Analysis of the performance of APC propellers in the engine O.S. 55 AX type glow

Importing relevant libraries

import pandas as pd
import numpy as np
import sys
import re
import matplotlib.pyplot as plt
import seaborn as sns
if sys.version_info[0] < 3: 
    from StringIO import S
    tringIO
else:
    from io import StringIO

Functions in use in the analysis

read_txt(filename, model)

def read_txt(filename, model):
    """Reads the .txt files from APC and returns a pandas Dataframe.

    Parameters
    ----------
    filename : str, path object or file-like object
        Name of the file to be read.
    model : str
        Name of the model to be read.

    Returns
    -------
    pandas.DataFrame
        A txt file is returned as a two-dimensional data structure with labeled axes.    
    """
    buffer=""
    model=str(model)
    num_lines = sum(1 for line in open(filename))
    line_on_focus = []
    rpm_lines = np.arange(14, num_lines, 37)
    count = 1
    headers=['Velocity (mph)', 'J (Adv Ratio)', 'Pe', 'Ct', 'Cp', 'PWR (Hp)', 'Torque (In-Lbf)', 'Thrust(Lbf)', 'RPM']
    np.arange(14, num_lines, 37)
    for i in range(18, num_lines+1):
        if (count==38):
            count=1
        if (count<=30):
            line_on_focus.append(i) 
        count=count+1
    count=1
    with open(filename) as f: 
        for line in f: 
            if count in rpm_lines:
                line = line.strip()
                line = re.sub(" +", " ",line)
                line = re.sub("PROP RPM = ", "", line)
                rpm=line
            if count in line_on_focus:
                line = line.strip()
                line = re.sub(" +", " ",line)
                buffer = buffer+line+' '+rpm+"\n"
            count+=1
        f.close()
    buffer=buffer.replace(" ", ",")
    buffer=StringIO(buffer)
    df=pd.read_csv(buffer, header=None, names=headers)
    print('Erros de NAN:\n',df.isna().sum(), '\n')
    df = df.dropna()
    df = df.apply(pd.to_numeric)
    df['Model'] = model
    return df

SI_dataframe(df)

def SI_dataframe (df):
    """Reads Dataframes with the data in the imperial system and returns a dataframe in the international measurement system.

    Parameters
    ----------
    df : Pandas DataFrame
        DataFrame to be converted to the SI system.

    Returns
    -------
    pandas.DataFrame
        Pandas DataFrame converted to the international measurement system.

    """
    df['Velocity (mph)'] = df['Velocity (mph)']*0.44704 
    df['PWR (Hp)'] = df['PWR (Hp)']*746
    df['Torque (In-Lbf)'] = df['Torque (In-Lbf)']*0.112984833333333
    df['Thrust(Lbf)'] = df['Thrust(Lbf)']*4.448222
    df['RPM'] = df['RPM']/60
    df.columns=['Velocity (m/s)', 'J (Adv Ratio)', 'Pe', 'Ct', 'Cp', 'PWR (W)', 'Torque (N/m)', 'Thrust (N)', 'Frequency (Hz)', 'Model'] 
    return df

APCPlot

APCplot(df, group, x, y, xlabel=None, ylabel=None, percent_x=False, percent_y=False, save_at=None)

def APCplot(df, group, x, y, xlabel=None, ylabel=None, percent_x=False, percent_y=False, save_at=None):
    """Receives a DataFrame with the data, groups it by model and plots a 2D figure with the columns specified in * x * and * y *. You can also save the picture to a file.

    Parameters
    ----------
    df : Pandas DataFrame
        DataFrame which is the source of the data.
    group : str with name of one of the df columns
        Used to group the DataFrame, also used for the legend of the plots
    x : str with name of one of the df columns
        Values to be line plotted on the x axis.
    y : str with name of one of the df columns
        Values to be line plotted on the y axis.
    xlabel : str, optional
        Title of the x axis (default is x).
    ylabel : str, optional
        Title of the y axis (default is y).
    percent_x : bool, optional
        Are values in x percentages stored as fractions of 1 (default is False).
    percent_y : bool, optional
        Are values in y percentages stored as fractions of 1 (default is False).
    save_at : str, optional
        Path wherein to save the figure on disk (default is None).
    
    """
    
    import matplotlib.ticker as mtick    
    grouped = df.groupby(group)

    plt.style.use('seaborn')
    fig = plt.figure()
    ax = fig.add_subplot(111)
    colors = list(sns.color_palette('husl', len(grouped.size().index)))
    if percent_x:
        ax.xaxis.set_major_formatter(mtick.PercentFormatter())
        count=0
        for i in grouped.size().index:
            ax.plot(100*grouped.get_group(i)[x], grouped.get_group(i)[y], label=i, color=colors[count])
            count+=1
    if percent_y:
        ax.yaxis.set_major_formatter(mtick.PercentFormatter())
        count=0
        for i in grouped.size().index:
            ax.plot(grouped.get_group(i)[x], 100*grouped.get_group(i)[y], label=i, color=colors[count])
            count+=1
    else:
        count=0
        for i in grouped.size().index:
            ax.plot(grouped.get_group(i)[x], grouped.get_group(i)[y], label=i, color=colors[count])
            count+=1

    if xlabel is not None:
        ax.set_xlabel(xlabel)
    
    if ylabel is not None:
        ax.set_ylabel(ylabel)
    
    else:
        ax.set_xlabel(x)
        ax.set_ylabel(y)    
    
    ax.legend()
    
    if save_at is not None:
        plt.savefig(save_at)

    return

Importing data

The data here comes from performance files on APC's website.

models = {'13x4' : '13x4.dat.txt',
          '12x6' : '12x6.dat.txt',
          '12x4' : '12x4.dat.txt',
          '12.25x3.75' : '1225x375.dat.txt',
          '11x7' : '11x7.dat.txt',
          '11.5x6' : '115x6.dat.txt',
          '11.5x4' : '115x4.dat.txt'}

data = []
count=1
for i in models.keys():
    print('Model ', i, 'imported succefully.')
    cache = read_txt(models[i], i)
    cache = SI_dataframe(cache)
    data.append(cache)
    if count==len(models):
        print('All files read succefully, Data Imported.')
    count+=1
df = pd.concat(data, ignore_index=True)

Model  13x4 imported succefully.
Erros de NAN:
 Velocity (mph)     0
J (Adv Ratio)      0
Pe                 0
Ct                 0
Cp                 0
PWR (Hp)           0
Torque (In-Lbf)    0
Thrust(Lbf)        0
RPM                0
dtype: int64 

Model  12x6 imported succefully.
Erros de NAN:
 Velocity (mph)     0
J (Adv Ratio)      0
Pe                 1
Ct                 1
Cp                 1
PWR (Hp)           1
Torque (In-Lbf)    1
Thrust(Lbf)        0
RPM                1
dtype: int64 

Model  12x4 imported succefully.
Erros de NAN:
 Velocity (mph)     0
J (Adv Ratio)      0
Pe                 1
Ct                 1
Cp                 1
PWR (Hp)           1
Torque (In-Lbf)    1
Thrust(Lbf)        0
RPM                1
dtype: int64 

Model  12.25x3.75 imported succefully.
Erros de NAN:
 Velocity (mph)     0
J (Adv Ratio)      0
Pe                 0
Ct                 0
Cp                 0
PWR (Hp)           0
Torque (In-Lbf)    0
Thrust(Lbf)        0
RPM                0
dtype: int64 

Model  11x7 imported succefully.
Erros de NAN:
 Velocity (mph)     0
J (Adv Ratio)      0
Pe                 5
Ct                 5
Cp                 5
PWR (Hp)           5
Torque (In-Lbf)    5
Thrust(Lbf)        0
RPM                5
dtype: int64 

Model  11.5x6 imported succefully.
Erros de NAN:
 Velocity (mph)     0
J (Adv Ratio)      0
Pe                 0
Ct                 0
Cp                 0
PWR (Hp)           0
Torque (In-Lbf)    0
Thrust(Lbf)        0
RPM                0
dtype: int64 

Model  11.5x4 imported succefully.
Erros de NAN:
 Velocity (mph)     0
J (Adv Ratio)      0
Pe                 0
Ct                 0
Cp                 0
PWR (Hp)           0
Torque (In-Lbf)    0
Thrust(Lbf)        0
RPM                0
dtype: int64 

All files read succefully, Data Imported.

df

	Velocity (m/s)	J (Adv Ratio)	Pe	Ct	Cp	PWR (W)	Torque (N/m)	Thrust (N)	Frequency (Hz)	Model
0	0.000000	0.00	0.0000	0.0738	0.0239	0.746	0.004858	0.289134	16.666667	13x4
1	0.089408	0.02	0.0545	0.0723	0.0240	0.746	0.004858	0.284686	16.666667	13x4
2	0.178816	0.04	0.1064	0.0708	0.0240	0.746	0.004971	0.275790	16.666667	13x4
3	0.312928	0.05	0.1557	0.0692	0.0241	0.746	0.004971	0.271342	16.666667	13x4
4	0.402336	0.07	0.2023	0.0675	0.0241	0.746	0.004971	0.266893	16.666667	13x4
...	...	...	...	...	...	...	...	...	...	...
4008	46.939200	0.46	0.3796	0.0216	0.0261	2916.860	1.325651	23.575577	350.000000	11.5x4
4009	48.816768	0.48	0.3302	0.0165	0.0239	2674.410	1.215491	18.077574	350.000000	11.5x4
4010	50.694336	0.50	0.2519	0.0112	0.0221	2466.276	1.121149	12.250403	350.000000	11.5x4
4011	52.571904	0.51	0.1442	0.0057	0.0202	2260.380	1.027484	6.196373	350.000000	11.5x4
4012	54.449472	0.53	-0.0018	-0.0001	0.0184	2057.468	0.935062	-0.066723	350.000000	11.5x4

4013 rows × 10 columns

Performance analysis

What is the maximum power delivered to the propellers by the O.S. 55 AX engine?

A Senior project was found that shows the maximum frequency values for two of these propellers on that engine, models 12.25 x 3.75 and 13x4, to be 206.25 Hz and 192.3834 Hz respectively.

With that, the maximum power, at these frequencies in these models, was found to be 736,302 W and 741,524 W and the average of the two, 738,913 W, was taken as the maximum power delivered by the engine.

df1=0
df1 = df.groupby('Model')
df1 = df1.get_group('12.25x3.75')
df1 = df1[(df1['Frequency (Hz)'] <= 206.25)]
max_power_1225x375 = df1['PWR (W)'].max()

df1=0
df1 = df.groupby('Model')
df1 = df1.get_group('13x4')
df1 = df1[(df1['Frequency (Hz)'] <= 192.3834)]
max_power_13x4 = df1['PWR (W)'].max()

max_power = np.mean([max_power_1225x375, max_power_13x4])

Afterwards, all entries with power greater than max power were removed from the dataframe.

dfs=[]
cache = 0
for i in models.keys():
    df1=0
    df1=df.groupby('Model')
    df1 = df1.get_group(i)
    df1=df1[(df1['PWR (W)'] <= max_power)]
    df1 = df1.groupby('Frequency (Hz)')
    cache = df1.size()
    cache = cache.sort_index(ascending=False)
    best_frequency = cache.idxmax()
    df1 = df1.get_group(best_frequency)
    dfs.append(df1)
df1 = pd.concat(dfs)

df1

	Velocity (m/s)	J (Adv Ratio)	Pe	Ct	Cp	PWR (W)	Torque (N/m)	Thrust (N)	Frequency (Hz)	Model
270	0.000000	0.00	0.0000	0.0782	0.0252	539.358	0.514646	30.697180	166.666667	13x4
271	0.983488	0.02	0.0557	0.0766	0.0248	531.898	0.507415	30.069981	166.666667	13x4
272	1.966976	0.04	0.1105	0.0749	0.0244	524.438	0.500297	29.407196	166.666667	13x4
273	2.950464	0.05	0.1642	0.0732	0.0241	516.978	0.493179	28.713273	166.666667	13x4
274	3.933952	0.07	0.2165	0.0713	0.0238	509.518	0.486174	27.988213	166.666667	13x4
...	...	...	...	...	...	...	...	...	...	...
3738	27.716480	0.47	0.7061	0.0173	0.0116	243.196	0.193204	6.187477	200.000000	11.5x4
3739	28.834080	0.49	0.6530	0.0131	0.0099	206.642	0.164393	4.679530	200.000000	11.5x4
3740	29.906976	0.51	0.5579	0.0088	0.0081	168.596	0.134339	3.144893	200.000000	11.5x4
3741	31.024576	0.53	0.3741	0.0044	0.0063	131.296	0.104511	1.583567	200.000000	11.5x4
3742	32.142176	0.55	-0.0021	0.0000	0.0046	95.488	0.076152	-0.004448	200.000000	11.5x4

210 rows × 10 columns

The following is the actual analysis

Thurst velocity plots for all the propellers

One of the fundamental ways to measure performance of propellers is to assess their performance in low velocities, (below takeoff velocity) since a higher thrust in these velocities can mean a faster acceleration a reduced runway length.

The figure below show that it is quite clear that the 12.25 x 3.75 model is the best performing one in this set.

APCplot(df=df1, group='Model', x='Velocity (m/s)', y='Thrust (N)')

Finding the coefficients of this curve

We can find the coefficients of the thrust curve (T) as a function of velocity (v) assuming a quadratic relationship and applying a least squares method, the following is the equation with the relationship and the sum of squares of the errors.

$$T(v) = -0.014436v^{2} + -0.845198v + 38.942274 \quad error = 0.010472$$

grouped = 0
grouped = df1.groupby('Model')
x=np.arange(0, 40, step=0.01)


coef = np.polyfit(grouped.get_group('12.25x3.75')['Velocity (m/s)'], grouped.get_group('12.25x3.75')['Thrust (N)'], 2, full=True)
#print('v^%f + v*%f + %f, erro = %f' % (coef[0][0], coef[0][1], coef[0][2], coef[1]))
fit = np.polyval(coef[0] , x)
fit = fit[fit>=0]
x = x[0:len(fit)]


plt.scatter(grouped.get_group('12.25x3.75')['Velocity (m/s)'], grouped.get_group('12.25x3.75')['Thrust (N)'], label='Data from propeller 12.25 x 3.75', color=sns.color_palette('husl', 7)[3])
plt.plot(x, fit, label='Fit', color='k')
plt.legend()
plt.xlabel('Velocity (m/s)')
plt.ylabel('Thrust (N)')
plt.savefig('fig_4.png', dpi=100)
plt.show()

Efficiency of a propeller

The formal definition of the efficiency (𝜂) of a propeller is fraction of the power it delivers in relation to the power recieved by the engine: $$\eta=\frac{{P}_{effective}}{{P}_{engine}}$$

In this case calculated with the available data, as defined by the bibliographic reference of the suplier UIUC:

D being the diameter of the propeller, 𝜌 the density of the air and n the frequency in revolutions per second, one can define the coefficients of thrust ($C_{t}$) and power ($C_{p}$) as: ${C}_{t} = \frac{T}{\rho {n}^{2} {D}^{4}}$ and ${C}_{p} = \frac{P}{\rho {n}^{3} {D}^{5}}$

With that, 𝜂 can be written as: $$\eta=\frac{{C}_{t}J}{{C}_{p}}$$

Efficiency as function of the advance ratio (J)

APCplot(df=df1, group='Model', x='J (Adv Ratio)', y='Pe',
       xlabel='J (Advance Ratio)', ylabel = 'η (Efficiency)',
       percent_y = True, save_at = 'fig_2.png')

Efficiency as a function of thrust

As you can see in the figure above, the propellers all have relatively high efficiencies (above 60%).

The 12.25 x 3.75 model was chosen, even though it had the lowest efficiency of all, 71.56%, for having an efficiency above the values set as acceptable in the literature (above 60%) and for having the largest area in the thrust-efficiency curve. This is due to an inverse relationship between these two variables as shown in the Figure below.

APCplot(df=df1, group='Model', x='Thrust (N)', y='Pe',
       xlabel='Empuxo (N)', ylabel = 'η (Eficiência)',
       percent_y = True, save_at = 'fig_3.png')