# Marine Propeller

**Libraries:**

Simscape /
Driveline /
Engines & Motors

## Description

The Marine Propeller block represents a propeller that converts a rotational mechanical motion into thrust for marine applications. You can configure the propeller with fixed or controllable blades. You can parameterize the propeller by using constants, polynomials, or tabulated data to characterize the thrust and torque coefficients. You can provide tabulated advance velocity data, or you can provide tabulated advance angle data to parameterize all four operational quadrants. Propellers that allow negative pitch or that can operate in reverse may include thrust and torque coefficient curves specific to the astern direction, which you can also specify in the block.

You can also include the wake effects of the vessel hull in the block. When you specify a constant wake fraction or enable a physical signal port, and the block calculates the wake effects automatically.

You can use a physical signal to control the blade pitch.

This terminology is helpful for understanding the block:

*Wake fraction*is the difference between the vessel velocity and the advance velocity expressed as a ratio of the vessel velocity.*Advance velocity*is the speed of the flow through the propeller,*V*._{a}*Advance ratio*is the speed of the flow through the propeller with respect to the propeller tip angular speed expressed as a ratio. The block uses this to determine*k*and_{T}*k*when you set_{Q}**Parameterization**to`Polynomial fit`

or`Tabulated data for advance ratio`

.*Advance angle*is the angular location of the propeller operational conditions on a four quadrant plot. The block uses this to determine*C*and_{T}*C*when you set_{Q}**Parameterization**to`Tabulated data for advance ratio`

.*Quadrant*is the relative two-dimensional location of the propeller operating condition where the vertical axis is*V*and the horizontal axis is_{a}*ω*.*Pitch*is the ideal translational propeller advance distance for a single revolution.*Open water*is when the effects of the hull are not present.

The block equations refer to these quantities:

*T*is the propeller thrust.*Q*is the propeller torque.*ρ*is the fluid density. You can specify the fluid density using the**Density**parameter or the**Rho**port.*P*is the pitch.*D*is the**Propeller diameter**parameter, where one represents*ω*is the propeller angular speed input at port**R**. For more information about using angular units in Simscape™, see Angular Units.*n*is the propeller angular speed in revolutions per second, which consistently nondimensionalizes the torque and thrust. The block defines*ω = 2πn*.*n*is the_{Thr}**Rotational speed threshold**parameter.*ε*is the**Propeller direction**parameter.*k*is the thrust coefficient with respect to the propeller rotational speed._{T}*k*is the resistive torque coefficient with respect to the propeller rotational speed._{Q}*p*is the polynomial thrust coefficient vector or 2-D matrix._{kT}*p*is the polynomial resistive torque coefficient vector or 2-D matrix._{kQ}*C*is the thrust coefficient with respect to the relative advance velocity._{T}*C*is the torque coefficient with respect to the relative advance velocity._{Q}*k*is the_{Thr}**Saturation threshold for nondimensional coefficients**parameter.*J*is the advance ratio.*V*is the advance velocity. Specify the advance velocity using the_{a}**Va**port.*V*is the relative advance velocity at a blade section at 70% of the blade radius._{R}*η*is the efficiency.*C*is the thrust coefficient based on the blade relative advance velocity at 70% blade radius._{T}^{*}*C*is the reference thrust coefficient vector or 2-D matrix._{T,TLU}^{*}*C*is the torque coefficient based on the blade relative advance velocity at 70% blade radius.^{*}_{Q}*C*is the reference torque coefficient vector or 2-D matrix.^{*}_{Q,TLU}*β*is the advance angle.*β*is the reference advance angle._{TLU}

### Parameterizations

The propeller performance depends on the thrust and resistive torque coefficients.
The **Parameterization** parameter gives you different options to
control these coefficients. The propeller output depends on the quadrant where the
propeller operates. The block defines the four quadrants as:

First quadrant:

*+V*,_{a}*+n*Second quadrant:

*+V*,_{a}*-n*Third quadrant:

*-V*,_{a}*-n*Fourth quadrant:

*-V*,_{a}*+n*

The figure shows a visual representation of the quadrants.

When you set **Parameterization** to
`Constant coefficients`

, you specify the thrust and
resistive torque coefficients directly. Otherwise, the block computes these
coefficients depending on the **Parameterization**
setting.

**Advance Ratio**

When you set **Parameterization** to ```
Polynomial
fit
```

or ```
Tabulated data for advance
ratio
```

, the block uses the advance ratio, *J*.
The block uses a numerically smoothed version of the fundamental thrust and
torque equations such that

$$\begin{array}{l}T={k}_{T}\rho {D}^{4}n\sqrt{{n}^{2}+{n}_{thr}^{2}}\\ Q={k}_{Q}\rho {D}^{5}n\sqrt{{n}^{2}+{n}_{thr}^{2}}\end{array}$$

The block defines the advance ratio as

$$J=\frac{{V}_{a}\epsilon n}{D({n}^{2}+{n}_{Thr}^{2})},$$

where the angular speed threshold
*n _{Thr}* linearizes the propeller
rotational speed,

*n*, for smoothing.

When you set **Parameterization** to:

`Polynomial fit`

—*k*and_{T}*k*vary with time according to the values you specify for the polynomial coefficient parameters. The block saturates_{Q}*J*to be between 0 and the first positive root of the polynomial and restricts*k*and_{T}*k*to always be positive. The block calculates the thrust and torque coefficients as_{Q}$$\begin{array}{l}{k}_{T}={\displaystyle \sum _{j=1:N}^{N}{p}_{kT,j}{J}^{j}}\\ {k}_{Q}={\displaystyle \sum _{j=1:N}^{N}{p}_{kQ,j}{J}^{j}}\end{array}$$

where

*p*and_{kT}*p*represent the polynomial coefficients._{kQ}`Tabulated data for advance ratio`

— You specify tabulated values for*k*and_{T}*k*for given values of_{Q}*J*or*J*and*P/D*, depending on the**Blade pitch type**parameter.

This basis of the propeller efficiency is the fundamental relationship

$$\eta =\frac{Powe{r}_{out}}{Powe{r}_{in}}=\frac{T{V}_{A}}{2\pi nQ}=\frac{J}{2\pi}\frac{{k}_{T}}{{k}_{Q}}.$$

When **Efficiency sensor** is on, and you set
**Parameterization** to
`Constant`

, the block calculates the smoothed
efficiency as

$$\eta =\frac{{k}_{T}\sqrt{{J}^{2}+{k}_{Thr}^{2}}}{2\pi {k}_{Q}}.$$

When **Efficiency sensor** is on, and you
set **Parameterization** to ```
Polynomial
fit
```

or ```
Tabulated data for advance
velocity
```

, the block calculates the smoothed efficiency as

$$\eta =\frac{{k}_{T}\sqrt{{J}^{2}+{k}_{Thr}^{2}}}{2\pi \sqrt{{k}_{Q}^{2}+{\left(0.01\xb7{k}_{Thr}\right)}^{2}}}.$$

**Advance Angle**

When you set **Parameterization** to ```
Tabulated
data for advance angle
```

, the block uses thrust and torque
coefficients with respect to relative advance angle. The block defines the
advance angle as

$$\beta =\mathrm{arctan}\left(\frac{{V}_{a}}{0.7\pi \epsilon nD}\right),$$

where *β* is cyclic. You must ensure that
the coefficient extrapolation and cycle wrapping occur as expected. The block
defines the thrust and torque coefficients for relative advance velocity as

$$\begin{array}{l}{C}_{T}^{*}=\frac{T}{{\scriptscriptstyle \frac{1}{8}}\rho {V}_{R}^{2}\pi {D}^{2}}\\ {C}_{Q}^{*}=\frac{Q}{{\scriptscriptstyle \frac{1}{8}}\rho {V}_{R}^{2}\pi {D}^{3}}\end{array}$$

where *V _{R}* is the
relative advance velocity at a blade section at 70% of the blade radius, such that

$${V}_{R}^{2}={V}_{A}^{2}+{\left(0.7D\pi \epsilon n\right)}^{2}.$$

Rearranging the coefficient equations yields the block equations for thrust and torque with respect to relative advance velocity:

$$\begin{array}{l}T={C}_{T}^{*}{\scriptscriptstyle \frac{1}{8}}\rho {V}_{R}^{2}{D}^{2}\\ Q={C}_{Q}^{*}{\scriptscriptstyle \frac{1}{8}}\rho {V}_{R}^{2}{D}^{3}\end{array}$$

When you set **Blade pitch** to:

`Constant`

, the block calculates the thrust and torque coefficients as$$\begin{array}{l}{C}_{T}^{*}=tablelookup\left({\beta}_{TLU},{C}_{T,TLU}^{*},\beta ,\text{interpolation=}interp\_method,\text{extrapolation=}extrap\_method\right)\\ {C}_{Q}^{*}=tablelookup\left({\beta}_{TLU},{C}_{Q,TLU}^{*},\beta ,\text{interpolation=}interp\_method,\text{extrapolation=}extrap\_method\right)\end{array}$$

`Controlled`

, the block calculates the thrust and torque coefficients as$$\begin{array}{l}{C}_{T}^{*}=tablelookup\left(P/{D}_{TLU},{\beta}_{TLU},{C}_{T,TLU}^{*},P/D,\beta ,\text{interpolation=}interp\_method,\text{extrapolation=}extrap\_method\right)\\ {C}_{Q}^{*}=tablelookup\left(P/{D}_{TLU},{\beta}_{TLU},{C}_{Q,TLU}^{*},P/D,\beta ,\text{interpolation=}interp\_method,\text{extrapolation=}extrap\_method\right)\end{array}$$

The basis of the propeller efficiency is the fundamental relationship

$$\eta =\frac{Powe{r}_{out}}{Powe{r}_{in}}=\frac{T{V}_{A}}{2\pi \epsilon nQ}=\frac{{V}_{A}{C}_{T}^{*}}{2\pi \epsilon nD{C}_{Q}^{*}}.$$

When **Efficiency sensor** is on, the block
calculates the smoothed efficiency as

$$\eta =\frac{1}{2\pi D}\sqrt{\frac{{V}_{A}^{2}{C}^{*2}+{\left(D\pi {n}_{Thr}{K}_{Thr}\right)}^{2}}{{n}^{2}{C}_{Q}^{*2}+{\left(0.1{n}_{Thr}{K}_{Thr}\right)}^{2}}}.$$

**Environment Interaction**

When you set **Translational connections** to
`Conserving`

, the block uses a constant wake
fraction to relate the vessel velocity to the advance velocity. You input the
thrust and velocity of the vessel by using the **R2** and
**C2** ports. The block computes the advance velocity
as:

$${V}_{A}=V(1-w),$$

where:

*V*is the vessel velocity. You can specify the vessel velocity relative to the reference by using the**R2**and**C2**ports, where*V*=*V*-_{R2}*V*._{C2}*w*is the**Wake fraction**parameter.

When you set **Translational connections** to
`Physical connections`

, you can use the
**Va** port to supply the advance velocity as a physical
signal. The block outputs the propeller thrust as a physical signal from the
**Th** port.

**Controlled Pitch**

When you set **Blade pitch type** to
`Controlled`

, you can parameterize the propeller
over a range of pitch-diameter ratios, *P/D*. You must specify
the *P/D* range as a vector in the **Pitch-diameter
ratio vector, P/D** parameter, where each element corresponds to a
row in the *k _{T}* and

*k*matrices.

_{Q}### Inertia

You can optionally include translational and rotational propeller inertia. To
simulate inertia, set **Rotational connections** or
**Translational connections** to
`Conserving`

, and select **Model
inertia**. When you select **Model inertia** and set
**Rotational connections** to
`Conserving`

, set the initial rotational velocity or
torque on the shaft in the **Initial Targets** section, or set an
algebraically linked variable to high priority to initialize the rotational inertia.
When you select **Model inertia** and set **Translational
connections** to `Conserving`

, set the initial
translational velocity or thrust in the **Initial Targets**
section, or set an algebraically linked variable to high priority to initialize the
translational mass.

**Note**

For rotational conserving connections, the block logs
`Q`

, the aerodynamic torque, and
`Inertia.t`

.

For translational conserving connections, the block logs
`thrust`

, the aerodynamic thrust, and
`mass.f`

.

You can use an Ideal Torque Sensor or Ideal Force Sensor blocks to log the sum of the inertia and the aerodynamic torque or force, respectively.

### Assumptions and Limitations

The block treats the fluid velocity and propeller rotational velocity as quasi-steady in time. Fluid flows uniformly over the propeller.

When you set

**Parameterization**to`Polynomial fit`

, the block assumes that the propeller torque and thrust coefficients are symmetric with the first quadrant.When you set

**Parameterization**to`Tabulated data for advance ratio`

, the block assumes the torque and thrust coefficients are identical in the first and third quadrants and the second and fourth quadrants.When you set

**Parameterization**to`Tabulated data for advance angle`

, the block removes the sign from*V*. To attain negative thrusts and torques, you must include the signs in the values of_{a}*C*and_{T}*C*._{Q}

### 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.

Nominal values provide a way to specify the expected magnitude of a variable in a model.
Using system scaling based on nominal values increases the simulation robustness. Nominal
values can come from different sources, one of which is the **Nominal
Values** section in the block dialog box or Property Inspector. For more
information, see Modify Nominal Values for a Block Variable.

## Examples

## Ports

### Inputs

### Outputs

### Conserving

## Parameters

## References

[1] Bernitsas, Michael M., D. Ray,
P. Kinley. "Kt, Kq and Efficiency Curves for the Wageningen B-Series Propellers."
*Report* 237. Department of Naval Architecture and Marine
Engineering. College of Engineering. University of Michigan, 1981.

[2] Carlton, J. S.
*Marine Propellers and Propulsion*. Second edition. Oxford:
Elsevier, 2007.

## Extended Capabilities

## Version History

**Introduced in R2021b**