Properties (Dense Medium Viscosity)

Description

This article describes the Shi model for estimating the apparent viscosity of dense media suspensions. Originally developed by Shi and Napier-Munn (1996a, 1996b), the model was later refined by Dunglison et al. (1999).

Dense medium suspensions exhibit non-Newtonian rheology, with shear-thinning behaviour and measurable yield stress. The Shi model accounts for these characteristics using a structured method that derives apparent viscosity flow curves from the observed behaviour of rotational viscometers.

The model is used to predict viscosity values for process models requiring dense media rheology estimates, and for practical metallurgical tasks such as plant design and optimisation.

Model theory

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MV model

Torque and shear stress

Shear rate calculation

Turbulence Correction (TC method)

Final apparent viscosity

A cubic spline is fitted to the corrected flow curve $({\dot {\gamma }},S_{\rm {r}})$. The apparent viscosity $\mu$ (kg/m.s) at a given shear rate is calculated from:

Dunglison TC curve

Excel

The Shi viscosity model may be invoked from the Excel formula bar with the following function call:

=mdProps_DMViscosity_Shi(Parameters as Range, Optional TCCurve as Range)

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

The input parameters and calculation results are defined below in matrix notation, along with an example image showing the selection of the same cells and arrays in the Excel interface:

${\begin{aligned}Parameters&={\begin{bmatrix}P_{80}{\text{ (mm)}}\\P_{20}{\text{ (mm)}}\\{\text{Medium type}}\\T{\text{ (deg. C)}}\\\rho _{\rm {f}}{\text{ (kg/m}}^{3}{\text{)}}\\\rho _{\rm {med}}{\text{ (kg/m}}^{3}{\text{)}}\\\rho _{\rm {L}}{\text{ (kg/m}}^{3}{\text{)}}\\{\dot {\gamma }}{\text{ (s}}^{-1}{\text{)}}\\\end{bmatrix}}\\\\{\mathit {TCCurve}}&={\begin{bmatrix}(C_{\rm {m}})_{1}{\text{ (-)}}&(C_{\rm {t}})_{1}{\text{ (-)}}\\\vdots &\vdots \\(C_{\rm {m}})_{n}{\text{ (-)}}&(C_{\rm {t}})_{n}{\text{ (-)}}\\\end{bmatrix}}^{*}\\\end{aligned}}$ ${\mathit {mdProps\_DMViscosity\_Shi}}={\begin{bmatrix}\mu {\text{ (kg/m.s)}}\\S_{\rm {r}}{\text{ (Pa)}}\\\psi {\text{ (Pa)}}\\\end{bmatrix}}$ where: $\rho _{\rm {med}}$ is density of the medium solids (kg/m3) $\rho _{\rm {L}}$ is density of the medium liquids (kg/m3) $n$ is the number of data points comprising the optional user-specified TC curve the superscript $^{*}$ indicates an optional input (default TC curve applied if omitted).

Figure 1. Example showing the selection of the Parameters (blue frame) Results (light blue frame) arrays in Excel.

SysCAD

The Shi viscosity model model is an optional calculation for dense medium separator units. If selected, the input and display parameters below are shown.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Viscosity
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
MediumSolids	Input / List	Specifies which species should be considered as the medium solids by the model. The drop down list will contain all solids species in the project, plus "-" for no selection. Medium properties will be user-specified (-) or extracted from the selected medium species in the feed stream.
MediumP80	Input / Display	The 80% passing size of the medium particles. User input value if MediumSolids is "-" or the selected medium solids species does not posses the size distribution property. Set by the properties of the selected medium solids species in the feed otherwise.
MediumP20	Input / Display	The 20% passing size of the medium particles. User input value if MediumSolids is "-" or the selected medium solids species does not posses the size distribution property. Set by the properties of the selected medium solids species in the feed otherwise.
MediumType	Atomised FeSi	The dense media is atomised ferrosilicon.
	Milled FeSi	The dense media is milled ferrosilicon.
	Magnetite	The dense media is magnetite.
MediumTemperature	Input / Display	Temperature of the dense medium. User input value if MediumSolids is "-" or SysCAD heat calculations are disabled. Set by the mixed temperature of the feed stream otherwise.
MediumDensity	Input / Display	Density of the medium. User input value if MediumSolids is "-". Set by the properties of the selected medium solids species in the feed otherwise.
MediumSolidsDensity	Input / Display	Density of the medium solids. User input value if MediumSolids is "-". Set by the properties of the selected medium solids species in the feed otherwise.
LiquidDensity	Input / Display	Density of the liquids in the medium. User input value if MediumSolids is "-". Set by the properties of the selected medium solids species in the feed otherwise.
ShearRate	Input	Shear rate applied to the medium.
ApparentViscosity	Display	Apparent viscosity of the medium under shear.
ShearStress	Display	Shear stress of the medium at the shear rate.
YieldStress	Display	Yield stress of the medium.