Pump (Centrifugal, Slurry): Difference between revisions

Latest revision as of 15:16, 27 June 2025

Description

The article describes a method for estimating the performance of a centrifugal slurry pump.

Centrifugal pumps are a critical component of metallurgical processing plants. The transport of solid particles in slurries presents additional complexity, as pumps are typically tested and rated for water duties only.

The centrifugal slurry pump modelling approach described below has the following features:

The reduction of typical vendor pump performance curves into generalised mathematical relationships for computational implementation
The de-rating of water duty pump performance for heterogenous slurries
Estimation of the effects of elevation, piping system properties and transport destination processing equipment (e.g. hydrocyclones)
An estimation of the settling velocity of slurries during transport

Note that the centrifugal slurry pump model described here is not intended to replace the more comprehensive design methods and tools currently available. Rather, it is intended to provide the user with a comparatively simpler tool which can be integrated with other unit models and provide useful estimates of pump speed, flow rate, motor power, and pressure head in the broader context of a mineral processing circuit. More sophisticated methods should be applied for engineering design or optimisation.

Model theory

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Generalised characteristic curves

Speed

Efficiency

Total dynamic head

Friction factor

Head ratio

Cave method

Warman method

Weir / HI method

Efficiency ratio

Motor Power

Froth Volume Factor

Settling velocity

Pumps in series and parallel

Excel

The centrifugal slurry pump model may be invoked from the Excel formula bar with the following function call:

=mdUnit_Pump_King(Parameters as Range)

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

The Parameters array and model results are defined below in matrix notation, along with example images showing the same arrays in the Excel interface:

Parameters={\begin{bmatrix}n_{\rm {stages}}\\n_{\rm {parallel}}\\{\text{Model mode}}\\{\text{f method}}\\{\text{HR method}}\\{\text{ER method}}\\{\text{FL method}}\\D_{\rm {imp}}{\text{ (m)}}\\A_{\rm {N}}\\B_{\rm {N}}\\C_{\rm {N}}\\A_{\rm {E}}\\B_{\rm {E}}\\C_{\rm {E}}\\(Q_{\rm {M}})_{\rm {S}}{\text{ (kg/s)}}\\(Q_{\rm {M}})_{\rm {L}}{\text{ (kg/s)}}\\\rho _{\rm {S}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\rho _{\rm {L}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\d_{50}{\text{ (m)}}\\{\rm {FVF}}{\text{ (v/v)}}\\\mu {\text{ (Pa.s)}}\\H_{\rm {s}}{\text{ (m)}}\\D{\text{ (m)}}\\L{\text{ (m)}}\\k{\text{ (m)}}\\H_{\rm {e}}{\text{ (kPa)}}\\\eta {\text{ (kW/kW)}}\\N{\text{ (Hz)}}\\f\\{\rm {HR}}{\text{ (m/m)}}\\{\rm {ER}}{\text{ (kW/kW)}}\\F_{\rm {L}}\\C_{\rm {fp}}{\text{ (frac)}}\\\end{bmatrix}},\;\;\;\;\;\;{\mathit {mdUnit\_Pump\_King}}={\begin{bmatrix}{\text{Iterations}}\\Q_{\rm {Feed}}{\text{ (m}}^{3}{\text{/s)}}\\Q{\text{ (m}}^{3}{\text{/s)}}\\\rho _{\rm {SL}}{\text{ (t/m}}^{3}{\text{)}}\\C_{\rm {W}}{\text{ (w/w)}}\\C_{\rm {V}}{\text{ (v/v)}}\\F_{\rm {L}}\\V_{\rm {L}}{\text{ (m/s)}}\\V{\text{ (m/s)}}\\\mathrm {Re} \\f\\H_{\rm {f}}{\text{ (m)}}\\H_{\rm {e}}{\text{ (m)}}\\H_{\rm {t}}{\text{ (m)}}\\{\rm {HR}}{\text{ (m/m)}}\\H_{\rm {w}}{\text{ (m)}}\\N{\text{ (Hz)}}\\N_{\rm {pu}}\\N_{\rm {Q}}\\E{\text{ (kW/kW)}}\\{\rm {ER}}{\text{(kW/kW)}}\\P_{\rm {shaft}}{\text{ (kW)}}\\P_{\rm {motor}}{\text{ (kW)}}\\\end{bmatrix}}\;\;\;\;\;\;

where:

${\text{Model mode}}$ indicates whether the pump model returns the flow rate at a given speed or the speed at a given flow rate, where
0 = Calculate speed at flow rate, 1 = Calculate flow rate at speed
${\text{f method}}$ is the friction factor estimation method, 0 = User, 1 = Colebrook, 2 = Churchill
${\text{HR method}}$ is the head ratio estimation method, 0 = User, 1 = Cave, 2 = Warman, 3 = Weir / HI
${\text{ER method}}$ is the efficiency ratio estimation method, 0 = User, 1 = ER equals HR, 2 = Warman, 3 = Weir / HI
${\text{FL method}}$ is the Durand factor estimation method, 0 = User, 1 = Equation
$(Q_{\rm {M}})_{\rm {S}}$ is the mass flow rate of solids through the pump (kg/s)
$(Q_{\rm {M}})_{\rm {L}}$ is the mass flow rate of liquids through the pump (kg/s)
${\text{Iterations}}$ is the number of internal model iterations required to resolve any dependence between friction head and pump flow rate
$Q_{\rm {Feed}}$ is the actual volumetric flow rate of slurry in the pump feed, computed from $(Q_{\rm {M}})_{\rm {S}}$ , $(Q_{\rm {M}})_{\rm {L}}$ , $\rho _{\rm {S}}$ , and $\rho _{\rm {L}}$

Figure 4. Example showing the selection of the Parameters (blue frame) array in Excel.

Figure 5. Example showing the Results (light blue frame) array in Excel.

When calculating speed at flow rate ( ${\text{Model mode}}=0$ ):

$Q=Q_{\rm {Feed}}$ and the value of $N$ is computed by the model and returned.
This mode might be used to simulate the operating point of a variable speed pump in a steady state model, for example.

When calculating flow rate at speed ( ${\text{Model mode}}=1$ ):

$N$ is a user input and $Q$ is computed and returned.
The return value of $Q$ may be different to $Q_{\rm {Feed}}$ .
This might indicate a quantity of make-up water is required to maintain the level of a tank feeding the pump, for example.

SysCAD

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

MD_Pump 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 no model calculations or actions are performed.
Options
ShowQIn	CheckBox	QIn and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream.
ShowQOut	CheckBox	QOut and associated tab pages (eg Sp) will become visible, showing the properties of the overflow 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.

Pump page

The Pump page is used to specify the input parameters for the centrifugal slurry pump model.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Pump
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
Iterations	Display	Shows the number of internal model iterations (per SysCAD step) required to converge the pump model.
Duty
NumStages	Input	The number of sequential duplicate pump stages.
NumParallel	Input	The number of duplicate pumps in parallel.
Config
CalculationMode	Speed	The model calculates the speed required to pump the volumetric flow rate of the feed stream, at the given head.
CalculationMode	Flow Rate	The model calculates the pump flow rate at the user-specified speed and head.
CharacteristicCurve
ImpellerDiameter / Dimp	Input	Diameter of the pump impeller.
AN	Input	Coefficient of the generalised characteristic curve equation.
BN	Input	Coefficient of the generalised characteristic curve equation.
CN	Input	Coefficient of the generalised characteristic curve equation.
EfficiencyCurve
AE	Input	Coefficient of the generalised efficiency curve equation.
BE	Input	Coefficient of the generalised efficiency curve equation.
CE	Input	Coefficient of the generalised efficiency curve equation.
Feed
FrothVolumeFactor / FVF	Input	Value of the Froth Volume Factor.
VolFlow / Qv	Display	Volumetric flow of solids and liquids in the pump feed stream, adjusted by the Froth Volume Factor.
SolidDensity / SRho	Display	Density of solids in the pump feed stream.
Liquids / LRho	Display	Density of liquids in the pump feed stream.
SlurryDensity / SLRho	Display	Density of slurry (solids plus liquids) in the pump feed stream.
SolidFrac / Sf	Display	Mass fraction of solids in the feed stream.
SolidVolFrac / Svf	Display	Volume fraction of solids in the feed stream.
LViscosity	Display	Viscosity of liquids in the feed stream.
Userd50	CheckBox	Indicates user-specified d50 value. Default is to use d50 computed from the feed stream.
d50	Input/Display	Mean size of particles in pump feed.
Head
StaticHead
StaticHead / Hs	Input	Static head component of the total dynamic head.
FrictionHead
Method	User Defined	The friction factor is specified by the user.
	Colebrook	The Colebrook equation is used to estimate the friction factor.
	Churchill	The Churchill equation is used to estimate the friction factor.
PipeDiameter / D	Input	Internal diameter of the pipe
EquivPipeLength / L	Input	Equivalent length of the pipe, including pipe fittings etc.
FrictionFactor / f	Input / Display	Friction factor used to estimate friction head.
Roughness / k	Input	Absolute surface roughness of the pipe internal wall.
ReynoldsNumber / Re	Display	Reynolds Number of the pipe flow stream.
FrictionHead / Hf	Display	Friction head component of the total dynamic head.
EquipmentHead
EquipmentPressure / Pe	Input	Pressure drop due to equipment at the end of the pipe.
EquipmentHead / He	Display	Pressure drop due to equipment at the end of the pipe, converted to head measurement units.
Total head
TotalHead / Ht	Display	Total dynamic head that the pump acts against. Sum of static, friction and equipment heads.
HeadRatio
Method	User Defined	The head ratio is specified by the user.
	Cave	Cave's method is used to estimate the head ratio.
	Warman	The Warman method is used to estimate the head ratio.
	Weir / HI	The Weir / HI method is used to estimate the head ratio.
HeadRatio / HR	Input / Display	Head ratio of the slurry stream.
EquivWaterHead / Hw	Display	Head of water equivalent to the slurry total dynamic head, as estimated via the head ratio.
EfficiencyRatio
Method	User Defined	The efficiency ratio is specified by the user.
	Equals HR	The efficiency ratio is set equal to the value of the head ratio.
	Warman	The Warman method is used to estimate the efficiency ratio.
	Weir / HI	The Weir / HI method is used to estimate the efficiency ratio.
EfficiencyRatio / ER	Input / Display	Efficiency ratio of the slurry stream.
SettlingVelocity
Method	User Defined	The Durand factor is specified by the user.
Method	Durand	The Durand factor is estimated by the Durand equation.
DurandFactor / FL	Input / Display	Durand factor to be used in the Durand equation.
SettlingVelocity / VL	Display	Settling velocity estimated by the Durand equation.
PipeVelocity / V	Display	Velocity of flow through the pipe at pump flow rate, Q.
Speed
Iterations	Display	The number of internal model iterations required to resolve any dependence between friction head and pump flow rate. Only applicable when CalculationMode = Flow Rate.
N	Input / Display	Rotational speed of the pump.
Q	Display	Volumetric flow rate of the pump at the given speed and head.
HeadNumber / NPU	Display	Dimensionless pump head number used in calculations.
FlowNumber / NQ	Display	Dimensionless pump flow number used in calculations.
Power
MotorFactor / Eta	Input	Efficiency factor of the pump. Fraction of power input to the motor which is useable by the pump at the shaft to move fluid.
Efficiency / E	Display	Efficiency of the pump.
ShaftPower / PShaft	Display	Power drawn by the pump at the shaft.
MotorPower / PMotor	Display	Power drawn by the pump motor, including inefficiencies.

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.

External links

References

@@ Line 186: / Line 186: @@
 </div><hide>
-[[File:Pump3.png|x600px|thumb|link={{filepath:Pump3.png}}|Figure 3. Weir Head and Efficiency Ratios nomograph, after Grizina et al. (2002).{{Grzina et al. (2002)}} Example progression through the nomograph is indicated by dashed arrows. (Click image for larger view).]]
+[[File:Pump3.png|x500px|thumb|link={{filepath:Pump3.png}}|Figure 3. Warman Head and Efficiency Ratios nomograph, after Grizina et al. (2002).{{Grzina et al. (2002)}} Example progression through the nomograph is indicated by dashed arrows. (Click image for larger view).]]
 Warman provides a ''nomograph'' method for approximating the head ratio of a pumped slurry.{{Grzina et al. (2002)}} The nomograph form is reproduced in Figure 3, including an example path through the approximation procedure. The property <math>C_{\rm V}</math> used by the nomograph is the concentration of solids by volume in the slurry (v/v).
@@ Line 193: / Line 193: @@
 </hide><div class="user-show">
+==== Weir / HI method ====
+</div><hide>
+Weir Minerals provide an updated method to estimate the head ratio based on the following equations:{{Roudnev and Loderer (2023)}}
+:<math>\mathrm{HR} = 1 - 0.01 \cdot \left [ A \cdot \left ( \dfrac{1.11}{D_{\rm imp}} \right )^{0.9} \cdot {d_{50}}^B \cdot \left ( \dfrac{\rho_{\rm S} - \rho_{\rm L}}{1.65} \right )^{0.65} \cdot \dfrac{C_{\rm V}}{0.15} \cdot (1 - {C_{\rm fp}})^2 \right ]</math>
+where:
+:<math>A =
+\begin{cases}
+.04, & D_{\rm imp} \leq 0.41 \text{ m}\\
+.005124 \, D_{\rm imp} + 1.9388, & 0.41 \text{ m} < D_{\rm imp} \leq 0.90 \text{ m}\\
+.5, & D_{\rm imp} > 0.89 \text{ m}\\
+\end{cases}
+</math>
+:<math>B = 0.4 \cdot \left ( d_{50} \right )^{-0.25}</math>
+and:
+* <math>C_{\rm V}</math> is the concentration of solids in the slurry by volume (v/v)
+* <math>C_{\rm fp}</math> is cumulative fraction passing 75 μm of particles in the feed (frac)
+* <math>\rho_{\rm L}</math> is the density of liquids (t/m<sup>3</sup>)
+Weir Minerals refer to this approach as the ''Weir Standard''. The method appears to be functionally equivalent to the Hydraulic Institute (HI) method described in ''ANSI/HI 12.1-12.6-2016 Rotodynamic Centrifugal Slurry Pumps''.{{Hydraulic Institute (2016)}}
+</hide><div class="user-show">
 === Efficiency ratio ===
 </div><hide>
@@ Line 202: / Line 230: @@
 where <math>E_{\mathit{eff}}</math> is the effective efficiency for the pump (kW/kW), accounting for slurry derating of the efficiency estimated from the characteristic curve.
-The efficiency ratio is sometimes reported to be approximately equal to the head ratio. {{Engin and Gur (2003)}} Alternatively, the Warman nomograph provides a method for estimating the efficiency ratio based on the head ratio and volume fraction solids of the slurry (Figure 3, bottom right).
+The efficiency ratio is sometimes reported to be approximately equal to the head ratio. {{Engin and Gur (2003)}}
+Alternatively, the Warman nomograph provides a method for estimating the efficiency ratio based on the head ratio and volume fraction solids of the slurry (Figure 3, bottom right). The updated Weir / HI method also applies the same Warman nomograph approach.
 </hide><div class="user-show">
 === Motor Power ===
 </div><hide>
@@ Line 324: / Line 355: @@
 {\rm ER}\text{ (kW/kW)}\\
 F_{\rm L}\\
+C_{\rm fp}\text{ (frac)}\\
 \end{bmatrix},\;\;\;\;\;\;
@@ Line 358: / Line 390: @@
 * <math>\text{Model mode}</math> indicates whether the pump model returns the flow rate at a given speed or the speed at a given flow rate, where <br>''0 = Calculate speed at flow rate, 1 = Calculate flow rate at speed''
 * <math>\text{f method}</math> is the friction factor estimation method, ''0 = User, 1 = Colebrook, 2 = Churchill''
-* <math>\text{HR method}</math> is the head ratio estimation method, ''0 = User, 1 = Cave, 2 = Warman''
+* <math>\text{HR method}</math> is the head ratio estimation method, ''0 = User, 1 = Cave, 2 = Warman, 3 = Weir / HI''
-* <math>\text{ER method}</math> is the efficiency ratio estimation method, ''0 = User, 1 = ER equals HR, 2 = Warman''
+* <math>\text{ER method}</math> is the efficiency ratio estimation method, ''0 = User, 1 = ER equals HR, 2 = Warman, 3 = Weir / HI''
 * <math>\text{FL method}</math> is the Durand factor estimation method, ''0 = User, 1 = Equation''
 * <math>(Q_{\rm M})_{\rm S}</math> is the mass flow rate of solids through the pump (kg/s)
@@ Line 598: / Line 630: @@
 ! colspan="3" style="text-align:left;" |''HeadRatio''
 |- style="vertical-align:top;"
-|rowspan="3" | Method
+|rowspan="4" | Method
 |User Defined
 |The head ratio is specified by the user.
@@ Line 607: / Line 639: @@
 |Warman
 |The Warman method is used to estimate the head ratio.
+|- style="vertical-align:top;"
+|Weir / HI
+|The Weir / HI method is used to estimate the head ratio.
 |-
 |HeadRatio / HR
@@ Line 618: / Line 653: @@
 ! colspan="3" style="text-align:left;" |''EfficiencyRatio''
 |- style="vertical-align:top;"
-|rowspan="3" | Method
+|rowspan="4" | Method
 |User Defined
 |The efficiency ratio is specified by the user.
@@ Line 627: / Line 662: @@
 |Warman
 |The Warman method is used to estimate the efficiency ratio.
+|- style="vertical-align:top;"
+|Weir / HI
+|The Weir / HI method is used to estimate the efficiency ratio.
 |-
 |EfficiencyRatio / ER

Pump (Centrifugal, Slurry): Difference between revisions

Latest revision as of 15:16, 27 June 2025

Contents

Description