Fine Wet Screen (Mwale): Difference between revisions

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== Description ==
== Description ==


This article describes an implementation of the '''Mwale''' (Mwale et al., 2016) model for fine wet screening.{{Mwale et al. (2016)}}
This article describes an implementation of the Mwale et al. (2016) model for fine wet screening.{{Mwale et al. (2016)}}


== Model theory ==
== Model theory ==


{{Under construction|section}}
=== Efficiency curve ===
 
Mwale et al. (2016) proposed a phenomenological model of fine wet screening where the partition to oversize, <math>E_{i{\rm o}}</math> (frac), is related to properties of both the screen and the feed:{{Mwale et al. (2016)}}
 
:<math>E_{i{\rm o}} = \exp \left [ - \dfrac{A_{\rm o} K}{F_{\rm f} s \left ( \dfrac{\bar d_i}{a} \right)^\alpha} \right ] + \dfrac{\delta . F_{\rm f}}{100(1-s)} \left [ \exp \left ( - \dfrac{\bar d_i}{a} \right ) \right ]^\alpha</math>
 
where:
* <math>A_{\rm o}</math> is the total screen open area (m<sup>2</sup>)
* <math>F_{\rm f}</math> is the solids feed rate (t/h)
* <math>s</math> is the mass fraction of solids in the feed (w/w)
* <math>x_{\rm a}</math> is the aperture size (mm)
* <math>\bar d_{i}</math> is the [[Conversions|geometric mean size]] of the size interval <math>i</math> (mm)
* <math>K</math> is the kinetic constant (t/hr.m<sup>2</sup>)
* <math>\alpha</math> is the sharpness of separation parameter (-)
* <math>\delta</math> is the fines bypass parameter (h/t)
 
=== Sharpness constant ===
 
Mwale et al. (2018) suggest the following equation to estimate the sharpness constant, <math>\alpha</math>, as a function of screen design and feed conditions:{{Mwale et al. (2018)}}
 
:<math>\alpha = k_1 \left ( \dfrac{{\rho_{\rm p}}^2 F_{\rm f} {A_{\rm o}}^{3.5}}{\mu a^2 {M_{\rm u}}^2} \right ) + k_2 \left ( \dfrac{A_{\rm o}}{a^2} \right ) + k_3</math>
 
where:
* <math>\rho_{\rm p}</math> is the density of the feed slurry (t/m<sup>3</sup>)
* <math>M_{\rm u}</math> is the mass flow rate of particles in the feed which are smaller than the aperture (t/h)
* <math>k_1</math>, <math>k_2</math> and <math>k_3</math> are coefficients of the sharpness equation
 
The apparent viscosity of slurry in the feed, <math>\mu</math> (Pa.s), is estimated by the classical Krieger-Dougherty (1959) equation:{{Krieger and Dougherty (1959)}}
 
:<math>\mu = \mu_{\rm 0} \left ( 1 - \dfrac{\phi_{\rm Eff}}{\phi_{\rm m}} \right)^{-k \phi_{\rm m}}</math>
 
where:
* <math>\mu_0</math> is the viscosity of the carrier fluid, i.e. water (Pa.s)
* <math>\phi_{\rm m}</math> is maximum packing fraction of particles (v/v), assumed to be 0.64
* <math>k</math> is a particle shape constant, assumed to be 2.5
 
The sharpness constant can be specified directly as the value of coefficient <math>k_3</math> when coefficients <math>k_1</math> and <math>k_2</math> are set to zero.
 
=== Bypass constant ===
 
Mwale et al. (2018) suggest the following simple linear relationship to estimate the bypass constant, <math>\delta</math>,:{{Mwale et al. (2018)}}
 
:<math>\delta = C_1.F_{\rm f} + C_2.s + C_3.x_{\rm a} + C_4</math>
 
where <math>C_1-C_4</math> are coefficients of the bypass equation.
 
The bypass constant can be specified directly as the value of coefficient <math>C_4</math> when coefficients <math>C_1-C_3</math> are set to zero.


== Excel ==
== Excel ==
{{Under construction|section}}


The Mwale fine wet screen model may be invoked from the Excel formula bar with the following function call:
The Mwale fine wet screen model may be invoked from the Excel formula bar with the following function call:


<syntaxhighlight lang="vb">=mdUnit_Screen_Mwale(Parameters as Range, Size as Range, Feed as Range)</syntaxhighlight>
<syntaxhighlight lang="vb">=mdUnit_Screen_Mwale(Parameters as Range, Size as Range, Feed as Range, OreSG as Range)</syntaxhighlight>


{{Excel (Text, Help, No Arguments)}}
{{Excel (Text, Help, No Arguments)}}
=== Inputs ===
{{Excel (Text, Inputs)}}
:<math>Parameters=
\begin{bmatrix}
x_{\rm a} \text{ (mm)}\\
A_{\rm o}\text{ (m}^2\text{)}\\
K\text{ (t/h.m}^2\text{)}\\
k_1\text{ (-)}\\
k_2\text{ (-)}\\
k_3\text{ (-)}\\
C_1\text{ (-)}\\
C_2\text{ (-)}\\
C_3\text{ (-)}\\
C_4\text{ (-)}\\
(Q_{\rm M,F})_{\rm L}\text{ (t/h)}\\
\rho_{\rm L}\text{ (t/m}^{\text{3}}\text{)}\\
\mu_0\text{ (mPa.S)}\\
\end{bmatrix},\;\;\;\;\;\;
Size = \begin{bmatrix}
d_{1}\text{ (mm)}\\
\vdots\\
d_n\text{ (mm)}\\
\end{bmatrix},\;\;\;\;\;\;
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},\;\;\;\;\;\;
\mathit{OreSG}= \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}
</math>
where:
* <math>n</math> is the number of size intervals
* <math>d_i</math> is the size of the square mesh interval that feed mass is retained on (mm)
* <math>d_{i+1}<d_i<d_{i-1}</math>, i.e. descending size order from top size (<math>d_{1}</math>) to sub mesh (<math>d_{n}=0</math> mm)
* <math>m</math> is the number of ore types
* <math>Q_{\rm M,F}</math> is feed solids mass flow rate by size and ore type (t/h)
* <math>(Q_{\rm M,F})_{\rm L}</math> is the mass flow feed rate of liquids in the feed (t/h)
* <math>\rho_{\rm S}</math> is the density of solids in the feed (t/m<sup>3</sup>)
* <math>\rho_{\rm L}</math> is the density of liquids in the feed (t/m<sup>3</sup>)
=== Results ===
The results are displayed in Excel as an array corresponding to the matrix notation below:
:<math>
\mathit{mdUnit\_Screen\_Mwale} =
\begin{bmatrix}
\begin{array}{c}
\begin{bmatrix}
\alpha\text{ (-)}\\
\delta\text{ ((t/h)}^{-1}\text{)}\\
F_{\rm f}\text{ (t/h)}\\
s\text{ (w/w)}\\
\rho_{\rm p}\text{ (t/m}^3\text{)}\\
\mu\text{ (Pa.s)}\\
M_{\rm u}\text{ (t/h)}\\
E_{\rm US}\text{ (frac)}\\
\end{bmatrix}
\end{array}
\begin{array}{cccccc}
\begin{bmatrix}
\bar d_1\text{ (mm)}\\
\vdots\\
\bar d_n\text{ (mm)}
\end{bmatrix}
&
\begin{bmatrix}
(Q_{\rm M,OS})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,OS})_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
(Q_{\rm M,OS})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,OS})_{nm}\text{ (t/h)}\\
\end{bmatrix}
&
\begin{bmatrix}
(Q_{\rm M,US})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,US})_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
(Q_{\rm M,US})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,US})_{nm}\text{ (t/h)}\\
\end{bmatrix}
&
\begin{bmatrix}
(E_{i \rm o})_1\text{ (frac)}\\
\vdots\\
(E_{i \rm o})_n\text{ (frac)}
\end{bmatrix}
\\
\\
\\
\\
\\
\\
\end{array}
\end{bmatrix}
</math>
where:
* <math>E_{\rm US}</math> is the efficiency of undersize removal achieved by the screen (frac)
* <math>Q_{\rm M,OS}</math> is mass flow rate of solids to the oversize stream (t/h)
* <math>Q_{\rm M,US}</math> is mass flow rate of solids to the undersize stream (t/h)
=== Example ===
The images below show the selection of input arrays and output results in the Excel interface.
{|
|- style="vertical-align:top;"
| [[File:FineWetScreenMwale1.png|left|frame|Figure 1. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]]
| [[File:FineWetScreenMwale2.png|left|frame|Figure 2. Example showing the selection of the '''Size''' (red frame), '''Feed''' (purple frame) and '''OreSG''' (green frame) arrays in Excel.]] 
| [[File:FineWetScreenMwale3.png|left|frame|Figure 3. Example showing the outline of the '''Results''' (light blue frame) array in Excel.]]
|}


== SysCAD ==
== SysCAD ==


{{Under construction|section}}
The sections and variable names used in the SysCAD interface are described in detail in the following tables.
 
Note that a '''Deck''' and '''Partition''' page is provided provided for each connected oversize discharge stream.
 
{{SysCAD (Page, Screen, DLL*Screen)}}
 
==== Deck page ====
 
The Deck page is used to specify the required model method and associated input parameters.
 
{{SysCAD (Text, Table Header)}}
 
{{SysCAD (Text, Screen, Deck)}}
 
{{SysCAD (Text, Help Link)}}
 
|-
! colspan="3" style="text-align:left;" |''Parameters''
|-
|Aperture / xa
|Input
|Size of the apertures in the deck.
|-
|OpenArea/ Ao
|Input
|Open area of the screening surface
|-
|KineticParameter / K
|Input
|Mwale model kinetic parameter
|-
|SharpnessCoeff1 / k1
|Input
|Mwale model sharpness coefficient
|-
|SharpnessCoeff2 / k2
|Input
|Mwale model sharpness coefficient
|-
|SharpnessCoeff3 / k3
|Input
|Mwale model sharpness coefficient
|-
|BypassCoeff1 / C1
|Input
|Mwale model bypass coefficient
|-
|BypassCoeff2 / C2
|Input
|Mwale mode3 bypass coefficient
|-
|BypassCoeff3 / C3
|Input
|Mwale model bypass coefficient
|-
|BypassCoeff4 / C4
|Input
|Mwale model bypass coefficient
 
{{SysCAD (Text, Screen, Liquids)|method=0}}
 
|-
! colspan="3" style="text-align:left;" |''Results''
|-
|SharpnessConstant / Alpha
|style="background: #eaecf0" | Display
|Mwale model sharpness constant
|-
|BypassConstant / Delta
|style="background: #eaecf0" | Display
|Mwale model bypass constant
|-
|Feed.SQm
|style="background: #eaecf0" | Display
|Mass flow rate of solids in the feed (excludes solids without PSD quality)
|-
|Feed.SQmSubAp
|style="background: #eaecf0" | Display
|Mass flow rate of particles in the feed which are smaller than the aperture size (sub-aperture)
|-
|Feed.Sf
|style="background: #eaecf0" | Display
|Mass fraction of solids in the feed (excludes solids without PSD quality)
|-
|Feed.SLRho
|style="background: #eaecf0" | Display
|Density of slurry in the feed
|-
|SlurryViscosity / mu
|style="background: #eaecf0" | Display
|Apparent viscosity of slurry in the feed
|-
|Efficiency
|style="background: #eaecf0" | Display
|Fraction of total sub-aperture sized material in feed that is actually recovered to the undersize stream.
|}
 
{{SysCAD (Page, Hydrocyclone, Partition)|ActionU=Partition|ActionL=partition|DestinationU=Oversize|DestinationL=oversize|UnitL=screen}}
 
{{SysCAD (Page, About)}}
 
==== Additional notes ====
 
{{SysCAD (Text, No PSD Splits)|gasstream=undersize}}


== References ==
== References ==

Latest revision as of 10:10, 30 January 2024

Description

This article describes an implementation of the Mwale et al. (2016) model for fine wet screening.[1]

Model theory

Efficiency curve

Mwale et al. (2016) proposed a phenomenological model of fine wet screening where the partition to oversize, (frac), is related to properties of both the screen and the feed:[1]

where:

  • is the total screen open area (m2)
  • is the solids feed rate (t/h)
  • is the mass fraction of solids in the feed (w/w)
  • is the aperture size (mm)
  • is the geometric mean size of the size interval (mm)
  • is the kinetic constant (t/hr.m2)
  • is the sharpness of separation parameter (-)
  • is the fines bypass parameter (h/t)

Sharpness constant

Mwale et al. (2018) suggest the following equation to estimate the sharpness constant, , as a function of screen design and feed conditions:[2]

where:

  • is the density of the feed slurry (t/m3)
  • is the mass flow rate of particles in the feed which are smaller than the aperture (t/h)
  • , and are coefficients of the sharpness equation

The apparent viscosity of slurry in the feed, (Pa.s), is estimated by the classical Krieger-Dougherty (1959) equation:[3]

where:

  • is the viscosity of the carrier fluid, i.e. water (Pa.s)
  • is maximum packing fraction of particles (v/v), assumed to be 0.64
  • is a particle shape constant, assumed to be 2.5

The sharpness constant can be specified directly as the value of coefficient when coefficients and are set to zero.

Bypass constant

Mwale et al. (2018) suggest the following simple linear relationship to estimate the bypass constant, ,:[2]

where are coefficients of the bypass equation.

The bypass constant can be specified directly as the value of coefficient when coefficients are set to zero.

Excel

The Mwale fine wet screen model may be invoked from the Excel formula bar with the following function call:

=mdUnit_Screen_Mwale(Parameters as Range, Size as Range, Feed as Range, OreSG as Range)

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 () x column () format:


where:

  • is the number of size intervals
  • is the size of the square mesh interval that feed mass is retained on (mm)
  • , i.e. descending size order from top size () to sub mesh ( mm)
  • is the number of ore types
  • is feed solids mass flow rate by size and ore type (t/h)
  • is the mass flow feed rate of liquids in the feed (t/h)
  • is the density of solids in the feed (t/m3)
  • is the density of liquids in the feed (t/m3)

Results

The results are displayed in Excel as an array corresponding to the matrix notation below:


where:

  • is the efficiency of undersize removal achieved by the screen (frac)
  • is mass flow rate of solids to the oversize stream (t/h)
  • is mass flow rate of solids to the undersize stream (t/h)

Example

The images below show the selection of input arrays and output results in the Excel interface.

Figure 1. Example showing the selection of the Parameters (blue frame) array in Excel.
Figure 2. Example showing the selection of the Size (red frame), Feed (purple frame) and OreSG (green frame) arrays in Excel.
Figure 3. Example showing the outline of the Results (light blue frame) array in Excel.

SysCAD

The sections and variable names used in the SysCAD interface are described in detail in the following tables.

Note that a Deck and Partition page is provided provided for each connected oversize discharge stream.

MD_Screen 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 MassFracToOS option appears below.
MassFracToOS Input Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the overflow stream when the model is off.
Options
ShowQFeed CheckBox QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed 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.

Deck page

The Deck page is used to specify the required model method and associated input parameters.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Deck
On Checkbox This enables the deck. If off, the feed to this deck passes directly to the next deck (or undersize) without partition.
Method Partition (User) The partition to oversize for each size interval is defined by the user. Different values can be used for different solids.
Partition (Reid-Plitt) The partition to oversize for each size interval is defined by a Reid-Plitt efficiency curve. Different parameters can be used for different solids.
Partition (Whiten-Beta) The partition to oversize for each size interval is defined by a Whiten-Beta efficiency curve. Different parameters can be used for different solids.
Vibrating (Karra) The Karra vibrating screen model is used to determine the partition of solids to oversize and undersize for each size interval.
Vibrating (Whiten) The Whiten vibrating screen model is used to determine the partition of solids to oversize and undersize for each size interval.
Vibrating (Metso) The Metso vibrating screen model is used to determine the partition of solids to oversize and undersize for each size interval.
Fine Wet (Mwale) The Mwale fine wet screen model is used to determine the partition of solids to oversize and undersize for each size interval.
HelpLink ButtonModelHelp.png Opens a link to this page using the system default web browser. Note: Internet access is required.
Parameters
Aperture / xa Input Size of the apertures in the deck.
OpenArea/ Ao Input Open area of the screening surface
KineticParameter / K Input Mwale model kinetic parameter
SharpnessCoeff1 / k1 Input Mwale model sharpness coefficient
SharpnessCoeff2 / k2 Input Mwale model sharpness coefficient
SharpnessCoeff3 / k3 Input Mwale model sharpness coefficient
BypassCoeff1 / C1 Input Mwale model bypass coefficient
BypassCoeff2 / C2 Input Mwale mode3 bypass coefficient
BypassCoeff3 / C3 Input Mwale model bypass coefficient
BypassCoeff4 / C4 Input Mwale model bypass coefficient
Liquids
LiquidsSeparMethod Split To OS (User) Liquids are split to oversize by a user-defined fraction of liquids in the feed.
OS Solids Fraction Sufficient liquids mass is recovered to the oversize stream to yield the user-defined oversize solids mass fraction value (if possible).
OS Liquids Fraction Sufficient liquids mass is recovered to the oversize stream to yield the user-defined oversize liquids mass fraction value (if possible).
OSSolidsFracReqd / OS.SfReqd Input Required value of the mass fraction of solids in the oversize stream. Only visible if OS Solids Fraction is selected.
OSLiquidsFracReqd / OS.LfReqd Input Required value of the mass fraction of liquids in the oversize stream. Only visible if OS Liquids Fraction is selected.
LiqSplitToOS / OS.LiqSplit Input/Display The fraction of feed liquids recovered to the oversize stream.
Results
SharpnessConstant / Alpha Display Mwale model sharpness constant
BypassConstant / Delta Display Mwale model bypass constant
Feed.SQm Display Mass flow rate of solids in the feed (excludes solids without PSD quality)
Feed.SQmSubAp Display Mass flow rate of particles in the feed which are smaller than the aperture size (sub-aperture)
Feed.Sf Display Mass fraction of solids in the feed (excludes solids without PSD quality)
Feed.SLRho Display Density of slurry in the feed
SlurryViscosity / mu Display Apparent viscosity of slurry in the feed
Efficiency Display Fraction of total sub-aperture sized material in feed that is actually recovered to the undersize stream.

Partition page

The Partition page is used to specify or display the partition by species and size values.

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.
Partition
Method Model/User Select model-calculated or user-defined partition to separate each solids species type.
Density Display Density of each solid species.
Size Display Size of each interval in mesh series.
MeanSize Display Geometric mean size of each interval in mesh series.
All (All column) Display
  • Actual overall partition to oversize of all solid species, for each size interval.
  • Excludes solid species not present in the screen feed.
Partition Display
  • Partition to oversize for each size interval, in each solid species, as determined by the selected model or user defined value.
  • Note: These values are displayed regardless of whether the solid species is present in the screen feed or not.
All (All row, All column) Display
  • Displays the actual, total, partition of all solids with a particle size distribution property in the feed to oversize.
  • Excludes solid species not present in the screen feed.
All (All row, per species) Display
  • Actual overall partition to oversize for each solid species, for all size intervals in that species.
  • Excludes solid species not present in the screen feed.

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 ButtonLicensingHelp.png Opens a link to the Installation and Licensing page using the system default web browser. Note: Internet access is required.
Information ButtonCopyToClipboard.png 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 ButtonBrowse.png 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.

Additional notes

  • Solid species that do not possess a particle size distribution property are split according to the overall mass split of the default particle size distribution species selected in the SysCAD Project Configuration.
  • If the default particle size distribution species is not present in the unit feed, the overall split of all other species with particle size distributions combined is used, as determined by the model.
  • Gas phase species report directly to the undersize stream without split.

References

  1. 1.0 1.1 Mwale, A.N., Mainza, A.N., Bepswa, P.A., Simukanga, S., Masinja, J.H., 2016. MODEL FOR FINE WET SCREENING. In XXVIII International Mineral Processing Congress Proceedings. Canadian Institute of Mining, Metallurgy and Petroleum.
  2. 2.0 2.1 Mwale, A.N., Mainza, A.N., Bepswa, P.A., Frausto, J.J., Ballantyne, G., Cruz, R. and Gomez, S., 2018. TESTING THE HIGH FREQUENCY FINE SCREEN MODEL USING INDUSTRIAL DATA. In XXIX International Mineral Processing Congress Proceedings. International Agency of Congress Management (MAKO LLC), Moscow, Russia, 2019.
  3. Krieger, I.M. and Dougherty, T.J., 1959. A mechanism for non‐Newtonian flow in suspensions of rigid spheres. Transactions of the Society of Rheology, 3(1), pp.137-152.