Introduction & Context

A particle-size distribution (PSD) is log-normal when the logarithm of the diameter is normally distributed. In spray drying, milling, crystallisation and classification, the median diameter \(x_{50}\) and the geometric standard deviation \(\sigma_g\) are the two parameters that completely describe the distribution. Knowing these parameters lets engineers predict:

heat- and mass-transfer surface area in dryers and reactors;
settling or elutriation cut sizes in cyclones;
packing density and flowability of powders;
sensor calibration for laser-diffraction instruments.

The conversion from the industrial descriptors (\(x_{50}\), \(\sigma_g\)) to the statistical descriptors (\(\mu\), \(\sigma\)) is required whenever population balances, Monte-Carlo simulations or maximum-likelihood fitting are performed.

Methodology & Formulas

Define the measurable inputs
\(x_{50}\): median diameter of the PSD, usually reported by instruments in µm.
\(\sigma_g\): geometric standard deviation, dimensionless, obtained from the ratio \[ \sigma_g = \frac{x_{84.13}}{x_{50}} = \frac{x_{50}}{x_{15.87}} \]
Convert to log-normal parameters
The natural logarithm of the diameters is normally distributed \( \mathcal{N}(\mu,\sigma) \), where \[ \mu = \ln x_{50} \] \[ \sigma = \ln \sigma_g \] These two scalars are the location and scale parameters of the underlying Gaussian.

Empirical range for dairy powders
Experience with spray-dried milk, whey and lactose shows that the log-spread \(\sigma\) is physically reasonable only inside the following window:

Parameter	Lower limit	Upper limit
\(\sigma = \ln\sigma_g\)	0.1	1.8

Values outside this range usually indicate agglomeration, fines removal or instrument artefacts.

Probability density function (PSD curve)
Once \(\mu\) and \(\sigma\) are known, the mass- or volume-density function is \[ f(x) = \frac{1}{x\sigma\sqrt{2\pi}}\exp\!\left[-\frac{(\ln x - \mu)^2}{2\sigma^2}\right] \] and the cumulative fraction undersize is \[ F(x) = \frac{1}{2}\left[1 + \mathrm{erf}\!\left(\frac{\ln x - \mu}{\sigma\sqrt{2}}\right)\right] \]

A log-normal fit smooths measurement noise, gives you two robust parameters (geometric mean D_g and geometric standard deviation σ_g) that are independent of bin width, and lets you calculate any percentile or moment analytically—handy for specs like Dv10, Dv50, Dv90 without re-running the full measurement.

Export the cumulative %Finer versus size data, convert to a probability plot on log-probit axes, and check linearity. If you see two straight segments or a clear tail, the sample is probably bimodal or has fines. Options:

Fit two log-normals and report each fraction by volume.
Truncate the tail at 99% and refit to get a single σ_g for the main mode.
Use a different model (e.g., Rosin-Rammler) only if the fit residual drops by >30%.

σ_g is very sensitive because the log transform stretches the fine end. Guard by:

Discarding channels below the optical model’s lower validity limit (usually 0.5% of the total volume).
Weighting each point by (V_i^0.5) in the non-linear regression to de-emphasize noisy tails.
Replicate measurements on the same slurry; if σ_g varies by >5%, clean the cell and re-measure.

Yes, compare directly—D_g and σ_g are model-based, not instrument constants. Make sure you use the same optical parameters (refractive index, absorption) for both runs. If the wet D_g is >10% finer, it usually means agglomerates broke up; if σ_g is wider, the dry run probably under-dispersed. Always report the dispersion method with the fit parameters.

Worked Example – Fitting a Log-Normal Distribution to a Ball-Mill Product PSD

A metallurgical plant has sieved the product from a ball-mill circuit and needs to describe the size distribution with a two-parameter log-normal model so the data can be fed to the flotation simulator. Only the median size x₅₀ and the geometric standard deviation σ_g were reported by the laboratory.

Median particle size, x₅₀ = 65 µm
Geometric standard deviation, σ_g = 1.6 (dimensionless)

The log-normal model uses the natural-log mean μ and log standard deviation σ. These are obtained from the known values as follows.

Convert x₅₀ to ln-space:
\[ \mu = \ln(x_{50}) = \ln(65) = 4.174 \]
Convert the geometric standard deviation:
\[ \sigma = \ln(\sigma_{g}) = \ln(1.6) = 0.470 \]
Normalisation constant for the probability density function:
\[ \frac{1}{x\sigma\sqrt{2\pi}} \quad \text{with} \quad \sqrt{2\pi}=2.507 \]

Final Answer: The log-normal distribution that represents the ball-mill product has ln-mean μ = 4.174 and ln-standard deviation σ = 0.470 (both dimensionless). These two parameters fully define the PSD model for further simulation work.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle