Centrifugal Pump

(To be removed) Centrifugal pump with choice of parameterization options

The Hydraulics (Isothermal) library will be removed in a future release. Use the Isothermal Liquid library instead. (since R2020a)

For more information on updating your models, see Upgrading Hydraulic Models to Use Isothermal Liquid Blocks.

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Libraries:
Simscape / Fluids / Hydraulics (Isothermal) / 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, 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

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, 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

ρref	Fluid density
c0, c1	Approximating coefficients. They can be determined either analytically from the Euler equation (Equations 1 and 2) or experimentally.
qref	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

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, p_D, 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:

$$p_{ref}={\rho }_{ref}\left(k(c_0-c_1{q}_{ref})-c_2{q}_{ref}{}^2-c_3{\left(q_D-q_{ref}\right)}^2\right)$$

(2)

The pump characteristics, approximated with four coefficients 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

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

(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:

$$p=\{\begin{array}{ll}-k_{leak}\cdot q\hfill & \text{for }\omega <=0\hfill \\ p_{\max }-k_{leak}\cdot q\hfill & \text{for }\omega >0,q<0\hfill \\ -k_{leak}\cdot \left(q-q_{\max }\right)\hfill & \text{for }\omega >0,q>q_{\max }\hfill \end{array}$$

(4)

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

$$a=\left(c_2+c_3\right)\cdot {\alpha }^2$$

$$b=\left(k\cdot c_1-2c_3\cdot q_D\right)\cdot \alpha$$

$$c=k\cdot c_0-c_3\cdot q_D^2$$

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

$$q_{\max }=\rho \frac{1}{{\alpha }^2}\left(k\cdot c_0-c_3\cdot q_D^2\right)$$

where

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

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 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

$$\eta =\frac{N_{hyd}}{N_{mech}}$$

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 ω_ref (Reference angular velocity) and the same fluid density ρ_ref (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 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 q_ref.

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 q_ref.

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 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 N_ref 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 than or equal to zero.

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

Ports

Conserving

T — Suction port

isothermal liquid

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

P — Outlet port

isothermal liquid

Hydraulic conserving port associated with the pump outlet.

S — Shaft port

isothermal liquid

Mechanical rotational conserving port associated with the pump driving shaft.

Parameters

Model parameterization — Model configurations by pressure differential, brake power, and pump capacity

By approximating polynomial (default) | By two 1D characteristics: P-Q and N-Q | By two 2D characteristics: P-Q-W and N-Q-W

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.

• 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 — Coefficient c₀ of the Euler pulse moment equation for pressure

326.8 Pa/(kg/m^3) (default)

Approximating coefficient c₀ in the block description preceding. This parameter is used if Model parameterization is set to By approximating polynomial.

Second approximating coefficient — Coefficient c₁ of the Euler pulse moment equation for pressure

3.104e4 Pa*s/kg (default)

Approximating coefficient c₁ in the block description preceding. This parameter is used if Model parameterization is set to By approximating polynomial.

Third approximating coefficient — Coefficient c₂ of the hydraulic losses calculation

1.097e7 Pa*s^2/(kg*m^3) (default)

Approximating coefficient c₂ in the block description preceding. This coefficient accounts for hydraulic losses in the pump. This parameter is used if Model parameterization is set to By approximating polynomial.

Fourth approximating coefficient — Coefficient c₃ that accounts for the difference in losses between nominal flow and flow through pump geometry

2.136e5 Pa*s^2/(kg*m^3) (default)

Approximating coefficient c₃ in the block description preceding. This coefficient accounts for additional hydraulic losses caused by deviation from the nominal delivery. This parameter is used if Model parameterization is set to By approximating polynomial.

Correction factor — Correction of Euler pressure calculation for reference pressure calculation

0.8 (default)

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

Pump design delivery — Nominal pump capacity

130 lpm (default)

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

Reference angular velocity — Shaft speed

1770 rpm (default)

Angular velocity of the driving shaft, at which the pump characteristics are determined. 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 — Reference density

920 kg/m^3 (default)

Fluid density at which the pump characteristics are determined.

Leak resistance — Coefficient for pressure outside of pump normal operating range

1e8 Pa/(m^3/s) (default)

Leakage resistance coefficient (see Equation 4). This parameter is used if Model parameterization is set to By approximating polynomial.

Drive shaft torque — Shaft friction torque

0.1 N*m (default)

The friction torque on the shaft at zero velocity. This parameter is used if Model parameterization is set to By approximating polynomial.

Torque-pressure coefficient — Relationship between pump pressure and shaft torque

0.1e-5 N*m/Pa (default)

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 Pressure differential — Capacity vector for tabular parameterization of pump pressure

[0, 28, 90, 130, 154, 182] lpm (default)

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

Pressure differential across pump vector — Pressure vector for tabular parameterization of pump pressure

[2.6, 2.4, 2, 1.6, 1.2, .8] bar (default)

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

Pump delivery vector for Brake power — Capacity vector for tabular parameterization of brake power

[0, 20, 40, 60, 80, 100, 120, 140, 160] lpm (default)

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

Brake power vector — Brake power vector for tabular parameterization of brake power

[220, 280, 310, 360, 390, 420, 480, 500, 550] W (default)

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

Angular velocity vector, w — Shaft speed vector for tabular parameterization of pump pressure

[3200, 3300, 3400, 3500] rpm (default)

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. 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 Pressure differential, Qp — Capacity vector for tabular parameterization of pump pressure

[0, 50, 100, 150, 200, 250, 300, 350] lpm (default)

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

Pressure differential table, p(Qp,w) — Pressure matrix for tabular parameterization of pump pressure

[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] bar (default)

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. 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 Brake power, Qb — Capacity vector for tabular parameterization of brake power

[0, 50, 100, 150, 200, 250, 300, 350] lpm (default)

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

Brake power table, Wb(Qb,w) — Brake power matrix for tabular parameterization of brake power

[1223, 1341, 1467, 1600; 1414, 1551, 1696, 1850; 1636, 1794, 1962, 2140; 1941, 2129, 2326, 2540; 2224, 2439, 2660, 2910; 2453, 2691, 2947, 3210; 2757, 3024, 3307, 3608; 2945, 3230, 3533, 3854] W (default)

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

Angular speed threshold for flow reversal — Shaft speed threshold region for flow reversal

1e-9 (default)

Shaft angular velocity that indicates the transition threshold between forward and reverse flow. A transition region is defined around 0 rad/s between the positive and negative values of the angular velocity threshold. Within this transition region, the computed leakage flow rate and friction torque are adjusted according to the transition term α to ensure smooth transition from one mode to the other.

Interpolation method — Tabular interpolation

Linear (default) | Smooth

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 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 — Tabular extrapolation

Linear (default) | Nearest

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 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.

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

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2007a

R2023a: To be removed

The Hydraulics (Isothermal) library will be removed in a future release. Use the Isothermal Liquid library instead.

For more information on updating your models, see Upgrading Hydraulic Models to Use Isothermal Liquid Blocks.