Torres-Casali HPGR

Summary

The Torres-Casali HPGR model represents high-pressure grinding rolls using the phenomenological model proposed by Torres and Casali in “A novel approach for the modelling of high-pressure grinding rolls”, Minerals Engineering, 2009. The model predicts HPGR performance from equipment geometry, operating conditions, ore bulk density, breakage parameters and selection parameters.

The model is suitable for HPGR simulations where throughput, power draw, specific energy consumption and product size distribution must be estimated from roll dimensions, operating pressure, operating gap and peripheral velocity.

In this DPSIM implementation, the HPGR product size distribution is calculated by combining single-particle compression for particles above a critical size and particle-bed compression solved through a block-discretized population balance. The model calculates edge, centre and total product size distributions for diagnostic output.

DPSIM model key: DPSIM.Comminution.TorresCasaliHPGR
Category: Comminution
Subcategory: HPGR
Display name: Torres-Casali HPGR

Parameters

#	Parameter	Description
1	Number of HPGRs in parallel	Number of HPGR units operating in parallel. This parameter is used to calculate total installed throughput, but the model preserves the feed stream flowrate in the product stream.
2	Diameter (m)	Roll diameter.
3	Length (m)	Roll length.
4	Operating pressure (bar)	Hydraulic operating pressure applied to the floating roll.
5	Operating gap (m)	Operating gap at the extrusion zone.
6	Peripheral velocity (m/s)	Roll peripheral velocity.
7	Bulk density (t/m3)	Feed bulk density at the beginning of the particle-bed compression zone.
8	Ore density at the extrusion zone (t/m3)	Ore density at the extrusion zone.
9	Number of blocks	Number of roll-width blocks used to discretize the pressure profile and particle-bed compression zone.
10	Edge product fraction	Fraction of product assigned to the edge zone of the rolls.
11	Specific selection function SE1	Reference value of the specific selection function.
12	Selection f1 exponent	First exponent used in the specific selection function.
13	Selection f2 exponent	Second exponent used in the specific selection function.
14	[Component] B function a1	Component-specific breakage distribution parameter.
15	[Component] B function a2 exponent	Component-specific breakage distribution exponent.
16	[Component] B function a3 exponent	Component-specific breakage distribution exponent.

Derived parameters

#	Derived parameter	Description
1	Nip angle (deg)	Calculated inter-particle compression angle.
2	Throughput per HPGR (tph)	Theoretical throughput capacity of one HPGR unit.
3	Compression force (kN)	Compression force applied to the particle bed.
4	Power draw (kW)	Calculated power draw per HPGR unit.
5	Specific energy consumption (kWh/t)	Calculated specific energy consumption.
6	Height of the particle bed compression zone (m)	Calculated vertical height of the particle-bed compression zone.
7	Critical size (m)	Critical size separating single-particle compression from particle-bed compression.
8	Block holdup (t)	Calculated holdup in each roll-width block.

Model Description

The Torres-Casali HPGR model calculates the operating state of the HPGR from roll geometry, operating pressure, operating gap, peripheral velocity and density parameters. The model first calculates the inter-particle compression angle:

$\cos\alpha_{IP} = \frac{\left( s_{0} + D \right) + sqrt\left( \left( s_{0} + D \right)^{2} - \frac{4s_{0\delta D}}{\rho_{a}} \right)}{2D}$

Where:

Symbol	Description	Unit
α_IP	Inter-particle compression angle.	rad
s_0	Operating gap.	m
D	Roll diameter.	m
δ	Ore density at the extrusion zone.	t/m3
ρ_a	Feed bulk density.	t/m3

The throughput per HPGR unit is calculated as:

$G_{S} = 3600\delta s_{0LU}$

Where:

Symbol	Description	Unit
G_S	Throughput per HPGR unit.	tph
L	Roll length.	m
U	Roll peripheral velocity.	m/s

The total installed throughput is calculated as:

$G_{S},total = N_{parallel}G_{S}$

Where:

Symbol	Description	Unit
G_S,total	Total installed throughput capacity.	tph
N_parallel	Number of HPGR units in parallel.	dimensionless

The compression force is calculated from operating pressure and projected roll area:

$F = \frac{100R_{PDL}}{2}$

The power draw is calculated as:

$P = 2F\sin\left( \frac{\alpha_{IP}}{2} \right)U$

The specific energy consumption is:

$W = \frac{P}{G_{S}}$

Where:

Symbol	Description	Unit
F	Compression force.	kN
R_P	Operating pressure.	bar
P	Power draw per HPGR unit.	kW
W	Specific energy consumption.	kWh/t

The height of the particle-bed compression zone is:

$z^{*} = \frac{D}{2}\sin\alpha_{IP}$

The critical size separating single-particle compression from particle-bed compression is:

$x_{C} = s_{0} + D\left( 1 - \cos\alpha_{IP} \right)$

Particles with representative size greater than x_C are first sent to the single-particle compression stage. Particles with representative size less than or equal to x_C enter directly into the particle-bed compression stage.

The single-particle compression product is calculated using the component breakage matrix:

$p_{i}^{SP} = sum_{l = 1}^{N}b_{il}f_{l}^{SP}$

Where:

Symbol	Description	Unit
$p_{i}^{SP}$	Product retained fraction in size class i after single-particle compression.	fraction
$b_{il}$	Breakage distribution from parent class l to product class i.	fraction
$f_{l}^{SP}$	Feed retained fraction to the single-particle compression stage.	fraction

The single-particle compression product is combined with the material that did not enter the single-particle stage, forming the feed to the particle-bed compression stage:

$f_{i}^{IP} = f_{i}^{x \leq x_{C}} + p_{i}^{SP}$

The component cumulative breakage function is:

$B_{c\left( x;y_{j} \right)} = a_{(1,c)\left( \frac{x}{y_{j}} \right)_{2,c}^{a}} + \left( 1 - a_{1,c} \right)\left( \frac{x}{y_{j}} \right)_{3,c}^{a}$

Where:

Symbol	Description	Unit
B_c(x;y_j)	Cumulative breakage function for component c.	fraction
x	Product size boundary.	µm
y_j	Parent class reference size.	µm
$a_{1,c}$	Component-specific breakage distribution parameter.	dimensionless
$a_{2,c}$	Component-specific breakage distribution exponent.	dimensionless
$a_{3,c}$	Component-specific breakage distribution exponent.	dimensionless

The differential breakage distribution is obtained from the cumulative breakage function:

$b_{ij,c} = 0,\ i \leq j$

$b_{ij,c} = B_{c\left( D_{i - 1};y_{j} \right)} - B_{c\left( D_{i};y_{j} \right)},\ i > j$

The implementation normalizes each breakage column so that the daughter fractions from each parent class sum to one when breakage products exist.

The specific selection function is calculated as:

$\ln\left( \frac{S_{i}^{E}}{S_{1}^{E}} \right) = f_{1}\ln\left( \frac{x_{i}}{x_{1}} \right) + f_{2}{\ln\left( \frac{x_{i}}{x_{1}} \right)}^{2}$

or equivalently:

$S_{i}^{E} = S_{1}^{E}\exp\left( f_{1}\ln\left( \frac{x_{i}}{x_{1}} \right) + f_{2}{\ln\left( \frac{x_{i}}{x_{1}} \right)}^{2} \right)$

Where:

Symbol	Description	Unit
$S_{i}^{E}$	Specific selection function for size class i.	model unit
$S_{1}^{E}$	Reference specific selection function.	model unit
x_i	Representative size of class i.	µm
x_1	Representative size of the first active size class.	µm
f_1	Selection f1 exponent.	dimensionless
f_2	Selection f2 exponent.	dimensionless

The roll is discretized into N_B blocks along its width. The centre position of block k is:

$y_{k} = \frac{L}{\left( 2N_{B} \right)\left( 2k - N_{B} - 1 \right)}$

The power assigned to block k is calculated from the parabolic pressure profile:

$P_{k} = P\frac{L^{2} - 4y_{k}^{2}}{sum_{{j = 1}_{B}^{N}\left( L^{2} - 4y_{j}^{2} \right)}}$

The holdup in each block is:

$H_{k} = \frac{G_{S}z^{*}}{3600\ U\ N_{B}}$

The selection rate for each size class and block is:

$S_{i,k} = \frac{\frac{P_{k}}{H_{k}S_{i}^{E}}}{3600}$

Where:

Symbol	Description	Unit
N_B	Number of blocks.	dimensionless
y_k	Centre position of block k along the roll width.	m
P_k	Power assigned to block k.	kW
H_k	Holdup in block k.	t
$S_{i,k}$	Selection rate for size class i in block k.	1/s

The particle-bed compression stage is solved as a steady-state plug-flow population balance. For each size class i and block k:

$v_{z}d\frac{m_{i,k}}{dz} = sum_{j = 1}^{i - 1}S_{(j,k)b_{(ij)m_{j,k}}} - S_{(i,k)m_{i,k}}$

The analytical solution used by the implementation is:

$p_{i,k} = sum_{j = 1}^{i}A_{ij,k}\exp\left( - \frac{S_{j,k}z^{*}}{v_{z}} \right)$

with:

$A_{ij,k} = 0,\ i < j$

$A_{ij,k} = \frac{sum_{l = j}^{i - 1}b_{il}S_{l,k}}{S_{i,k} - S_{j,k}}A_{lj,k},\ i > j$

$A_{ii,k} = f_{i}^{IP} - sum_{l = 1}^{i - 1}A_{il,k}$

Where:

Symbol	Description	Unit
$m_{i,k}$	Mass fraction retained in size class i at position z in block k.	fraction
$p_{i,k}$	Product retained fraction in size class i from block k.	fraction
v_z	Vertical velocity in the particle-bed compression zone.	m/s
$A_{ij,k}$	Analytical solution coefficient.	fraction

The total HPGR product is calculated as the average of the block products:

$p_{i}^{HPGR} = \frac{1}{N_{B}sum_{{k = 1}_{B}^{N}p_{i,k}}}$

The edge product is calculated from the edge blocks. The number of edge blocks considered on one side is:

$E = 0.5\ a\ N_{B}$

The edge product distribution is:

$p_{i}^{E} = \frac{1}{E}\left( sum_{{k = 1}_{i,k}^{floor(E)p}} + \left( E - floor(E) \right)p_{i,ceil(E)} \right)$

The implementation averages both left and right edges of the roll.

The centre product is calculated by mass balance:

$p_{i}^{C} = \frac{p_{i}^{HPGR} - a\ p_{i}^{E}}{1 - a}$

Where:

Symbol	Description	Unit
$p_{i}^{HPGR}$	Total HPGR product retained fraction in size class i.	fraction
$p_{i}^{E}$	Edge product retained fraction in size class i.	fraction
$p_{i}^{C}$	Centre product retained fraction in size class i.	fraction
a	Edge product fraction.	fraction

The model solves the PSD calculation independently for each component using the component-specific breakage parameters. The component product retained masses are then recombined to form the total product size distribution and the product component-by-size matrix.

The derived parameter “Throughput per HPGR” reports the theoretical throughput of one HPGR unit.

The breakage parameters are component-specific. The selection and operating parameters are common to all components.

The model requires calibrated breakage and selection parameters. Torres and Casali fitted these parameters from pilot-scale HPGR test data; therefore, the model should be calibrated against pilot, vendor or plant survey data before being used for design conclusions.

The edge product fraction is a calibration parameter. The model calculates edge and centre products for diagnostics, but the product stream receives the total HPGR product distribution.

The model assumes plug flow through the compression zone and uses a parabolic pressure profile across the roll width. These assumptions should be checked when applying the model outside the calibration range.