By Dr. Richard Grenville, Philadelphia Mixing Solutions Ltd.
Most viscous fluids processed in industry are nonNewtonian, i.e., their viscosities are dependent on the shear rates exerted on them by agitator impellers or their velocities flowing through pipes. If a fluid is “shearthinning,” its apparent viscosity decreases as the impeller speed or velocity increases. Many products are formulated to have nonNewtonian rheologies. Examples would include ketchup (Koochecki et al., 2009) and paint (Eley, 2005).
A single viscosity value cannot fully describe a fluid’s nonNewtonian behavior. Simple devices like the Stormer viscometer should not be used to generate the data required for engineeringdesign purposes paint (Eley, 2005).
If the rheology has not been correctly specified, one possible consequence is that the blend time will be longer than expected, i.e., it will take longer than expected to reach the product’s required degree of homogeneity. With cycle times longer than expected, fewer batches can be processed each day.
Detailing the consequences
With a Newtonian fluid, viscosity is constant at any shear rate — at any impeller speed or pipe velocity. Calculations of blend time or pipeflow pressure drop are relatively easy to perform provided that the fluid’s viscosity is known.
In contrast, the viscosity of a nonNewtonian fluid is not constant. Instead, it must be measured over a range of shear rates, chosen as representative of those to be encountered in the equipment. For example, for design of a paintspray system, the shearrate range must be much higher than that required to generate data for agitator design.
Viscometers actually measure shear stress exerted on a fluid by the viscometer bob or spindle, over a range of shear rates, or measure the pressure drop of a fluid flowing through a capillary, over a range of flow rates. There are many good reviews of viscosity measurement techniques and equipment, for example Hiemenz(1986) and Patton (1966).
Typical plots of shear stress versus shear rate are shown in figure 1 for four model fluids.

Figure 1. Shear Stress versus Shear Rate Curves 
The simplest model used to relate the shear stress to the shear rate is the power law:
(1)
If the plot is a straight line (fluid 1), n = 1 and the fluid is Newtonian. If the plot is curved (fluids 2 and 3) the fluid is nonNewtonian and shearthinning. Fluid 3 exhibits greater shearthinning behavior with n = 0.25 while fluid 2 is moderately shearthinning with n = 0.75.
The dynamic viscosity is the shear stress divided by the shear rate, so:
(2)
Figure 2 takes the data plotted in figure 1 and replots them according to equation 2. For fluid 1, the exponent n1 is zero, the viscosity is independent of shear rate and its rheology is Newtonian. For fluids 2 and 3 the exponent n1 is negative. Thus, as the shear rate increases the apparent viscosity decreases. Hence the fluids are shearthinning.

Figure 2. Apparent Viscosity versus Shear Rate Curves 
Some fluids, especially slurries with high solids concentrations, exhibit a yield stress. This means that the agitator or pump must exert a minimum shear stress on the fluid before it will start to move. In this case the power law is modified with a constant, the yield stress, and this is the HerschelBulkley model:
(3)
(4)
In figures 1 and 2, fluid 4 exhibits a yields stress of 25 Pascals with n = 0.5.
If n = 1 in equations 3 and 4, the HerschelBulkely model reduces to the Bingham plastic model:
(5)
(6)
Here, μ∞ is the infinite shear viscosity. When the shear rate is high, the second term on the right hand side of equation 6 is small and the apparent viscosity approaches a constant value of μ∞.
The power law K values of the four model fluids plotted in figures 1 and 2 have been adjusted so that, at a shear rate of 200 s^{1}, all four have an apparent viscosity of 0.50 Pa s or 500 cP. At any other shear rate, the apparent viscosities are different and these differences are amplified at low shear rates.
The method proposed by Metzner and Otto (1957) is commonly used to estimate the shear rate in a stirred tank. The shear rate is proportional to the impeller’s rotational speed:
(7)
The constant, k_{S}, is dependent on impeller type and ranges from 10 for Rushton and pitched blade turbine impellers (figures 3a and 3b) and hydrofoils (figure 3c) to 30 for helical ribbon impellers (figure 3d). So for a turbine impeller operating at 120 RPM, the shear rate will be approximately 20 s^{1}.
For the four model fluids in figures 1 and 2, at a shear rate of 20 s1, the viscosities range from 500 cP (fluid 1) to 2800 cP (fluid 3).
Impeller selection
Accurate measurement and analysis of a fluid’s rheology must support the impeller selection decision.
If the fluid is shearthinning and so viscous that the impeller operates in the laminar regime, a helical ribbon would commonly be used. This is essentially a “positive displacement” impeller. If the fluid is moderately viscous, so that the impeller operates in the transitional or turbulent regime, standard turbine (figure 3b) or hydrofoil (figure 3c) impellers can be used.
There is a range of Reynolds numbers which are too high for a helical ribbon to be used and too low for turbine and hydrofoil impellers. In this region, the CounterFlow^{TM} impeller (figure 3e) developed by Philadelphia Mixing Solutions, Ltd. is very effective. The number, diameter and spacing of the impellers are dependent on the viscosity of the fluid.
These impellers can operate in the laminar regime, but the number and diameter of the impellers needed typically will make the helical ribbon agitator is the most costeffective solution.
Final words
A single value of viscosity cannot describe the rheological properties of a nonNewtonian fluid.
The shear stress versus shearrate curve must be measured over the shearrate range which the fluid will experience in the process. Shear rates experienced in pipe and spraynozzle flow are much higher than in a stirred tank.
The data can be fitted to the appropriate rheological model which can then be used to estimate the apparent viscosity of the fluid.
The rheology is also used to determine which impeller geometry is going to be most effective for the process. For moderately viscous fluids pitched blade turbines or hydrofoils can be used, but as the viscosity and degree of nonNewtonian behavior increase, other impellers should be considered. For fluids with high viscosity where the impeller operates in the laminar regime, helical ribbon impellers should be considered.
CounterFlow^{TM} impellers operate in the range of Reynolds numbers that are too high for a helical ribbon and too low for turbines and hydrofoils.
Nomenclature
K 
Power law consistency 
Pa s^{n} 
N 
Impeller rotational speed 
RPS 
n 
Power law index 
 

Shear rate 
s^{1} 
μ 
Viscosity 
Pa s 
μ_{A} 
Apparent viscosity 
Pa s 
μ∞ 
Infinite shear viscosity 
Pa s 
ρ 
Density 
kg m^{3} 
τ 
Shear stress 
Pa 
τ_{0} 
Yield stress 
Pa 
References
Koochecki, A., A. Ghandi, S. M. A. Razavi, S. A. Mortazavi & T. Vasiljevic, The Rheological Properties of Ketchup as a Function of Different Hydrocolloids and Temperature, Int. J. Food. Sci. Tech., 44, 596 – 602, 2009.
Eley, R. E., Applied Rheology in the Protective and Decorative Coatings Industry, Rheo. Rev., 173 – 240, 2005.
Hiemenz, P. C., Principles of Colloid and Surface Chemistry, 2nd edition, chapter 4, Marcel Dekker Inc., 1986.
Patton, T. C., Paint Flow and Pigment Dispersion, 2nd edition, chapter 2, Interscience, 1966.
Metzner, A. B. & R. E. Otto, Agitation of nonNewtonian Fluids, AIChEJ, 3, 3 – 10, 1957.