Losses through long annulus. 1

[dbecker]

Hello,

If I have an annulus that has a large ratio of annulus radius to annulus gap (meaning the gap is small compared to the radius) what is the method of calculating K?

I am assuming that K = f (L/D) where D is hydraulic diameter.

The problem I see though is that the flow area is not well represented with hydraulic diameter and this will be problematic if i use equation 3-20 of cranes because it requires a value for "D^4". And D is hydraulic diameter which may be many times smaller than what is represented by the actual flow area. Do you see what I mean?

Thank you!!

 

[moltenmetal]

Perry's Chemical Engineers Handbook refers to the hydraulic radius/hydraulic diameter concept but lists absolutely no caveats in relation to annular flow, unlike the Crane paper already listed.

Perry's claims that the use of the hydraulic diameter in the Fanning or Darcy equation F= 4fL/D is valid for ALL long, noncircular channels in turbulent flow, full or not (as long as variation in height is negligible).

Perry's makes no reference to how one is supposed to calculate the velocity, and hence the Reynolds Number, in order to calculate f. That's the crux of the issue.

Perrys lists its underlying references, but I don't have easy access to them: Brightton and Jones, J. Basic Eng. 86, p835-842, and Lawn and Elliot, J.Mech.Eng. Sci., 14, 195-254. Does anybody have or have access to these, to determine which velocity they're using? Or am I, as usual, missing something obvious?!

 

[25362]

Perry's ChE Handbook VIII stresses the difference between hydraulic and equivalent diameters when dealing with non-circular channels, saying that

the hydraulic diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter.

It also gives formulas for equivalent diameters for ellipses, rectangles and annulus, as well as references for regular polygons.

 

[moltenmetal]

...but makes no such distinction for TURBULENT flow, at least not that I've seen- and I re-read the section recently.

 

[25362]

Not so, for turbulent flow in rectangular ducts of large aspect ratio, the manual suggests the value of the friction factor from the Blasius formula for 4,000<Re<10⁵ to be increased by about 10%.

It also says that the [&Delta;]P_f in noncircular channels is a bit larger than the one estimated by the hydraulic diameter method due to secondary flows perpendicular to the axial flow direction.

This makes one wonder whether other methods like the Petroleum Method exemplified above wouldn't be underestimating [&Delta;]P_f values.

 

[moltenmetal]

If we're only talking about the 10-15% difference in the friction factors which IS noted in Perry's, then I'm perfectly happy to keep using the hydraulic diameter method. Remembering that these are all correlative methods, I have a dim view of their precision and design accordingly.

katmar: it sounds like that is what you've concluded, comparing the "petroleum method" to the hydraulic diameter method using the correct velocity, you get pretty good agreement. Did I sum that up correctly?

We're no longer talking about, as was implied in the previous discussion, a total non-applicability of the hydraulic diameter method to annuli- we're just clearing up the difference between effective/equivalent diameter of circular pipe, and the hydraulic diameter used in the Fanning equation. Correct? The two are clearly different.

As katmar points out, the correct velocity to use with the hydraulic diameter method is the actual velocity in the annulus, ie. the actual volumetric flow divided by the true cross sectional area of the annulus. The only non-circular channel example in Crane uses the actual velocity if I recall correctly.

If we're talking about really narrow annuli, it would appear that the system reduces to flow between wide parallel plates, which should be pretty well understood both in laminar and turbulent flow conditions.

 

[dbecker]

Thank you for the helpful information.

I have studied your responses.

I may end up using the Petroleum method.

The reason I CANNOT use hydraulic diameter is because i am calculating flow rate. I do not know flow rate at this point. I only know upstream, downstream pressure and pipe geometry (annulus).

Therefore, hydraulic diameter is not suited for this calculation.

Hydraulic diameter is better suited for calculated dimensionless numbers like Reynolds number for an annulus, because it is geared more towards how turbulent the flow is versus what the flow rate should be.

The flow rate of an annulus is not well represented by hydraulic diameter because of the simple fact that the cross sectional area of Dh is going to be very small compared to actual Dh, which will yeild smal flow rates when the annulus is short (low friction).

I do believe that when the annulus gap becomes smaller, we could consider that system as two parallel plates with width = 2 pi (R) and gap = radial gap.

I am going to run some calculations, stay tuned.

 

[katmar]

If you are trying to calculate the flowrate you will need to do it by trial and error - irrespective of whether you use the hydraulic diameter or the Petroleum Engineering method. You have to guess a flowrate and calculate the pressure drop, compare it with the known pressure drop, and repeat the calculation with an adjusted flowrate until the calculated pressure drop matches the actual. I described this process more completely in thread378-235298

Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[dbecker]

katmar,

I am aware of this.

I am designing software to do these iterations for me.

But before I can do that I have to make sure all my variables for equation 3-20 of Crane's make sense with what I am trying to calculate.


Thanks,