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Slurry viscosity

How much does suspending the particulate thicken the carrier?

The formula
η=η0⁢1+2.5⁢φ+6.2⁢φ2
η
Mixture viscosity, Pa·s
η0
Carrier viscosity, Pa·s
φ
Volume fraction, %
LaTeX 
η = η0 \left( 1 + 2.5 \cdot φ + 6.2 \cdot φ^2 \right)

Work it out

The liquid the particulate is carried in. Handbook viscosity at 20 °C.

%

The solid's share of the total volume.

Compare with a variation

Result

η Mixture viscosity 1.143 mPa·s

The suspension's dynamic viscosity. One mPa·s is one centipoise; water is about 1.

k Relative viscosity 1.141

How many times thicker than the carrier alone.

With your numbers
1.143 mPa·s=1.002 mPa·s⁢1+2.5⁢5 %+6.2⁢5 %2
LaTeX 
1.143\,\mathrm{mPa·s} = 1.002\,\mathrm{mPa·s} \left( 1 + 2.5 \cdot 5\,\mathrm{%} + 6.2 \cdot 5\,\mathrm{%}^2 \right)

This result is a link — the address bar holds your numbers, so it can be pasted into a post and opened to the same answer.

What this looks like

Mixture viscosity against particulate loading Particulate loading swept from 2.5 % to 7.5 % with everything else held at your numbers. The dashed lines cross where you are.
Mixture viscosity against particulate loadingMixture viscosity rises from 1.07 mPa·s to 1.22 mPa·s as particulate loading rises from 2.5 % to 7.5 %. At your particulate loading of 5 % it is 1.14 mPa·s.1.051.11.151.21.2524685 %1.14 mPa·sParticulate loading (%)Mixture viscosity (mPa·s)
The formula behind the curve
η=η0⁢1+2.5⁢φ+6.2⁢φ2
η
Mixture viscosity, Pa·s
η0
Carrier viscosity, Pa·s
φ
Volume fraction, %
LaTeX 
η = η0 \left( 1 + 2.5 \cdot φ + 6.2 \cdot φ^2 \right)
What moves the answer Each input moved 10% either way, with the others held still, and the effect on mixture viscosity.
What moves the answerMixture viscosity is most sensitive to Particulate loading, which moves it by about 1.38% for a 10% change. It is the only input.Change in the answer when each input moves by 10%-2%-1%0%1%2%Particulate loading±1.38
The formula behind the curve
η=η0⁢1+2.5⁢φ+6.2⁢φ2
η
Mixture viscosity, Pa·s
η0
Carrier viscosity, Pa·s
φ
Volume fraction, %
LaTeX 
η = η0 \left( 1 + 2.5 \cdot φ + 6.2 \cdot φ^2 \right)

Method

  1. Look up the carrier's viscosity at 20 °C — about 1 mPa·s for water, 30 for a light mineral oil, 1400 for glycerol.
  2. Convert the loading to a fraction by dividing by 100.
  3. Einstein's term: each particle disturbs the flow around it, and for dilute rigid spheres the viscosity rises by 2.5 times the volume fraction. Batchelor's term, 6.2 φ², accounts for pairs of particles interacting, and matters from a few per cent upward.
  4. Multiply the carrier's viscosity by the bracket. The ratio itself — the bracket — is the relative viscosity.

Assumptions

  • Rigid, spherical, non-interacting particles in a dilute suspension. Einstein's coefficient is exact for isolated spheres and Batchelor's extends it to about 10 % by volume. Beyond that the formula falls increasingly short of the real thickening, and above roughly 40 % the suspension stops behaving as a Newtonian liquid at all.
  • Particles far larger than molecules and far smaller than the tube — the continuum picture. Iron filings in a two-centimetre tube qualify; a coarse grit does not.
  • The carrier's viscosity is its 20 °C value. Water thins by about 2 % per degree of warming, oils by more; a medium running warm is thinner than this.
  • No magnetic field. A magnetised slurry of ferromagnetic particles thickens sharply along the field — the magnetorheological effect — which is a large part of what an EPG does to its medium and is outside this formula.