Model Reference / Herbst-Fuerstenau PBM Mill

Herbst-Fuerstenau PBM Mill

Summary

The Herbst-Fuerstenau PBM Mill model represents continuous open-circuit grinding using the population balance scale-up procedure proposed by Herbst and Fuerstenau in “Scale-up procedure for continuous grinding mill design using population balance models”, International Journal of Mineral Processing, 1980. The model is based on a linear lumped-parameter population balance formulation in which breakage kinetics, material transport and size distribution response are represented explicitly.

The model should be used when mill product size distribution must be estimated from a feed size distribution, specific power input, mean residence time, a residence-time-distribution approximation, size-dependent breakage rates and component-specific breakage distribution parameters.

In this DPSIM implementation, the model applies the open-circuit Herbst-Fuerstenau response to each component independently. The continuous mill response is calculated using an analytical matrix solution and an N-mixer residence-time factor.

DPSIM model key: DPSIM.Comminution.HerbstFuerstenauPBMMill
Category: Comminution
Subcategory: Mills
Display name: Herbst-Fuerstenau PBM Mill

Parameters

#	Parameter	Description
1	Specific power P/H (kW/t)	Net specific power input used to scale the breakage rate function.
2	Mean residence time (min)	Mean residence time of solids in the mill.
3	RTD mixers in series	Number of ideal mixers in series used to approximate the residence time distribution.
4	Specific breakage rate at reference size (t/kWh)	Energy-specific breakage rate at the reference particle size.
5	Breakage rate size exponent	Exponent controlling the variation of breakage rate with particle size.
6	Reference size for breakage rate (um)	Reference particle size used in the breakage-rate scaling equation.
7	[Component] breakage distribution phi	Component-specific mixing factor between the two breakage distribution terms. This value is limited between 0 and 1.
8	[Component] breakage distribution gamma	Component-specific exponent of the first breakage distribution term.
9	[Component] breakage distribution beta	Component-specific exponent of the second breakage distribution term.

Model Description

The Herbst-Fuerstenau PBM Mill receives one feed stream and generates one product stream. The product stream preserves the feed solids flowrate and water flowrate. The product size distribution and component-by-size matrix are recalculated from the open-circuit population balance model.

For each component, the open-circuit mill response is calculated as:

$m_{c}^{P} = T\ J_{C}T^{- 1}m_{c}^{F}$

Where:

Symbol	Description
$m_{c}^{P}$	Product retained-fraction vector for component c.
$m_{c}^{F}$	Feed retained-fraction vector for component c.
T	Eigenvector transform matrix generated from the breakage rates and breakage distribution matrix.
J_C	Diagonal open-circuit residence-time modal matrix.
$T^{- 1}$	Inverse of the transform matrix.

The size-dependent breakage rate is calculated from specific power and reference-size breakage rate:

$k_{i} = k_{ref}\left( \frac{P_{H}}{60} \right)\left( \frac{x_{i}}{x_{ref}} \right)^{a}$

Where:

Symbol	Description	Unit
k_i	Breakage rate for active size class i.	1/min
k_ref	Specific breakage rate at reference size.	t/kWh
P_H	Specific power input P/H.	kW/t
x_i	Representative particle size of active size class i.	µm
x_ref	Reference size for breakage rate.	µm
a	Breakage rate size exponent.	dimensionless

The last active size class is treated as terminal and is assigned zero breakage rate:

$k_{N} = 0$

The open-circuit residence-time modal factor is calculated using the N-mixer approximation:

$J_{C,ii} = \left( \frac{N_{m}}{N_{m} + k_{i}\tau} \right)_{m}^{N}$

Where:

Symbol	Description	Unit
$J_{C,ii}$	Diagonal modal residence-time factor for size class i.	dimensionless
N_m	Number of RTD mixers in series.	dimensionless
τ	Mean residence time.	min

For each component c, the cumulative breakage distribution is calculated as:

$B_{c\left( x;y_{j} \right)} = \varphi_{c}\left( \frac{x}{y_{j}} \right)_{c}^{\gamma} + \left( 1 - \varphi_{c} \right)\left( \frac{x}{y_{j}} \right)_{c}^{\beta}$

Where:

Symbol	Description	Unit
B_c(x;y_j)	Cumulative breakage distribution for component c.	fraction
x	Product size boundary.	same unit as size mesh
y_j	Parent size-class reference boundary.	same unit as size mesh
φ_c	Component breakage distribution phi.	fraction
γ_c	Component breakage distribution gamma.	dimensionless
β_c	Component breakage distribution beta.	dimensionless

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

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

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

The daughter fractions in each parent-size column are normalized when breakage products exist.

Where:

Symbol	Description	Unit
$b_{ij,c}$	Fraction of broken material from parent class j reporting to product class i for component c.	fraction
$D_{i - 1}$	Upper boundary of product size class i.	same unit as size mesh
D_i	Lower boundary of product size class i.	same unit as size mesh

The transform matrix is calculated recursively:

$T_{ij} = 0,\ i < j$

$T_{ii} = 1$

$T_{ij} = \frac{sum_{k = j}^{i - 1}b_{ik}k_{k}T_{kj}}{k_{i} - k_{j}},\ i > j$

Where:

Symbol	Description
$T_{ij}$	Element of the transform matrix.
k_i	Breakage rate of size class i.
$b_{ik}$	Differential breakage fraction from parent class k to product class i.

The model solves the open-circuit PBM equation independently for each component. The calculated component retained masses are recombined into the total product retained distribution:

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

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
$m_{c,i}^{P}$	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
$M_{S}^{F}$	Feed dry solids flowrate.	tph

The model is an open-circuit PBM mill model. It does not implement the closed-circuit classifier matrix formulation from Herbst and Fuerstenau.

The residence-time distribution is represented by an N-mixer approximation. The original Herbst-Fuerstenau formulation allows a general residence-time distribution, so this implementation should be treated as a simplified open-circuit version.

The transform matrix requires distinct breakage rates. If two active size classes produce equal or nearly equal breakage rates, the numerical transform may fail and the affected component is bypassed.

The model requires calibrated breakage-rate and breakage-distribution parameters. The component-specific breakage parameters allow different components to have different breakage behavior, but the breakage-rate size exponent and residence-time parameters are common to all components.