## The Joukowsky Equation

Many engineers feel like they can do steady-state analysis in a spreadsheet. Still, if they want to look at anything transient - for example, the pressure surge from closing a valve suddenly or from a rupture – a simulator is a must. There is actually an easy way to do that sort of transient analysis by hand, which works for both gas and liquid pipelines and which is quite accurate in many situations. It's always good to have a pencil-and-paper calculation to fall back on as a sanity check on a complex piece of software, even if it's Atmos SIM.

The Joukowsky Equation lets us perform surge calculations by hand. It is named after Nikolay Zhukovsky, a founding father of aerodynamics and also hydrodynamics. (In this blog entry, I will stick with the more common spelling of the equation, "Joukowsky"). In addition to coming up with this equation, Zhukovsky was the first person to physically explain the lift on an airplane wing and produced many other important results in engineering and mathematics. I won't try to derive the Joukowksy equation here. Still, it follows from Newton's Second Law, force = mass times acceleration, combined with the physical fact that no disturbance can travel through a fluid any faster than the wave speed (the wave speed being roughly 300 m/s in gas pipelines and around 1-1.5 km/s in liquid pipelines).

The Joukowsky Equation is ∆P = - ρ c ∆V. Where ∆P is the magnitude of the pressure surge caused by the velocity change and is what we are trying to calculate: ΔV is the velocity change causing the surge: p is the density of the fluid and c is the wave speed. (For there not to be any unit conversion factors, these quantities must be in a consistent set of units, like SI: pressure in Pascals, density in kg/m3, wave speed and velocity change in m/s).

The Joukowsky equation can be applied either way: you can calculate the pressure surge accompanying a given velocity change: or the velocity change to be expected from a sudden pressure change.

For example, let's say we have a gasoline pipeline flowing at 2 m/s and I shut a valve quickly: how much pressure surge can I expect? Checking my reference books (or Google), I see that the density of gasoline is about 800 kg/m3 and the speed of sound is about 1380 m/s. The gasoline stops suddenly from the valve closure, so ΔV is -2 m/s. Then the pressure change on the upstream side is ΔP = - 1380 m/s * 800 kg/m3 * -2 m/s = 22 bar (= 320 psi) and on the downstream side is -22 bar. If the valve really closes instantaneously, then a pressure surge of 22 bar shoots back up the pipe and a negative surge shoots down the pipe, both at 1380 m/s. This is a perfectly valid transient model result: we would expect SIM to give a similar answer, and indeed, it does, as we see below:

If you want an extremely accurate solution or a solution when the pressure surge has traveled more than 5 or 10 km, then there are several caveats:

- Eventually, friction will cause the height of the surge to decrease: the width of the surge (meaning, the time it takes the surge to pass a point) will remain at however long it took to close the valve. Thus, in our gasoline example, if the valve closed in 1 second, then the pressure will rise 22 bar over 1380 m and that rise will pass a point in the pipeline - even a point many kilometers away - over 1 second.
- In a gas pipeline with a large enough pressure change that ΔV is close to c - meaning that the gas velocity gets close to choked flow, as occurs in a pipeline rupture - nonlinear processes become important and the Joukowsky result is going to be less accurate. This is because the true pressure wavefront will be more spread out.
- For this result to be exact, you have to use the right wave speed c. That means it needs to be adjusted for pipe expansion if you're looking at a very-large-diameter pipe, a particularly thin-walled pipe, or a plastic pipe (the correction for this can be found online or in any engineering handbook.)
- The wave speed in a system held at constant temperature (for example, an uninsulated subsea pipe) will be different from in a pipe with good insulation.
- Eventually, you will start seeing reflected waves off any other closed valves in the system or anything held at constant pressure like a tank or gas reservoir.
- In a liquid system, a negative-pressure surge might drop the pressure below vapor: in that case, slack flow will develop and the dynamics can become very complicated.

Atmos SIM automatically handles all of these more complex situations. But for a quick back-of-the-envelope estimate, the Joukowsky equation is still very useful!

See Jason’s previous blog on handling bad data here. You can also read some papers by Jason and blogs below: