Pipe Rupture Sonic Flow 2

[TangoCleveland]

We are trying to evaluate consequence of a pipe rupture for hazard analysis work. We want to determine what length of pipe will contribute flow through a rupture up to the point that sonic flow is achieved.

Fluid is a gas (air, nitrogen, etc.) At time t=zero, we have a pressurized pipe with no flow. Suddenly, the end of the pipe ruptures and flow starts. Flow area is equal to pipe inside diameter. How long will it take the fluid to accelerate to Mach 1 at the discharge? What length of the flowing fluid will disperse up to that time? We're talking pipe sizes of 1/2" OD or so, and pressures around 2400 psig.

We're determining sonic flow velocity, then equating the Darcy formula (with f * L / D) to the isentropic pressure relation with the k-1 terms in it, then solving for L.

I'm having trouble grasping the concept, although other engineers say it's correct. The numbers that come out seem sensible. Can anyone posit an alternate solution?


Larry

 

[BigInch]

This is relatively easy to do with a transient gas simulation program.

Why are you limiting the length of time to that until sonic flow is reached. Is the break closed off then? Why not just take the volume in the length of pipe from the break to the shut down valve?

What happens if the pipe breaks in the middle? Twice as much gas escapes during your sonic time limit, no?

Going the Big Inch!

http://virtualpipeline.spaces.msn.com

 

[TangoCleveland]

I'm still trying to puzzle this out, so maybe explaining it will help. You're right, if the pipe breaks in the middle, we'll have twice the flow.

We don't have a shutdown valve. We're looking at pipe failure in a pressurized system (infinite pressure source), and my manager thinks that something less than the entire pipe volume will contribute to the overpressure wave generated by the rupture. Can you recommend software or a consultant for assistance on this?

First, some background. For a number of years, we've been using an old NASA Safety Manual procedure to determine personnel exclusion zones for pneumatic pressure tests. It's based on potential energy from pressure and volume, and converting ft-lbf of potential energy into lbm of TNT equivalent. From lbm of TNT, we use old Army blast data to determine blast overpressure at various distances. The exclusion zone is the distance where blast overpressure is less than 0.5 psig. The spreadsheet we use for this evaluation is attached as "Piping Pneumatic Test Exclusion Zone."

Now, we are looking at blast overpressure in hazard analyses of operating systems. We are trying to evaluate an "equivalent length" of piping that will contribute blast energy in the initial pipe rupture, not during the entire flowing event. We're assuming a constant source of pressure, and trying to determine the volume of gas that will discharge until Mach 1 is achieved at the pipe exit. I attached our first try in another spreadsheet, "Piping Over Pressure Blast Energy....". The spreadsheet was developed by my manager, but I'm not sure if he's on the right track. The idea behind this is that the entire length of pipe would not be involved in the generation of overpressure, which would give a smaller exclusion zone than considering the entire length of pipe.

Larry

 

[insult2injury]

Is it imperitive that the exclusion zone is kept to a minimum area? If not, it'd be my opinion that using the total volume and possibly erring a little on the high end is better than the alternative.

I2I

 

[TangoCleveland]

BigInch,
Thanks for the idea about transient gas simulation. I found a couple papers from the pipeline world that might help out. I hadn't thought of the pipeline sources before.

insult2injury,
It is better to err conservatively and use the total volume, but we're trying to fine-tune our hazard analysis procedures a little. We've noticed that there could be a difference on tubing size runs.

Larry

 

[BigInch]

There will be a lot of volume difference from different tube sizes.

 

[sailoday28]

How long will it take the fluid to accelerate to Mach 1 at the discharge? Instantenaeously, if the back pressure is low enough and adiabatic conditions exist.

Choke pressure,M=1 (for a perfect gas, constant spec heats) at the exit will initially be
P/Po= [2/(gamma-1)]^[2*gamma/(gamma-1)] where Po is source pressure and p, the exit pressure.

Further, if the pipe is considered frictionless, following M=1 at the exit, and if the source pressure is maintained constant, there will be a short period of time in which there will be supersonic flow in the pipe at at its exit.

Regards

 

[mechprocess]

I have done many of these calculations.

A guillotine line break will cause flow from the high pressure source to the low pressure area. Gas flow will occur. The amount of gas flow rate is based on the pressure difference which in your case will be limited by sonic conditions at the exit of the flow regime. Why? Because the gas flows towards the lower pressure and when it does, it expands. Since the flow area is limited, the gas velocity must increase as it flows towards the lower pressure area. The further it flows, the lower the pressure is and the greater the gas velocity becomes. Conservation of enthalpy and mass dictates all this stuff. The funny thing is that as the gas velocity increases with decreasing pressures, the velocity of sound in the medium is approached and this velocity is the limit at which a pressure difference can be sensed by the upstream flow. This is the choked pressure.

There is a chart in Crane TP 410 page A-22 that shows the maximum pressure ratio for gas flows and is related to the total system loss coefficient K (inward projecting entrance = 0.78, pipe = fL/d and exit = 1). Assume f = 0.027 for 1/2 inch pipe (Crane A-26)

First you need to calculate the compressibility factor of your fluid at 2400 psig. This will determine your upstream specific volume.

W = 1891 Y d^2 SQRT(upstream density x DP/K) This formula is in Crane TP-410 equations 3-19

Were

W = lbm/hr
Y is interpolated from chart on page A-22
K is the sum of your fL/d (entrance, Exit, and pipe losses)
d is the diameter of your flow path inches

Use your ideal gas law to determine the upstream density, don’t forget the compressibility factor since your upstream pressure is so great!

For a given system total K value, there is a limited pressure ratio: (upsteam pressure psia - downstream pressure psia)/upstream pressure psia). From page A-22 you can determine the maximum value of DP in Equation 3-19. It won't be 2400. Most likely 50% of that.

This is based on the adiabatic assumption which assumes that no heat is lossed in the flow path. The heat of friction is absorbed by the fluid.

Note, the more pipe you can take credit for, the less your choked flow will be.

I hope this helps. If you have access to API 520 Appendix E there is a set of equations that curve fit Crane 410 A-22 charts.

I have a spread sheet for this but I don’t know how to get it to you.

In your case.

You may have an entrance loss of 0.5
10 feet of pipe 0.027*10*12/0.546 = 6
An exit loss of 1

Total K = 7.5
Upstream pressure = 2414.7 psia
Z = 1.0001 hmmm surprizing beter check that
Choked mass flow rate = 23,624 lbm/hr

dP/Dp sonic = 0.71
Y sonic = 0.71
Your actual pressure drop is 0.71*2414.7 = 1714 psi

The exit pressure is 2414.17-1714 = 700 psia

 

[TangoCleveland]

We are working from a NASA document:
Workbook for predicting pressure wave and fragment effects of exploding propellant tanks and gas storage vessels
Baker, W. E.; Kulesz, J. J.; Ricker, R. E.; Bessey, R. L.; Westine, P. S.; Parr, V. B.; Oldham, G. A.
NASA Center for AeroSpace Information (CASI)
NASA-CR-134906; REPT-02-4130 , 19751101; Nov 1, 1975
Technology needed to predict damage and hazards from explosions of propellant tanks and bursts of pressure vessels, both near and far from these explosions is introduced. Data are summarized in graphs, tables, and nomographs.
Accession ID: 76N19296
Document ID: 19760012208
View PDF File
Updated/Added to NTRS: 2005-08-25

I'm still struggling with the concept, but we're working out a method.


BigInch,
I tried to get to your web page, but our firewall told me
"Access Denied for client 139.88.109.26
Restricted Categories: Dating/Social;Media Downloads"
I'll try from home later this week.

thanks all for your contributions.

Larry

 

[Montemayor]

mechprocess:

Excellent display of know-how and practicality. While reading your response, it suddenly dawned on me that I'd done something similar to that and when I checked my Crane on p. A-22, there was my scribbled notation. Thank you for reminding me and helping us all with your explanation. You deserve an applause.

 

[sailoday28]

Most individuals recognize how to calculate the pressure rise from the sudden change in velocity of a highly incompressible fluid. delta velocity = delta P/(rho a)
where a is sound speed.

The above hammer equation is an integration of
dp/(rho a)+(-) du=0 along characteristic paths dx/dt=u+(-)a.
My previous post is an application of frictionless adiabatic flow of a perfect gas, constant specific heats and integration of the above differential equation. It is obtained from solutions of the conservation equations and generally known as solution by method of characteristics.

Integration of the above equations from the specified initial condtions will yield the initial pressure ratio that I have posted. If the actual pressure ratio is less, than the initial exit flow is not at M=1. And in time will build up as steady state approaches to M=1.

As I stated previously, there will be a flow within and at the exit for a short period of time at M>1. As time goes on the flow conditions (without friction ) will approach M=1 as steady state approaches.

Do not use steady state conservation equations for the solution of a rapid transient.

I suggest that if one wants the solution to a transient gas flow problem, s/he should refer to a text such as
Vol II written by Ascher Shapiro. An excellent example of the rupture described above is illustrated in his text.

Another text which includes the method of characteristics and is well described in it's chaper 8 is Introdution to Unsteady Thermofluid Mechanics by Frederick J. Moody.

Should anyone want references to ASME publications which also describe the transient as I have described it,

contact sailoday_28@yahoo.com