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Variable-displacement bidirectional thermal liquid pump

**Library:**Simscape / Fluids / Thermal Liquid / Pumps & Motors

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**

A total of four operation modes are possible. The working mode depends on the pressure
gain from port A to port B (Δ*p*), the angular velocity at port R relative to port C
(*ω*), and the instantaneous displacement input at port D
(*D*). The Operation Modes figure maps the modes to
the octants of a Δ*p*-*ω*-*D* chart.
The modes are labeled 1–4:

Mode

**1**: forward pump — A positive shaft angular velocity generates a positive pressure gain.Mode

**2**: reverse motor — A negative pressure drop (shown in the figure as a positive pressure gain) generates a negative shaft angular velocity.Mode

**3**: reverse pump — A negative shaft angular velocity generates a negative pressure gain.Mode

**4**: forward motor — A positive pressure drop (shown in the figure as a negative pressure gain) generates a positive shaft angular velocity.

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.

The pump model accounts for power losses due to leakage and friction. Leakage is
internal and occurs between the pump inlet and outlet only. The block computes the
leakage flow rate and friction torque using your choice of five loss
parameterizations. You select a parameterization using block variants and, in the
`Analytical or tabulated data`

case, the
**Friction and leakage parameterization** parameter.

**Loss Parameterizations**

The block provides three Simulink^{®} variants to select from. To change the active block variant,
right-click the block and select **Simscape** > **Block choices**. The available variants are:

`Analytical or tabulated data`

— Obtain the mechanical and volumetric efficiencies or losses from analytical models based on nominal parameters or from tabulated data. Use the**Friction and leakage parameterization**parameter to select the exact input type.`Input efficiencies`

— Provide the mechanical and volumetric efficiencies directly through physical signal input ports.`Input losses`

— Provide the mechanical and volumetric losses directly through physical signal input ports. The mechanical loss is defined as the internal friction torque. The volumetric loss is defined as the internal leakage flow rate.

The mass flow rate generated at the pump is

$$\dot{m}={\dot{m}}_{\text{Ideal}}-{\dot{m}}_{\text{Leak}},$$

where:

$$\dot{m}$$ is the actual mass flow rate.

$${\dot{m}}_{\text{Ideal}}$$ is the ideal mass flow rate.

$${\dot{m}}_{\text{Leak}}$$ 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.*τ*_{Ideal}is the ideal driving torque.*τ*_{Friction}is the friction torque.

The ideal mass flow rate is

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

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**.*D*_{Sat}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:

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

where:

*D*is the displacement specified at physical signal port**D**.*D*_{Threshold}is the specified value of the**Displacement threshold for motor-pump transition**block parameter.

The internal leakage flow rate and friction torque calculations depend on the
block variant selected. If the block variant is ```
Analytical or
tabulated data
```

, the calculations depend also on the
**Leakage and friction parameterization** parameter
setting. There are five possible permutations of block variant and
parameterization settings.

**Case 1: Analytical Efficiency Calculation**

If the active block variant is ```
Analytical or tabulated
data
```

and the **Leakage and friction
parameterization** parameter is set to
`Analytical`

, the leakage flow rate is

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

and the friction torque is

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

where:

*K*_{HP}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*_{TP}is the specified value of the**Friction torque vs pressure drop coefficient**block parameter.*D*_{Nom}is the specified value of the**Nominal Displacement**block parameter.*τ*_{0}is the specified value of the**No-load torque**block parameter.*ω*_{Nom}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:

*ω*_{Nom}is the specified value of the**Nominal shaft angular velocity**parameter. This is the angular velocity at which the nominal volumetric efficiency is specified.*μ*_{Nom}is the specified value of the**Nominal Dynamic viscosity**block parameter. This is the dynamic viscosity at which the nominal volumetric efficiency is specified.*Δp*_{Nom}is the specified value of the**Nominal pressure drop**block parameter. This is the pressure drop at which the nominal volumetric efficiency is specified.*η*_{v,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.

**Case 2: Efficiency Tabulated Data**

If the active block variant is ```
Analytical or tabulated
data
```

and the **Leakage and friction
parameterization** parameter is set to ```
Tabulated data
— volumetric and mechanical efficiencies
```

, the leakage
flow rate is

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

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

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

*τ*_{Friction,Motor}is the friction torque in motor mode.*τ*_{Friction,Pump}is the friction torque in pump mode.

The smoothing parameter *α* is given by the hyperbolic function

$$\alpha =\text{tanh}\left(\frac{4\Delta p}{\Delta {p}_{\text{Threshold}}}\right)\xb7\text{tanh}\left(\frac{4\omega}{{\omega}_{\text{Threshold}}}\right)\xb7\mathrm{tanh}\left(\frac{4D}{{D}_{\text{Threshold}}}\right),$$

where:

*Δp*_{Threshold}is the specified value of the**Pressure gain threshold for pump-motor transition**block parameter.*ω*_{Threshold}is the specified value of the**Angular velocity threshold for pump-motor transition**block parameter.*D*_{Threshold}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
*Δp*–*ɷ*–*D* domain via
the **Volumetric efficiency table** block parameter. When
operating in pump mode (quadrants **1** and
**3** of the
*Δp*–*ɷ*–*D* chart shown
in the Operation Modes figure), the
leakage flow rate is:

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

where *η*_{v} 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
*Δp*–*ɷ*–*D* chart), the
leakage flow rate is:

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

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

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

where *η*_{m} 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
*Δp*–*ɷ*–*D* chart):

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

**Case 3: Loss Tabulated Data**

If the active block variant is ```
Analytical or tabulated
data
```

and the **Leakage and friction
parameterization** parameter is set to ```
Tabulated data
— volumetric and mechanical losses
```

, the leakage
(volumetric) flow rate is specified directly in tabulated form over the
*Δp*–*ɷ*–*D* 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:

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

The friction torque is similarly specified in tabulated form:

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

where *q*_{Leak}(*Δp*,*ω*) and *τ*_{Friction}(*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.

**Case 4: Efficiency Physical Signal Inputs**

If the active block variant is `Input efficiencies`

,
the leakage flow rate and friction torque calculations are as described for
efficiency tabulated data (case 2). The volumetric and mechanical efficiency
lookup tables are replaced with physical signal inputs that you specify through
ports EV and EM.

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

**Case 5: Loss Physical Signal Inputs**

If the block variant is `Input losses`

, the leakage
flow rate and friction torque calculations are as described for loss tabulated
data (case 3). The volumetric and mechanical loss lookup tables are replaced
with physical signal inputs that you specify through ports LV and LM.

The signs of the inputs are ignored. The block sets the signs automatically
from the operating conditions established during simulation—more precisely, from
the *Δp*–*ɷ* quadrant in which the component
happens to be operating. In other words, whether an input is positive or
negative is irrelevant to the block.

Fluid compressibility is negligible.

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