Process Viscometry in Flows of Non-Newtonian Fluids Using An

Korea-Australia Rheology Journal, 29(4), 317-323 (November 2017) www.springer.com/13367 DOI: 10.1007/s13367-017-0031-1

Process viscometry in flows of non-Newtonian fluids using an anchor agitator Hae Jin Jo1, Hye Kyeong Jang1, Young Ju Kim2,† and Wook Ryol Hwang1,* 1School of Mechanical Engineering, Gyeongsang National University, Jinju 52828, Republic of Korea 2Exploration System Research Department, Korea Institute of Geoscience and Mineral Resources, Pohang 37559, Republic of Korea (Received Octrober 18, 2017; final revision received November 5, 2017; accepted November 13, 2017)

In this work, we present a viscosity measurement technique for inelastic non-Newtonian fluids directly in flows of anchor agitators that are commonly used in highly viscous fluid mixing particularly with yield stress. A two-blade anchor impeller is chosen as a model flow system and Carbopol 940 solutions and Xan- than gum solutions with various concentrations are investigated as test materials. Following the Metzner- Otto correlation, the effective shear rate constant and the energy dissipation rate constant have been estimated experimentally by establishing (i) the relationship between the power number and the Reynolds number using a reference Newtonian fluid and (ii) the proportionality between the effective shear rate and the impeller speed with a reference non-Newtonian fluid. The effective viscosity that reproduces the same amount of the energy dissipation rate, corresponding to that of Newtonian fluid, has been obtained by measuring torques for various impeller speeds and the accuracy in the viscosity prediction as a function of the shear rate has been compared with the rheological measurement. We report that the process viscometry with the anchor impeller yields viscosity estimation within the relative error of 20% with highly shear-thinning fluids. Keywords: agitators, process viscometry, torque measurement, Metzner-Otto correlation, non-Newtonian fluids

1. Introduction measurement on the shaft. The Metzner-Otto method is a heuristic approach in quantifying the total energy dissipa- Mechanical agitation involving physical or chemical tion rate in non-Newtonian flows in an agitator by intro- changes in a stirred vessel is commonly encountered in a ducing the effective shear rate concept (Metzner and Otto, wide variety of process industries such as food, plastics, 1957). It is popular in the mixing community and indeed rubbers, mining and waste water treatment (Paul et al., is considered as a ‘single major practical technique for 2003). Among others, fluid mixing in a laminar regime is incorporating non-Newtonian effect’ in agitator analyses. considered as a difficult operation particularly with highly It has been mainly used in the field of mixing for non- viscous yield stress fluids such as gels, slurries and drill- Newtonian fluids relating to power requirement, heat ing muds, due to the formation of nearly unyielding region transfer, blend time and viscosity measurement (Doraiswamy away from the rotating impeller. In this particular case, the et al., 1994). anchor-type impeller with close clearance between impel- As was indicated by Doraiswamy et al. (1994), the ler blades and vessel walls is frequently employed in spite Metzner-Otto method has been frequently used in quan- of its poor mixing performance, as it minimizes unyield- tifying the energy dissipation rate as a function of the ing region over the entire vessel (Pedrosa and Nunhez, Reynolds number and a representative (effective) shear 2000). Process characterization and monitoring are neces- rate in non-Newtonian fluid flows inside an agitator. sary to control the mixing process in agitator system espe- Nagata et al. (1971) studied the power consumption of cially with rheologically complex fluids in a laminar shear-thinning liquids with various impellers using the regime and the process viscosity measurement would be a Metzner-Otto correlation. Similar works were reported in right choice in this purpose. literatures: The power consumption with thixotropic liq- In this work, we present an experimental method to uids (Edwards et al., 1976); the power requirement pre- measure the process viscosity for a flow in an anchor agi- diction from experimental data with a helical ribbon tator with non-Newtonian fluids. The present work fol- impeller (Shamlou and Edwards, 1985); analytical mod- lows the Metzner-Otto correlation in estimating the eling the energy dissipation rate with a helical ribbon viscosity of rheologically complex fluids using the torque impeller and effects of fluid viscoelasticity on the power characterization (Carreau et al., 1993); the power consumption with the anchor impeller by 3D numerical sim- *Corresponding author; E-mail: [email protected] †Co-corresponding author; E-mail: [email protected] ulation and experimental validation (Tanguy et al., 1996).

3 5 Literatures on the application of the Metzner-Otto method Np = P/ρN D (2) to the viscosity measurement using the agitator are rela- tively rare compared to the power characterization. Brito with ρ and D being the fluid density and the impeller De La Fuente et al. (1998) measured the process viscosity diameter, respectively. The flow regime can be identified using the helical ribbon impellers experimentally and with the Reynolds number and it is defined as Re = ρND2/ reported highly accuracy viscosity measurements with μ conventionally in the mixing community. In case of a complex fluids. Similar studies on food materials with laminar flow (Re≤10 )of a Newtonian fluid with a viscos- helical ribbon impellers were investigated by Eriksson et ity μ, the torque on the shaft and the power draw scale al. (2002). with μND3 and μN2D3, respectively, and therefore the In this work, we present an experimental process vis- power number Np appears reciprocal to the Reynolds num- cometry based on the Metzner-Otto correlation for flows ber Re with a constant Kp: in anchor agitators with inelastic non-Newtonian fluids. A Np = Kp/Re.(3) two-blade anchor impeller is chosen as a model flow system and Carbopol 940 solutions and Xanthan gum solu- The symbol Kp is the energy dissipation rate constant, tions are investigated as test fluid materials. Following the which is a function of flow geometries only and measures Metzner-Otto correlation, the effective shear rate constant the amount of the rotational drag on the impeller. From 3 and the energy dissipation rate constant have been esti- Eq. (3) the torque on the shaft becomes KpμND /2π inside mated experimentally by establishing (i) the relationship the agitator in a laminar regime. between the power number and the Reynolds number and The Metzner-Otto correlation claims that the reciprocal (ii) the proportionality between the effective shear rate and relationship between the power number and the Reynolds the impeller speed. The effective viscosity has been number (Eq. (3)) can be reproduced even for non-New- obtained by measuring torques for various impeller speeds tonian fluids with the same constant Kp, if the Reynolds and the accuracy in the viscosity measurement as a func- number is properly corrected (Metzner and Otto, 1957). tion of the shear rate has been compared with the rheo- The corrected Reynolds number is the effective Reynolds logical measurement. number Reeff, defined with the effective viscosity μeff; and the effective viscosity μeff is the viscosity at the effective · 2. The Metzner-Otto Method shear rate γeff , which was defined in Eq. (1). That is, 2 · Np = Kp/Reeff, Reeff = ρND /μeff , and μeff = μ()γeff .(4) In this section, we briefly review the Metzner-Otto method for the completeness of the present work. Flow in Once the two flow constants Kp and Ks are known, one an agitator is complex and highly non-uniform even in a can predict the viscosity as a function of the flow rate by laminar regime due to the rotation of impellers of non- measuring the torque in a complex flow like an agitator trivial shapes such as the anchor or helical ribbon impel- system as follows. Firstly, for a given set of the torque and lers particular for highly viscous fluids (Paul et al., 2003) the impeller speed, one can determine the power number and the complexity in the flow field becomes much con- Np as well as the corresponding effective Reynolds num- siderable with the presence of non-Newtonian fluids. ber. Then the effective viscosity is determined directly Metzner and Otto (1957) proposed a constant factor to from the effective Reynolds number and, plotting the define a representative (or effective) shear rate, multiplied effective viscosity as a function of the effective shear rate by the impeller speed N in characterizing the power draw (Eq. (1)), the flow curve of a non-Newtonian fluid is then for a flow in an agitator: obtained. The above procedures can be easily imple- · mented experimentally. γ eff = Ks N .(1)

The effective shear rate constant Ks is hypothesized to 3. Experimental Methods depend on impeller shapes and tank geometries only, inde- pendent of rheological behaviors of fluids. In fact, Ks has In this work, we employed an anchor impeller in a flat turned out to be ‘nearly’ a constant for a given agitator bottomed vessel and a motor-driven stirrer with the torque geometry, primarily with the impeller type and there are a measurement unit as a model viscosity measurement sys- couple of known constants for some frequently used tem. The geometry and dimensions of the anchor agitator impellers in industry (Paul et al., 2003): Ks = 10 (propel- in the experiment are indicated in Fig. 1. The vessel diam- ler), 12 (Rushton), 30 (helical ribbon) and 25 (anchor). eter is T = 86 mm and the liquid height is H =90 mm. The The power draw in agitators is often described by a anchor impeller geometry is defined as follows: the dimensionless power number Np, the ratio of the total anchor diameter D =77.4 mm, the anchor height Hi = 81.4 energy dissipation rate P inside agitator to the character- mm, the width and thickness of the anchor blade W = 11.6 istic energy dissipation rate in a turbulent regime: and Th =7 mm and the clearance between anchor blades

318 Korea-Australia Rheology J., 29(4), 2017 Process viscometry in flows of non-Newtonian fluids using an anchor agitator

Fig. 2. (Color online) The stress as a function of the shear rate for five non-Newtonian fluids (0.5 wt%, 1 wt%, and 2 wt% Car- bopol 940 solutions and 1 wt% and 2 wt% Xanthan gum solu- Fig. 1. The geometry of the anchor impeller in a vessel. tions). and vessel walls C =4.3 mm. The dimensionless geomet- the same time its viscosity is high enough to allow the rical ratios are then D/T = 0.9, W/D = 0.15 and C/D = torque measurement accurately. (A small amount of a 0.056. Impeller rotation is driven by a high precision stir- NaOH aqueous solution was added to increase the viscos- rer (IS600, Trilab Co., Japan) whose speed can be con- ity of 0.5 wt% Carbopol 940 solution in experiments.) trolled from 3 to 600 rpm along with the torque Elasticity in a fluid may yield significant deviation from measurement up to 1.4 N·m with the resolution around 1.4 reciprocal relationship of the power number and the effec- mN·m. The impeller is rotated in the clockwise direction tive Reynolds number (Carreau et al., 1993). (viewed from the top) and the maximum measurable The remaining 1 and 2 wt% Carbopol 940 solutions torque has been set by 1 N·m to protect the measuring were employed as test fluids in predicting the viscosity unit. In measuring the torque, we first measured unloaded using the predetermined flow constants Kp and Ks from a torque on the shaft as a function of the impeller speed Newtonian fluid (silicone oil) and the reference low inside the empty vessel and the net torque on the shaft that concentration Carbopol solution, following the method actually generates the flow in the agitator is obtained by described in Sec. 2. Two different concentration of Xan- subtracting the unloaded torque from the measured torque than gum solutions have been also employed as test fluids at the corresponding impeller speed. in measuring the viscosity with the Metzner-Otto correla- In the experiment, three different kinds of fluids are tion. Figure 2 presents the shear stress as a function of the employed as working fluids. Highly viscous silicone oil shear rate for five different non-Newtonian fluids in this (KF-96-10000cs, Shinetsu Co., Japan) of the density ρ = work measured by the cone-and-plate geometry (50 mm, 3 975 kg/m and the viscosity μ =9.75 Pa·s is used to obtain MCR301, Anton Paar) at 20°C. The shear stress is mea- the reference power draw characteristics (the relationship sured at the shear rate 0.008~100 [1/s] and as expected between the power number and the Reynolds number for stress behaviors of the test fluids appear distinctive. a Newtonian fluid), from which the energy dissipation rate During the procedure in determining the effective shear constant Kp (Eq. 3) can be determined. The use of a highly rate constant with a reference non-Newtonian fluid, one viscous fluid is preferred to reduce errors in the torque needs an explicit viscosity equation as a function of the measurement. shear rate. In this regard, the shear stress data in Fig. 2 has As for non-Newtonian fluids, we tested Carbopol solu- been fitted with the regularized Herschel-Bulkley model tions (Carbopol 940, Lubrizol Co., USA) and Xanthan for the lowest concentration Carbopol solution (0.5 wt%). gum solutions (Xanthan gum, Sigma-Aldrich Co., USA). The model is selected due to the presence of a yield stress Three different concentrations of Carbopol solutions are (Fig. 2) and is written as prepared: 0.5, 1 and 2 wt%. The 0.5 wt% Carbopol solu- · mγ· · n τγ() = τy()1–e + Kγ .(5) tion has been used as a reference non-Newtonian fluid to determine the constant Ks (Eq. (1)). As for the reference The stiffness parameter m is introduced to avoid abrupt fluid in finding Ks, a low concentration Carbopol solution change in the shear stress, which is necessary in compar- is chosen, since fluid elasticity can be minimized and at ison with numerical simulations in future investigations

Korea-Australia Rheology J., 29(4), 2017 319 Hae Jin Jo, Hye Kyeong Jang, Young Ju Kim and Wook Ryol Hwang

−1 −1 (Alexandrou et al., 2003; Mitsoulis, 2007). The parame- with Re and, by taking the average of the product NPRe n ters were found to be K = 17.69 Pa·s , n = 0.2792, τy = at each data set, the energy dissipation rate constant Kp has −1 16.5 Pa and m = 5000. The fitted curve for 0.5 wt% Car- found to be 226. The averaged correlation NP = 226 Re bopol 940 solution with Eq. (5) is presented also in Fig. 2, is presented also in Fig. 3b for comparison. which shows the accuracy in fitting the stress behavior in Having determined the energy dissipation rate constant this case. Kp, one can proceed to estimate the effective shear rate constant Ks using a non-Newtonian fluid as a reference 4. Results and Discussion fluid. As mentioned, 0.5 wt% Carbopol 940 solution has been employed as a reference non-Newtonian fluid, with 4.1. Experimental determination of the flow constants which the torque measurement has been carried out for Figure 3a shows the net torque T as a function of the seven different impeller speeds from 30 rpm to 210 rpm. impeller speed N using the silicone oil as a working fluid. At each impeller speed, we computed the power P and the The impeller speed ranges from 30 rpm to 300 rpm and power number NP by multiplying the torque and the angu- the resultant torque resides from 0.08 to 1 N·m, which is lar velocity, as was done in the previous Newtonian case. much larger than the tolerance in the torque measurement. Then the effective Reynolds number is estimated by the From the data in Fig. 3a, the total energy dissipation rate power number characteristics (Fig. 3b along with the –1 P inside the vessel is estimated by P =2πNT and the equation NP = 226 Reeff ). The effective viscosity is now 2 power number NP is then computed by Eq. (2). Plotted in expressed as μeff = ρND /Reeff, from which the effective Fig. 3b is the power number NP as a function of the Reyn- shear rate is obtained by the viscosity function (Eq. (5)) olds number for a Newtonian fluid (silicone oil). As for the reference non-Newtonian fluid. The effective shear expected, the power number NP has turned out to scale rate constant Ks for each impeller speed is then determined

Fig. 3. (Color online) Flow characteristics of the anchor agitator Fig. 4. (Color online) Flow characterization of the anchor agi- system with a Newtonian fluid (10,000 cs silicone oil): (a) the tator with a reference non-Newtonian fluid (0.5 wt% Carbopol torque measurement as a function of the impeller speed; (b) the 940 solution): (a) the determination of the effective shear rate relationship between the power number and the Reynolds num- constant; (ii) the relationship between the power number and the ber. effective Reynolds number with Ks =25.

320 Korea-Australia Rheology J., 29(4), 2017 Process viscometry in flows of non-Newtonian fluids using an anchor agitator by Eq. (1). Figure 4a shows the effective shear rate con- correlating the torque and the impeller speed in an agitator stant Ks for the seven different impeller speeds with 0.5 appear similar to rheometric data in Fig. 2 that relates the wt% Carbopol 940 solution. The constant Ks ranges from shear stress to the shear rate. Since the torque is the inte- 21.70 to 33.78 and the largest value appears with the high- gral of the wall shear stress and the shear rate scales lin- est impeller speed (210 rpm) and the smallest one from early with the impeller speed, this similarity is obvious, the slowest rotation (30 rpm). Fluctuation in Ks seems to but it provides physical insights related with the Metzner- be large, around 13% variation from the average value of Otto correlation. That is, although flows in agitators are 25. However, this amount of variation in Ks is believed not complex due to complexity in geometries such as impel- to affect a lot the measured viscosity, by recalling the fact lers, vessels, baffles, there seems to be representative that the viscosity dependence on shear rate is best shear stress and shear rate that characterize the flow sys- described in a log-log plot. In this regard, we selected the tem and Metzner and Otto (1957) proposed the method for effective shear rate constant Ks simply by 25. this characterization based on the energy balance. Plotted in Fig. 4b is the relationship between the power From the data on torques and impeller speeds for each number and the (effective) Reynolds number for both a fluid material, one can compute the power and power Newtonian fluid (silicone oil) and the reference non-New- number KP for each impeller speed, as was done previ- tonian fluid (0.5 wt% Carbopol 940). In constructing the ously. Then the effective Reynolds number is determined –1 correlation, the effective shear rate coefficient Ks = 25 has by Reeff = 226 N p and the effective viscosity is then μeff 2 been employed in re-evaluating the effective viscosity of = ρND /Reeff. The corresponding shear rate is the effective the corresponding Reynolds number in all data for the ref- shear rate γ· eff , which was determined by Eq. (1) using the erence non-Newtonian fluid. Figure 4b shows nearly a predetermined KS = 25. Figures 6a and 6b show the effec- single master curve even with the non-Newtonian fluid tive viscosity as a function of the effective shear rate for with the reciprocal relationship, which is identical to that Carbopol 940 solutions and Xanthan gum solutions, of the Newtonian reference fluid. respectively. To make assessment on the accuracy in vis-

4.2. Viscosity measurement

Once the two flow constants Kp and Ks are fixed, one can perform the process viscometry for non-Newtonian fluids. As example non-Newtonian fluids, we tested two difference concentrations of Carbopol 940 solutions (1 wt% and 2 wt%) and of Xanthan gum solutions (1 wt% and 2 wt%). Plotted in Fig. 5 are the torque data for several values of impeller speed of an anchor agitator for four different example fluids. For comparison, we presented also the torque measurement data for the Newtonian and non-Newtonian reference fluids. We note that the curve

Fig. 6. (Color online) Process viscometry using the anchor agitator: (a) viscosity measurement with 1 wt% and 2 wt% Car- Fig. 5. (Color online) Data on the torque measurement as a func- bopol solutions; (b) viscosity measurement with 1 wt% and 2 tion of the impeller speed of the anchor impeller system for all wt% Xanthan gum solutions. Viscosity measured with the rhe- the fluids in this study. ometer is also plotted for comparison.

Korea-Australia Rheology J., 29(4), 2017 321 Hae Jin Jo, Hye Kyeong Jang, Young Ju Kim and Wook Ryol Hwang

bined effects of inertia and fluid elasticity and is a subject of the future investigation.

5. Conclusions

In this work, we present a viscosity measurement technique for inelastic non-Newtonian fluids in flows of anchor agitators that are commonly used in highly viscous fluid mixing particularly with yield stress, following the Metzner-Otto method. A two-blade close clearance anchor in a flat bottom vessel is chosen as a model agitator system. Using a highly viscous Newtonian fluid (10,000 cs silicone oil), we first established the reciprocal relation- Fig. 7. (Color online) The relationship between the power num- ship between the power number and the Reynolds number ber and the (effective) Reynolds number for all the test fluids in by measuring the torque at several impeller rotation speed the present work. and the energy dissipation rate constant Kp. Then 0.5 wt% Carbopol 940 solution is introduced as a reference non- Newtonian fluid and, by matching the power number as a cosity measurement, the viscosity curves from the steady function of the Reynolds number as in the Newtonian sys- shear experiment with the rheometer have been also pre- tem, the effective shear rate constant Ks is found. Once the sented. As expected, the process viscosity measured by two flow number Kp and Ks are fixed, four example non- the torque in an agitator appears quite similar to that from Newtonian fluids were tested for the process viscometry, the rheometer. The maximum relative errors between pro- measuring the viscosity from the torque in the agitator, cess viscometry and rheological measurement are 13.7% and the accuracy of the measurement were limited within (for 1 wt% Carbopol 940), 21.2% (2 wt% Carbopol 940), 20% relative error. 14.3% (1 wt% Xanthan gum) and 21.4% (for 2 wt% Xan- than gum), which may be considered acceptable in the Acknowledgement process viscometry. The discrepancy between the two measurements seems to be natural, because the flow field This work is supported by Korea Agency for Infrastruc- in the anchor agitator is not viscometric: i.e., it is com- ture Technology Advancement grant funded by Ministry posed of extension/compression, separation/recombina- of Land, Infrastructure and Transport (17IFIP-B133614- tion, acceleration/deceleration, etc. There are two interesting 01, The Industrial Strategic Technology Development Pro- observation in the error with the viscosity measurement in gram) and by the National Research Foundation of Korea Fig. 6: (i) the error is likely to increase with the impeller (NRF-2016R1A2B4014326). Y.J.K. acknowledges the speed; and (ii) the viscosity of Xanthan gum solution is financial supports from the Ministry of Trade, Industry & always over predicted and the opposite phenomena hap- Energy (Korea) (The Industrial Strategic Technology pens with Carbopol solution. The first observation seems Development Program, 10062514). to be related with the flow regime: the effective Reynolds number for large impeller speeds is close to or even larger References than 10, which is considered as the laminar limit in flows in the agitator. The reason for the second observation is Alexandrou, A.N., P. L. Menn, G. Georgiou, and V. Entov, 2003, not clear and is a subject of future investigation, though Flow instabilities of Herschel-Bulkley fluids, J. Non-Newton. fluid elasticity seems to be responsible for increase of the Fluid Mech. 116, 19-32. torque, according to Carreau et al. (1993). Brito-De La Fuente, E., J.A. Nava, L.M. Lopez, L. Medina, G. Finally, we present the relationship between the power Ascanio, and P.A. Tanguy, 1998, Process viscometry of com- number and the (effective) Reynolds number for all the plex fluids and suspensions with helical ribbon agitators, Can. J. Chem. Eng. 76, 689-695. fluid materials investigated in this work in Fig. 7. For the Carreau, P.J., R.P. Chhabra, and J. Cheng, 1993, Effect of rheo- four test fluids (1 wt% and 2 wt% Xanthan gum and Car- logical properties on power consumption with helical ribbon bopol solutions), the effective Reynolds number is calcu- agitators, AIChE J. 39, 1421-1430. lated based on the process viscometry result. All the Doraiswamy, D., R.K. Gremville, and A.W. Etchells III, 1994, curves are found to collapse into a single master line NP = −1 Two-scores years of the Metzner-Otto correlation, Ind. Eng. 226 Re , which is identical to that of a Newtonian fluid Chem. Res. 33, 2253-2258. (silicone oil). There is a small amount of discrepancy near Edwards, M.F., J.C. Godfrey, and M.M. Kashani, 1976, Power the Reynolds number around 10, which seems to be com- requirement for the mixing of thixotropic liquids, J. Non-New-

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