Is the rate of pressure change in a piping system limited by the speed of sound? That is, if in a piping system the pump suddenly stops will the pressure at the end of the line be constant until a pressure wave travelling at the speed of sound reaches that point?
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Is rate of pressure change limited by the speed of sound?
- Thread starter Andy22
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Pump stopping, starting and pressure changes all initiate pressure waves from the pump, as does any flow or pressure change. The waves travel both up and down the pipeline from the point of origin at the "effective" speed of sound of the liquid inside. The effective speed of sound is basically a function of the fluid density and pipe properties.
I also have a "blog" on waterhammer and surge (with a bit more detail about surge in oil pipelines) on my webspace page that you might find useful.
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I also have a "blog" on waterhammer and surge (with a bit more detail about surge in oil pipelines) on my webspace page that you might find useful.
Going the Big Inch!
sailoday28
Mechanical
The rate of pressure change for highly incompressible fluids is dependent upon density, sound speed and pipe friction.
The pressure pulse travels along characteristics paths defined by dx/dt= fluid velocity plus (or minus)sound speed. For liquids, the local velocity is usually small compared to the sound speed and dx/dt is reasonably approximated as dx/dt = sound speed. Further, if sound speed is constant, the integration is simple. Note that sound speed and density are a function of pressure and temperature.
The above assumes the pipe to have rigid walls.
Regards
The pressure pulse travels along characteristics paths defined by dx/dt= fluid velocity plus (or minus)sound speed. For liquids, the local velocity is usually small compared to the sound speed and dx/dt is reasonably approximated as dx/dt = sound speed. Further, if sound speed is constant, the integration is simple. Note that sound speed and density are a function of pressure and temperature.
The above assumes the pipe to have rigid walls.
Regards
Rereading your question carefully, actually you ask about the, "rate of pressure change", not the limiting speed of a pressure wave.
I would think the build up of pressure is dependent on how fast you can release enough energy to increase the density of the fluid sufficiently to cause a pressure increase.
Since increasing density would require the molecules to physically move together, we would be looking at the limitation of F = m x a. Taking things to an extreme here, in a nuclear physics sense, the limit on compacting molecules and individual atoms must be at the speed of light, since nothing can move faster than that, so no, I'd say that the "rate of pressure increase" could effectively be instantaneous (or at the speed of light if you will), at least initially. BUT, I'm sure not a nuclear physicist, so don't count on it.
Going the Big Inch!
I would think the build up of pressure is dependent on how fast you can release enough energy to increase the density of the fluid sufficiently to cause a pressure increase.
Since increasing density would require the molecules to physically move together, we would be looking at the limitation of F = m x a. Taking things to an extreme here, in a nuclear physics sense, the limit on compacting molecules and individual atoms must be at the speed of light, since nothing can move faster than that, so no, I'd say that the "rate of pressure increase" could effectively be instantaneous (or at the speed of light if you will), at least initially. BUT, I'm sure not a nuclear physicist, so don't count on it.
Going the Big Inch!
sailoday28
Mechanical
The rate of pressure change is determined from solution of the "hammer equations" from the method of characteristics (moc)
along a + characteristic
dV/dt+1/(rho*c)* dxdt +friction=0 (1)
dx/dt = V +c (1a)
where V= velocity, x distance, t =time c=sound velocity
along a - characteristic
dV/dt-1/(rho*c)* dxdt +friction=0 (2)
dx/dt = V -c (2a)
For highly incompressible fluids and where V<<c (liquids)
1a and 1b become
dx/dt = + (or)- c
and generally it is assumed that rho and c are constant such that the equations 1,1a, 2 and 2a are easily integrated.
Solutions of the integrated equations 1 and 2 from known conditions of a previous time yields the pressure and velocity and a new time.
The new time is determined from the integrated equations 1a and 2a.
Neglecting friction yields the well know hammer equations of change in velocity = pressure rise/rho/c
If for example the presssure/velocity vs. time is known at say a location x=0, then the information traveling along the characteristics in say the + direction is known and the solution obtained at new time steps from information on the downstream -(minus) chracteristics. Rate of pressure rise at other locations is then known.
Use of F=ma as suggested from a previous post on this thread could be used if a pump is starting and the idea is to move a pig. MOC would be applied to the liquid with the charateristics having F=ma as a boundary condition.
F=ma could also be used as an approximation, if a relatively long slug of water at one temperatre in a pipe is separated from a relatively small slug at a different temperature. MOC being used for the long slug.
If there were for example a pipe rupture to the atmosphere, then the solution of the characteristic equations is simple, since the end condition of pressure is known and the break exit velocity easily determined.
Similarly, if there is a rapid valve closure,solution of the characteristic equation is simple, since the velocity at the end of characteristic is zero and the pressure easily determined.
Regards
along a + characteristic
dV/dt+1/(rho*c)* dxdt +friction=0 (1)
dx/dt = V +c (1a)
where V= velocity, x distance, t =time c=sound velocity
along a - characteristic
dV/dt-1/(rho*c)* dxdt +friction=0 (2)
dx/dt = V -c (2a)
For highly incompressible fluids and where V<<c (liquids)
1a and 1b become
dx/dt = + (or)- c
and generally it is assumed that rho and c are constant such that the equations 1,1a, 2 and 2a are easily integrated.
Solutions of the integrated equations 1 and 2 from known conditions of a previous time yields the pressure and velocity and a new time.
The new time is determined from the integrated equations 1a and 2a.
Neglecting friction yields the well know hammer equations of change in velocity = pressure rise/rho/c
If for example the presssure/velocity vs. time is known at say a location x=0, then the information traveling along the characteristics in say the + direction is known and the solution obtained at new time steps from information on the downstream -(minus) chracteristics. Rate of pressure rise at other locations is then known.
Use of F=ma as suggested from a previous post on this thread could be used if a pump is starting and the idea is to move a pig. MOC would be applied to the liquid with the charateristics having F=ma as a boundary condition.
F=ma could also be used as an approximation, if a relatively long slug of water at one temperatre in a pipe is separated from a relatively small slug at a different temperature. MOC being used for the long slug.
If there were for example a pipe rupture to the atmosphere, then the solution of the characteristic equations is simple, since the end condition of pressure is known and the break exit velocity easily determined.
Similarly, if there is a rapid valve closure,solution of the characteristic equation is simple, since the velocity at the end of characteristic is zero and the pressure easily determined.
Regards
The OP askes about the limiting factor in a pressure increase, which I take to be asking how fast can pressure rise. I do not consider that he's asking about what you see at the end of a pipeline when you turn on a pump at the beginning. If he is asking, how fast can pressure rise, forget the pipe and consider an explosive scenario, or if you still want to consider pipe, consider how fast pressure can rise at the point of a violent collapse of a vapor pocket. We all know by now that the typical pressure rise in a long pipe is just the way you say, but I do not believe that is the question.
What is the limit to how fast a pressure can rise anywhere? That limit is certainly not the speed of sound, although the wave front would travel at the speed of sound.
Going the Big Inch!
What is the limit to how fast a pressure can rise anywhere? That limit is certainly not the speed of sound, although the wave front would travel at the speed of sound.
Going the Big Inch!
sailoday28
Mechanical
The question is related to a pump suddenly stopping. IF the pressure flow charateristics are know are can be modeled, then the question asked relates to what happens at the downstream end of the piping.
The pressure at the downstream end will be constant until the first new characteristic generated by dx/dt = u + c (if the + direction is the downstream direction)reaches that point.
Regards
The pressure at the downstream end will be constant until the first new characteristic generated by dx/dt = u + c (if the + direction is the downstream direction)reaches that point.
Regards
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