# Variable-Displacement Pump (TL)

Variable-displacement bidirectional thermal liquid pump

• Library:
• 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 signal input specified at port D.

Ports A and B represent the pump inlets. Ports R and C represent the motor 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 also. This operation mode is referred to here as forward pump.

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

### Energy Balance

Mechanical work done by the pump is associated with an energy exchange. The governing energy balance equation is:

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

where:

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

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

The mechanical power is generated 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

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

and the ideal generated torque is

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

where:

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

• DSat is a smoothed displacement computed so as to remove numerical discontinuities between negative and positive displacements.

• ω is the shaft angular velocity.

• Δp is the pressure drop from inlet to outlet.

The saturation displacement is defined as:

${D}_{\text{Sat}}=\left\{\begin{array}{ll}\sqrt{{D}^{2}+{D}_{\text{Threshold}}^{2}},\hfill & D\ge 0\hfill \\ -\sqrt{{D}^{2}+{D}_{\text{Threshold}}^{2}},\hfill & D<0\hfill \end{array}.$

where:

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

• DThreshold is the specified value of the Displacement threshold for motor-pump transition block parameter.

### Leakage and friction parameterization

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

Analytical

When you set Leakage and Friction Parameterization 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

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

where:

• KHP is the Hagen-Poiseuille coefficient for laminar pipe flows. This coefficient is computed from the specified nominal parameters.

• μ is the dynamic viscosity of the thermal liquid, taken here as the average of its values at the thermal liquid ports.

• 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:

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

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

• DNom is the specified value of the Nominal Displacement block parameter.

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

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

The Hagen-Poiseuille coefficient is determined from nominal fluid and component parameters through the equation

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

where:

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

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

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

Tabulated Efficiencies

When you set Leakage and friction parameterization 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.

The smoothing parameter α is given by the hyperbolic tangent function

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

where:

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

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

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

The leakage flow rate is calculated from the volumetric efficiency, a quantity that is specified in tabulated form over the ΔpɷD domain via the Volumetric efficiency table block parameter. When operating in pump mode (quadrants 1 and 3 of the ΔpɷD chart shown in the Operation Modes figure), 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, obtained either by interpolation or extrapolation of the tabulated data. Similarly, when operating in motor mode (quadrants 2 and 4 of the ΔpɷD chart), the leakage flow rate is:

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

The friction torque is similarly calculated from the mechanical efficiency, a quantity that is specified in tabulated form over the ΔpɷD domain via the Mechanical efficiency table block parameter. When operating in pump mode (quadrants 1 and 3 of the ΔpɷD chart):

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

where ηm is the mechanical efficiency, obtained either by interpolation or extrapolation of the tabulated data. Similarly, when operating in motor mode (quadrants 2 and 4 of the ΔpɷD chart):

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

Tabulated Losses

When you set Leakage and friction parameterization to Tabulated data - volumetric and mechanical losses, the leakage (volumetric) flow rate is specified directly in tabulated form over the ΔpɷD domain:

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

The mass flow rate due to leakage is calculated from the volumetric flow rate:

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

The friction torque is similarly specified in tabulated form:

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

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

Input Efficiencies

When you set Leakage and friction parameterization 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 volumetric and mechanical efficiency lookup tables are replaced with physical signal inputs that you specify through ports EV and EM.

The efficiencies are positive quantities with value between 0 and 1. Input values outside of these bounds are set equal to the nearest bound (0 for inputs smaller than 0, 1 for inputs greater than 1). In other words, the efficiency signals are saturated at 0 and 1.

Input Losses

When you set Leakage and friction parameterization 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 volumetric and mechanical loss lookup tables are replaced with physical signal inputs that you specify through ports LV and LM.

The block ignores the sign of the input. The block sets the signs automatically from the operating conditions established during simulation—more precisely, from the Δpɷ quadrant in which the component happens to be operating.

### Assumptions and Limitations

• Fluid compressibility is negligible.

• Loading on the motor shaft due to inertia, friction, and spring forces is negligible.

### Variables

To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

## Ports

### Input

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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, defined as 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, defined as 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

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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

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Method to compute flow-rate and torque losses due to internal leaks and friction. When you select Analytical, 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 drop, 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, and the block enables the physical signal ports, EV and EM. You use these ports to specify the volumetric and mechanical efficiency.

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. 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.

#### Dependencies

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

Pressure drop that corresponds to 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.

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. 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.

Ratio of actual mechanical power to ideal 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 drops for the corresponding tabular efficiency data. The vector must be at least two elements in strictly increasing order.

You can specify the data for a single quadrant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

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

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

You can specify the data for a single quadrant, (ɷ, Δp,D). Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

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

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

You can specify the data for a single quadrant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

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

Volumetric efficiencies at the specified fluid pressure drops, shaft angular velocities, and displacements. The efficiencies must be in the range of 01. M, N, and L are the sizes of the specified lookup-table vectors:

• M is the number of vector elements in the Pressure drop vector for efficiencies, dp parameter.

• N is the number of vector elements in the parameter.

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

You can specify the data for a single quadrant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

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

Mechanical efficiencies corresponding to the specified fluid pressure drops, shaft angular velocities, and displacements. The efficiencies must be in the range of 01. M, N, and L are the sizes of the specified lookup-table vectors:

• M is the number of vector elements in the Pressure drop vector for efficiencies, dp parameter.

• N is the number of vector elements in the parameter.

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

You can specify the data for a single quadrant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

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

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

You can specify the data for a single quadrant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

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

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

You can specify the data for a single quadrant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

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

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

You can specify the data for a single quadrant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

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

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

• M is the number of vector elements in the Pressure drop vector for losses, dp parameter.

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

You can specify the data for a single quadrant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

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

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

• M is the number of vector elements in the Pressure drop vector for losses, dp parameter.

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

You can specify the data for a single quadrant, (ɷ, Δp). Refer to the block description for the operation modes corresponding to the various quadrants. 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 drop from inlet to outlet below which the block begins to transition between motoring and pumping 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 motoring and pumping 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. Select Warning to be notified when the fluid pressure drop, shaft angular velocity, or instantaneous displacement cross outside the specified tabular data. The warning does not cause simulation to stop.

#### Dependencies

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

• Tabulated data - volumetric and mechanical efficiencies

• Tabulated data - volumetric and mechanical losses

• Input signal - volumetric and mechanical efficiencies

• Input signal - volumetric and mechanical losses

Simulation warning mode for operating conditions outside the motoring mode. The block generates a warning if the motor transitions to pumping mode. Select Warning to be notified when this transition occurs. The warning does not cause simulation to stop.

## Version History

Introduced in R2017b