Centrifugal pump with choice of parameterization options

Pumps and Motors

The Centrifugal Pump block represents a centrifugal pump of any type as a data-sheet-based model. Depending on data listed in the manufacturer's catalog or data sheet for your particular pump, you can choose one of the following model parameterization options:

`By approximating polynomial`

— Provide values for the polynomial coefficients. These values can be determined analytically or experimentally, depending on the data available. This is the default method.`By two 1D characteristics: P-Q and N-Q`

— Provide tabulated data of pressure differential P and brake power N versus pump delivery Q characteristics. The pressure differential and brake power are determined by one-dimensional table lookup. You have a choice of three interpolation methods and two extrapolation methods.`By two 2D characteristics: P-Q-W and N-Q-W`

— Provide tabulated data of pressure differential P and brake power N versus pump delivery Q characteristics at different angular velocities W. The pressure differential and brake power are determined by two-dimensional table lookup. You have a choice of three interpolation methods and two extrapolation methods.

These parameterization options are further described in greater detail:

Connections P and T are hydraulic conserving ports associated with the pump outlet and inlet, respectively. Connection S is a mechanical rotational conserving port associated with the pump driving shaft. The block positive direction is from port T to port P. This means that the pump transfers fluid from T to P as its driving shaft S rotates in the globally assigned positive direction.

If you set the **Model parameterization** parameter
to `By approximating polynomial`

, the pump
is parameterized with the polynomial whose coefficients are determined,
analytically or experimentally, for a specific angular velocity depending
on the data available. The pump characteristics at other angular velocities
are determined using the affinity laws.

The approximating polynomial is derived from the Euler pulse moment equation [1, 2], which for a given pump, angular velocity, and fluid can be represented as the following:

$${p}_{ref}=k\cdot {p}_{E}-{p}_{HL}-{p}_{D}$$ | (1-1) |

where

p_{ref} | Pressure differential across the pump for the reference regime, characterized by the reference angular velocity and density |

k | Correction factor. The factor is introduced to account for dimensional fluctuations, blade incongruity, blade volumes, fluid internal friction, and so on. The factor should be set to 1 if the approximating coefficients are determined experimentally. |

p_{E} | Euler pressure |

p_{HL} | Pressure loss due to hydraulic losses in the pump passages |

p_{D} | Pressure loss caused by deviations of the pump delivery from its nominal (rated) value |

The Euler pressure, *p _{E}*,
is determined with the Euler equation for centrifugal machines [1, 2] based on known pump dimensions. For an
existing pump, operating at constant angular velocity and specific
fluid, the Euler pressure can be approximated with the equation

$${p}_{E}={\rho}_{ref}\left({c}_{0}-{c}_{1}\cdot {q}_{ref}\right)$$

where

ρ_{ref} | Fluid density |

c_{0}, c_{1} | Approximating coefficients. They can be determined either analytically from the Euler equation [1, 2] or experimentally. |

q_{ref} | Pump volumetric delivery at reference regime |

The pressure loss due to hydraulic losses in the pump passages, *p*_{HL},
is approximated with the equation

$${p}_{HL}={\rho}_{ref}\cdot {c}_{2}\cdot {q}_{ref}{}^{2}$$

where

ρ_{ref} | Fluid density |

c_{2} | Approximating coefficient |

q_{ref} | Pump volumetric delivery at reference regime |

The blade profile is determined for a specific fluid velocity,
and deviation from this velocity results in pressure loss due to inconsistency
between the fluid velocity and blade profile velocity. This pressure
loss, *p*_{D}, is estimated with
the equation

$${p}_{D}={\rho}_{ref}\cdot {c}_{3}{\left({q}_{D}-{q}_{ref}\right)}^{2}$$

where

ρ_{ref} | Fluid density |

c_{3} | Approximating coefficient |

q_{ref} | Pump volumetric delivery at reference regime |

q_{D} | Pump design delivery (nominal delivery) |

The resulting approximating polynomial takes the form:

$${p}_{ref}={\rho}_{ref}\left(k({c}_{0}-{c}_{1}{q}_{ref})-{c}_{2}{q}_{ref}{}^{2}-{c}_{3}{\left({q}_{D}-{q}_{ref}\right)}^{2}\right)$$ | (1-2) |

The pump characteristics, approximated with four coefficients *c*_{0}, *c*_{1}, *c*_{2},
and *c*_{3}, are determined for
a specific fluid and a specific angular velocity of the pump's driving
shaft. These two parameters correspond, respectively, to the **Reference
density** and **Reference angular velocity** parameters
in the block dialog box. To apply the characteristics for another
velocity ω or density ρ, the affinity laws are used. With
these laws, the delivery at reference regime, which corresponds to
given pump delivery and angular velocity, is computed with the expression

$${q}_{ref}=q\frac{{\omega}_{ref}}{\omega}$$ | (1-3) |

where *q* and ω are the instantaneous
values of the pump delivery and angular velocity. Then the pressure
differential *p*_{ref} at reference
regime computed with Equation 1-2 and
converted into pressure differential *p* at current
angular velocity and density

$$p={p}_{ref}\cdot {\left(\frac{\omega}{{\omega}_{ref}}\right)}^{2}\cdot \frac{\rho}{{\rho}_{ref}}$$

Equation 1-2 describes
pump characteristic for ω > 0 and *q* >=
0. Outside this range, the characteristic is approximated with the
following relationships:

$$p=\{\begin{array}{ll}-{k}_{leak}\cdot q\hfill & \text{for}\omega =0\hfill \\ {p}_{\mathrm{max}}-{k}_{leak}\cdot q\hfill & \text{for}\omega 0,q0\hfill \\ -{k}_{leak}\cdot \left(q-{q}_{\mathrm{max}}\right)\hfill & \text{for}\omega 0,q{q}_{\mathrm{max}}\hfill \end{array}$$ | (1-4) |

$${q}_{\mathrm{max}}=\frac{-b+\sqrt{{b}^{2}+4ac}}{2a}$$

$$a=\left({c}_{2}+{c}_{3}\right)\cdot {\alpha}^{2}$$

$$b=\left(k\cdot {c}_{1}-2{c}_{3}\cdot {q}_{D}\right)\cdot \alpha $$

$$c=k\cdot {c}_{0}-{c}_{3}\cdot {q}_{D}^{2}$$

$$\alpha =\frac{\omega}{{\omega}_{ref}}$$

$${q}_{\mathrm{max}}=\rho \frac{1}{{\alpha}^{2}}\left(k\cdot {c}_{0}-{c}_{3}\cdot {q}_{D}^{2}\right)$$

where

k_{leak} | Leakage resistance coefficient |

q_{max} | Maximum pump delivery at given angular velocity. The delivery
is determined from Equation 1-2 at p =
0. |

p_{max} | Maximum pump pressure at given angular velocity. The pressure
is determined from Equation 1-2 at q =
0. |

k | Correction factor, as described in Equation 1-1. |

The hydraulic power at the pump outlet at reference conditions is

$${N}_{hyd}={p}_{ref}\cdot {q}_{ref}$$

The output hydraulic power at arbitrary angular velocity and density is determined with the affinity laws

$$N={N}_{ref}\left(\frac{\omega}{{\omega}_{ref}}\right)\cdot \frac{\rho}{{\rho}_{ref}}$$

The power at the pump driving shaft consists of the theoretical hydraulic power (power before losses associated with hydraulic loss and deviation from the design delivery) and friction loss at the driving shaft. The theoretical hydraulic power is approximated using the Euler pressure

$${N}_{hyd0}={p}_{Eref}\cdot {q}_{ref}\cdot {\left(\frac{\omega}{{\omega}_{ref}}\right)}^{3}$$

where

N_{hyd0} | Pump theoretical hydraulic power |

p_{Eref} | Euler pressure. The theoretical pressure developed by the pump before losses associated with hydraulic loss and deviation from the design delivery. |

The friction losses are approximated with the relationship:

$${N}_{fr}=\left({T}_{0}+{k}_{p}\cdot p\right)\cdot \omega $$

where

N_{fr} | Friction loss power |

T_{0} | Constant torque at driving shaft associated with shaft bearings, seal friction, and so on |

k_{p} | Torque-pressure relationship, which characterizes the influence of pressure on the driving shaft torque |

The power and torque at the pump driving shaft (brake power *N*_{mech} and
brake torque *T*) are

$${N}_{mech}={N}_{hyd0}+{N}_{fr}$$

$$T=\frac{{N}_{mech}}{\omega}$$

The pump total efficiency *η* is computed
as

$$\eta =\frac{{N}_{hyd}}{{N}_{mech}}$$

If you set the **Model parameterization** parameter
to `By two 1D characteristics: P-Q and N-Q`

,
the pump characteristics are computed by using two one-dimensional
table lookups: for the pressure differential based on the pump delivery
and for the pump brake power based on the pump delivery. Both characteristics
are specified at the same angular velocity ω* _{ref}* (

To compute pressure differential at another angular velocity,
affinity laws are used, similar to the first parameterization option.
First, the new reference delivery *q _{ref}* is
computed with the expression

$${q}_{ref}=q\frac{{\omega}_{ref}}{\omega}$$

where *q* is the current pump delivery. Then
the pressure differential across the pump at current angular velocity
ω and density ρ is computed as

$$p={p}_{ref}\cdot {\left(\frac{\omega}{{\omega}_{ref}}\right)}^{2}\cdot \frac{\rho}{{\rho}_{ref}}$$

where *p _{ref}* is the
pressure differential determined from the P-Q characteristic at pump
delivery

Brake power is determined with the equation

$$N={N}_{ref}\cdot {\left(\frac{\omega}{{\omega}_{ref}}\right)}^{3}\cdot \frac{\rho}{{\rho}_{ref}}$$

where *N _{ref}* is the
reference brake power obtained from the N-Q characteristic at pump
delivery

The torque at the pump driving shaft is computed with the equation *T* = *N* /
ω .

If you set the **Model parameterization** parameter
to `By two 2D characteristics: P-Q-W and N-Q-W`

,
the pump characteristics are read out from two two-dimensional table
lookups: for the pressure differential based on the pump delivery
and angular velocity and for the pump brake power based on the pump
delivery and angular velocity.

Both the pressure differential and brake power are scaled if
fluid density ρ is different from the reference density ρ_{ref},
at which characteristics have been obtained

$$p={p}_{ref}\cdot \frac{\rho}{{\rho}_{ref}}$$

$$N={N}_{ref}\cdot \frac{\rho}{{\rho}_{ref}}$$

where *p _{ref}* and

Fluid compressibility is neglected.

The pump rotates in positive direction, with speed that is greater or equal to zero.

The reverse flow through the pump is allowed only at still shaft.

**Model parameterization**Select one of the following methods for specifying the pump parameters:

`By approximating polynomial`

— Provide values for the polynomial coefficients. These values can be determined analytically or experimentally, depending on the data available. The relationship between pump characteristics and angular velocity is determined from the affinity laws. This is the default method.`By two 1D characteristics: P-Q and N-Q`

— Provide tabulated data of pressure differential and brake power versus pump delivery characteristics. The pressure differential and brake power are determined by one-dimensional table lookup. You have a choice of three interpolation methods and two extrapolation methods. The relationship between pump characteristics and angular velocity is determined from the affinity laws.`By two 2D characteristics: P-Q-W and N-Q-W`

— Provide tabulated data of pressure differential and brake power versus pump delivery characteristics at different angular velocities. The pressure differential and brake power are determined by two-dimensional table lookup. You have a choice of three interpolation methods and two extrapolation methods.

**First approximating coefficient**Approximating coefficient

*c*in the block description preceding. The default value is_{0}`326.8`

Pa/(kg/m^3). This parameter is used if**Model parameterization**is set to`By approximating polynomial`

.**Second approximating coefficient**Approximating coefficient

*c*in the block description preceding. The default value is_{1}`3.104e4`

Pa*s/kg. This parameter is used if**Model parameterization**is set to`By approximating polynomial`

.**Third approximating coefficient**Approximating coefficient

*c*in the block description preceding. This coefficient accounts for hydraulic losses in the pump. The default value is_{2}`1.097e7`

Pa*s^2/(kg*m^3). This parameter is used if**Model parameterization**is set to`By approximating polynomial`

.**Fourth approximating coefficient**Approximating coefficient

*c*in the block description preceding. This coefficient accounts for additional hydraulic losses caused by deviation from the nominal delivery. The default value is_{3}`2.136e5`

Pa*s^2/(kg*m^3). This parameter is used if**Model parameterization**is set to`By approximating polynomial`

.**Correction factor**The factor, denoted as

*k*in the block description preceding, accounts for dimensional fluctuations, blade incongruity, blade volumes, fluid internal friction, and other factors that decrease Euler theoretical pressure. The default value is`0.8`

. This parameter is used if**Model parameterization**is set to`By approximating polynomial`

.**Pump design delivery**The pump nominal delivery. The blades profile, pump inlet, and pump outlet are shaped for this particular delivery. Deviation from this delivery causes an increase in hydraulic losses. The default value is

`130`

lpm. This parameter is used if**Model parameterization**is set to`By approximating polynomial`

.**Reference angular velocity**Angular velocity of the driving shaft, at which the pump characteristics are determined. The default value is

`1.77e3`

rpm. This parameter is used if**Model parameterization**is set to`By approximating polynomial`

or`By two 1D characteristics: P-Q and N-Q`

.**Reference density**Fluid density at which the pump characteristics are determined. The default value is

`920`

kg/m^3.**Leak resistance**Leakage resistance coefficient (see Equation 1-4). The default value is

`1e+8`

Pa/(m^3/s). This parameter is used if**Model parameterization**is set to`By approximating polynomial`

.**Drive shaft torque**The friction torque on the shaft at zero velocity. The default value is

`0.1`

N*m. This parameter is used if**Model parameterization**is set to`By approximating polynomial`

.**Torque-pressure coefficient**The coefficient that provides relationship between torque and pump pressure. The default value is

`1e-6`

N*m/Pa. This parameter is used if**Model parameterization**is set to`By approximating polynomial`

.**Pump delivery vector for P-Q table**Specify the vector of pump deliveries, as a one-dimensional array, to be used together with the vector of pressure differentials to specify the P-Q pump characteristic. The vector values must be strictly increasing. The values can be nonuniformly spaced. The minimum number of values depends on the interpolation method: you must provide at least two values for linear interpolation, at least three values for cubic or spline interpolation. The default values, in lpm, are

`[0 28 90 130 154 182]`

. This parameter is used if**Model parameterization**is set to`By two 1D characteristics: P-Q and N-Q`

.**Pressure differential across pump vector**Specify the vector of pressure differentials across the pump as a one-dimensional array. The vector will be used together with the pump delivery vector to specify the P-Q pump characteristic. The vector must be of the same size as the pump delivery vector for the P-Q table. The default values, in bar, are

`[2.6 2.4 2 1.6 1.2 0.8]`

. This parameter is used if**Model parameterization**is set to`By two 1D characteristics: P-Q and N-Q`

.**Pump delivery vector for N-Q table**Specify the vector of pump deliveries, as a one-dimensional array, to be used together with the vector of the pump brake power to specify the N-Q pump characteristic. The vector values must be strictly increasing. The values can be nonuniformly spaced. The minimum number of values depends on the interpolation method: you must provide at least two values for linear interpolation, at least three values for cubic or spline interpolation. The default values, in lpm, are

`[0 20 40 60 80 100 120 140 160]`

. This parameter is used if**Model parameterization**is set to`By two 1D characteristics: P-Q and N-Q`

.**Brake power vector for N-Q table**Specify the vector of pump brake power as a one-dimensional array. The vector will be used together with the pump delivery vector to specify the N-Q pump characteristic. The vector must be of the same size as the pump delivery vector for the N-Q table. The default values, in W, are

`[220 280 310 360 390 420 480 500 550]`

. This parameter is used if**Model parameterization**is set to`By two 1D characteristics: P-Q and N-Q`

.**Pump delivery vector for P-Q and W table**Specify the vector of pump deliveries, as a one-dimensional array, to be used together with the vector of angular velocities and the pressure differential matrix to specify the pump P-Q-W characteristic. The vector values must be strictly increasing. The values can be nonuniformly spaced. The minimum number of values depends on the interpolation method: you must provide at least two values for linear interpolation, at least three values for cubic or spline interpolation. The default values, in lpm, are

`[0 50 100 150 200 250 300 350]`

. This parameter is used if**Model parameterization**is set to`By two 2D characteristics: P-Q-W and N-Q-W`

.**Angular velocity vector for P-Q and W table**Specify the vector of angular velocities, as a one-dimensional array, to be used for calculating both the pump P-Q-W and N-Q-W characteristics. The vector values must be strictly increasing. The values can be nonuniformly spaced. The minimum number of values depends on the interpolation method: you must provide at least two values for linear interpolation, at least three values for cubic or spline interpolation. The default values, in rpm, are

`[3.2e+03 3.3e+03 3.4e+03 3.5e+03]`

. This parameter is used if**Model parameterization**is set to`By two 2D characteristics: P-Q-W and N-Q-W`

.**Pressure differential matrix for P-Q and W table**Specify the pressure differentials across pump as an

`m`

-by-`n`

matrix, where`m`

is the number of the P-Q-W pump delivery values and`n`

is the number of angular velocities. This matrix will define the pump P-Q-W characteristic together with the pump delivery and angular velocity vectors. Each value in the matrix specifies pressure differential for a specific combination of pump delivery and angular velocity. The matrix size must match the dimensions defined by the pump delivery and angular velocity vectors. The default values, in bar, are:This parameter is used if[ 8.3 8.8 9.3 9.9 ; 7.8 8.3 8.8 9.4 ; 7.2 7.6 8.2 8.7 ; 6.5 7 7.5 8 ; 5.6 6.1 6.6 7.1 ; 4.7 5.2 5.7 6.2 ; 3.4 4 4.4 4.9 ; 2.3 2.7 3.4 3.6 ; ]

**Model parameterization**is set to`By two 2D characteristics: P-Q-W and N-Q-W`

.**Pump delivery vector for N-Q and W table**Specify the vector of pump deliveries, as a one-dimensional array, to be used together with the vector of angular velocities and the brake power matrix to specify the pump N-Q-W characteristic. The vector values must be strictly increasing. The values can be nonuniformly spaced. The minimum number of values depends on the interpolation method: you must provide at least two values for linear interpolation, at least three values for cubic or spline interpolation. The default values, in lpm, are

`[0 50 100 150 200 250 300 350]`

. This parameter is used if**Model parameterization**is set to`By two 2D characteristics: P-Q-W and N-Q-W`

.**Brake power matrix for N-Q and W table**Specify the pump brake power as an

`m`

-by-`n`

matrix, where`m`

is the number of the N-Q-W pump delivery values and`n`

is the number of angular velocities. This matrix will define the pump N-Q-W characteristic together with the pump delivery and angular velocity vectors. Each value in the matrix specifies brake power for a specific combination of pump delivery and angular velocity. The matrix size must match the dimensions defined by the pump delivery and angular velocity vectors. The default values, in W, are:This parameter is used if[ 1.223e+03 1.341e+03 1.467e+03 1.6e+03 ; 1.414e+03 1.551e+03 1.696e+03 1.85e+03 ; 1.636e+03 1.794e+03 1.962e+03 2.14e+03 ; 1.941e+03 2.129e+03 2.326e+03 2.54e+03 ; 2.224e+03 2.439e+03 2.66e+03 2.91e+03 ; 2.453e+03 2.691e+03 2.947e+03 3.21e+03 ; 2.757e+03 3.024e+03 3.307e+03 3.608e+03 ; 2.945e+03 3.23e+03 3.533e+03 3.854e+03 ; ]

**Model parameterization**is set to`By two 2D characteristics: P-Q-W and N-Q-W`

.**Interpolation method**Select one of the following interpolation methods for approximating the output value when the input value is between two consecutive grid points:

`Linear`

— For one-dimensional table lookup (`By two 1D characteristics: P-Q and N-Q`

), uses a linear interpolation function. For two-dimensional table lookup (`By two 2D characteristics: P-Q-W and N-Q-W`

), uses a bilinear interpolation algorithm, which is an extension of linear interpolation for functions in two variables.`Cubic`

— For one-dimensional table lookup (`By two 1D characteristics: P-Q and N-Q`

), uses the Piecewise Cubic Hermite Interpolation Polynomial (PCHIP). For two-dimensional table lookup (`By two 2D characteristics: P-Q-W and N-Q-W`

), uses the bicubic interpolation algorithm.`Spline`

— For one-dimensional table lookup (`By two 1D characteristics: P-Q and N-Q`

), uses the cubic spline interpolation algorithm. For two-dimensional table lookup (`By two 2D characteristics: P-Q-W and N-Q-W`

), uses the bicubic spline interpolation algorithm.

This parameter is used if

**Model parameterization**is set to`By By two 1D characteristics: P-Q and N-Q`

or`By two By two 2D characteristics: P-Q-W and N-Q-W`

. For more information on interpolation algorithms, see the PS Lookup Table (1D) and PS Lookup Table (2D) block reference pages.**Extrapolation method**Select one of the following extrapolation methods for determining the output value when the input value is outside the range specified in the argument list:

`From last 2 points`

— Extrapolates using the linear method (regardless of the interpolation method specified), based on the last two output values at the appropriate end of the range. That is, the block uses the first and second specified output values if the input value is below the specified range, and the two last specified output values if the input value is above the specified range.`From last point`

— Uses the last specified output value at the appropriate end of the range. That is, the block uses the last specified output value for all input values greater than the last specified input argument, and the first specified output value for all input values less than the first specified input argument.

This parameter is used if

**Model parameterization**is set to`By By two 1D characteristics: P-Q and N-Q`

or`By two By two 2D characteristics: P-Q-W and N-Q-W`

. For more information on extrapolation algorithms, see the PS Lookup Table (1D) and PS Lookup Table (2D) block reference pages.

Parameter determined by the type of working fluid:

**Fluid density**

Use the Hydraulic Fluid block or the Custom Hydraulic Fluid block to specify the fluid properties.

The block has the following ports:

`T`

Hydraulic conserving port associated with the pump suction, or inlet.

`P`

Hydraulic conserving port associated with the pump outlet.

`S`

Mechanical rotational conserving port associated with the pump driving shaft.

[1] T.G. Hicks, T.W. Edwards, *Pump Application Engineering*,
McGraw-Hill, NY, 1971

[2] I.J. Karassic, J.P. Messina, P. Cooper, C.C. Heald, *Pump
Handbook*, Third edition, McGraw-Hill, NY, 2001

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