Energy absorption by aircraft fuselage

[Ortix]

Homework Statement

1. When an aircraft flies above the clouds during the daylight period it is continuously exposed to sunlight. This sunlight heats up the cabin. Although the outside air temperature at cruise altitude is – 56 degrees this is still one of the determining factors for the sizing of the cooling of the air conditioning system. Suppose that the sun is directly overhead, the solar constant is 1367 W/m2 and the albedo is 30%. What is the incoming power when the fuselage of the aircraft is 65 m long and has a diameter of 9 m. (Hint: the sunlight only reaches the top half of the fuselage and only the component perpendicular to the surface of the fuselage has to be taken into account. The flight altitude has no influence on the solar constant.)
a) 35 kW
b) 70 kW
c) 140 kW
d) 280 kW
e) 350 kW
f) 560 kW
g) 720 kW

Homework Equations

Energy absorption = (1-a)S*A
a = albedo (30 procent in my case)
S = 1367 W/m^2
A= area of the fuselage which has dimension : R=9m length = 65m

The Attempt at a Solution

I tried plugging in the numbers but I didn't get the right result.. This got never explained during the lecture... It's on a practice exam

Help is greatly appreciated!

 

[berkeman]

Homework Statement

1. When an aircraft flies above the clouds during the daylight period it is continuously exposed to sunlight. This sunlight heats up the cabin. Although the outside air temperature at cruise altitude is – 56 degrees this is still one of the determining factors for the sizing of the cooling of the air conditioning system. Suppose that the sun is directly overhead, the solar constant is 1367 W/m2 and the albedo is 30%. What is the incoming power when the fuselage of the aircraft is 65 m long and has a diameter of 9 m. (Hint: the sunlight only reaches the top half of the fuselage and only the component perpendicular to the surface of the fuselage has to be taken into account. The flight altitude has no influence on the solar constant.)
a) 35 kW
b) 70 kW
c) 140 kW
d) 280 kW
e) 350 kW
f) 560 kW
g) 720 kW

Homework Equations

Energy absorption = (1-a)S*A
a = albedo (30 procent in my case)
S = 1367 W/m^2
A= area of the fuselage which has dimension : R=9m length = 65m

The Attempt at a Solution

I tried plugging in the numbers but I didn't get the right result.. This got never explained during the lecture... It's on a practice exam

Help is greatly appreciated!

What is the surface area of the top half of the structure?

But then you also have to derate that number, since only the component of sunlight that is perpendicular to the surface counts. Can you think of a trig function to use for that derating?