# 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 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.

### Operating Modes

The block has four modes of operation, as shown by this image.

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}:

The quadrant labeled

**1**represents the forward pump mode. In this mode, the positive shaft angular velocity causes a pressure increase from port**A**to port**B**and flow from port**A**to port**B**The quadrant labeled

**2**represents the reverse motor mode. In this mode, the flow from port**B**to port**A**causes a pressure decrease from**B**to**A**and negative shaft angular velocity.The quadrant labeled

**3**represents the reverse pump mode. In this mode, the negative shaft angular velocity causes a pressure increase from port**B**to port**A**and flow from port**B**to port**A**The quadrant labeled

**4**represents the forward motor mode. In this mode, the flow from port**A**to port**B**causes a pressure decrease from**A**to**B**and positive shaft angular velocity.

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

### Energy Balance

The block associates the mechanical work done by the pump with an energy exchange. The governing energy balance equation is

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

where:

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

The block generates mechanical power 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 required 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 gain from inlet to outlet.

### Leakage and Friction Parameterization

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

**Analytical**

When you set the **Leakage and friction parameterization**
parameter 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. The block computes this coefficient from the specified nominal parameters.*μ*is the dynamic viscosity of the fluid, which is the average of the values at the ports.*K*_{TP}is the friction torque vs. pressure gain coefficient at nominal displacement, which the block determines from the**Mechanical efficiency at nominal conditions**parameter,*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 value of the**Nominal pressure gain**parameter. This value is the pressure gain at which the nominal volumetric efficiency is specified.*τ*_{0}is the value of the**No-load torque**parameter.*ω*_{Nom}is the value of the**Nominal shaft angular velocity**parameter.

The block determines the Hagen-Poiseuille coefficient from the nominal fluid and component parameters

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

where:

*ω*_{Nom}is the value of the**Nominal shaft angular velocity**parameter. This value is the angular velocity at which the block specifies the nominal volumetric efficiency.*μ*_{Nom}is the value of the**Nominal Dynamic viscosity**parameter. This value is the dynamic viscosity at which the block specifies the nominal volumetric efficiency.*η*_{v,Nom}is the value of the**Volumetric efficiency at nominal conditions**parameter. This is the volumetric efficiency that corresponds to the specified nominal conditions.

**Tabulated Efficiencies**

When you set the **Leakage and friction parameterization**
parameter 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.

This hyperbolic function describes the smoothing parameter
*α*

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

where:

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

The block calculates the leakage flow rate from the volumetric efficiency,
which the ** Volumetric efficiency table** parameter specifies
over the *Δp*–*ɷ* domain. When operating in
pump mode, 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, which the block obtains either by interpolation or
extrapolation of the tabulated data. Similarly, when operating in motor mode,
the leakage flow rate is:

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

The block calculates the friction torque from the mechanical efficiency, which
the **Mechanical efficiency table** parameter specifies over
the *Δp*–*ɷ* domain. When operating in pump
mode, the friction torque is

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

where *η*_{m}is the
mechanical efficiency, which the block obtains either by interpolation or
extrapolation of the tabulated data. Similarly, when operating in motor mode,
the friction torque is

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

**Tabulated Losses**

When you set the **Leakage and friction parameterization**
parameter to ```
Tabulated data - volumetric and mechanical
losses
```

, the block specifies the leakage volumetric flow rate
in tabulated form over the *Δp*–*ɷ* domain:

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

The block calculates the mass flow rate due to leakage from the volumetric flow rate:

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

The block calculates the friction torque 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, specified through
interpolation or extrapolation of the tabulated data in the **Volumetric
loss table** and **Mechanical loss table**
parameters.

**Input Efficiencies**

When you set the **Leakage and friction parameterization**
parameter 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 block replaces the
volumetric and mechanical efficiency lookup tables with the physical signal
input ports **EV** and **EM**.

The efficiencies are positive quantities with values between
`0`

and `1`

.The block sets input values
outside of these bounds to `0`

for inputs smaller than
`0`

or `1`

for inputs greater than
`1`

. The block saturates the efficiency signals at the
value of the **Minimum volumetric efficiency** or
**Minimum mechanical efficiency ** parameter and the
**Maximum volumetric efficiency** or **Maximum
mechanical efficiency** parameter.

**Input Losses**

When you set the **Leakage and friction parameterization**
parameter to ```
Input signal - volumetric and mechanical
losses
```

, the leakage flow rate and friction torque calculations
are identical to the ```
Tabulated data - volumetric and mechanical
efficiencies
```

setting. The block replaces the volumetric and
mechanical loss lookup tables with the physical signal input ports
**LV** and **LM**.

The block expects the inputs to be positive and sets the signs automatically
from the *Δp*–*ɷ* quadrant where the component
is operating. If you provide a negative signal, the block returns zero
losses.

### Assumptions and Limitations

The block treats the pump as a quasi-steady component.

The block ignores the effects of fluid inertia and elevation.

The pump wall is rigid.

The block ignores external leakage.

## Ports

### Input

### Conserving

## Parameters

## Model Examples

## Extended Capabilities

## Version History

**Introduced in R2016a**