A scale-up model for High-Pressure Grinding Rolls (HPGR) in hard rock grinding involves predicting the performance of larger HPGR units based on data from smaller pilot-scale or laboratory tests. The goal is to ensure that the full-scale HPGR achieves similar efficiency, product size distribution, and throughput as the smaller units. Below is a structured approach to developing a reliable scale-up model:
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1. Key Parameters for HPGR Scale-Up
The following parameters must be considered when scaling up an HPGR for hard rock grinding:
– Operating Pressure (P) – Specific grinding force (N/mm²)
– Roll Speed (v) – Peripheral roll speed (m/s)
– Gap Size (S) – Minimum distance between rolls (mm)
– Feed Size Distribution (F80) – 80% passing size of feed material
– Product Size Distribution (P80) – 80% passing size of product
– Throughput Capacity (Q) – Tonnage per hour (t/h)
– Power Draw (W) – Energy consumption (kWh/t)
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2. Empirical and Phenomenological Scale-Up Models
Several models exist for scaling up HPGR performance:
# (a) Bond’s Law-Based Model
– Relates energy consumption to particle size reduction.
– Modified Bond work index can be used for HPGR.
\[
E = W_i \left( \frac{10}{\sqrt{P_{80}}} – \frac{10}{\sqrt{F_{80}}} \right)
\]
where \(E\) = specific energy consumption, \(W_i\) = Bond work index.
# (b) Population Balance Model (PBM)
– Uses breakage distribution and selection functions.
– Requires experimental data from pilot tests.
\[
P(x) = \int_0^x B(x,y) S(y) F(y) \, dy
\]
where:
– \(P(x)\) = product size distribution,
– \(B(x,y)\) = breakage function,
– \(S(y)\) = selection function,
– \(F(y)\) = feed size distribution.
# (c) Morrell’s Power Model
– Estimates power draw based on roll dimensions and operating conditions.
\[
W = C \cdot D^{2.5} \cdot L \cdot v \cdot P
\]
where:
– \(C\) = empirical constant,
– \(D\) = roll diameter,
– \(L\) = roll length,
– \(v\) = roll speed,