14th European Conference on Mixing Warszawa, 10-13 September 2012

MIXING OF SHEAR-THINNING FLUID WITH YIELD STRESS IN A VESSEL WITH UNSTEADILY ROTATING IMPELLER

Szymon Woziwodzki, Lubomira Broniarz-Press

a Poznań University of Technology, Faculty of Chemical Technology, Department of

Chemical Engineering and Equipment, pl. M. Skłodowskiej-Curie 2, 60-965 Poznań, Poland szymon.woziwodzki@put.poznan.pl

Abstract. Power characteristics and cavern formation for shear-thinning fluids with a yield stress fluids in an unbaffled vessel stirred by turbine impeller was studied. The effect of operating conditions such as oscillation frequency and the liquid physical properties (viscosity and yield stress) on the cavern dimensions and power requirements were evaluated experimentally. According to experimental data it was found that unsteadily mixing caused enlargement of cavern dimensions (diameter and height) in comparison to unidirectional mixing. Moreover, it was found that greater effect of oscillation frequency on cavern dimensions was obtained for PBT impeller. For RT impeller this effect was negligible. Basing on results obtained it was found that unsteadily mixing enhances mixing efficiency in comparison to standard mixing. Keywords: Laminar mixing, cavern, yield stress, power requirements.

1. INTRODUCTION Shear-thinning fluids with a yield stress are commonly encountered in industrial mixing

operations. Mixing of such fluids results in formation of caverns (well mixed regions) in vicinity of rotating impeller [1-4]. Various models allow predicting cavern size. According to Solomon et al. [1] cavern diameter Dc can be estimated using power number Po and yield stress τy

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅⋅⋅⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

y

c DNPoDD

τρ

π

22

3

3 4 (1)

That model assumed that the cavern was spherical and stress at cavern boundary was equal to fluid yield stress. The above model was modified by Elson et al. [2] assuming that cavern was a cylinder with height Hc

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅⋅⋅⋅

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅

=⎟⎠⎞

⎜⎝⎛

yt

c

c

c DNPo

DHD

Dτ

ρ

π

22

2

3

31

1 (2)

The cavern may be also described by a torus [3] with a cross-sectional radius rc. In this case cavern radius can be determined using power law model and velocity on the cavern boundary [4] as follows

)/2(1/12

0))/2(1(

4412 nn

nc

TF

Kn

r−

− ⎟⎠⎞⎜

⎝⎛+⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛⎟⎠⎞⎜

⎝⎛ −= πν (3)

where v0 is the velocity at the cavern boundary and F total force imparted by impeller Taking into account a yield stress approach Eq. (3) simplifies to Eq. (4) [4]

515

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅⋅⋅⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

y

c DNPoDD

τρ

π

22

2

3 36.1 (4)

The prediction of the cavern size is important in the mixing of yield stress fluids because it can give evidence that good mixing is in the whole fluid and there are no regions with poor mixing. Shear-thinning fluids with a yield stress are rheologically complex. There exist several rheological models which describe relation between shear stress and shear rate as well as yield stress. The Herschel-Bulkley model

nKγττ += 0 (5) is one of the most used.

Taking into account the efficiency of mixing, it is advisable to increase the volume of caverns. To increase volume of them, it is advisable to generate chaotic mixing. One of the methods to generate such flow is a mixing mode when impeller periodically changes the direction or rate of rotation (unsteadily mixing). The knowledge about unsteadily agitation with forward-reverse rotating impellers is still incomplete. The forward-reverse mixing could be performed in turbulent flow to achieve better homogeneity in shorter time compared to standard mixing, but the high shear level of forward-reverse mixing mode could increase the mixing time and mixing power. Previous investigations have shown [5-6] that in turbulent flow regime power requirements are greater compared to standard mixing. In addition mixing time was higher despite higher energy dissipation rates. Therefore advantage of forward-reverse mixing mode was limited to multiphase mixing processes.

The main objective of this study was to investigate an effect of unsteadily mixing on power characteristics and cavern formation for shear-thinning fluids with a yield stress in a vessel stirred by turbine impeller as well as to estimate an effect of frequency of unsteadily rotation.

2. EXPERIMENTAL PART Experimental set-up is presented at Figure 1. The vessel with diameter T = 0.19 m

(H/T = 1) had a flat bottom. Two types of impellers were used: Rushton turbine (RT) and six pitched-up blade turbine (PBT). The ratio of impeller diameter (D = 0.065 m) to vessel diameter was equal to D/T = 0.342. The working viscous non-Newtonian fluid with a yield stress were Xanthan gum aqueous solutions 0.4%, 0.6% and 1.0% (Table 1).

2.1. Flow visualization experiment

Observation of cavern dimensions was performed using flow visualization technique. Visualization experiment was investigated using the following procedure: the vessel was filled with about 3 L of the Xanthan gum solution. Approximately 10 ml of 2 % ultramarine solution was injected into the impeller vicinity. The experiments were recorded using Canon EOS 1D Mark III photographic camera (pictures with the resolution of 3888x2552 pixels). The next step of the studies was to analyze obtained pictures. The analysis was carried out according to the following procedure: change the color mode from RGB (24 bits) to HSB (8 bit grayscale, scale image (a diameter of shaft as the well-known dimension), convert image to a 2-bit-system (the H channel).

Using the ImageJ software the cross sectional area and size (diameter and height) have been identified.

2.2. Analytical method

Impeller rotational speed N was changed in time. Type of time-course of N was the same as in the previous investigation [5]. The maximal impeller rotational speed Nmax and the absolute value of minimal impeller speed |Nmin| were the same Nmax = |Nmin|. The maximal impeller rotational speed Nmax was ranged from 4 to 16 s-1 (Reynolds number for forward-

516

reverse mixing ReFR ranged from 4 to 150). The oscillation frequency f was as follows: f = 0.115 Hz, f = 0.23 Hz and f = 0.46 Hz

On the basis of changes in torque the average value of the torque for forward-reverse mixing TFR, average impeller rotational speed NFR, forward-reverse mixing power PFR and power number PoFR for unsteady forward-reverse agitation were determined. The method of calculation of TFR, PFR and PoFR has been presented in the previous publication [5].

3. RESULTS

3.1. Rheology Xanthan gum is a high molecular weight polymer. Several studies have shown that

Xanthan gum solution is a pseudoplastic fluid with yield stress. Therefore, its rheology can be described by the Herschel–Bulkley model (5).

Rheological properties of Xanthan gum were estimated using a computer-controlled rheometer AntonPaar Physica MCR501. A plate and plate geometry was used (PP50). The diameter of the plate was 5 cm. All measurements were made at 20 oC. Flow curves (Figure 2) were obtained at controlled shear rate in the range of 0.001– 100 s-1. Results of rheological measurements are presented in Table 1. Table 1. Rheological properties of Xanthan gum solutions

Xanthan gum mass concentration (%)

Yield stress τ0 (Pa)

Consistency index

K (Pa sn)

Flow index n

Regression coefficient

(R2)

Density ρ (kg m-3)

0.4% 4.66 0.87 0.64 0.966 994 0.6% 7.35 3.00 0.47 0.974 990 1.0% 14.53 13.08 0.30 0.958 988

Figure 1. Experimental set-up: 1- motor, 2 - inverter, 3 - torqmeter, 4 - injection device, 5 -PC, 6 - lighting, 7 - Canon EOS 1D Mark III

Figure 2. Flow curves of Xanthan gum solutions

The apparent viscosity of the fluid at the wall was determined.

The Metzner and Otto model [7] can be used to determine the shear and the apparent viscosity:

FRsw Nk=γ (6)

( ) 10 −+= nFRs

FRsw NkKNk

τμ (7)

The Reynolds number can be defined as:

( )nFRs

FRsFR NkK

DNkRe+

=0

22

τρ (8)

517

The Metzner and Otto constants for RT and PBT impellers were as follows: ks = 11.5 (RT) and ks = 13 (PBT).

3.2. Power requirements Mixing power measurements have been performed, using strain gauge technique, for

unsteadily forward-reverse mixing as well as for unidirectional standard mixing. The relation between power number, for forward-reverse mixing PoFR, and Reynolds number ReFR is presented in Figures 3a and 3b.

Figure 3a. Power characteristics for RT impeller Figure 3b. Power characteristic for PBT impeller Experimental data indicated that the power number PoFR was not constant. It is dependent

on Reynolds number. Moreover it was found that PoFR was independent of oscillation frequency. Comparison of power numbers for forward-reverse mixing and unidirectional mixing showed that greater power requirements needs forward-reverse mixing. Increase in power number for PBT impeller was about 20%. This relationship was valid in the range of Reynolds number from 4 to 150.

Another relationship was observed for RT impeller. For ReFR < 40 there was no difference between power requirements for forward-reverse and unidirectional mixing. It suggests that in Reynolds number range ReFR < 40 region of disturbed flow behind the blades and vortex shedding phenomena are minimized. At ReFR > 40 differences in power numbers were increasing. The disturbed flow region occurred because the direction change of circumferential flow in a vessel is delayed in relation to changes of direction of impeller rotation. Thus, for a moment, the liquid flows around the rotor blades. This causes the vortex shedding phenomena [5]. Vortex shedding can be modified by impeller oscillations. A matching of impeller movement with the flow velocity results in the resonance phenomenon, where vortices are shed in synchronization with impeller movement [8-10]. This phenomenon requires further study.

3.3. Cavern dimensions

The next step of studies was to evaluate an effect of forward-reverse mixing on cavern diameter and height. Moreover the effect of oscillation frequency was analyzed as well as comparison to unidirectional mixing was carried out.

The experimental data have shown that cavern reached vessel wall firstly (Dc/D = 3). When the cavern reached the wall the height of the cavern increases with increase of the forward-reverse impeller speed NFR. It has been shown that for radial impeller RT the effect of oscillation frequency on cavern diameter as well as on cavern height was negligible in Reynolds number range from ReFR = 4 to ReFR = 152 (Figures 4-5). Another effect has been found for pitched blade turbine. For ReFR > 60 the effect of oscillation frequency was negligible (similar to RT impeller), but for ReFR < 60 this effect was evident (Figures 6-7). The cavern diameter as well as cavern height were increased with increase of oscillation

518

frequency f. The highest cavern dimensions were obtained for f = 0.46 Hz. The divergences in cavern dimensions were increased with decreasing of Reynolds number values ReFR. The cavern diameters were about twice larger for f = 0.46 Hz and ReFR = 4 (Xanthan gum solution of 1.0%) and about 10% for f = 0.46 Hz and ReFR = 152 (Xanthan gum solution of 0.4%) in comparison to oscillation frequency f = 0.115 Hz.

Figure 4. Relation between cavern dimensions and Reynolds number for RT impeller

Figure 5. Relation between cavern height and Reynolds number for RT impeller

Figure 6. Relation between cavern dimensions and Reynolds number for PBT impeller

Figure 7. Relation between cavern height and Reynolds number for PBT impeller

Figure 8. Relation between cavern dimensions and Reynolds number for PBT impeller

Figure 9. Relation between cavern height and Reynolds number for RT impeller and Xanthan gum solution 0.6%

The next step of our investigation was to compare cavern dimensions for RT and PBT impellers. It has been found that highest cavern were achieved in vessel with RT impeller for Xanthan gum solution of 0.4% (about 10% highest) while for remaining solutions (0.6% and 1.0%) the highest caverns were in vessel with PBT impeller and oscillation frequency f = 0.46 Hz (about 25% highest). In the last step, it has been performed a comparison of results obtained for forward-reverse mixing with the results for the unidirectional mixing. Figures 8 and 9 present the exemplary

519

comparison of the cavern diameters Dc in a vessel with PBT impellers and the cavern height Hc in vessel equipped with RT impellers. It has been noticed that cavern dimensions were higher in vessel with forward-reverse rotating impeller. That effect was valid in Reynolds number values range ReFR < 40 at f � (0.115,0.46) Hz for RT impeller and at f = 0.46 Hz for PBT impeller. Greatest effect of oscillation frequency on cavern dimensions for PBT impeller has been observed.

4. CONSLUSIONS Shear-thinning fluids with a yield stress are commonly encountered in industrial mixing

operations. Mixing of yield stress fluids results in formation of caverns (well mixed regions) in vicinity of rotating impeller. The prediction of the cavern size is important in the mixing of yield stress fluids because it can give evidence that good mixing is in the whole fluid and there are no regions with poor mixing. According to experimental data it was found that unsteadily mixing caused enlargement of cavern dimensions (diameter and height) in comparison to unidirectional mixing. Enlargement of dimensions was observed for Rushton turbine (RT) as well as for PBT impeller. Moreover, it was found that greater effect of oscillation frequency on cavern dimensions was obtained for PBT impeller. For RT impeller this effect was negligible. Basing on results obtained it was found that unsteadily mixing enhances mixing efficiency in comparison to unidirectional mixing.

5. REFERENCES [1] Solomon J., Nienow, A.W., Pace G.W., 1981. “Flow patterns in agitated plastic and pseudo-plastic fluids”, Fluid Mixing IChemE Symp. Ser., 64, A1-A13. [2] Elson, P.T.,Cheesman, D.J., Nienow, A.W., 1986. “X-ray studies of cavern sizes and mixing performance with fluid possessing a yield stress”, Chem. Eng. Sci., 41, 2555-2562. [3] Galindo, E., Nienow, A.W., 1992. “Mixing of highly viscous simulated Xanthan fermentation broths with the Lightnin A315 impeller”, Biotechnol. Prog., 8, 233-239. [4] Amanullah A., Hjorth S.A., Nienow A.W., 1998. “A new mathematical model to predict cavern diameters in highly shear thinning, power law liquids using axial flow impellers”, Chem. Eng. Sci., 53, 455-469. [5] Woziwodzki, S., 2011. „Unsteady mixing characteristics in a vessel with forward-reverse rotating impeller”, Chem. Eng. Tech. 34, 767-774. [6] Woziwodzki S., Szatkowska K., 2011. “Effect of eccentricity on mixing power of forward-reverse rotating impellers”, Przem. Chem. 90, 1702-1704. [7] Metzner A.B., Otto R.E., 1957. “Agitation of non-Newtonian fluids”, AIChE J., 3, 3–10. [8] Lam K.M., 2009. “Vortex shedding flow behind a slowly rotating circular cylinder”, J. Fluids. Struct. 25, 245-262. [9]. Dol S.S., Kopp G.A., Martinuzzi R.J., 2008. “The suppression of periodic vortex shedding from a rotating circular cylinder”, J. Wind Eng. Ind. Aerodyn., 6, 1164-1184. [10]. Badr H.M., Coutanceau M. , Dennis S.C.R., Menard C., 1990. “Unsteady flow past a rotating circular cylinder at Reynolds numbers 103 and 104”, J. Fluid. Mech., 220,459-489.

520