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Fixed-displacement pump in isothermal liquid system

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

The Fixed-Displacement Pump (IL) block models a pump with constant-volume
displacement. The fluid may move from port **A** to port
**B**, called *forward mode*, or from port
**B** to port **A**, called *reverse
mode*. Pump mode operation occurs when there is a pressure gain in the
direction of the flow. Motor mode operation occurs when there is a pressure drop in the
direction of the flow.

Shaft rotation corresponds to the sign of the fluid volume. Positive fluid displacement corresponds to positive shaft rotation in forward mode. Negative fluid displacement corresponds to negative shaft angular velocity in forward mode.

**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 pump block has analytical, lookup table, and physical signal parameterizations. When using tabulated data or an input signal for parameterization, you can choose to characterize pump operation based on efficiency or losses.

The threshold parameters **Pressure gain threshold for pump-motor
transition** and **Angular velocity threshold for pump-motor
transition** identify regions where numerically smoothed flow transition
between the pump operational modes can occur. For the pressure and angular velocity
thresholds, choose a transition region that provides some margin for the transition
term, but which is small enough relative to the typical pump pressure gain and angular
velocity so that it will not impact calculation results.

If you set **Leakage and friction parameterization** to
`Analytical`

, the block calculates leakage and friction
from constant values of shaft velocity, pressure gain, and friction torque. The
leakage flow rate, which is correlated with the pressure differential over the pump,
is calculated as:

$${\dot{m}}_{leak}=K{\rho}_{avg}\Delta p,$$

where:

*Δp*_{nom}is*p*_{B}–*p*_{A}.*ρ*_{avg}is the average fluid density.*K*is the Hagen-Poiseuille coefficient for analytical loss,$$K=\frac{D{\omega}_{nom}\left(1-{\eta}_{v,nom}\right)}{\Delta {p}_{nom}},$$

where:

*D*is the**Displacement**.*ω*_{nom}is the**Nominal shaft angular velocity**.*η*_{v, nom}is the**Volumetric efficiency at nominal conditions**.*Δp*_{nom}is the**Nominal pressure gain**.

The friction torque, which is related to the pump pressure differential, is calculated as:

$${\tau}_{fr}=\left({\tau}_{0}+k\left|\Delta p\right|\right)\mathrm{tanh}\left(\frac{4\omega}{5\times {10}^{-5}{\omega}_{nom}}\right),$$

where:

*τ*_{0}is the**No-load torque**.*k*is the friction torque vs. pressure gain coefficient at nominal displacement, which is determined from the**Mechanical efficiency at nominal conditions**,*η*:_{m, nom}$$k=\frac{{\tau}_{fr,nom}-{\tau}_{0}}{\Delta {p}_{nom}}.$$

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

*ω*is the relative shaft angular velocity, or $${\omega}_{R}-{\omega}_{C}$$.

When using tabulated data for pump efficiencies or losses, you can provide data for one or more of the pump operational modes. The signs of the tabulated data determine the operational regime of the block. When data is provided for less than four operational modes, the block calculates the complementing data for the other mode(s) by extending the given data into the remaining quadrants.

```
Tabulated data - volumetric and mechanical
efficiencies
```

parameterizationThe leakage flow rate is calculated as:

$${\dot{m}}_{leak}={\dot{m}}_{leak,pump}\left(\frac{1+\alpha}{2}\right)+{\dot{m}}_{leak,motor}\left(\frac{1-\alpha}{2}\right),$$

where:

$${\dot{m}}_{leak,pump}=\left(1-{\eta}_{\upsilon}\right){\dot{m}}_{ideal}$$

$${\dot{m}}_{leak,motor}=\left({\eta}_{v}-1\right)\dot{m}$$

and *η*_{v} is the volumetric efficiency,
which is interpolated from the user-provided tabulated data. The transition
term, *α*, is

$$\alpha =\mathrm{tanh}\left(\frac{4\Delta p}{\Delta {p}_{threshold}}\right)\mathrm{tanh}\left(\frac{4\omega}{{\omega}_{threshold}}\right),$$

where:

*Δp*is*p*_{B}–*p*_{A}.*p*_{threshold}is the**Pressure gain threshold for pump-motor transition**.*ω*is*ω*_{R}–*ω*_{C}.*ω*_{threshold}is the**Angular velocity threshold for pump-motor transition**.

The friction torque is calculated as:

$${\tau}_{fr}={\tau}_{fr,pump}\left(\frac{1+\alpha}{2}\right)+{\tau}_{fr,motor}\left(\frac{1-\alpha}{2}\right),$$

where:

$${\tau}_{fr,pump}=\left(1-{\eta}_{m}\right)\tau $$

$${\tau}_{fr,motor}=\left({\eta}_{m}-1\right){\tau}_{ideal}$$

and *η*_{m} is the
mechanical efficiency, which is interpolated from the user-provided tabulated
data.

```
Tabulated data - volumetric and mechanical
losses
```

parameterizationThe leakage flow rate is calculated as:

$${\dot{m}}_{leak}={\rho}_{avg}{q}_{loss}\left(\Delta p,\omega \right),$$

where *q*_{loss} is interpolated from the
**Volumetric loss table, q_loss(dp,w)** parameter, which is
based on user-supplied data for pressure drop, shaft angular velocity, and fluid
volumetric displacement.

The shaft friction torque is calculated as:

$${\tau}_{fr}={\tau}_{loss}\left(\Delta p,\omega \right),$$

where *τ*_{loss} is interpolated from the
**Mechanical loss table, torque_loss(dp,w)** parameter,
which is based on user-supplied data for pressure drop and shaft angular
velocity.

When you select ```
Input signal - volumetric and mechanical
efficiencies
```

, ports **EV** and
**EM** are enabled. The internal leakage and shaft friction are
calculated in the same way as the ```
Tabulated data - volumetric and
mechanical efficiencies
```

parameterization, except that
*η*_{v} and
*η*_{m} are received directly at ports
**EV** and **EM**, respectively.

When you select ```
Input signal - volumetric and mechanical
losses
```

, ports **LV** and **LM**
are enabled. These ports receive leakage flow and friction torque as positive
physical signals. The leakage flow rate is calculated as:

$${\dot{m}}_{leak}={\rho}_{avg}{q}_{LV}\mathrm{tanh}\left(\frac{4\Delta p}{{p}_{thresh}}\right),$$

where:

*q*_{LV}is the leakage flow received at port**LV**.*p*_{thresh}is the**Pressure gain threshold for pump-motor transition**parameter.

The friction torque is calculated as:

$${\tau}_{fr}={\tau}_{LM}\mathrm{tanh}\left(\frac{4\omega}{{\omega}_{thresh}}\right),$$

where

*τ*_{LM}is the friction torque received at port**LM**.*ω*_{thresh}is the**Angular velocity threshold for pump-motor transition**parameter.

The volumetric and mechanical efficiencies range between the user-defined specified minimum and maximum values. Any values lower or higher than this range will take on the minimum and maximum specified values, respectively.

The pump flow rate is:

$$\dot{m}={\dot{m}}_{ideal}-{\dot{m}}_{leak},$$

where $${\dot{m}}_{ideal}={\rho}_{avg}D\cdot \omega .$$

The pump torque is:

$$\tau ={\tau}_{ideal}+{\tau}_{fr},$$

where $${\tau}_{ideal}=D\cdot \Delta p.$$

The mechanical power delivered by the pump shaft is:

$${\phi}_{mech}=\tau \omega ,$$

and the pump hydraulic power is:

$${\phi}_{hyd}=\frac{\Delta p\dot{m}}{{\rho}_{avg}}.$$

If you would like to know if the block is operating beyond the
supplied tabulated data, you can set **Check if operating beyond the
quadrants of supplied tabulated data** to
`Warning`

to receive a warning if this occurs, or
`Error`

to stop the simulation when this occurs. For
parameterization by input signal for volumetric or mechanical losses, you can be
notified if the simulation surpasses operating modes with the **Check if
operating beyond pump mode** parameter.

You can also monitor pump functionality. Set **Check if pressures are less
than pump minimum pressure** to `Warning`

to
receive a warning if this occurs, or `Error`

to stop the
simulation when this occurs.

Pre-parameterization of the Fixed-Displacement Pump (IL) block with manufacturer data is available. This data allows you to model a specific supplier component.

To load a predefined parameterization,

Click the "Select a predefined parameterization" hyperlink in the Fixed-Displacement Pump (IL) block dialog description.

Select a part from the drop-down menu and click

**Update block with selected part**.If you change any parameter settings after loading a parameterization, you can check your changes by clicking

**Compare block settings with selected part**. Any difference in settings between the block and pre-defined parameterization will display in the MATLAB command window.

**Note**

Predefined parameterizations of Simscape components use available data sources for supplying parameter values. Engineering judgement and simplifying assumptions are used to fill in for missing data. As a result, deviations between simulated and actual physical behavior should be expected. To ensure requisite accuracy, you should validate simulated behavior against experimental data and refine component models as necessary.

Variable-Displacement Pump (IL) | Fixed-Displacement Pump (TL) | Pressure-Compensated Pump (IL) | Fixed-Displacement Motor (IL)