidmotion or corresponding velocities are approximately the same inboth cases).

• Equivalent suspension of solids, where the levels of suspensionare identical.

• Equivalent rates of mass transfer, where mass transfer is occurringbetween a liquid and a solid phase, between liquid-liquid phases,or between gas and liquid phases, and the rates are identical.

A scale ratio R is used for scale-up from the standard configurationas shown in Table 8-2. The procedure is:

1. Determine the scale-up ratio R, assuming that the original vesselis a standard cylinder with DT1 = H1. The volume V1 is

VD

HDT T

11

2

11

3

4 4= =π π

• (8-35)

mixing of fluids

33

Page 3: SCALE-UP OF MIXING SYSTEMS

The ratio of the volumes is then

VV

DD

DD

T

T

T

T

2

1

23

13

23

13

44

= =ππ (8-36)

The scale-up ratio R is

RDD

VV

T

T

T

T

= =

2

1

2

1

13

(8-37)

Using the value of R, calculate the new dimensions for allgeometric sizes. That is,

DA2 = RDA1, J2 = RJ1, W2 = RW1

E2 = RE1, L2 = RL1, H2 = RH1

or

RDD

DD

WW

HH

JJ

EE

A

A

T

T

= = = = = =2

1

2

1

2

1

2

1

2

1

2

1

2. The selected scale-up rule is applied to determine the agitatorspeed N2 from the equation:

N NR

NDD

nT

T

n

2 1 11

2

1=

=

(8-38)

• where n = 1 for equal liquid motion• n = 3/4 for equal suspension of solids• n = 2/3 for equal rates of mass transfer (corresponding equivalent

power per unit volume, which results in equivalent interfacialarea per unit volume)

The value of n is based on theoretical and empirical considera-tions and depends on the type of agitation problem.

3. Knowing the value of N2, the required power can be determinedusing Equation 8-17 and the generalized Power number correlation.

mixing of fluids

34

Page 4: SCALE-UP OF MIXING SYSTEMS

Other possible ways of scaling up are constant tip speed uT(πNDA),and a constant ratio of circulating capacity to head Q/h.

Since P ∝ N3D5A and V ∝ D3

A then

PV

N DA∝ 3 2 (8-39)

For scale-up from system 1 to system 2 involving geometricallysimilar tanks and same liquid properties, the following equations canbe applied:

N1DA1 = N2DA2

For a constant tip speed,

NN

DD

A

A

2

1

1

2

= (8-40)

For a constant ratio of circulating capacity to head, Q/h,

N D N DA A13

12

23

22= (8-41)

Example 8-3

Scraper blades set to rotate at 35 rpm are used for a pilot plantaddition of liquid ingredients into a body-wash product. What shouldthe speed of the blades be in a full-scale plant, if the pilot and thefull-scale plants are geometrically similar in design? Assume scale-up is based on constant tip speed, diameter of the pilot plant scraperblades is 0.6 m, and diameter of the full-scale plant scraper bladesis 8 ft.

Solution

The diameter of the full scale plant scraper blades = 8.0 × 0.3048= 2.4384 m (2.4 m).

Assuming constant tip speed,

NN

DD

A

A

2

1

1

2

= (9-42)

mixing of fluids

35

Page 5: SCALE-UP OF MIXING SYSTEMS

where N1 = scraper speed of pilot plantN2 = scraper speed of full-scale plant

DA1 = diameter of pilot plant scraper bladesDA2 = diameter of full-scale plant scraper blades

NN D

D

rpm

A

A2

1 1

2

35 0 62 4

8 75

=

= ( )( )( )

=

..

.

Example 8-4

During liquid makeup production, color pigments (i.e., solid havingidentical particle size) are added to the product via a mixer. In thepilot plant, this mixer runs at 6,700 rpm and has a diameter head of0.035 m. Full-scale production is geometrically similar and has a mixerhead diameter of 0.12 m. Determine the speed of the full-scale produc-tion mixer head. What additional information is required for the motorto drive this mixer? Assume that power curves are available for thismixer design, and the scale-up basis is constant power/unit volume.

Solution

For constant power per unit volume, Equation 8-39 is applied: P/V∝ N3D2

A or N13D2

A1 = N23D2

A2. Therefore,

N NDD

A

A2 1

1

2

2 3

=

where N1 = 6,700 rpmDA1 = 0.035 mDA2 = 0.12 m

N2

2 3

6 7000 0350 12

=

,

..

mixing of fluids

36

Page 6: SCALE-UP OF MIXING SYSTEMS

N2 = 2,946.7 rpm

N2 ≈ 2,950 rpm

The power required for mixing is P = NpρN3D5A, where the Power

number (NP) is a function of the Reynolds number [i.e., NP = f(NRe)]:

NNDA

Re = ρµ

2

The plant must be provided with the viscosity of the product and itsdensity after addition of the pigments.

Example 8-5

A turbine agitator with six flat blades and a disk has a diameter of0.203 m. It is used in a tank with a diameter of 0.61 m and height of0.61 m. The width is W = 0.0405 m. Four baffles are used with awidth of 0.051 m. The turbine operates at 275 rpm in a liquid havinga density of 909 kg/m3 and viscosity of 0.02 Pas.

Calculate the kW power of the turbine and kW/m3 of volume. Scaleup this system to a vessel whose volume is four times as large, forthe case of equal mass transfer rate.

Solution

The Reynolds number for mixing is NRe. The number of revolutionsper sec, N = 275/60 = 4.58 rev/sec.

NND

kgm

revm

mkg

ARe

. .

.•

sec• •

• sec

, .

=

= ( )( )( )

=

ρµ

2

2

32909 4 58 0 203

0 02

8 578 1

NRe ≈ 8,600

mixing of fluids

37

Page 7: SCALE-UP OF MIXING SYSTEMS

590 Modeling of Chemical Kinetics and Reactor Design

Using curve 6 in Figure 8-14, the Power number NP = 6.0. Thepower of the turbine P = NpρN3D5

A:

P = (6.0)(909)(4.583)(0.2035) kgm

revm3

3

35•

sec•

= 0.1806 kW (0.24 hp)

The original tank volume V1 = πD3T1/4. The tank diameter DT1 = 0.61:

V1

30 614

= ( )( )π .

V1 = 0.178m3

The power per unit volume is P/V

PV

kW m

=

=

0 18060 178

1 014 3

..

. /

For the scale-up of the system, the scale-up ratio R is

RVV

DD

DD

T

T

T

T

= = =2

1

23

13

23

13

44

ππ

RVV

DD

T

T

=

=2

1

13

2

1(8-37)

where V2 = 4V1

V

m

2

3

4 0 178

0 712

= ( )

=

.

.

R = ( ) =4 1 58713 .

mixing of fluids

38

Page 8: SCALE-UP OF MIXING SYSTEMS

The dimensions of the larger agitator and tank are:

DA2 = RDA1 = 1.587 × 0.203 = 0.322 m

DT2 = RDT1 = 1.587 × 0.61 = 0.968 m

For equal mass transfer rate n = 2/3.

N NR2 1

231=

(8-38)

=

4 58

11 587

23

..

= 3.37 rev/sec

The Reynolds number NRe is

NN DA

Re = ρµ2 2

2

= ( )( )( )

909 3 37 0 3220 02

2

32. .

.•

sec• •

• seckgm

revm

mkg

= 15,880.9

NRe ≈ 16,000

Using curve 6 in Figure 8-14, NP = 6.0. Power required by theagitator is P2 = NpρN2

3D5A2

Pkgm

revm2

3 53

3

356 0 909 3 37 0 322=( )( )( ) ( )

. . . •sec

P2 = 722.57 W

= 0.723 kW (0.97 hp)

mixing of fluids

39

Page 9: SCALE-UP OF MIXING SYSTEMS

The power per unit volume P/V is:

PV

2

2

0 7230 712

= ..

= 1.015 kW/m3

MIXING TIME SCALE-UP

Predicting the time for obtaining concentration uniformity in a batchmixing operation can be based on model theory. Using the appropriatedimensionless groups of the pertinent variables, a relationship can bedeveloped between mixing times in the model and large-scale systemsfor geometrically similar equipment.

Consider the mixing in both small and large-scale systems to occurin the turbulent region, designated as S and L respectively. Using theNorwood and Metzner’s correlation [26], the mixing time for bothsystems is

t N D g D

H D

t N D g D

H DS S AS AS

S TS

L L AL AL

L TL

2 2 3 1 6 1 2

1 2 3 2

2 2 3 1 6 1 2

1 2 3 2

( )=

( )• •

(8-43)

Applying the scale-up rule of equal mixing times, and rearrangingEquation 8-43, yields

NN

DD

DD

DD

HH

L

S

TL

TS

AS

AL

AS

AL

L

S

=

23

32

43

12

12

(8-44)

Assuming geometric similarity,

HH

DD

L

S

AL

AS

= (8-45)

DD

DD

TL

TS

AL

AS

= (8-46)

mixing of fluids

40

Page 10: SCALE-UP OF MIXING SYSTEMS

Substituting Equations 8-45 and 8-46 into Equation 8-44 gives

NN

DD

DD

DD

DD

L

S

AL

AS

AS

AL

AS

AL

AL

AS

=

23

32

43

12

12

(8-47)

NN

DD

L

S

AL

AS

=

23

16

or

NN

DD

L

S

AL

AS

=

14

(8-48)

The exponent n for the mixing time scale-up rule is 0.25.The power P of the agitator for both large and small systems is

PN D

P

N DL

L AL

S

S ASρ ρ3 5 3 5= (8-49)

where

PP

NN

DD

L

S

L

S

AL

AS

=

3 5

(8-50)

Substituting Equation 8-48 into Equation 8-50 yields

PP

DD

DD

L

S

AL

AS

AL

AS

=

0 75 5.

(8-51)

or

PP

DD

L

S

AL

AS

=

5 75.

(8-52)

mixing of fluids

41

Page 11: SCALE-UP OF MIXING SYSTEMS

The power per unit volume P/V for both large- and small-scalesystems is:

P VP V

PD

PD

PP

DD

L L

S S

LTL

STS

L

S

TS

TL

=

=

π

π

3

3

3

4

4

• (8-53)

Substituting Equations 8-46 and 8-52 into Equation 8-53 gives

P V

P VDD

DD

DD

L

S

AL

AS

AS

AL

AL

AS

( )( ) =

=

5 75 3

2 75

.

.

(8-54)

Table 8-7 summarizes correlations for the effects of equipmentsize on the rotational speed needed for the same mixing time byvarious investigators.

The relationships in Table 8-7 show that the rotational speed toobtain the same batch mixing time is changed by a small power ofthe increase in linear equipment dimension as equipment size ischanged. Equation 8-49 shows that greater power is required for alarge-scale system compared to a smaller system. Often, the powerrequired for a larger system may be prohibitive, thus modificationof the scale-up rule is needed (e.g., tL = 10tS or tL = 100tS) toobtain a lower power requirement. It should be noted that relaxa-tion of mixing time requirements may not pose other problems. Forexample, if the mixing is accompanied by a chemical reaction in aCSTR, assuming that the Norwood-Metzner [26] correlation for mixingtime (t) is still applicable, it must be ensured that the mixing time inthe larger scale (tL = 10tS or tL = 100tS) is less than 5% of the averageresidence time of the liquids in the reactor, otherwise the conversion

mixing of fluids

42

Page 12: SCALE-UP OF MIXING SYSTEMS

Tabl

e 8-

7ab

ler

Effe

ctE

f fe

ct o

f eq

uip

men

t si

ze o

n r

ota

tio

nal

sp

eed

nee

ded

fo

r th

e sa

me

mix

ing

tim

e

Rel

atio

nsh

ipb

etw

een

N a

nd

DE

qu

ipm

ent

∆ρE

qu

atio

nIn

vest

igat

or

N ∝

D–1

/6P

rope

lle r

s, n

o ba

ffle

sN

ot z

e ro

θρ

ρN

Dp

V

DN

gZL

22

20

25

9

=∆

.

van

de V

usse

[17

]

N ∝

D–0

.1 t

o –0

.2P

a ddl

e s,

turb

ine s

Not

ze r

ρρ

Q V

DN

gZL

22

03

.

van

de V

usse

[17

]

N =

con

sta n

tP

rope

lle r

s, p

a ddl

e s,

turb

ine s

Ze r

ova

n de

Vus

se(1

7)

N ∝

D–1

/5P

rope

lle r

sZ

e ro

θ=

()

CZ

T

NN

Dg

L1

12

16

24

61

6R

e

Fox

and

Gex

[18

]

N ∝

D1/

4T

urbi

nes

Ze r

oN

orw

ood

a nd

Me t

z ne r

[26

]

Sour

c e:

Gra

y , J

. B

., M

ixin

g I

The

ory

a nd

Pra

c tic

e , V

. W

. U

hl a

nd J

. B

. G

ray ,

Eds

. A

c ade

mic

Pre

ss I

nc.,

1966

.

mixing of fluids

43

Page 13: SCALE-UP OF MIXING SYSTEMS

The power per unit volume P/V for both large- and small-scalesystems is:

P VP V

PD

PD

PP

DD

L L

S S

LTL

STS

L

S

TS

TL

=

=

π

π

3

3

3

4

4

• (8-53)

Substituting Equations 8-46 and 8-52 into Equation 8-53 gives

P V

P VDD

DD

DD

L

S

AL

AS

AS

AL

AL

AS

( )( ) =

=

5 75 3

2 75

.

.

(8-54)

Table 8-7 summarizes correlations for the effects of equipmentsize on the rotational speed needed for the same mixing time byvarious investigators.

The relationships in Table 8-7 show that the rotational speed toobtain the same batch mixing time is changed by a small power ofthe increase in linear equipment dimension as equipment size ischanged. Equation 8-49 shows that greater power is required for alarge-scale system compared to a smaller system. Often, the powerrequired for a larger system may be prohibitive, thus modificationof the scale-up rule is needed (e.g., tL = 10tS or tL = 100tS) toobtain a lower power requirement. It should be noted that relaxa-tion of mixing time requirements may not pose other problems. Forexample, if the mixing is accompanied by a chemical reaction in aCSTR, assuming that the Norwood-Metzner [26] correlation for mixingtime (t) is still applicable, it must be ensured that the mixing time inthe larger scale (tL = 10tS or tL = 100tS) is less than 5% of the averageresidence time of the liquids in the reactor, otherwise the conversion

mixing of fluids

44

Page 14: SCALE-UP OF MIXING SYSTEMS

scale-up chart only applies to systems of similar geometry. Whenthe geometry is different, special and specific analyses of the systemare required.

Samant and Ng [28] compared various scale-up rules for agitatedreactors. They suggested that a scale-up rule of power per unit volumeand constant average residence time (where the power per unit volumeand average residence time cannot be increased) is the most suited inmany operations. However, this still may not improve or preserve theperformance of the systems. Therefore, adequate consideration mustbe given to a tradeoff between performance and operating constraints.

mixing of fluids

45

REFERENCES 2. Levenspiel, O., Chemical Reaction Engineering, 3rd ed., John

Wiley & Sons, New York 1999.3. Penny, W. R., “Guide to trouble free mixers,” Chem. Eng., 77 (12),

171, 1970.4. Holland, F. A. and Chapman, F. S., Liquid Mixing and Processing

in Stirred Tanks, Reinhold, New York, 1966.5. Myers, K. J., Reeder, M., and Bakker, A., “Agitating for success,”

The Chemical Engineer, pp. 39–42, 1996.