Estimation models with dimensional analysis and linear regressions (student version)

Written by Marc Budinger (INSA Toulouse), Scott Delbecq (ISAE-SUPAERO) and Félix Pollet (ISAE-SUPAERO), Toulouse, France.

Propellers

The application chosen for this notebook are UAV propellers. Propellers characteristics can be expressed by \(C_T\) and \(C_P\) coefficients. These coefficients are function of dimensions and conditions of use of propellers. Dimensional analysis and linear regression of suppliers data can be used for the generation of \(C_T\) and \(C_P\) prediction models.

Fig. 6 APC MR (Multi-Rotor) propellers

Dimensional analysis and \(\pi\) numbers

The propeller performances can be expressed with 2 aerodynamic coefficients:

• The thrust: \(F = C_{T} \rho_{air} n^2 D^4\)

• The power: \(P = C_{P} \rho_{air} n^3 D^5 \)

The dimensional analysis and especially the Buckingham \(\pi\) theorem enable to find this results.

Dimensional analysis of the propeller thrust

The thrust \(F\) of a propeller depends of multiple parameters (geometrical dimensions, air properties, operational points):
\(F=f(\rho_{air},n,D,p,V,\beta_{air})\)
with the parameters express in the following table.

Parameter	M	L	T
Thrust \(T\) [N]	1	1	-2
Mass volumic (Air) \(\rho_{air}\) [kg/m\(^3\)]	1	-3	0
Rotational speed \(n\) [Hz]	0	0	-1
Diameter \(D\) [m]	0	1	0
Pitch \(p\) [m]	0	1	0
Drone speed \(V\) [m/s]	0	1	-1
Bulk modulus (Air) \(\beta_{air}\) [Pa]	1	-1	-2
\(=\pi_0\)
\(=\pi_1\)
\(=\pi_2\)
\(=\pi_3\)

Remark: The dimension of a parameter \(x\) is function of dimensions L, M and T : \([x]=M^aL^bT^c\). The previous table gives the value of \(a\), \(b\) and \(c\) for each parameter of the problem.

Exercise 7

Complete the table with 4 dimensionless \(\pi\) numbers possible for the given problem. Explain the number of dimensionless number.

Parameter	M	L	T
Thrust \(T\) [N]	1	1	-2
Mass volumic (Air) \(\rho_{air}\) [kg/m\(^3\)]	1	-3	0
Rotational speed \(n\) [Hz]	0	0	-1
Diameter \(D\) [m]	0	1	0
Pitch \(p\) [m]	0	1	0
Drone speed \(V\) [m/s]	0	1	-1
Bulk modulus (Air) \(\beta_{air}\) [Pa]	1	-1	-2
\(=\pi_0\)
\(=\pi_1\)
\(=\pi_2\)
\(=\pi_3\)

Effect of the rotational speed

APC suppliers give complete propeller data for all their propellers. From the file APC_STATIC-data-all-props.csv, we find all static data provided by APC:

import pandas as pd

# Read the .csv file with bearing data
path = "https://raw.githubusercontent.com/SizingLab/sizing_course/main/laboratories/Lab-multirotor/assets/data/"
df = pd.read_csv(path + "APC_STATIC-data-all-props.csv", sep=";")
# Print the head (first lines of the file)
df.head()

	LINE	COMP	TYPE	RPM	DIAMETER(IN)	PITCH(IN)	BLADE(nb)	THRUST(LBF)	POWER(HP)	TORQUE(IN.LBF)	Cp	Ct	AREA(m^2)	THRUST(N)	POWER(W)	ANGLE	EFF	N.D
0	1	1	NaN	1000	10.5	4.5	2	0.03	0.01	0.02	0.03	0.08	0.06	0.1335	7.457	0.43	60.180222	10500.0
1	2	1	NaN	2000	10.5	4.5	2	0.13	0.01	0.08	0.03	0.08	0.06	0.5785	7.457	0.43	60.180222	21000.0
2	3	1	NaN	3000	10.5	4.5	2	0.29	0.01	0.17	0.03	0.08	0.06	1.2905	7.457	0.43	60.180222	31500.0
3	4	1	NaN	4000	10.5	4.5	2	0.52	0.02	0.30	0.03	0.08	0.06	2.3140	14.914	0.43	60.180222	42000.0
4	5	1	NaN	5000	10.5	4.5	2	0.81	0.04	0.47	0.03	0.08	0.06	3.6045	29.828	0.43	60.180222	52500.0

For next steps, we keep only the Multi-Rotor type propellers (MR).

# Data Filtering
dfMR = df[df["TYPE"] == "MR"]
dfMR.head()

	LINE	COMP	TYPE	RPM	DIAMETER(IN)	PITCH(IN)	BLADE(nb)	THRUST(LBF)	POWER(HP)	TORQUE(IN.LBF)	Cp	Ct	AREA(m^2)	THRUST(N)	POWER(W)	ANGLE	EFF	N.D
135	147	8	MR	2000	10.0	4.5	2	0.14	0.01	0.09	0.04	0.11	0.05	0.6230	7.457	0.45	72.772802	20000.0
146	148	8	MR	3000	10.0	4.5	2	0.32	0.01	0.20	0.04	0.11	0.05	1.4240	7.457	0.45	72.772802	30000.0
147	149	8	MR	4000	10.0	4.5	2	0.57	0.02	0.36	0.04	0.11	0.05	2.5365	14.914	0.45	72.772802	40000.0
148	150	8	MR	5000	10.0	4.5	2	0.90	0.04	0.56	0.04	0.11	0.05	4.0050	29.828	0.45	72.772802	50000.0
149	151	8	MR	6000	10.0	4.5	2	1.29	0.08	0.79	0.04	0.11	0.05	5.7405	59.656	0.45	72.772802	60000.0

We plot the \(C_p\) and \(C_t\) for the a 10x4.5 propeller (COMP n° 8 in the previous table). We can notice that these coefficients are constant up to a certain value of speed of rotation. The manufacturer recommends using these propellers for a product speed of rotation x diameter less than a limit (depending on the type of propeller technology) and given here:
Maximum speed(RPM) x prop diameter (inches) = 105,000
for MR type which gives a blade tip speed of 135 m/s.

Question: Explain the origin of this operating limit comes from and the \(\pi\) number that can express it.

# Keep only the component n°8
df8 = dfMR[dfMR["COMP"] == 8]

# Extract forbidden ND product
df8ND = df8[df8["N.D"] > 105000]

import numpy as np
import matplotlib.pyplot as plt

# plot the data
plt.plot(
    df8["RPM"],
    df8["Cp"],
    "bo",
    df8["RPM"],
    df8["Ct"],
    "ro",
)
plt.plot(
    df8ND["RPM"],
    df8ND["Cp"],
    "ko",
    df8ND["RPM"],
    df8ND["Ct"],
    "ko",
)
plt.xlabel("Rotational Speed [RPM]")
plt.ylabel("Cp (blue) and Ct (red)")
plt.grid()
plt.show()

Linear regression

For next calculations, we keep only data with following criteria:

• Type ‘MR’ (Multi-Rotor)

• Maximum RPM < 105,000/prop diameter (inches)

# Keep only operating points with ND<105000
dfMRND = dfMR[dfMR["N.D"] < 105000]

The APC static data correspond to the hover operational point where the speed V=0. The aerodynamic coefficients are thus only a function of \(p/D\) (called ‘ANGLE’ in the .csv file) dimensionless number.

\(C_t=\frac{F}{\rho_{air} n^2 D^4}=f(\frac{p}{D})\)
\(C_p=\frac{P}{\rho_{air} n^3 D^5}=g(\frac{p}{D})\)

The following code uses the Scikit-learn package in order to set up a \(C_t\) estimator for the static case (\(V=0\) or \(J=0\)).

# Import packages
from sklearn import linear_model
from sklearn.metrics import r2_score
import matplotlib.pyplot as plt


# Data
x = df_MR_ND["ANGLE"].values
y_Ct = df_MR_ND["Ct"].values

# Matrix X and Y
X = x.reshape(-1, 1)

Y_Ct = y_Ct.reshape(-1, 1)

# Create a new object for the linear regression
reg_Ct = linear_model.LinearRegression()

reg_Ct.fit(X, Y_Ct)


# Y vector prediction
Ct_est = reg_Ct.predict(X)

# Ct Parameters
# ----
coef = float(reg_Ct.coef_)
intercept = float(reg_Ct.intercept_)
r2 = r2_score(Y_Ct, Ct_est)


# Plot the data
plt.plot(x, Y_Ct, "o", label="Reference data")
plt.plot(x, Ct_est, "-g", label="Data prediction")
plt.xlabel("Pitch/Diameter  ratio")
plt.ylabel("Ct")
plt.title("Comparison of reference data and regression")
plt.legend()
plt.grid()
plt.show()

print(f"Ct estimation model : Ct = {intercept:.2f} + {coef:.2f} * p/D with R2={r2:.3f}")

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[5], line 8
      4 import matplotlib.pyplot as plt
      7 # Data
----> 8 x = df_MR_ND["ANGLE"].values
      9 y_Ct = df_MR_ND["Ct"].values
     11 # Matrix X and Y

NameError: name 'df_MR_ND' is not defined

Exercise 8

Perform a linear regression of \(C_p\) data.

y_Cp =

reg_Cp = 



# Cp Parameters 
# -----
coef = float(reg_Ct.coef_)
intercept = float(reg_Ct.intercept_)
r2 = r2_score(Y_Ct, Ct_est)


# Plot the data 
plt.plot(x, Y_Cp, 'o', label='Reference data')
plt.plot(x, Cp_est, '-g', label='Data prediction')
plt.xlabel('Pitch/Diameter  ratio')
plt.ylabel('Cp')
plt.title('Comparison of reference data and regression')
plt.legend()
plt.grid()
plt.show()

print(f"Cp estimation model : Cp = {intercept:.2f} + {coef:.2f} * p/D with R2={r2:.3f}")

  Cell In[6], line 1
    y_Cp =
          ^
SyntaxError: invalid syntax