Shear rate calculation 1

[quark]

I am doing some design work for batch concentration using ultra filtration. Just struck up with the shear rate calculation as the flow is turbulent. I am not ready with final pressure drop calculation as the process is still under trial and error method (of module selection).

So, assuming laminar flow, corresponding to the design velocity, gives the maximum shear rate and assuming maximum velocity equals twice the average velocity and profiling the velocity to drop to zero near pipe wall, I am calculating the maximum shear rate as V_max/r, where 'r' is radius of the pipe section. The objective is to check whether the shear rate is less than the recommended shear rate for our biological product. Suppose, the maximum calculated shear rate is less than the recommendation, is it OK to have such redundancy in the calculation?

Or do you guys have any thumb rules for turbulent flows?

 

[BigInch]

Not sure what you mean, but I think you might be referring to the transition range to fully turbulent flow???

For pipe flow the profile is still parabolic around Re = 2,000 and transitions to fully turbulent somewhere around 8,000 to 10,000. The actual points of the beginning and the ending of transition flow usually need to be verified experimentally, if high accuracy is needed.

http://virtualpipeline.spaces.msn.com

 

[Latexman]

If the fluid is Newtonian:

d[γ]_wall/dt = 4Q/([π]R³)

If it's non-Newtonian, it depends on the rheology model you are using. For the most common model, the Power-Law, it is:

d[γ]_wall/dt = ((3n + 1)/n) x Q/([π]R³)

Reference: Isothermal Flow of Non-Newtonian Fluids in Pipes, Martin H. Wohl, Chemical Engineering Magazine, April 8, 1968, page 143-146.

Good luck,
Latexman

 

[Latexman]

As for extending the above laminar flow equations to turbulent flow, most literature takes the out that "there is a viscous sublayer at the wall in which the effects of turbulence is negligible. Viscous shear predominates and eddy diffusion, if present, is minor."

Good luck,
Latexman

 

[quark]

I am sorry for the ambiguous post. I am actually trying to calculate shear rate in UF membranes (ceramic). The recommended velocity is from 3 to 6m/s (for self cleaning purpose) and there are some restrictions to the maximum allowable shear rate. The flow is essentially turbulent but I don't have the required data to calculate the shear rate as the selection is still under trial and error method.

So, I am presuming that the flow is laminar (first redundancy in calculating the actual shear rate) to do the calculation. Secondly, I am assuming that the velocity profile is triangular (this give me further redundancy over parabolic profile as the slope is steeper). Now the shear rate(du/dy) is simply (maximum velocity-0)/(0-r) or -V_max/r.

My question is, will this be the maximum possible shear rate occuring in a pipe of fixed diameter and with a fixed velocity?

 

[BigInch]

Still a bit nebulous.

Since you have assumed a triangular velocity profile, your shear is highest at the wall and the shear rate is thereby maximum and constant with r, consistant with your assumption.

Considering the distinction between shear and shear rate in other velocity profiles, the shear would be max at the wall, but not necessarily the shear rate, where the transition between 0 or very low wall flow velocities to interior velocities could occur over a very small change in r, giving a high change in shear rate.

At those velocities, I wouldn't think this case is laminar flow and the greatest change in velocity with change of r may be higher elsewhere. A nonuniform temperature profile may drastically affect the velocity profile as well, depending on how viscosity varies with temperature. With a warm fluid entering a cool pipe it might be possible that laminar flow exists close to the wall, but turbulent flow at interior locations, or visa versa.

Are we getting any closer?

http://virtualpipeline.spaces.msn.com

 

[quark]

Yep, partly. The temperature of the fluid is pretty constant. The reason why I averaged out the shear rate is because that gives you the maximum slope. For example, the max. velocity at the center is 3m/s and at pipe wall it is zero. So, at half radius, the assumed value can be 1.5m/s. If I consider a parabolic profile, since the slope is lower (than that of a triangle), velocity at half radius can be higher than 1.5m/s and thus shear rate should be low.

 

[BigInch]

But ... shear rate is d_Velocity/d_Radius, no? So in the extreme it might get pretty high as you move from the wall flow layer (velocity ~0) to an interior layer over a very thin transition thickness between the two (dR).

http://virtualpipeline.spaces.msn.com

 

[quark]

Oh... you are talking about the laminar sublayer in case of turbulent flows. But I think I can ignore it as the cross membrane fluid is the reject stream (that is the fluid crossing the pipe wall). Nevertheless, you have a point here. I will see what should be done when the selection of the membranes is completed. Thanks and have a nice weekend.

 

[Latexman]

With a little substitution and manipulation, the Newtonian equation I gave above can be rearranged to:

d[γ]_wall/dt = 4V_ave/r

This agrees with the equation I use most of the time:

d[γ]_wall/dt = 8V_ave/d

For laminar flow, V_max = 2 x V_ave, so:

d[γ]_wall/dt = 2V_max/r

I believe you are off by a factor of 2 if you assume the laminar equations are representative.

Also, give some thought to using the experimental data of Rothfus, Archer, Klimas, and Sikchi which is plotted in many, many reference books, like Figure 5-14 in Perry's ChEs' Handbook (mine is the 6th Ed.) or Figure 5-7 in McCabe and Smith's Unit Ops of ChEing (mine is the 3rd Ed.), to estimate your V_max in turbulent flow. Then, use the above laminar flow V_max equation to calculate your d[γ]_wall/dt. Putting a turbulent flow maximum velocity into a laminar flow maximum velocity equation does carry some implied assumptions, so it needs to be thought out first.

Good luck,
Latexman

 

[btrueblood]

I'm having trouble with the triangular profile giving you a higher shear rate. The term du/dy will be greater (steeper gradient) at the wall for a parabolic distribution (laminar profile) than for linear distribution.

 

[Latexman]

btrueblood is right! In addition to what he said about the triangular velocity profile, the velocity distribution for turbulent flow is more spherical near the axial centerline and du/dy near the wall is greater than those for laminar flow. So, it appears a laminar flow assumption will be closer to reality than a triangular velocity profile, and a turbulent flow assumption will be closest of all, if we could just figure out how to easily apply it.

Based on this, the last paragraph in my previous post is rubbish. Plugging a turbulent maximum velocity into an equation derived for a laminar flow parabolic velocity profile is just not right the more I think about it. Sorry about that

Good luck,
Latexman

 

[quark]

Thank you for the input and replies. I couldn't reply in time as I was away from the office.

btrueblood,

You are right and a triangular profile is not correct even for laminar flow. Perhaps, I was trying to make things too simple (I think, I was looking only at the vortex of the parabola and some portion down to that).

Latexman,

I did refer McCabe&Smith before posting the question. My Reynold's number was a bit smaller than those for which the graphs were made, but I was safe as per charts. I tried some emperical relations based on total pressure drop and the pipe length. These values put me under safe cover but those are average values.

Various methods, checked by me and suggested by you guys increased the confidence level of the calculation.

Interestingly, the take of McCabe & Smith on laminar sublayer seems to be only limited to academic interest and not a practical one. However, I have to bother less, in my case, as there is a cross flow across the membrane.

BigInch,

Thanks. You pointed out right from the beginning but I overlooked it.

Thank you all once again.