Mechanical-to-hydraulic power conversion device

**Library:**Hydraulics (Isothermal) / Pumps and Motors

The Fixed-Displacement Pump block represents
a device that extracts power from a mechanical rotational network
and delivers it to a hydraulic (isothermal liquid) network. The pump
displacement is fixed at a constant value that you specify through
the **Displacement** parameter.

Ports **T** and **P** represent the pump inlets.
Port **S** represents the pump drive shaft. During normal operation,
the pressure gain from port **T** to port **P** is
positive if the angular velocity at port **S** 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 **T** to port **P** (Δ*p*) and on the angular velocity at port **S**
(*ω*). The Operation Modes figure maps the modes
to the quadrants of a Δ*p*-*ω* 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 volumetric flow rate generated at the pump is

$$q={q}_{\text{Ideal}}+{q}_{\text{Leak}},$$

*q*is the net volumetric flow rate.*q*_{Ideal}is the ideal volumetric flow rate.*q*_{Leak}is the internal leakage volumetric flow rate.

The driving torque required to power the pump is

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

*τ*is the net driving torque.*τ*_{Ideal}is the ideal driving torque.*τ*_{Friction}is the friction torque.

The ideal volumetric flow rate is

$${q}_{\text{Ideal}}=D\omega ,$$

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

*D*is the specified value of the**Displacement**block parameter.*ω*is the instantaneous angular velocity of the rotary shaft.*Δp*is the instantaneous pressure gain from inlet to outlet.

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

$${q}_{\text{Leak}}={K}_{\text{HP}}\Delta p,$$

$${\tau}_{\text{Friction}}=\left({\tau}_{0}+K{\text{}}_{\text{TP}}\left|\Delta p\right|\right)\mathrm{tanh}\left(\frac{4\omega}{{\omega}_{\text{Threshold}}}\right),$$

*K*_{HP}is the Hagen-Poiseuille coefficient for laminar pipe flows. This coefficient is computed from the specified nominal parameters.*K*_{TP}is the specified value of the**Friction torque vs pressure gain coefficient**block parameter.*τ*_{0}is the specified value of the**No-load torque**block parameter.*ω*_{Threshold}is the threshold angular velocity for the motor-pump transition. The threshold angular velocity is an internally set fraction of 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

$${K}_{\text{HP}}=\frac{{\nu}_{\text{Nom}}}{\rho v}\frac{\text{\hspace{0.17em}}{\rho}_{\text{Nom}}{\omega}_{\text{Nom}}D}{\Delta {p}_{\text{Nom}}}\left(1-{\eta}_{\text{v,Nom}}\right),$$

*ν*_{Nom}is the specified value of the**Nominal kinematic viscosity**block parameter. This is the kinematic viscosity at which the nominal volumetric efficiency is specified.*ρ*_{Nom}is the specified value of the**Nominal fluid density**block parameter. This is the density at which the nominal volumetric efficiency is specified.*ω*_{Nom}is the specified value of the**Nominal shaft angular velocity**block parameter. This is the angular velocity at which the nominal volumetric efficiency is specified.*ρ*is the actual fluid density in the attached hydraulic (isothermal liquid) network. This density can differ from the specified value of the**Nominal fluid density**block parameter.*v*is the kinematic viscosity of the fluid associated with the fluid network.*Δp*_{Nom}is the specified value of the**Nominal pressure gain**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

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

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

*α*is a numerical smoothing parameter for the pump-pump transition.*q*_{Leak,Pump}is the leakage flow rate in pump mode.*q*_{Leak,Motor}is the leakage flow rate in motor mode.*τ*_{Friction,Pump}is the friction torque in pump mode.*τ*_{Friction,Motor}is the friction torque in motor mode.

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

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

*Δp*_{Threshold}is the specified value of the**Pressure drop threshold for motor-pump transition**block parameter.*ω*_{Threshold}is the specified value of the**Angular velocity threshold for motor-pump transition**block parameter.

The leakage flow rate is computed from efficiency tabulated data through the equation

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

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

*η*_{v}is the volumetric efficiency obtained through interpolation or extrapolation of the**Volumetric efficiency table, e_v(dp,w)**parameter data.

Similarly, the friction torque is computed from efficiency tabulated data through the equation

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

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

*η*_{m}is the mechanical efficiency obtained through interpolation or extrapolation of the**Mechanical efficiency table, e_m(dp,w)**parameter data.

**Case 3: Loss Tabulated Data**

```
Analytical or tabulated
data
```

and the ```
Tabulated data
— volumetric and mechanical losses
```

, the leakage flow
rate equation is$${q}_{\text{Leak}}={q}_{\text{Leak}}\left(\Delta p,\omega \right).$$

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

**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.

**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.

If the block variant is set to ```
Analytical or tabulated
data
```

, you can plot a variety of performance, efficiency,
and loss curves from simulation data and component parameters. Use
the context-sensitive menu of the block to plot the characteristic
curves. Right-click the block to open the menu and select **Fluids** > **Plot characteristic**. A test harness opens with instructions on how to generate
the curves. See Pump and Motor Characteristic Curves.

Fluid compressibility is negligible.

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

Fixed-Displacement Motor | Fixed-Displacement Motor (TL) | Fixed-Displacement Pump (TL) | Variable-Displacement Motor | Variable-Displacement Pump

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