Behavior of a Power-Law Fluid in a Mixer

Mixers are often used to process non-Newtonian fluids and many of those fluids can be modeled using a power law for the relation between viscosity and shear rate. This model simulates the flow in a flat-bottom tank with a single four-blade pitched impeller. The simulated fluid is a solution of 1% Carbopol in water, which is a shear-thinning fluid that can be modeled by the power law. The calculated power numbers are compared with experimental results.

General Description

Geometry

The model is one of the cases considered in Ref. 1. The complete model geometry is shown in Figure 1. It is a flat bottom tank with diameter T = 0.45 m and height H = 0.44 m. It has four baffles with width W = 1/12·T. In the tank, there is an impeller which in Ref. 1 is specified to be an A200 impeller (Ref. 2). That means, that the impeller is a four-blade impeller with flat, 45° pitched blades. The impeller diameter, Da, is 0.41·T and the clearance (the distance between the bottom of the tank and the impeller), C, is 0.75·Da.

Figure 1: Flat-bottom tank with a four-blade, pitched impeller.

Liquid

The liquid is a 0.1 % Carbopol solution in water. It is a shear-thinning fluid and was selected in Ref. 1 since it is transparent enough for laser Doppler velocimetry. Rheological measurements suggest that the solution can be modeled as a power-law fluid under the range of shear rates expected in this set-up. The power law prescribes the following relation between the viscous stress K and the strain rate tensor S

(1)

where

is the shear rate, m is the power-law consistency coefficient and n is the flow-behavior index. According to Ref. 1, for shear rates in the range of 0.1–1000 s−1, m = 9 and n = 0.2.

Reynolds Number and Power Number

This model computes the power number, Np, as a function of the impeller Reynolds number, NRe.

The power number is defined by (Ref. 3)

where ρ is the fluid density, N is the rotations per second, Da is the impeller diameter and P is the power draw which in turn is determined by

(2)

In Equation 2, the integral is taken over the surface of the impeller. Ω is the rotational speed vector and Γ is the torque acting on a point r = {x, y, z} is defined by

where F is the force acting on each point of the surface.

For a Newtonian fluid, the impeller Reynolds number is defined as

(3)

where μ is the (Newtonian) dynamic viscosity. Equation 3 is however not valid for a shear-thinning liquid where the viscosity may vary by several order of magnitudes throughout the model. In the laminar flow regime, the impeller Reynolds number is instead calculated using the Metzner and Otto method (Ref. 1 and Ref. 3) where the viscosity μ is replaced by an average, near-impeller viscosity, μavg, defined by

(4)

where m is the power-law consistency coefficient and n is the flow-behavior index.

in Equation 4 is an average, near-impeller shear rate that can be calculated as a function of the impeller rotational speed:

(5)

where Ks is a constant that is assumed to depend only on impeller geometry. For the A200 impeller, Ks = 8.58 (Ref. 1). Combining Equation 3, Equation 4, and Equation 5 gives

In this model, the Reynolds number NRe = 50, 100, 200, and 400 which corresponds to N = 1.6337, 2.4001, 3.529, and 5.187 respectively.

Model Setup

A fully correct flow pattern can be obtained from a time-dependent simulation of the complete geometry shown in Figure 1. This would however be a very time-consuming procedure. Ref. 1 therefore uses a method equivalent to the frozen-rotor approach available in COMSOL. In a frozen-rotor approach, the impeller does not actually rotate. Instead, the rotational effect is included by the means of Coriolis and centrifugal forces. With a frozen-rotor approach, it is possible to utilize the symmetry of the geometry and only solve for a quarter of the mixer as shown in Figure 2. The geometry can be seen to have two domains. The domain enclosing the impeller is specified as rotating while the outer domain is stationary. This decomposition is not necessary when using a frozen-rotor approach, but it simplifies the setup of the boundary conditions.

Figure 2: Geometry used in the simulation.

Both baffles and impeller blades are very thin and are therefore treated as walls with zero thickness. This is a reasonable assumption for axial impellers (Ref. 4).

The power law (Equation 1) predicts infinite effective viscosity at zero shear rate, which of course is non-physical. The power law is therefore supplemented with a lower limit on the shear rate,

. In this model,

is set to 0.1 s−1 which is the lowest shear rate for which the power law is a valid viscosity model for the Carbopol solution.

To calculate the power, notice that, since the impeller shaft is parallel to the z-axis, Ω has only one component and is equal to {0, 0, 2πN}. This means that Equation 2 reduces to

(6)

Results and Discussion

Figure 3 shows the flow pattern for NRe = 400. Fluid is ejected downward from the impeller and is recirculated back along the outer wall. The movement of the fluid is almost entirely restricted to a high-shear region close to the impeller where the viscosity is low. The effective viscosity in the rest of the mixer is very high and the fluid there behaves more like a solid. The logarithm of the effective viscosity is shown in Figure 4.

Figure 3: Velocity vectors and velocity magnitude for NRe = 400.

Figure 4: Logarithm of the effective viscosity for NRe = 400. Arrows show the velocity field.

The calculated power numbers are plotted in Figure 5 together with the experimental results from Ref. 1. The agreement is at least as good as the agreement between experiments and simulations reported in Ref. 1.

Figure 5: Power number as a function of Reynolds number.

References

1. W. Kelly and B. Gigas, “Using CFD to predict the behavior of power law fluids near axial-flow impellers operating in the transitional flow regime,” Chem. Eng. Science, vol. 58, pp. 2141–2152, 2003.

2. SPX Lightnin A200 impeller, http://www.spx.com/en/lightnin/pc-impellers/

3. E.L. Paul, V.A. Atiemo-Obeng, and S.M. Dresta, eds., Handbook of Industrial Mixing, Science and Practice, John Wiley & Sons, 2004.

4. D. Chapple, S.M. Kresta, A. Wall, and A. Afacan, “The Effect of Impeller and Tank Geometry on Power Number for a pitched blade Turbine,” Trans IChemE, vol. 80, pp. 364–372, 2002.

Polymer_Flow_Module/Verification_Examples/power_law_mixer

Modeling Instructions

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Model Wizard

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Load the geometry sequence from file. Note that parameters used in the geometry are included with the sequence.

Geometry 1

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Browse to the model’s Application Libraries folder and double-click the file power_law_mixer_geom_sequence.mph.

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Also load parameters for the fluid properties and the simulation conditions.

Global Definitions

Parameters 1

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Browse to the model’s Application Libraries folder and double-click the file power_law_mixer_parameters.txt.

Moving Mesh

Rotating Domain 1

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Select Domain 2 only.

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In the f text field, type -N0.

Materials

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Laminar Flow (spf)

Fluid Properties 1

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Materials

Material 1 (mat1)

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In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Density	rho	rho_u	kg/m³	Basic
Fluid consistency coefficient	m_pow	m_powerlaw	Pa·s	Power law
Flow behavior index	n_pow	n_powerlaw	1	Power law

Laminar Flow (spf)

Fluid Properties 1

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Definitions

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Integration 2 (intop2)

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Variables 1

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In the table, enter the following settings:


Name	Expression	Unit	Description
tau_riw	x(spf.T_stress_uy+spf.T_stress_dy)-y(spf.T_stress_ux+spf.T_stress_dx)	N/m	Torque per area (rotating interior walls)
tau_rw	x(spf.T_stressy)-y(spf.T_stressx)	N/m	Torque per area (rotating walls)
P_riw	tau_riw*rot1.alphat	W/m²	Power draw per area (rotating interior walls)
P_rw	tau_rw*rot1.alphat	W/m²	Power draw per area (rotating walls)
mu_avg	m_powerlaw(KsN0*1[s])^(n_powerlaw-1)	kg/(m·s)	Average effective viscosity
NRe	Da^2N0rho_u/mu_avg		Impeller Reynolds number
Np	4(intop1(P_riw)+intop2(P_rw))/(N0^3Da^5*rho_u)		Impeller power number