Pump (Centrifugal, Slurry): Difference between revisions
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== Model theory == | == Model theory == | ||
{{Restricted content}} | |||
<hide> | |||
[[File:Pump1.png|450px|thumb|Figure 1. A pump characteristic curve in the format typically provided by vendors, after Weir (2009).{{Weir (2009)}}]] | [[File:Pump1.png|450px|thumb|Figure 1. A pump characteristic curve in the format typically provided by vendors, after Weir (2009).{{Weir (2009)}}]] | ||
[[File:Pump2.png|450px|thumb|Figure 2. The speed and | [[File:Pump2.png|450px|thumb|Figure 2. The speed and efficiency curve intersection points in Figure 1 are collapsed and regressed to generalised quadratic functions via the method described by King (2002).{{King (2002)}}]] | ||
The centrifugal slurry pump modelling methodology outlined in this section applies to pump duties where:{{Griffiths (2003)}} | The centrifugal slurry pump modelling methodology outlined in this section applies to pump duties where:{{Griffiths (2003)}} | ||
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The volumetric '''flow rate''' provided by a pump operating at a given speed and pressure head can be computed via the quadratic formula: | The volumetric '''flow rate''' provided by a pump operating at a given speed and pressure head can be computed via the quadratic formula: | ||
:<math> N_{\rm Q} = \dfrac{ | :<math> N_{\rm Q} = \dfrac{B_{\rm N} - \sqrt{{B_{\rm N}}^2 + 4 C_{\rm N} (A_{\rm N} - N_{\rm pu})}}{-2 C_{\rm N}}</math> | ||
and <math>Q = N_{\rm Q} \cdot N \cdot {D_{\rm imp}}^3</math>. | and <math>Q = N_{\rm Q} \cdot N \cdot {D_{\rm imp}}^3</math>. | ||
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where <math>\rho_{\rm SL}</math> is the density of slurry (t/m<sup>3</sup>) and <math>\mu</math> is the dynamic viscosity of the liquid (cP). | where <math>\rho_{\rm SL}</math> is the density of slurry (t/m<sup>3</sup>) and <math>\mu</math> is the dynamic viscosity of the liquid (cP). | ||
The Colebrook equation is valid for Reynolds numbers in excess of 4,000. Furthermore, the Colebrook equation requires solution by iterative numerical means due to <math>f</math> appearing in both | The Colebrook equation is valid for Reynolds numbers in excess of 4,000. Furthermore, the Colebrook equation requires solution by iterative numerical means due to <math>f</math> appearing in both sides of the equation. | ||
The '''Churchill''' approximation to Colebrook's formula is: | The '''Churchill''' approximation to Colebrook's formula is: | ||
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Centrifugal slurry pump derating is undertaken by approximating the head of water that is equivalent to the head of a particular slurry being pumped, via the equation: | Centrifugal slurry pump derating is undertaken by approximating the head of water that is equivalent to the head of a particular slurry being pumped, via the equation: | ||
:<math>H_{\rm w} = \dfrac{H_{\rm t}}{\ | :<math>H_{\rm w} = \dfrac{H_{\rm t}}{{\rm HR}}</math> | ||
where <math>H_{\rm w}</math> is the equivalent head of water (m) and <math>\ | where <math>H_{\rm w}</math> is the equivalent head of water (m) and <math>{\rm HR}</math> is the head ratio (m/m) for the slurry. | ||
The equivalent water head, <math>H_{\rm w}</math>, is then used in the characteristic curve equations above in place of the total head, <math>H_{\rm t}</math>. | The equivalent water head, <math>H_{\rm w}</math>, is then used in the characteristic curve equations above in place of the total head, <math>H_{\rm t}</math>. | ||
The head ratio, <math>\ | The head ratio, <math>{\rm HR}</math> can be estimated by many means. Two approaches are included in the centrifugal slurry pump model, the ''Cave'' and ''Warman'' approaches. | ||
==== Cave method ==== | ==== Cave method ==== | ||
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The Cave equation is:{{Engin and Gur (2003)}} | The Cave equation is:{{Engin and Gur (2003)}} | ||
:<math>\ | :<math>{\rm HR} = 1 - 0.0385 (\rho_{\rm S} - 1) \left ( \dfrac{\rho_{\rm S} + 4}{\rho_{\rm S}} \right ) C_{\rm W} \ln \left ( \dfrac{d_{50}}{22.7} \right )</math> | ||
where: | where: | ||
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=== Efficiency ratio === | === Efficiency ratio === | ||
Similarly to the head ratio, the efficiency ratio <math>\ | Similarly to the head ratio, the efficiency ratio <math>{\rm ER}</math> (kW/kW) derates pump efficiency for slurry duty, i.e.: | ||
:<math>E_{\mathit{eff}} = \ | :<math>E_{\mathit{eff}} = {\rm ER} \cdot E</math> | ||
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. | 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. | ||
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The power drawn by the pumping duty at the drive shaft, <math>P_{\rm shaft}</math> (kW), is:{{King (2002)}} | The power drawn by the pumping duty at the drive shaft, <math>P_{\rm shaft}</math> (kW), is:{{King (2002)}} | ||
:<math>P_{\rm shaft} = \dfrac{Q \cdot \rho_{\rm SL} \cdot g \cdot H_{\rm t}}{E \cdot \ | :<math>P_{\rm shaft} = \dfrac{Q \cdot \rho_{\rm SL} \cdot g \cdot H_{\rm t}}{E \cdot {\rm ER}}</math> | ||
Due to electrical and mechanical inefficiencies, the power drawn by the motor, <math>P_{\rm motor}</math> (kW), is: | Due to electrical and mechanical inefficiencies, the power drawn by the motor, <math>P_{\rm motor}</math> (kW), is: | ||
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where <math>\eta</math> is the motor factor (kW/kW). | where <math>\eta</math> is the motor factor (kW/kW). | ||
=== Froth Volume Factor === | |||
When determining pump performance, the volumetric flow rate of slurry may need to be adjusted for the presence of ''froth'':{{Weir (2009b)}} | |||
:<math>Q = {\rm FVF} \cdot \big [(Q_{\rm V})_{\rm S} + (Q_{\rm V})_{\rm L} \big ]</math> | |||
where: | |||
* <math>{\rm FVF}</math> is the Froth Volume Factor, the ratio of frothed slurry volume to the original (un-frothed) slurry volume (v/v) | |||
* <math>(Q_{\rm V})_{\rm S}</math> and <math>(Q_{\rm V})_{\rm L}</math> are the volumetric flow rates of solids and liquids, respectively, in the slurry (m<sup>3</sup>/s) | |||
Slurry density is similarly adjusted: | |||
:<math>\rho_{\rm F} = \dfrac{\rho_{\rm SL}}{{\rm FVF}}</math> | |||
where <math>\rho_{\rm F}</math> (t/m<sup>3</sup>) is the density of the frothed slurry, used in place of <math>\rho_{\rm SL}</math> in the pump performance equations when froth is present. | |||
=== Settling velocity === | |||
The settling velocity of a heterogenous slurry can be estimated by using the Durand equation:{{Dunne et al. (2019)}} | |||
:<math>V_{\rm L} = F_{\rm L} \sqrt{\dfrac{2gD \rho_{\rm S} - \rho_{\rm L}}{\rho_{\rm L}}}</math> | |||
where <math>\rho_{\rm L}</math> is the liquid density (t/m<sup>3</sup>), and <math>F_{\rm L}</math> is the dimensionless Durand factor which may be approximated by:{{Griffiths (2003)}} | |||
:<math>F_{\rm L} = 0.4794 + 0.5429 {C_{\rm V}}^{0.1058} \log (d_{50}) - 1</math> | |||
The settling velocity may be compared to the actual pipe velocity to asses whether settling may be an issue for the application in question. | |||
=== Pumps in series and parallel === | === Pumps in series and parallel === | ||
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Note that the assumption of identical performance for pumps within series or parallel configurations is a major simplification, which may not be strictly valid in practice due to differing piping system properties, pump wear, control configuration etc. | Note that the assumption of identical performance for pumps within series or parallel configurations is a major simplification, which may not be strictly valid in practice due to differing piping system properties, pump wear, control configuration etc. | ||
</hide> | |||
== Excel == | == Excel == | ||
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\rho_{\rm L}\text{ (t/m}^\text{3}\text{)}\\ | \rho_{\rm L}\text{ (t/m}^\text{3}\text{)}\\ | ||
d_{50}\text{ (m)}\\ | d_{50}\text{ (m)}\\ | ||
{\rm FVF}\text{ (v/v)}\\ | |||
\mu\text{ (Pa.s)}\\ | \mu\text{ (Pa.s)}\\ | ||
H_{\rm s}\text{ (m)}\\ | H_{\rm s}\text{ (m)}\\ | ||
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N\text{ (Hz)}\\ | N\text{ (Hz)}\\ | ||
f\\ | f\\ | ||
\ | {\rm HR}\text{ (m/m)}\\ | ||
\ | {\rm ER}\text{ (kW/kW)}\\ | ||
F_{\rm L}\\ | F_{\rm L}\\ | ||
\end{bmatrix},\;\;\;\;\;\; | \end{bmatrix},\;\;\;\;\;\; | ||
mdUnit\_Pump\_King= | \mathit{mdUnit\_Pump\_King} = | ||
\begin{bmatrix} | \begin{bmatrix} | ||
\text{Iterations}\\ | \text{Iterations}\\ | ||
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H_{\rm e}\text{ (m)}\\ | H_{\rm e}\text{ (m)}\\ | ||
H_{\rm t}\text{ (m)}\\ | H_{\rm t}\text{ (m)}\\ | ||
\ | {\rm HR}\text{ (m/m)}\\ | ||
H_{\rm w}\text{ (m)}\\ | H_{\rm w}\text{ (m)}\\ | ||
N\text{ (Hz)}\\ | N\text{ (Hz)}\\ | ||
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N_{\rm Q}\\ | N_{\rm Q}\\ | ||
E\text{ (kW/kW)}\\ | E\text{ (kW/kW)}\\ | ||
\ | {\rm ER}\text{(kW/kW)}\\ | ||
P_{\rm shaft}\text{ (kW)}\\ | P_{\rm shaft}\text{ (kW)}\\ | ||
P_{\rm motor}\text{ (kW)}\\ | P_{\rm motor}\text{ (kW)}\\ | ||
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* <math>(Q_{\rm M})_{\rm S}</math> is the mass flow rate of solids through the pump (kg/s) | * <math>(Q_{\rm M})_{\rm S}</math> is the mass flow rate of solids through the pump (kg/s) | ||
* <math>(Q_{\rm M})_{\rm L}</math> is the mass flow rate of liquids through the pump (kg/s) | * <math>(Q_{\rm M})_{\rm L}</math> is the mass flow rate of liquids through the pump (kg/s) | ||
* <math>Iterations</math> is the number of internal model iterations required to resolve any dependence between friction head and pump flow rate | * <math>\text{Iterations}</math> is the number of internal model iterations required to resolve any dependence between friction head and pump flow rate | ||
* <math>Q_{\rm Feed}</math> is the actual volumetric flow rate of slurry in the pump feed, computed from <math>(Q_{\rm M})_{\rm S}</math>, <math>(Q_{\rm M})_{\rm L}</math>, <math>\rho_{\rm S}</math>, and <math>\rho_{\rm L}</math> | * <math>Q_{\rm Feed}</math> is the actual volumetric flow rate of slurry in the pump feed, computed from <math>(Q_{\rm M})_{\rm S}</math>, <math>(Q_{\rm M})_{\rm L}</math>, <math>\rho_{\rm S}</math>, and <math>\rho_{\rm L}</math> | ||
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The sections and variable names used in the SysCAD interface are described in detail in the following tables. | The sections and variable names used in the SysCAD interface are described in detail in the following tables. | ||
==== {{ | ==== {{SysCAD (Text, UnitType Prefix)}}Pump page ==== | ||
The first tab page in the access window will have this name. | The first tab page in the access window will have this name. | ||
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|Input | |Input | ||
|Coefficient of the generalised efficiency curve equation. | |Coefficient of the generalised efficiency curve equation. | ||
|- | |||
! colspan="3" style="text-align:left;" |''Feed'' | |||
|- | |||
|FrothVolumeFactor / FVF | |||
|Input | |||
|Value of the Froth Volume Factor. | |||
|- | |||
|VolFlow / Qv | |||
|style="background: #eaecf0" | Display | |||
|Volumetric flow of solids and liquids in the pump feed stream, adjusted by the Froth Volume Factor. | |||
|- | |||
|SolidDensity / SRho | |||
|style="background: #eaecf0" | Display | |||
|Density of solids in the pump feed stream. | |||
|- | |||
|Liquids / LRho | |||
|style="background: #eaecf0" | Display | |||
|Density of liquids in the pump feed stream. | |||
|- | |||
|SlurryDensity / SLRho | |||
|style="background: #eaecf0" | Display | |||
|Density of slurry (solids plus liquids) in the pump feed stream. | |||
|- | |||
|SolidFrac / Sf | |||
|style="background: #eaecf0" | Display | |||
|Mass fraction of solids in the feed stream. | |||
|- | |||
|SolidVolFrac / Svf | |||
|style="background: #eaecf0" | Display | |||
|Volume fraction of solids in the feed stream. | |||
|- | |||
|LViscosity | |||
|style="background: #eaecf0" | 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 | |||
|style="background: #eaecf0" | Input/Display | |||
|Mean size of particles in pump feed. | |||
|- | |- | ||
! colspan="3" style="text-align:left;" |''Head'' | ! colspan="3" style="text-align:left;" |''Head'' | ||
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|Equivalent length of the pipe, including pipe fittings etc. | |Equivalent length of the pipe, including pipe fittings etc. | ||
|- | |- | ||
| | |FrictionFactor / f | ||
|style="background: #eaecf0" | Input / Display | |style="background: #eaecf0" | Input / Display | ||
|Friction factor used to estimate friction head. | |Friction factor used to estimate friction head. | ||
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== References == | == References == | ||
[[Category:Excel]] | |||
[[Category:SysCAD]] | |||
[[Category:Dynamic]] |
Latest revision as of 10:17, 4 December 2024
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
This content is available to registered users. Please log in to view. |
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:
|
When calculating speed at flow rate ():
- and the value of 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 ():
- is a user input and is computed and returned.
- The return value of may be different to .
- 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. |
Flow Rate | The model calculates the pump flow rate at the user-specified speed and head. | |
TransferPull | CheckBox |
|
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. | |
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. | |
EfficiencyRatio / ER | Input / Display | Efficiency ratio of the slurry stream. |
SettlingVelocity | ||
Method | User Defined | The Durand factor is specified by the user. |
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 | ||
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. |