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

 

[vzeos]

sailoday28 (Mechanical)

I don’t know why you would introduce length in calculating the average pressure. I don’t mean to be sarcastic, BUT(%-)) if you were calculating the arithmetic average, would it be (P₁ + P₂)/L ? I don’t think so. Uhl is taking a weighted average of the flowing pipe pressure. He introduced P_m via the simplified compressibility equation. (I think Prof Weymouth did it the same way.) Uhl then used the simplified compressibility equation with both P_m and PdP, integrated both sides and solved for P_m.

 

[sailoday28]

RGasEng (Mechanical)
If the pressure distribution is integrated over the length,
the resulting units are pressure*length units. If one divides by a length the resulting units are in terms of pressure. ---The mean pressure--

 

[vzeos]

sailoday28 (Mechanical)

You are indeed correct….. I did my last calculus more than 30 years ago, so I may not be of much help to you. I think to do what you are suggesting you need to specify P as a function of x before integrating. You could probably use the flow equation as your pressure distribution function but one purpose of evaluating the average pressure is to plug it into the flow equation to correct for elevation. I don’t know if your method would make this any easier.

 

[sailoday28]

RGasEng (Mechanical)Please refer to my prior post of June 25. If those questions are answered then the pressure distribution as a function of pipe length may be obtained and if necessary, numerically integrated.

 

[vzeos]

sailoday28 (Mechanical)

I have seen some of your posts on heat transfer and I have learned a great deal from them. I also saw your prior post in this thread and paid it little attention to it because it did not contribute to the original subject. I always try to help when I can but allow me to suggest respectfully that if you require answers to your questions or if you wish to pursue tangential issues you should start new threads.

 

[sailoday28]

RGasEng (Mechanical)The originator of the post stated"

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."
I am under the impression that you now agree that the avg or mean pressure involves integration of pressure over length. Further to have the pressure distribution,one then can use the formulas and assumptions used in obtaining end pressures. Where are the tangential issues?

Respectfully
soday28

 

[silouane]

every know the formula PV=nRT.
at the first stage, it is right the average of the two ends.
But, that can change if there is:
- temperature variation
- Speed variation
- Pipes dimension variation (bends or ventury parts)
- modification of the nitrogen.

the is no statistics on that. We can have an exact calcul.

 

[sailoday28]

Can someone give me a simple lesson on placing equations in a response?

 

[vzeos]

sailoday28,

There really is no way to insert equations per se. The equations that you see are formatted text. Eng-Tips uses the TGML compiler to format text. If you look at the bottom of the reply box, you'll see a check box that says Process TGML. Leave the check box checked but click on Process TGML. You'll get a menu of TGML formats. Formatting text is similar to writing computer code; just follow the rules and it gets easier with practice. In most cases you can copy and paste the format examples and insert your text. As an example, I formatted the Colebrook equation below to show what it looks like before and after it is processed. Note, in the before posting example, I substituted | for [ for illustrative purposes so that the TGML compiler would not process the formatting code.

Colebrook Equation before posting
1/|√]f= -2Log{(k|sub]e[/sub]/3.7D)+(2.51/N|sub]Re[/sub])(1/|√]f)}

Colebrook Equation after posting (replacing | with [ )
1/[√]f= -2Log{(k_e/3.7D)+(2.51/N_Re)(1/[√]f)}

 

[sailoday28]

Z=1/(1+aP) Pv=RT/(1+aP) v=RT/p/(1+aP)
1/v= P(1+aP)/RT

For isothermal process dP/v= P(1+aP)/RT dP (****)
integral of dP/v from P1 to P=
1/(RT)[ P^2/2+aP^3/3 ] (%%%%%)

Mean pressure
u velocity, x distance p pressure f friction factor
v specif volume D diameter G mass flux flow/unit area
subscirpt 1 conditions at pipe inlet
Dynamic eq. of motion
udu/dx + v dp/dx +fu^2/(2D)=0 (1)
conserv. of mass u=Gv (2)
Divide (1) by U^2 and substitute (2) into (1)

du/u + 1/v(G^2) dP +fdx /(2D) =0 (3)
after multiplying by dx

du/u from (2)= dv/v

and (3) upon integration with f a constant becomes with substitution of (%%%%%) becomes
ln (v/v1)+1/(RTG^2 [P^2/2-P1^2/2+aP^3/3-aP1^3/3
+ f(x-x1)/2/D=0 or

-ln[p(1+aP)]-ln[p1(p11+aP1)]
+1/(RTG^2 [P^2/2-P1^2/2+aP^3/3-aP1^3/3+ f(x-x1)/2/D=0

The above is difficult to solve for P vs x.
However pick P substitute and get x
Use numberical integ and mean pressure may be obtained.
Hopefully others will check for algebraic mistakes in above formulae.

 

[vzeos]

sailoday28
I'm a little too rusty to check for algebraic mistakes. But one question: How do you deal with a ?

 

[sailoday28]

Note:In my formulas, I believe my algebra is correct, however, I have been careless with some parenthesis. Also, the analysis is ISOTHERMAL.

To deal with a, there are charts of Z vs. reduced pressure (P/Pcritical) with reduced temp T/Tcritical as a varying parameter.

For 0<Reduced pressure< from 0 to about 0.8
along a constant reduce pressure line, the curves seem to be fairly straigt with a negative slope, ie "a" is minus

Reference, for example Gouq-Jen Su, Ind. Eng. Chem., 38, 802 (1946)
This refernce inclues Nitrogen Methane, Ethane, Ethylene Propane, n-Butane, Isopentane, n-Heptane, CO2, H20 AND
Reduced isotherms of 1, 1.1, 1.2, 1.5 and 2.

Regards

 

[DrCaddy]

Sailoday28:
Sorry for the late reply to your post on the 25th. I was trying to familiarise myself with the equations and calculations related to pressure. I think it might take me ages to learn them all.

Pressure at the inlet and density of the gas at the inlet are known. The flow rate of the gas is assumed to be constant and the value of flow rate is also available. Temperature is also assumed to be constant(Room temperature). I used the Darcy-Weisbach equation to calculate the exit pressure of the pipe.

With the above data in hand, I needed the average pressure in order to calculate the average density of the gas in the pipe section(Average density is needed in the Darcy-Weisbach equation). Since average pressure is dependent on the end pressures, one of which is already known, the Darcy-Weisbach equation will contain only one unknown(the other end pressure), and can be solved for that unknown. This was my theory.

But I did not realise that the compressibility of gas will make things so complicated. I need to mention that the nitrogen cylinders are custom-made high pressure nitrogen cylinders(usually 3000 psi, but sometimes upto to 4500 psi). I think even the sophisticated equtions of state are not suitable for this kind of situation(Please correct me if I am wrong). I am reading through books to see if there is some way out. If anyone could help

 

[sailoday28]

DrCaddy (Mechanical) Room temp of the nitrogen gives it a relatively high reduced temperature, since Tcritical is about 227 Rankine. The reduced pressure with pcritical approx 33.5 atmospheres is a possible concern.
I suggest looking into equations of state (eos)which are valid in the approximate range of temperatures and pressures of your interest.

Then the flow equations will have to be numerically solved--if the eos is to cubersome to handle with regard to integration, etc.

 

[vzeos]

DrCaddy,

What will your actual temperature range be?

 

[sailoday28]

The IR Compressed Air and Gas Data 2nd Ed. 5th printing
pg 34-36 illustrates Z vs P for N2. In the temp/press ranges that are specified, it appears that a linear curve fit is applicable with "a" being positive.

 

[vzeos]

sailoday28 (Mechanical):

See thread687-127478 (4 Jul 05 17:59), the ALLPROPS program may interest you. You can calculate compressibility factors.

 

[sailoday28]

DrCaddy (Mechanical),RGasEng (Mechanical)
With reference to
The IR Compressed Air and Gas Data 2nd Ed. 5th printing
in the range of 3000-5000psi, and room temperature, the compressibility can be approximated in the form
Z=ap^2+bP +c
Substiture into pv=zRT and solve for v
v=RT(ap+b+c/P)
Referring to my previous posts relating to the integral of v*dP or integral of RT(ap+b+c/p)dp
which yields RT(ap^2/2+bp+c ln(p) [you must include limits of upstream and downsteam pressures].

If further info is needed contact me at sailoday_28@yahoo.com