Introduction & Context

In process engineering, Particle Size Distribution (PSD) analysis quantifies the range and frequency of particle diameters in a bulk solid or slurry. A statistically valid sampling plan is required because the measured PSD is only an estimate of the true population; without enough samples the estimate may mis-classify fines or coarse fractions, leading to incorrect cyclone cut-point settings, mill over-grinding, or off-spec product. The worksheet below calculates the minimum number of grab samples (n) that must be collected to keep the confidence interval of the mean diameter within a user-specified margin of error (E). It also converts the time between successive grabs into an hourly sampling frequency so that field teams can schedule the collection campaign.

Methodology & Formulas

Step 1 – Define the precision target
The accepted error in the mean particle size is set by the engineer. A smaller E tightens precision but increases n.
Step 2 – Estimate population variability
The standard deviation of the particle size, σ, is obtained from historical data or a pilot study. It must be positive.
Step 3 – Compute required sample size
For a two-sided confidence interval at level 1−α, the minimum sample count is \[ n = \frac{Z^{2}\sigma^{2}}{E^{2}} \] where Z is the critical value of the standard normal distribution corresponding to the chosen confidence level.
Step 4 – Convert time interval to sampling frequency
If successive samples are separated by Δt minutes, the hourly rate is \[ f = \frac{60}{\Delta t} \quad \text{[samples h}^{-1}\text{]} \]

Input validity criteria
Parameter	Condition	Remark
σ	> 0	Population standard deviation must be positive
E	> 0 and < σ	Margin of error must be positive and smaller than variability
Δt	> 0	Time between grabs must be positive

Collect a sample at least every 4–6 hours during steady operation and every 30–60 minutes during grade changes or upsets. Tie the frequency to your control-loop dead time: if the loop responds in 20 minutes, sample at half that interval. Store results in 30-minute rolling averages to smooth noise while still detecting true shifts within two control actions.

Install an isokinetic sampler in a straight pipe run at least 5 pipe diameters downstream of elbows or valves.
Position the probe tip at one-third of the radius from the wall where velocity profile is flattest.
Use a 360° slot sampler for pneumatic lines; use a vertical gravity drop with a cutter for belt or chute discharge.
Flush the sample line for 30 seconds before collecting the test portion to clear settled fines.

Choose online when you need <1% variation detection within 5 minutes; it pays for itself in 6–9 months on variable-grade lines. Use lab analysis if your target D50 changes less than 5 µm per day or if the stream temperature exceeds 80 °C and would foul the optical window. Hybrid: run online for control and send a weekly grab sample to the lab for calibration drift checks.

For laser diffraction, 2 g of powder with 1 g cm⁻³ density is enough; for sieve stacks, use 100 g minimum to keep the largest particle below 1% of total mass. When top size exceeds 1 mm, apply the “10% rule”: sample mass (g) ≥ 10 × (top size mm)². Always split with a rotary riffling divider to 1.5× the required mass to allow for repeat tests.

Worked Example: Determining the Required Number of Averages for PSD Analysis

A process engineer needs to estimate the power spectral density (PSD) of pressure fluctuations in a reactor. The engineer wants the estimate to be within a ±5.0% relative error at a 95% confidence level. The standard deviation of the measured signal is known to be 10.0 Pa. The measurement system records data at a sampling frequency of 2 Hz over a total observation window of 30.0 seconds.

σ (standard deviation): 10.0 Pa
E (desired relative error): 5.0% (0.05 as a fraction)
Z (confidence factor for 95% confidence): 1.96
Δt (total observation time): 30.0 s
f_s (sampling frequency): 2.0 Hz

Calculate the required number of independent averages (n) using the formula \[ n = \left(\frac{Z \, \sigma}{E \, \mu}\right)^{2} \] where μ is the mean value of the signal. For a relative error specification, the mean cancels, giving \[ n = \left(\frac{Z \, \sigma}{E}\right)^{2} \] Substituting the known values: \[ n = \left(\frac{1.96 \times 10.0}{5.0}\right)^{2} = (3.92)^{2} = 15.366 \] Rounded to three decimal places, n = 15.366.
Determine the total number of samples collected during the observation window: \[ N_{\text{samples}} = f_{s} \times \Delta t = 2.0 \,\text{Hz} \times 30.0 \,\text{s} = 60.0 \] Rounded, N = 60 samples.
Check that the available samples are sufficient to achieve the required number of averages. Each average typically uses a segment of the data; assuming non-overlapping segments, the maximum number of independent averages is \[ n_{\max} = \frac{N_{\text{samples}}}{\text{samples per segment}} \] If one segment contains the entire record (30 s), then only one average is possible, so the engineer must either increase the observation time or accept a larger error. For illustration, if the engineer splits the record into 5-second segments (10 samples per segment), the achievable number of averages is \[ n_{\max} = \frac{60}{10} = 6 \] Since 6 < 15.366, the current setup does not meet the desired confidence.
Adjust the observation time to meet the requirement. Required total samples: \[ N_{\text{required}} = n \times \text{samples per segment} \] Using 10 samples per segment: \[ N_{\text{required}} = 15.366 \times 10 = 153.66 \] Required observation time: \[ \Delta t_{\text{required}} = \frac{N_{\text{required}}}{f_{s}} = \frac{153.66}{2.0} = 76.830 \,\text{s} \] Rounded, the engineer needs 76.830 seconds of data.

Final Answer: To achieve a ±5% relative error at 95% confidence, the PSD analysis requires at least 15.366 independent averages. With a sampling frequency of 2 Hz, this translates to a minimum observation time of 76.830 seconds (≈ 154 samples). The original 30-second record (60 samples) is insufficient.

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