# Variable-Displacement Pump (IL)

Variable-displacement pump in an isothermal liquid network

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• Simscape / Fluids / Isothermal Liquid / Pumps & Motors

## Description

The Variable-Displacement Pump (IL) block models a pump with variable-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.

The shaft rotation corresponds to the sign of the fluid volume moving through the pump, which is received as a physical signal at port D. Positive fluid displacement at D corresponds to positive shaft rotation in forward mode. Negative fluid displacement at D corresponds to negative shaft angular velocity in forward mode.

Operation Modes

The block has eight modes of operation. The working mode depends on the pressure gain from port A to port B, Δp = pBpA; the angular velocity, ω = ωRωC; and the fluid volumetric displacement at port D. The figure above maps these modes to the octants of a Δp-ω-D chart:

• 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.

• Mode 5, Reverse Motor: Flow from port B to port A causes a pressure decrease from B to A and positive shaft angular velocity.

• Mode 6, Forward Pump: Negative shaft angular velocity causes pressure increase from A to B and flow from A to B.

• Mode 7, Forward Motor: Flow from port A to B causes a pressure decrease from A to B and negative shaft angular velocity.

• Mode 8, Reverse Pump: Positive shaft angular velocity causes a pressure increase from port B to port A and flow from B to A.

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, Angular velocity threshold for pump-motor transition, and Displacement 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. For the displacement threshold, choose a threshold value that is smaller than the typical displacement volume during normal operation.

### Analytical Leakage and Friction Parameterization

If you set Leakage and friction parameterization to `Analytical`, the block calculates internal leakage and shaft friction from constant nominal values of shaft velocity, pressure gain, volumetric displacement, and volumetric efficiency. 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,

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

where:

• Dnom is the Nominal displacement.

• ωnom is the Nominal shaft angular velocity.

• ηnom 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:

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

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:

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

• 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 eight operational modes, the block calculates the complementing data for the other mode(s) by extending the given data into the remaining octants.

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

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

where:

• Δp is pBpA.

• Δpthreshold 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:

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

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

The shaft friction torque is calculated as:

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

where τloss is interpolated from the Mechanical loss table, torque_loss(dp,w,D) parameter, which is based on user-supplied data for pressure gain, shaft angular velocity, and fluid volumetric displacement.

### 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:

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

To be notified if the block is operating beyond the supplied tabulated data, you can set Check if operating beyond the octants of supplied tabulated data to `Warning` to receive a warning if this occurs, or `Error` to stop the simulation when this occurs. when using input signals 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.

## 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 reference angular velocity and torque.

### Input

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Volumetric displacement of the pump, in m^3/rad, specified as a 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```.

Pump efficiency for the mechanical delivery of power, 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, Shaft angular velocity vector, w, and Displacement vector, D parameters and interpolated from the 3-D dependent Volumetric efficiency table, e_v(dp,w,D) and Mechanical efficiency table, e_m(dp,w,D) 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; Shaft angular velocity vector, w; and Displacement vector, D parameters and interpolated from the 3-D dependent Volumetric loss table, q_loss(dp,w,D) and Mechanical loss table, torque_loss(dp,w,D) 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 friction torque are received as physical signals at ports LV and LM, respectively.

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

#### Dependencies

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

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 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 mechanical power to ideal mechanical power 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 and the parameters for the 3-D dependent Volumetric efficiency table, e_v(dp,w,D) 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 Pressure gain vector, dp and the parameters for the 3-D dependent Volumetric efficiency table, e_v(dp,w,D) 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 fluid volumetric displacement data for the tabular parameterization of leakage and torque friction. This vector forms an independent axis with the Shaft angular velocity vector, w and the parameters for the 3-D dependent Volumetric efficiency table, e_v(dp,w,D) 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```

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

• P 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-by-P matrix of mechanical efficiencies at the specified fluid pressure gain, shaft angular velocity, and displacement. Linear interpolation is employed between table elements. M, N, and P are the sizes of the corresponding 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.

• P 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-by-P matrix of volumetric leakage at the specified fluid pressure gain, shaft angular velocity, and displacement. Linear interpolation is employed between table elements. M, N, and P are the sizes of the corresponding 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.

• P 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 losses```.

M-by-N-by-P matrix of friction torque at the specified fluid pressure gain, shaft angular velocity, and displacement. Linear interpolation is employed between table elements. M, N, and P are the sizes of the corresponding 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.

• P 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```.

Pump pressure gain that indicates the transition threshold 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```

Shaft angular velocity that indicates the transition threshold between motor and pump 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 loss```

Volumetric displacement that indicates the transition threshold between pump and motor functionality. A transition region is defined around 0 cm^3/s between the positive and negative values of the displacement threshold. Within this transition region, the computed leakage flow rate and friction torque is adjusted according to the transition term α to ensure smooth transition from one mode to the other. It is also used to transition the ideal mass flow rate when the sign of D changes.

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 the block operates outside of the pump mode functionality. 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 and B experiences low pressure. Select `Warning` to be notified when the pressure falls below a minimum specified value. Select `Error` to stop the simulation when the pressure falls below a minimum specified value.

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

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

#### Dependencies

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

• `Warning`

• `Error`

## Version History

Introduced in R2020a