Thickener (Ideal Kynch)
Description
This article describes a continuous thickener model based on ideal Kynch sedimentation theory, as presented by King (2002, 2012).[1][2].
The model predicts the achievable underflow concentration and flowrate, based on the thickener size and the settling characteristics of the slurry.
Model theory
Settling flux
Settling velocity
Richardson-Zaki
Michaels-Bogler
Extended Wilhelm-Naide
Feed flux
Underflow solids concentration
Operating modes
Concentration units
Limitations
Batch settling velocity curve extraction
Multi-test method
Single-test method
Flocculant dosage and suspension conditions
Excel
The King thickener model may be invoked from the Excel formula bar with the following function calls:
=mdUnit_Thickener_IdealKynch(Parameters as Range, Size As Range, Feed as Range, Density as Range, SettlingVelocity as Range, Optional NumChartPoints as Integer)
Invoking the function with no arguments will print Help text associated with the model, including a link to this page.
Inputs
The required inputs are defined below in matrix notation with elements corresponding to cells in Excel row ([math]\displaystyle{ i }[/math]) x column ([math]\displaystyle{ j }[/math]) format:
- [math]\displaystyle{ \mathit{Parameters} = \begin{bmatrix} D\text{ (m)}\\ \text{Settling velocity method}\\ \text{Underflow mode}\\ \mathit{CD}_{\rm Target}\text{ (v/v)}\\ \mathit{QD}_{\rm Target}\text{ (m}^3\text{/h)}\\ \rho_{\rm L}\text{ (t/m}^3\text{)}\\ (Q_{\rm M,F})_{\rm L}\text{ (t/h)}\\ \end{bmatrix},\;\;\;\;\;\; \mathit{Size} = \begin{bmatrix} d_{1}\text{ (mm)}\\ \vdots\\ d_n\text{ (mm)}\\ \end{bmatrix},\;\;\;\;\;\; \mathit{Feed}= \begin{bmatrix} (Q_{\rm M,F})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,F})_{1m}\text{ (t/h)}\\ \vdots & \ddots & \vdots\\ (Q_{\rm M,F})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,F})_{nm}\text{ (t/h)}\\ \end{bmatrix},\;\;\;\;\;\; }[/math]
- [math]\displaystyle{ \mathit{Density}= \begin{bmatrix} (\rho_{\rm s})_1\text{ (t/m}^\text{3}\text{)} & \dots & (\rho_{\rm s})_m\text{ (t/m}^\text{3}\text{)}\\ \end{bmatrix},\;\;\;\;\;\; \mathit{SettlingVelocity}= \begin{bmatrix} v_{\rm {TF}}\text{ (mm/s)}\\ n\text{ (-)}\\ \end{bmatrix}\quad \text{or}\quad\begin{bmatrix} v_{\rm {TF}}\text{ (mm/s)}\\ n\text{ (-)}\\ \phi_{\rm max}\text{ (v/v)}\\ \end{bmatrix}\quad \text{or}\quad\begin{bmatrix} v_{\rm {TF}}\text{ (mm/s)} & 0.0\\ \alpha_1\text{ (-)} & \beta\text{ (-)}\\ \vdots & \vdots\\ \alpha_n\text{ (-)} & \beta\text{ (-)}\\ \end{bmatrix} }[/math]
where:
- [math]\displaystyle{ \text{Settling velocity method} }[/math] specifies the settling velocity equation applied, 0 = Richardson-Zaki, 1 = Michaels-Bogler, 2 = Extended Wilhelm-Naide
- [math]\displaystyle{ \text{Underflow mode} }[/math] specifies the targeted underflow condition, 0 = Target underflow concentration, 1 = Target underflow flow rate, 2 = Limiting capacity
- [math]\displaystyle{ \mathit{CD}_{\rm Target} }[/math] is the underflow concentration operating target for the thickener (v/v)
- [math]\displaystyle{ \mathit{QD}_{\rm Target} }[/math] is the underflow flow rate operating target for the thickener (m{sup}3{/sup}/h)
- [math]\displaystyle{ \rho_{\rm L} }[/math] is the density of liquids to the thickener (t/m3)
- [math]\displaystyle{ (Q_{\rm M,F})_{\rm L} }[/math] is the mass flow feed rate of liquids to the thickener (t/h)
- [math]\displaystyle{ m }[/math] is the number of solids species types with distinct densities
- [math]\displaystyle{ n }[/math] is the number of size intervals
- [math]\displaystyle{ d_i }[/math] is the size of the square mesh interval that feed mass is retained on (mm)
- [math]\displaystyle{ Q_{\rm M,F} }[/math] is feed solids mass flow rate by size and solid species (t/h)
- [math]\displaystyle{ \rho_{\rm S} }[/math] is the density of each solid species (t/m3)
Note that the ideal Kynch thickener model does not explicitly model particle size effects, as the formulation is based solely on solids concentration and settling flux relationships. However, particle size-level information is included to maintain consistency with the data structures used by other unit models in flowsheet simulations.
Results
The results are displayed in Excel as an array corresponding to the matrix notation below:
- [math]\displaystyle{ \mathit{mdUnit\_Thickener\_IdealKynch} = \begin{bmatrix} \begin{array}{c} \begin{bmatrix} A\text{ (m}^2\text{)}\\ Q_{\rm F}\text{ (m}^3\text{/s)}\\ \rho_{\rm {s,F}}\text{ (t/m}^3\text{)}\\ C_{\rm F}\text{ (v/v)}\\ f_{\rm F}\text{ (m}^3\text{/m}^2\text{s)}\\ C_{\rm I}\text{ (v/v)}\\ \psi(C_{\rm I})\text{ (m}^3\text{/m}^2\text{s)}\\ f_{\rm Fmax}\text{ (m}^3\text{/m}^2\text{s)}\\ W_{\rm Fmax}\text{ (t/h)}\\ C_{\rm M}\text{ (v/v)}\\ q\text{ (m}^3\text{/m}^2\text{s)}\\ Q_{\rm D}\text{ (m}^3\text{/s)}\\ C_{\rm D}\text{ (v/v)}\\ \rho_{\rm {SL,UF}}\text{ (t/m}^3\text{)}\\ W_{\rm S,UF}\text{ (w/w)}\\ R_{\rm f}\text{ (frac)}\\ (Q_{\rm M,UF})_{\rm L}\text{ (t/h)}\\ (Q_{\rm M,OF})_{\rm L}\text{ (t/h)}\\ \end{bmatrix} \end{array} \begin{array}{ccc} \begin{bmatrix} d_1\text{ (mm)}\\ \vdots\\ d_n\text{ (mm)} \end{bmatrix} & \begin{bmatrix} (Q_{\rm M,UF})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,UF})_{1m}\text{ (t/h)}\\ \vdots & \ddots & \vdots\\ (Q_{\rm M,UF})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,UF})_{nm}\text{ (t/h)}\\ \end{bmatrix} & \begin{bmatrix} (Q_{\rm M,OF})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,OF})_{1m}\text{ (t/h)}\\ \vdots & \ddots & \vdots\\ (Q_{\rm M,OF})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,OF})_{nm}\text{ (t/h)}\\ \end{bmatrix} & \begin{bmatrix} C_1\text{ (v/v)}\\ \vdots\\ C_k\text{ (v/v)} \end{bmatrix}^* & \begin{bmatrix} V(C_1)\text{ (m/s)}\\ \vdots\\ V(C_k)\text{ (m/s)} \end{bmatrix}^* & \begin{bmatrix} \psi(C_1)\text{ (m}^3\text{/m}^2\text{s)}\\ \vdots\\ \psi(C_k)\text{ (m}^3\text{/m}^2\text{s)} \end{bmatrix}^* \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \end{array} \end{bmatrix} }[/math]
where:
- [math]\displaystyle{ q }[/math] is the downward volumetric flux due to underflow discharge (m3/m2.s)
- [math]\displaystyle{ \rho_{\rm {S,F}} }[/math] is the density of solids in the feed (t/m3)
- [math]\displaystyle{ \rho_{\rm {SL,UF}} }[/math] is the density slurry in the underflow (t/m3)
- [math]\displaystyle{ W_{\rm S,UF} }[/math] is the mass fraction of solids in the underflow (w/w)
- [math]\displaystyle{ R_{\rm f} }[/math] is the fraction of feed liquid reporting to the underflow (frac)
- [math]\displaystyle{ (Q_{\rm M,UF})_{\rm L} }[/math] is the mass flow rate of liquid in the underflow stream (t/h)
- [math]\displaystyle{ (Q_{\rm M,OF})_{\rm L} }[/math] is the mass flow rate of liquid in the overflow stream (t/h)
- [math]\displaystyle{ Q_{\rm M,UF} }[/math] is mass flow rate of solids to the underflow stream (t/h)
- [math]\displaystyle{ Q_{\rm M,OF} }[/math] is mass flow rate of solids to the overflow stream (t/h)
- [math]\displaystyle{ k }[/math] is the number of [math]\displaystyle{ C }[/math], [math]\displaystyle{ V }[/math], [math]\displaystyle{ \psi }[/math] and data points returned, i.e. [math]\displaystyle{ k = \mathit{NumChartPoints} }[/math]
- [math]\displaystyle{ ^* }[/math] indicates values returned only when [math]\displaystyle{ \mathit{NumChartPoints}\gt 0 }[/math]
Example
The images below show the selection of input arrays and output results in the Excel interface.
SysCAD
The sections and variable names used in the SysCAD interface are described in detail in the following tables.
MD_Thickener page
The first tab page in the access window will have this name.
| Tag (Long/Short) | Input / Display | Description/Calculated Variables/Options |
|---|---|---|
| Tag | Display | This name tag may be modified with the change tag option. |
| Condition | Display | OK if no errors/warnings, otherwise lists errors/warnings. |
| ConditionCount | Display | The current number of errors/warnings. If condition is OK, returns 0. |
| GeneralDescription / GenDesc | Display | This is an automatically generated description for the unit. If the user has entered text in the 'EqpDesc' field on the Info tab (see below), this will be displayed here.
If this field is blank, then SysCAD will display the unit class ID. |
| Requirements | ||
| On | CheckBox | This enables the unit. If this box is not checked, then the MassFracToUF option appears below. |
| MassFracToUF | Input | Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the underflow stream when the model is off. |
| Method | Solid-Liquid Split | The fractional split of solids and liquids to the underflow stream is defined by the user. |
| Ideal Kynch | The Ideal Kynch sedimentation model is used to determine the split of solids and liquids to the underflow and overflow streams. | |
| Options | ||
| ShowQFeed | CheckBox | QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream. |
| ShowQOF | CheckBox | QOF and associated tab pages (eg Sp) will become visible, showing the properties of the overflow stream. |
| ShowQUF | CheckBox | QUF and associated tab pages (eg Sp) will become visible, showing the properties of the underflow stream. |
| SizeForPassingFracCalc | Input | Size fraction for % Passing calculation. The size fraction input here will be shown in the Stream Summary section. |
| FracForPassingSizeCalc | Input | Fraction passing for Size calculation. The fraction input here will be shown in the Stream Summary section. |
| Stream Summary | ||
| MassFlow / Qm | Display | The total mass flow in each stream. |
| SolidMassFlow / SQm | Display | The Solids mass flow in each stream. |
| LiquidMassFlow / LQm | Display | The Liquid mass flow in each stream. |
| VolFlow / Qv | Display | The total Volume flow in each stream. |
| Temperature / T | Display | The Temperature of each stream. |
| Density / Rho | Display | The Density of each stream. |
| SolidFrac / Sf | Display | The Solid Fraction in each stream. |
| LiquidFrac / Lf | Display | The Liquid Fraction in each stream. |
| Passing | Display | The mass fraction passing the user-specified size (in the field SizeForPassingFracCalc) in each stream. |
| Passes | Display | The user-specified (in the field FracForPassesSizeCalc) fraction of material in each stream will pass this size fraction. |
Thickener page
The Thickener page is used to specify the input parameters for the thickener model.
Profiles page
| Tag (Long/Short) | Input / Display | Description/Calculated Variables/Options |
|---|---|---|
| Profiles | ||
| NumPoints | Input | Number of profile data points to show. |
| SolidVolFrac / Svf | Display | Volume fraction of solids profile data points. |
| SettlingVelocity / vs | Display | Solids vettling velocity profile data points. |
| SettlingFlux / Psi | Display | Solids settling flux profile data points. |
CSz page
The CSz page provides composition and constituent information by particle size for QFeed, QOF, and QUF. It appears after the standard SysCAD stream information when the corresponding stream display option is selected.
- Results are available for Species, Component, Element, and Individual Phase, with views for Composition, Distribution by Size, and Stream Contribution.
- The calculations are described in Stream Analyser - Composition by Size.
- These combined composition-by-size results are not available in the standard SysCAD stream pages. They are particularly useful for comparing simulated results with measured size-by-size data, such as elemental assays or mineral grades by size.
| Tag (Long/Short) | Input / Display | Description/Calculated Variables/Options |
|---|---|---|
| Distribution | ||
| Name | Display | Shows the name of the SysCAD Size Distribution (PSD) quality associated with the feed stream. |
| IntervalCount | Display | Shows the number of size intervals in the SysCAD Size Distribution (PSD) quality associated with the feed stream. |
| SpWithPSDCount | Display | Shows the number of species in the feed stream assigned with the SysCAD Size Distribution (PSD) quality. |
| CSz | ||
| Property | Composition | Displays the mass fraction of each species, component, element, or individual phase within each size interval. The All row displays the overall composition over all size intervals. |
| Distribution by Size | Displays the fraction of each constituent's total mass that occurs in each size interval. The All column displays the distribution by size of all solids with a particle size distribution property in the selected stream. | |
| Stream Contribution | Displays each constituent and size interval mass as a fraction of the total mass of all solids with a particle size distribution property in the selected stream. | |
| Comp | ||
| Size | Display | Size of each interval in the mesh series. |
| All (All column) | Display | Total composition of all solids with a particle size distribution property in each size interval. The result is 100% when the interval contains solids with a particle size distribution property and 0 when it does not. |
| Comp | Display | Mass fraction of each solid species with a particle size distribution property within each size interval. |
| All (All row, All column) | Display | Total composition of all solids with a particle size distribution property over all size intervals. The result is 100% when the selected stream contains solids with a particle size distribution property and 0 when it does not. |
| All (All row, per species) | Display | Overall mass fraction of each solid species with a particle size distribution property in the selected stream for all size intervals. |
| CmpComp | ||
| Components | Hides or shows the component composition table. | |
| Size | Display | Size of each interval in the mesh series. |
| All (All column) | Display | Total composition of all solids with a particle size distribution property in each size interval. The result is 100% when the interval contains solids with a particle size distribution property and 0 when it does not. |
| CmpComp | Display | Mass fraction of each displayed solid component within each size interval. |
| All (All row, All column) | Display | Total composition of all solids with a particle size distribution property over all size intervals. |
| All (All row, per component) | Display | Overall mass fraction of each displayed solid component in the selected stream over all size intervals. |
| EleComp | ||
| Elements | Hides or shows the element composition table. | |
| Size | Display | Size of each interval in the mesh series. |
| All (All column) | Display | Total composition of all solids with a particle size distribution property in each size interval. The result is 100% when the interval contains solids with a particle size distribution property and 0 when it does not. |
| EleComp | Display | Mass fraction of each displayed element within each size interval. |
| All (All row, All column) | Display | Total composition of all solids with a particle size distribution property over all size intervals. |
| All (All row, per element) | Display | Overall mass fraction of each displayed element in the selected stream over all size intervals. |
| IPhComp | ||
| IPhases | Hides or shows the individual phase composition table. | |
| Size | Display | Size of each interval in the mesh series. |
| All (All column) | Display | Total composition of all solids with a particle size distribution property in each size interval. The result is 100% when the interval contains solids with a particle size distribution property and 0 when it does not. |
| IPhComp | Display | Mass fraction of each displayed individual phase within each size interval. |
| All (All row, All column) | Display | Total composition of all solids with a particle size distribution property over all size intervals. |
| All (All row, per individual phase) | Display | Overall mass fraction of each displayed individual phase in the selected stream over all size intervals. |
| Dist | ||
| Size | Display | Size of each interval in the mesh series. |
| All (All column) | Display | Fraction of the total mass of all solids with a particle size distribution property in the selected stream that occurs in each size interval. |
| Dist | Display | Fraction of the total mass of each solid species with a particle size distribution property that occurs in each size interval. |
| All (All row, All column) | Display | Total distribution by size of all solids with a particle size distribution property. The result is 100% when the selected stream contains solids with a particle size distribution property and 0 when it does not. |
| All (All row, per species) | Display | Total distribution by size of each solid species with a particle size distribution property. The result is 100% when that species is present and 0 when it is not. |
| CmpDist | ||
| Components | Hides or shows the component distribution-by-size table. | |
| Size | Display | Size of each interval in the mesh series. |
| All (All column) | Display | Fraction of the total mass of all solids with a particle size distribution property in the selected stream that occurs in each size interval. |
| CmpDist | Display | Fraction of each displayed solid component's total mass that occurs in each size interval. |
| All (All row, All column) | Display | Total distribution by size of all solids with a particle size distribution property. |
| All (All row, per component) | Display | Total distribution by size of each displayed solid component. The result is 100% when that component is present and 0 when it is not. |
| EleDist | ||
| Elements | Hides or shows the element distribution-by-size table. | |
| Size | Display | Size of each interval in the mesh series. |
| All (All column) | Display | Fraction of the total mass of all solids with a particle size distribution property in the selected stream that occurs in each size interval. |
| EleDist | Display | Fraction of each displayed element's total mass that occurs in each size interval. |
| All (All row, All column) | Display | Total distribution by size of all solids with a particle size distribution property. |
| All (All row, per element) | Display | Total distribution by size of each displayed element. The result is 100% when that element is present and 0 when it is not. |
| IPhDist | ||
| IPhases | Hides or shows the individual phase distribution-by-size table. | |
| Size | Display | Size of each interval in the mesh series. |
| All (All column) | Display | Fraction of the total mass of all solids with a particle size distribution property in the selected stream that occurs in each size interval. |
| IPhDist | Display | Fraction of each displayed individual phase's total mass that occurs in each size interval. |
| All (All row, All column) | Display | Total distribution by size of all solids with a particle size distribution property. |
| All (All row, per individual phase) | Display | Total distribution by size of each displayed individual phase. The result is 100% when that individual phase is present and 0 when it is not. |
| Contr | ||
| Size | Display | Size of each interval in the mesh series. |
| All (All column) | Display | Fraction of the total mass of all solids with a particle size distribution property in the selected stream that occurs in each size interval. |
| Contr | Display | Stream Contribution of each solid species with a particle size distribution property and size interval to the total mass of all solids with a particle size distribution property in the selected stream. |
| All (All row, All column) | Display | Total stream contribution of all solids with a particle size distribution property. The result is 100% when the selected stream contains solids with a particle size distribution property and 0 when it does not. |
| All (All row, per species) | Display | Overall stream contribution of each solid species with a particle size distribution property for all size intervals. |
| CmpContr | ||
| Components | Hides or shows the component stream contribution table. | |
| Size | Display | Size of each interval in the mesh series. |
| All (All column) | Display | Fraction of the total mass of all solids with a particle size distribution property in the selected stream that occurs in each size interval. |
| CmpContr | Display | Stream Contribution of each displayed solid component and size interval to the total mass of all solids with a particle size distribution property in the selected stream. |
| All (All row, All column) | Display | Total stream contribution of all solids with a particle size distribution property. |
| All (All row, per component) | Display | Overall stream contribution of each displayed solid component for all size intervals. |
| EleContr | ||
| Elements | Hides or shows the element stream contribution table. | |
| Size | Display | Size of each interval in the mesh series. |
| All (All column) | Display | Fraction of the total mass of all solids with a particle size distribution property in the selected stream that occurs in each size interval. |
| EleContr | Display | Stream Contribution of each displayed element and size interval to the total mass of all solids with a particle size distribution property in the selected stream. |
| All (All row, All column) | Display | Total stream contribution of all solids with a particle size distribution property. |
| All (All row, per element) | Display | Overall stream contribution of each displayed element for all size intervals. |
| IPhContr | ||
| IPhases | Hides or shows the individual phase stream contribution table. | |
| Size | Display | Size of each interval in the mesh series. |
| All (All column) | Display | Fraction of the total mass of all solids with a particle size distribution property in the selected stream that occurs in each size interval. |
| IPhContr | Display | Stream Contribution of each displayed individual phase and size interval to the total mass of all solids with a particle size distribution property in the selected stream. |
| All (All row, All column) | Display | Total stream contribution of all solids with a particle size distribution property. |
| All (All row, per individual phase) | Display | Overall stream contribution of each displayed individual phase for all size intervals. |
About page
This page is provides product and licensing information about the Met Dynamics Models SysCAD Add-On.
| Tag (Long/Short) | Input / Display | Description/Calculated Variables/Options |
|---|---|---|
| About | ||
| HelpLink | Opens a link to the Installation and Licensing page using the system default web browser. Note: Internet access is required. | |
| Information | Copies Product and License information to the Windows clipboard. | |
| Product | ||
| Name | Display | Met Dynamics software product name |
| Version | Display | Met Dynamics software product version number. |
| BuildDate | Display | Build date and time of the Met Dynamics Models SysCAD Add-On. |
| License | ||
| File | This is used to locate a Met Dynamics software license file. | |
| Location | Display | Type of Met Dynamics software license or file name and path of license file. |
| SiteCode | Display | Unique machine identifier for license authorisation. |
| ReqdAuth | Display | Authorisation level required, MD-SysCAD Full or MD-SysCAD Runtime. |
| Status | Display | License status, LICENSE_OK indicates a valid license, other messages report licensing errors. |
| IssuedTo | Display | Only visible if Met Dynamics license file is used. Name of organisation/seat the license is authorised to. |
| ExpiryDate | Display | Only visible if Met Dynamics license file is used. License expiry date. |
| DaysLeft | Display | Only visible if Met Dynamics license file is used. Days left before the license expires. |


