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I am an architect in need help with an aerodynamics problem for a design that I am developing. I need to know the the potential velocity of air moving from one end of a pipe to the other if any with a constant 20° difference in temperature. The temperature at one end of the pipe would be uniformly 20°F colder than the temperature at the opposite end. I'm assuming the temperature difference would cause warm air to move to the cold end. The pipe /tunnel would be 30 feet in diameter and 5000 feet in length. I am interested in using any potential work or force from this volume of air, moving due to temperature difference. Thank you for any help that you can provide.

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    $\begingroup$ Natural Ventilation of a Tunnel is the topic you're looking for. You'll need to determine if it is something you're competent to conduct the analysis or you'll need to hire an engineer who has this within their competency. You also will need to figure out how exact of a figure you will need. dependent upon how exact of a velocity you need the construction of the tunnel could be very important, concrete vs steel, D shape vs round vs square, etc. But as far as internet answers I can tell you the velocity will be somewhere between 2 ft per second and speed of sound. $\endgroup$ – Dopeybob435 Nov 18 '16 at 19:49
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    $\begingroup$ @Ken_Miller: warm air rises. Your assumption about warm air flowing to the cold end of the tunnel is not valid. The opposite will occur, if there is a sufficient difference in air pressures between each end of the tunnel. You need to consult a ventilation engineer, preferably a mine ventilation engineer. $\endgroup$ – Fred Nov 19 '16 at 14:18
  • $\begingroup$ As Paul says, you need to know the boundary conditions at the two ends. So you're missing either the density or the pressure at each end. If the ends are open to the atmosphere, you can just use the standard atmosphere values as a first approximation. $\endgroup$ – jjack Nov 20 '16 at 15:21
  • $\begingroup$ Then you need to make some assumptions about the tunnel and compute the pressure drop due to friction. Then you have to solve for the velocity in the tunnel, provided you have enough equations. $\endgroup$ – jjack Nov 20 '16 at 15:28
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First of all, the flow of air is caused by the difference in pressure, and not in temperature, although they are related. So you need to know the pressure drop. You have to guess the pressure difference, or use the ideal gas law for finding the pressure at the ends:

$P=\rho RT$, where $\rho$ is the density of air at that specific temperature (see here), $T$ is the temperature that you must know and $R$ is a gas constant for air ($R$=53.35 ft lbf/lb/$^{\circ}$R).

There are many ways to find the velocity, but that would require to know the roughness of the pipe, flow regime etc. I recommend that you do your calculations for velocity here once you know your pressures.

And use the properties of air from here.

You can fine the conversion in velocity (from Imperial to SI) here.

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