Spline Partition
Summary
The Spline Partition model is an empirical component-specific separator that uses user-defined size-recovery points to build a partition curve for each component. For each component, the model fits a spline through five size-recovery points and uses the fitted curve to calculate the recovery of each size class to the concentrate stream.
The model should be used when the separation response is known from testwork, plant survey data, vendor information, or calibrated performance curves, and when a flexible size-by-size recovery curve is required instead of a fixed analytical partition equation.
DPSIM model key:
DPSIM.Classification.SplinePartition
Category: Classification
Subcategory: Partition curves
Display name: Spline Partition
Parameters
| # | Parameter | Description |
|---|---|---|
| 1 | Concentrate stream solids (%) | Target solids percentage of the concentrate stream. The model calculates the water assigned to the concentrate stream from this value, limited by the water available in the feed. |
| 2 | [Component] Point 1 size (µm) | Particle size of the first recovery point for the component. |
| 3 | [Component] Point 1 recover (%) | Recovery to concentrate at Point 1 for the component. |
| 4 | [Component] Point 2 size (µm) | Particle size of the second recovery point for the component. |
| 5 | [Component] Point 2 recover (%) | Recovery to concentrate at Point 2 for the component. |
| 6 | [Component] Point 3 size (µm) | Particle size of the third recovery point for the component. |
| 7 | [Component] Point 3 recover (%) | Recovery to concentrate at Point 3 for the component. |
| 8 | [Component] Point 4 size (µm) | Particle size of the fourth recovery point for the component. |
| 9 | [Component] Point 4 recover (%) | Recovery to concentrate at Point 4 for the component. |
| 10 | [Component] Point 5 size (µm) | Particle size of the fifth recovery point for the component. |
| 11 | [Component] Point 5 recover (%) | Recovery to concentrate at Point 5 for the component. |
Model Description
The Spline Partition model receives one feed stream and generates two product streams. In DPSIM, the product port represents the concentrate stream and the tail port represents the tail stream.
For each component c, the user defines five size-recovery points:
(d_(c,1),R_(c,1)), (d_(c,2),R_(c,2)), (d_(c,3),R_(c,3)), (d_(c,4),R_(c,4)), (d_(c,5),R_(c,5))
Where:
| Symbol | Description | Unit |
|---|---|---|
| d_(c,k) | Size coordinate of spline point k for component c. | µm |
| R_(c,k) | Recovery to concentrate at spline point k for component c. | % |
| k | Spline point index. | dimensionless |
Before fitting the spline, the model sorts the points by particle size. Recovery values are limited to the range from 0 to 100%. If two size points are equal or not increasing after sorting, the model applies a small increment to keep the spline input strictly increasing.
The fitted spline is used to estimate the recovery to concentrate at the representative size of each DPSIM size class:
R_(c,i)=Spline_c(d_i)
The partition to concentrate is:
E_(c,i)=R_(c,i)/100
The partition is limited internally between 0 and 1:
E_(c,i)^*=min(1,max(0,E_(c,i)))
Where:
| Symbol | Description | Unit |
|---|---|---|
| R_(c,i) | Spline-estimated recovery to concentrate for component c in size class i. | % |
| E_(c,i) | Partition to concentrate before limiting. | fraction |
| E_(c,i)^* | Partition to concentrate after limiting. | fraction |
| d_i | Representative particle size of size class i. | µm |
The first size interval, corresponding to the DPSIM top-size class, is forced to report to the concentrate stream:
E_(c,0)^*=1
If the spline fit fails for a component, the recovery for the affected component is set to zero for the evaluated size classes, except for the top-size class.
For each component and size class, the feed component retained mass is calculated as:
M_(F,c,i)=z_(F,c,i) M_(F,i)
The concentrate and tail component retained masses are then:
M_(C,c,i)=E_(c,i)^* M_(F,c,i)
M_(T,c,i)=(1-E_(c,i)^*)M_(F,c,i)
Where:
| Symbol | Description | Unit |
|---|---|---|
| M_(F,c,i) | Feed mass flowrate of component c in size class i. | tph |
| M_(C,c,i) | Concentrate mass flowrate of component c in size class i. | tph |
| M_(T,c,i) | Tail mass flowrate of component c in size class i. | tph |
| z_(F,c,i) | Fraction of component c in feed size class i. | fraction |
| M_(F,i) | Feed retained mass flowrate in size class i. | tph |
The total retained mass in each product size interval is calculated by summing over components:
M_(C,i)=sum_c M_(C,c,i)
M_(T,i)=sum_c M_(T,c,i)
The product retained size distributions are then:
p_(C,i)=M_(C,i)/sum_i M_(C,i)
p_(T,i)=M_(T,i)/sum_i M_(T,i)
The component fractions in each product size interval are:
z_(C,c,i)=M_(C,c,i)/M_(C,i)
z_(T,c,i)=M_(T,c,i)/M_(T,i)
Where:
| Symbol | Description | Unit |
|---|---|---|
| M_(C,i) | Total concentrate mass flowrate in size class i. | tph |
| M_(T,i) | Total tail mass flowrate in size class i. | tph |
| p_(C,i) | Concentrate retained fraction in size class i. | fraction |
| p_(T,i) | Tail retained fraction in size class i. | fraction |
| z_(C,c,i) | Fraction of component c in concentrate size class i. | fraction |
| z_(T,c,i) | Fraction of component c in tail size class i. | fraction |
The model calculates the water assigned to the concentrate stream from the requested concentrate stream solids percentage:
W_C=M_S^C(1-X_C)/X_C
with:
X_C=S_C/100
Where:
| Symbol | Description | Unit |
|---|---|---|
| W_C | Water flowrate assigned to the concentrate stream. | tph |
| M_S^C | Concentrate solids flowrate. | tph |
| X_C | Target concentrate solids fraction. | fraction |
| S_C | Target concentrate solids percentage. | % |
If the calculated concentrate water is greater than the feed water, all feed water is assigned to the concentrate stream and the tail stream receives no water. If the target solids percentage is zero, the model also assigns all feed water to the concentrate stream.
The model is fully empirical. Its accuracy depends on the quality and representativeness of the five size-recovery points defined for each component. It should be calibrated against measured size-by-size recovery data whenever possible.