# Fixed-Displacement Pump (TL)

Mechanical-hydraulic power conversion device

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

## Description

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

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

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.

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

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

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

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

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.

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

*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.*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.

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

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

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**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**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

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

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

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

where:

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

The leakage flow rate is calculated from the volumetric efficiency, a quantity
that is specified in tabulated form over the
*Δp*–*ɷ* domain via the
**Volumetric efficiency table** block parameter. When
operating in pump mode (quadrants **1** and
**3** of the *Δp*–*ɷ*
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* 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* domain via the
**Mechanical efficiency table** block parameter. When
operating in pump mode (quadrants **1** and
**3** of the *Δp*–*ɷ* 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* 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 *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`

). 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 *Δp*–*ɷ* quadrant in which
the component happens to be operating.

### Assumptions and Limitations

The pump is treated as a quasi-steady component.

The effects of fluid inertia and elevation are ignored.

The pump wall is rigid.

External leakage is ignored.

### 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

### Conserving

## Parameters

## Model Examples

## Extended Capabilities

## Version History

**Introduced in R2016a**