Force generated by liquid meeting a restrictor

[c2sco]

Hi folks,

I'm looking at a pipe filling with liquid that has a restrictor orifice in it, part way along but some distance from the pump. The line is initially empty and a pump is started at its inlet end, so the liquid runs along the line, filling it. Just before the liquid meets the RO, it is running at an average velocity V1. Immediately after it has passed through the RO, the flow has dropped to a new value V2 due to the pressure drop caused by the RO. What I'm interested in is the transient forces caused on the pipe carrying the RO as the liquid meets it.

Method 1: the force generated is (pipe area minus hole area)x (pressure drop across the RO after the liquid has passed through it).

Method 2: the force generated is the Joukowski force due to the drop in velocity, ie (density)x(wave [sonic] velocity)x(V1-V2)x (area of pipe)

Any suggestions which is correct?

Thanks,

Stuart

 

[bewdley]

Simply, the force is the delta P in the RO times the cross area, whichever unit you choose.

 

[jalvarez]

bewdley:
What you concluded is applicable only in a steady state situation. The liquid has already passed through the orifice. And the "cross area" must be the cross section of the pipe, minus the orifice area.
Stuart is proposing a transient situation. In the moment the liquid reachs the plate there appears what is called a "water hammer". This phenomenum is caused due to the sudden drop in velocity (due to the restriction). So, it is the second concept which applies. I humbling recognize that I don't know about Joukowski, but the water hammer calculation -pressure developed in the plate- will depend on the liquid desity, its speed and sonic velocity, but also on the total volume of water in the line.
This "water hammer" is very serious in pipelines, appearing e.g. when closing a valve. A hydraulic calculation for these "water hammers" could be obtained from

http://www.lmnoeng.com/WaterHammer/WaterHammer.htm

Your case is a simplification of the extensive one presented in the above site.
Obviously, this is a problem in very big and long pipes, in a normal plant is enough precaution to open the discharge valve of the pump(s) slowly...
Have a safe and healthy day
J.Alvarez

 

[c2sco]

Thanks for your posts.

Joukowski was the first guy to investigate this phenomonon of water hammer, and his equation for the pressure generated by a fast transient event such as a sudden valve closure in a long, running pipeline is
P=rho.c.v
where
rho=density,
c=velocity of sound in the liquid in the pipe (allowing for pipe elasticity as well as fluid compressibility), and
v= velocity before the event.

For this to be true the event has to happen in a time close to or less than the pipeline period T where T=2L/c where L = length of pipe upstream of the restriction. Slower events produce lower peak pressures. The pressure transient produces high forces on the pipe supports which is often more of a problem than the risk of bursting the pipe due to exceeding its bursting pressure.

Clearly the liquid flooding the orifice hole will be pretty fast, and the pipe is sufficiently long, so this condition applies, and I'm in favour of the second option, and not the first [or the third (suggested after I entered the thread) which is to use (pipe area - hole area) rather than (pipe area)]. My client thinks the first equation is right (his equation) and I'm still trying to convince him otherwise!!

Surge throws up sone interesting problems and I'm always interested in others' views.

Regards,

Stuart

 

[jalvarez]

Stuart:
The equation presented as from Joukowsy is perhaps a simplification, and doesn't take into account the lenght of the pipe. So, I assume it is not correct, it doesn't applies to your case. In common short pipes, nothing happens, the normal times for closing or opening valves are enough to avoid these effects. The same applies to your case.
The methodology presented in

http://www.lmnoeng.com/WaterHammer/WaterHammer.htm

is very clear, and reinforce the issue of needness of the lenght.
Have a healthy and safe day.
J.Alvarez

 

[c2sco]

Thanks for your comment.

LMNO only looks at a simple case, but still seems to make it look more complicated than it is. Seems a good site, though.

In fact, the Joukowski head is independent of pipe length (yes, the length makes NO difference to the answer)when the event time (T) [eg how long it takes a valve to shut assuming its characteristic is linear] is short enough compared to the pipeline period. The pipeline period depends on length (2L/c), so basically for short pipes or slow event, only momentum change forces exist on the pipe. However as the "relative event time" 2L/cT gets bigger, you approach the Joukowski situation, until at 2L/cT>=1, the full Joukowski pressure and force are generated. It doesn't matter if 2L/cT>>>1 - the same pressure and consequent force results, and is due to the sonic wave travelling upstream.

Hope this helps your understanding. Surge takes a while to get your head round and I'm frequently scratching mine on unusual problems!

Stuart