A circular pipe of 25-mm outside diameter is placed in an airstream at 25 °C and 1-atm pressure. The air moves in cross flow ove
r the pipe at 15 m/s, while the outer surface of the pipe is maintained at 100 °C. a) What is the drag force exerted on the pipe per unit length of the pipe? (50 pts) b) What is the rate of heat transfer from the pipe per unit length of the pipe? (50 pts)
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The classical motion for an oscillator that starts from rest at location x₀ is
x(t) = x₀ cos(ωt)
The probability that the particle is at a particular x at a particular time t
is given by ρ(x, t) = δ(x − x(t)), and we can perform the temporal average
to get the spatial density. Our natural time scale for the averaging is a half
cycle, take t = 0 → π/ ω
Thus,
ρ =
Limit is 0 to π/ω
We perform the change of variables to allow access to the δ, let y = x₀ cos(ωt) so that
ρ(x) =
Limit is x₀ to -x₀
Limit is -x₀ to x₀
This has as expected. Here the limit is -x₀ to x₀
The expectation value is 0 when the ρ(x) is symmetric, x ρ(x) is asymmetric and the limits of integration are asymmetric.