Initial sizing

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:
​TW=f(WS)\frac{T}{W}=f\left(\frac{W}{S}\right)WT​=f(SW​)
.
In the above equation TW\frac{T}{W}WT​
is referred to as the thrust to weight ratio and the WS\frac{W}{S}SW​
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 PW\frac{P}{W}WP​
expressed in W/kgW/kgW/kg
. This is more convenient when designing propeller-powered aircraft.
P/W for a level, constant-velocity turn
​PW=qcruise(CD0WSg+k(ng)2WSg)Vcruiseηg\frac{P}{W}=q_{cruise}\left(\frac{C_{D0}}{\frac{W}{S}g}+k\left(\frac{n}{g}\right)^2\frac{W}{S}g\right)\frac{V_{cruise}}{\eta}gWP​=qcruise​(SW​gCD0​​+k(gn​)2SW​g)ηVcruise​​g
​
P/W for a desired rate of climb
​PW=(ROCVclimb+qrefWSgCD0+kqrefWSg)Vclimbηg\frac{P}{W}=\left(\frac{ROC}{V_{climb}}+\frac{q_{ref}}{\frac{W}{S}g}C_{D0}+\frac{k}{q_{ref}}\frac{W}{S}g\right)\frac{V_{climb}}{\eta}gWP​=(Vclimb​ROC​+SW​gqref​​CD0​+qref​k​SW​g)ηVclimb​​g
​
P/W for a desired cruise airspeed
​PW=(qcruiseCD0WSg+kqcruiseWSg)Vcruiseηg\frac{P}{W}=\left(\frac{q_{cruise}C_{D0}}{\frac{W}{S}g}+\frac{k}{q_{cruise}}\frac{W}{S}g\right)\frac{V_{cruise}}{\eta}gWP​=(SW​gqcruise​CD0​​+qcruise​k​SW​g)ηVcruise​​g
​
P/W for a desired takeoff distance
​PW=[Vtakeoff22gStakeoff+qtakeoffCD,takeoffWSg+Cf,takeoff(1−qtakeoffCL,takeoffWSg)]Vclimbηg,Vtakeoff=1.1Vstall\frac{P}{W}=\left[\frac{V_{takeoff}^2}{2gS_{takeoff}}+\frac{q_{takeoff}C_{D,takeoff}}{\frac{W}{S}g}+C_{f,takeoff}\left(1-\frac{q_{takeoff}C_{L,takeoff}}{\frac{W}{S}g}\right)\right]\frac{V{climb}}{\eta}g,\\ V_{takeoff}=1.1V_{stall}WP​=[2gStakeoff​Vtakeoff2​​+SW​gqtakeoff​CD,takeoff​​+Cf,takeoff​(1−SW​gqtakeoff​CL,takeoff​​)]ηVclimb​g,Vtakeoff​=1.1Vstall​
​
​Vstall=2ρCL,maxWSgV_{stall}=\sqrt{\frac{2}{\rho C_{L,max}}\frac{W}{S}g}Vstall​=ρCL,max​2​SW​g​
​
P/W for a best range
​PW=(qrangeCD0WSg+kqrangeWSg)Vrangeηg,Vrange=2gρWSkCD0\frac{P}{W}=\left(\frac{q_{range}C_{D0}}{\frac{W}{S}g}+\frac{k}{q_{range}}\frac{W}{S}g\right)\frac{V_{range}}{\eta}g,\quad V_{range}=\sqrt{\frac{2g}{\rho}\frac{W}{S}\sqrt{\frac{k}{C_{D0}}}}WP​=(SW​gqrange​CD0​​+qrange​k​SW​g)ηVrange​​g,Vrange​=ρ2g​SW​CD0​k​​​
​
P/W for best endurance
​PW=(qenduranceCD0WSg+kqenduranceWSg)Venduranceηg,Vendurance=2gρWSk3CD0\frac{P}{W}=\left(\frac{q_{endurance}C_{D0}}{\frac{W}{S}g}+\frac{k}{q_{endurance}}\frac{W}{S}g\right)\frac{V_{endurance}}{\eta}g,\quad V_{endurance}=\sqrt{\frac{2g}{\rho}\frac{W}{S}\sqrt{\frac{k}{3C_{D0}}}}WP​=(SW​gqendurance​CD0​​+qendurance​k​SW​g)ηVendurance​​g,Vendurance​=ρ2g​SW​3CD0​k​​​
​
CLmax for desired stall speed
​CL,max=1qstallWSgC_{L,max}=\frac{1}{q_{stall}}\frac{W}{S}gCL,max​=qstall​1​SW​g
​
The expressions above depend on the following parameters: 
Parameter
Description
​nnn
​
Load factor
​CL,maxC_{L,max}CL,max​
​
Maximum lift coefficient
​CD0C_{D0}CD0​
​
Minimum drag coefficient
​kkk
​
Induced drag coefficient
​η\etaη
​
Propulsive efficiency
​Vcruise,qcruiseV_{cruise}, q_{cruise}Vcruise​,qcruise​
​
Cruise airspeed and dynamic pressure
​Vclimb,qclimb,ROCV_{climb}, q_{climb}, ROCVclimb​,qclimb​,ROC
​
Climb airspeed, dynamic pressure and rate of climb
​Vtakeoff,qtakeoff,StakeoffV_{takeoff}, q_{takeoff}, S_{takeoff}Vtakeoff​,qtakeoff​,Stakeoff​
​
Takeoff airspeed, dynamic pressure, distance
​VstallV_{stall}Vstall​
​
Stall airspeed
​ρ\rhoρ
​
Density
​ggg
​
Acceleration due to gravity
​CfC_{f}Cf​
​
Ground friction coefficient
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 CL,maxC_{L,max}CL,max​
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 25 kg/m225\text{ }kg/m^225 kg/m2
the power loading should be approximately 200 W/kg200\text{ }W/kg200 W/kg
. In addition for a design having a wing loading of 25 kg/m225\text{ }kg/m^225 kg/m2
and a desired stall speed of 15 m/s15\text{ }m/s15 m/s
the maximum lift coefficient CL,maxC_{L,max}CL,max​
of the design should be approximately 1.81.81.8
.  Consider you would like your design to have a mass of 20kg20 kg20kg
. For a wing loading of 25 kg/m225\text{ }kg/m^225 kg/m2
and power loading of 200 W/kg200\text{ }W/kg200 W/kg
you will need a reference area of approximately Sref=0.8 m2S_{ref}=0.8\text{ }m^2Sref​=0.8 m2
and power system of approximately 4000 W4000\text{ }W4000 W
.
Example
Please refer to the Python code below which plots the constraint analysis for specific combination of parameters.
import math
import matplotlib.pyplot as plt
import numpy as np
​
​
k=0.0593
CD0=0.0181
eta=0.6
m=20
S=3.2200
V_turn=25
V_cruise=25
V_climb=20
ROC=10
ROC_ceiling=0.5
CL_CD0=0.2784
CL_max=1.7
rho=1.1116
rho_takeoff=1.225
rho_ceiling=0.8191
g=9.80665
phi=math.radians(30)
n=1/math.cos(phi)
S_takeoff=50
Cf_takeoff=0.025
​
N=100
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)
​
i=0
for ws in wss:
​
    #Turn
    q=1/2*rho*V_turn**2
    pw_turn[i]=q*(CD0/(ws*g)+k*(n/q)**2*ws*g)*V_turn/eta*g
    P_turn = pw_turn[i] * m
​
    #Climb
    q=1/2*rho*V_climb**2
    pw_climb[i]=(ROC/V_climb+q/(ws*g)*CD0+k/q*ws*g)*V_climb/eta*g
    P_climb = pw_climb[i] * m
​
    #Cruise
    q=1/2*rho*V_cruise**2
    pw_cruise[i]=(q*CD0*(1/(ws*g))+k*(1/q)*ws*g)*V_cruise/eta*g
    P_cruise = pw_cruise[i] * m
​
    #Takeoff
    V_stall=math.sqrt(2/(rho_takeoff*CL_max)*ws*g)
    V_takeoff=1.1*V_stall
​
    CL_takeoff = 0.8*CL_max
    CD_takeoff = CD0 + k*(CL_takeoff-CL_CD0)**2
​
    q=1/2*rho_takeoff*(0.7*V_takeoff)**2
    pw_takeoff[i]=(V_takeoff**2/(2*g*S_takeoff)+q*CD_takeoff/(ws*g)+Cf_takeoff*(1-(q*CL_takeoff)/(ws*g)))*V_takeoff/eta*g
    P_takeoff=pw_takeoff[i] * m
​
    #Ceiling
    pw_ceiling[i]=(ROC_ceiling/math.sqrt(2/rho_ceiling*ws*g*math.sqrt(k/(3*CD0)))+4*math.sqrt((k*CD0)/3))*V_climb/eta*g
    P_cng=pw_ceiling[i]*m
​
    #Best endurance
    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
​
    #Best range
    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
​
    i=i+1
​
plt.figure()
plt.plot(wss,pw_cruise,label='Cruise')
plt.plot(wss,pw_climb,label='Climb')
plt.plot(wss,pw_turn,label='Turn')
plt.plot(wss,pw_endurance,label='Endurance')
plt.plot(wss,pw_range,label='Range')
plt.plot(wss,pw_ceiling,label='Ceiling')
plt.plot(wss,pw_takeoff,label='Takeoff')
plt.title('Constraints')
plt.legend()
plt.xlabel('Wing loading (kg/m^2)')
plt.ylabel('Power loading (W/kg)')
plt.show()

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Parameter	Description
$n$	Load factor
$C_{L,max}$	Maximum lift coefficient
$C_{D0}$	Minimum drag coefficient
$k$	Induced drag coefficient
$\eta$	Propulsive efficiency
$V_{cruise}, q_{cruise}$	Cruise airspeed and dynamic pressure
$V_{climb}, q_{climb}, ROC$	Climb airspeed, dynamic pressure and rate of climb
$V_{takeoff}, q_{takeoff}, S_{takeoff}$	Takeoff airspeed, dynamic pressure, distance
$V_{stall}$	Stall airspeed
$\rho$	Density
$g$	Acceleration due to gravity
$C_{f}$	Ground friction coefficient