The drive calculation for a ball mill jar involves determining the necessary motor power, speed, and torque to rotate the jar efficiently while ensuring proper grinding action. Below is a step-by-step guide:
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1. Critical Speed Calculation
The ball mill must operate below its critical speed (the speed at which centrifugal force prevents grinding media from cascading).
\[
N_c = \frac{42.3}{\sqrt{D – d}}
\]
Where:
– \( N_c \) = Critical speed (RPM)
– \( D \) = Internal diameter of the jar (meters)
– \( d \) = Diameter of grinding balls (meters)
For effective grinding, operate at 60–75% of critical speed.
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2. Required Motor Power
The power required depends on:
– Jar size
– Mass of grinding media & material
– Friction losses
# (a) Empirical Formula (Bond’s Law)
For dry grinding:
\[
P = 10.6 \cdot C \cdot G \cdot D^{0.3} \cdot (1 – 0.937J) \cdot \phi_c \cdot N
\]
Where:
– \( P \) = Power (kW)
– \( C \) = Empirical constant (~0.2–0.5 for lab mills)
– \( G \) = Mass of grinding media + material (kg)
– \( J \) = Fractional filling ratio (~0.3–0.5)
– \( \phi_c \) = Fraction of critical speed (~0.6–0.75)
– \( N \) = Mill speed (RPM)
# (b) Simplified Power Estimation
For small-scale mills:
\[
P ≈ 0.05 \, \text{to} \, 0.5 \, \text{kW per liter capacity}
\]
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3. Torque Calculation
Torque (\( T \)) required to overcome friction & inertia:
\[
T = F_r \cdot R + I\alpha
\]
Where:
– \( F_r \) = Frictional force (~10–20% of total load weight)
– \( R \) = Jar radius (m)
– \( I \) = Moment of inertia (\( I ≈ mR^2 \))
– \( α \) = Angular acceleration (\( α ≈ Δω/Δt \))
Alternatively, torque can be estimated as:
\[
T ≈ P / ω
\]
Where:
– \( ω \) = Angular velocity (\( ω ≈ 2πN/60 \, rad/s\))
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4. Motor Selection Criteria
Choose a motor with:
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