# Fixed-Displacement Pump (IL)

Fixed-displacement pump in isothermal liquid system

• Library:
• Simscape / Fluids / Isothermal Liquid / Pumps & Motors

## 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 = pBpA and the angular velocity, ω = ωRωC:

• 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 friction torque. The leakage flow rate, which is correlated with the pressure differential over the pump, is calculated as:

`${\stackrel{˙}{m}}_{leak}=K{\rho }_{avg}\Delta p,$`

where:

• Δp is pBpA.

• ρavg is the average fluid density.

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

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

where:

• D is the Displacement.

• ωnom is the Nominal shaft angular velocity.

• ηv, nom is the Volumetric efficiency at nominal conditions.

• Δpnom is the Nominal pressure gain.

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

`${\tau }_{fr}=\left({\tau }_{0}+k|\Delta p|\right)\mathrm{tanh}\left(\frac{4\omega }{5×{10}^{-5}{\omega }_{nom}}\right),$`

where:

• τ0 is the No-load torque.

• k is the friction torque vs. pressure gain coefficient at nominal displacement, which is determined from the , ηm, nom:

`$k=\frac{{\tau }_{fr,nom}-{\tau }_{0}}{\Delta {p}_{nom}}.$`

τfr,nom is the friction torque at nominal conditions:

`${\tau }_{fr,nom}=\left(\frac{1-{\eta }_{m,nom}}{{\eta }_{m,nom}}\right)D\Delta {p}_{nom}.$`

• 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:

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

where:

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

• ${\stackrel{˙}{m}}_{leak,motor}=\left({\eta }_{v}-1\right)\stackrel{˙}{m}$

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

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

where:

• Δp is pBpA.

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

• ω is ωRωC.

• ωthreshold 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 ηm 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:

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

where qloss 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 τloss 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 ηv and ηm 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:

`${\stackrel{˙}{m}}_{leak}={\rho }_{avg}{q}_{LV}\mathrm{tanh}\left(\frac{4\Delta p}{{p}_{thresh}}\right),$`

where:

• qLV is the leakage flow received at port LV.

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

The friction torque is calculated as:

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

where

• τLM is the friction torque received at port LM.

• ωthresh 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:

`$\stackrel{˙}{m}={\stackrel{˙}{m}}_{ideal}-{\stackrel{˙}{m}}_{leak},$`

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

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:

`${\phi }_{hyd}=\frac{\Delta p\stackrel{˙}{m}}{{\rho }_{avg}}.$`

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

Pre-parameterization of the Fixed-Displacement Pump (IL) block with manufacturer data is available. This data allows you to model a specific supplier component.

1. Click the "Select a predefined parameterization" hyperlink in the Fixed-Displacement Pump (IL) block dialog description.

2. Select a part from the drop-down menu and click Update block with selected part.

3. If you change any parameter settings after loading a parameterization, you can check your changes by clicking Compare block settings with selected part. Any difference in settings between the block and pre-defined parameterization will display in the MATLAB command window.

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

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Liquid entry or exit port to the pump.

Liquid entry or exit port to the pump.

Rotating shaft angular velocity and torque.

Pump casing angular velocity and torque.

### Input

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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```.

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```.

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```.

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

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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.

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

Angular velocity of the shaft under nominal operating conditions.

#### Dependencies

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

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`.

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

#### Dependencies

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

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

#### Dependencies

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

Ratio of actual torque to ideal torque generated at nominal conditions.

#### Dependencies

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

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```

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```

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 parameter.

#### Dependencies

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

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 parameter.

#### Dependencies

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

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 parameter.

#### Dependencies

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

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 parameter.

#### Dependencies

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

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 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 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 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```.

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```

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```

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```

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```.

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.

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`

## Version History

Introduced in R2020a