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Rheological properties of coarse food suspensionsin tube flow at high temperatures

Krittalak Chakrabandhu 1, Rakesh K. Singh *

Department of Food Science, Purdue University, W. Lafayette, IN 47907-1160, USA

Received 3 April 2001; received in revised form 27 January 2004; accepted 24 February 2004

Abstract

The effects of particle concentration and carrier fluid temperature on rheological behavior of model food suspensions consisting

of 1.5% CMC solution and green peas (1530% v/v) were investigated using a tube viscometer. The flow behavior of the suspensionswas represented by the power law model. The suspension consistency coefficient (m) increased with particle concentration and

decreased with temperature, whereas the opposite trends were observed for the suspension flow behavior index ( n). Among various

theoretical, semi-empirical, and empirical equations tested for suspension apparent viscosity (l) estimation, the third order

expansion of Einstein equation, which was derived via the hydrodynamic approach, provided the best estimates for l. Of equations

tested for m estimation, those in which n was included offered better estimates of experimental values, with an empirical equation

obtained based on the Einstein equation and the incorporation of n term providing the best m estimation. These findings suggest

that, for concentrated coarse suspensions subjected to conditions presented here, the dependence between m and n is of importance

and should be considered in order to achieve a better m estimation. Besides, better representations for power law parameters of such

suspensions may be obtained based on a theoretical expression derived for l via the hydrodynamic approach. The study presented

here provides a much-needed insight toward the flow behavior of concentrated coarse food suspensions at high temperature,

information of which is vital for various food processes.

2004 Elsevier Ltd. All rights reserved.

Keywords: Non-Newtonian fluids; Rheological properties; Coarse suspension; Particulate foods

1. Introduction

Designing an aseptic process requires information

about flow properties of the foods. Flow properties have

significant effects on the residence time distribution and

heat transfer of the particulate food system as reported

by several researchers (Awuah, Ramaswamy, Simpson,

& Smith, 1996; Ramaswamy, Awuah, & Simpson, 1996;

Sandeep & Zuritz, 1994). A number of studies have been

done on rheological behavior of fluid foods and manyare summarized by Holdsworth (1971). However, only a

few studies on rheological behavior of coarse food sus-

pensions can be found in the literature, especially with

the carrier fluids that exhibit non-Newtonian behavior

(Bhamidipati & Singh, 1990; Martinez-Padilla, Cornejo-

Romero, Cruz-Cruz, Jaquez-Huacuja, & Barbosa-Ca-

novas, 1999; Pordesimo, Zuritz, & Sharma, 1994).

Due to the larger particle size, only few types of

viscometers can be used in characterizing the flow

behavior of coarse suspensions. These include tube

viscometers (Bhamidipati & Singh, 1990), wide-gap

parallel plate viscometer (Pordesimo et al., 1994), wide-

gap rotational viscometer (Martinez-Padilla et al.,1999). The above-mentioned equipments are although

functional, difficulties still exist especially in the studies

for dense suspensions. Some of the difficulties could be

caused by the interference to the equipment parts by the

large particles as reported by Martinez-Padilla et al.

(1999), centrifugal effects in the parallel plate viscometer

as pointed out by Pordesimo et al. (1994), or simply

large amount of materials and floor space required to

run the tests using the tube viscometer. This leads to the

scarcity of the knowledge on the flow behavior of coarse

food suspensions. More insight on the flow behavior of

Journal of Food Engineering 66 (2005) 117128

www.elsevier.com/locate/jfoodeng

* Corresponding author. Address: Department of Food Science and

Technology, University of Georgia, Food Science Building, Athens

30602, USA. Tel.: +1-706-542-2286; fax: +1-706-542-1050.

E-mail address: [email protected] (R.K. Singh).1 Current address: Death Receptor Laboratory, Institute of Signal-

ing, Developmental Biology and Cancer Research, CNRS UMR 6543,

33 Ave. de Valombrose 06189, Nice, France.

0260-8774/$ - see front matter 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jfoodeng.2004.02.039
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/


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coarse food suspensions particularly in the conditionsapplicable to aseptic processing is clearly desirable. The

objectives of this study were (1) to characterize the flow

behavior of model food suspensions consisting of non-

Newtonian carrier fluid and high concentration of large

particulates at high temperatures, (2) to investigate the

effect of particle concentration and temperature on flow

behavior of non-Newtonian coarse suspensions, and (3)

to investigate the applicability of several mathematical

expressions in explaining the effect of particle concen-

tration on the flow behavior of coarse suspensions.

2. Materials and methods

2.1. Food system

Green peas were used as model particulate phase

because of their commercial availability and their

applications in real food in the market. Frozen green

peas were purchased from Purdue University food store.

Green peas were thawed and drained prior to each

experimental run.

An aqueous solution (1.5% w/v) of sodium carboxy-

methylcellulose (CMC) (TIC Gum, Inc., Belcamp, MD)

was selected as the carrier fluid. The CMC solution wasprepared by slowly adding the desired amount of CMC

powder into a mixing tank filled with water continu-

ously agitated by a mechanical mixer. The mixture was

agitated for approximately 45 min then left overnight

(1215 h) for the CMC to dissolve. Prior to the experi-

mental run, the CMC solution was mixed with green

peas to attain the specified particle volume fractions

(1530%v/v). Due to the delicate texture of green peas,

the CMC-pea suspension was made in several small

batches of 0.0379 m3 to achieve the desired particle

concontration. The suspension was mixed manually to

ensure the uniform distribution of green peas in the

suspension and then delivered into the pump feeder

during the experimental run. The physical properties of

the food materials are summarized in Table 1.

2.2. Experimental setup

The rheological characterization of the fluid food,

with or without particulates in aseptic processing con-

ditions was performed using an in-line tube viscometer.

The schematic diagram of the experimental setup is

shown in Fig. 1. The system consisted of a 150 l feed

tank, a moving pocket type pump (Moyno Product

Progressive cavity pump, Robbins and Myers, Inc.,

Springfield, OH), a helical double tube heat exchanger

(46.3 m long, 0.032 m I.D.) as a heater, a helical doubletube heat exchanger (33.5 m long, 0.032 m I.D.) as a

cooler (Stork, Amsterdam, Netherlands), an electro-

magnetic flow meter (Promag 33, Endress and Hauser

Inc. Greenwood, IN), an insulated straight stainless steel

tube (0.022 m I.D.), a differential pressure transducer

(Rosemount Model 3051 Smart Pressure Transmitter,

Rosemount Inc., Eden Prairie, MN), resistance tem-

perature devices (RTDs), and a programmable logic

controller (PLC, 5-15, Allen-Bradley Co., Inc. Milwau-

kee, WI) for control and data acquisition. The tube

viscometer (2.89 m long) is located in the straight tube

Nomenclature

D tube diameter, m

L length, m

Le entrance length, m

m fluid consistency coefficient, Pa sn

m suspension consistency coefficient, Pa sn

n fluid flow behavior index

n suspension flow behavior index

P pressure, Pa

Q volumetric flow rate, m3/s

Qws volumetric flow rate without slip, m3/s

Qm measured volumetric flow rate, m3/s

r tube radius, m

Re Reynolds number

T temperature, C

Greek charactersb slip coefficient, m Pa1 s1

/ particle volume fraction

/m maximum particle volume fraction

_c shear rate, s1

_cw wall shear rate, s1

l suspension apparent viscosity, Pa s

lf fluid apparent viscosity, Pa s

lr relative viscosity (l=lf)

s shear stress, Pa

sw wall shear stress, Pa

Table 1Physical properties of model food system

Property Value

Green peas

Average diameter (m) 0.0084 0.0025

Density (kg/m3) 1026 9.770

CMC solution (1.5% w/v)

Density (kg/m3) 1022 5.000

118 K. Chakrabandhu, R.K. Singh / Journal of Food Engineering 66 (2005) 117128


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section of the system. The entrance length (Le) of 3.44 m

(Le > 100 I.D.) was used to avoid the entrance effect.The entrance length was verified against the equation

given by Kays (1966).

Le

D

Re

20 1

2.3. Experimental run

Water was pumped through the system until the

system reached the desired temperature. Then the test

food suspension was delivered to the positive displace-

ment pump. Once the flow and temperature stabilized,

the data of flow rate and pressure drop were collected

with the data acquisition system. The suspension was

not reused after heating due to the change in rheological

properties of CMC solution being heated at high tem-perature. The experimental parameters are summarized

in Table 2.

2.4. Wall shear stress and wall shear rate calculations for

tube viscometry

The relationship between shear rate and shear stress

was obtained from the measurement of pressure gradi-

ent and volumetric flow rate based on the following

assumptions: (a) steady flow, (b) time independent fluid

properties, (c) fluid velocity has no radial or tangential

components, (d) no slip condition at pipe wall, (e)incompressible fluid, and (f) isothermal flow. Wall shear

stress was calculated from:

sw rDP

2L2

Shear rate was calculated using RabinowitschMooney

equation

_cw 3Q

pr3 sw

dQ=pr3

dsw3

The derivative term in Eq. (3) was obtained by taking

the derivative of a function from curve fitting the plot ofQ

pr3vs. sw.

When a tube viscometer is used to study the flow

behavior of dense coarse suspension, wall effects would

be present. The slip at wall occurs when fluid, having

lower viscosity than the bulk suspension, forms a thin

lubricating layer at the wall of the tube. Since the Ra-

binowitsch-Mooney equation was derived based on the

assumption of no slip at wall, the effect of the slippagemust be corrected in order to use the equation to obtain

the true shear rate. The classical method for slip cor-

rection can be found in the literature (Mooney, 1931;

Jastrzebski, 1967; Steffe, 1996). The correction is done

by introducing a term to account for the added flow due

to wall slippage to the overall flow rate. This results in

the following equation:

Qm

swpr3

Qws

swpr3b

r4

Generally, the effective slip coefficient, b, is evaluated

using tubes of different radii. However, to determine theslip coefficient by varying the tube radius would require

large amount of food materials in this study. Recog-

nizing the similar effect of varying tube radius and

varying particle concentration on the wall shear stress,

an alternative procedure based on the utilization of

variable particle concentration was used to determine

the slip coefficient. The procedure has been described by

Chakrabandhu (2000). The slip coefficient was then used

for flow rate correction. The corrected volumetric flow

rate was subsequently used to calculate the true wall

shear rate.

Table 2

Experimental parameters for study of rheological properties of coarse

food suspensions

Parameter Value

Carrier fluid temperature (C) 85, 110, 135

Particle concentration (% v/v) 0, 15, 20, 25, 30

Volumetric flow rate (m3/s) 1.261043.15 104

Fig. 1. Schematic diagram of the experimental setup for the study of coarse suspension flow behavior at high temperature.

K. Chakrabandhu, R.K. Singh / Journal of Food Engineering 66 (2005) 117128 119


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2.5. Flow behavior characterization of coarse food

suspensions

Once the true shear rate was calculated, flow behavior

parameters of the suspension were computed. As several

studies have reported that the flow behavior of suspen-

sions follows power law pattern, the power law para-

meters of the coarse suspension were investigated. The

suspension consistency coefficient (m) and flow behav-

ior index (n) were estimated by fitting the experimental

data in the power law model (Eq. (5)) using linear

regression (SAS, 1989).

ln s ln m n ln _c 5

The suspension apparent viscosity (l) was then calcu-

lated from

l m _cn1 6

An analysis of variance (ANOVA) was performed for

the effects of particle concentration and suspensiontemperature on the consistency coefficient and flow

behavior index of suspensions.

2.6. Mathematical representation of apparent viscosity for

coarse suspensions

Selected theoretical expressions from the literature

developed from various approaches, i.e. hydrodynamic

approach (Einstein, 1905; Ting & Luebbers, 1957), en-

ergy approach (Frankel & Acrivos, 1967) as well as

semi-empirical (Mooney, 1951; Thomas, 1965) expres-

sions were considered for the prediction of the apparentviscosity of suspensions that contain high concentra-

tions of coarse particulates. A statistical test (v2-test)

was performed to determine the goodness of fit of the

mathematical expressions to the experimental results.

The v2 value is expressed as:

v2 Xni1

Ymeasured Ypredicted2

Ypredicted

!7

The null hypothesis of no significant differences between

the predicted and experimental results is rejected if the

calculated v2 value is larger than the tabulated critical

value (c) depending on the confidence level (a) and de-gree of freedom (df). The expressions tested are listed

below.

Einsteins equation

lr l

lf 1 2:5/ 8

Second-order expanded Einsteins equation

lr 1 2:5/ 10:05/2 9

The second-order coefficient of 10.05 arrived after con-

sidering the hydrodynamic interaction of spheres and

the effect of double formation due to collisions (Thomas,

1965).Third-order expanded Einsteins equation

lr 1 2:5/ 10:05/2 a/3 10

Thomass equation

lr 1 2:5/ 10:05/2 aeb/ 11

Instead of expanding the power series to the third order,

Thomas proposed that the fourth term which represents

the particle interaction to be an exponential term as

suggested by Eyring, Henderson, Stover, and Eyring

(1964). The empirical constants, a and b, in Eq. (11)

were given by Thomas as 0.00273 and 16.6, respectively.

Mooneys equation:

log lr 2:5/

1 k/12

where k is an empirically determined parameter sug-

gested by Mooney to be in the range of 1.351.91. Sincek is a parameter related to the packing geometry, several

numerical values that represent theoretical packing

geometries along with some that have been presented in

the literature were used to investigate the fit of the

Mooney equation to the experimental data from this

study. The values and their significance are summarized

in Table 3. The value of k was also empirically deter-

mined to obtain the best fit to the experimental data.

FrankelAcrivoss equation

lr b/=/m

1=3

1 /=/m

1=3" # 13The FrankelAcrivos equation was derived based on the

assumption that the particle shell is spherical and b takes

the value of 9/8. Instead, if a cubical particle shell was

assumed, b would take the value of 3p=16. In the veri-fication of their equation, Frankel and Acrivos used the

/m value of 0.625 which was obtained by Thomas (1965)

from available data in the literature. In this study, the

values of b and /m were varied (Table 4). The /m value

was also empirically determined to obtain the best fit to

the experimental data.

Table 3

Values of k tested for best fit of Mooneys equation to experimental

data

k Value Significance

1.350 Lower bound for k suggested by Mooney

1.910 Upper bound for k suggested by Mooney

0.340 Maximum volume fraction for completely tetrahe-

dral packing

0.524 Maximum volume fraction for completely cubical

packing

0.625 Maximum volume fraction used by Thomas (1965)

and Frankel and Acrivos (1967)



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TingLuebbers equation

lr /m

/m /14

In the same manner as the tests for other expressions

mentioned above, the values of /m in the TingLueb-

bers equation was also varied (0.34 and 0.524) and an

empirical value of/m was also determined to obtain the

best fit for the equation.

2.7. Mathematical representation of power law para-

meters for coarse suspensions

Compared to the expressions for apparent viscosity

of suspension, smaller number of mathematical expres-

sions relating the power law parameters to the solid

fraction exist. In this study, the following equations

proposed by Jarzebski (1981) were tested for the fit to

experimental results.

Modified FrankelAcrivoss equation

m

m

9

8

/=/m1=3

1 /=/m1=3

" #n15

m

m

3p

16

/=/m1=3

1 /=/m1=3

" #n16

Modified Mooneys equation:

lnm

m

2:5/

1 k/

n17

Empirical equations of simple forms listed in Table 5

were also fitted to the experimental data to describe the

dependence of power law parameters on the particle

volume fraction in the conditions covered in this study.

Due to the fact that the consistency coefficient and the

flow behavior index together represent the flow curve of

suspensions, it was considered that a better prediction of

consistency coefficient could be obtained if the descrip-

tive equation also contained the flow behavior index

term or vice versa. Also, most of the previously de-

scribed mathematical expressions, which have been

shown in the literature to describe well the l and m

values, contain the particle fraction terms in the form of

the volume fraction-to-maximum volume fraction ratio

(/=/m) or those similar to the what given in the Einsteinequation. Therefore the following equations were also

investigated:

m

m a

/

/m

bn

n

c18

m

m

a1 2:5/b n

n

c

19

3. Results and discussion

3.1. Effects of particle concentration and temperature on

power law parameters of coarse suspensions

From the known value of slip velocity, the corrected

volumetric flow rate was computed and used in calcu-

lation of the true shear rate according to the Rabi-

nowitschMooneys equation. The flow curves were

plotted for suspensions at 85, 110, and 135 C in Figs. 2

4. The suspensions exhibited power law behavior. Flow

behavior parameters of the suspensions were determined

according to the power law model. An analysis of var-

iance (ANOVA) was performed for the effects of particle

concentration and temperature on the consistency

coefficient (m) and flow behavior index (n) of suspen-

sions. The power law parameters of the suspensions are

summarized in Table 6.

Table 5

List of empirical equations of simple forms used to describe thedependence of power law parameters on the particle volume fraction

Power law

parameter

described

Basis of empirical equation

Volume fraction to

maximum volume

fraction ratio

Einstein equation

Consistency

coefficient

m

m a /

/m

bm

m a1 2:5/

b

m

m a /

/m

bnc m

m a1 2:5/

bnc

Flow Behavior

Index

n

n a /

/m

bn

n a1 2:5/

b

a, b and c are empirical constants.

Fig. 2. Shear stressshear rate relationships for suspensions at 85 C

and various particle concentrations.

Table 4

Constants tested for best fit of FrankelAcrivos equation to experi-

mental data

Parameter Values

B 9/8, 3p=16/m 0.340, 0.524, 0.625



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Particle concentration and temperature showed sig-

nificant effects on the consistency coefficient and the flow

behavior index (P < 0:0001). In general, the consistencycoefficient increased with particle concentration and

decreased with temperature as shown in Fig. 5. For all

experimental temperatures, as the particle concentration

was increased from 0% to 15%, the consistency coeffi-

cient increased significantly (a 0:05). However, thesuspension at 85 C, which possessed highest carrier

fluid viscosity, exhibited a much higher rate of increasein consistency coefficient than what was observed for the

suspensions at higher temperatures. As the particle

concentration level increased from 15% to 20% and

25%, although slight increase in consistency coefficient

was observed for suspensions at 110 and 135 C, no

significant change in the consistency coefficient was

observed at 95% confidence level for all experimental

temperatures. As the particle concentration level in-

creased from 25% to 30%, the consistency coefficient

increased significantly (a 0:05) for suspensions at85 C and 110 C whereas no significant change in the

consistency coefficient was observed at 135 C. For all

particle concentrations, suspensions at 85 C showedsignificantly higher consistency coefficient than those at

higher temperatures. When comparing the suspensions

at 110 C and 135 C at the same level of particle con-

centration, however, the difference between the suspen-

Fig. 5. Effects of particle concentration and temperature on the con-

sistency coefficient of coarse suspension.

Table 6

Power law parameters for CMC solution with and without particulates

Temperature (C) Particle concentration (%v/v) Consistency coefficient, m (Pasn) Flow behavior index, n Shear rate, _c (1/s)

85 0 5.1 0.47a 0.510 017a 159374

15 26 4.1b 0.30 0.016b 44.6125

20 23 4.0b 0.35 0.038b 35.7115

25 26 1.7b

0.32 0.023c

69.124830 39 8.7c 0.25 0.050d 33.0118

110 0 1.6 0.21d 0.58 0.020e 115372

15 2.1 0.62e 0.58 0.082e 73.4205

20 1.8 0.09e 0.62 0.008e 114190

25 3.0 0.35e 0.56 0.019f 71.8126

30 7.9 0.88f 0.36 0.017g 151247

135 0 0.35 0.085g 0.75 0.047h 125209

15 1.4 0.11e;h 0.55 0.017e;i 108266

20 2.0 0.71e;h 0.51 0.081e;i 99.1263

25 1.5 0.29h 0.59 0.056f;i 82.8158

30 1.5 0.10h 0.56 0.014i 155229

* Values having same superscripts (ai) are not significantly different at a 0:05.







7/12

sions at these temperatures was not as discernable. Al-

though the suspensions at 110 C generally had higher

consistency coefficient than those at 135 C, as can be

seen at 25% and 30% particle concentration levels

(P < 0:01), the difference was insignificant at the particleconcentration levels of 15% and 20%.

The flow behavior indices indicate shear-thinning

behavior for all suspensions (n < 1). The effects of par-ticle concentration and temperature on flow behavior

index of coarse suspensions were in opposite trends of

the effects on the consistency coefficients. That is, in

general, the flow behavior index decreased with an in-

crease in particle concentration and increased with

temperature as shown in Fig. 6. As for the effect of

temperature, even though the difference between the

flow behavior indices of the suspensions at 110 and 135

C was not evident at the particle concentration levels of

2025%, the difference at 30% particle concentration

level was very noticeable. As the particle concentration

was increased from 0% to 15% the flow behavior indexdecreased significantly (a 0:05) for suspensions at 85C and 135 C. At 110 C, however, the flow behavior

index did not exhibit a significant difference as the par-

ticle concentration increased from 0% to 15%. When the

particle concentration was increased from 15% to 20%,

no significant changes in the flow behavior index was

observed at all temperature tested. As the particle con-

centration was increased from 20% to 25% and 30%, an

evidently decreasing trend (a 0:05) could be observedfor suspensions at 85 C and 110 C. The suspensions at

135 C, however, did not exhibit any significant changes

as the particle concentration increased.

It should be noted that for mixtures containing high

concentration of coarse particulates (i.e., 1530%), as

the viscosity of the carrier fluid increased, the influence

of particle concentration was more pronounced. Also, as

the particle concentration increased, the influence of the

carrier fluid viscosity was more pronounced. This indi-

cates a significant interaction (P < 0:005) between par-

ticle concentration and temperature (which reflects the

effect of carrier fluid viscosity) on the power law

parameters of coarse suspensions. Also, the effect of the

interaction between these variables was more observable

on the consistency coefficient than on the flow behavior

index.

The effect of temperature on the consistency coeffi-

cient of CMC solution containing no particulates (m)

can described satisfactorily by an Arrhenius-type equa-

tion

m AeEa=RT R2 0:986 20

where the pre-exponential constant (A) and the activa-

tion energy are 0.0000014 mPa sn and 66 kJ/(gmol),

respectively.

For the flow behavior index (n) of the CMC solution,

the effect of temperature, can be described well by an

exponential equation,

n 0:2138 exp0:0092T R2 0:996 21

Exponential function have been used successfully to

express the effect of temperature on CMC solution in the

temperature range below 70 C (Bhamidipati & Singh,

1990).

3.2. Mathematical expressions for apparent viscosity of

coarse suspensions

Variations of Einstein, Mooney, FrankelAcrivos,

and TingLuebber expressions, as well as the expression

presented by Thomas (1965) were tested for the good-

ness of fit to the experimental data. The results aresummarized in Table 7. It was found that most expres-

sions tested adequately represented the apparent vis-

cosity for the entire range of particle fraction covered in

this study. The original Einstein equation (Eq. (8)) ex-

plained the suspensions apparent viscosity quite well for

the entire range of/. As a matter of fact, the Einsteins

equation was the best in explaining the suspension

apparent viscosity at the higher temperature range (110

and 135 C), i.e., for the lower levels of fluid apparent

viscosity. Expanding the Einsteins equation to second-

order (Eq. (9)) improved the prediction of suspension

apparent viscosity at the lower temperature level (85

C), i.e., higher fluid apparent viscosity. However, it didnot appear to improve the prediction for the entire range

of particle fraction. When the Einsteins equation was

expanded to the third-order, with the second order

coefficient of 10.05 and the best-fit third-order coefficient

(a 20:84), i.e.,

lr 1 2:5/ 10:05/2 20:84/3 22

the prediction for the entire range of lr improved

markedly when compared to the prediction by the ori-

ginal Einsteins equation. Thomas (1965) noted that

the fourth term in the expanded Einsteins equationFig. 6. Effects of particle concentration and temperature on the flow

behavior index of coarse suspension.



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Table 7

Summary of goodness of fit test values (v2) for apparent viscosity expressions of coarse suspensions

Equation Variation v2 value

0:15 < / < 0:30 (df 37, c 24.08) 0:15 < / < 0:20 (df17, c 8.68) 0:25

viscosimetr a-11

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