# Fixed-Displacement Motor (TL)

Hydraulic-mechanical power conversion device

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

## Description

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

Ports **A** and **B** represent the motor inlets.
Ports **R** and **C** represent the motor drive shaft
and case. During normal operation, a pressure drop from port **A** to
port **B** causes a positive flow rate from port **A**
to port **B** and a positive rotation of the motor shaft relative to
the motor case. This operation mode is referred to here as *forward
motor*.

**Operation Modes**

The block has four modes of operation. The working mode depends on the pressure drop
from port **A** to port **B**, *Δp =
p*_{B} – *p*_{A}
and the angular velocity, *ω = ω*_{R} –
*ω*_{C}:

Mode 1,

*Forward Motor*: Flow from port**A**to port**B**causes a pressure decrease from**A**to**B**and a positive shaft angular velocity.Mode 2,

*Reverse Pump*: Negative shaft angular velocity causes a pressure increase from port**B**to port**A**and flow from**B**to port**A**.Mode 3,

*Reverse Motor*: Flow from port**B**to port**A**causes a pressure decrease from**B**to**A**and a negative shaft angular velocity.Mode 4,

*Forward Pump*: Positive shaft angular velocity causes a pressure increase from port**A**to port**B**and flow from**A**to**B**.

The response time of the motor is considered negligible in comparison with the system response time. The motor 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:

*Φ*_{A}and*Φ*_{B}are energy flow rates at ports**A**and**B**, respectively.*P*_{hydro}is the motor hydraulic power. It is a function of the pressure difference between the motor ports: $${P}_{hydro}=\Delta p\frac{\dot{m}}{\rho}$$

The motor mechanical power is generated from the motor torque,
*τ* and angular velocity, *ω*:

$${P}_{mech}=T\omega .$$

### Flow Rate and Driving Torque

The mass flow rate generated at the motor 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 torque generated at the motor is

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

where:

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

**Ideal Flow Rate and Ideal Torque**

The ideal mass flow rate is

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

and the ideal generated torque is

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

where:

*ρ*is the average of the fluid densities at thermal liquid ports**A**and**B**.*D*is the**Displacement**parameter.*ω*is the shaft angular velocity.*Δp*is the pressure drop from inlet to outlet.

### 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

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

and the friction torque is

$${\tau}_{\text{Friction}}=\left({\tau}_{0}+{K}_{\text{TP}}\left|\Delta p\right|\text{tanh}\frac{4\omega}{\left(5\cdot {10}^{-5}\right){\omega}_{\text{Nom}}}\right),$$

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 friction torque vs. pressure gain coefficient at nominal displacement, which is determined from the**Mechanical efficiency at nominal conditions**,*η*:_{m}$$k=\frac{{\tau}_{fr,nom}-{\tau}_{0}}{\Delta {p}_{nom}}.$$

*τ*is the friction torque at nominal conditions:_{fric}$${\tau}_{fr,nom}=\left(1-{\eta}_{m,nom}\right)D\Delta {p}_{nom}.$$

*τ*_{0}is the specified value of the**No-load torque**parameter.*ω*_{Nom}is the specified value of the**Nominal shaft angular velocity**parameter.*Δp*_{Nom}is the specified value of the**Nominal pressure drop**parameter. This is the pressure drop at which the nominal volumetric efficiency is specified.

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

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

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.*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.

**Tabulated Efficiencies**

When you set **Leakage and friction parameterization** to
```
Tabulated data - volumetric and mechanical
efficiencies
```

, the leakage flow rate is

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

and the friction torque is

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

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 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.*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 motor mode (quadrants **1** and
**3** of the
*Δp*–*ɷ*–*D* chart shown
in the Operation Modes figure), the
leakage flow rate is:

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

where *η*_{v} is the
volumetric efficiency, obtained either by interpolation or extrapolation of the
tabulated data. Similarly, when operating in pump mode (quadrants
**2** and **4** of the
*Δp*–*ɷ*–*D* chart), the
leakage flow rate is:

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

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 motor mode (quadrants **1** and
**3** of the
*Δp*–*ɷ*–*D* chart):

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

where *η*_{m} is the
mechanical efficiency, obtained either by interpolation or extrapolation of the
tabulated data. Similarly, when operating in pump mode (quadrants
**2** and **4** of the
*Δp*–*ɷ*–*D* chart):

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

**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 *Δp*–*ɷ* domain:

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

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:

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

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.

**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`

). The efficiency signals are saturated at the
**Minimum volumetric efficiency** or **Minimum
mechanical efficiency ** and **Maximum volumetric
efficiency** or **Maximum mechanical efficiency
**.

**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 expects the inputs to be positive. It 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.

### Assumptions and Limitations

The motor is treated as a quasi-steady component.

The effects of fluid inertia and elevation are ignored.

The motor wall is rigid.

External leakage is ignored.

## Ports

### Input

### Conserving

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2016a**