Results
4.2 Systematic Investigation
4.2.1 Influence of Power Input on the Fluid Dynamics in Stirred Vessel
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For these reasons, the investigation with 3D-PIV is not done for all the impellers, but only for the four presented. The two-dimensional PIV is used for the whole investigation. However, in the next comparisons the results obtained with the 3D-PIV are presented whenever they are available and coherent within the set of chosen data points.
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(A) (B) (C)
Velocity magnitude|v nno| / m s–1
Height h / mm
Radius r / mm Radius r / mm Radius r / mm
Fig. 4-9 Flow field obvserved with PROPRing-h/d=0.33 with increasing power input: (A) 19 W m--3, (B) 98 W m--3, (C) 488 W m--3.
With the increase of power input, also the intensity of the velocity field increases, since the impeller rotational frequency is increased. The CDF for the normalized velocity with the five values of power input shows slight differences, since all the curves are overlapped with each other, see Fig. 4-10 (A). This outcome of the CDF demonstrate that the ratio of the velocity with the tip speed maintains a scale factor for the different values of power input, exhibiting the self-similarity of the system.
Providing a higher power input to the fluid means operating at a higher impeller frequency, hence generating a higher Re. As a consequence, the fluid flows with a higher turbulence, that is translated to higher values of TKE in the stirred vessel. This is confirmed by the CDF of the TKE, shifted on the x axis for different power input values, see Fig. 4-10 (B).
Cumulative distribution function Q0 / – Cumulative distribution function Q0 / –
10–3 10–2 10–1 100 10–4 10–3 10–2 10–1 100
Normalized velocity magnitude |vno|utip–1 / – Turbulence kinetic energy k / m2 s–2 Fig. 4-10 Influence of power input with PROPRing-h/d=0.33, measured in water with 3DA: (A) CDF of normalized velocity, (B) CDF of TKE.
The local EDR is an important as well as challenging quantity to determine, especially in the impeller region, where highest values are found. The determination of local EDR is usually based on the assumption of isotropy, which is far from describing the reality of a stirred vessel. Furthermore, the calculation of local EDR from the velocity gradients obtained by PIV
0 0.2 0.4 0.6 0.8 1
(A)
0 0.2 0.4 0.6 0.8 1
19 W/m³ 49 W/m³ 98 W/m³ 195 W/m³ 488 W/m³ (B)
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deeply depends on the spatial resolution of the analysis (Delafosse et al., 2011). This should be equal or smaller than the Kolmogorov microscale, where the dissipation occurs with the eddies of smallest size. The finite grid sizes used by PIV to calculate the velocity gradients, the IWs, often exceed the smallest size of the eddies, generating an unreal distribution of EDR in the space of the stirred vessel. A suggested way to determine the local EDR is the large eddy method, suggested by Sagaut and Lee (2002).
In this experimental work, with the preliminary tests on the measuring technique and the processing of the great amount of different results, the local EDR is not calculated. However, by calculating the average EDR with Eq. (2-48), it is possible to see the influence of the power input on this quantity according to
`T q A
iS (2-48)
with ρ = 997.66 kg m--3 and V = 0.00299 m3, the density and volume of the fluid, respectively.
Moreover, the determination of the Kolmogorov microscale with Eq.°(2-47), gives an idea on the change of the eddies’ size with the change of power input.
b q wh `⁄ xT ⁄' (2-47)
with ν = µρ--1 = 1.0023 × 10-6 m2 s--1. The values calculated this way, are reported in Tab. 4-4 with the corresponding values of power input and Reynolds number.
Tab. 4-4 Average energy dissipation rate and Kolmogorov scale calculated for different values of power input.
PV–1 / W m–3 Re / – ε / m2s–3 η / µm
19 20,964 0,019 85
49 28,453 0,049 67
98 35,849 0,098 57
195 45,166 0,195 48
488 61,300 0,489 38
The increase of power input produces a higher turbulence in the fluid, in which the TKE is dissipated at a higher rate by eddies of smaller size.
Considering that the size of the smallest IW used for the calculation is equal to 24 × 24 pixels, which corresponds to 1.4 mm = 1.4 × 103 µm, it is possible to say that the spatial resolution of this PIV analysis is not able to detect the smallest turbulent eddies. The typical Kolmogorov scale of a tank bioreactor is around the range of 50--200 µm (Delafosse et al., 2011), as estimated in the calculation of the mean value of η, in Tab. 4-4.
Another valuable result from the analysis of the data obtained by PIV is represented by the gradients of the forces that act on the particles. The shear and normal gradients are calculated with the formulas explained and provided by Wollny (2010), see Section 2.3. In the derivation of the formulas, a coordinate transformation is operated in the velocity tensor, in order to have the tangential and the normal components of the velocity with respect to the direction of the flow. The values calculated for each IW result in the vector field of shear and normal gradients and the field of the strain tensor’s magnitude. From the statistical analysis of
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the field it is possible to calculate the distribution of the values in the area of interest, by using the histogram and the CDF. These representations are given in Fig. 4-11.
(A) (B) (C)
Strain tensor magnitude|\]| / s–1
Height h / mm Shear gradien
t γ]
nt / s–1 Height h / mm Normal gradien
t γ]
nn / s–1 Height h / mm
Radius r / mm Radius r / mm Radius r / mm
q0 Q0 q0 Q0
Shear gradient γ]nt / s–1 Normal gradient γ]nn / s–1
Fig. 4-11 Strain tensor calculated for PROPRing-h/d=0.33, PV--1 = 98 W m--3: (A) Shear and normal gradient fields, (B) Shear and normal gradients histograms and CDFs, (C) Strain tensor’s magnitude field.
The high shear rate zone corresponds to the region, below the impeller, that delimits the points of high velocity. Here the velocity gradients in the direction that is normal to the flow stream are the highest, since the velocity decreases from the highest to a very low value, and the other way around. In this case, considering the flow field in Fig. 4-9 (B), the velocity goes from about 0.6 m s--1 to 0.2 m s--1 in a reduced length of some mm: the shear gradient, in Fig. 4-11 (A), in this region shows the maximum value of shear rate, which is around 150 s--1.
However, the vector field is not the best option to use when a direct, quantitative comparison is required. The representation of several distribution functions, when easy to interprete, can give a good and intuitive idea of the analysis. The CDFs of the shear and normal gradients, for the five values of power input, are represented in Fig. 4-12.
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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.