Fluid Flow Equations With Real Values

[shevir1]

Hi, all
I am working on a project that involves analyzing fluid flow behavior in oil & gas pipelines.
I am interested in calculating and plotting the velocity profile, and also(pressure distribution, temp distribution)

Having studies fluid mechanics last year I am aware of the EOS for fluid flow.
Ie continuity equation, energy equation , momentum equation.
My problem is, i dealt with the above equations using constants and unknown variables making it quite straight forward to derive.

I am trying to apply these equations to my project but I am getting a bit confused on how to approach setting up the equations, and my knowledge is a bit rusty on the subject. I was hoping anybody could assist me on the matter? I have randomly selected data from a case study that provides the
Volumetric flow rate : 0.00479m^3/s
pipe diameter: 0.406m
therefore velocity (v)=0.0420m/s
(also other more data which i dnt think is relevant for now)

Description of flow
-Horizontal pipe, single phase flow(for simplicity)
-Medium : water
-Steady state
-Incompressible
-2 dimensional

with the velocity above can anyone run through an example of how I would apply say the conservation of momentum. And moving onto plotting a velocity profile, and also pressure distribution profile.

Will appreciate any answers or suggestions.
Thnks

 

[shevir1]

see pic

 

[Chestermiller]

Why do you say that the flow is 2D? Is the flow laminar or turbulent?

 

[shevir1]

HI,
The flow is turbulent.
I say the flow is 2D because of the viscous effects the flow will face. Or is this example better carried out in 1D?

 

[Chestermiller]

HI,
The flow is turbulent.
I say the flow is 2D because of the viscous effects the flow will face. Or is this example better carried out in 1D?

Is the fluid velocity varying with both radial position and axial position in the pipe, or just radial position?

 

[shevir1]

Just in the radial position

 

[OrangeDog]

If you assume the flow is turbulent then your analysis will be intractable. Turbulence only exists in fully 3D systems. Actually, turbulence cannot exist in anything less than 3 dimensions. This is due to the Poinicare-Bendixon theorem:

Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. In contrast, for continuous dynamical systems, the Poincaré–Bendixson theorem shows that a strange attractor can only arise in three or more dimensions. Finite-dimensional linear systems are never chaotic; for a dynamical system to display chaotic behavior, it has to be either nonlinear or infinite-dimensional.

So from that paragraph alone you can see that assuming turbulent flow won't be helpful for finding an analytical solution. Emperical correlations or conducting your own experiments might be a really fun alternative to using the N-S equations directly. Additionally, large oil pipes usually have open channel flow inside them. The civil engineering section of a good fluids book will probably discuss that in more detail.

 

[shevir1]

Thank you for your response and recommendations, I initially chose the 2D flow to simplify the analysis and was pointed in the direction that 1D or 2D was most commonly used for fluid dlow analysis within pipelines.

 

[Chestermiller]

Thank you for your response and recommendations, I initially chose the 2D flow to simplify the analysis and was pointed in the direction that 1D or 2D was most commonly used for fluid dlow analysis within pipelines.

If you know the Reynolds number, you can determine the friction factor, and, from the friction factor, you can determine the shear stress at the wall, and, from the shear stress at the wall, you can determine the pressure drop. So you don't have to worry about the "Poinicare-Bendixon theorem," whatever the heck that is. We are going to be using the experimental correlation of friction factor as a function of Reynolds number (a dimensionless correlation based on actual experimental data) to get the pressure drop. So, what is the value of the Reynolds number for your flow?

Chet

 

[shevir1]

I see, this approach sounds more familiar.
I calculate a reynolds number of: 36686
this was found from velocity and diameter previously stated, and with density and dynamic visc for water at an elevated temp.

 

[Chestermiller]

I see, this approach sounds more familiar.
I calculate a reynolds number of: 36686
this was found from velocity and diameter previously stated, and with density and dynamic visc for water at an elevated temp.

Do you remember how to determine the friction factor, using either a graphical correlation or an equation?

 

[shevir1]

yes I recall by Darcy equation, but can this be used on turbulent flows.
I also noted other methods, such as blasius and moodys diagram but am a bit rusty on those.

By using Blasius equation I equate a friction factor of : 0.0057

which method would be best to implement/provide greatest accuracy?

 

[Chestermiller]

yes I recall by Darcy equation, but can this be used on turbulent flows.
I also noted other methods, such as blasius and moodys diagram but am a bit rusty on those.

By using Blasius equation I equate a friction factor of : 0.0057

which method would be best to implement/provide greatest accuracy?

This value is fine. The equation you used is 0.079/Re^0.25, correct? Now determine the shear stress at the wall.

(Just curious. How long ago did you have this Fluid Mechanics course that you no longer remember how to determine the pressure drop in a tube?)

 

[OrangeDog]

So you don't have to worry about the "Poinicare-Bendixon theorem," whatever the heck that is.

I took a class on nonlinear dynamics (Differential Equations II basically) and it was one of the most important things we learned in our semester. Rather than just say "turbulence doesn't exist in 2-d" I thought it would be useful to provide insight into why that is the case. I used Strogatz book, Nonlinear Dynamics and Chaos, for Diff Eq II. Essentially, the mathematics of a 2D continuous linear system will never be able to have anything like turbulence.

 

[shevir1]

Yes that's the equation.
Was last year. Sorry just wanted guidance on my approach to this situation. My main outcome was to find out how this distribution in pressure will look like as a plot analytically, not only to find the pressure drop.

 

[Chestermiller]

I took a class on nonlinear dynamics (Differential Equations II basically) and it was one of the most important things we learned in our semester. Rather than just say "turbulence doesn't exist in 2-d" I thought it would be useful to provide insight into why that is the case. I used Strogatz book, Nonlinear Dynamics and Chaos, for Diff Eq II. Essentially, the mathematics of a 2D continuous linear system will never be able to have anything like turbulence.

What the OP is trying to do is predict the pressure drop in a pipe for turbulent flow. As you correctly point out, this is a very intricate and complicated problem if one is trying to attack it by solving the transient Navier Stokes equations in 3D. However, over a century ago, engineers figured out how to circumvent this difficulty by making use of dimensionless correlations of actual real-world experimental data. The correlations apply to all viscous fluids at any flow rates. The only thing your original comment accomplished was to confuse the OP such that he almost gave up on solving his problem, which, in reality,can be solved rather easily using the dimensionless correlations.

Chet

 

[Chestermiller]

Yes that's the equation.
Was last year. Sorry just wanted guidance on my approach to this situation. My main outcome was to find out how this distribution in pressure will look like as a plot analytically, not only to find the pressure drop.

So, if you get your book out, you will see the rest of the steps required (actually, only two). The pressure should be a linear function of distance down the tube.

I don't think you will need any more help with incompressible flow, but we're here if you need us. For compressible flow, the integration of the force balance is a little different, and involves the square of the pressure. Your book should have that too.

Chet

 

[shevir1]

ok, will do so this evening. I will incorporate compressible flow at later stages of the project.
Will be in touch if i need any more assistance.

 

[OrangeDog]

What the OP is trying to do is predict the pressure drop in a pipe for turbulent flow. As you correctly point out, this is a very intricate and complicated problem if one is trying to attack it by solving the transient Navier Stokes equations in 3D. However, over a century ago, engineers figured out how to circumvent this difficulty by making use of dimensionless correlations of actual real-world experimental data. The correlations apply to all viscous fluids at any flow rates. The only thing your original comment accomplished was to confuse the OP such that he almost gave up on solving his problem, which, in reality,can be solved rather easily using the dimensionless correlations.

Chet

No need to be rude, and read more carefully next time:

Empirical correlations or conducting your own experiments might be a really fun alternative to using the N-S equations directly. Additionally, large oil pipes usually have open channel flow inside them. The civil engineering section of a good fluids book will probably discuss that in more detail.

 

[boneh3ad]

I took a class on nonlinear dynamics (Differential Equations II basically) and it was one of the most important things we learned in our semester. Rather than just say "turbulence doesn't exist in 2-d" I thought it would be useful to provide insight into why that is the case. I used Strogatz book, Nonlinear Dynamics and Chaos, for Diff Eq II. Essentially, the mathematics of a 2D continuous linear system will never be able to have anything like turbulence.

The "nice" thing about turbulence, however, is that while the full, unsteady velocity field can only truly exist in three dimensions, the average fields will fairly rapidly reach some steady state that is often 1-D or 2-D. Any treatment of the fluctuations can then typically be treated statistically if necessary. This is sort of the crux of turbulence modeling.

Strogatz is a great book, by the way. You can almost read it like a novel.

 

[OrangeDog]

The "nice" thing about turbulence, however, is that while the full, unsteady velocity field can only truly exist in three dimensions, the average fields will fairly rapidly reach some steady state that is often 1-D or 2-D. Any treatment of the fluctuations can then typically be treated statistically if necessary. This is sort of the crux of turbulence modeling.

Strogatz is a great book, by the way. You can almost read it like a novel.

I didn't like his writing style to tell you the truth. I suppose he did a good job writing an introduction given the subject matter is so complex. I originally wanted to research turbulence, but I discovered (ironically) through my "free time" investigations into the subject that I enjoy mathematical modeling in general.
The program I may enter (if I get information on funding - got accepted but still waiting for a second letter detailing TA/RA situation) will be nonlinear control of UAVs, which is fluids, engineering, and mathematical modeling all-in-one.