It was already determined that the jet velocity of this engine is 376 metres per second. Suppose that this Boeing 777 flies at 11 kilometres at M=0.82, what is then the jet efficiency of this engine (in percent)?
To find the jet efficiency of the engine, we need to calculate the ratio of the actual velocity of the jet to the ideal velocity of the jet.
First, let's convert the speed of the Boeing 777 from kilometers per hour to meters per second.
11 kilometers is equal to 11,000 meters.
So, the speed of the Boeing 777 is 11,000 meters divided by 1 hour, which is 11,000 meters/3,600 seconds, which is approximately 3.06 meters per second.
Next, we need to calculate the ideal velocity of the jet.
The ideal velocity of the jet is the speed of sound, which is approximately 343 meters per second.
So, the ratio of the actual velocity to the ideal velocity is:
3.06 meters per second / 343 meters per second = 0.00893
Finally, we multiply this ratio by 100 to find the jet efficiency as a percentage:
0.00893 * 100 = 0.893%
Therefore, the jet efficiency of this engine is approximately is 0.893%.
To calculate the jet efficiency of this engine, we need to determine the thrust produced by the engine and the power consumed by it.
First, we need to convert the speed of the aircraft from kilometers per hour to meters per second.
Given that the aircraft is flying at 11 kilometers per hour, we can convert this to meters per second by multiplying by a conversion factor of 1000/3600.
11 kilometers/hour * (1000 meters/kilometer) * (1 hour/3600 seconds) = 3.06 meters/second
Now, since the Mach number is given as M=0.82, we can calculate the speed of the aircraft relative to the speed of sound (v/V) using the formula:
v/V = M * sqrt(gamma * R * T)
where:
- v/V is the velocity ratio,
- M is the Mach number,
- gamma is the specific heat ratio of air (approximately 1.4),
- R is the universal gas constant for air (approximately 287 J/(kg·K)),
- T is the temperature in Kelvin.
Assuming standard temperature (approximately 288 K), plugging in the values:
v/V = 0.82 * sqrt(1.4 * 287 * 288) = 0.82 * 327.14 = 267.75
Now, we can calculate the thrust produced by the engine using the following equation:
Thrust = mass flow rate * jet velocity
The mass flow rate can be calculated using the formula:
mass flow rate = thrust / jet velocity
Given that the jet velocity is 376 meters per second, we can calculate the mass flow rate:
mass flow rate = Thrust / jet velocity = Thrust / 376
To find the thrust, we need to know the mass flow rate. Unfortunately, the given information does not provide the mass flow rate or the thrust directly. Without this information, it is not possible to calculate the jet efficiency of the engine in this specific case.
To determine the jet efficiency of the engine, we need to calculate the effective exhaust velocity of the engine (Ve) first.
Step 1: Convert the jet velocity from meters per second to kilometers per second.
- Jet velocity = 376 meters per second
- 1 kilometer = 1000 meters
Jet velocity in kilometers per second = 376 meters per second / 1000 = 0.376 kilometers per second
Step 2: Calculate the velocity of the airplane (Va).
- Velocity of the airplane = 11 kilometers
- Mach number (M) = 0.82
Velocity of the airplane (Va) = 11 kilometers * M = 11 kilometers * 0.82 = 9.02 kilometers per second
Step 3: Calculate the effective exhaust velocity (Ve).
- Effective exhaust velocity (Ve) = Jet velocity - Velocity of the airplane (Va)
Ve = 0.376 kilometers per second - 9.02 kilometers per second = -8.644 kilometers per second
Note: The negative sign indicates that the airplane is traveling faster than the jet exhaust velocity.
Step 4: Calculate the jet efficiency (η).
- Jet efficiency (η) = (Ve / Va) * 100
η = (-8.644 kilometers per second / 9.02 kilometers per second) * 100 = -95.78%
The jet efficiency of this engine (in percent) is approximately -95.78%. Note that a negative value indicates that the jet efficiency is less than 100%.