Model Reference / Specific Energy Comminution

Specific Energy Comminution

Summary

The Specific Energy Comminution model estimates the product size distribution from an imposed operating specific energy and component-specific Bond Work Index values. The model is based primarily on the Bond energy-size relationship originally proposed by Bond in “The third theory of comminution” Transactions AIME, 1952. The user-defined exponent allows the same generalized energy-size form to represent Bond-like behavior at n=1.5 and Rittinger-like behavior at n=2.0.

The model should be used when the comminution product size must be estimated from a known or assumed operating specific energy, rather than from available installed power or a fixed product P80.

In this DPSIM implementation, the model calculates a product P80 for each component, reconstructs each component product PSD using a Rosin-Rammler-like curve, and recombines the component products into the final product stream.

DPSIM model key: DPSIM.Comminution.SpecificEnergy
Category: Comminution
Subcategory: Energy
Display name: Specific Energy

Parameters

#	Parameter	Description
1	Operation specific energy (kWh/t)	Operating specific energy applied to the feed solids. This value is used directly in the energy-size equation and is also multiplied by the feed solids flowrate to calculate required power.
2	Model exponent (1.5: Bond, 2.0)	Exponent used in the generalized energy-size relationship. A value of 1.5 gives Bond-like behavior, while a value of 2.0 gives Rittinger-like behavior.
3	[Component] WI(kWh/short ton)	Component-specific Bond Work Index. The implementation converts this value to metric-ton energy basis before using it with the operating specific energy in kWh/t.

Derived parameters

#	Derived parameter	Description
1	Required Power(kW)	Calculated power required to apply the specified operating specific energy to the feed solids flowrate.

Model Description

The Specific Energy Comminution model receives one feed stream and generates one product stream. The product stream preserves the feed solids flowrate and water flowrate. The product size distribution is recalculated from the imposed specific energy.

The required power is calculated as:

$P_{req} = E_{g}M_{S}^{F}$

Where:

Symbol	Description	Unit
P_req	Required power calculated by the model.	kW
E_g	Operation specific energy.	kWh/t
$M_{S}^{F}$	Feed dry solids flowrate.	tph

For each component c, the model calculates the feed component P80:

$F_{80,c}$

The component Work Index is entered as kWh/short ton and converted internally to the metric-ton basis used by the specific energy:

$C_{c} = 10\ W_{i,c}1.102$

Where:

Symbol	Description	Unit
C_c	Component energy-size constant used by the implementation.	model unit
$W_{i,c}$	Bond Work Index of component c.	kWh/short ton

The product P80 for each component is calculated from the generalized energy-size relationship:

$E_{g} = C_{c\left( P_{80,c}^{1 - n} - F_{80,c}^{1 - n} \right)}$

Solving for product P80 gives:

$P_{80,c} = \left( F_{80,c}^{1 - n} + \frac{E_{g}}{C_{c}} \right)^{\frac{1}{1 - n}}$

Where:

Symbol	Description	Unit
E_g	Operation specific energy.	kWh/t
$F_{80,c}$	Feed P80 of component c.	µm
$P_{80,c}$	Calculated product P80 of component c.	µm
n	Model exponent.	dimensionless

After calculating the component product P80, the model reconstructs the component product passing curve using the feed Rosin-Rammler sharpness and feed fines bypass:

$Y_{c,i} = b_{c} + \left( 1 - b_{c} \right)\left( 1 - \exp\left( \ln{0.2}\left( \frac{d_{i}}{P_{80,c}} \right)_{c}^{s} \right) \right)$

Where:

Symbol	Description	Unit
$Y_{c,i}$	Cumulative passing fraction of component c at size class i.	fraction
b_c	Feed fines bypass fraction of component c.	fraction
d_i	Size opening of class i.	µm
$P_{80,c}$	Calculated product P80 of component c.	µm
s_c	Rosin-Rammler sharpness estimated from the feed component PSD.	dimensionless

The cumulative passing curve is converted to retained fractions in descending size order:

$r_{c,0} = 1 - Y_{c,0}$

$r_{c,i} = Y_{c,i - 1} - Y_{c,i}$

The component product retained mass in each size interval is then calculated as:

$m_{c,i}^{P} = M_{c}^{F}r_{c,i}$

Where:

Symbol	Description	Unit
$r_{c,i}$	Retained fraction of component c in size interval i.	fraction
$m_{c,i}^{P}$	Product retained mass flowrate of component c in size interval i.	tph
$M_{c}^{F}$	Feed solids mass flowrate of component c.	tph

The total product retained mass in each size interval is calculated by summing the component retained masses:

$m_{i}^{P} = sum_{c}m_{c,i}^{P}$

The product retained fraction is then:

$p_{i}^{P} = \frac{m_{i}^{P}}{M_{S}^{F}}$

The product component fraction in each size interval is:

$z_{c,i}^{P} = \frac{m_{c,i}^{P}}{m_{i}^{P}}$

Where:

Symbol	Description	Unit
$m_{i}^{P}$	Total product retained mass flowrate in size interval i.	tph
$p_{i}^{P}$	Product retained fraction in size interval i.	fraction
$z_{c,i}^{P}$	Fraction of component c in product size interval i.	fraction

The first retained interval is calculated as 1 minus the first cumulative passing value. Any remaining component mass not assigned by the retained curve is added to the last size interval to close the component mass balance.

The Work Index parameter is entered in kWh/short ton. The implementation multiplies this value by 1.102 to convert it to the metric-ton energy basis used with the operation specific energy in kWh/t.

The model preserves the feed solids flowrate and water flowrate. It recalculates only the product size distribution and the component-by-size matrix.

The model predicts product P80 from energy and then reconstructs the complete PSD with a Rosin-Rammler-like curve. Therefore, the detailed product PSD is an empirical reconstruction and should be calibrated against measured data whenever possible.

The model assumes that the specific energy is applied to all components according to the same operating energy input, while the component response is differentiated by the component Work Index and feed PSD.

The model exponent is limited to the range from 1.5 to 2.0. Values near 1.5 represent Bond-like behavior, while values near 2.0 represent Rittinger-like behavior.