Consider a light propeller driven aircraft flying in the lower troposphere. Use the following data:
S=15.9m^{2}  b=10.3m  W=11,800N 

C_{D0}=0.024  e=0.82  ρ=1.045Kg/m^{3} 
Calculations
Aspect ratio (AR)=$ frac{b^{2}}{S}$
=$frac{{10.3}^{2}}{15.9}$
= 6.67

Induced Drag factor (K):
K=$ frac{1}{pitext{eAR}}$
=$ frac{1}{pi*0.82*6.67}$
= 0.058
Chord length:
Assuming the shape of the wing to be rectangular, we can calculate the chord length by following formula:
$$AR = frac{S}{c}$$
$$c = frac{S}{text{AR}}$$
$$= frac{15.9}{6.67} = 2.38m$$
The chord length is 2.38m.
b)
$$D = frac{1}{2}rho V^{2}Sleft( C_{D_{0}} + kC_{L}^{2} right)$$
$$= frac{1}{2}rho V^{2}SC_{D_{0}} + frac{2KW^{2}}{text{ρS}}V^{ 2}$$
In the above equation, the first portion is parasitic drag and second is induced drag.
Parasitic Drag$ = frac{1}{2}rho V^{2}SC_{D_{0}}$
Induced Drag =$frac{2KW^{2}}{text{ρS}}V^{ 2}$
Total Drag= Parasitic Drag + Induced Drag
[CHART]
Figure 1: Plot of parasitic drag, induced drag, and total drag against air speed
c)
The given speed 50knots to 190knots at sea level are equivalent speed. To find the true speed we can use following relations:
$$frac{1}{2}rho{V_{text{TAS}}}^{2} = frac{1}{2}rho_{0}{V_{text{EAS}}}^{2}$$
$$V_{text{TAS}} = V_{text{EAS}}sqrt{frac{rho_{0}}{rho}}$$
$${= V}_{text{EAS}}sqrt{frac{1.225}{1.045}}$$
= 1.083V_{EAS}
Where, V_{TAS}= True air speed
V_{EAS}= Equivalent air speed
ρ= density at certain level
ρ_{0} = density at sea level
For, V_{EAS}= 50knots,
V_{TAS} = = 54.15knots
For, V_{EAS}= 190knots,
V_{TAS} = = 205.77knots
d)
To find the minimum drag we need to find the velocity at which the minimum drag occurs. This can be calculated by solving the partial derivatives of drag force with respect to velocity as:
$$frac{partial D}{partial V} = 0$$
$$or, frac{partial}{partial V}left( frac{1}{2}rho{V_{m}}^{2}SC_{D_{0}} + frac{2KW^{2}}{text{ρS}}{V_{m}}^{ 2} right)$$
$$or, rho V_{m}SC_{D_{0}} – frac{4KW^{2}}{text{ρS}}{V_{m}}^{ 3} = 0$$
or, 0.39877V_{m} − 1944189V_{m}^{ − 3} = 0
or, 0.39877V_{m}^{4} − 1944189 = 0
$$or, {V_{m}}^{4} = frac{1944189}{0.39877}$$
or, V_{m} = 4875441.259
V_{m} = 46.98m/s
V_{m} = 91.34knots
Hence, the velocity at which the drag force is minimum is 91.34 knots. This suggests that the minimum drag occurs at the velocity where the parasitic and induced drag intersects as shown in figure 1.
Minimum Drag is:
$$D_{m} = frac{1}{2}rho{V_{m}}^{2}SC_{D_{0}} + frac{2KW^{2}}{text{ρS}}{V_{m}}^{ 2}$$
$$= frac{1}{2}*1.045*{46.98}^{2}*15.9*0.024 + frac{2*0.058*11800^{2}}{1.045*15.9}{46.98}^{ 2}$$
= 440.07 + 440.44
= 880.51N
The minimum drag is 880.51N which occurs at velocity 91.34knots and can be located at the velocity corresponding to the intersection of parasitic and induced drag plot in figure 1.
e)
To calculate the altitude of the flight it is necessary to understand the density altitude and pressure altitude. The variation of temperature with altitude plays vital role. To include all these necessary entities, we need to observe the standard table that shows the variation of pressure and density including lapse rates. Considering available conditions such as:
Light propeller driven aircraft flying in the lower troposphere.
Density at sea level (ρ_{0}) = 1.225kg/m^{3}
Density at required altitude (ρ)= 1.045kg/m^{3}
density ratio (σ):
$$sigma = frac{rho}{rho_{0}}$$
$$= frac{1.045}{1.225}$$
= 0.85
To calculate the required altitude, observing the table as shown in figure 2.
For density ratio 0.85, altitude= 1600m
= 1600 Χ 3.28084
= 5249.344 ft
Figure 2:Atmospheric properties in ISA [1]
f)
The required power can be calculated as the product of drag and air speed.
P = DV=$left{ frac{1}{2}rho V^{2}SC_{D_{0}} + frac{text{KW}left( frac{W}{S} right)}{frac{1}{2}rho V^{2}} right} V$
$$= left( frac{1}{2}text{ρS}C_{D_{0}} right)V^{3} + left( frac{2KW^{2}}{text{ρS}}V^{ 1} right)$$
To calculate the minimum power required, we have to find the partial derivative of P with respect to V and solve.
$$frac{partial P}{partial V} = 0$$
$$or, frac{partial}{partial V}left( frac{1}{2}rho{V_{text{mp}}}^{3}SC_{D_{0}} + frac{2KW^{2}}{text{ρS}}{V_{text{mp}}}^{ 1} right) = 0$$
$$or, left( frac{3}{2}rho{V_{text{mp}}}^{2}SC_{D_{0}} – frac{2KW^{2}}{text{ρS}}{V_{m}}^{ 2} right) = 0$$
$$or, left( frac{3}{2}rho{V_{text{mp}}}^{4}SC_{D_{0}} = frac{2KW^{2}}{C} right)$$
$$or, left( {V_{text{mp}}}^{4} = frac{4KW^{2}}{{3rho}^{2}S^{2}C_{D_{0}}} right)$$
$$or, V_{text{mp}} = sqrt{frac{2W}{text{ρS}}sqrt{frac{k}{3C_{D_{0}}}}}$$
$$or, V_{text{mp}} = sqrt{frac{2W}{text{ρS}}sqrt{beta}}$$
Where $beta = frac{k}{3C_{D_{0}}} = frac{0.058}{3*0.024} = 0.81$
Hence, the minimum required power can be calculated using the following expression.
$$ V_{text{mp}} = sqrt{frac{2W}{text{ρS}}sqrt{beta}}$$
Calculation of V_{mp}:
$$V_{text{mp}} = sqrt{frac{2W}{text{ρS}}sqrt{beta}}$$
$$= sqrt{frac{2*11800}{1.045*15.9}sqrt{0.81}}$$
= 35.75m/s
Comparing V_{mp}=35.75m/s with the air speed for minimum drag V_{m}=46.98m/s, we found V_{mp} is slower by 11.23m/s.
g)
air speed for minimum power:
V_{mp} = 35.75m/s
=35.75*1.94384
= 64.49knots
Minimum power:
$$P_{m} = left( frac{1}{2}text{ρS}C_{D_{0}} right)V^{3} + left( frac{2KW^{2}}{text{ρS}}V^{ 1} right)$$
$$= left( frac{1}{2}1.045*15.9*0.024 right){35.75}^{3} + left( frac{2*0.58{*11800}^{2}}{1.045*15.9}{35.75}^{ 1} right)$$
= 9110.09 + 271914.61
= 281024.7watt
= 281.024.7KW
h)
Parasitic power =Parasitic drag Χ air speed=$left( frac{1}{2}text{ρS}C_{D_{0}} right)V^{3}$
Induced Power= Induced drag Χ air speed = $frac{2KW^{2}}{text{ρS}}V^{ 1}$
Total Power= Parasitic power + Induced Power
[CHART]
Figure 3: Plot of parasitic, induced, and total power against air speed
Figure 3 shows the plot of parasitic power, induced power, and total power against air speed. The induced power decreases as the speed increases.
i)
The ratio of lift force to the drag force is the dimensionless quantity which represents the aerodynamic efficiency. i.e.
$$eta = frac{Lift force(L)}{Drag force (D)}$$
$$eta = frac{frac{1}{2}rho V^{2}SC_{L}}{frac{1}{2}rho V^{2}SC_{D}}$$
$$eta = frac{C_{L}}{C_{D}}$$
[CHART]
Figure 4: Plot of efficiency vs coefficient of lift
The climb ratio can be defined as the ratio of distance travelled along the ground to the altitude. The rate of climb can be defined as the vertical component of the velocity.
References
[1] P. E. Tulapurkara, “NPTEL,” [Online]. Available: https://nptel.ac.in/courses/101/106/101106035/. [Accessed 15 06 2021].