How to Measure Propeller Performance Coefficients

By Adam Kahraman

What are the propeller performance coefficients and why are they important?

Comparing propeller performance coefficients allows us to identify which propeller designs are more aerodynamically efficient and better suited to a given operating condition. This enables comparison of propeller geometries and designs independent of scale or rotation speed.

In this article we explore the key performance coefficients and how to measure them using the Input Transformations feature in the Flight Stand software.

These coefficients can help you make more meaningful comparisons between propellers, allowing you to save time and resources in UAV development.

Table of Contents

1. Introduction
2. Performance Coefficients
   1. Propulsion system components
   2. Thrust Coefficient Formula
   3. Torque Coefficient Formula
   4. Power Coefficient Formula
   5. Advance Ratio Formula
3. How to Measure Propeller Performance Coefficients
   1. Propeller Coefficient Formulas Setup
   2. Propeller Test Setup
4. Results
   1. Reynolds Number vs RPM
   2. Thrust Coefficient Graph
   3. Torque Coefficient Graph
   4. Power Coefficient Graph
   5. Advance Ratio Graph
5. Conclusion

Introduction

While static propeller efficiency is an important and useful metric to study, relying on it alone provides an incomplete picture on your performance. To compare how different propellers perform across various operating conditions, a more in-depth analysis can be achieved through the use of dimensionless coefficients. 

Propeller performance coefficients enable fair comparison of propeller efficiency and design suitability independent of scale or rotation speed.

This guide provides an overview of these propeller coefficients with a demonstration of how to compute them using the Input Transformations feature in the Flight Stand software.

It is important to note that some of these propeller coefficients are limited in their ability to assess performance at hover when forward velocity = 0. That said, for VTOL or tilt-rotor drones, they can still offer useful insights on performance during phases of flight with forward velocity.

For a guide more focused on VTOL hover performance, check out our article on How to Increase Drone Flight Time.

If you prefer to skip the math and jump straight into the step-by-step guide, head to Part 3 of this article.

Figure 1: Visual representation of the Advance Ratio equation

Performance Coefficients

In order to make comparisons between propellers of different sizes easier, engineers have come up with dimensionless (normalized) coefficients. For propellers, some of these key coefficients include the thrust coefficient, torque coefficient, power coefficient and advance ratio.

These coefficients are helpful when comparing propellers of different designs tested under similar operating conditions.

Reynolds Number Explained

In order to have a fundamental understanding of these coefficients, we first need to understand the relationship between propeller performance and Reynolds number.

When an object moves through a fluid like air, it faces aerodynamic forces governed by two properties of the fluid; the inertial forces driving it forward (momentum) and the viscosity of the fluid resisting this movement (friction). These two parameters are affected by the temperature, density and speed of the fluid. 

Reynolds number represents a relative measure of the inertial forces to viscous forces in a flow. In the context of airfoils, a higher Reynolds number indicates improved aerodynamic efficiency because the inertial forces dominate over the viscous effects.

While many aerodynamic analyses focus on the Reynolds number on a specific section of a propeller, the formula here calculates the Tip Reynolds number - the flow regime at the outermost and fastest spinning point on the propeller.

 

Small propellers typically operate at relatively low Reynolds numbers, where boundary-layer behavior is more sensitive to operating conditions and separation can occur more readily. Larger propellers operate at higher Reynolds numbers, where boundary layers are generally more robust and performance is less sensitive to small changes in operating conditions. This relationship can be seen in the formula itself; the larger the diameter (D), the larger the Reynolds number (Re).

When analyzing propeller performance, the Reynolds number must be taken into account as it dictates the specific air regime a propeller is operating in. For different propeller sizes and rotation speeds, the Reynolds number changes, and as it changes, the efficiency and performance change as well. If two propellers operate at very different Reynolds numbers, differences in performance may be due to flow regime effects rather than geometry or design quality.

That’s why, to ensure a fair comparison between different propeller designs, the representative Reynolds number should be the same or similar. That said, for fair comparison, Reynolds number should be consideredalong with other parameters such as advance ratio, tip Mach number, and blade loading.

Thrust Coefficient Formula

The thrust coefficient depends on propeller shape (pitch, diameter), air density and the rotation speed of the propeller. It can be calculated using the formula below:

• T is the thrust generated (N)
• ⍴_a is the density of air (kg/m³)
• n is the rotation speed of the propeller (Hz)
• D is the propeller diameter (m)

The formula shows us that there is a linear relationship between the thrust coefficient and thrust, therefore a higher coefficient means the thrust output per rotation is higher. So what does this mean in practice?

Because the thrust coefficient is unitless, you can compare propellersindependently of scale and thrust.

For example, you are comparing a 10 inch propeller and a 15 inch propeller. If you were to compare the thrust output only, the 15 inch would clearly win.But if the thrust coefficient were used, you can determine which propeller is actually moving more air for its size, given they are both operating at the same regimes. The same goes for same-size propellers operating at different RPMs.

Here's where it gets even better. For propellers operating in similar conditions (comparable Reynolds numbers and advance ratios), the thrust coefficient remains relatively stable. This means you can compare different propeller designs across a range of operating speeds, making it easier to identify which design is fundamentally more efficient. 

Check out the Results section later in this article to see how the thrust coefficient behaves at different RPMs.

Torque Coefficient Formula

The torque coefficient describes the aerodynamic torque load on the motor using the same nondimensional framework as the thrust coefficient. You can calculate it using the formula below:

 

• Q is the torque generated (Nm)
• ⍴_a is the density of air (kg/m³)
• n is the rotation speed of the propeller (Hz)
• D is the propeller diameter (m)

Similar to the thrust coefficient, this metric allows designers to compare the torque requirements of different propellers, normalized for their size and operating speed.

While C _T tells you the thrust output, C_Q shows the mechanical cost to the motor for this output.By using the torque coefficient to compare different propellers, you can select a propeller that matches the torque ranges your motor operates the most efficiently.

Power Coefficient Formula

The power coefficient is a measure of the mechanical energy required to make a propeller spin, normalized for its size and operating speed.

• P is the mechanical power (W)
• ⍴_a is the density of air (kg/m³)
• n is the rotation speed of the propeller (Hz)
• D is the propeller diameter (m)

A higher C_P corresponds to greater aerodynamic power absorbed by the propeller at a given size and rotation speed. This coefficient allows you to compare the power requirements of different propeller designs.

Advance Ratio Formula

The advance ratio is a nondimensional parameter that relates axial inflow velocity to propeller rotation speed and diameter, describing how fast the propeller moves through the air relative to how fast it rotates.

By combining the flight velocity and rotation speed into a single unitless formula, the advance ratio allows you to evaluate the propeller’s ability to travel forward for every revolution. To measure air velocity during testing, you can capture data using a pressure sensor or WindProbe that integrates with the Flight Stand software.

• V_a is the flight velocity (m/s)
• n is the rotation speed of the propeller (Hz)
• D is the propeller diameter (m)

A higher ratio indicates that the aircraft is moving fast relative to how quickly the blades are spinning. A lower ratio indicates that the propeller is spinning fewer times while covering a shorter distance.

The tables below show the typical advance ratios for a smaller propeller and a larger propeller respectively. It is important to note that this range depends on the specific pitch and diameter of the blade. The values for what is considered a “low” or “high” advance ratio differs for larger propellers.

Figure 4: Typical advance ratio range for a small scale propeller

Figure 5: Typical advance ratio range for a large scale propeller

A lower advance ratio occurs during takeoff when static thrust is required, while hovering represents zero advance ratio. However, maintaining a low ratio while aiming to fly fast is an indicator of an inefficient propulsion system with a propeller that is spinning excessively without sufficient forward movement.

How to Measure Propeller Performance Coefficients

To measure the propeller coefficients in real-time, you can use the Flight Stand software’s Input Transformations feature. This allows you to take raw data from the software and apply custom formulas to calculate any values you need for the propellers you are testing.

Propeller Coefficient Formulas Setup

The formulas we are generating in this article require us to input constants such as the air density, propeller diameter, and viscosity of air. We will first create these constants as new transformations (variable inputs) in the Flight Stand software.

Step 1: Go to the “Input transformations” tab in your Flight Stand software.

Figure 6: Input transformations tab on the Flight Stand software

Step 2: Click “Add new transformation”. Name it “Air density” and type the value you want to use in the formula box. Here we're using the standard sea-level air density of 1.225 kg/m³, but you can use a value more relevant to your local temperature, altitude, and conditions. Click “Save” to turn this value into a constant. You can repeat this as many times as you wish to add other constants.

Figure 7: Defining the air density constant for input transformations

Step 3: Click “Add new transformation” and name it “Propeller Diameter” and type in the diameter of the propeller you are testing. We’re using a 4” propeller which is 0.1016 meters. Click “Save” to save this transformation.

Figure 8: Defining propeller diameter for input transformations

Further reading: How to Connect Analog Sensors to Your Flight Stand

Step 4: To build the formula for the thrust coefficient, click “Add new transformation” and select your required variables as “Formula inputs”, including the ones we just created: air density, diameter, thrust and RPM.

Note on units: Check the units shown next to each input to ensure they match the units that the formula expects.

Figure 9: Assigning input parameters from the dropdown menu

Step 5: Type your thrust coefficient formula into the “Formula” box using the variables selected in the last step. Use the formula preview to verify that your equation is correct.

 Figure 10: Final formula preview with input values

Note on math expressions: The “Formula” text box expects “ ** ” for the power sign. A full list of math expressions can be found by clicking the question mark next to the formula box.

Figure 11: Reference guide for formula math syntax

Step 6: You can go back to the Input transformations tab to verify your saved formulas and hover your mouse over the expressions to see a mathematical preview. Under the “Current value” column, the software will output the value of the coefficients in real-time alongside the input data from the software.

Figure 12: Overview of the completed formula setup

Propeller test setup

Next we will set up an automated test to observe and record our newly created coefficients.

Step 1: Open the Flight Stand software and connect your Flight Stand to your computer via USB.
Step 2: Navigate to the “Powertrain mappings” tab and locate “Extra mappings”. 
Step 3: Under “Add extra input/output” select the extra mapping you previously defined in the input transformation tab. 

Figure 13: Mapping transformations from dropdown menu

Step 4: Repeat step 2 for all the transformations you wish for the software to record the values while performing the tests.

Figure 14: Summary of all active transformations in the software

Next, we can set up an automated test to run. In this case, we are building a simple linear throttle ramp.

Step 5: Open the “Automatic control” tab and select “Ramp” to begin your sequence.

Figure 15: Automatic control wizards

Step 6: Input your start and end values and the rate to define the throttle increase over time. 

Figure 16: Defined linear ramp parameters for the test sequence

Step 7: Use the “Sequence preview” window to verify the timing and throttle pattern.
Step 8: Name your test and enable “Continuous recording” to ensure all data points are captured.

Figure 17: Naming and executing the test sequence

Step 9: Navigate to “Manual Control” to activate your ESC.
Step 10: Return to the “Automatic control” tab and click “Execute sequence”. Save the results to export as a .csv file.

An in-depth and visual guide to this process can be found in the video below:

How to Automate Thrust Stand - Flight Stand Software

Results

Reynolds Number vs RPM

First, let’s look at the Reynolds number that was observed in this demonstration. The following graph shows the relationship between the Reynolds number and the rotation speed across the tested range.

Figure 18: Reynolds Number vs Rotation Speed (RPM)

This visualization can be used to identify the specific flow regime your propeller is in at different rotation speeds and ensure you are making comparisons between different propellers at similar Reynolds numbers.

This is important because the aerodynamic behavior changes across different Reynolds numbers. To have a fair comparison between propellers, they must be tested under similar conditions so that the observed performance differences can be attributed to propeller design. This is especially important for smaller propellers that are more sensitive to changes in Reynolds number.

Thrust Coefficient Graph

With our collected data, we observe the relationship between thrust, RPM, and the thrust coefficient:

Figure 19: Thrust coefficient vs Rotation Speed (RPM) vs Thrust (kgf)

As RPM increases from 5,500 to 16,000, we can see that the thrust, represented by the white line, increases steadily. The thrust coefficient, represented by the orange line, starts at approximately 0.15 and gradually rises to about 0.18 as the propeller spins faster. For a small scale propeller, this gradual increase in thrust coefficient is expected because higher rotational speeds increase the Reynolds number, improving the aerodynamic performance of the blade sections and allowing for greater thrust generation per rotation. The sharp drop at 16,000 RPM marks the end of the test where the propeller stops spinning.

Torque Coefficient Graph

Performing the same analysis for the torque coefficient we get the following graph:

Figure 20: Torque Coefficient vs Rotation Speed (RPM) vs Torque (Nm)

As RPM increases, the torque, represented by the white line, also increases steadily, showing the increasing mechanical load on the motor. The torque coefficient, represented by the orange line, remains relatively constant.

This demonstrates the usefulness of the torque coefficient as a normalized metric. While the torque that the motor has to provide increases with RPM, the ability for the motor to overcome that torque remains relatively constant throughout the range of RPM that we have tested. Which means that in this scenario, the motor is not struggling to provide the torque required to spin the propeller throughout the RPM range that we have tested. Monitoring the torque coefficient can help you determine if the motor is being utilized in its optimal torque range for the selected propeller.

Power Coefficient Graph

The graph below shows how the power coefficient and mechanical power changes across rotation speeds. This visualization quickly shows the power requirements across the operating RPM range, helping to determine if the motor can handle the propeller’s load at the target rotation speed.

Figure 21: Power Coefficient vs Rotation Speed (RPM) vs Mechanical Power (W)

In this scenario, the power coefficient remains relatively constant throughout the tested RPM range. For smaller propellers, the power coefficient stabilizes more quickly with increasing RPM, and the tested range shows we have already surpassed this transition point, operating entirely within the plateau region.

Advance Ratio Graph

Another key coefficient to explore is the advance ratio, representing the relationship between the speed at which an aircraft is moving forward and the speed at which the propeller is rotating.

Figure 22: Advance Ratio vs Rotation Speed (RPM) at 10 m/s flight velocity

For the sake of this experiment, a constant flight velocity of 10 m/s is assumed, a typical cruise speed for a small drone.

This visualization helps you understand how your propeller performs at different flight speeds. A propeller with good performance at low advance ratios performs better at hover and takeoff, while one that performs well at high advance ratios is better suited for fast forward flight and cruising.

This allows you to select or match your propeller for your typical flight characteristics and helps you identify potential issues - some propellers might struggle to maintain high efficiency at your typical cruise speed, others might not generate sufficient thrust for takeoff and hover.

Conclusion

Propeller performance coefficients, which nondimensionalize thrust, torque and power, allow propeller designs to be evaluated for aerodynamic efficiency and operating suitability independent of scale or rotational speed.

The Input Transformations feature in the Flight Stand software lets you calculate any performance metric you need in real-time during your tests. By inputting the formula once, you can automatically generate performance data for every test, export it as a CSV file, and analyze them.

Throughout this process, we looked at how to set up an automated test, collect data, and use the software to turn raw inputs into useful values through the formulas we inputted. This approach was used to find unitless performance coefficients for thrust, torque and power.

However, analyzing different propellers may not always be an apples-to-apples comparison. Aerodynamic effects like the Reynolds number and advance ratio play a major role in how a propeller performs at different flow regimes. In our results, the coefficients behaved as expected for a small-scale propeller.

While this article focused on specific coefficients and advance ratios, the Input Transformations feature can be used for many other formulas relevant to your project. A complementary article would be to perform the test with different propeller designs which would help how these coefficients compare against each other in different conditions.

If you've enjoyed this article, you will also enjoy our free eBook: Drone Building and Optimization: How to Increase Your Flight Time.

Another quick informative read would be to check out our article on Propeller Efficiency at Airspeeds from 0 to 38 mph.