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Centrifugal pump with choice of parameterization options

**Library:**Simscape / Fluids / Hydraulics (Isothermal) / 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 two 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 two 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, Equations 1 and 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) |

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 in Equations 1 and 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 (Equations 1 and 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:

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

$${q}_{ref}=q\frac{{\omega}_{ref}}{\omega}$$ | (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
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 2 describes pump characteristic for ω > 0 and *q* >= 0.
Outside this range, the characteristic is approximated with the following
relationships:

$$p=\{\begin{array}{ll}-{k}_{leak}\cdot q\hfill & \text{for}\omega =0\hfill \\ {p}_{\mathrm{max}}-{k}_{leak}\cdot q\hfill & \text{for}\omega 0,q0\hfill \\ -{k}_{leak}\cdot \left(q-{q}_{\mathrm{max}}\right)\hfill & \text{for}\omega 0,q{q}_{\mathrm{max}}\hfill \end{array}$$ | (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 2 at p = 0. |

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

k | Correction factor, as described in Equation 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.

[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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