Introduction & Context

The Rosin–Rammler distribution is the most widely used particle-size model in comminution, spray drying, pneumatic classification and other particulate processes. It predicts the cumulative mass fraction R that is retained on a sieve of aperture x. From two measured retention points (R₁, x₁) and (R₂, x₂) the two empirical parameters—n (uniformity coefficient) and x′ (characteristic size)—are extracted. These parameters feed directly into mill-scale-up, burner design, cyclone cut-size calculations and quality-control specifications.

Methodology & Formulas

Linearise the Rosin–Rammler equation
The cumulative retained mass fraction is
\[ R(x) = 1 - \exp\!\left[-\left(\frac{x}{x'}\right)^{n}\right] \]
Re-arranging and taking logarithms twice gives the straight-line form
\[ \ln\!\bigl(-\ln(1-R)\bigr) = n\ln x - n\ln x' \]
which is solved from two data points.
Build simultaneous equations
For each measured point i = 1, 2 define
\[ y_i = \ln\!\bigl(-\ln(1-R_i)\bigr), \quad X_i = \ln x_i \]
The two equations
\[ y_1 = nX_1 - n\ln x' \]
\[ y_2 = nX_2 - n\ln x' \]
are solved explicitly for the slope n and intercept -n ln x'.
Explicit solution
Uniformity coefficient
\[ n = \frac{y_1 - y_2}{X_1 - X_2} \]
Characteristic size
\[ x' = x_1 \Big/\!\left(-\ln(1-R_1)\right)^{1/n} \]

Empirical validity window for crushed and ground materials
Parameter	Lower limit	Upper limit
n	0.70	3.0
x′ / µm	15	2000

The Rosin-Rammler equation is F(D) = 1 – exp[–(D/D₅₀)ⁿ].

D₅₀ is the characteristic diameter at which 63.2% of the total mass is below that size; it sets the horizontal position of the curve.
n is the distribution uniformity index; higher n values steepen the curve (narrower spread), while lower n values flatten it (wider spread).

Export the cumulative mass-undersize data.
Transform to linear form: ln[–ln(1 – F)] = n ln(D) – n ln(D₅₀).
Perform linear regression of ln[–ln(1 – F)] vs ln(D); slope = n, intercept = –n ln(D₅₀).
Back-calculate D₅₀ = exp(–intercept / n).

Increase atomizer wheel speed or nozzle pressure to raise n and reduce fines.
Raise inlet temperature to speed droplet drying, minimizing secondary fragmentation.
Reduce feed solids to shift D₅₀ downward while maintaining n.
Check for cyclone or bag-house leakage that may add extraneous fines to the sample.

Yes. Use the closed-form moments for a Rosin-Rammler distribution:

D[3,2] = D₅₀ Γ(1 + 1/n) / Γ(1 – 1/n) (Sauter mean)
D[4,3] = D₅₀ Γ(1 + 2/n) / Γ(1 + 1/n) (De Brouckère mean)

where Γ is the gamma function. Most spreadsheet packages provide Γ(x) directly.

Worked Example – Determining Rosin-Rammler Parameters from Two Sieve Tests

A pneumatic conveying line is being designed to transport ground limestone. To specify the cyclone efficiency, the particle-size distribution is required. Plant data from a laser-diffraction analyser give two reliable points on the cumulative undersize curve:

90% of the solids are finer than 850 µm (i.e., R = 0.90 retained on that sieve).
30% of the solids are finer than 212 µm (i.e., R = 0.30 retained on that sieve).

Fit the Rosin-Rammler equation to these points and report the uniformity coefficient n and the size constant x′.

Convert the retained mass fractions to cumulative oversize fractions:
\( R_1 = 0.900 \) at \( x_1 = 850\ \mu m \)
\( R_2 = 0.300 \) at \( x_2 = 212\ \mu m \)
Compute the double-logarithmic terms required by the Rosin-Rammler linearisation:
\( \ln[-\ln(R)] \)
For point 1: \( \ln[-\ln(0.900)] = 0.834 \)
For point 2: \( \ln[-\ln(0.300)] = -1.031 \)
Take natural logarithms of the corresponding sizes:
\( \ln x_1 = \ln(850) = 6.745 \)
\( \ln x_2 = \ln(212) = 5.357 \)
Evaluate the slope of the Rosin-Rammler straight line (uniformity coefficient n):
\[ n = \frac{\ln[-\ln R_1] - \ln[-\ln R_2]}{\ln x_1 - \ln x_2} \]
\[ n = \frac{0.834 - (-1.031)}{6.745 - 5.357} = \frac{1.865}{1.388} = 1.343 \]
Back-substitute to find the size constant x′ (the size at which 36.8% is retained):
Using point 1:
\[ x' = x_1 \cdot \exp\left( \frac{-\ln[-\ln R_1]}{n} \right) \]
\[ x' = 850 \cdot \exp\left( \frac{-0.834}{1.343} \right) = 850 \cdot 0.538 = 457\ \mu m \]

Final Answer: The limestone follows a Rosin-Rammler distribution with uniformity coefficient n = 1.343 and size constant x′ = 457 µm.

"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