Fluid mechanics friction factor formula

[billybob70]

hopefully this will be my last question...

My textbook says friction factor = 64/reynold's number for laminar flow.

But it doesn't say how this equation is derived. does anyone know where this equation comes from?
is it an approximation (i.e. will be more accurate for certain Re's and less accurate at others), or is it exact?

Thanks.

 

[Q_Goest]

It's an empirical relationship that only holds for very low Re and is only an approximation. The approximation could be significantly off (ie: 10% or even more). I've seen some very large errors in pressure drop when these formulas are applied in real life.

 

[Clausius2]

It's an empirical relationship that only holds for very low Re and is only an approximation. The approximation could be significantly off (ie: 10% or even more). I've seen some very large errors in pressure drop when these formulas are applied in real life.

I don't think it is an empirical approximation. I remeber to have seen how to derive it for low Reynolds (Stokes) flow. I'll take a look at the derivation.

 

[Q_Goest]

Hi Clausius. I had to look that one up. You're absolutely right, for laminar flow the friction factor is actually a derivation from the Hagen-Poiseuille equation. http://www.transferencia.cl/transferencia/libro01/libro/ftmfv1_005j.doc" See the top of page 72.

It is only for turbulent flow that we need the emperical data. I had always assumed the entire Moody diagram was developed from emperical data. Now I know better, good catch!

 

[Clausius2]

Hi Clausius. I had to look that one up. You're absolutely right, for laminar flow the friction factor is actually a derivation from the Hagen-Poiseuille equation. http://www.transferencia.cl/transferencia/libro01/libro/ftmfv1_005j.doc" See the top of page 72.
It is only for turbulent flow that we need the emperical data. I had always assumed the entire Moody diagram was developed from emperical data. Now I know better, good catch!

Ahhh, just there. Thanx for the link.

 

[billybob70]

so you really think its an exact equation?
The reason i ask is that i was doing pipe flow simulations on flow lab for different Re's (different velocities, everything else the same).
the friction factor in the fully developed region was never quite equal to f=64/Re. the percentage of error was usually less than 1% for laminar flow. but if the formula was that simple, i would think the software could just plug in the numbers and get the right answer.

if friction factor formula is an exact equation, then the head loss must be an exact equation also, right?

i am thinking this is all related to kinematic velocity, and that some of the variables then cancel out. but i haven't studied this very much yet.

 

[FredGarvin]

The head loss is going to be \Delta p =f \frac{L}{D} \frac{\rho V^2}{2}

Well, I guess for the derivation, it's going to depend on how far back you want to go. Ultimately, the derivation for friction factor starts at equating Newton's 2nd law and the definition for a Newtonian fluid. I definitely could not remember all of this so I had to go back to my fluids book (Munson, Young and Okiishi). If I get all of the LATEX right on this it will be a miracle:

\frac{du}{dr} = -(\frac{\Delta p}{2 \mu L})r

\int du = -\frac{\Delta p}{2 \mu L} \int r dr

u = -(\frac{\Delta p}{2 \mu L}) r^2 + C_1

If you apply the BC's that u=0 at the wall and r=D/2, C_1 will then become

C_1 = (\frac{\Delta p}{16 \mu L})D^2 which is the centerline velocity of the profile.

Now the velocity profile can be rewritten as:

u(r) = (\frac{\Delta p D^2}{16 \mu L}) (1 - (\frac{2r}{D})^2)

The volumetric flowrate can be obtained by integrating across the pipe:

Q = \int u(r)2\pi r dr = 2 \pi V_c \int (1 - (\frac{2r}{D})^2)

Q = \frac{\pi R^2 V_c}{2}

The average velocity, V = Q/A so:

V = \frac{\pi R^2 V_c}{2 \pi R^2} = \frac{V_c}{2} = \frac{\Delta p D^2}{32 \mu L}

and to get pressure drop: \Delta p = (\frac{32 \mu L V}{D^2})

Now divide both sides by the dynamic pressure to get the dimensionless form

\frac{\Delta p}{.5 \rho V^2} = (\frac{32 \mu L V}{D^2}) (\frac{1}{.5 \rho V^2})

\frac{\Delta p}{.5 \rho V^2} = 64(\frac{\mu}{\rho V D}(\frac{L}{D}) = \frac{64}{Re}(\frac{L}{D})

Now, the one thing I can not comment on is the leap of reasoning to eliminate the L/D component for laminar flow. Perhaps Clausius can enlighten us there. However, you should now be able to see where that 64/Re came from.

 

[Clausius2]

The head loss is going to be \Delta p =f \frac{L}{D} \frac{\rho V^2}{2}
Now, the one thing I can not comment on is the leap of reasoning to eliminate the L/D component for laminar flow. Perhaps Clausius can enlighten us there. However, you should now be able to see where that 64/Re came from.

Nice job, Fred. But why do you want to eliminate L/D?? Compare your final formula with the first one you posted, and see that f=64/Re. In general, L is non equal D.

 

[FredGarvin]

Doh! It was staring me right in the face. Thanks.

 

[Q_Goest]

Hi Fred, nice derivation there.

Just a few thoughts. If I'm not mistaken, this is only for an incompressible fluid, correct? Alternatively, one can use this for situations where the change of density is small. The way to handle large variations of density is to break the flow up into smaller lengths of pipe where density variation is small and use the nominal fluid properties for each section.

Also, what seems interesting to me is that surface roughness is not part of the equation in the laminar flow range. I presume the single equation used for laminar flow makes the assumption that the walls are smooth. Thoughts?

Another thought that comes to mind is that fluid flow is generally turbulent, not laminar, so the friction factor taken from the Moody diagram (or Colebrook? formulas) requires emperical correlations. The emperical data I believe is mostly taken from water and air testing. From all this, we have a workable method of calculating flow and pressure drop. The software I use and similar software used in industry will generally use this information to calculate pressure drop and flow. Other correlations are used for two phase flow.

From what I've experienced, these formulas are no where near the 1% accuracy we'd like to have. For example, both myself and a few other engineers did a flow analysis on a 2 phase liquid helium line a few years back using a computer model, and all got about the same results. When the system was installed, it experienced about 40% less flow than calculated. I think it is important to understand what the assumptions and limitations are on any flow analysis. In the case of the helium line, the correlations derived from water, air and other data didn't seem to fit very well at all.

 

[FredGarvin]

Hey Q.
Yeah, that is all for incompressible flow. I can't say I have gone through the motions not keeping \rho constant. I can't imagine it would add too much more complexity, but then again, I haven't done it, so I can't really say for sure.

I agree that it's kind of funny that surface roughness never enters the mix in laminar flow. If I am not mistaken, it is because the inertial forces are so much more dominant than viscous that the effects of surface roughness are negligible. The actual curve for a smooth pipe on the Moody chart starts pretty far to the right at higher Re (about 4000 on my chart). The whole left side is the laminar flow range. I'll do a little more hunting around to see if I can find a definitive answer though.

You do also bring up another good point about the need for correlations. I don't know if it's my way with numbers or what, but I think the only time I have been that close to the expected was in a school lab. I wonder if that means I need some work on my calculations.

 

[Clausius2]

.
Also, what seems interesting to me is that surface roughness is not part of the equation in the laminar flow range. I presume the single equation used for laminar flow makes the assumption that the walls are smooth. Thoughts?

I think it is completely coherent. Maybe the problem is to understand what is laminarity. IFF the walls would have a non negligible roughness in laminar flow the flow would not be longer laminar, how can one write a parameter which destroys the theory?. Fluid magnitude variations in laminar flow only depend on MACROSCALES. As Re increases, the convection plays a more important role instabilizing flow field, depending magnitude variations on (Kolmogorov)MICROSCALES.
In fully turbulent flow f=f(\epsilon/D) only because inertia does not play any role, but an slight modification of wall geometry has an extraordinary effect on fluid magnitudes. The characteristic length scale induced by the wall rugosity affects eddy reflecting phenomena and dumping properties. As you may know, there are a couple of interesting boundary layers in turbulent flow: the boundary layer itself dominated by molecular viscosity, the buffer layer in which reflection-dumping of eddies is important, and the external turbulent layer dominated by the turbulent viscosity (Reynolds Stresses). The buffer layer has an extraordinary dependence on the wall geometry (if you have coped with CFD you will know about wall functions, Karman universal constant----not as universal as I thought after the Barenblatt seminar I attended a week ago---), and it is because the characteristic eddy length (Kolmogorov \eta microscale) is comparable to the characteristic rugosity. For that reason, an small variation on \epsilon would have a great impact in \eta, and this last dimension is vital in the development of a turbulent flow because it is the responsible of transporting and dissipating the kinetic energy into heat.
In laminar flow, the perturbations induced by wall rugosity are literally dumped by viscosity, because the flow (the convective terms) are stable enough for allowing such dumping. It is for that reason that \epsilon must not appear in any formulation of laminar flow. On the other hand, altering the wall geometry such that provoking boundary layer separation etc is made via macroscale alteration, and would have a great impact in flow, producing perhaps the transition to turbulence (the flow is unable to dump such a perturbation).
I think I have never talked about this, but this is a great place for doing so. The laplacian terms in N-S equations \nabla^2 \overline{u} come from the viscous stresses. These terms have an Elliptic-Parabolic behavior, such that are afraid about what is happening in all the flow domain, intercepting information of what happens both upstream, downstream, up and down. The viscous terms generate stability via diffusion-dissipation of perturbations. On the other hand, the convective terms \overline{u}\nabla\overline{u} are instable per se. These terms have an Hyperbolic behavior, such that they only transmit information along priviledged lines (characteristic lines). This terms are responsible of setting up shock waves and sharp discontinuities of fluid variables, because they don't intercept the information multidirectionally. As Re is increased the convectives terms become stronger. Ideally one may assume \mu=0 (Ideal flow) but that's not physically realizable. In fact the viscous terms are negligible at very high Re but --->not just zero<----. One needs to incorporate laplacian terms in turbulent flow besides an additional viscosity generated by turbulent shear stresses which come precisely from the convective terms. The molecular viscosity (the laplacian terms) gives an small dosis (but real) stability to the flow. Finally, when the flow is laminar, the laplacian terms enhance enough stability because they are not negligible compared with the convective instabilizers. Any microscopic variation is instantaneously dumped and dissipated by the laplacian term.

Hope everybody (included me) understood the big picture.
Javier.

 

[Q_Goest]

Clausius (or is it Javier) - I have a headache now... please send a shot of vodka.

Well, I think I could agree with the general atmosphere of your post Let me play that back to you in my own words: The point is that if Re is low enough to maintain laminar flow, any turbulence which might be generated near the walls due to surface roughness is dampened (you said "dumped" and a few other things) by the fluid's viscosity such that any minor turbulence, if it could develop, would remain at or very near the wall and not propagate into the flow stream.

That was basically what I had imagined when writing the question. Thanks!

 

[Clausius2]

Clausius (or is it Javier) - I have a headache now... please send a shot of vodka.
Well, I think I could agree with the general atmosphere of your post Let me play that back to you in my own words: The point is that if Re is low enough to maintain laminar flow, any turbulence which might be generated near the walls due to surface roughness is dampened (you said "dumped" and a few other things) by the fluid's viscosity such that any minor turbulence, if it could develop, would remain at or very near the wall and not propagate into the flow stream.
That was basically what I had imagined when writing the question. Thanks!

Sorry, I meant "dampened". Yeah, your conclusion and the express abstract of my post are right. I supposed it would be interesting to look at it from a mathematical point of view. Fluid Mech begineers usually read equations scared and without having any idea what do they mean. If some of those read this thread maybe we bring light to them.