A supersonic transport has a mass of 65 Mg and cruises at 1...

A supersonic transport has a mass of $65 \mathrm{Mg}$ and cruises at 1 1-km standard altitude at a Mach number of 2.25. If the angle of attack is $2^{\circ}$ and its wings can be approximated by flat plates, estimate $(a)$ the required wing area in $\mathrm{m}^{2}$ and (b) the thrust required in $\mathrm{N}$.

A supersonic transport has a mass of $65 \mathrm{Mg}$ and cruises at 1 1-km standard altitude at a Mach number of 2.25. If the angle of attack is $2^{\circ}$ and its wings can be approximated by flat plates, estimate
$(a)$ the required wing area in $\mathrm{m}^{2}$ and (b) the thrust required in $\mathrm{N}$.

Key Concepts

Drag and Thrust Requirements

Calculating the thrust required involves understanding and quantifying drag forces acting against the aircraft's motion. In supersonic flight, drag comprises wave drag and skin friction, among others, and the thrust needed must overcome these losses to maintain level flight. This concept links the aerodynamic drag calculated from the aircraft’s geometry, angle of attack, and flow conditions to the propulsion system’s output.

Standard Atmosphere

The standard atmosphere model provides the necessary properties of the air, such as density, at different altitudes. At a specified altitude, these atmospheric parameters are critical for accurately calculating dynamic pressure, which in turn directly impacts the lift and drag forces experienced by the aircraft. It ensures that the analysis reflects realistic conditions encountered during flight.

Thin Airfoil Theory Approximations

For wings that can be approximated as flat plates, thin airfoil theory simplifies the analysis of lift and drag. With small angles of attack, linear approximations for the lift coefficient can be applied. This concept is particularly useful in supersonic flow where the behavior of a flat plate can represent the performance of a thin wing, thereby facilitating estimates of aerodynamic forces.

Lift Generation

This concept involves generating enough aerodynamic force to oppose the weight of the aircraft. It is based on the relationship between lift, dynamic pressure, wing area, and the lift coefficient, and it is essential in determining the required wing area so that the aircraft can maintain level flight. The calculation typically equates the lift produced by the wing to the gravitational force acting on the mass of the aircraft.

Supersonic Aerodynamics

Supersonic aerodynamics is concerned with the behavior of airflow when an aircraft moves faster than the speed of sound. In this regime, compressibility effects become significant, altering the pressure distribution around the aircraft. This affects both lift and drag, and requires special considerations, such as shock waves and changes in flow properties, which are critical when estimating performance parameters like required thrust and wing area.

Transcript

00:01 In this problem, we're given a 4 ,000 kilogram or 4 megapilogram airplane with wings of length 5 meters and with 1 .75 meters flying into speed of 80 meters per second.

00:12 And we're asked to find the smallest angle of lift, assuming the wind is an naca 2409 section.

00:20 And we're given the density of air as i have written it.

00:24 Okay, so we're going to do a little bit of math here to figure out what we're going to find.

00:31 And we know that our sum, i already did this for another problem.

00:38 I'm going to follow this along with a highlighter.

00:48 Falling right here.

00:50 So we know that our sum of first is ma, and we can write this equation next.

00:59 That is provided in our text.

01:01 Here's a quick free body diagram, and we can figure out that our sum of our forces here will be m -a -y.

01:22 That was sum of f -y.

01:33 Okay.

01:37 Then we're going to combine a couple of equations.

01:44 Let me get over here to combine my equations.

01:48 Okay, so let's do, okay, unless rearranges, cl will be equal to 2mg over apl.

02:24 Okay.

02:27 Then we'll have okay, then we'll get cl is 2mg over 2bl density times v squared, which will equal mg times bl density v squared, density v squared...

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