Slurry density and solids concentration: converting mass %, volume % and density

A mineral-processing slurry may be described using several different quantities:

slurry density;
solids percentage by mass;
solids percentage by volume;
volumetric flow;
dry-solids mass flow.

These quantities are linked by straightforward mass and volume relationships when the slurry matches the assumptions of the selected model.

The Minerallum calculator applies those relationships only to user-entered values. It does not use Minerallum product properties, supplier databases or proprietary standards tables.

Direct answer

Let:

ρs = solid-particle density;
ρl = carrier-liquid density;
ρm = slurry density;
w = solids mass fraction;
φ = solids volume fraction.

When w is known:

\rho_m= \frac{1}{ \frac{w}{\rho_s} + \frac{1-w}{\rho_l} }

Solids volume fraction then is:

\phi= \frac{w\rho_m}{\rho_s}

When slurry density is known:

w= \frac{\rho_s(\rho_m-\rho_l)} {\rho_m(\rho_s-\rho_l)}

provided:

\rho_s\ne\rho_l

and the entered mixture density is physically compatible with the two phase densities.

Derivation

Take one unit of total slurry mass.

Solids mass:

w

Liquid mass:

1-w

Solid volume:

V_s=\frac{w}{\rho_s}

Liquid volume:

V_l=\frac{1-w}{\rho_l}

For the two-phase additive-volume model:

V_m=V_s+V_l

therefore:

V_m= \frac{w}{\rho_s} + \frac{1-w}{\rho_l}

Density is mass divided by volume. Because total mass was chosen as one unit:

\rho_m= \frac{1}{ \frac{w}{\rho_s} + \frac{1-w}{\rho_l} }

This relationship is part of classical mineral-processing slurry calculations, including the treatment in Wills’ Mineral Processing Technology.

Mass percent and volume percent are not interchangeable

“40% solids” is incomplete unless the basis is given.

Solids by mass

w=\frac{m_s}{m_s+m_l}

40 wt% means 40% of the total slurry mass is dry solids.

Solids by volume

\phi=\frac{V_s}{V_s+V_l}

40 vol% means 40% of total mixture volume is occupied by solid particles.

The two percentages are usually different when the phase densities differ.

Volume fraction from density

The mixture density can also be written:

\rho_m= \phi\rho_s+(1-\phi)\rho_l

so:

\phi= \frac{\rho_m-\rho_l} {\rho_s-\rho_l}

Peer-reviewed slurry-density research uses this volume-fraction weighted relationship.

Mass fraction from density

Since:

w= \frac{\phi\rho_s}{\rho_m}

substituting the expression for φ gives:

w= \frac{\rho_s(\rho_m-\rho_l)} {\rho_m(\rho_s-\rho_l)}

This is useful when slurry density is measured and representative phase densities are known.

Which solids density should be entered?

The calculation needs the density connecting solid mass to the actual particle volume represented in the slurry.

Do not enter loose or tapped bulk density.

Bulk density includes inter-particle void space and depends strongly on packing.

EPA technical material explicitly distinguishes particle density from bulk density.

For porous or absorptive particles, the appropriate density basis can depend on the measurement method and on which pores are accessible to the carrier liquid. The calculator cannot resolve that material-specific question automatically.

Why the user must enter carrier-liquid density

Some mineral-processing examples approximate water as 1000 kg/m³.

Minerallum deliberately does not hard-code that value.

Actual carrier-liquid density may depend on:

temperature;
dissolved salts;
dissolved reagents;
use of a non-water carrier.

If dissolved species are present, their contribution should be reflected in the liquid-density input.

Mass flows from volumetric slurry flow

When volumetric slurry flow Q is known:

\dot m_m=Q\rho_m

Dry-solids flow:

\dot m_s=Q\rho_mw

equivalently:

\dot m_s=Q\phi\rho_s

Liquid mass flow:

\dot m_l=Q\rho_m(1-w)

These equations convert volumetric slurry flow into dry-solids and liquid throughput when the input density and concentration are representative.

Dilution ratio

Liquid mass per unit dry-solids mass is:

D=\frac{1-w}{w}

At 40 wt% solids:

D=\frac{0.60}{0.40}=1.5

When w=0, the ratio is not finite and should be shown as undefined.

Physical consistency check

For a simple two-phase mixture:

\min(\rho_s,\rho_l) \le \rho_m \le \max(\rho_s,\rho_l)

If the entered slurry density falls outside this interval, the three density values are not compatible with the locked model.

Minerallum should return an error rather than force a concentration into the 0–100% range.

What if solid and liquid densities are identical?

If:

\rho_s=\rho_l

mixture density is independent of composition.

Slurry density alone therefore cannot identify solids concentration.

The tool should explicitly report that the inverse problem is not identifiable.

Entrained air is a major limitation

The calculator contains two phases:

solid + liquid

It does not include gas.

Entrained air lowers measured mixture density. Wills’ Mineral Processing Technology specifically notes that air bubbles can cause density and flow errors in some mineral-processing pulps, particularly after flotation.

With a gas phase:

\rho= \phi_s\rho_s+ \phi_l\rho_l+ \phi_g\rho_g

The v1 calculator does not solve this three-phase problem.

A substantially aerated density measurement should therefore not be inserted blindly into the two-phase model.

Slurry may not be homogeneous

Particles can segregate or settle.

PNNL studies of slurry mixing and transport show that particle and density distributions can create concentration stratification when mixing is insufficient.

The calculator assumes the entered values actually represent the stream or sample being analysed.

It does not correct sampling error.

What the Minerallum tool calculates

It can calculate:

slurry density;
solids wt%;
solids vol%;
liquid fractions;
dilution ratio;
dry-solids and liquid mass flows when Q is supplied.

What it does not calculate

It does not calculate:

viscosity;
rheology;
settling velocity;
pressure drop;
critical transport velocity;
pump head or power;
NPSH;
cavitation;
pipe size;
erosion;
suspension stability;
process safety.

Those require additional properties and models.

Privacy

The calculator is designed to run locally in the browser.

Technical inputs should not be sent to analytics, APIs or Minerallum servers.

The contact link should not automatically transmit densities, concentrations or flow rates.

Quick example

For:

ρs = 2650 kg/m³;
ρl = 1000 kg/m³;
w = 40 wt%;

\rho_m= \frac{1}{ 0.40/2650+0.60/1000 }

which gives approximately:

1331.6583\ kg/m^3

Then:

\phi= \frac{0.40\times1331.6583}{2650}

giving approximately:

20.1005\%

The companion worked-example article closes all mass and volume flows for Q = 100 m³/h.

References

Wills, B. A.; Finch, J. Wills’ Mineral Processing Technology, 8th edition, Chapter 3. https://doi.org/10.1016/C2010-0-65478-2
Scelzo, M. T.; Eneren, P.; Sakamoto, Y.; Peveroni, L. (2021). Design and validation of a capacitance-based sensor for slurry density measurement. Experimental Thermal and Fluid Science, 122, 110299. https://doi.org/10.1016/j.expthermflusci.2020.110299
Bamberger, J. A.; Greenwood, M. S. (2004). Using ultrasonic attenuation to monitor slurry mixing in real time. Ultrasonics, 42, 145–148. https://doi.org/10.1016/j.ultras.2004.02.016
Bamberger, J. A.; Greenwood, M. S. (2004). Measuring fluid and slurry density and solids concentration non-invasively. Ultrasonics, 42, 563–567. https://doi.org/10.1016/j.ultras.2004.01.032
U.S. EPA. Fundamentals of Soil Science as Applicable to Management of Hazardous Wastes - particle-density versus bulk-density definitions.

Key point

The tool exposes:

inputs → equations → substitutions → result

so the user can independently verify the calculation.

Slurry density and solids concentration: converting mass %, volume % and density

Direct answer

Derivation

Mass percent and volume percent are not interchangeable

Solids by mass

Solids by volume

Volume fraction from density

Mass fraction from density

Which solids density should be entered?

Why the user must enter carrier-liquid density

Mass flows from volumetric slurry flow

Dilution ratio

Physical consistency check

What if solid and liquid densities are identical?

Entrained air is a major limitation

Slurry may not be homogeneous

What the Minerallum tool calculates

What it does not calculate

Privacy

Quick example

References

Key point

Sources and references

Wills? Mineral Processing Technology, Chapter 3

Scelzo et al. (2021), Slurry density measurement

Bamberger & Greenwood (2004), Slurry mixing

Bamberger & Greenwood (2004), Density and concentration measurement

Fundamentals of Soil Science: particle density and bulk density

Direct answer

Derivation

Mass percent and volume percent are not interchangeable

Solids by mass

Solids by volume

Volume fraction from density

Mass fraction from density

Which solids density should be entered?

Why the user must enter carrier-liquid density

Mass flows from volumetric slurry flow

Dilution ratio

Physical consistency check

What if solid and liquid densities are identical?

Entrained air is a major limitation

Slurry may not be homogeneous

What the Minerallum tool calculates

What it does not calculate

Privacy

Quick example

References

Key point

Sources and references

Wills? Mineral Processing Technology, Chapter 3

Scelzo et al. (2021), Slurry density measurement

Bamberger & Greenwood (2004), Slurry mixing

Bamberger & Greenwood (2004), Density and concentration measurement

Fundamentals of Soil Science: particle density and bulk density

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