Results
4.2 Systematic Investigation
4.2.2 Influence of Fluid Rheology on the Fluid Dynamics in Stirred Vessel
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Cumulative distribution function Q0 / – Cumulative distribution function Q0 / –
10–1 100 101 102 103 10–1 100 101 102 103
Shear gradient γ]nt / s–1 Normal gradient γ]nn / s–1
Fig. 4-12 Influence of power input, PROPRing-h/d=0.33: (A) shear gradients, (B) normal gradients.
The increase of power input produces a stronger hydomechanical stress on the particles, that results with the increase of both shear and normal gradients. The fluid flows with a higher turbulence and it exerts a higher strain, so the particles in the fluid feel more shear, the tangential component, and more normal, elongational forces. However, from the CDFs plotted in Fig. 4-12, it results clear that the shear components are prevalent on the normal ones, which contrarily to the former do not exceed 102 s--1.
4.2.2 Influence of Fluid Rheology on the Fluid Dynamics in Stirred
Systematic Investigation
Appendix B. The flow patterns developed with PROPRing-h/d=0.33 show a more axial behaviour in water, with respect to glycerin and xanthan, where the flow presents also a radial feature.
(A) (B) (C)
Velocity magnitude|v nno| / m s–1
Height h / mm
Radius r / mm Radius r / mm Radius r / mm
Fig. 4-13 Influence of rheology on the flow patterns developed with PROPRing-h/d=0.33, for PV--1 = 100 W m--3: (A) water, (B) glycerin, (C) xanthan.
If the nature of the flow regime is taken into account, this result is reasonable. Water is here characterized by a Reynolds number Re = 35,849 and glycerin by Re = 3,136. For xanthan, due to its non-Newtonian behaviour, the determination of Re is more difficult, since the viscosity has not a constant value. The Reynolds number for a non-Newtonian fluid can be determined with the appartent viscosity µa. Xanthan is described with Ostwald-de-Waele model, which uses two characteristic coefficients of the fluid, n and K, see Section 2.2. For xanthan, these coefficient are n = 0.3845 and K = 0.17185 kg m--1 sn--2. The apparent viscosity of the fluids that are described with this model can be calculated by
ga q 0\]2 (2-6)
and, considering that the shear rate locally changes, µa does not have a constant value. A possible solution may consist in taking only those into regions account with the highest and the (quasi) lowest shear. In the shear gradient field, showed in Fig. 4-14, it is possible to take 200 s--1 and 20 s--1 as the maximum and minimum values of shear, respectively. The corresponding maximum and minimum values of apparent viscosity are µa,max = 0.00659 Pa s and µa,min = 0.027187 Pa s. The calculation of Re gives a maximum and a minimum value:
Remax = 5,624 and Remin = 1,276. The impeller region is characterized by a Re that is 4 times the Re in the rest of the vessel. The flow is in a transitional flow. The full turbulent regime of water determines a more axial flow, with respect to the transitional flow of glycerin and xanthan.
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Shear gradien
t γ]
nt / s–1
Height h / mm
Radius r / mm
Fig. 4-14 Shear gradient observed in xanthan at PV--1 = 100 W m--3 with PROPRing-h/d=0.33.
The CDF of the normalized velocity does not show big differences between glycerin and xanthan, for both power input values, see Fig. 4-15 (A). In water the velocities are generally higher, because at the same power input, the rotational frequency of the impeller in water is higher as compared to the other two fluids. The lower viscosity of water allows the inertial forces to overcome the viscous forces with lower power input.
Cumulative distribution function Q0 / – Cumulative distribution function Q0 / –
10–3 10–2 10–1 100 10–5 10–4 10–3 10–2 10–1
Normalized velocity magnitude |vno|utip–1 / – Turbulence kinetic energy k / m2 s–2 Fig. 4-15 Influence of rheology for PROPRing-h/d=0.33, with two constant values of power input:
(A) CDF of the normalized velocities of the vector field, (B) CDF of the TKE field.
The TKE generated by the motion of the impeller is higher when it stirs a less viscous fluid, see Fig. 4-15 (B). In a fluid with low viscosity, like water, the dominance of the inertial forces upon the viscous forces is translated with higher values of Re, hence to a higher turbulence.
With the CDFs of TKE, it is possible to appreciate a significant difference between the three fluids. Here, xanthan and glycerin curves do not overlap, but they are distinct and different.
This means that, even if the distribution of the normalized velocities is comparable, the fluctuating velocities are significantly different between the two fluids. The values of TKE are smaller for xanthan because the high viscous forces contrast the turbulence, as compared to water and glycerin. The fields of TKE relative to the lowest value of power input are reported in Appendix B.
0 0.2 0.4 0.6 0.8 1
Xanthan 100 W/m³ Xanthan 200 W/m³ Glycerin 100 W/m³ Glycerin 200 W/m³ Water 100 W/m³ Water 200 W/m³
(A)
0 0.2 0.4 0.6 0.8 1
(B)
Systematic Investigation
Like for the previous comparison, the average EDR can be calculated with Eq. (2-48). The results are reported in Tab. 4-6.
Tab. 4-6 Calculation of ε with V = 2.99 L, volume of the fluid in the vessel.
Fluid PV–1 / W m–3 ε / m2 s–3 water
ρ = 997.66 kg m--3
98 195
0.098 0.195 glycerin
ρ = 1150 kg m--3
100 200
0.087 0.174 xanthan
ρ = 998.37 kg m--3
100 200
0.100 0.200
These values are calculated by using the density of the fluids: the differences between the values obtained for the three fluids are comparable and no substantial discrepancy is remarked. However, the averaged EDR is an approximate value, from which is not possible to have information on the local intensity of this quantity in the stirred vessel.
Cumulative distribution function Q0 / – Cumulative distribution function Q0 / –
10–1 100 101 102 103 10–1 100 101 102 103
Shear gradient γ]nt / s–1 Normal gradient γ]nn / s–1
Fig. 4-16 Influence of rheology for PROPRing-h/d=0.33 at PV--1 = 98 W m–3: (A) shear gradient, (B) normal gradient.
The shear produced in the three fluids is found very similar in range and distribution, see Fig. 4-16 (A). Considering the similarity between the values of the average EDR calculated for water, glycerin and xanthan, this result for the shear gradient seems to be confirmed.
However, it has to be considered that for this comparison the values of average EDR are used, which are not able to provide local information inside the vessel. A little difference is visible for the normal gradient, Fig. 4-16 (B), which shows slightly higher values for water, in the middle range. This suggests that the decrease of the viscosity produces a higher turbulence which exerts a higher normal stress on the particles (compression and elongation forces in the fluid). However, the differences generally are small when comparing both types of gradients, the shear and the normal gradient.
0 0.2 0.4 0.6 0.8 1
(A)
0 0.2 0.4 0.6 0.8 1
Xanthan Glycerin Water (B)
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