Fixed-Displacement Pump (IL)

Fixed-displacement pump in isothermal liquid system

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• Simscape / Fluids / Isothermal Liquid / Pumps & Motors

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Description

The Fixed-Displacement Pump (IL) block models a pump with constant-volume displacement. The fluid may move from port A to port B, called forward mode, or from port B to port A, called reverse mode. Pump mode operation occurs when there is a pressure gain in the direction of the flow. Motor mode operation occurs when there is a pressure drop in the direction of the flow.

Shaft rotation corresponds to the sign of the fluid volume. Positive fluid displacement corresponds to positive shaft rotation in forward mode. Negative fluid displacement corresponds to negative shaft angular velocity in forward mode.

Operation Modes

The block has four modes of operation. The working mode depends on the pressure gain from port A to port B, Δp = p_B – p_A and the angular velocity, ω = ω_R – ω_C:

• Mode 1, Forward Pump: Positive shaft angular velocity causes a pressure increase from port A to port B and flow from port A to port B.

• Mode 2, Reverse Motor: Flow from port B to port A causes a pressure decrease from B to A and negative shaft angular velocity.

• Mode 3, Reverse Pump: Negative shaft angular velocity causes a pressure increase from port B to port A and flow from B to A.

• Mode 4, Forward Motor: Flow from port A to B causes a pressure decrease from A to B and positive shaft angular velocity.

The pump block has analytical, lookup table, and physical signal parameterizations. When using tabulated data or an input signal for parameterization, you can choose to characterize pump operation based on efficiency or losses.

The threshold parameters Pressure gain threshold for pump-motor transition and Angular velocity threshold for pump-motor transition identify regions where numerically smoothed flow transition between the pump operational modes can occur. For the pressure and angular velocity thresholds, choose a transition region that provides some margin for the transition term, but which is small enough relative to the typical pump pressure gain and angular velocity so that it will not impact calculation results.

Analytical Leakage and Friction Parameterization

If you set Leakage and friction parameterization to Analytical, the block calculates leakage and friction from constant values of shaft velocity, pressure gain, and torque. The leakage flow rate, which is correlated with the pressure differential over the pump, is calculated as:

$${\dot{m}}_{leak}=K{\rho }_{avg}\Delta p,$$

where:

• Δp_nom is p_B – p_A.

• ρ_avg is the average fluid density.

• K is the Hagen-Poiseuille coefficient for analytical loss,

  $$K=\frac{D_{nom}{\omega }_{nom}\left(1-{\eta }_{v,nom}\right)}{\Delta p_{nom}},$$

  where:

  • D is the Displacement.

  • ω_nom is the Nominal shaft angular velocity.

  • η_{v, nom} is the Volumetric efficiency at nominal conditions.

The friction torque, which is related to the pump pressure differential, is calculated as:

$${\tau }_{fr}=\left({\tau }_0+k\left|\Delta p\right|\right)\tanh \left(\frac{4\omega }{5\times {10}^{-5}{\omega }_{nom}}\right),$$

where:

• τ₀ is the No-load torque.

• k is the Friction torque vs. pressure gain coefficient.

• ω is the relative shaft angular velocity, or $${\omega }_R-{\omega }_C$$.

Tabulated Data Parameterizations

When using tabulated data for pump efficiencies or losses, you can provide data for one or more of the pump operational modes. The signs of the tabulated data determine the operational regime of the block. When data is provided for less than four operational modes, the block calculates the complementing data for the other mode(s) by extending the given data into the remaining quadrants.

The Tabulated data - volumetric and mechanical efficiencies parameterization

The leakage flow rate is calculated as:

$${\dot{m}}_{leak}={\dot{m}}_{leak,pump}\left(\frac{1+\alpha }{2}\right)+{\dot{m}}_{leak,motor}\left(\frac{1-\alpha }{2}\right),$$

where:

• $${\dot{m}}_{leak,pump}=\left(1-{\eta }_{\upsilon }\right){\dot{m}}_{ideal}$$

• $${\dot{m}}_{leak,motor}=\left({\eta }_v-1\right)\dot{m}$$

and η_v is the volumetric efficiency, which is interpolated from the user-provided tabulated data. The transition term, α, is

$$\alpha =\tanh \left(\frac{4\Delta p}{\Delta p_{threshold}}\right)\tanh \left(\frac{4\omega }{{\omega }_{threshold}}\right),$$

where:

• Δp is p_B – p_A.

• p_threshold is the Pressure gain threshold for pump-motor transition.

• ω is ω_R – ω_C.

• ω_threshold is the Angular velocity threshold for pump-motor transition.

The friction torque is calculated as:

$${\tau }_{fr}={\tau }_{fr,pump}\left(\frac{1+\alpha }{2}\right)+{\tau }_{fr,motor}\left(\frac{1-\alpha }{2}\right),$$

where:

• $${\tau }_{fr,pump}=\left(1-{\eta }_m\right)\tau$$

• $${\tau }_{fr,motor}=\left({\eta }_m-1\right){\tau }_{ideal}$$

and η_m is the mechanical efficiency, which is interpolated from the user-provided tabulated data.

The Tabulated data - volumetric and mechanical losses parameterization

The leakage flow rate is calculated as:

$${\dot{m}}_{leak}={\rho }_{avg}{q}_{loss}\left(\Delta p,\omega \right),$$

where q_loss is interpolated from the Volumetric loss table, q_loss(dp,w) parameter, which is based on user-supplied data for pressure drop, shaft angular velocity, and fluid volumetric displacement.

The shaft friction torque is calculated as:

$${\tau }_{fr}={\tau }_{loss}\left(\Delta p,\omega \right),$$

where τ_loss is interpolated from the Mechanical loss table, torque_loss(dp,w) parameter, which is based on user-supplied data for pressure drop and shaft angular velocity.

Input Signal Parameterization

When you select Input signal - volumetric and mechanical efficiencies, ports EV and EM are enabled. The internal leakage and shaft friction are calculated in the same way as the Tabulated data - volumetric and mechanical efficiencies parameterization, except that η_v and η_m are received directly at ports EV and EM, respectively.

When you select Input signal - volumetric and mechanical losses, ports LV and LM are enabled. These ports receive leakage flow and friction torque as positive physical signals. The leakage flow rate is calculated as:

$${\dot{m}}_{leak}={\rho }_{avg}{q}_{LV}\tanh \left(\frac{4\Delta p}{p_{thresh}}\right),$$

where:

• q_LV is the leakage flow received at port LV.

• p_thresh is the Pressure gain threshold for pump-motor transition parameter.

The friction torque is calculated as:

$${\tau }_{fr}={\tau }_{LM}\tanh \left(\frac{4\omega }{{\omega }_{thresh}}\right),$$

where

• τ_LM is the friction torque received at port LM.

• ω_thresh is the Angular velocity threshold for pump-motor transition parameter.

The volumetric and mechanical efficiencies range between the user-defined specified minimum and maximum values. Any values lower or higher than this range will take on the minimum and maximum specified values, respectively.

Pump Operation

The pump flow rate is:

$$\dot{m}={\dot{m}}_{ideal}-{\dot{m}}_{leak},$$

where $${\dot{m}}_{ideal}={\rho }_{avg}D\cdot \omega .$$

The pump torque is:

$$\tau ={\tau }_{ideal}+{\tau }_{fr},$$

where $${\tau }_{ideal}=D\cdot \Delta p.$$

The mechanical power delivered by the pump shaft is:

$${\phi }_{mech}=\tau \omega ,$$

and the pump hydraulic power is:

$${\phi }_{hyd}=\frac{\Delta p\dot{m}}{{\rho }_{avg}}.$$

If you would like to know if the block is operating beyond the supplied tabulated data, you can set Check if operating beyond the quadrants of supplied tabulated data to Warning to receive a warning if this occurs, or Error to stop the simulation when this occurs. For parameterization by input signal for volumetric or mechanical losses, you can be notified if the simulation surpasses operating modes with the Check if operating beyond pump mode parameter.

You can also monitor pump functionality. Set Check if pressures are less than pump minimum pressure to Warning to receive a warning if this occurs, or Error to stop the simulation when this occurs.

Predefined Parameterization

There are multiple available built-in parameterizations for the Fixed Displacement Pump (IL) block.

This pre-parameterization data allows you to set up the block to represent a specific supplier component. To load a predefined parameterization, click on the "Select a predefined parameterization" hyperlink in the Fixed Displacement Pump (IL) block dialog and select the specific part you want to upload from the list of available components.

Note

Predefined parameterizations of Simscape components use available data sources for supplying parameter values. Engineering judgement and simplifying assumptions are used to fill in for missing data. As a result, deviations between simulated and actual physical behavior should be expected. To ensure requisite accuracy, you should validate simulated behavior against experimental data and refine component models as necessary.

Ports

Conserving

A — Liquid port

isothermal liquid

Liquid entry or exit port to the pump.

B — Liquid port

isothermal liquid

Liquid entry or exit port to the pump.

R — Mechanical port

mechanical rotational

Rotating shaft angular velocity and torque.

C — Mechanical port

mechanical rotational

Pump casing reference angular velocity and torque.

Input

EV — Volumetric efficiency

physical signal

Pump efficiency for fluid displacement, specified as a physical signal. The value must be between 0 and 1.

Dependencies

To enable this port, set Leakage and friction parameterization to Input signal - volumetric and mechanical efficiencies.

EM — Mechanical efficiency

physical signal

Pump efficiency for the mechanical supply of energy, specified as a physical signal. The value must be between 0 and 1.

Dependencies

To enable this port, set Leakage and friction parameterization to Input signal - volumetric and mechanical efficiencies.

LV — Leakage flow rate, m^3/s

physical signal

Pump volumetric losses in m^3/s, specified as a physical signal.

Dependencies

To enable this port, set Leakage and friction parameterization to Input signal - volumetric and mechanical losses.

LM — Friction torque, N*m

physical signal

Pump mechanical losses in N*m, specified as a physical signal.

Dependencies

To enable this port, set Leakage and friction parameterization to Input signal - volumetric and mechanical losses.

Parameters

Leakage and friction parameterization — Method of calculating leakage flow rate and friction torque

Analytical (default) | Tabulated data - volumetric and mechanical efficiencies | Tabulated data - volumetric and mechanical losses | Input signal - volumetric and mechanical efficiencies | Input signal - volumetric and mechanical losses

Parameterization of the leakage and friction characteristics of the pump.

• In the Analytical parameterization, the leakage flow rate and the friction torque are calculated by analytical equations.

• In the Tabulated data - volumetric and mechanical efficiencies parameterization, the volumetric and mechanical efficiencies are calculated from the user-supplied Pressure gain vector, dp and Shaft angular velocity vector, w parameters and interpolated from the 2-D dependent Volumetric efficiency table, e_v(dp,w) and Mechanical efficiency table, e_m(dp,w) tables.

• In the Tabulated data - volumetric and mechanical loss parameterization, the leakage flow rate and friction torque are calculated from the user-supplied Pressure gain vector, dp and Shaft angular velocity vector, w parameters and interpolated from the 2-D dependent Volumetric loss table, q_loss(dp,w) and Mechanical loss table, torque_loss(dp,w) tables.

• In the Input signal - volumetric and mechanical efficiencies parameterization, the volumetric and mechanical efficiencies are received as physical signals at ports EV and EM, respectively.

• In the Input signal - volumetric and mechanical loss parameterization, the leakage flow rate and torque friction are received as physical signals at ports LV and LM, respectively.

Displacement — Fixed-volume fluid displacement

30 cm^3/rev (default) | scalar

Amount of fluid displaced by shaft rotating under nominal or typical operating conditions.

Nominal shaft angular velocity — Shaft angular velocity

188 rad/s (default) | scalar

Angular velocity of the shaft under nominal operating conditions.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Analytical.

Nominal pressure gain — Pressure gain between fluid entry and exit

10 MPa (default) | scalar

Pump pressure gain between the fluid entry and exit under nominal or typical operating conditions.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Analytical.

Volumetric efficiency at nominal conditions — Ratio of actual flow rate to ideal flow rate

0.92 (default) | positive scalar in the range of (0,1]

Ratio of actual flow rate to ideal flow rate at nominal conditions.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Analytical.

No-load torque — Baseline torque

0.05 N*m (default) | scalar

Minimum value of torque to overcome seal friction and induce shaft motion.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Analytical.

Friction torque vs. pressure gain coefficient — Constant of proportionality between friction torque and pump pressure gain

0.6 m*N/MPa (default) | positive scalar

Constant of proportionality between friction torque and pump pressure gain.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Analytical.

Pressure gain vector, dp — Vector of pressure data for tabulated data parameterization

[.1, 1, 2] MPa (default) | 1-by-n vector

Vector of pressure differential values for the tabular parameterization of leakage and torque friction. This vector forms an independent axis with the Shaft angular velocity vector, w parameter for the 2-D dependent Volumetric efficiency table, e_v(dp,w) and Mechanical efficiency table, e_m(dp,w,D) parameters. The vector elements must be listed in ascending order.

Dependencies

To enable this parameter, set Leakage and friction parameterization to either:

• Tabulated data - volumetric and mechanical efficiencies

• Tabulated data - volumetric and mechanical losses

Shaft angular velocity vector, w — Vector of angular velocity data for tabulated data parameterization

[25, 50, 100, 200, 300, 400] rad/s (default) | 1-by-n vector

Vector of angular velocity data for the tabular parameterization of leakage and torque friction. This vector forms an independent axis with the Shaft angular velocity vector, w parameter for the 2-D dependent Volumetric efficiency table, e_v(dp,w) and Mechanical efficiency table, e_m(dp,w) parameters. The vector elements must be listed in ascending order.

Dependencies

To enable this parameter, set Leakage and friction parameterization to either:

• Tabulated data - volumetric and mechanical efficiencies

• Tabulated data - volumetric and mechanical losses

Volumetric efficiency table, e_v(dp,w) — Volumetric efficiency for tabulated data parameterization

[.816, .908, .954, .977, .981, .984; .325, .663, .831, .916, .925, .946; .137, .568, .78, .892, .893, .91] (default) | M-by-N-by-P matrix

M-by-N matrix of volumetric efficiencies at the specified fluid pressure gain and shaft angular velocity. Linear interpolation is employed between table elements. M and N are the sizes of the correlated vectors:

• M is the number of vector elements in the Pressure gain vector, dp parameter.

• N is the number of vector elements in the Shaft angular velocity vector, w parameter.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Tabulated data - volumetric and mechanical efficiencies.

Mechanical efficiency table, e_m(dp,w) — Mechanical efficiency for tabulated data parameterization

[.996, .996, .996, .996, .996, .996; .988, .989, .989, .989, .989, .99; .981, .981, .982, .982, .983, .984] (default) | M-by-N-by-P matrix

M-by-N matrix of mechanical efficiencies at the specified fluid pressure gain and shaft angular velocity. Linear interpolation is employed between table elements. M and N are the sizes of the correlated vectors:

• M is the number of vector elements in the Pressure gain vector, dp parameter.

• N is the number of vector elements in the Shaft angular velocity vector, w parameter.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Tabulated data - volumetric and mechanical efficiencies.

Volumetric loss table, q_loss(dp,w) — Volumetric losses for tabulated data parameterization

[.023, .023, .023, .023, .0285, .032; .0844, .0842, .0845, .084, .1125, .108; .1079, .108, .11, .108, .1605, .18] * 1e-3 m^3/s (default) | M-by-N-by-P matrix

M-by-N matrix of volumetric efficiencies at the specified fluid pressure gain and shaft angular velocity. Linear interpolation is employed between table elements. M and N are the sizes of the correlated vectors:

• M is the number of vector elements in the Pressure gain vector, dp parameter.

• N is the number of vector elements in the Shaft angular velocity vector, w parameter.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Tabulated data - volumetric and mechanical loss.

Mechanical loss table, torque_loss(dp,w) — Mechanical losses for tabulated data parameterization

[.002, .002, .002, .002, .002, .002; .0607, .0556, .0556, .0556, .0556, .0505; .1937, .1937, .1833, .1833, .1729, .1626] N*m (default) | M-by-N-by-P matrix

M-by-N matrix of mechanical losses at the specified fluid pressure gain and shaft angular velocity. Linear interpolation is employed between table elements. M and N are the sizes of the correlated vectors:

• M is the number of vector elements in the Pressure gain vector, dp parameter.

• N is the number of vector elements in the Shaft angular velocity vector, w parameter.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Tabulated data - volumetric and mechanical loss.

Minimum volumetric efficiency — Minimum value of volumetric efficiency

1e-3 (default) | positive scalar in the range of (0,1]

Minimum value of volumetric efficiency. If the input signal is below this value, the volumetric efficiency is set to the minimum volumetric efficiency.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Input signal - volumetric and mechanical efficiencies.

Maximum volumetric efficiency — Maximum value of volumetric efficiency

1 (default) | positive scalar in the range of (0,1]

Maximum value of volumetric efficiency. If the input signal is above this value, the volumetric efficiency is set to the maximum volumetric efficiency.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Input signal - volumetric and mechanical efficiencies.

Minimum mechanical efficiency — Minimum value of mechanical efficiency

1e-3 (default) | positive scalar in the range of (0,1]

Minimum value of mechanical efficiency. If the input signal is below this value, the mechanical efficiency is set to the minimum mechanical efficiency.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Input signal - volumetric and mechanical efficiencies.

Maximum mechanical efficiency — Maximum value of mechanical efficiency

1 (default) | positive scalar in the range of (0,1]

Maximum value of mechanical efficiency. If the input signal is above this value, the mechanical efficiency is set to the maximum mechanical efficiency.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Input signal - volumetric and mechanical efficiencies.

Pressure gain threshold for pump-motor transition — Pressure threshold for mode transition

1e-3 MPa (default) | positive scalar

Threshold pressure gain value for the transition between pump and motor functionality. A transition region is defined around 0 MPa between the positive and negative values of the pressure gain 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.

Dependencies

To enable this parameter, set Leakage and friction parameterization to either:

• Tabulated data - volumetric and mechanical efficiencies

• Input signal - volumetric and mechanical efficiencies

• Input signal - volumetric and mechanical losses

Angular velocity threshold for pump-motor transition — Angular velocity threshold for mode transition

10 rad/s (default) | positive scalar

Threshold angular velocity value for the transition between pump and motor functionality. 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.

Dependencies

To enable this parameter, set Leakage and friction parameterization to:

• Tabulated data - volumetric and mechanical efficiencies

• Input signal - volumetric and mechanical efficiencies

• Input signal - volumetric and mechanical losses

Check if operating beyond the quadrants of supplied tabulated data — Whether to notify if extents of supplied data are surpassed

None (default) | Warning | Error

Whether to notify if the extents of the supplied data are surpassed. Select Warning to be notified when the block uses values beyond the supplied data range. Select Error to stop the simulation when the block uses values beyond the supplied data range.

Dependencies

To enable this parameter, set Leakage and friction parameterization to:

• Tabulated data - volumetric and mechanical efficiencies

• Tabulated data - volumetric and mechanical losses

Check if operating outside of pump mode — Whether to notify if block operates outside of the pump mode functionality

None (default) | Warning | Error

Whether to notify if block operates outside of the pump mode functionality. This block has four operation modes: forward pump, reverse pump, reverse motor, and forward motor. Select Warning to be notified when the block operates in the forward or reverse motor modes. Select Error to stop the simulation when the block operates in the forward or reverse motor modes.

Dependencies

To enable this parameter, set Leakage and friction parameterization to Input signal - volumetric and mechanical losses.

Check if pressures are less than pump minimum pressure — Whether to notify if fluid exit pressure is low

None (default) | Warning | Error

Whether to notify if the fluid at port A or B experiences low pressure. Select Warning to be notified when the outlet pressure falls below a minimum specified value. Select Error to stop the simulation when the outlet pressure falls below a minimum specified value.

The parameter helps to identify potential conditions for cavitation, when the fluid pressure falls below the fluid vapor pressure.

Pump minimum pressure — Lower threshold to acceptable pressure at the pump outlet

0.101325 MPa (default) | positive scalar

Lower threshold of acceptable pressure at the pump inlet and outlet.

Dependencies

To enable this parameter, set Check if pressures are less than pump minimum pressure to:

• Warning

• Error