Model Reference / Known Product P80 Comminution

Known Product P80 Comminution

Summary

The Known Product P80 Comminution model estimates the required power and product size distribution for a user-defined target product P80. The energy calculation is based on the Bond energy-size relationship originally proposed by Bond in “The third theory of comminution”, Transactions AIME, 1952. The product size distribution is reconstructed using a Rosin-Rammler-type cumulative passing curve.

The model should be used when the target product P80 is known and the simulation must estimate the corresponding power requirement and product PSD from the feed P80, component-specific Work Index values, product sharpness and fines parameters.

In this DPSIM implementation, the model first calculates the specific energy required to reach the target global product P80. It then calculates a component-specific product P80 using each component Work Index and reconstructs each component product PSD using the specified product sharpness and fines fraction.

DPSIM model key: DPSIM.Comminution.KnownP80
Category: Comminution
Subcategory: Energy
Display name: Target P80

Parameters

#	Parameter	Description
1	Product PSD Sharpness	Sharpness exponent used in the Rosin-Rammler-type product PSD reconstruction. Higher values generate a steeper product size distribution.
2	Product PSD P80	Target product P80 used to calculate the required specific energy.
3	Product PSD %Fines	Fines or bypass fraction added to the reconstructed product passing curve.
4	[Component] WI(kWh/short ton)	Component-specific Bond Work Index. The implementation converts this value to metric-ton energy basis before using it with kWh/t.

Derived parameters

#	Derived parameter	Description
1	Required Power(kW)	Calculated power required to reach the target product P80 at the current feed solids flowrate.

Model Description

The Known Product P80 Comminution model receives one feed stream and generates one product stream. The product stream preserves the feed solids flowrate and water flowrate. The model recalculates the product retained size distribution and component-by-size matrix.

The average feed Work Index is calculated from the component masses in the feed:

$W_{i,avg} = \frac{sum_{c}M_{c}^{F}W_{i,c}}{M_{S}^{F}}$

Where:

Symbol	Description	Unit
$W_{i,avg}$	Feed-mass-weighted average Work Index.	kWh/short ton
$M_{c}^{F}$	Feed solids mass flowrate of component c.	tph
$W_{i,c}$	Bond Work Index of component c.	kWh/short ton
$M_{S}^{F}$	Total feed dry solids flowrate.	tph

The Bond specific energy required to reach the target product P80 is calculated as:

$E_{g} = 10\ W_{i,avg}1.102\ \left( \frac{1}{sqrt\left( P_{80,target} \right)} - \frac{1}{sqrt\left( F_{80}^{F} \right)} \right)$

If the calculated value is negative, the implementation sets it to zero:

$E_{g} = \max\left( 0,E_{g} \right)$

Where:

Symbol	Description	Unit
E_g	Specific energy required by the model.	kWh/t
$P_{80,target}$	User-defined target product P80.	µm
$F_{80}^{F}$	Feed stream P80.	µm
1.102	Conversion factor from kWh/short ton to kWh/t metric basis.	dimensionless

The required power is then calculated as:

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

Where:

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

For each component c, the component energy-size constant is calculated as:

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

The component product P80 is then calculated from the Bond energy relationship:

$P_{80,c} = \left( F_{80,c}^{- \frac{1}{2}} + \frac{E_{g}}{C_{c}} \right)^{- 2}$

Where:

Symbol	Description	Unit
C_c	Component energy-size constant used by the implementation.	model unit
$F_{80,c}$	Feed P80 of component c.	µm
$P_{80,c}$	Calculated product P80 of component c.	µm

After the component product P80 is calculated, the model reconstructs the component product passing curve as:

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

Where:

Symbol	Description	Unit
$Y_{c,i}$	Cumulative passing fraction of component c at size class i.	fraction
b	Product fines fraction.	fraction
d_i	Size opening of class i.	µm
$P_{80,c}$	Calculated product P80 of component c.	µm
s	Product PSD sharpness.	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:

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

Any positive residual component mass not assigned by the retained curve is added to the last size interval to close the component mass balance.

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:

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

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
$r_{c,i}$	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
$m_{i}^{P}$	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 Work Index parameter is entered in kWh/short ton. The implementation multiplies this value by 1.102 to convert it to metric-ton energy basis before using it with specific energy in kWh/t.

If the target product P80 is coarser than the feed P80, the calculated Bond energy may become negative. In this case, the implementation sets the energy to zero.

The model calculates the energy required to reach a target global product P80. In multi-component simulations, the final product P80 may not match the target exactly because each component responds according to its own Work Index and feed P80.

The complete product PSD is reconstructed using a Rosin-Rammler-type curve from the calculated component P80, product sharpness and fines fraction. Therefore, the detailed shape of the PSD is empirical and should be calibrated against measured product size distributions whenever possible.