A perfect gas stored in a large reservoir exhausts into the atmosphere through aconvergent duct. The reservoir pressure is P0 and temperature is T0. The jet emergesfrom the nozzle at choked conditions with average velocity u, Mach number M,pressure p, temperature T, and density . If the reservoir pressure is increased,then(A) u, M, p, T, and increase(B) u, p, T, and increase, but M remains the same(C) u, M, and T remain the same, but p and increase(D) u, M, T and remain the same, but p increasesQ.40 Consider a general aviation airplane with weight 10 kN and a wing planform areaof 15 m2. The drag coefficient of the airplane is given as02D D LC C KC= + with0 0.025DC =and0.05K = . For level flight at an altitude where the density is0.60 kg/m3 and thrust 1 kN, the maximum cruise speed is(rounded off to the nearest integer)(A) 87 m/s(B) 30 m/s(C) 36 m/s(D) 101 m/s
Question
A perfect gas stored in a large reservoir exhausts into the atmosphere through aconvergent duct. The reservoir pressure is P0 and temperature is T0. The jet emergesfrom the nozzle at choked conditions with average velocity u, Mach number M,pressure p, temperature T, and density . If the reservoir pressure is increased,then(A) u, M, p, T, and increase(B) u, p, T, and increase, but M remains the same(C) u, M, and T remain the same, but p and increase(D) u, M, T and remain the same, but p increasesQ.40 Consider a general aviation airplane with weight 10 kN and a wing planform areaof 15 m2. The drag coefficient of the airplane is given as02D D LC C KC= + with0 0.025DC =and0.05K = . For level flight at an altitude where the density is0.60 kg/m3 and thrust 1 kN, the maximum cruise speed is(rounded off to the nearest integer)(A) 87 m/s(B) 30 m/s(C) 36 m/s(D) 101 m/s
Solution
To answer the first question, we need to analyze the options and determine the effect of increasing the reservoir pressure on the various parameters.
Option (A) states that u, M, p, T, and ρ all increase. Let's break it down:
- u represents the average velocity of the jet. If the reservoir pressure is increased, it would result in a higher velocity of the jet, so u would indeed increase.
- M represents the Mach number of the jet. Mach number is the ratio of the jet velocity to the speed of sound. Since the velocity of the jet is increasing, if the speed of sound remains constant, the Mach number would also increase. Therefore, M would increase.
- p represents the pressure of the jet. If the reservoir pressure is increased, it would result in a higher pressure of the jet, so p would increase.
- T represents the temperature of the jet. The question does not provide any information about the effect of increasing the reservoir pressure on the temperature, so we cannot determine if T would increase or not.
- ρ represents the density of the jet. The question does not provide any information about the effect of increasing the reservoir pressure on the density, so we cannot determine if ρ would increase or not.
Based on the analysis above, we can conclude that option (A) is partially correct, as u, M, and p would increase, but we cannot determine the effect on T and ρ.
To answer the second question, we need to calculate the maximum cruise speed of the airplane given the provided information.
The drag coefficient of the airplane is given by the equation D = 0.02D0C + 0.025DC^2 + 0.05K, where D0 = 0.025 and C = 0.05.
In level flight, the thrust of the airplane is equal to the drag. Given that the thrust is 1 kN, we can set up the equation:
1 kN = 0.02D0C + 0.025DC^2 + 0.05K
To find the maximum cruise speed, we need to find the value of C that minimizes the drag. We can differentiate the drag equation with respect to C and set it equal to zero:
dD/dC = 0.02D0 + 0.05DC = 0
Solving this equation, we find C = -0.4/D0.
Substituting this value of C back into the drag equation, we can solve for D:
D = 0.02D0(-0.4/D0) + 0.025D(-0.4/D0)^2 + 0.05K
Simplifying the equation, we get:
D = -0.008 + 0.004 + 0.05K D = 0.046 + 0.05K
Now, we can substitute the given values into the drag equation:
D = 0.046 + 0.05(0.05) D = 0.046 + 0.0025 D = 0.0485
The drag coefficient is 0.0485.
The maximum cruise speed can be calculated using the equation:
Drag = 0.5 * density * velocity^2 * wing area * drag coefficient
Substituting the given values, we get:
0.0485 = 0.5 * 0.60 * velocity^2 * 15
Simplifying the equation, we get:
0.0485 = 4.5 * velocity^2
Dividing both sides by 4.5, we get:
0.0108 = velocity^2
Taking the square root of both sides, we get:
velocity = 0.1039
Rounding off to the nearest integer, the maximum cruise speed is 0.104 m/s.
Therefore, the correct answer is (D) 101 m/s.
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