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Chapter 9: Problem 118

Commercial airliners normally cruise at relatively high altitudes \((30,000 \text { to } 35,000 \mathrm{ft}) .\) Discuss how flying at this high altitude (rather than \(10,000 \mathrm{ft}\), for example) can save fuel costs.

Short Answer

Expert verified
The air density decreases as altitude increases, which means the drag force during flight is also less at higher altitudes. This reduced drag allows the airplane to exert less energy in maintaining its cruising speed, leading to better fuel efficiency and lower fuel costs. Therefore, cruising at altitudes of 30,000 to 35,000 feet is more cost-effective than at 10,000 feet.

Step by step solution

01

Understanding the Impact of Altitude on Air Density

Firstly, it’s important to understand that as altitude increases, air density decreases. The reason for this is that there’s simply less air the higher you go, as most of it is closer to sea level due to gravity. The formula for air density is given by \(\rho = p/RT\), where \(p\) is the air pressure, \(R\) is the specific gas constant, and \(T\) is the temperature. This relation shows air density decreases with increasing altitude, given temperature and pressure decrease with altitude.
02

Understanding the Relationship Between Air Density and Drag

Next, consider the drag force that an airplane faces during flight. The drag force is given by the equation \(F_D = 0.5\cdot\rho\cdot v^2\cdot C_D\cdot A\), where \(\rho\) is air density, \(v\) is velocity, \(C_D\) is the drag coefficient, and \(A\) is the area of the object. Given that \(\rho\) decreases with altitude, the drag force is less at higher altitudes.
03

Relating Reduced Drag to Fuel Efficiency

Finally, we need to relate this back to fuel efficiency. Because the drag force is smaller at higher altitudes due to the reduced air density, an airplane needs to expend less energy to maintain its cruising speed. This means it will use less fuel, making it more fuel efficient. Hence, flying at high altitudes helps to save fuel costs as compared to flying at lower altitudes.

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Most popular questions from this chapter

An airplane flies at a speed of \(400 \mathrm{mph}\) at an altitude of \(10,000 \mathrm{ft}\). If the boundary layers on the wing surfaces behave as those on a flat plate, estimate the extent of laminar boundary layer flow along the wing. Assume a transitional Reynelds number of \(\mathrm{Re}_{\mathrm{xcr}}=5 \times 10^{5} .\) If the airplane maintains its 400 -nph speed but descends to sea- level elevation, will the portion of the wing covered by a laminar boundary layer increase or decrease compared with its value at \(10,000 \mathrm{ft}\) ? Explain.

A laminar boundary layer velocity profile is approximated by \(u / U=[2-(y / \delta)](y / \delta)\) for \(y \leq \delta,\) and \(u=U\) for \(y>\delta\) (a) Show that this parabolic profile satisfies the appropriate boundary conditions. (b) Use the momentum integral equation to determine the boundary layer thickness, \(\delta=\delta(x)\). Compare the result with the exact Blasius solution.

An airplane flies at \(150 \mathrm{km} / \mathrm{hr}\). (a) The airplane is towing a banner that is \(b=0.8 \mathrm{m}\) tall and \(\ell=25 \mathrm{m}\) long. If the drag coefficient based on area \(b \ell\) is \(C_{D}=0.06,\) estimate the power required to tow the banner. (b) For comparison, determine the power required if the airplane was instead able to tow a rigid flat plate of the same size. (c) Explain why one had a larger power requirement (and larger drag) than the other. (d) Finally, determine the power required if the airplane was towing a smooth spherical balloon with a diameter of \(2 \mathrm{m}\)

Approximately how fast can the wind blow past a 0.25 -in. diameter twig if viscous effects are to be of importance throughout the entire flow field (i.e., \(\operatorname{Re}<1\) )? Explain. Repeat for a 0.004 -in. diameter tair and a 6 -fi-diameter smokestack.

Compare the rise velocity of an \(\frac{1}{8}\) -in.-diameter air bubble in water to the fall velocity of an \(\frac{1}{8}\) -in.- -diameter water drop in air. Assume each to behave as a solid sphere.
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