# Centrifugal Pump

Centrifugal pump with choice of parameterization options

Pumps and Motors

## Description

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.

### Parameterizing the Pump by Approximating Polynomial

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

 pref 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. pE Euler pressure pHL Pressure loss due to hydraulic losses in the pump passages pD Pressure loss caused by deviations of the pump delivery from its nominal (rated) value

The Euler pressure, pE, 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

 ρref Fluid density c0, c1 Approximating coefficients. They can be determined either analytically from the Euler equation [1, 2] or experimentally. qref Pump volumetric delivery at reference regime

The pressure loss due to hydraulic losses in the pump passages, pHL, is approximated with the equation

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

where

 ρref Fluid density c2 Approximating coefficient qref 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, pD, is estimated with the equation

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

where

 ρref Fluid density c3 Approximating coefficient qref Pump volumetric delivery at reference regime qD Pump design delivery (nominal delivery)

The resulting approximating polynomial takes the form:

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

The pump characteristics, approximated with four coefficients c0, c1, c2, and c3, 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 pref 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:

 (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

 kleak Leakage resistance coefficient qmax Maximum pump delivery at given angular velocity. The delivery is determined from Equation 1-2 at p = 0. pmax 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

 Nhyd0 Pump theoretical hydraulic power pEref 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

 Nfr Friction loss power T0 Constant torque at driving shaft associated with shaft bearings, seal friction, and so on kp 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 Nmech and brake torque T) are

`${N}_{mech}={N}_{hyd0}+{N}_{fr}$`
`$T=\frac{{N}_{mech}}{\omega }$`

The pump total efficiency η is computed as

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

### Parameterizing the Pump by Pressure Differential and Brake Power Versus Pump Delivery

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 ωref (Reference angular velocity) and the same fluid density ρref (Reference density).

To compute pressure differential at another angular velocity, affinity laws are used, similar to the first parameterization option. First, the new reference delivery qref 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 pref is the pressure differential determined from the P-Q characteristic at pump delivery qref.

Brake power is determined with the equation

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

where Nref is the reference brake power obtained from the N-Q characteristic at pump delivery qref.

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

### Parameterizing the Pump by Pressure Differential and Brake Power Versus Pump Delivery at Different Angular Velocities

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 ρref, at which characteristics have been obtained

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

where pref and Nref are the pressure differential and brake power obtained from the plots.

## Basic Assumptions and Limitations

• 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.

## Parameters

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 two 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 two interpolation methods and two extrapolation methods.

First approximating coefficient

Approximating coefficient c0 in the block description preceding. The default value is `326.8` Pa/(kg/m^3). This parameter is used if Model parameterization is set to `By approximating polynomial`.

Second approximating coefficient

Approximating coefficient c1 in the block description preceding. The default value is `3.104e4` Pa*s/kg. This parameter is used if Model parameterization is set to `By approximating polynomial`.

Third approximating coefficient

Approximating coefficient c2 in the block description preceding. This coefficient accounts for hydraulic losses in the pump. The default value is `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 c3 in the block description preceding. This coefficient accounts for additional hydraulic losses caused by deviation from the nominal delivery. The default value is `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 smooth 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 smooth 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 smooth 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 smooth 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:

```[ 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 ; ]```
This parameter is used if 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 smooth 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:

```[ 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 ; ]```
This parameter is used if 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` — Select this option to get the best performance.

• `Smooth` — Select this option to produce a continuous curve or surface with continuous first-order derivatives.

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:

• `Linear` — Select this option to produce a curve or surface with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region.

• `Nearest` — Select this option to produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data.

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.

## Global Parameters

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.

## Ports

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.

## References

[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

## See Also

#### Introduced in R2007a

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