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 efference curve intersection points in Figure 1 are collapsed and regressed to generalised quadratic functions via the method described by King (2002).{{King (2002)}}]]
[[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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==== Speed ====  
==== Speed ====  


King's procedure begins by specifying the pump head and flow rate in terms of two dimensionless groups, the pump head number, <math>N_{pu}</math>, and the pump flow number, <math>N_Q</math>,:
King's procedure begins by specifying the pump head and flow rate in terms of two dimensionless groups, the pump head number, <math>N_{\rm pu}</math>, and the pump flow number, <math>N_{\rm Q}</math>,:


:<math>N_{pu} = \dfrac{H_{t} g}{N^2{D_{imp}}^2}</math>
:<math>N_{\rm pu} = \dfrac{H_{\rm t} g}{N^2{D_{\rm imp}}^2}</math>


:<math>N_{Q} = \dfrac{Q}{N \cdot {D_{imp}}^3}</math>
:<math>N_{Q} = \dfrac{Q}{N \cdot {D_{\rm imp}}^3}</math>


where:
where:
* <math>H_t</math> is the total pressure head of the pump (m)
* <math>H_{\rm t}</math> is the total pressure head of the pump (m)
* <math>g</math> is acceleration due to gravity (m/s<sup>2</sup>)
* <math>g</math> is acceleration due to gravity (m/s<sup>2</sup>)
* <math>N</math> is the pump rotation speed (revolutions per second, or Hz)
* <math>N</math> is the pump rotation speed (revolutions per second, or Hz)
* <math>D_{imp}</math> is the pump impeller diameter (m)
* <math>D_{\rm imp}</math> is the pump impeller diameter (m)
* <math>Q</math> is the volumetric flow rate through the pump (m<sup>3</sup>/s)
* <math>Q</math> is the volumetric flow rate through the pump (m<sup>3</sup>/s)


The pump head number and pump flow number may then be linked by the following relation:
The pump head number and pump flow number may then be linked by the following relation:


:<math>N_{pu} = A_N - B_N N_Q - C_N {N_Q}^2</math>
:<math>N_{\rm pu} = A_{\rm N} - B_{\rm N} N_{\rm Q} - C_{\rm N} {N_{\rm Q}}^2</math>


where <math>A_N</math>, <math>B_N</math>, and <math>C_N</math> are the coefficients of the ''generalised speed characteristic curve'' function.
where <math>A_{\rm N}</math>, <math>B_{\rm N}</math>, and <math>C_{\rm N}</math> are the coefficients of the ''generalised speed characteristic curve'' function.


The coefficients of the above equation may be obtained from a set of characteristic curves (e.g. Figure 1) via regression, as demonstrated by the upper chart of Figure 2.
The coefficients of the above equation may be obtained from a set of characteristic curves (e.g. Figure 1) via regression, as demonstrated by the upper chart of Figure 2.
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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_Q = \dfrac{- B_N - \sqrt{{B_N}^2 + 4 C_N (A_N - N_{pu})}}{2 C_N}</math>
:<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_Q \cdot N \cdot {D_{imp}}^3</math>.
and <math>Q = N_{\rm Q} \cdot N \cdot {D_{\rm imp}}^3</math>.


Alternatively, the '''speed''' of a pump providing a given volumetric flow rate at a given pressure head may be computed by substituting the equations for <math>N_{pu}</math> and <math>N_Q</math> into the generalised characteristic speed equation and solving for <math>N</math>:
Alternatively, the '''speed''' of a pump providing a given volumetric flow rate at a given pressure head may be computed by substituting the equations for <math>N_{\rm pu}</math> and <math>N_{\rm Q}</math> into the generalised characteristic speed equation and solving for <math>N</math>:


:<math>\dfrac{H_t g}{N^2{D_{imp}}^2} = A_N - B_N \left (\dfrac{Q}{N \cdot {D_{imp}}^3} \right ) - C_N \left( \dfrac{Q}{N \cdot {D_{imp}}^3} \right)^2 \implies N = \dfrac{B_N Q + \sqrt{(B_NQ)^2 - 4 A_N (-g{D_{imp}}^4H_t-C_NQ^2)}}{2 {D_{imp}}^3 A_N}</math>
:<math>\dfrac{H_{\rm t} g}{N^2{D_{\rm imp}}^2} = A_{\rm N} - B_{\rm N} \left (\dfrac{Q}{N \cdot {D_{\rm imp}}^3} \right ) - C_{\rm N} \left( \dfrac{Q}{N \cdot {D_{\rm imp}}^3} \right)^2 \implies N = \dfrac{B_{\rm N} Q + \sqrt{(B_{\rm N}Q)^2 - 4 A_{\rm N} (-g{D_{\rm imp}}^4H_{\rm t}-C_{\rm N}Q^2)}}{2 {D_{\rm imp}}^3 A_{\rm N}}</math>


==== Efficiency ====
==== Efficiency ====
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Although not explicitly described by King, the efficiency characteristic curves are also amenable to the same generalisation procedure as the speed curves, i.e.:
Although not explicitly described by King, the efficiency characteristic curves are also amenable to the same generalisation procedure as the speed curves, i.e.:


:<math>E = A_{E} - B_{E} N_Q - C_{E} {N_Q}^2</math>
:<math>E = A_{\rm E} - B_{\rm E} N_{\rm Q} - C_{\rm E} {N_{\rm Q}}^2</math>


where <math>E</math> is the pump efficiency (kW/kW), and <math>A_E</math>, <math>B_E</math>, and <math>C_E</math> are the coefficients of the ''generalised efficiency characteristic curve'' function.
where <math>E</math> is the pump efficiency (kW/kW), and <math>A_{\rm E}</math>, <math>B_{\rm E}</math>, and <math>C_{\rm E}</math> are the coefficients of the ''generalised efficiency characteristic curve'' function.


As with the speed curves, regression techniques may be applied to extract the coefficients of the above equation, an example of which is shown in the lower part of Figure 2.
As with the speed curves, regression techniques may be applied to extract the coefficients of the above equation, an example of which is shown in the lower part of Figure 2.
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The ''total dynamic head'' is the equivalent pressure head that a pump acts against, taking into account static head, friction losses in the pipe, and pressure drops across connected equipment at the piping terminus, i.e.:
The ''total dynamic head'' is the equivalent pressure head that a pump acts against, taking into account static head, friction losses in the pipe, and pressure drops across connected equipment at the piping terminus, i.e.:


:<math>H_t = H_s + H_f + H_e</math>
:<math>H_{\rm t} = H_{\rm s} + H_{\rm f} + H_{\rm e}</math>


where:
where:
* <math>H_s</math> is the static head arising from elevation changes between the source and destination, including fluid head above the pump suction intake (e.g. a tank) (m)
* <math>H_{\rm s}</math> is the static head arising from elevation changes between the source and destination, including fluid head above the pump suction intake (e.g. a tank) (m)
* <math>H_f</math> is the head caused by friction losses through the connected piping system (m)
* <math>H_{\rm f}</math> is the head caused by friction losses through the connected piping system (m)
* <math>H_e</math> is the head due to pressure drops across equipment at the end of the piping system, such as hydrocyclones etc (m)
* <math>H_{\rm e}</math> is the head due to pressure drops across equipment at the end of the piping system, such as hydrocyclones etc (m)


Friction losses may be estimated by the Darcy-Weisbach equation:{{Dunne et al. (2019)}}
Friction losses may be estimated by the Darcy-Weisbach equation:{{Dunne et al. (2019)}}


:<math>H_f = \dfrac{f L V^2}{2 g D}</math>
:<math>H_{\rm f} = \dfrac{f L V^2}{2 g D}</math>


where:
where:
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where <math>k</math> is the absolute surface roughness of the pipe interior (m), and <math>\mathrm {Re}</math> is the Reynolds number:
where <math>k</math> is the absolute surface roughness of the pipe interior (m), and <math>\mathrm {Re}</math> is the Reynolds number:


:<math>\mathrm {Re} = \dfrac{D V \rho_{SL} \cdot 10^6} {\mu}</math>
:<math>\mathrm {Re} = \dfrac{D V \rho_{\rm SL} \cdot 10^6} {\mu}</math>


where <math>\rho_{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 side of the equation.
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_w = \dfrac{H_t}{\mathit{HR}}</math>
:<math>H_{\rm w} = \dfrac{H_{\rm t}}{{\rm HR}}</math>


where <math>H_w</math> is the equivalent head of water (m) and <math>\mathit{HR}</math> is the head ratio (m/m) for the slurry.  
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_w</math>, is then used in the characteristic curve equations above in place of the total head, <math>H_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>\mathit{HR}</math> can be estimated by many means. Two approaches are included in the centrifugal slurry pump model, the ''Cave'' and ''Warman'' approaches.
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>\mathit{HR} = 1 - 0.0385 (\rho_S - 1) \left ( \dfrac{\rho_S + 4}{\rho_S} \right ) C_W \ln \left ( \dfrac{d_{50}}{22.7} \right )</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:
* <math>C_W</math> is the concentration of solids in the slurry by weight (w/w)
* <math>C_{\rm W}</math> is the concentration of solids in the slurry by weight (w/w)
* <math>d_{50}</math> is the mean particle size diameter (μm), defined here as the 50% mass fraction passing size.
* <math>d_{50}</math> is the mean particle size diameter (μm), defined here as the 50% mass fraction passing size.
* <math>\rho_S</math> is the specific gravity or density of solids (- or t/m<sup>3</sup>)
* <math>\rho_{\rm S}</math> is the Specific Gravity or density of solids (- or t/m<sup>3</sup>)


==== Warman method ====
==== Warman method ====
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[[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|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).]]


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_V</math> used by the nomograph is the concentration of solids by volume in the slurry (v/v).
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).


The Warman nomograph has been digitised and an interpolative calculation procedure is integrated into the centrifugal slurry pump model as an option for estimating the head ratio.
The Warman nomograph has been digitised and an interpolative calculation procedure is integrated into the centrifugal slurry pump model as an option for estimating the head ratio.
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=== Efficiency ratio ===
=== Efficiency ratio ===


Similarly to the head ratio, the efficiency ratio <math>\mathit{ER}</math> (kW/kW) derates pump efficiency for slurry duty, i.e.:
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}} = \mathit{ER} \cdot E</math>
:<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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=== Motor Power ===
=== Motor Power ===


The power drawn by the pumping duty at the drive shaft, <math>P_{\mathit{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_{\mathit{shaft}} = \dfrac{Q \cdot \rho_{SL} \cdot g \cdot H_t}{E \cdot \mathit{ER}}</math>
:<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_{motor}</math> (kW), is:
Due to electrical and mechanical inefficiencies, the power drawn by the motor, <math>P_{\rm motor}</math> (kW), is:


:<math>P_{motor} = \dfrac{P_{\mathit{shaft}}}{\eta}</math>
:<math>P_{\rm motor} = \dfrac{P_{\rm shaft}}{\eta}</math>


where <math>\eta</math> is the motor factor (kW/kW).
where <math>\eta</math> is the motor factor (kW/kW).


=== Pumps in series and parallel ===
=== 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.


Centrifugal pumps configured in '''series''' (stages) add the total head capability of each individual pump:
=== Settling velocity ===


:<math>H_t = \sum_{i=1}^{n_{stages}} H_i</math>
The settling velocity of a heterogenous slurry can be estimated by using the Durand equation:{{Dunne et al. (2019)}}


where <math>n_{stages}</math> is the number of stages in series and <math>H_i</math> is pump head at stage <math>i</math> (m).
:<math>V_{\rm L} = F_{\rm L} \sqrt{\dfrac{2gD \rho_{\rm S} - \rho_{\rm L}}{\rho_{\rm L}}}</math>


If all the pumps are identical, then the head experienced by any pump is:
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>H_i = \dfrac{H_t}{n_{stages}}</math>
:<math>F_{\rm L} = 0.4794 + 0.5429 {C_{\rm V}}^{0.1058} \log (d_{50}) - 1</math>


Pumps operating in '''parallel''' simply multiply the volumetric flow rate:
The settling velocity may be compared to the actual pipe velocity to asses whether settling may be an issue for the application in question.


:<math>Q = \sum_{i=1}^{n_{parallel}} Q_i</math>
=== Pumps in series and parallel ===


where <math>n_{parallel}</math> is the number of pumps in parallel and <math>Q_i</math> is the volumetric flow rate provided by the pump at stage <math>i</math> (m).
Centrifugal pumps configured in '''series''' (stages) add the total head capability of each individual pump:


If all the pumps are identical, then the volumetric flow rate of any pump is:
:<math>H_{\rm t} = \sum_{i=1}^{n_{\rm stages}} H_i</math>


:<math>Q_i = \dfrac{Q}{n_{parallel}}</math>
where <math>n_{\rm stages}</math> is the number of stages in series and <math>H_i</math> is pump head at stage <math>i</math> (m).


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.
If all the pumps are identical, then the head experienced by any pump is:


=== Settling velocity ===
:<math>H_i = \dfrac{H_{\rm t}}{n_{\rm stages}}</math>


The settling velocity of a heterogenous slurry can be estimated by using the Durand equation:{{Dunne et al. (2019)}}
Pumps operating in '''parallel''' simply multiply the volumetric flow rate:


:<math>V_L = F_L \sqrt{\dfrac{2gD \rho_S - \rho_L}{\rho_L}}</math>
:<math>Q = \sum_{i=1}^{n_{\rm parallel}} Q_i</math>


where <math>\rho_L</math> is the liquid density (t/m<sup>3</sup>), and <math>F_L</math> is the dimensionless Durand factor which may be approximated by:{{Griffiths (2003)}}
where <math>n_{\rm parallel}</math> is the number of pumps in parallel and <math>Q_i</math> is the volumetric flow rate provided by the pump at stage <math>i</math> (m).


:<math>F_L = 0.4794 + 0.5429 {C_V}^{0.1058} \log (d_{50}) - 1</math>
If all the pumps are identical, then the volumetric flow rate of any pump is:


The settling velocity may be compared to the actual pipe velocity to asses whether settling may be an issue for the application in question.
:<math>Q_i = \dfrac{Q}{n_{\rm parallel}}</math>


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 ==


Line 246: Line 265:
:<math>Parameters=
:<math>Parameters=
\begin{bmatrix}
\begin{bmatrix}
n_{stages}\\
n_{\rm stages}\\
n_{parallel}\\
n_{\rm parallel}\\
\text{Model mode}\\
\text{Model mode}\\
\text{f method}\\
\text{f method}\\
Line 253: Line 272:
\text{ER method}\\
\text{ER method}\\
\text{FL method}\\
\text{FL method}\\
D_{imp}\text{ (m)}\\
D_{\rm imp}\text{ (m)}\\
A_N\\
A_{\rm N}\\
B_N\\
B_{\rm N}\\
C_N\\
C_{\rm N}\\
A_E\\
A_{\rm E}\\
B_E\\
B_{\rm E}\\
C_E\\
C_{\rm E}\\
(Q_{M})_{Solids}\text{ (kg/s)}\\
(Q_{\rm M})_{\rm S}\text{ (kg/s)}\\
(Q_{M})_{Liquids}\text{ (kg/s)}\\
(Q_{\rm M})_{\rm L}\text{ (kg/s)}\\
\rho_S\text{ (t/m}^\text{3}\text{)}\\
\rho_{\rm S}\text{ (t/m}^\text{3}\text{)}\\
\rho_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_s\text{ (m)}\\
H_{\rm s}\text{ (m)}\\
D\text{ (m)}\\
D\text{ (m)}\\
L\text{ (m)}\\
L\text{ (m)}\\
k\text{ (m)}\\
k\text{ (m)}\\
H_e\text{ (kPa)}\\
H_{\rm e}\text{ (kPa)}\\
\eta\text{ (kW/kW)}\\
\eta\text{ (kW/kW)}\\
N\text{ (Hz)}\\
N\text{ (Hz)}\\
f\\
f\\
\mathit{HR}\text{ (m/m)}\\
{\rm HR}\text{ (m/m)}\\
\mathit{ER}\text{ (kW/kW)}\\
{\rm ER}\text{ (kW/kW)}\\
F_L\\
F_{\rm L}\\
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


mdUnit\_Pump\_King=
\mathit{mdUnit\_Pump\_King} =
\begin{bmatrix}
\begin{bmatrix}
\text{Iterations}\\
\text{Iterations}\\
Q_{Feed}\text{ (m}^3\text{/s)}\\
Q_{\rm Feed}\text{ (m}^3\text{/s)}\\
Q\text{ (m}^3\text{/s)}\\
Q\text{ (m}^3\text{/s)}\\
\rho_{SL}\text{ (t/m}^3\text{)}\\
\rho_{\rm SL}\text{ (t/m}^3\text{)}\\
C_W\text{ (w/w)}\\
C_{\rm W}\text{ (w/w)}\\
C_V\text{ (v/v)}\\
C_{\rm V}\text{ (v/v)}\\
F_L\\
F_{\rm L}\\
V_L\text{ (m/s)}\\
V_{\rm L}\text{ (m/s)}\\
V\text{ (m/s)}\\
V\text{ (m/s)}\\
\mathrm {Re}\\
\mathrm {Re}\\
f\\
f\\
H_f\text{ (m)}\\
H_{\rm f}\text{ (m)}\\
H_e\text{ (m)}\\
H_{\rm e}\text{ (m)}\\
H_t\text{ (m)}\\
H_{\rm t}\text{ (m)}\\
\mathit{HR}\text{ (m/m)}\\
{\rm HR}\text{ (m/m)}\\
H_w\text{ (m)}\\
H_{\rm w}\text{ (m)}\\
N\text{ (Hz)}\\
N\text{ (Hz)}\\
N_{pu}\\
N_{\rm pu}\\
N_Q\\
N_{\rm Q}\\
E\text{ (kW/kW)}\\
E\text{ (kW/kW)}\\
\mathit{ER}\text{(kW/kW)}\\
{\rm ER}\text{(kW/kW)}\\
P_{\mathit{shaft}}\text{ (kW)}\\
P_{\rm shaft}\text{ (kW)}\\
P_{motor}\text{ (kW)}\\
P_{\rm motor}\text{ (kW)}\\
\end{bmatrix}\;\;\;\;\;\;
\end{bmatrix}\;\;\;\;\;\;
</math>
</math>
Line 314: Line 334:
* <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''
* <math>\text{FL method}</math> is the Durand factor estimation method, ''0 = User, 1 = Equation''
* <math>\text{FL method}</math> is the Durand factor estimation method, ''0 = User, 1 = Equation''
* <math>(Q_{M})_{Solids}</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_{M})_{Liquids}</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_{Feed}</math> is the actual volumetric flow rate of slurry in the pump feed, computed from <math>(Q_{M})_{Solids}</math>, <math>(Q_{M})_{Liquids}</math>, <math>\rho_S</math>, and <math>\rho_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>


| [[File:Pump4.png|frame|Figure 4. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]]
| [[File:Pump4.png|frame|Figure 4. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]]
Line 324: Line 344:


When calculating '''speed at flow rate''' (<math>\text{Model mode} = 0</math>):
When calculating '''speed at flow rate''' (<math>\text{Model mode} = 0</math>):
* <math>Q = Q_{Feed}</math> and the value of <math>N</math> is computed by the model and returned.  
* <math>Q = Q_{\rm Feed}</math> and the value of <math>N</math> 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.
* 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''' (<math>\text{Model mode} = 1</math>):
When calculating '''flow rate at speed''' (<math>\text{Model mode} = 1</math>):
* <math>N</math> is a user input and <math>Q</math> is computed and returned.  
* <math>N</math> is a user input and <math>Q</math> is computed and returned.  
* The return value of <math>Q</math> may be different to <math>Q_{Feed}</math>.  
* The return value of <math>Q</math> may be different to <math>Q_{\rm Feed}</math>.  
* This might indicate a quantity of make-up water is required to maintain the level of a tank feeding the pump, for example.
* This might indicate a quantity of make-up water is required to maintain the level of a tank feeding the pump, for example.


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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.


==== {{Name (Text, mdModels2, SysCAD, DLL)}}*Pump page ====
==== {{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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|Diameter of the pump impeller.
|Diameter of the pump impeller.
|-
|-
|aN
|AN
|Input
|Input
|Coefficient of the generalised characteristic curve equation.
|Coefficient of the generalised characteristic curve equation.
|-
|-
|bN
|BN
|Input
|Input
|Coefficient of the generalised characteristic curve equation.
|Coefficient of the generalised characteristic curve equation.
|-
|-
|cN
|CN
|Input
|Input
|Coefficient of the generalised characteristic curve equation.
|Coefficient of the generalised characteristic curve equation.
Line 433: Line 453:
! colspan="3" style="text-align:left;" |''EfficiencyCurve''
! colspan="3" style="text-align:left;" |''EfficiencyCurve''
|-
|-
|aE
|AE
|Input
|Input
|Coefficient of the generalised efficiency curve equation.
|Coefficient of the generalised efficiency curve equation.
|-
|-
|bE
|BE
|Input
|Input
|Coefficient of the generalised efficiency curve equation.
|Coefficient of the generalised efficiency curve equation.
|-
|-
|cE
|CE
|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.
|-
|-
|FractionFactor / f
|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

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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:


where:

  • 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
  • is the friction factor estimation method, 0 = User, 1 = Colebrook, 2 = Churchill
  • is the head ratio estimation method, 0 = User, 1 = Cave, 2 = Warman
  • is the efficiency ratio estimation method, 0 = User, 1 = ER equals HR, 2 = Warman
  • is the Durand factor estimation method, 0 = User, 1 = Equation
  • is the mass flow rate of solids through the pump (kg/s)
  • is the mass flow rate of liquids through the pump (kg/s)
  • is the number of internal model iterations required to resolve any dependence between friction head and pump flow rate
  • is the actual volumetric flow rate of slurry in the pump feed, computed from , , , and
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 ():

  • 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 ButtonModelHelp.png 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
  • Only available when in SysCAD Dynamic mode and the pump is set to Flow Rate calculation.
  • Sets the pump to SysCAD Transfer Pull mode, where material is drawn from an upstream surge unit (e.g. tank) at the calculated pump flowrate (Q).
  • See SysCAD Simulation Modes for more information.
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 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.

References

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