Reference ID: MET-FC50 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Number Average Diameter (dn) is a fundamental statistical metric in process engineering used to characterize the central tendency of particle size distributions (PSD). In industrial applications, such as fermentation monitoring or reactor feed analysis, determining the dn is critical for assessing biological viability, surface area availability, and reaction kinetics. While reactor residence time distribution (RTD) models provide the framework for flow behavior, the dn calculation provides the geometric foundation required to translate mass-based sensor data into actionable population statistics.
Methodology & Formulas
The calculation of the number average diameter relies on the arithmetic mean of discrete size classes. The process involves calculating the weighted sum of diameters relative to the total particle count.
The primary formula for the number average diameter is defined as:
\[ d_n = \frac{\sum (n_i \cdot d_i)}{\sum n_i} \]
When converting from mass fraction (wi) to number fraction (ni), assuming constant density and spherical geometry, the relationship is defined as:
\[ n_i \propto \frac{w_i}{d_i^3} \]
To ensure the validity of the calculation, the following empirical constraints and thresholds must be applied:
Parameter
Constraint/Threshold
Sphericity (Aspect Ratio)
≤ 1.2
Minimum Diameter
≥ 0.1 µm
Maximum Diameter
≤ 100.0 µm
Input Integrity
Length of di must equal length of ni
Division Safety
Total count must be > 1e-9
The computational procedure follows these logical steps:
Calculate the weighted sum: weighted_sum = ∑ (ni · di)
Calculate the total count: total_count = ∑ ni
Compute the average: dn = weighted_sum / total_count
The number average diameter, often denoted as D10, is calculated by taking the arithmetic mean of the diameters of all particles in a sample. To perform this calculation, follow these steps:
Measure the individual diameter of each particle in the representative sample.
Sum the diameters of all individual particles.
Divide the total sum by the total number of particles counted.
The number average diameter is heavily weighted toward the smallest particles in a distribution because it treats every particle as having equal importance regardless of its volume or mass. Even a small number of sub-micron particles can significantly shift the average downward, which is why this metric is often used to characterize nucleation or the presence of fines in a process stream.
Process engineers should select the number average diameter when the specific application depends on the count of particles rather than their total mass or volume. Common scenarios include:
Monitoring the efficiency of filtration systems where particle count is the primary failure metric.
Assessing the potential for particle agglomeration in colloidal suspensions.
Evaluating the kinetics of crystal growth or dissolution where surface area and count are critical.
Worked Example: Number Average Diameter in Fermentation Broth Analysis
In a biotechnology process, a sample from a yeast fermentation broth is analyzed to assess cell population homogeneity using particle size data. The goal is to compute the number average diameter to evaluate biological viability. The particles are assumed spherical, and the sample is dilute to avoid coincidence errors.
Knowns:
Diameter of size class 1, \( d_1 = 4.000 \ \mu m \), with particle count \( n_1 = 60 \).
Diameter of size class 2, \( d_2 = 6.000 \ \mu m \), with particle count \( n_2 = 40 \).
Aspect ratio of particles: 1.050 (from data validation).
Maximum allowable aspect ratio for sphericity assumption: 1.200.
Empirical diameter bounds: minimum 0.100 \( \mu m \), maximum 100.000 \( \mu m \).
Step-by-Step Calculation:
Input Validation: Check sphericity and size bounds. Aspect ratio (1.050) ≤ 1.200, so no shape correction needed. All diameters are within 0.100 to 100.000 \( \mu m \).
Weighted Summation: Compute \( \sum (n_i \cdot d_i) \). From the data, the calculated weighted sum is 480.000.
Total Count: Compute \( \sum n_i \). From the data, the total particle count is 100.000.
Division for Average: Calculate the number average diameter using \( d_n = \frac{\sum (n_i \cdot d_i)}{\sum n_i} = \frac{480.000}{100.000} = 4.800 \ \mu m \).
Final Answer: The number average diameter \( d_n = 4.800 \ \mu m \).
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
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