© 2020 The Korean Society of Rheology and Springer 137

Korea-Australia Rheology Journal, 32(2), 137-144 (May 2020)DOI: 10.1007/s13367-020-0012-7

www.springer.com/13367

pISSN 1226-119X eISSN 2093-7660

Comparative numerical and experimental investigation of process viscometry for

flows in an agitator with a flat blade turbine impeller

Hae Jin Jo1, Young Ju Kim

2,* and Wook Ryol Hwang1,*

1School of Mechanical Engineering, Research Center for Aircraft Parts Technology (ReCAPT), Gyeongsang National University, Jinju 52828, Republic of Korea

2Resources Engineering Plant Research Department, Korea Institute of Geoscience and Mineral Resources, Pohang 37559, Republic of Korea

(Received September 21, 2019; final revision received February 8, 2020; accepted February 10, 2020)

This paper presents a method for measuring the viscosity of generalized Newtonian fluid directly in flowsgenerated by flat-blade turbine impellers, which are commonly used for moderate mixing and dispersion.A flat-blade turbine with four blades is defined as a model system and analyzed through numerical sim-ulations with experimental verification. Carbopol 940 solution, a high viscosity non-Newtonian fluid witha yield stress, and a bentonite based drilling mud solution were selected as test fluids. Numerical simulationtechniques for flow in agitators with a yield stress was established using the rotating coordinate system andflow solutions were validated with experiments by comparing the torque on the impeller shaft. TheMetzner-Otto constant and the energy dissipation rate constant were predicted by numerical simulationsusing the Metzner-Otto correlation and validated via experiments. The effective viscosity that reproducestotal energy dissipation rate identical to that of a Newtonian fluid was obtained from both numerical andexperimental methods at different impeller speeds, from which the material viscosity curve was establishedas a function of the shear rate. The accuracy of viscosity prediction was compared with a rheological mea-surement and the average relative error was below 12% and 7% in the experiment and simulation, respec-tively. This method has the advantage of being able to measure the in-situ viscosity, where a drilling mudneeds to transport more and heavier cuttings and careful preparation of the mud is key issue to a successfuldrilling process.

Keywords: agitators, process viscometry, torque measurement, numerical simulation, Metzner-Otto correla-

tion

1. Introduction

Mechanical stirring vessels in which physical or chem-

ical changes occur through agitation are used in process

industries such as the chemical processing, biochemical

processing and environmental improvement industries

(Paul et al., 2003). Among others, flat-blade turbines are

common choices in the process industry for rheologically

generalized Newtonian fluid due to its simplicity in man-

ufacturing paints or polymer solution, in spite of their rela-

tively weak mixing performance: for examples, the formation

of a nearly unyielding region far from the rotating impeller

for highly viscous fluids, like yield stress fluids such as

gels, slurries, and drilling muds (Thakur et al., 2004).

Process characterization and monitoring are necessary

for controlling the mixing process in an agitator system

and the most important property that needs to be moni-

tored for systems with rheologically generalized Newto-

nian fluids is the viscosity especially for in a laminar

regime. This paper present combined numerical and exper-

imental methods for measuring the process viscosity of

non-Newtonian fluids directly in flows generated by a

flat-blade turbine impeller. Using the torque data at dif-

ferent impeller speeds from both numerical simulations and

experiments, the Metzner-Otto correlation was employed to

measure the in-situ viscosity of rheologically generalized

Newtonian fluids and the accuracy of numerical simula-

tion was verified through experimental results.

The Metzner-Otto method is a heuristic method for

quantifying total energy dissipation rate in non-Newtonian

flows in an agitator, and is based on effective shear rate

concept (Metzner and Otto, 1957). In this method a con-

stant factor was introduced to correlated the mean (effec-

tive) shear rate with the impeller speed such that

(1)

where , KS and N are the effective shear rate, the

Metzner-Otto constant and the impeller speed (in revolu-

tion per second). In their method, the Metzner-Otto con-

stant KS was found to depend mostly on the geometry of

the agitator system and independent of rheological behav-

ior of fluids. That is, there is a representative shear rate

,eff sK N �

·eff

*Corresponding authors; E-mail: W.R. Hwang (wrhwang@gnu.ac.kr)and Y.J. Kim (kyjp7272@kigam.re.kr)

Hae Jin Jo, Young Ju Kim and Wook Ryol Hwang

138 Korea-Australia Rheology J., 32(2), 2020

that characterizes the flow system despite the complexity

of flow in agitators owing to the complex geometry of

components such as the impeller, vessel, and baffle. In

order to understand how the Metzner-Otto method is

employed, flow characterization must be introduced first

that is commonly used in mixing community. The power

draw in agitators is often described by a dimensionless

power number Np that is the ratio of the total energy dis-

sipation rate, P inside agitator to the characteristic turbu-

lent energy dissipation rate: with and D

being the fluid density and the impeller diameter, respec-

tively. The Reynolds number is traditionally defined in a

mixing community as Re = ND2/ with being the vis-

cosity. In a laminar flow , Np is inversely pro-

portional to :

(2)

where Kp is the energy dissipation rate constant, which is

again only a function of agitator geometries. The laminar

regime corresponds to for many impellers, but

laminar flow and it may persist until or greater

for stirrers with very small wall-clearance such as the

anchor and helical-ribbon mixer. Details of discussion on

flow regime in agitators can be found Wichterle and Wein

(1975) and Hoogendorn and den Hartog (1967). Metzner

and Otto (1957) showed that the reciprocal relationship

between the power number and the Reynolds number Eq.

(2) can be even for non-Newtonian fluids established in

the same was as a Newtonian fluid with same constant Kp,

with the corrected (effective) Reynolds number. The effec-

tive Reynolds number Reeff is defined using the effective

viscosity eff at the effective shear rate . In summary,

the power-number and Reynolds number relationship Eq.

(2) can be expressed as follows:

(3)

A large amount of studies on the Metzner-Otto method

in literatures are the power consumption characterization

using various types of impellers for rheologically gener-

alized Newtonian fluid (Carreau et al., 1993; Edwards et

al., 1976; Furukawa et al., 2012; Nagata et al., 1971; Tan-

guy et al., 1996; Thakur et al., 2004; Woziwodzki et al.,

2010). A complete review on the Metzner-Otto method

and its applications was presented by Doraiswamy et al.

(1994). However, its application to viscosity measurement

is relatively rare. Brito De La Fuente et al. (1998) and

Eriksson et al. (2002) experimentally measured the pro-

cess viscosity of generalized Newtonian fluids with high

accuracy using helical ribbon impellers. Recently, Jo et al.

(2017) presented a method that experimentally measures

the viscosity of generalized Newtonian fluid in flows gen-

erated by anchor agitators.

In the present work, we investigated the viscosity mon-

itoring of a flat blade turbine impeller with numerical

methods and its accuracy was validated with comparative

experimental study. Eq. (3) can be rearranged to find the

effective viscosity at the effective shear rate as a function

of the impeller speed and torque:

(4)

where is the mean energy dissipation rate. Therefore,

the effective viscosity can be predicted as only a function

of the torque data T at different impeller speeds N in com-

plex flows such as in an agitator system, once the two

flow constants Kp and Ks are known. For a given set of

values of the torque and the impeller speed, one can deter-

mine Np and the corresponding effective Reynolds number

Reeff, from which the effective viscosity eff can be deter-

mined directly Eq. (3). All these procedures are readily

available numerically and experimentally. As indicated in

Eq. (4), the effective viscosity is the averaged viscosity in

terms of the energy dissipation rate with Kp as a geometric

factor: .

The paper is organized as follows. First, a numerical

method is introduced to solve the flow field in an agitator

with a flat-blade turbine, which employ a rotating coor-

dinate system such that the impeller is considered fixed in

the rotating coordinate and the vessel wall without baffles

is rotating in the opposite direction. Then governing equa-

tions, boundary conditions and solution methods for flow

simulations are presented. Then experimental details were

presented for the purpose of validation. A flat-blade tur-

bine impeller with four blades was chosen as a model flow

system, and Carbopol 940 solutions and bentonite-based

drilling mud solutions were used as test fluid materials. In

petroleum engineering, drilling mud is a viscous fluid

mixture used to perform rock cutting on surfaces and to

lubricate and cool drill bits in oil and gas drilling opera-

tions. In the drilling industry, viscous mud can carry a

greater amount of cutting, and careful control of the mud's

properties is a key issue for a successful drilling process

(Dyke, 1998). Finally, the effective viscosity was esti-

mated by measuring the torque for various impeller speeds

in both numerical and experimental methods, and the

accuracy of the viscosity measurement as a function of the

shear rate was compared with a rheological measurement.

This method has the advantage of being able to measure

viscosity in the industrial field, which can be used as a

useful method for in-situ identification with non-Newto-

nian fluids in agitator flows with various geometries of

industrial processes.

2. Numerical Methods

Np P N3D

5=

Re 10

,

p pN K Re

Re 10

Re 100

·eff

2, and .p p eff eff eff eff effN K Re Re ND �

2 2 3

1 2,eff

P app P

NT

K K N D

�

2

p eff appK �

Comparative numerical and experimental investigation of process viscometry for flows in an agitator ...

Korea-Australia Rheology J., 32(2), 2020 139

In this work, highly viscous fluid mixing with a flat

blade turbine in a flat-bottomed vessel was selected as a

model problem, as depicted in Fig. 1. We are particularly

interested in yield stress fluids such as Carbopol solutions

or drilling mud. In this case, the flow is considered lam-

inar and vertical baffles must be avoided to circumvent the

formation of Moffat eddies between the baffle and the

wall. In the absence of the baffle, the flow can be viewed

steady independent of time, once the coordinate system is

chosen to rotate along with the impeller shaft. Of course,

the flow is unsteady, if observed from the fixed reference

coordinate. If vertical baffles are not installed, the free sur-

face between liquid and air may form to convex shape,

which is called the free surface swirling. The net force of

gravity and centrifugal acceleration is responsible for the

swirling near the impeller shaft. According to Rieger et al.

(1979), the Froude number is a key factor to determine the

amount of the surface suppression and it is defined as

Fr = N2D/g with g being the gravity, which is the ratio of

the centrifugal force to the gravity force. Free surface

swirling can be neglected for Fr < O(1). In the present

study, the Froude number was found 0.332 for the max-

imum impeller speed 600 rpm with an impeller diameter

D = 65 mm. Therefore, the free surface swirling can be

neglected in this problem and the free surface can be con-

sidered flat with negligible stress from the air side.

To solve the velocity field inside the vessel, a moving

coordinate was introduced that rotates at the same angular

velocity of the impeller in the present work. Denoting the

relative velocity field ur with respect to the moving coor-

dinate, the total (or absolute) velocity viewed from a fixed

reference frame can be expressed as follows:

(5)

where = k, k and r are the angular velocity of the

rotating frame, a unit vector along the axis and the local

position vector inside the rotating frame, respectively. The

second term in Eq. (5) is the velocity contribution from

the rotating frame.

Flow problems can be described easily with respect to

the rotating coordinate system, considering the relative

velocity ur as a primary variable. Taking the time deriv-

atives to Eq. (5) yields an expression of the total acceler-

ation consisting of the Coriolis and centrifugal accelerations

for a constant angular velocity (See for example Hous-

ner and Hudson, 1959). The same procedure can be done

with the Navier-Stokes equation and, rewriting the equa-

tion in terms of the relative velocity ur, the momentum con-

servation equations in the rotating coordinate can be

written as

(6)

In Eq. (6), the second term on the left-hand side rep-

resents the Coriolis force including the primitive unknown

variable ur, and the last term on the right-hand side is the

centrifugal force from the rotating frame. In addition, the

continuity equation in terms of the relative velocity ur

reduces to its original form, since the divergence of the

rigid-body motion vanishes.

(7)

As for the boundary condition, the rotational speed is

assigned to the vessel wall in the opposite direction and a

no-slip boundary condition is set for the flat blade turbine

with shaft surfaces. As mentioned earlier, a free slip

boundary condition can be introduced on the upper flat

free surface: i.e., and with , n and tdenoting the stress, the normal vector and tangential vec-

tor to the free surface, respectively. The former indicates

no-penetration condition and the latter one is for the free

slip condition.

A commercial software, COMSOL Multiphysics 5.3,

was employed to solve the flow problem by modification

of the Coriolis terms in Eq. (6) in a weak form. The com-

putational domain is discretized into tetrahedral elements

with quadratic velocity and linear pressure interpolation.

To check the mesh refinement, we tested four different

meshes with degrees of freedom 8,410 (denoted by M1),

146,303 (M2), 635,838 (M3) and 1,252,754 (M4). Con-

sidering flow solutions from mesh M4 as the reference,

the relative error in the total energy dissipation rate, or the

power draw, for each mesh is shown in Fig. 2 and the error

,

tot r u u ω r

2 .r r r r

p u u ω u u g ω ω r

0r

u

ur n 0= t n 0=

Fig. 1. Geometry of the flat blade turbine in a flat-bottomed ves-

sel.

Hae Jin Jo, Young Ju Kim and Wook Ryol Hwang

140 Korea-Australia Rheology J., 32(2), 2020

shows uniform convergence, as the number of unknowns

increases. Power draw was computed by integrating the

local energy dissipation rate over the fluid volume.

A Newtonian fluid with the viscosity of 9.75 Pa·s and

density 975 kg/m3 was employed for the mesh refinement

test. One can observe from Fig. 2 that the accuracy of flow

simulation can be guaranteed up to three significant digits

with the mesh M3. Although not presented here, the accu-

racy of the rotating frame approach has been verified in

comparison with simulation with a rotating impeller.

In order to solve the flow problem with a viscoplastic

fluid, we introduced a simple regularization method after

Papanastasiou (1987) and selected the regularized Her-

schel-Bulkley model to represent the viscosity behavior

for Carbopol 940 solutions and drilling mud. The model

can be written as

(8)

The stiffness parameter m was introduced in Eq. (8) to

prevent abrupt changes in the shear stress, with which a

priori estimation of the yield boundary can be avoided

(Alexandrou et al., 2003; Mitsoulis, 2007).

The mesh M3 was employed in remaining simulations

in this study and is shown in Fig. 3a. As an example result

of flow solutions, the yield volume in the flat blade tur-

bine system at the impeller speed 30 rpm was presented in

Fig. 3b, where the fluid was Carbopol 940 2 wt.% aque-

ous solutions with the yield stress of 105 Pa. (See the

experimental section for the rheological characterization

and parameters for the regularized Herschel-Bulkley

model for this solution as well as the geometry informa-

tion for the model agitator system.) In this case, the yield

region was defined by the relative velocity magnitude

larger than 1%. The relative velocity is defined as the

velocity normalized by the impeller tip velocity (ND).

3. Experimental Methods

For comparative experiments, the flat blade turbine in a

flat-bottomed vessel was built, where the vessel diameter

T was 86 mm, the liquid height is H was 90 mm, and the

impeller diameter D was 65 mm with the clearance from

vessel bottom to impeller 35.5 mm. The geometry and

dimensions of the flat blade turbine in the experiment and

simulation are shown in Fig. 1. The impeller was driven

by a high-precision stirrer (IS600, Trilab Co., Japan) with

controllable speed from 3 rpm to 600 rpm and an embed-

·2

1 .m n

ye K

�

� �

Fig. 2. (Color online) Relative error in the mesh refinement test

performed for predicting the total energy dissipation rate.

Fig. 3. (Color online) (a) Geometry and the finite-element mesh

of the flat-blade turbine system studied in this work; (b) Yielded

region in the flat-blade turbine system in 30 rpm when 2 wt.%

Carbopol was used as test fluid.

Comparative numerical and experimental investigation of process viscometry for flows in an agitator ...

Korea-Australia Rheology J., 32(2), 2020 141

ded torque sensor. Torques up to 1.4 N∙m could be mea-

sured with about 1 mN∙m resolution. Relatively large

impeller diameter with D/T = 0.756 was employed, in

order to obtain sufficiently large torque, larger than the

minimum resolution of the torque sensor, which enables

comparison between numerical and experimental results

for a wide range of the impeller speed for a given fluid.

In the experiment, four different working fluids were

employed. Highly viscous silicone oil (KF-96-10000cs,

Shinetsu Co., Japan) with the density of 975 kg/m3 and

a viscosity of 9.75 Pa∙s was selected as a reference New-

tonian fluid. Again, a highly viscous fluid was preferred to

minimize errors in torque measurement. The relationship

between power number and Reynolds number Eq. (2) for

a Newtonian fluid is determined by the reference Newto-

nian fluid, from which the constant Kp were determined

Eq. (2). For the non-Newtonian fluids, three different flu-

ids were prepared: 0.5 wt.% and 2 wt.% concentrations of

Carbopol aqueous solutions (Carbopol 940, Lubrizol Co.,

USA) and a bentonite-based drilling mud solution (Ben-

tonite, Duksan Co., Korea). Among three, 0.5 wt.% Car-

bopol solution was used as a reference non-Newtonian

fluid to determine the Metzner-Otto constant KS experi-

mentally, by avoiding effects of elasticity with low con-

centration. A small amount of NaOH solution was added to

the Carbopol 940 solutions in the experiments to increase

the viscosity. Drilling mud was a 7 wt.% bentonite aque-

ous solution with 0.5 wt.% xanthan gum and xanthan gum

was added to increase the viscosity. Xanthan gum solution

of 0.5 wt.% in the present study may show elasticity in

moderate agitation conditions. However the quantification

method is based on the balance of energy dissipation rate

which concern the external power is dissipated as viscous

dissipation within a system. Since the energy inside is dis-

sipated by viscous dissipation, elastic dissipation is not

considered in this case. The flow quantification can be

accomplished accurately by two flow numbers, only for

liquids with negligible elastic behaviors. 2 wt.% Carbopol

940 solution and the bentonite-based drilling mud solution

were used as test fluids for predicting the viscosity from

the predetermined flow constants Kp and KS in both

numerical and experimental analyses.

Figure 4 shows rheological behaviors of three different

non-Newtonian fluids as a function of the shear rate,

which were measured using a cone-and-plate geometry

(50 mm, MCR301, Anton Paar, Austria) at 20oC and at a

shear rate in the range 0.008-100 [1/s]. The shear stress

data in Fig. 4 were fitted with the regularized Herschel-

Bulkley model to be incorporated with numerical simula-

tions. Fitted parameters of each working fluid, K = 27

Pa·sn, n = 0.37, y = 27 Pa, m = 5000 for 0.5 wt.% Car-

bopol 940, K = 30 Pa·sn, n = 0.35, y= 105 Pa, m = 5000

for 2 wt.% Carbopol 940, K = 10 Pa·sn, n = 0.26, y = 7.61 Pa,

m = 100 for bentonite based drilling mud and fitted equa-

tions were plotted in Fig. 4 as well.

4. Results and Discussion

4.1 Determination for the flow constants using

numerical and experimental conditionsFigure 5a shows the torque T at the impeller shaft as a

function of the impeller speed N using silicone oil as a

working fluid from both numerical simulations and exper-

iments. The impeller speed was from 30 rpm to 300 rpm,

in which the torque varies from 0.014 to 0.14 N∙m that is

larger than the resolution of the torque sensor. In the case

of the simulation, the torque was calculated by evaluating

the integral of the cross product of the position vector r

and traction force t over the impeller surface including the

shaft: i.e, with and n being the out-

ward normal vector on the surface. As presented in Fig.

5a, the torque results from both numerical simulations and

experiments shows good agreement with the average error

of 1.46%, which proves the validity and accuracy of

numerical simulations. Plotted in Fig. 5b are the relation-

ships between the power number MP and the Reynolds

number for a Newtonian fluid (silicone oil) calculated

through experiments and numerical simulations. The

power number NP was computed by Np = P/N3D5 and the

total energy dissipation rate P with flat blade turbine sys-

tem was calculated by P = 2NT. As shown in Fig. 5b, the

power number NP scales with Re1 and, taking the average

of the product NpRe1 at each data set, the energy dissi-

pation rate constant KP is found to be 64.78 in experiments

and 63.92 from numerical results. Now we can define the

effective Reynolds number of the non-Newtonian fluid by

using the energy dissipation rate constant KP.

After determining the energy dissipation rate constant

KP from a Newtonian fluid, the Metzner-Otto constant KS

T r td= t n=

Fig. 4. (Color online) Stress as a function of the shear rate for the

three different non-Newtonian fluids (0.5 wt.% and 2 wt.% Car-

bopol 940 solutions and 7 wt.% bentonite based drilling mud

solution).

Hae Jin Jo, Young Ju Kim and Wook Ryol Hwang

142 Korea-Australia Rheology J., 32(2), 2020

is estimated using a reference non-Newtonian fluid, 0.5

wt.% Carbopol 940 solution. Note that the energy dissi-

pation rate constant KP from a Newtonian fluid is still

valid for non-Newtonian fluids. We calculated the power

P and the power number NP analogous to the Newtonian

fluid case, for ten different impeller speeds from 30 rpm

to 300 rpm both in experiments and simulations. The

effective Reynolds number Reeff was then identified by the

corresponding Reynolds number that is determined by the

power number characteristics of a Newtonian fluid

Reeff = KpNP1 (Fig. 5b). The effective viscosity is deter-

mined as eff = ND2/Reeff , from which the effective shear

rate is identified by the corresponding shear rate of the

effective viscosity using the fitted viscosity curve. The

Metzner-Otto constant KS is then determined for each

impeller speed N by Eq. (1): .

Plotted in Fig. 6a is the Metzner-Otto constant KS for the

ten different impeller speeds with the reference non-New-

tonian fluid from both experiments and numerical simu-

lations. The Metzner-Otto constant KS ranges from 12.84

to 16.21 in experiment and 11.1 to 12.44 in numerical

simulation. The discrepancy between numerical and

experimental values of KS in Fig. 6a was mainly caused by

the generalized Newtonian fluid model in the present

study. Effects of fluid elasticity and extensional flow con-

tribution cannot be treated correctly with the present vis-

cosity model. Variability of KS seems to be large, about

15.23% (in experiment) and 6.87% (in simulation) varia-

tion from the average value of each method. However, the

variability of KS does not significantly affect the measured

viscosity because of the viscosity dependence of shear rate

is best shown in a log-log plot. For example, the maxi-

mum 10% error in the effective shear rate yields less than

10% error in the viscosity estimation (for a power-law

fluid with the power-law index less than one) and this

amount of error might be considered negligible in viscos-

ity prediction of a non-Newtonian fluid in practice, as the

viscosity usually changes over hundreds or even thou-

sands times as a function of the shear rate. Therefore, we

set the Metzner-Otto constant KS to 15.15 in experiment

KS ·eff1–

N=

Fig. 5. (Color online) Flow characteristics of the flat blade tur-

bine system with a Newtonian fluid (10,000 cs silicone oil): (a)

Comparison of the torque measured as a function of the impeller

speed between the experiment and the simulation; (b) The rela-

tionship between the power number and the Reynolds number.

Fig. 6. (Color online) Flow characterization of the flat blade tur-

bine for a reference non-Newtonian fluid (0.5 wt.% Carbopol

940 solution): (a) Values of the Metzner-Otto constant KS deter-

mined in this study; (ii) The relationship between the power

number and the effective Reynolds number

Comparative numerical and experimental investigation of process viscometry for flows in an agitator ...

Korea-Australia Rheology J., 32(2), 2020 143

and 11.59 in numerical simulation.

Figure 6b shows the relationship between the power

number and effective Reynolds number for both a New-

tonian fluid (silicone oil 10,000 cs) and the reference non-

Newtonian fluid (0.5 wt.% Carbopol 940 solution). The

data for Carbopol solution is obtained by using the two

different Metzner-Otto constants 15.15 (denoted by ‘exp’)

and 11.59 (denoted by ‘num’) in Fig. 6b. In Fig. 6b, an

almost single master curve can be observed even with the

different Metzner-Otto constants, indicating a reciprocal

relationship that is analogous to the Newtonian reference

fluid.

4.2 Viscosity measurementOnce both flow constants KP and KS are fixed, the pro-

cess viscosity measurement can be performed with non-

Newtonian fluids. We tested two different fluids: Carbopol

940 solutions 2 wt.% and bentonite-based drilling mud

solution. Figure 7 shows torque data for various impeller

speeds of flat blade turbine, from both numerical simula-

tions and experiments, including the data from Newtonian

and non-Newtonian reference fluids.

From the data on torques and impeller speeds for each

fluid material, one can calculate the power and power

number NP for each impeller speed, as earlier discussed.

The effective Reynolds number is determined by

Reeff = 64.78 (in experiment), Reeff = 63.92 (in

simulation) and the effective viscosity is then eff = ND2/

Reeff. The corresponding shear rate is the effective shear

rate that was determined by Eq. (1) using predeter-

mined KS = 15.15 (in experiment) and KS = 11.59 (in sim-

ulation).

Plotted in Figs. 8a and 8b is the effective viscosity as a

function of the effective shear rate for Carbopol 940 solu-

tion 2 wt.% and bentonite-based drilling mud solution,

respectively. To assess the accuracy of viscosity measure-

ments, a viscosity curve from a steady shear test using a

rheometer is plotted together. As expected, the process

viscosity measurement with torque on the agitator is anal-

ogous to the data from the rheometer. The mean relative

errors of process viscometry and rheological measurement

are 8.45% (in experiment) and 1.52% (in simulation) for

Carbopol 940 2 wt.%; and 12.36% (in experiment) and

7.18% (in simulation) drilling mud.

5. Conclusions

In this study, we present a viscosity measurement tech-

nique by numerical simulations and validated through

experiments for generalized Newtonian fluid directly in

flows with flat blade turbine impeller that are commonly

used for moderate mixing and dispersion. Flow simulation

for agitated vessel in the absence of the baffle was per-

formed by a relatively simple steady state simulation by

introducing the rotating coordinate along with additional

Coriolis and centrifugal accelerations. We perform the

Np

1–Np

1–

·eff

Fig. 7. (Color online) Torque measurements for different impeller

speeds in the flat-blade turbine system for all the fluids consid-

ered in this study.

Fig. 8. (Color online) Process viscometry for the flat blade tur-

bine: (a) Viscosity measurement with 2 wt.% the Carbopol solu-

tions; (b) Viscosity measurement with the drilling mud solution

(7 wt.% bentonite and 0.5 wt.% Xanthan gum). Viscosity mea-

sured with the rheometer is also plotted for comparison.

Hae Jin Jo, Young Ju Kim and Wook Ryol Hwang

144 Korea-Australia Rheology J., 32(2), 2020

process viscometry in two different methods: purely

numerical and purely experimental methods. We deter-

mined the reciprocal relationship with the energy dissipa-

tion rate constant KP between the power number and the

Reynolds number by measuring the torque for each impel-

ler rotation speed for a 10,000 cs silicone oil (Newtonian

reference fluid) in both numerical and experimental meth-

ods. Then a 0.5 wt.% Carbopol 940 solution as the ref-

erence non-Newtonian fluid is introduced to determine the

Metzner-Otto constant KS by matching the power number

to the Reynolds number. After the two flow number KP

and KS were fixed, two non-Newtonian fluid examples are

tested for the process viscometry in both numerical sim-

ulations and experiments. Results from numerical simula-

tions for the process viscometry was compared, along

with experimental process viscometry, with a rheological

measurement and the average relative error was below

12% and 7% in the experiment and numerical simulations,

respectively. The proposed method is applicable to in-situ/

on-line viscosity monitoring of non-Newtonian fluids in

agitator flows of a wide variety of industrial processes.

The present work is the first step in developing the on-

line viscosity monitoring system in the drilling mixing pit.

Using the two flow numbers for a specific mixing pit, data

set of torque and impeller rotation speed can be converted

into the viscosity and the shear rate. We are now devel-

oping a wireless torque monitoring system with strain

gauges (half bridge) attached on the agitator shaft to per-

form the in-situ/on-line viscosity monitoring with the mud

mixing pit.

Acknowledgement

This work is supported by Korea Agency for Infrastruc-

ture Technology Advancement grant funded by Ministry

of Land, Infrastructure and Transport (20IFIP-B133614-

04, Investigation and assessement of mud flow prediction

and simulation in a drill hole).

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