# Calculation of pressure variation due to fluid discharge

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1. Oct 31, 2018

### vepexu

Dear all,

I have a headache and I need you help, with the following problem.

I need to create excel sheet for the calculation (estimate) of the size of the orifice on the orifice plate that will produce a pressure reduction rate in pipeline at 0,5 bar per minute. It is not important to maintain pressure reduction rate during entire depressurization at 0,5 per minute, but it is essential that it starts at that level since it is maximum allowed.

I would like to create a template in MS excel for this problem.

Input data :
Steel pipeline with outer diameter of 0,1524 meters (m), wall thickness 0,012 m, pipe length 100 m, at depth of 300 m (seawater), pipeline is filled with seawater and pressurized to 150 bar.

Input data is just for reference and numbers are made up, I would like to know the methodology of solving this problem so I can create a template for calculation in the future.

2. Oct 31, 2018

### Staff: Mentor

Can you please provide a diagram?

3. Nov 2, 2018

### vepexu

Since it is not a issue right now and I'm preparing for when it might arise I have no detailed diagram to send.
It should look like example (very crude) below:

Pipeline has a diameter of D, heads can have a diameter equal or greater than D. Heads are considerably shorter than pipeline, for example Head length is 5 meters (m), pipeline length 50000 m. Heads are pressurized to the same pressure as the pipeline.

Let's say the pipeline will be depressurized from head 1 and on one of those side pipes orifice plate will be installed, by opening the valve fully pressure will be released through the orifice plate to ambient pressure of 30 bar (300 m depth).

Since we are talking only about estimate, we can disregard pressure losses across head equipment.

4. Nov 2, 2018

### Staff: Mentor

What assumptions are you making regarding the pipe expansion under pressure? Do you know how to determine the volumetric expansion of the water as the pressure is released? Is your main problem calculating the flow rate through the orifice plate as a function of the pressure drop across the plate? In one case, you have a 100 m pipeline and in the other case, you have a 50 km pipeline. The situations are somewhat different because of the speed of sound time delay. Which situation do you really want to analyze?

5. Nov 2, 2018

### vepexu

I started with the following expressions:

Pipe expansion is taken into consideration with above expression.
Volumetric expansion of liquid due to reduction in pressure is calculated with the following expression: V1 = V0 (1 - (p1 - p0) / E), but I have trouble incorporating it with expressions stated above, I assumed ΔV was needed reduction volume.
Problem is that the pressure reduction rate inside of the pipeline should not exceed 0,5 bar per minute, and since the valve for pressure release will be open fully from the start, orifice plate should restrict flow enough to ensure the maximum allowed rate of depressurization is not exceeded.
Since I'm trying to make a template (pipelines will range from 50m to over 50000 meters) for this calculation and it is an estimate, can we disregard the speed of sound time delay? Does it have a big influence on the overall results?

P.S. Thank You very much for showing interest in solving this issue.

6. Nov 3, 2018

### Staff: Mentor

I see what your problem is. From your second equation, we get the density of the liquid in the pipe as:
$$\rho=\rho_0+\rho_0\left[\frac{D}{EW}(1-\nu^2)+\frac{1}{B}\right](P-P_0)$$where $\rho_0$ is the density at $P_0=30$ bars. Your first equation can be expressed as $$q=AC_D\sqrt{\frac{2(P-P_0)}{\rho}}$$ (although usually, the $C_D$ is inside the square root sign). The mass flow rate through the orifice is $$\dot{m}=q\rho=AC_D\sqrt{2\rho(P-P_0)}$$The total mass of water in the pipe at any time is $$M=\rho \frac{\pi D_i^2}{4}L$$So, the rate of change of mass in the pipe is
$$\frac{dM}{dt}=-\dot{m}$$So,$$\frac{\pi D_i^2}{4}L\rho_0\left[\frac{D}{EW}(1-\nu^2)+\frac{1}{B}\right]\frac{d(P-P_0)}{dt}=-AC_D\sqrt{2\rho_0(P-P_0)}$$This assumes that all the fluid in the pipe is at the same pressure, which is an OK assumption for a fairly short pipe. However, for a long pipe (e.g., 50 km), that will not be the case.

7. Nov 5, 2018

### vepexu

I'm beginning to understand what was the issue.
Let's say we know how much cubic meters of seawater was necessary to fill the pipeline completely.
Now we start to pressurize the pipeline until desired pressure px is reached, and we know that we had to pump additional Vx cubic meters of seawater into the pipeline to reach desired pressure px, whatever it may be.
So we know starting pressure and volume of seawater inside the pipeline, we know final pressure and volume of seawater inside the pipeline, we know pressure outside the pipeline. Could all of this be used to simplify the calculation above and get the size of the orifice for a controlled discharge of seawater at 0,5 bars per minute?

8. Nov 5, 2018

### Staff: Mentor

The final equation already gives the result you are looking for the case of a short pipe. But, for the case of a 50 km long pipe, the pressure within the pipe will not be uniform, and the analysis will be more complicated, involving a traveling pressure/density wave with the fluid along the pipe.

9. Nov 5, 2018

### gmax137

What do your results look like when you plug values into the equation? As a sanity check, I did the calc using steam tables (ordinary water, I can't find my seawater tables). Converting to foot pound units (to match my steam tables), I get a contained volume (in your 50 km pipe) of 22,700 ft3. Assuming 40F temperature, the density at 150 barg (2190 psia) is 62.881 lbm/ft3; the density one minute later (at 149.5 barg or 2183 psia) is 62.879. Neglecting the pipe dimension changes, that means in one minute the orifice discharges 45.5 lbm or 0.72 ft3, or 5.5 gpm. The surrounding fluid at 300 meters is ~445 psia, so the delta-P is 1745 psi.

Are your results similar? That delta-P is pretty high for an orifice, it is going to wear out quickly. And, you might see choking in the orifice. Consider several orifices in series, or a small "drag" valve instead of an orifice.

10. Nov 5, 2018

### Staff: Mentor

For a 50 km pipe, the equation I derived does not apply. There will be a traveling expansion wave traveling along the fluid at the speed of sound (taking into account the pipe expansion also). Ahead of the wave front, the pressure will be unchanged. The equation I derived only applies to a short pipe.

11. Nov 6, 2018

### vepexu

Discharge manifold contains pipes with orifice plate flange and two valves specifically designed for each pipeline, so once the pressure is released, head together with manifold is removed so longevity is not an issue. One would even argue that theoretical fast wear would expand the orifice which in turn would cause less restriction in the flow and maintain pressure drop at x bar/min as pressure in pipeline decreases, but that is a question for an another day.
What is the issue is that I have some numbers received from 3rd party that are supposedly calculated from those 2 equations I gave in my post before, it was used for an actual pipeline and it worked as planned.
Here are the numbers changed a bit, I can't be too specific...
It was approximately a 900 km pipeline that was pressurized to 230 bar.
From my second equation ΔV/Δp was 40 m3/bar.
From first it was calculated that orifice diameter is 32 mm, for Δp = 0,16 bar/min.
ΔV/Δp I got easily from the second equation, but from the first equation or from Yours Chestermiller I can't get diameter of 32 mm. Not even close.

12. Nov 6, 2018

### Staff: Mentor

Like I said, my analysis applies only to a very short pipeline. For an (essentially) infinite pipeline (900 km is essentially infinite), the solution is more complex, involving speed of sound effects. If you would like to work with me on that problem, I would be glad to help.

13. Nov 6, 2018

### JBA

Just a comment regarding the effect of the intended purpose of the restriction in the pressure drop rate.

From a macro view, i.e. the entire length of the pipeline, the effect of the decompression wave is pertinent to the the rate of pressure decay; but, from a micro view, the decompression rate at the downstream portion experiencing the active pressure reduction, it would appear that the pressure retention beyond the decompression wave is irrelevant and the short pipe solution could still apply because that sector will experience the immediate pressure reduction upon the opening of the venting valve. So it would appear that which solution that is applicable depends upon the purpose of the rate restriction.

For example, if buckling of the pipeline due to a sudden drop in internal pressure were to be the issue, then the pipe sector nearest the discharge point would be the one to experience the immediate pressure drop and the delay of pressure reduction upstream of the decompression wave would be of little concern. On the other hand, if the reduction in the rate of pressure drop is intended to allow some upstream item, such as an isolation valve to react then the macro case would be more relevant.

14. Nov 6, 2018

### Staff: Mentor

I still think that the long pipe effect can be taken into account in a rather straightforward way and, if we are not satisfied with the short pipe accuracy, we should proceed with this.

15. Nov 6, 2018

### gmax137

Are these pipe dimensions fictional as well? I'm having a hard time imagining a 900 km pipe only 6 inches in diameter. Plus, 6.00 inch OD is not a standard pipe size. How fictional is the inside diameter?

16. Nov 6, 2018

### JBA

I agree that that, regardless of my comment, the long pipe analysis is required if a program inclusive of all pipe lengths is to be developed; but, that does not mean that the short pipe solution should not also an appropriate element of that program since both cases can simultaneously occur during a pipeline venting.

17. Nov 7, 2018

### vepexu

Thank You for offering help with additional calculations, but since we are working with really slow depressurization rates and flow speeds this estimate should suffice. If you have an itch and want to resolve it with infinite pipeline calculations, I will be glad to try and resolve it together but my knowledge of that kind of calculations is limited.

I tend to agree, but only when talking about an estimate. Since the idea is to start depressurization at the specified maximum rate of bar/min, and not maintain it we can observe local pressure drop and initial depressurization rate in short section of pipeline as a governing case.
I would be happy to discuss this.

Somewhat fictional, first set of data I gave in my original post is for a short section of the pipeline that will be installed in the future. And You would be right the 900 km pipeline is 42 inch, with wall thickness 29,6 mm in average.

18. Nov 7, 2018

### Staff: Mentor

I'm interested in the reason why the calculation using the equation I presented in post #6 was not even close. Can you please show the details of your calculation. Also, please describe how you used your first equation to get the rate of pressure decrease, including details of the calculation. Thanks.

19. Nov 7, 2018

### Staff: Mentor

I've done an analysis of the infinitely long pipe case, assuming release of compression in an inviscid fluid. See the figure below.

The analysis shows three regions:

The first region is outside the pipe, where the density and pressure are that of the surrounding sea water, at $\rho_0$ and $P_0$.

The second region is within the pipe, far from the orifice plate, where the pressure and density of the fluid have not yet changed from the original pressure and density of $P_i$ and $\rho_i$.

The third region is within the pipe, starting at the orifice plate. This region is one of uniform lower (reduced) density and pressure than the initial pressure and density, but higher pressure and density $P_r$ and $\rho_r$ than outside the pipe. . The pressure and density in this region do not change with time, but the region grows larger with time. The interface between the third and second regions propagates to right at the speed of sound in the system c, where $$C=\sqrt{\frac{B^*}{\rho}}$$with $$\frac{1}{B^*}=\left[\frac{D}{EW}(1-\nu^2)+\frac{1}{B}\right]$$
All the increase in volume resulting from releasing the pressure in zone 3 exits through the hole in the orifice plate. The analysis shows that the volume rate of flow out of the orifice plate is given by the equation $$q=\frac{(P_i-P_r)}{B^*}A_PC=\frac{(P_i-P_r)}{\sqrt{\rho B^*}}A_P$$where $A_P$ is the cross sectional area of the pipe. This volume flow rate must match that calculated from the discharge equation for the orifice plate, which is driven by the pressure difference between region 3 and the region 1 outside the pipe:
$$q=AC_D\sqrt{\frac{2(P_r-P_0)}{\rho}}$$where A is the area of the orifice plate. Therefore, $$\frac{(P_i-P_r)}{\sqrt{B^*}}A_P=AC_D\sqrt{2(P_r-P_0)}$$ This equation can be solved for $P_r$ to determine the pressure in region 3.

Basically, what this idealized analysis shows is that, after the orifice is opened, the pressure in the pipe region immediately adjacent to the orifice plate immediately drops to a new lower value, and then stays at that value for all time (or, basically, until the expansion wave reaches the far end of the pipe, hundreds of km away).

20. Nov 7, 2018

### Staff: Mentor

I made a calculation for the idealized model in the previous post. The inputs were 230 bars initial pressure in pipe, 30 bars outside pipe, hole diameter 32 mm. The calculated speed of sound in the water was about 1.4 km/sec. The calculated pressure for zone 3 came out to about 140 bars. The calculated discharge rate comes out close to 120 kg/sec, and velocities in the pipe in zone 3 come out close to 6.5 m/s. It looks like a 32 mm hole is too large for this case (unless I made a mistake in arithmetic).