Reference ID: MET-AE04 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Size-reduction equipment (mills, grinders, crushers) is first tested at laboratory scale to establish achievable throughput under controlled conditions.
To predict the full-scale production throughput, engineers apply a geometric scale-up law that relates the mass flow rate to a characteristic linear dimension of the unit.
The method is widely used in pharmaceutical, food, mineral, and chemical industries for budgetary quotes, motor sizing, and downstream line balancing.
Methodology & Formulas
Characteristic dimension
Let \(D_{\text{lab}}\) be the representative diameter of the laboratory unit and \(D_{\text{prod}}\) that of the production unit (both in metres).
Scale factor
The empirical scale factor \(SF\) is defined by
\[
SF = \left( \frac{D_{\text{prod}}}{D_{\text{lab}}} \right)^{n}
\]
where the exponent \(n\) is obtained from pilot-plant regressions; typical values range from 1.5 to 2.5 for size-reduction devices.
Throughput projection
The expected production throughput \(Q_{\text{prod}}\) (kg h−1) is then
\[
Q_{\text{prod}} = Q_{\text{lab}} \cdot SF
\]
with \(Q_{\text{lab}}\) the measured laboratory throughput.
Regime
Scale Factor Range
Interpretation / Action
Empirical
\(1 \le SF \le 10\)
Linear geometric scaling considered reliable.
Extrapolation
\(SF > 10\) or \(D_{\text{prod}} > 3\,D_{\text{lab}}\)
Non-linear effects (dead zones, classification efficiency) likely; pilot validation required.
Throughput typically scales with the inverse square root of the target size reduction ratio (F80/P80). For example, reducing feed from 1000 µm to 100 µm (ratio 10) yields about 3.2× lower throughput than 1000 µm to 316 µm (ratio 3.16). This relationship is driven by the energy required to create new surface area and the mill’s classification efficiency at finer sizes.
Vertical stirred mills (e.g., VXP, SMD) retain 70–85% of baseline throughput at 30 µm P80.
High-pressure grinding rolls (HPGR) followed by ball milling sustain 60–70% throughput versus single-stage ball mills at 40 µm P80.
IsaMill with ceramic media holds 90% throughput down to 20 µm P80, but requires 30–40% more specific energy.
Raise mill speed by 3–5% while monitoring bearing temperatures.
Increase ball charge by 1–2% volume to boost breakage probability.
Tighten recycle load from 250% to 300% to raise mill density and residence time.
Add 0.05% mill-feed moisture to reduce coating and improve slurry rheology.
Use the Bond throughput model: \(Q2/Q1 = (Wi1/Wi2) \times (P80\_1/P80\_2)^{0.5} \times (F80\_2/F80\_1)^{0.5}\). Insert the Bond work indices at both sizes (from plant data or literature) and solve for \(Q2\). Calibrate with two recent plant surveys; accuracy is ±8% for ball mills and ±12% for stirred mills.
Worked Example – Throughput Scaling for Size Reduction
A pilot-scale laboratory mill processes 100 kg h−1 of a material at a target particle size of 0.5 mm. The commercial plant must produce the same material at a larger target size of 1.5 mm. The scale-up follows the power-law relationship \(Q \propto D^{\,n}\) with an exponent \(n = 2\). A safety factor of 9 is specified for equipment sizing, and the laboratory operating size (0.5 mm) is already the safe limit.
Knowns
Laboratory throughput, \(Q_{\text{lab}} = 100\) kg h−1
Laboratory target size, \(D_{\text{lab}} = 0.5\) mm
Production target size, \(D_{\text{production}} = 1.5\) mm
Scale-up exponent, \(n = 2\)
Laboratory safe size, \(D_{\text{lab\_safe}} = 0.5\) mm
Safety factor, \(SF = 9\)
Desired production throughput, \(Q_{\text{production}} = 900\) kg h−1 (to be verified)
Scale the laboratory throughput to the production size (ignoring safety factor for the base estimate):
\[ Q_{\text{scaled}} = Q_{\text{lab}} \times 9 = 100 \times 9 = 900\ \text{kg h}^{-1} \]
Check the result against the target production throughput:
\(Q_{\text{scaled}} = 900\ \text{kg h}^{-1}\) matches the specified \(Q_{\text{production}}\).
Apply the safety factor to confirm equipment capacity (optional):
\[ Q_{\text{required\_capacity}} = Q_{\text{scaled}} \times SF = 900 \times 9 = 8\,100\ \text{kg h}^{-1} \]
This indicates that the plant equipment should be rated for at least 8,100 kg h−1 to provide the required safety margin.
Final Answer
The throughput required to meet the production size target is 900 kg h−1. Accounting for the safety factor, the equipment should be sized for a minimum capacity of 8,100 kg h−1.
"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
