Comment on page
The aim of the initial sizing is to come up with a design which is light, performs well, and is inexpensive to manufacture and operate.
A popular method for initial sizing is the so called constraint analysis method. The method can be used to asses the required wing area and power for an aircraft such that the aircraft meets all performance requirements.
The performance requirements are defined by mathematical expressions of the following form:
In the above equation
is referred to as the thrust to weight ratio and the
is referred to as the wing loading. The expressions relating the wing loading to the thrust to weight ratio are dependent on the performance requirements. Below are commonly used performance requirements equations. They can be found in any aircraft performance textbook.
The performance equations below are rewritten as a function of the power loading
. This is more convenient when designing propeller-powered aircraft.
The expressions above depend on the following parameters:
When performing initial sizing it is difficult to determine the values for the above parameters. After all we haven't even started designing the aircraft! You have probably guessed already - aircraft design is an iterative process.
The image below shows all the performance requirements equations. The required
for a desired stall speed is also shown.
Wing vs power loading graph
In order to meet all performance requirements, the design point should be above all performance requirement equations. In the above image for a design having a wing loading of
the power loading should be approximately
. In addition for a design having a wing loading of
and a desired stall speed of
the maximum lift coefficient
of the design should be approximately
. Consider you would like your design to have a mass of
. For a wing loading of
and power loading of
you will need a reference area of approximately
and power system of approximately
Please refer to the Python code below which plots the constraint analysis for specific combination of parameters.
import matplotlib.pyplot as plt
import numpy as np
wss = np.linspace(1,30,N)
pw_turn = np.zeros(N)
pw_climb = np.zeros(N)
pw_cruise = np.zeros(N)
pw_takeoff = np.zeros(N)
pw_ceiling = np.zeros(N)
pw_endurance = np.zeros(N)
pw_range = np.zeros(N)
for ws in wss:
P_turn = pw_turn[i] * m
P_climb = pw_climb[i] * m
P_cruise = pw_cruise[i] * m
CL_takeoff = 0.8*CL_max
CD_takeoff = CD0 + k*(CL_takeoff-CL_CD0)**2
P_takeoff=pw_takeoff[i] * m
V_endurance = math.sqrt(2 * g * ws * math.sqrt(k / (3*CD0)) / rho)
q = 1/2 * rho * V_endurance**2
pw_endurance[i] = (q*CD0/(ws*g)+k/q*ws*g)*V_endurance/eta*g
P_endurance = pw_endurance[i] * m
V_range = math.sqrt(2 * g * ws * math.sqrt(k / CD0) / rho)
q = 1/2 * rho * V_range**2
pw_range[i] = (q*CD0/(ws*g)+k/q*ws*g)*V_range/eta*g
P_range = pw_range[i] * m
plt.xlabel('Wing loading (kg/m^2)')
plt.ylabel('Power loading (W/kg)')