Studies on heat transfer non-newtonian fluids

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Studies on Heat Transfer to Agitated Non-Newtonian Fluids

Introduction

Many of the fluids encountered in the process industries, in which there are polymeric materials, petroleum products, two phase fluids, etc., exhibit non-Newtonian behaviour; that is to say, the shear rate is not directly porportional to the applied stress. There are many situations where such fluids have to be heated or cooled in agitated vessels, hence the non-Newtonian characteristics can be of considerable significance. The difficulty of studies on heat transfer to non-Newtonian fluids in agitated vessels arise in adopting the "representative non-Newtonian viscosity (apparent or

modified viscosity)". The matter is due to the point that

the relationship between shear stress and shear rate is

non-linear, and leads to problems in specifying the Rey-nolds number, Prandtl number and viscosity ratio.

The present paper deals with time-independent

pseudo-plastic fluids which are encountered very often in the

chemical processes.

Background

The apparent viscosities, which have been well used on heat transfer in agitated vessels, are shown in Table 1.

For non-Newtonian fluids it

is possible to define an apparent viscosity u* as follows.

p* = T (2.1)

The methods developed so far in order to estimate an apparent viscosity may be grouped largely into two classes.

*Technical Administration Department,

Mitsubishi Chemical Plant Engineering Center

Yoshio Miyairi*

In the present paper, modified (or apparent) viscosities in heat transfer to agitated non-Newtonian fluids are derived for various flow patterns. The three constant Ellis model is used to characterize the given non-Newtonian fluids. Heat transfer correlation

equations are proposed based on these modified viscosities.

The experimental studies are carried out with several types of agitators: (1) an agitator with a 2-bladed paddle, (2) an agitator

with a helical screw fitted in a draft tube, (3) an agitator with a helical ribbon. The non-Newtonian fluids tested are various

concentrations of polymer aqueous solutions of CMC, HEC, PO and SPA.

As a result of using the modified viscosities proposed in this paper, the correlations for non-Newtonian heat transfer in the three types of agitators agree very well with those for Newtonian heat transfer. For the heat transfer of highly viscous agitated liquids, the agitators with a helical screw and a helical ribbon may be found to be very favorable for industrial uses. Heat transfer for impeller

surfaces seems to be effective.

The experimental data and their correlating methods in this paper may be valuable for industrial designs since such studies have not been carried out adequately. It seems that this evaluating methods of modified viscosities give a useful approach for future

studies in this field.

One is the method of using directly flow characteristic curve, to estimate a representative shear rate in an agitated vessel.

The method was firstly

proposed by

Metzner & Otto". They have developed it basing on

measurements of agitation power for both Newtonian and non-Newtonian fluids. Many of the investigators have de-rived experimentally a following relation(' 1(15)(20.

= kN (2.2)

By subsequent investigators the ways considered flow behaviour in the vicinity of heat transfer surface have been

proposed.(13)(14) It should be noted that these apparent

viscosities have been mainly evaluated from power input measurements and hence they are more appropriate to the fluid in the region of the impeller rather than that near the heat transfer surface.

The other

is the method of obtaining analytically a

modified Reynolds number and a resultant Prandtl number and viscosity ratio, supposing a flow pattern in an agitated

vessel(3)(6)(18). In the case the flows of power law fluids

in a circular tube and on a flat plate have been applied.

Many investigators have dealt with the limited fluids

exhibited by power law models or relatively low viscous fluids! There are very often cases that the concepts of agita-tion power have been applied in evaluating the viscosity p* and that the characteristics of non-Newtonian flow have been neglected in deriving the shear rate Therefore, it may be considered that each correlation of data obtained by each investigator is consistent itself, but it cannot be evaluated as a general expression.

Table 1 Apparent viscosities proposed recently on heat transfer in agitated vessels

3. Flow Characteristics of Non-Newtonian High Polymer Solutions

The aqueous polymer solutions such as CMC (carboxy-methyl cellulose), HEC (hydroxyethyl cellulose), PO (poly-ethylene oxide) and SPA (sodium polyacrylate) treated in the present experiments exhibit in general the flow charac-teristics shown in Fig. 1.(9)(") These solutions behave Newtonian in either lower or higher shear rate ranges and non-Newtonian in the intermediate field, where they are possible to be approximated to power law models. The data measured by Meter et al., as an example of those in a relatively high shear rate region are shown in Fig. 1, but the cases operated in the above region have been rarely found in chemical industries, and most cases have been operated generally below the intermediate region (i." <2001/sec). As seen from the data about PO and CMC dilute solutions, the flow characteristics of polymer aqueous solutions used in this experiments can be expressed by the three constant Ellis model. On the other hand, it might be problematical to use a power law model in the wide range of shear rate, because which is only able to express linear relation.

4. Correlations of Heat Transfer in an Agitated Vessel The present experiments were carried out on three types of agitators: (1) agitators with 2-bladed paddle (cooling through an impeller and heating through a jacketed vessel

wall), (2) an agitator with a hilical screw in a draft tube

(cooling through a

draft tube and heating through a

jacketed vessel wall, (3) an agitator with a helical ribbon

(cooling through a helical ribbon and heating through a jacketed vessel wall). The paddle agitator has been used very well as a typical one. The helical screw and ribbon agitators were selected, considering the validity for agitation systems with highly viscous fluids. The experimental data may be correlated by a following dimensionless expression developed firstly by Chilton et al.(4), which has been also used well by many subsequent investigators.

Dih

Nu =

= C (Re*)'

(Pr*)3

(Vis*Y0J4

where,

Re* = pArd'hi*

Pr* =Cpp*/X Vis*- us* hi*

The 11* is a non-Newtonian modified (or apparent)

viscosity. As mentioned later (in Sec. 7.1), heat transfer data for Newtonian fluids in the present experiments were well

correlated by Eq.

(4.1). For non-Newtonian heat transfer, however, the choice of p* presents difficulties.

5. Non-Newtonian Viscosity in an Agitated System Since there are a lot of non-Newtonian fluids which can not be expressed by a power law model as shown in Fig. 1 as an example, the three constant Ellis model may be used for the non-Newtonian fluids.

No. Authors Apparent viscosity Geometrical flow _{characteristics}Flow Applied surface Agitators Ref. No.

1 Carreau, Charest K + 2 1

(6n ). Tubular flow Power law model Jacket Turbine (3)

&Corneille 8 n 2 Mizushina, Ito

Murakami& Tanaka

K

!si r'

=14 Couette flow Power law model Jacket Turbine

Anchor

(14)

L'i1

d-3 Mizushina, Ito,

Murakami &Kini

[ 1- exp(-aRec)] Power law model Coil Anchor (13)

N,

N2-n =1.4 Rec Pd a x104,

4 Pavlushenko K Couette flow Power law model Jacket Paddle (5)

& Gluz K(47rNr (Newtonian) Turbine

Anchor

Screw

Ribbon

5 Hagedorn &

Salam one KNn-1

Flat plate flow Power law model Jacket Paddle

Turbine (6) Propeller Anchor 6 Pollard & Kantyka =18N(N< 50) at _{= 32N335(N>so)} (flow behav. curve)

Based on power Flow data

requirement Jacket & coil Anchors (15) 7 Skelland& Dimmick

A* (.) at i*

= 10N

(flow behav. curve)

Bosed on power Flow data

requirement

Coil Propellers (20)

8 Sandal!& K _{(3n +}1

)11

Modified tubular Power law model Jacket Turbine (18)

Patel (BN)1-n 4,7 flow Anchor

(4.1)

(4.2) (4.3) (4.4)

-103 102 10' 10° 10 10-2 1 1

(1+1

2 ) Ti Tio T%

where, a, T, and no are rheological parameters. The master plot of Ellis model is shown in Fig. 2.

This superposition of Newton's law of viscosity and the power law may be the simplest and most generally useful three constant model.

It describes properly the lower-limiting viscosity 77g. The quantity T% is the shear stress at which the non-Newtonian viscosity has dropped to 1/2 no. This model seems to have

sufficient flexibility to fit the data'. Furthermore, the

combination ( 710/7% ) is a characteristic time for the

non-Newtonian fluids. On the other hand the power model has the dimension and zero shear objections and has not a characteristic time and was pointed to have some defects by

Reiner (17).

The method developed in the present paper is necessary to evaluate an appropriate average flow pattern at the flow region contributing to heat transfer, in order to obtain a modified viscosity

which define a modified

Reynolds number [Eq. (4.2)], a modified Prandtl number [Eq. (4.3)], and a modified viscosity ratio [Eq. (4.4)] in heat transfer. The method deriving the various modified viscosities in agitated equipments with heat transfer treated in this experiment will be developed below. This way is carried out on the agitating systems with a paddle, a helical screw in a draft tube and a helical ribbon. Based on the concept for a

laminar flow, the procedure will be more applicable in

laminar operations.

(5.2)

Fig. 1 Flow characteristics of polymer aqueous solutions

10° 10 10-2 10 MTB94 October 1974 111m. _a=1.0

Fig. 3 Tangential flow profile between two coaxial

cylind-ers

5.1 An agitator with 2-bladed paddle

To evaluate the modified viscbsity at the jacketed vessel wall in an agitator with 2-bladed paddle, the behaviour of Ellis model fluid between two concentric cylinders will be considered as shown in Fig. 3. This is a _simplified

representation of the complicated fluid flow in an agitated

vessel.

Solving the equation of motion leads to a following

relation between an angular velocity cui and the shear stress exerted on the wall of an outer cylinder To.

( 1 1)

(T,Z (

1 1)

2 K2 2ct 1<20

where,

(-112- )

T E

T,A _{° 7%}

By defining a friction factor and a Reynolds number as follows:

f

= shear stress on an outer cylinder

kinetic energy of an inner cylinder

...,N....,__

.

lir

_RN

Ellis for viscometric model 11.1 101 1

ri+1,1-1

Iri21

,_,ii+,/,,,,,

flows: r .,2 J

k

III. ill

-,

V mikraaria'l

ETI9Mig

V

WilliNlill

I

ilik

lin

'ill!"

0

111 T1,

ill

CMC

12*--"rati1111111-soln. 11111 data Power approximation

111111.21h011ili

law

1

-will

I

lol

id

Cl) '5

_ ..

_{_}

.

st

_VII

ian

'4/

4,sse,.(2o,,

A

°o'

I

.

MI,

Expe rimentalcurve

Newton llie .1,s,,,/ 474

IL,

ERN

Nat cited type ,pe inRef. H , (9) )-al.

i.--ii

11

virj.mh'

p; _{''' i,}

-./

.

mbliaMEL

data

aiS12.

ihg'11

..

(Meter et

MEL

PO soln. _{... 2nd Newtonian}

ler

E s parametersar".reV cmc

PO N aa

general r.

= a[-]

1.8 2.2 3.1

=me.

.11

Iii

befiaviour cigne2] 600 46 173 \

.

dyne ] [ nose ] 380 0.9 20.3

1*,,I

C re 1

IIHIIHI

10-2 10-1 10° 101 10°

Fig. 2 Master plot of Ellis model

10° 101 102 103 104

(5.3)

(5.4) (

Ellismodel curve

II

To

1 2KRa.)- 2

2P

( 27r

Re_

P ( 27) (2KR)2 (5.6)

the following equation for the Newtonian fluid flow in a

concentric annulus is obtained:

10= Re (87) ( 1 K2)

1

(5.7)

Where, K

is the diameter ratio of the inner and outer

cylinders. If the Reynolds number is defined by Eq. (5.6)

for the Ellis model fluid, the following expressions are

obtained from Eqs. (5.3), (5.5) and (5.7).

Retan

pNd2 (5.8) tan

T),

(1 -OW

if-

tan

=(5.9)

_{2 (To1"-1 1}

1(1-1)+

(

2 K2 2a K2'

Eqs. (5.8) and (5.9) define a modified Reynolds number and a modified viscosity for the tangential flow between two vertical coaxial cylinders, and the definitions would be used to correlate data for the heat transfer to the vessel wall of an agitator with a paddle or an anchor since in these cases the tangential flow has a significant effect on theheat

transfer.

For highly viscous fluids, almost all of the energy of

agitation may be dissipated near the impeller wall, and, since the fluid flow near the impeller is closely related to the power consumption and the heat transfer, the following modified non-Newtonian viscosity for the power

consump-tion may be employed in correlating data for the heat

transfer to the impeller wall 00.

(5.5) Tio (5.10) where, F- --

( - 7

.4Q) (Q--ri)

- (

4<v> )(a-ri ), (5.13) 7rR T% R 7,4

For a Newtonian fluid in a circular tube, the friction factor:

fw = (5.14)

p<v>2

=

Ty

and the Reynolds number are related as follows:

fw=

_Re16 (5.15)

and the following expression may be obtained by the

method described in Sec. 5.1 using Eq. (5.15).

D<v>p

Repipe

-' pipe 11

Me

₄

8<v> n

1 +a + 3 1 Tw (a'

D

To apply Eqs. (5.16) and (5.17) to the fluid flow in an agitator, pND2 for D<v> p in Eq. (5.16) is introduced and for <v> and D in Eq. (5.17),ND and the following hydrau-lic equivalent diameter for the axial flow in a concentric annulus respectively: 4rh

De

-(K

(1 102

0

_1-K4

_1-K2

1 K2 In(-1-c)

The validity of these way

is shown in the

litera-ture(7)(19). The modified

Reynolds number and the

modified viscosity are expressed as

R Re _ax

pNd2

ax

This approach may be applied in correlating data for the heat transfer to the vessel walls and to the outer wall of a draft tube in which the flow is primarily similar to the fluid flow in an annulus.

Applying the method developed in Sec. 5.1 to the flow shown in Fig. 4 (b) gives the following expression:

1 1 1 1 I ( 1 )]

V° = TR [ 21n (

21a TR'

0-1

(5.22) where, v0 77. o V 2R Ty

The modified Raynolds number and a modified viscosity are given by g*power= 1+ a + 3

T'(a'

8N

("--4 4 T%

T,, a, 8N(770/r) in the above equation is

a function

expressed by Eq. (5.12) of the following section. Where,

r

(

4<v>

1 (IL') = 8N (21E-1,

Tw=

T%

(5.11)

and <v> and 7, are the average velocity and the shear rate applied to the wall in a circular piper flow.

5.2 An agitator with a helical screw in a draft tube For this type of agitator, the axial flow of an Ellis model fluid in a concentric annulus is considered as the model of flow [See Fig. 4 (a)]. Initially, for the laminar flow of the Ellis model fluid in a circular tube, the relation between the average fluid velocity <v> and the shear stress 7,, at the wall is given by the following equation.

4

r = T 4 , (1 + ₃ T (5.12)

a+

w

J1ax =

1+ 0,1

a+3

T (a'

De Ti,

4 8Nd TR (5.23) (5.16) (5.17) (5.18) (5.19) (5.20) (5.21)

-R . = T = + no TR Tyz * a II

pNd2 Re*QX = P.ax 1 In (--i-c-) 71.

Pax' -

_{2 1} 1 1 1 2 2(ce-1) I TR(a,Vo)P-1( 1)

1n ()

(5.25) where, vo and R should be replaced to (rrNp) and (d/2) respectively. This approach may be applied to the flow behaviour near the wall of a helical screw.

(b) Fluid flow profile due to pressure difference Fluid flow profile due to inner wall motion

Fig. 4 Axial flow profile in a concentric annulus

(5.24)

Table 2 Summary of modified non-Newtonian viscosities for heat transfer correlations on various agitated 5.3 An agitator with a helical ribbon systems

In the case of small clearance, the flow between a vessel _{Modified Characteristic}

Wall of heat

wall and a ribbon may be approximated to a flow between No. Impeller non-Newtonian length used in

transfer _{viscosity Nusselt number}

two parallel flat plates. In the case of vertical fluid flow

under a condition of complete slip on the helical ribbon, 1 2-bladed Jacketed vessel wall Eq. ( 5.9 /

paddle

the fluid velocity is given by rrNp ( = rrNd; with the helical Impeller wall Eq. 15.101

ribbon used in this experiment). For completely adhesive 2 Helical Jacketed vessel wall Eq. (5.21)

flow on the ribbon wall, it

is rrNp in the tip section. As screw in

a draft Inner wall of a draft

indicated in Fig. 5, the fluid flow is generally represented tube tube Eq. 15.25/ D di

by a vector from A to an arbitrary point on the line B, C Outer wall of a draft Eq. (5.21) Ddo

and lies between the following two limited values: tube

3 Helical Jacketed vessel wall Flow curve

irNd

_{< [}

the fluid velocity at the tip section ribbon and Eq. (5.27)

I

of the helical ribbon < rrNd Impeller wall Flow curve

and Eq. 15.281

The shear rate at the vessel wall seems to lie within the

following range. 6. Experimental Work

) _(5.26)

N/27IN

(

D )

27N(D

The following expression correlated very well with this

experimental data.

= 27r N (

D

dd)

(5.27)

Using the 7* obtained from Eq. (5.27), the modified

viscosity at the vessel wall may be estimated from a flow

curve. On the other hand, it is very difficult to estimate the modified viscosity at the helical ribbon wall, because the helical

ribbon has four side walls. Hence, the present

experimental data were tried to correlate by an equation multiplied Eq. (5.27) by a factor (3. It was found that (3 0.5 gave the best correlation for the experimental data. The modified viscosity may be estimated using the following expression from the flow curve.

rrN d

2 D

Table 2 gives a summary of the modified non-Newtonian viscosities for heat transfer correlations on various agitated

systems.

7rNp

irNd)

rNd

Fig. 5 Peripheral velocity of a helical ribbon

MTB 94 October 1974

(5.28)

6.1 Apparatus

Fig. 6 is

a schematic diagram of the experimental

apparatus. As shown in Fig. 7, the jacket has two compartments like those described by Mizushina et

al. (13) The amount of the condensate on the vessel wall is devised to be measurable. Experiments were carried out under a condition of steady-state heat transfer in which steam was fed into the jacket and cooling water into an impeller or a draft tube. A summary of the experimental

xR

(a)

conditions is shown in Table 3.

(1) Heat transfer on a 2-bladed paddle system

Two kinds of 2-bladed paddles, as shown in Fig. 8 were used in the experiment. Cooling water enters the paddle through the pipe which runs along the shaft. The paddle has several compartments as shown in Fig. 8 in order to

distribute the cooling water. The temperature of the

impeller wall was measured by thermocouples which were connected to the rotating impeller. The generated voltages of thermocouples were recorded through a silver slip-ring device. Most of the experiments were carried out with an agitated vessel with baffles, but some data were obtained for agitation without baffles in order to

6 5 14

c?-/

2 Cooling water in

(heat transfer runs for

Cooling draft tube)

water out 13 5000 5000 4550 4000 Steam 9 .Condensate 11 Lri. 12 4. Cooling water in

(heat transfer runs for impellers)

1 agitated vessel

2 slip ring

3 revolution counter

4 v-velt and pulley

5 reducer Lquid level Flange(bakelite) Steam inlet Pressure gauge Vessel wall (on, 6mm thick) Jacket Insulating material (asbestus)

Fig. 6 Schematic diagram of the experimental apparatus

0

Fig. 7 Jacketed agitated vessel

In

investigate the effect of the baffles on the rate of heat transfer.

(2) Heat transfer on a screw system

The cross-sectional areas of the inside and the outside of Table 3 Summary of this experimental work

Agitated vessel

material copper diameter ID) - 40cm

depth of Liquid (H) - 65.2cm type of bottom dished

Types of heat transfer runs and agitators

type A cooling from a paddle and heating from a jacketed

vessel wall

impellers paddle A: 20cm diameter paddle

paddle B: 32cm diameter paddle

type B cooling from a draft tube and heating from a

jacketed vessel wall

impeller helical screw (21cm diameter)

type C cooling from a helical ribbon and heating from a

jacketed vessel wall

impeller helical ribbon (38cm)

(31 Area of heat transfer surfaces and impeller rotational speed

Paddle A (Paddle B) _{200( or 324)} Cooling water out 28.50 fCooling water in

Thermocouples for measuring wall temperature

Fig. 8 2-bladed paddle used for measurement of heat trans-fer to an impeller (unit: mm)

6 motor 11 steam trap

7 burner 12 balance

8 boiler 13 rotameter

9 mist separator 14 druggist's scale

10 pressure gauge

cooling area Ern2 _{area [rn° ]}heating _{speed [rpm]}rotational

type A 0.0697 for paddle A 0.865 10-150

0.1370 for paddle B

type B 0.373 for inside

of a draft tube 0.865 10-180 0.473 for outside of a draft tube type C 0.480 0.865 10-80 Drain outlet measuring condensate

Steam & water drain outlet

(1)

0

A'Bearing &

sea(

Fig. 10 Helical ribbon

Fig. 9 Apparatus used for measurement of heat transfer to a draft tube (unit: mm)

1 guide rod for a draft tube 2 inlet for cooling water

3 leg of a draft tube

4 outside wall of a draft tube

5 partition plate

6 conduit for cooling water

7 heat insulator (asbestos)

8 inside wall of a draft tube

r T

3140H-H

2100

Draft tube

(b) Helical screw agitator

M71394 October 1974

the draft tube were designed to be the same. The heat transfer characteristics to the inner and the outer walls of the draft tube may be obtained experimentally. This

is possible because the

cooling system forms two

independent streams of cooling water in two conduits which are separated by a thermally insulated space, as shown in the detailed drawing of Fig. 9 (a). The screw type impeller used in the experiment, is shown in Fig. 9 (b). The flow directions in the inside of the draft tube were carried out both downward and upward.

(3) Heat transfer on a helical ribbon system

The schematic diagram of the helical ribbon used in this experiment is shown in Fig. 10.

The cooling water is

fed into ducts arranged in the

ribbon in

the same way as a paddle. The methods

introducing water and measuring the skin temperatures of a ribbon are also the same as those for a 2-bladed paddle.

6.2 Agitated liquids

As non-Newtonian fluids,

various concentrations of aqueous solutions of CMC, HEC, PO and SPA were used. In the heat transfer correlations for Newtonian fluids, water and the CMC aqueous solutions which exhibited Newtonian behaviour in case of prolonged violent agitation were used. The measuring methods of physical properties adopted here are as follows.

6.3 Method of setting a thermocouple on the surface of heat transfer

Strek(21) has discussed on the temperature measure-ments of the surface of the cylindrical vessel wall in an agitated vessel. He showed that the overall average ture coincided with the arithmetic mean of the

tempera-tures at the four locations in the vertical or longitudinal

direction. The subsequent investigators, Mizushina et

, adopted, on the basis of the experimental fact, to measure the temperatures at the four places. Therefore, this

experiment have also followed Streks' method for the

measurement of the temperature of the cylindrical vessel wall.

The accuracy of the temperature measurement in the range of high Reynolds numbers may depend generally on

how the

thermocouple is set. Konno et

al.0) have

investigated on four methods of thermocouple setting. In the present study the experimental results for the various methods of temperature measurement were investigated. The experiment was done by setting thermocouples at five places in the longitudinal direction of the cylindrical wall, using an agitated vessel with a draft tube for a range of high Reynolds numbers (See Fig. 11). Three different methods of setting the thermocouples were employed as follows:

setting a thermocouple by soldering, 2 places

setting a thermocouple with silicon rubber (a binding agent) and putting a piece of aluminum leaf on the agent,

2 places

setting a thermocouple with silicon rubber (a binding agent) and putting a copper plate on the agent, then soldering the plate, 1 place.

An exemple of this experimental results is shown in Fig. 11. As shown in Fig. 11, methods (b) and (c) show almost the same temperature.

The temperature

of method

(a), however, deviates slightly from the other methods. This seems to be due to the difficulty in finishing the surface of wall.

In any case, the error of measured temperatures by these methods may be very small since the temperature variation among them is within 3%.

Method (b) is less durable than the others. Therefore, in the present experiment method (a) for temperature meas-urements of the wall for the 2-bladed paddles was adopted and method (c) for the other walls.

Liquid level Bottom 94 95 96 Surface temp. (°C I Embedded methods Aluminum leaf /Copper Silicon / rubber

Fig. 11 Dependence of the surface temperature on the method of setting the thermocouples (Reynolds number = 2.15 x 105, driving potential = 20°C)

Liquid a l-1 T [dyne/crn2] 'no [poise] 5-6% CMC 1.6-2.0 600-2550 13-2400 3-4.5% CMC 1.4-2.0 380-1100 0.5-170 1-2.5% CMC 1.1-2.0 100-540 0.13-2.6 8% HEC 1.4-1.6 80-240 23-500 4% H EC 1.1-1.2 10 2-6 1.5-2% PO 1.2-2.5 40-150 0.5-20 0.5% SPA 2.8 25 0.5-20

Density Pycnometers and Berne densimeters

Viscosity Weissenberg rheogoniometer,

ad-verse type Cannon-Fenske visco-meters and Ostwald viscovisco-meters Thermal conductivity Unsteadly comparative method

with a hot wire

Heat capacity Adiabatically calorimetric method The characteristics of non-Newtonian viscosities in this experiment are shown in Table 4. These were exhibited in the form of the rheological parameters of the Ellis model.

For each experimental liquid a and

Ty, were constant without temperature dependency. no was only dependent

on temperature and can be correlated with a following

equation.

no = AeBIT

(6.1)

where, A and B were constant and obtained experimentally. Both the thermal conductivity and the heat capacity of polymer aqueous solutions were turned out lower than those of water at the same conditions.

For example, the experimental data of the thermal

conductivities of 2%, 4% and 6% by weight CMC solutions

were about 5%, 8% and 10% less than those of water

respectively. The decreasing

ratio was independent of

temperature.

The heat capacity decreased with increase of polymer concentration, and the tendency of decrease is

a little

greater in the range of lower temperature.

Table 4 Rheological characteristics of agitated liquids used in this experiment

al.(13) ,

(c)

/

Fuse

7. Experimental Results 7.1 Newtonian correlations

The experiments of heat transfer to agitated Newtonian fluids were carried out as a special case of non-Newtonian ones. This results are shown in Table 5.

Firstly, an agitation system with 2-bladed paddle will be mentioned. The experimental data for heat transfer to a jacketed vessel wall are slightly higher than those of Chilton

et al.() (2-bladed paddle, d xb = 0.6 x 0.1 ft, D = 1 ft) and Table 5 Summary of empirical constants in heat transfer

correlation equation [Eq. (4.1)1 in this work

10^ 10. 10 10' 10" 10' 100 10 100 I I IIIIIII I I IIIII1 I 11 100 10' 102 C7,0,0 Downward flow

,I1, Upward flow

IIIId I

10° 10s

7.-,

Fig. 12 Heat transfer correlations to a draft tube and jack-eted vessel wall (helical screw)

MTB94 October 1974

Uhl(23) (2-bladed paddle, d x b = 14 x 2.5 in., D = 23 in.).

Nevertheless, the present experimental results may be appropriate if one considers the difference in geometry. Such differences have been demonstrated using Pursell's correlation(16), which was considered the geometry of the vessel. The results of the 2-bladed paddle type B are slightly lower than those of the 2:bladed paddle type A. This may be due to an adverse flow condition because of the larger impeller diameter. Both correlations for the two types of paddles may be expressed by a following equation.

Nu = 0.36Re% PrY' Vis-°.14

(Did

)0.4 (7.1)

The heat transfer coefficients to the impeller are 2.5 to 3 times as large as those to the jacketed vessel wall over a wide range

of Reynolds numbers. Therefore, the flow

behaviour near the impellers has a direct effect upon the heat transfer. Both heat transfer characteristics of agitation systems with a helical screw and a ribbon are better in a low Reynolds region, these systems seem to be valid as the devices for heat removal from highly viscous liquids. The characteristics for downward discharged flows were some-what better than those for upward discharged ones.

Heat transfer for a helical ribbon was not much effective in a turbulent region, compared with the past data(12). It was also found by the visual check in a turbulent region that a cowandering behaviour with a rotating helical ribbon was induced and that the original movement of the impeller was not performed. The phenomena were guessed to be caused by the effect of impeller thickness. This seems to originate the lower coefficients of heat transfer.

As an example of heat transfer correlations, the data for a helical screw agitation system are shown in Fig. 12.

7.2 Non-Newtonian correlations

Non-Newtonian correlations were developed for the three types of agitation systems described in Sec. 7.1, in which Newtonian heat transfer correlations were obtained;

10. 10"

IIsInVis*

1- Pr' '/'

V1e"e 1:1 * 10 - )". AO OA .0 PN,P

ill

I 11 I I 1

II

101 10? 10P ,0 Newtonian 2% CMC 3% CMC 4.5% CMC 5.5% CMC 1.5% PO '10"

Fig. 13 Correlations of non-Newtonian heat transfer for an agitator with 2-bladed paddle (d = 20 cm)

Agitator _{of heat transfer}Surface a Re 2-bladed paddle

Paddle A Impeller 1.80 1/2 8-200

0.78 2/3 200-700,000

Jacketed vessel 0.49 2/3 10-700,000

Paddle B Impeller 0.78 2/3 100-70,000

Jacketed vessel wall 0.40 2/3 100-700,000

Helical screw Inside of a draft-tube 0.50 1/2 10-60

0.255 2/3 60-100,000

Outside of a draft-tube 0.82 1/2 10-400

0.30 2/3 400-100,000

Jacketed vessel wall 0.94 1/2 10-700

0.35 2/3 700-100,000

Helical ribbon Impel ler 6.6 1/3 1-100

3.1 1/2 100-4,000

Jacketed vessel wall 0.78 1/3 1.5-10

0.53 1/2 10-180

0.23 2/3 180-4,000

I

IC,

Fig. 114 Correlations of non-Newtonian heat transfer for an agitator with 2-bladed paddle (d = 32.2 cmi

1

that is,

Agitators with 2-braded paddles: Figs. 13 and 1

An agitator with a helical screw in a draft tube: [Fig. 15 p An agitator with a helical ribbon: Fig. 16

The solid lines drawn in each figure are Newtonian

correlation ones discussed in Sec. 7.1. By using the various modified non Newtonian viscosities 'developed in Chap'. 5, the experimental data for the non-Newtonian heat transfer

could be correlated' in terms of the 'Newtonian heat

transfer.

11 Discussion

8.1 'Usefulness 'Of agitated heat transfer equipments If a heat transfer coefficient is

far lower land

a

dissipation, energy of agitation increases very much,

con-10.

Fig. 15 Correlations of non-Newtonian heat transfer for an Fig. 16 Correlations of non-Newtonian heat transfer f r-an

Table 6 Comparison of heat transfer coefficients

Nu V i"

_.,

_{Ri,..--= pAirP11.4}

Pr -1

Pr

- Newtonian _ CMC ..

AI

__

0 2.5% 3.5% CMC 4.5% CMC -e., I 1 C Re 1 Pr VI s

11

I'

110,7!,,

/

1111

I

Nu-11

Emommmumpticsove

Au

II

j_., MIIV

A1111

1

i.'

4**74...

la.

I

''''' .

A.

I

eecso : A

I

I

''''

_.. Aita _xe'

MM. ..M111111111Mi go,'' 2 1111MMA._111111111111MMIIIMI

CalliMIIIIIIM

NMNIPEIMMIIIIIII

Alt

1111.01.;7°'' AIM

I -

Newtonian

III

II

- cji II

111 0 2-6%6.

li

II

.-- ---- PrNI

'

i.::: 4 ' -5.--0 , A 1.5-2% Po 0

,,,,,,tee .. /-2.0 _

imm.m.

0.596 SPA

in

IMMINIMMEN=Mnidll

'

U.

1

MN

II =ME

NI

III

Lower Reynolds region(Re 1`5)1

Heat transfer Re h1PrY" IRatio Heat transfer

No.. Run type _{surface i[kcal/m' hrt] ![-] area Ern9}

1i PaddleA Jacketed, war! 15.11 4.99, 1,200 0.865

2 Screw in a

draft tube

Jac ketedl wall

14.25.62

1.13 0.865,

Inside wallof a draft tube

14.23.76

0175 0.39 'Outside wallof a draft tube 14.2 5.17' 1,04 0A9

.1Cliq Higher Reynolds region (Re 3.0x lOsT

'Paddle A Jacketed wall 2.9xtos 3.72x1034 1.00

0865

4 Screw in a draft tube ,Jacketed wall 2.7 2.54 0.68, 0'.865 Inside wall of a draft tube, Outside wall of a draft tube

17

2.7 2.90 2.27 0.78, 061 0.39 0.49 10° 10' 10. 10° 102 10

agitator with a helicall screw in a draft tube agitator with a helical ribbon

10' 10" 10° 10: 6 * 4 6 810. Newtonian 8% HEC 4% 'HEC 3% CMC 4% CMC 6% CMC I I 100 4 6 2 102 6 10. -10' 1-] 3 102 10'

trary to additional surface of heat transfer, the heat transfer equipment is not said to be better. These points will be

discussed below, basing on the experimental data for

Newtonian liquids.

(1) Heat transfer coefficient

The comparison of heat transfer coefficients in various experimental runs would be possible

to be done,

considering the correction of the Prandtl number. The resultant numerical values are compared in Table 6. Heat transfer area reaches about double by setting the draft tube in a jacketed vessel. Furthermore, at Re 15, the heat transfer coefficients of jacketed wall in the case of setting the draft tube are 1.13 times as high as those

without the draft tube. At

Re:---.3 x 105, the decreasing of coefficients were no more than about 30 percentages

Table 7 Dissipation energy of agitation Agitator with a helical screw in a draft tube

-* = experimental values

Table 8 Comparison of r and p* estimated from various methods

Jacketed vessel wall heat transfer with 2-bladed paddle ( dID = 0.5)

relatively higher Reynolds number region

-GP method MIMT method HS method Present method

[rpm]

7.

1,* [sec' ] [poise] 0.67 2.1 1.33 2.1 2.0 2.1

Inside wall heat transfer of a draft tube with helical screw relatively lower Reynolds number region

-GP method MIMT method

where, GP method: 4/TN, MIMT method: i* = 14

(Pi

)

HS method: =N, Present method: Eq. (5.9) for Table 8 (a), Eq. (5.25) for Table 8 (b)

HS method Present method

i.

AL*

'ax'

g*ax'

[sec'I [poise] [sec' ] [poise]

0.33 250 1.00 220 1.67 193 MTB94 October 1974 [rpm] Re Np [--] I-1

F5) _{Qv Net heat transfer}

rate* [kcal/hr] [Joule/sec] [kcal/hrl (1) 30 72 100 150 200 6.5 54 15.6 23 21.7 17 32.5 12.5 43.3 9.0 4.84 4.15 28.6 24.5 56.5 48.5 140 120 239 205 2.24x103 3.36 3.76 5.00 5.67 where, _{Re = pNaP} =

(1) (21)2(-)

= 11.0N 40 [rpm] ^I... [sec ]

If'

[poise] [sec')

i.

1.1* [poise] 40 8.4 1.9 59 0.91 80 16.8 1.7 11] 0.52 120 25 1.5 176 0.35

[sec' ] [poise] [sec-11 [poise]

20 4.2 150 162 38 60 12.6 104 490 25 100 21 86 810 18.8 il-an [sec' ] Atan [poise] 71 0.87 103 0.74 127 0.68 8.5 120 33 72 52 60

of the case without the draft tube. Therefore, the

agitation equipment with a helical screw in a draft tube is useful in a wide range of Reynolds numbers without saying a highly viscous region.

(2) Energy dissipation of agitation

The power requirements in this experiments with a

helical screw in a draft tube are evaluated in Table 7, by using the power requirement curve obtained for a small scale agitator which was nearly similar to the agitator used in this experiment. The viscosity used in calculation is

a 40 poise, which

is

nearly comparable to the

maximum value in this experiment. It is assumed that all of dissipated energy converts to viscous generated heat and its heat is given to an agitated liquid. Though the generated heat increases with a rotational speed of agitator, all of the quantities calculated are no more than about 4 percentage of the net quantities of heat transfer. The viscous heat generation in this experiment seems to be very little and be out of the question.

8.2 Modified non-Newtonian viscosities

Comparison of the present methods and some recent

methods

The comparison of the present methods with the three past methods [by Gluz et al. (GP)5, Mizushina et al. (MIMT)(14) and Hagedorn et al. (HS)6] is shown in Table 8.Table 8 (a) exhibits the jacketed vessel wall heat

transfer of agitator with a 2-bladed paddle in a high

Reynolds numbers. Table 8

(b) concerns with heat

(a)

-2ir

2 10° 9 8 7 I 1

1111

Correlation equation:

Nuc(PIVPd")Cr1/3(

K)-0.14 1 )Mizushina et al. /.;=K(14 27rN Yac* DI 2)Metzner et al.

_K 6.

2 )" A A 0 A Rc.

(a) Low Reynolds number region

A

transfer of the inside wall of a draft tube in a relatively low Reynolds number region.

The modified non-Newtonian viscosities p* estimated by the methods of GP and HS are too high. The reason seems that the geometry of an agitated system in the derivation of the apparent shear rate .1,* was neglected.

On the other hand, the viscosities by MIMT method

agree with those of the present method in a turbulent region, but in a laminar region their values are very high. MIMT method has been applied mainly for a turbulent system in their paper, and so the discharging effect of an impeller is probably to be taken into account in their

But a higher r and the resultant lower p* would be

estimated in their method as the discharge effect is sure to decrease in a laminar region. Where, the 'tan and the fax. in the present method are calculated backward by the /4, and the /fax, respectively. The comparison of heat transfer correlations obtained by some methods is shown in Fig. 17. It seems that the methods developed in this paper are valid both quantitatively and qualitatively.

(2) Relation between the characteristic shear rate r and

the rotational speed of impellerN

Some recent studies on heat transfer to agitated non-Newtonian fluids have presented a following

rela-tion 5)(20)

= kN (8.1)

The factor k has been obtained experimentally as a

constant value in the given fluid and geometry of a heat transfer agitation system. Fig. 18 shows that the above

4 3 2 10° 1 1 (b) High Reynolds number region

fact is valid. In the case of this methods, the and

ax, are calculated backward from the corresponding

pta and ifax, respectively in the same manner of Sec. 8.1(1) It would be comprehended easily that the

2 r-t

6%CMC

0.2 04 06 0.8 10

Fig. 18 Relation between the characteristic shear rate and the rotational speed of impeller

N(rps) 2 10' 9 8 7 6 5 4 3 Newtonian 0.0 , A this method Mizushina et al. fb,. , A Metzner et al. 3 4

56 7

8 9 10' 2 3 4

56

Fig. 17 Comparison of heat transfer correlations obtained by some methods

10' 9 8 7 6 5 10' 9 8 7 4 3 6 5 2 6 7 8 910 2 3 4 5 2 7 6 5 4 3 10' 8 10 8

j

a = 1.8 ) dyne r _cm2 -=-170 (poise) at 70°C

10° 10 10 10-0

..._-..warsn=.====:

--...

--...

CUM --

.7474.2117.7.-milll

...11

r...9... ',ma"

From

...imum. masonc...-Emer

aim

MO di= 11111110011. 10.0.!

No lo

IIII IIIMEIIIIIIIIMEISINIONINSI

IF

lighliElltlid

_,,,==-....

_CM MEM MINIMMI,UMB.UMMIMMI0.1 =MMEMIENIMMII211117110112 MIIIIMEME111~.1111.

1111116.-ril 111W111.111111

1111111111111

1

...

...

...

...CM

"AM:111.1 1111:11111111==.111...

=mum

imm...IIImEn...mmbh

minnummiumilmomummumminun

11111111111111111111.1=110

111111E11111111

Sutterby model: arc sinh BV17N-1 r 77,1,1-710 Bil/2 II for viscometric flows:

77_71or arc sinh i^

77/

Fig. 19 Master plot of Sutterby model

model is convenient for variational methods as its viscosity is expressed in the form of a function of shear

rate.

(2) Derivation of the modified non-Newtonian viscosity

As an example, the derivation method of modified

viscosity on an outside wall of a draft tube and jacketed vessel wall heat transfers with a helical screw in a draft tube will be discussed below.

For a laminar flow in circular tube the following relation can be derived by solving the equation of motion,

4Q 4 arc sinh -2A

B (

) =--(B'') [1

I 7rR 3 1 _A

xf

arc sinh

F)

I

t3d t

(8.4)

B'-y'

where, (8.5)

7,,

By the balanced equation of forces and the constitutive equation,

(B)

(

ApR arc sinh (Biw ) A

(8.6)

(B71 [

77° 2L Therefore, ( ( 2L

-

e I A ,B ( irR3 ) B APR

F

4Q , no (8.7)

In the same way in Sec. 5.2, the modified Reynolds number and the viscosity are given by

pNd2 Re*ax = 11*.x Fe (A, B-8Nd1 De ax =77 8Nd D, 9. Design Procedure

Heat transfer coefficients for pseudoplastic fluids in a wide range of laminar to turbulent flows can be estimated as follows. As an example of the methods proposed in the present paper, let us illustrate on heat transfer of a jacketed vessel wall with a 2-bladed paddle.

(a) Determine the flow characteristic curve ( T vs. 71) for a

given fluid by measurements in a suitable viscometer at the some appropriate temperatures and fix the para-meters of Ellis model by superposing measured data on the master plot of Ellis model (See Fig. 2).

Illustration of measured flow characteristics is shown in Figs. 20 and 21. In this test a and 7-,A of a given liquid could be regarded as constant values in wide range of temperature. And no could be exhibited in the form of Eq. (6.1).

MTB 94 October 1 974

methods proposed in this paper are taken into account the matter which k varies with a given geometry and a flow pattern. Therefore, this methods would suggest that the rescent methods, which were not considered the

above matter, have limitations in actual applications

though they are convenient, because of briefness, for practical uses. It seems that the present methods suggest a direction of future studies on heat transfer of agitation

systems.

8.3 Application of the Sutterby model to a non-New-tonian fluid

The three constant Sutterby model has been proposed as a model which can express enough various flow data of high ploymer aqueous solutions with the same accuracy as the Ellis modelP2) As a result of applying the Sutterby

model to the arrangement of the data obtained in

this

experiments, the data coincided nearly with those induced by the Ellis model. Therefore, it seems that the methods are

also valid to be applied to heat transfer correlations in

agitated non-Newtonian fluids. The result shouid be skipped here, as it is much the same to the result of the Ellis model.

The Sutterby model and an example of the derivating

methods of modified non-Newtonian viscosities will be

developed below.

(1) The Sutterby model

The Sutterby model generalized the Eyring model is expressed as follows:

= --T/ (II)

a

(8.2)

arcsinh B II

n ( II) = z 1A (8.3)

It contains both a lower limiting viscosity 77. and a

characteristic time B. Fig. 19 shows the master plot of

Sutterby model. Since the functional dependence is

somewhat complicated,

it

is not so easy to use in

analytical studies as the Ellis model. But the Sutterby

10° 10' 10. 10. (8.8) (8.9) [

/

1 .. ,

Prepare a

chart graphed the flow equation (flow

velocity vs. wall shear stress: Eq. (5.3)) for a idealized flow pattern, which must be guessed from a given agitat-ing system (See Fig. 22). Obtain the modified viscosity at the required rotational speed of impeller, using the corresponding equation Eq.

(5.9) and the chart (the

curve of a = 2.2 in Fig. 22). Caluculate the Reynolds and Prandtl numbers.

Obtain the viscosity ratio by means of reading zero shear viscosities no at the temperatures of jacketed vessel wall and bulk fluid respectively, using Fig. 21.

Evaluate the heat transfer coefficient on the jacketed vessel wall by Eq. OIL

This evaluating method may be applicable for many Newtonian heat transfer correlations of various agitation

4 2 101 8 6 4 10° 6 4 2 46 810°

Fig. 20 Flow characteristic curve (an example of this ex-perimental data)

Nomenclature

systems,

if the estimation of idealized flow pattern of a

given agitation system is possible roughly.

10'

10°

10

34

Fig. 21 Zero shear viscosity (an example of this experimen-tal data)

Fig. 22 Master plot of Ellis model for tangential flows in an annulus with K = 0.5 Aqueous solution of 1,5 wt% PO Ellis model : I a=2.2

.r,,,.

dyne

Milill

r1/2-46_cm. (pose) V

Aril

A

1/TX 10. (1/°K) - 30°C o,,, Aqueous of I solution 1.5 wt %PO Experimental 6, data kt Curve of the 50°C

!---...,Nume 'N.,,--5 Po'se IP'Ellis

n 1,2=46a=22 model d Y-,

'''

V (poise)

il,

11/11/111.

1111 80C

vi.

_e="1.3 pose

imilime...

1111=111MMIIIIMMLIIIIII

11111=111111111111MEMMI11117

mou

d )-0.5

EMMOPirApPSIICMpt_

41.0alf10401111111-x (

A"OPE

to...11/111 0

APii.

MI "" II ( Toe , ,:,..

,

1 1 x 1 a

/

12, i

A =

parameter in Sutterby model

[I

K =

parameter in power law [g.secn-2 /cm ]

=constant in Eq. (6.1) [poise] k =constant in Eqs. (2.2) or (8.1)

[]

a =constant in Table1 (= 1.4x 104)

[]

L =

axial length of a circular tube [cm]

B =

parameter in Sutterby model [sec]

N =

rotational speed of impeller [rps]

=constant in Eq. (6.1) [°K] n =parameter in power law

[]

b = impeller width [cm]

n' =

flow behaviour index

[]

C =

constant in Eq. (4.1)

[]

P, =

power consumption [erg/sec]

=heat capacity

[cal/gt]

Ap =

pressure drop [dyne/cm2]

D =

vessel diameter [cm]

p

=impeller patch [cm]

Ddi =

inner diameter of a draft tube [cm]

Q =

volumetric flow rate through [cm' /sec]

Ddo = outer diameter of a draft tube [cm] a circular tube

De =

hydraulic equivalent diameter [cm]

Qv = viscous

dissipation rate [kcal/sec]

=characteristic length [cm]

R =

radius [cm]

d =

impeller diameter [cm] rh = hydraulic equivalent radius [cm]

F,

=defined by Eq. (8.7) T = dimensionless shear stress

[]

f

=friction factor

[]

=temperature

['K]

H =

liquid depth in an agitated vessel [cm]

V° =

defined in Eq. (5.23)

[]

h =heat transfer coefficient [cal/sec cm2°C]

<v> =

average velocity [cm/sec]

= NuPr*-Y Vis"."

[]

=vertically moving velocity [cm/sec]

28 29 30 3.1 3.2 3.3 2 4 6 8100 2 4 6 8 102 2 10' 10° 10' 10° 1 40 20 10 8 6 4 2 0.8 0. 2 =' = = = = = = = = : 1) = = = = = = = = 4

[]

[]

[]

[]

[1/sec) [1/sec] [poise] [poise]

[]

Bird, R.B.: Can. Jour. Chem. Eng.,q, 161 (1965) Calderbank, P.H. and M.B. Moo-Young: Trans. lnstn. Chem. Engrs.,37, 26 (1959)

Carreau, P., G. Charest and J.L. Corneille: Can. Jour. Chem. Eng.,44, 3(1966)

Chilton, J.H., J.B. Drew and R.H. Jebens: Ind. Eng. Chem.,36, 510 (1944)

Gluz, M.D. and I.S. Pavlushenko: Jour. Appl. Chem. USSR, 39, 2323 (1966)

Hagedorn, D. and J.J. Salamone: I & EC., Process Design & Development, 6, 469 (1967)

Knudsen, J.G. and D.L. Katz: "Fluid dynamics and heat transfer", McGraw-Hill Inc. (1965)

Konno, H., K. Okada, K. Sasabayashi and S. Otani: Kagaku Kogaku, 31,872 (1967)

Meter, D.M. and R. B. Bird: A. I. Ch. E. Jour., 10, 878 (1964)

Metzner, A.B. and R.E. Otto: A. I. Ch. E. Jour., 3,3 (1957)

Mitsuishi, N. and N. Hirai: Jour. Chem. Eng. Japan, 2, 217 (1969)

Mizushina, T., R. Ito, S. Hiraoka, T. Murata and T.

References

Np = power number

Pr

= Prandtl number

Re = Reynolds number

Re, = pN2

d211<

Vis = viscosity ratio <Subscripts>

ax = quantity for axial f,low in a concentric annulus ax' = quantity for an axial flow in a concentric annulus

when an inner cylinder moves vertically

i

= quantity evaluated at an inner surface

o = quantity evaluated at an outer surface pipe = quantity for a circular pipe flow

power = quantity used for estimation of power consumption

R = quantity evaluated at a radiusR

s = quantity evaluated at a surface tan = quantity for a tangential flow w = quantity evaluated at a wall

<Superscripts>

* = quantity for modified (or apparent) non-Newtonian viscosity

NI TB 94 October 1974

Yamanaka: Kagaku Kogaku, 34 , 1213 (1970)

Mizushina, T., R.

Ito, Y. Murakami and M. Kin:

Kagaku Kogaku, 30, 719 (1966)

Mizushina, T., R. Ito, Y. Murakami and S. Tanaka: Kagaku Kogaku, 30, 819 (1966)

Pollard,

J. and T.A. Kantyka: Trans. lnstn. Chem.

Engrs., 47, T21 (1969)

Pursell, H.P.: M. S. Thesis, Newark College of Engi-neering, Newark (1954)

Reiner, M.: "Deformation, strain and flow", Lewis & Co. (1960)

Sandall, O.C. and K.G. Patel: I & EC., Process Design

& Development, 9, 139 (1970)

Schlichting, H.: "Boundary layer theory," McGrow-Hill Inc. (1960)

Skelland, A.H.P. and G.R. Dimmick: I & EC., Process Design & Development, 8, 167 (1969)

Strek, F.: Int. Chem. Eng., 3, 533 (1963) Sutterby, J.L.: A.I. Ch. E. Jour., 12, 63 (1966) Uhl,. V.W.: Chem. Eng. Prog. Sym. Series, 51, 93 (1954)

of an inner cylinder

a = parameter in Ellis model = constant in Eq. (4.1) = constant in Table 1 = defined by Eq. (5.13)

= dimensionless shear rate (= = shear rate

= shear rate (tensor) = non-Newtonian viscosity = zero shear viscosity

= ratio of radius of inner cylinder to that of outer cylinder

X = thermal conductivity [cal/cm sec°C]

p = viscosity [poise]

P = density [g/cm3]

7 = shear stress (tensor) [dyne/cm2]

= shear stress [dyne/cm2] = parameter in Ellis model [dyne/cm2]

= defined in Eq. (5.19)

[]

= dimensionless angular velocity

[]

=,angular velocity [rad/sec] II = second invariant of shear rate (tensor) [1 /sec]

Nu = Nusselt number

[]

, 16) 9(8) , (12) 1(15) 06)! (17) 1 81 I

1[]