ME3834 ‐ Fluid Mechanics Problem 9.79 Given: Mass of the aircraft = 9500 Kg The plane lands at 350 km/hr and the parachute system alone must slow the airplane to 100 km/hr in less than 1200 m. Find: (a) Find the minimum diameter required for a single parachute, and for three non‐interfering parachutes. (b) Plot speed against distance and time; maximum "g''s Solution: 1
.
Newton's second law for the aircraft is
2
Where A and
are the single parachute area and drag coefficient
2
Separating variables
V t
Integrating, with IC V = Vi
.
-------------------------------- (1)
Integrating again with respect to t
x t
2M/ C ρA ln 1
V.t
Eliminating t from Eqs. 1 and 2
x
-------------------------------------- (2)
2M/ C ρA ln V /V
-------------- (3)
To find the minimum parachute area we must solve Eq 3 for A
with x = xf when V = Vf
A
ln
------------------------------------- (4)
For three parachutes, the analysis is the same except A is replaced with
3A, leading to
1 ME3834 ‐ Fluid Mechanics Problem 9.79 A
ln
------------------------------------------------- (5)
The "g" s are given by
which has a maximum at the initial instant (V = Vi)
Given data:
M = 9500 kg
V i = 350 km/hr
V f = 100 km/hr
x f = 1200 m
C D = 1.42 (Table 9.3)
ρ= 1.23 kg/m3
Computed Results:
Single:
Triple:
A = 11.4 m2
D = 3.80 m
A = 3.8 m2
D = 2.20 m
"g "'s = -1.01 Max
2 ME3834 ‐ Fluid Mechanics Problem 9.79 3