# Variable-Displacement Pump (TL)

Variable-displacement bidirectional thermal liquid pump

Libraries:
Simscape / Fluids / Thermal Liquid / Pumps & Motors

## Description

The Variable-Displacement Pump block represents a device that extracts power from a mechanical rotational network and delivers it to a thermal liquid network. The pump displacement varies during simulation according to the physical input signal at port D.

Ports A and B represent the pump inlets. Ports R and C represent the drive shaft and case. During normal operation, the pressure gain from port A to port B is positive if the angular velocity at port R relative to port C is positive.

### Operating Modes

The block has four modes of operation, as shown by this image.

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:

• The quadrant labeled 1 represents the forward pump mode. In this mode, the positive shaft angular velocity causes a pressure increase from port A to port B and flow from port A to port B

• The quadrant labeled 2 represents the reverse motor mode. In this mode, the flow from port B to port A causes a pressure decrease from B to A and negative shaft angular velocity.

• The quadrant labeled 3 represents the reverse pump mode. In this mode, the negative shaft angular velocity causes a pressure increase from port B to port A and flow from port B to port A

• The quadrant labeled 4 represents the forward motor mode. In this mode, the flow from port A to port B causes a pressure decrease from A to B and positive shaft angular velocity.

• The quadrant labeled 5 represents the reverse motor mode. In this mode, the flow from port B to port A causes a pressure decrease from B to A and positive shaft angular velocity.

• The quadrant labeled 6 represents the forward pump mode. In this mode, the negative shaft angular velocity causes a pressure increase from port A to port B and flow from port A to port B

• The quadrant labeled 7 represents the forward motor mode. In this mode, the flow from port A to port B causes a pressure decrease from A to B and negative shaft angular velocity.

• The quadrant labeled 8 represents the reverse pump mode. In this mode, the positive shaft angular velocity causes a pressure increase from port B to port A and flow from port B to port A

The response time of the pump is negligible in comparison with the system response time. The pump reaches steady state nearly instantaneously and is treated as a quasi-steady component.

### Energy Balance

The block associates the mechanical work done by the pump with an energy exchange. The governing energy balance equation is:

`${\varphi }_{A}+{\varphi }_{B}+{P}_{hydro}=0,$`

where:

• ΦA and ΦB are the energy flow rates at ports A and B, respectively.

• Phydro is the pump hydraulic power, which is a function of the pressure difference between the pump ports: ${P}_{hydro}=\Delta p\frac{\stackrel{˙}{m}}{\rho }$.

The block generates mechanical power due to torque, τ, and angular velocity, ω:

`${P}_{mech}=\tau \omega .$`

### Flow Rate and Driving Torque

The mass flow rate generated at the pump is

`$\stackrel{˙}{m}={\stackrel{˙}{m}}_{\text{Ideal}}-{\stackrel{˙}{m}}_{\text{Leak}},$`

where:

• $\stackrel{˙}{m}$ is the actual mass flow rate.

• ${\stackrel{˙}{m}}_{\text{Ideal}}$ is the ideal mass flow rate.

• ${\stackrel{˙}{m}}_{\text{Leak}}$ is the internal leakage mas flow rate.

The driving torque required to power the pump is

`$\tau ={\tau }_{\text{Ideal}}+{\tau }_{\text{Friction}},$`

where:

• τ is the actual driving torque.

• τIdeal is the ideal driving torque.

• τFriction is the friction torque.

Ideal Flow Rate and Ideal Torque

The ideal mass flow rate is

`${\stackrel{˙}{m}}_{\text{Ideal}}=\rho D\omega ,$`

and the ideal required torque is

`${\tau }_{\text{Ideal}}=D\Delta p,$`

where:

• ρ is the average of the fluid densities at thermal liquid ports A and B.

• ω is the shaft angular velocity.

• Δp is the pressure gain from inlet to outlet.

• D is the Displacement specified at physical signal port D.

### Leakage and friction parameterization

You can parameterize leakage and friction analytically, by using tabulated efficiencies or losses, or by using input efficiencies or input losses.

Analytical

When you set the Leakage and Friction Parameterization parameter to `Analytical`, the leakage flow rate is

`${\stackrel{˙}{m}}_{\text{Leak}}=\frac{{K}_{\text{HP}}{\rho }_{\text{Avg}}\Delta p}{{\mu }_{\text{Avg}}},$`

and the friction torque is

`${\tau }_{\text{Friction}}=\left({\tau }_{0}+{K}_{\text{TP}}|\Delta p|\frac{|D|}{{D}_{\text{Nom}}}\text{tanh}\frac{4\omega }{\left(5e-5\right){\omega }_{\text{Nom}}}\right),$`

where:

• KHP is the Hagen-Poiseuille coefficient for laminar pipe flows. The block computes this coefficient from the specified nominal parameters.

• μ is the dynamic viscosity of the fluid, which is the average of the values at the ports.

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

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

τfr,nom is the friction torque at nominal conditions:

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

• ΔpNom is the value of the Nominal pressure gain parameter. This value is the pressure gain at which the nominal volumetric efficiency is specified.

• DNom is the value of the Nominal Displacement parameter.

• τ0 is the value of the No-load torque parameter.

• ωNom is the value of the Nominal shaft angular velocity parameter.

The block determines the Hagen-Poiseuille coefficient from the nominal fluid and component parameters

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

where:

• ωNom is the value of the Nominal shaft angular velocity parameter. This value is the angular velocity at which the block specifies the nominal volumetric efficiency.

• μNom is the value of the Nominal Dynamic viscosity parameter. This value is the dynamic viscosity at which the block specifies the nominal volumetric efficiency.

• ηv,Nom is the value of the Volumetric efficiency at nominal conditions parameter. This is the volumetric efficiency that corresponds to the specified nominal conditions.

Tabulated Efficiencies

When you set the Leakage and friction parameterization parameter to ```Tabulated data - volumetric and mechanical efficiencies```, the leakage flow rate is

`${\stackrel{˙}{m}}_{\text{Leak}}={\stackrel{˙}{m}}_{\text{Leak,Pump}}\frac{\left(1+\alpha \right)}{2}+{\stackrel{˙}{m}}_{\text{Leak,Motor}}\frac{\left(1-\alpha \right)}{2},$`

and the friction torque is

`${\tau }_{\text{Friction}}={\tau }_{\text{Friction,Pump}}\frac{1+\alpha }{2}+{\tau }_{\text{Friction,Motor}}\frac{1-\alpha }{2},$`

where:

• α is a numerical smoothing parameter for the motor-pump transition.

• ${\stackrel{˙}{m}}_{\text{Leak,Motor}}$ is the leakage flow rate in motor mode.

• ${\stackrel{˙}{m}}_{\text{Leak,Pump}}$ is the leakage flow rate in pump mode.

• τFriction,Motor is the friction torque in motor mode.

• τFriction,Pump is the friction torque in pump mode.

This hyperbolic function describes the smoothing parameter α

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

where:

• ΔpThreshold is the value of the Pressure gain threshold for pump-motor transition block parameter.

• ωThreshold is the value of the Angular velocity threshold for pump-motor transition block parameter.

• DThreshold is the value of the Angular velocity threshold for motor-pump transition block parameter.

The block calculates the leakage flow rate from the volumetric efficiency, which the Volumetric efficiency table parameter specifies over the ΔpɷD domain. When operating in pump mode, the leakage flow rate is:

`${\stackrel{˙}{m}}_{\text{Leak,Pump}}=\left(1-{\eta }_{\text{v}}\right){\stackrel{˙}{m}}_{\text{Ideal}},$`

where ηv is the volumetric efficiency, which the block obtains either by interpolation or extrapolation of the tabulated data. Similarly, when operating in motor mode, the leakage flow rate is:

`${\stackrel{˙}{m}}_{\text{Leak,Motor}}=-\left(1-{\eta }_{\text{v}}\right)\stackrel{˙}{m}.$`

The block calculates the friction torque from the mechanical efficiency, which the Mechanical efficiency table parameter specifies over the ΔpɷD domain. When operating in pump mode, the friction torque is

`${\tau }_{\text{Friction,Pump}}=\left(1-{\eta }_{\text{m}}\right)\tau ,$`

where ηm is the mechanical efficiency, which the block obtains either by interpolation or extrapolation of the tabulated data. Similarly, when operating in motor mode, the friction torque is

`${\tau }_{\text{Friction,Motor}}=-\left(1-{\eta }_{\text{m}}\right){\tau }_{\text{Ideal}}.$`

Tabulated Losses

When you set the Leakage and friction parameterization parameter to ```Tabulated data - volumetric and mechanical losses```, the block specifies the leakage volumetric flow rate in tabulated form over the ΔpɷD domain:

`${q}_{\text{Leak}}={q}_{\text{Leak}}\left(\Delta p,\omega ,D\right).$`

The block calculates the mass flow rate due to leakage from the volumetric flow rate:

`${\stackrel{˙}{m}}_{\text{Leak}}=\rho {q}_{\text{Leak}}.$`

The block calculates the friction torque in tabulated form:

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

where qLeak(Δp,ω,D) and τFriction(Δp,ω,D) are the volumetric and mechanical losses, specified through interpolation or extrapolation of the tabulated data in the Volumetric loss table and Mechanical loss table parameters.

Input Efficiencies

When you set the Leakage and friction parameterization parameter to ```Input signal - volumetric and mechanical efficiencies```, the leakage flow rate and friction torque calculations are identical to the ```Tabulated data - volumetric and mechanical efficiencies``` setting. The block replaces the volumetric and mechanical efficiency lookup tables with the physical signal input ports EV and EM.

The efficiencies are positive quantities with values between `0` and `1`.The block sets input values outside of these bounds are to `0` for inputs smaller than `0` or `1` for inputs greater than `1`. The block saturates the efficiency signals at the value of the Minimum volumetric efficiency or Minimum mechanical efficiency parameter and the Maximum volumetric efficiency or Maximum mechanical efficiency parameter.

Input Losses

When you set the Leakage and friction parameterization parameter to ```Input signal - volumetric and mechanical losses```, the leakage flow rate and friction torque calculations are identical to the ```Tabulated data - volumetric and mechanical efficiencies``` setting. The block replaces the volumetric and mechanical loss lookup tables with the physical signal input ports LV and LM.

The block expects the inputs to be positive and sets the signs automatically from the Δpɷ quadrant where the component is operating. If you provide a negative signal, the block returns zero losses.

### Assumptions and Limitations

• The block treats the pump as a quasi-steady component.

• The block ignores the effects of fluid inertia and elevation.

• The pump wall is rigid.

• The block ignores external leakage.

## Ports

### Input

expand all

Physical signal input port for the volume of fluid displaced rotation. A smoothing function eases the transition between positive and negative input values.

Physical signal input port for the volumetric efficiency coefficient. The input signal has an upper bound at the Maximum volumetric efficiency parameter value and a lower bound at the Minimum volumetric efficiency parameter value.

#### Dependencies

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

Physical signal input port for the mechanical efficiency coefficient. The input signal has an upper bound at the Maximum mechanical efficiency parameter value and a lower bound at the Minimum mechanical efficiency parameter value.

#### Dependencies

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

Physical signal input port for the volumetric loss which is the internal leakage flow rate between the pump inlets.

#### Dependencies

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

Physical signal input port for the mechanical loss which is the friction torque on the rotating pump shaft.

#### Dependencies

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

### Conserving

expand all

Thermal liquid conserving port associated with the pump inlet.

Thermal liquid conserving port associated with the pump outlet.

Mechanical rotational conserving port associated with the pump case.

Mechanical rotational conserving port associated with the rotational pump shaft.

## Parameters

expand all

Method to compute the flow-rate and torque losses due to internal leaks and friction. Use the `Analytical` if the block parameters are generally available from component data sheets. When you select ```Tabulated data - volumetric and mechanical efficiencies``` or ```Tabulated data - volumetric and mechanical losses```, the block uses lookup tables to map pressure gain, angular velocity, and displacement to component efficiencies or losses.

When you select ```Input signal - volumetric and mechanical efficiencies``` or ```Input signal - volumetric and mechanical losses```, the block performs the leakage flow rate and friction torque calculations the same as the ```Tabulated data - volumetric and mechanical efficiencies``` or ```Tabulated data - volumetric and mechanical losses``` settings, respectively. When you select ```Input signal - volumetric and mechanical efficiencies```, the block enables the EV and EM ports. You use these ports to specify the volumetric and mechanical efficiency. When you select ```Input signal - volumetric and mechanical losses```, the block enables the LV and LM ports. You use these ports to specify the volumetric and mechanical losses.

Fluid displacement for a given volumetric efficiency. These values are typically available at standard operating conditions in the manufacturer data sheet. The block uses this parameter to calculate the leakage flow rate and friction torque.

Angular velocity of the rotary shaft that corresponds to the given volumetric efficiency. The standard operating conditions in the manufacturer data sheet typically list these values. The block uses this parameter to calculate the leakage flow rate and friction torque.

#### Dependencies

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

Pressure gain that corresponds to the given volumetric efficiency. The standard operating conditions in the manufacturer data sheet typically list these values. The block uses this parameter to calculate the internal leakage flow rate.

#### Dependencies

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

Dynamic viscosity of the hydraulic fluid for the given volumetric efficiency. These values are typically available at standard operating conditions in manufacturer data sheet. The block uses this parameter to calculate the internal leakage flow rate.

#### Dependencies

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

Volumetric efficiency for the given conditions. The block defines the volumetric efficiency as the ratio of actual to ideal volumetric flow rates. The standard operating conditions in the manufacturer data sheet typically list these values. The block uses this parameter to calculate the internal leakage flow rate.

#### Dependencies

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

Ratio of ideal mechanical power to actual mechanical power at nominal conditions.

#### Dependencies

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

Torque to overcome the seal friction and induce rotation of the mechanical shaft. This torque is the load-independent component of the total friction torque.

#### Dependencies

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

Pressure gains for the corresponding tabular efficiency or loss data. The vector must be at least two elements in strictly increasing order.

Specify the data for a single octant, (ɷ, Δp, D). Refer to Operating Modes for the operation modes corresponding to parameter specifications and the resulting octants.

#### Dependencies

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

Shaft angular velocities for the corresponding tabular efficiency or loss data. The vector must be at least two elements in strictly increasing order.

Specify the data for a single octant, (ɷ, Δp, D). Refer to Operating Modes for the operation modes corresponding to parameter specifications and the resulting octants.

#### Dependencies

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

Displacements at which to specify the efficiencies or losses tabular data. The vector must be at least two elements in strictly increasing order.

Specify the data for a single octant, (ɷ, Δp, D). Refer to Operating Modes for the operation modes corresponding to parameter specifications and the resulting octants.

#### Dependencies

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

Volumetric efficiencies at the specified fluid pressure gains, shaft angular velocities, and displacements. The efficiencies must be in the range of `0``1`. M, N, and L are the sizes of the specified lookup-table 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.

• L is the number of vector elements in the Displacement vector, D parameter.

Specify the data for a single octant, (ɷ, Δp, D). Refer to Operating Modes for the operation modes corresponding to parameter specifications and the resulting octants.

#### Dependencies

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

Mechanical efficiencies that correspond to the specified fluid pressure gains, shaft angular velocities, and displacements. The efficiencies must be in the range of `0``1`. M, N, and L are the sizes of the specified lookup-table 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.

• L is the number of vector elements in the Displacement vector, D parameter.

Specify the data for a single octant, (ɷ, Δp, D). Refer to Operating Modes for the operation modes corresponding to parameter specifications and the resulting octants.

#### Dependencies

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

Volumetric losses at the specified fluid pressure gains, shaft angular velocities, and displacements. The block defines volumetric loss as the internal leakage volumetric flow rate between port A and port B. M, N, and L are the sizes of the specified lookup-table 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.

• L is the number of vector elements in the Displacement vector, D parameter.

Specify the data for a single octant, (ɷ, Δp, D). Refer to Operating Modes for the operation modes corresponding to parameter specifications and the resulting octants.

#### Dependencies

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

Mechanical losses for the given pressure gains, shaft angular velocities, and displacements. The block defines mechanical loss as the friction torque due to seals and internal components. M, N, and L are the sizes of the specified lookup-table 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.

• L is the number of vector elements in the Displacement vector, D parameter.

Specify the data for a single octant, (ɷ, Δp, D). Refer to Operating Modes for the operation modes corresponding to parameter specifications and the resulting octants. The tabulated data for the mechanical losses must obey the convention in the figure, with positive values at positive angular velocities and negative values at negative angular velocities.

#### Dependencies

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

Smallest allowed value of the volumetric efficiency. The input from the physical signal port EV saturates inputs below this value.

#### Dependencies

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

Largest allowed value of the volumetric efficiency. The input from the physical signal port EV saturates inputs above this value.

#### Dependencies

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

Smallest allowed value of the mechanical efficiency. The input from the physical signal port EM saturates inputs below this value.

#### Dependencies

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

Largest allowed value of the mechanical efficiency. The input from the physical signal port EM saturates inputs above this value.

#### Dependencies

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

Pressure gain from the inlet to the outlet below which the block begins to transition between the motor and pump modes. The block uses a hyperbolic tangent function to smooth the leakage flow rate and friction torque.

#### Dependencies

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

Shaft angular velocity below which the block begins to transition between the motor and pump modes. The block uses the hyperbolic tangent function to smooth the leakage flow rate and friction torque.

#### Dependencies

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

Flow area at the component inlet and outlet. The areas are assumed equal.

Simulation warning mode for operating conditions outside the range of tabulated data. If you select `Warning`, the block generates a warning when the fluid pressure gain or the shaft angular velocity cross outside the specified tabular data. The warning does not cause simulation to stop. If you select `Error`, the simulation will stop for conditions outside the range of tabulated data.

#### Dependencies

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

• ```Tabulated data - volumetric and mechanical efficiencies```

• ```Tabulated data - volumetric and mechanical losses```

Simulation warning mode for operating conditions outside the pump mode. When you set this parameter to `Warning`, the block generates a warning if the pump transitions to motoring mode. The warning does not cause simulation to stop. If you select `Error`, the simulation will stop for conditions outside the pump mode.

#### Dependencies

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

## Version History

Introduced in R2017b