Average pressure in gas pipe 5

[DrCaddy]

Hi Firends,

This is my first post. Please tell me if made any mistake in the explanation of my problem.

There is a high pressure gas pipe of uniform CSA - fixed installation - carrying nitrogen from a cylinder to nozzle. I need to calculate the average pressure in a certain section AB of the pipe. The parameters that I have in hand are:

* The pressure at both the ends of section AB
* Temperature of the gas
* Design flow rate through the section AB

I assume that the average pressure is simply the average of the pressure at the two ends. I am a beginner in this area. So I am not sure if my assumption is correct. When I googled, I saw how to calculate the average pressure inside a container, but I think that would not suit my case.

Your advice on this is greatly needed.
TIA,
Caddy

 

[metengr]

I would say that your approach for calculating average pressure over length AB is reasonable for fluid flow. For most fluid mechanics applications, flow of a fluid (gas or liquid) is described by a very simplified version of Bernoulli's equation where

delta P/g + 1/2 velocity ^2/g + height = constant

Thus, to have flow thru a horizontal pipe (height = 0) of constant cross-sectional area, velocity of the fluid is related to pressure drop and density of the fluid.

Since you have the delta P, the average pressure over the length of section AB would be (Pa + Pb)/2

 

[DrCaddy]

Dear metengr,

I was not quite sure if the average of the terminal pressures will give the average pressure in the pipe. Since the system involves high pressure, I thought, the change in pressure over the length of the pipe might be non-linear and hence some complicated formula might be involved in the calculation. Thank you for the reply.

Regards,
Caddy

 

[TD2K]

As the pressure of the gas decreases due to flowing losses, the density will decreases and the velocity increases and the head loss per foot will increase (and there is another effect here from temperature). If you plotted the pressure loss down the pipe, it would increase the farther you get from the inlet which I believe is your concern.

IF the losses are however reasonably small you can neglect the effect I've just described and use the approach you asked about.

 

[JStephen]

It seems to me that any application that required the "average pressure" is probably also approximate enough that it wouldn't matter how you actually averaged it.

We're assuming you don't have a shock wave in the section in question, in which case, you'd have big variations in pressure.

 

[DrCaddy]

Thank you TD2K and JStephen for your ideas.

Shock waves are not a concern because the calculations are for the system in steady-state. Moreover the system does not have an external source to feed it other than its own nitrogen cylinders.

As TD2K said, I was concerned how the plot of pressure loss versus the distance from the inlet would look like. If the effect of temperature is neglected, I think the plot will be a straight line(I am not sure, though). Then one could use the average of the end pressures as the average pressure of the section. Otherwise the formula must be modified based on the shape of the curve.

The length of the pipe section could vary from just a few feet upto about 200 feet. I needed the average pressure to calculate the average density of the gas in that section. Since the pressure loss for longer pipe sections are not negligible, I thought it might be appropriate to clarify my approach to find the average pressure.

 

[zdas04]

When I work on problems that require density, I calculate the density at the head of the pipe and then at the tail. If they are "close enough" to each other I use the average, if not I find a different equation. "Close enough" depends on what you're trying to do. Some problems give you good enough answers if the values are the same in the second decimal place, others matter if there is a difference in the fifth decimal place. It all depends on the quality of your data (e.g., if you have a +/-5 psi gauge then taking density to 7 decimal places is silly), and the importance of precision in your answer (sometimes you want to know whether a volume flow rate is thousands of standard cubic feet per day or millions of standard cubic feet per day and other times it matters if you have 159 or 175 MCF/d).

David Simpson, PE
MuleShoe Engineering

http://www.muleshoe-eng.com

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips Fora.

The harder I work, the luckier I seem

 

[vzeos]

The correct equation to calculate the average pressure in a gas pipeline flowing under steady state conditions is:

P_avg=(2/3)*(P₁³-P₂³)/(P₁²-P₂²)

This equation is quite common in the natural gas industry, especially in the natural gas transmission field.

About 25 years ago, I looked up the original paper that introduced the Weymouth equation. I found that Prof Weymouth also derived the average pressure equation in the same paper. I guess you can call this the Weymouth Aerage Pressure equation . (If anyone is interested, the reference is: Problems in Natural-Gas Engineering: Trans. Am. Soc. Mech. Eng., vol. 34, 1912, pp 185-206.)

 

[metengr]

RGasEng;
Interesting information. Here is another equation that was referenced by AGA and generally accepted for determination of average pressure in a pipe, which accounts for non-linearity of pressure drop over distance;

Pavg = 2/3[P1+P2 - (P1*P2)/(P1+P2)]

 

[sailoday28]

zdas04 (Mechanical)states"It all depends on the quality of your data .....'I concur
Are the both end pressures measured?
OR is one pressure measured and the other calculated--If so, what formulas/assumtions are used for the calculation?
Are the temperature measured or calculated--If calculated what formulas, etc?

 

[vzeos]

metengr:
I have seen that equation before. Although I have not done the algebra in a long time, I believe that they can be shown to be algebraically equal.

 

[quark]

They don't seem to be algebraically equal.

a³-b³ = (a-b)(a²+ab+b²)

So, (P1³-P2³)/(P1²-P2²) = (P1-P2)(P1²+P1P2+P2²)/(P1-P2)(P1+P2) = (P1²+P1P2+P2²)/(P1+P2)

So the algebraic sign of P1P2 should be + rather than -

I wish I had the paper with me.

Regards,

 

[DrCaddy]

quark:
If you multiply and divide the right-hand side of the equation metengr wrote by (P1-P2) then you will come to the equation RGasEng has written.

Thank you everybody for the posts. Since the result of average pressure calculations are used for further calculation of other parameters of the system, I wanted it as precise as possible and at the same time trying to not go beyond what is needed. Since the latest equation cares for non-linearity also which is exactly what I was looking for and also is an accepted standard practice in the industry I am very happy.

 

[quark]

I am sorry, it is wrong on my part. I thought the denomnator was for entire RHS expression.

 

[25362]

Metengr's formula for the average has apparently been evaluated at 4/3 the arithmetic average less 1/3 the harmonic average between P₁ and P₂, as follows:

4/3 (P₁+P₂)/2 - 1/3 [2P₁P₂/(P₁+P₂)] =​

2/3 (P₁+P₂) - 2/3 P₁P₂/(P₁+P₂) =​

2/3 [P₁+P₂ - P₁P₂/(P₁+P₂)]​

 

[quark]

Can you be a bit descriptive? I am poor in statistics. Why 4/3rd of A.M and 1/3rd of H.M? what is the idea behind it?

Thanks in advance.

 

[25362]

To quark,

I got the results I mentioned by playing around with metengr's formula, having noticed the arithmetic mean (AM) and the harmonic mean (HM) of P₁ and P₂ are both involved.

The average pressure I found = 4/3(AM)-1/3(HM) may be (I'm hypothesizing) the result of applying numerical methods such as the trapezium rule or Simpson's rule to treat either experimental results or a friction drop formula and its curve.

 

[vzeos]

Attached is the derivation of the average pressure equation per Uhl, A. E., et al., Steady Flow in Gas Pipelines, Technical Report No. 10, Institute of Gas Technology, Chicago, 1965, pages 85 and 86.


For the operating ranges usually encountered in transmission practice; i. e., pressures below 1000 psia and temperatures of 60° F or higher, the variation of z with P is nearly linear, so that the compressibility factor may be represented by a function of the form

Z=1/(1+aP)

where a is a Constant dependent on T. The mean pressure (P_m) which must be used can be determined as follows:

(1/Z_avg)[∫]_P1^P2 PdP = [∫]_P1^P2 (P/Z)dP

or

(1+aP_m) [∫]_P1^P2 PdP = [∫]_P1^P2 P (1+aP) dP

Upon integration

(1+aP_m)(P₁ ² - P₂ ²)/2 = (P₁ ² - P₂ ²)/2 + a(P₁ ³ - P₂ ³)/3

or

P_m = 2/3(P₁ ² - P₂ ²)/( P₁ ³ - P₂ ³)

If full integration of the general energy balance is desired, however, for the sake of extreme accuracy or where a computer program is involved, the variation of P and T and the consequent variation of z along the line can be accounted for by graphical or numerical techniques. Graphical integration could be effected by plotting P/ZT against P, and subsequently determining the area under the curve between the initial and terminal values of P. An additional possibility would involve the use of an analytical or virial equation of state.

 

[vzeos]

Correction

Last equation should be:

P_m = 2/3(P₁ ³ - P₂ ³)/( P₁ ² - P₂ ²)

 

[sailoday28]

where a is a Constant dependent on T. The mean pressure (Pm) which must be used can be determined as follows:

(1/Zavg)?P1P2 PdP = ?P1P2 (P/Z)dP

Isn't the mean pressure the integaral of Pdx (limits x1 and x2) divided by x2-x1 where x is the length ?