Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 11, Problem 43P

During major windstorms, high vehicles such as RVs and semis may be thrown off the road and boxcars off their tracks, especially when they are empty and in open areas. Consider a 6000-kg semi that is
10 m long, 2.5 m high, and 2 in wide. The distance between the bottom of the truck and the road is 0.8 in. Now the truck is exposed to winds from its side surface. Determine the wind velocity that will tip the truck over to its side. Take the air density to be 1.1 kg/m3 and assume the weight to be uniformly distributed.

Chapter 11, Problem 43P, During major windstorms, high vehicles such as RVs and semis may be thrown off the road and boxcars

Expert Solution & Answer
Check Mark
To determine

The wind velocity that will tip the truck over to its side.

Answer to Problem 43P

The wind velocity that will tip the truck over to its side is 41.98m/s.

Explanation of Solution

Given information:

The mass of the truck is 6000kg, the length of the truck is 10m,the height of the truck is 2.5m, the width of the truck is 2m, the distance between the bottom of the truck and the road is 0.80m and the air density is 1.1kg/m3.

Write the expression of weight of the truck.

  W=mg...... (I)

Here, the mass of the truck is m and the acceleration due to gravity is g.

Write the expression of frontal area.

  A=L×h...... (II)

Here, the length of the truck is L and the height of the truck is h.

The below figure represent the forces acting on the truck.

Fluid Mechanics: Fundamentals and Applications, Chapter 11, Problem 43P

Figure-(1)

Write the expression of equilibrium of momentum about point P as shown in Figure-(1).

  FD(h2+b)=W(a2)...... (III)

Here, the width of the truck is a, the distance between the bottom of the truck and the road is b, and the drag force is FD.

Refer table 11.2 "Representative drag coefficient CD for various three dimensional bodies based on the frontal area for Re>104 unless stated otherwise" to obtain the drag coefficient for rectangular plate as.

  CD=1.10+0.02(Lh+hL)...... (IV)

Write the expression of drag force acting on the truck.

  FD=12CDρAV2...... (V)

Here, the density of the air is ρ and the velocity of the wind is V.

Write the expression of characteristics length of the truck.

  LC=AP...... (VI)

Here, the perimeter of the truck is P.

Write the expression of perimeter of the truck.

  P=2(L+h)...... (VII)

Write the expression of Reynolds number.

  Re=VLCν...... (VIII)

Here, the kinematic viscosity is ν.

Calculation:

Substitute 6000kg for m and 9.81m/s2 for g in Equation (I).

  W=(6000kg)×(9.81 m/s 2)=(58860 kgm/s 2)×( 1N 1 kgm/s 2 )=58860N

Substitute 10m for L and 2.5m for h in Equation (II).

  A=(10m)×(2.5m)=25m2

Substitute 2.5m for h, 0.8m for b, 58860N for W and 2m for a in Equation (III).

  FD( 2.5m2+0.8m)=(58860N)( 2m2)FD×2.05m=(58860N)×(1m)FD=( 58860N)×( 1m)( 2.05m)FD=28712.195N

Substitute 10m for L and 2.5m for h in Equation (IV).

  CD=1.10+0.02( 10m 2.5m+ 2.5m 10m)=1.10+0.02×4.25=1.185

Substitute 1.185 for CD, 1.1kg/m3 for ρ, 25m2 for A and 28712.195N for FD in Equation (V).

  (28712.195N)=12×(1.185)×(1.1kg/ m 3)×(25m2)V2V2=2×( 28712.195N)( 1.185)×( 1.1 kg/ m 3 )×( 25 m 2 )V= 2×( 28712.195N )×( ( 1 kgm/s 2 ) 1N ) ( 1.185 )×( 1.1 kg/ m 3 )×( 25 m 2 )V=41.98m/s

Substitute 10m for L and 2.5m for h in Equation (VII).

  P=2(10m+2.5m)=2×12.5m=25m

Substitute 25m for P and 25m2 for A in Equation (VI).

  LC=( 25 m 2 )( 25m)=5m

Refer table A-9 "Properties of air" to find the value of kinematic viscosity of air as 1.75×105m2/s corresponding to density of air is 1.1kg/m3.

Substitute 1.75×105m2/s for ν, 5m for LC and 41.98m/s for V in Equation (VIII).

  Re=( 41.98m/s )×( 5m)( 1.75× 10 5 m 2 /s )=209.9 m 2/s1.75× 10 5 m 2/s=119.94×105

So, the Equation used for CD is correct because the obtained Reynolds number is greater than 104.

Conclusion:

The wind velocity that will tip the truck over to its side is 41.98m/s.

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