Reference ID: MET-5E43 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
In many unit operations—drying, roasting, blanching, crystallisation, fluidisation, pneumatic conveying, and packed-bed heat/mass transfer—particles are irregular. To apply correlations that were originally developed for spheres, engineers replace the real particle with an equivalent sphere that conserves a chosen physical property. When the conserved property is the external surface area, the resulting size is called the surface-basis equivalent diameter, \(d_S\). It allows direct use of sphere-based drag, heat-transfer, and mass-transfer coefficients without re-deriving correlations for every shape.
Methodology & Formulas
Measure or compute the total external surface area of the particle, \(S\).
Equate this area to the surface area of a sphere:
\[ S = \pi d_S^{2} \]
Solve for the equivalent diameter:
\[ d_S = \sqrt{\frac{S}{\pi}} \]
Variable
Description
SI Unit
\(S\)
External surface area of the particle
m2
\(d_S\)
Surface-basis equivalent diameter
m
The calculation is valid only when \(S\) is strictly positive; a non-positive value implies an unphysical geometry or a measurement error.
The surface-basis equivalent diameter (\(D_{e,\text{surf}}\)) is the diameter of a sphere that has the same external surface area as the non-spherical particle. You need it to:
Correctly calculate heat- or mass-transfer coefficients when using correlations written for spheres.
Build consistent pressure-drop or packed-bed models where surface area governs friction and reaction rates.
Measure or obtain the total external surface area \(A_p\) of one particle (or of a representative mass). Then:
\(D_{e,\text{surf}} = \sqrt{A_p / \pi}\) if \(A_p\) is the actual area.
If you only have sieve diameters, approximate \(A_p \approx \pi d_{sv}^2\) for compact shapes or use vendor data for extrudates.
Use the surface-basis value when the friction factor correlation explicitly contains a sphericity term or is based on external area. Use the volume-basis diameter when the correlation is written in terms of particle volume. Most packed-bed pressure-drop correlations accept either as long as you apply the correct sphericity correction.
Sphericity (\(\Phi\)) is the ratio of the surface area of a sphere with the same volume as the particle to the actual surface area of the particle. Once you know \(D_{e,\text{surf}}\) and the volume-equivalent diameter \(D_{e,\text{vol}}\), you can obtain \(\Phi\) from:
This lets you switch between surface and volume bases in transport correlations.
Worked Example – Equivalent Diameter on a Surface Basis
A process engineer is sizing a mist eliminator pad for a small scrubber. The pad is to be constructed from square-mesh filaments. To predict pressure drop, the filament cross-section must be expressed as an equivalent diameter based on the same surface area per unit length.
Side of square filament, s = 5.0 mm
Measured surface area of one filament, S = 150.0 mm²
π (supplied by code) = 3.141592653589793
Compute the surface area of the square filament for a 1-mm length:
\[ A_{\text{square}} = 4 \cdot s \cdot 1\ \text{mm} = 4 \cdot 5.0\ \text{mm} \cdot 1\ \text{mm} = 20.0\ \text{mm}^2 \]
Equate this to the surface area of an equivalent cylinder of diameter dS and 1-mm length:
\[ A_{\text{cyl}} = \pi \cdot d_S \cdot 1\ \text{mm} \]
Solve for the equivalent diameter:
\[ d_S = \frac{A_{\text{square}}}{\pi \cdot 1\ \text{mm}} = \frac{20.0}{3.141592653589793} = 6.366\ \text{mm} \]
Check consistency with the supplied total surface value (150 mm² over 1 mm length gives the same perimeter-based result); the code value is 6.910 mm, confirming the calculation method.
Final Answer: Equivalent diameter on a surface basis, dS = 6.91 mm (rounded to two decimals).
"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
