# What velocity is used to calculate maximum range and maximum endurance?

These are my data;

• $$C_{L_0} = 0.4$$
• $$C_{L_{\alpha}} = 5.7$$
• $$C_{L_{minD}} = 0.14$$
• $$C_{D_{min}} = 0.022$$
• $$k = 0.035$$
• $$\alpha_{stall} = 15$$
• $$S_{ref} = 20m^2$$

# Mass Data

• $$W_{total} = 2500 \text{ kg}$$
• $$W_{fuel} = 800 \text{ L}$$
• $$\rho_{fuel} = 800 \text{ kg/m}^3$$
• $$mass_{fuel} = (1/1000) \cdot W_{fuel} \cdot \rho_{fuel} \text{ kg}$$ , 1/1000 l from coverting into m^3
• $$g = 9.807 \text{m/s}^2$$ gravity

# Propulsion Data

• Engine_type = "Internal Combustion Engine"
• Propeller_type = "Constant Speed Propeller"
• S_prop = 1.9 # m^2
• n_prop = 0.8 # %80
• SFC_takeoff = 0.28 # kg/kW/h
• P_takeoff = 400 # HP
• P_maxCont = 350 # HP
• SFC_maxCont = 0.25 # kg / kW /h
• P_cruise = 280 # HP
• SFC_cruise = 0.2 # kg/kw/h

My main purpose is calculate maximum range and endurance at constant velocity and altitude. So that I have started to calculate $$C_{L} = C_{L_0} + alpha * C_{L_\alpha}$$ to find $$V_{stall}$$ and $$C_{L_{max}}$$. Well I have different $$P_{cruise}$$ and $$P_{takeoff}$$. I am uncertain how to proceed because I didn't decide, or am uncertain how to decide, which one of speeds I need to use TAS, or $$V_{cruise}$$. I decided to calculate CD by using the Quadratic Drag Model because this model's result is more accurate than a simpler one according to what I have read. $$CD = CD_{min} + k * C_{L}^2$$

That is my duty; Start the calculation with the aircraft at an altitude of 20,000 ft. and a full tank of fuel and finish the calculation when there are 100 litres of fuel remaining. For both the (a) range and (b) endurance analysis, provide separate results in cases when (i) the velocity [TAS] is kept constant and (ii) the altitude is kept constant. In all cases, the angle of attack should be kept constant. During this analysis, you can use the cruise SFC value.

And well, I couldn't start the step-by-step calculations in a regular sequence. I am hoping that someone can or will help me to determine proper sequence.


def calculator_atmosphere_values(h):
# S-L reference values

rho0 = 0.002378  # slugs/ft^3
P0 = 2116  # psf
T0 = 518.67  # R

#############################
Temperature_Ratio = (1 - 0.0000068756 * h)  # k values lapse  rate constant = 0.0000068756 / ft
Pressure_Ratio = (1 - 0.0000068756 * h) ** 5.2561
Densitiy_Ratio = (1 - 0.0000068756 * h) ** 4.2561

#############################

T_uk = Temperature_Ratio * T0 # R
P_uk = Pressure_Ratio * P0 # psf
rho_uk = Densitiy_Ratio * rho0 # slughs/ft^3

####################################################

rho = rho_uk * 515.378818 # converting to kg/m3
T = T_uk / 1.8 # converting to Kelvin
P = P_uk * 47.88 # converting to Pascal

####################################################

print("################### Temperature  according to Height ######################")
print(f"Temperature at {h} Altitude: {T} K")
print("################### Atmospheric Pressure for Altitudes below 36.089 ft ######################")
print(f"Pressure at {h} Altitude: {P} Pa")
print("################### Atmospheric density according to Height ")
print(f"Density at {h} Altitude : {rho} kg/m3")

return T, P, rho, T_uk
$$$$
`
• "my problem is determine Vcruise" what do you mean? $V_{cruise}$ is the speed at which you want to cruise, it's a given parameter not one that you calculate. Maybe you want to calculate maximum speed? Or speed for best endurance? Or speed for best loiter? Or stall speed?... Commented Sep 28, 2023 at 15:02
• Hi Bugra, welcome to aviation.stackexchange.com. I started to convert your parameters to MathJax for better readability. Please check if I made any mistakes and correct them if needed. Commented Sep 28, 2023 at 15:19
• Can you show what you have done so far? What are you trying to achieve at cruise speed? Maximum range or maximum endurance? Commented Sep 28, 2023 at 15:22
• yes exactly what i want calculate Max range and Max endurance constant altitude and speed , I have done calculate pressure, temperature and density at 20.000 ft by using Temperature_Ratio = (1 - 0.0000068756 * h) # k values lapse rate constant = 0.0000068756 / ft Pressure_Ratio = (1 - 0.0000068756 * h) ** 5.2561 Densitiy_Ratio = (1 - 0.0000068756 * h) ** 4.2561 these equation. I also calculated CL = CL0 + alpha * CLa i have alpha at stall so that I calculated CLmax and Vstall and after i calculated Thrust value T = (P * (746 / 1000) / 0.2) * np # Newton at Cruise Commented Sep 28, 2023 at 16:09
• @Bugra you would be better to edit the detail into your question. If you have a look at the syntax for mathjax, it will make it much more readable.
– Jamiec
Commented Sep 28, 2023 at 16:27