Introduction & Context
The Crystal Size Distribution (CSD) analysis quantifies the size spectrum of dry granular solids such as granulated sugar, powders, or crystals. Accurate CSD data are essential for process engineers because particle size influences flowability, packing density, dissolution rate, and downstream equipment performance (e.g., hoppers, conveyors, reactors). Mechanical sieve analysis is the most common laboratory technique for obtaining a mass‑weighted size distribution, from which key percentiles (D₁₀, D₅₀, D₉₀) and the coefficient of variation (CV) are derived. These metrics are routinely used in specification compliance, process control, and scale‑up studies.
Methodology & Formulas
1. Data Preparation
For each sieve size \(d_i\) (ordered from largest opening to smallest) the retained mass \(m_i\) is measured. The total sample mass is denoted \(M_{\text{total}}\).
Mass fraction for each sieve:
\[ w_i = \frac{m_i}{M_{\text{total}}} \]Cumulative undersize (percent retained finer than or equal to \(d_i\)) is obtained by summing mass fractions from the finest sieve upward:
\[ F(d_i) = \left( \sum_{k=i}^{N} w_k \right) \times 100 \]where \(N\) is the number of sieves (including the pan).
2. Validity Checks (implemented in the Python routine)
| Check | Mathematical Condition | Action on Failure |
|---|---|---|
| Total mass fraction | \( \left| \sum_{i=1}^{N} w_i - 1 \right| \le 10^{-3} \) | Raise error – sample mass mismatch |
| Sieve size range | \( d_i \in [d_{\min}, d_{\max}] \) with \( d_{\min}=20\;\mu\text{m},\; d_{\max}=125\,000\;\mu\text{m} \) | Raise error – size outside empirical validity |
| Monotonicity of cumulative distribution | \( F(d_{i}) \ge F(d_{i+1}) \;\; \forall i \) | Raise error – non‑monotonic curve |
3. Percentile Determination (Linear Interpolation)
For a target percentile \(P\) (e.g., 10, 50, 90 %), locate the two adjacent sieves that bound \(P\): lower sieve size \(d_{\ell}\) with cumulative undersize \(F_{\ell}\) and upper sieve size \(d_{u}\) with cumulative undersize \(F_{u}\). The interpolated particle size \(D_{P}\) is
\[ D_{P} = d_{\ell} + (d_{u} - d_{\ell}) \frac{P - F_{\ell}}{F_{u} - F_{\ell}} \]Applying this formula yields the standard percentiles:
- \( D_{10} \) – 10 % undersize
- \( D_{16} \) – 16 % undersize (used for CV)
- \( D_{50} \) – median (50 % undersize)
- \( D_{84} \) – 84 % undersize (used for CV)
- \( D_{90} \) – 90 % undersize
4. Coefficient of Variation (CV)
The CV expresses the spread of the distribution relative to the median size:
\[ CV = \frac{D_{84} - D_{16}}{2\,D_{50}} \times 100\% \]5. Summary of Computational Steps (mirroring the Python code)
- Compute each \(w_i\) from measured \(m_i\) and \(M_{\text{total}}\).
- Generate \(F(d_i)\) by cumulative summation from the finest sieve upward and multiply by 100.
- Perform the three validity checks listed above; abort if any fail.
- For each required percentile (10, 16, 50, 84, 90) invoke the interpolation formula to obtain \(D_{P}\).
- Calculate CV using the interpolated \(D_{16}\), \(D_{50}\), and \(D_{84}\).
- Round results to the desired precision (commonly three decimal places for micrometer values and one decimal place for percent).
6. Typical Interpretation
In industrial crystallization, a CV < 30 % denotes a narrow distribution, 30–50 % a moderate spread, and > 50 % a broad distribution. The percentile values are compared against product specifications (e.g., D50 = 500 µm, D90 < 900 µm, D10 > 200 µm) to assess compliance.