Ratio of wetted perimeter to partial volume/area

[phosty]

I am looking for an equation which can relate the ratio of the wetted perimeter to the partial area (WPA) of a horizontal pipe part filled with liquid.

I am familiar with the usual partial volume equations and use them regularly however the starting point with these is that you typically know 'h' (height of liquid) and 'D' (internal diameter). Whilst I can simply determine the ratio WPA if I use trial and error for 'h' for a given 'D' it is not an elegant solution which can go into a spreadsheet cell. I have tried and failed miserably to rearrange the formulas in GPSA to give theta as a function of the liquid area and even MathCAD won't turn the equations around for me.

I have created a lookup table to do this in excel (attached) but again it is not elegant.

Is there a simple equation (or even a complicated equation which I could at least put in a spreadsheet cell)?

 

[25362]

Both are functions of the radius R and the central angle [θ] expressed in radians:

Area, A = R²([θ] - sin[θ])
Perimeter, P = 2[π]R([θ]/2[π]) = R[θ]

A/P = R[1- (sin[θ]/[θ])]

Besides, if needed:

the chord C = 2R sin([θ]/2)
the sagitta S = R[1-cos([θ]/2)]

I took these equations from old notes. I may be wrong, please check and confirm.

 

[phosty]

25362, many thanks for your reply. I actually already had these formulas - the problem is solving the first equation for theta so that one can then determine the perimeter. I did not know how to rearrange for theta=????.

The problem I am trying to solve is this:

knowing the volumetric flows of gas and liquid in a horizontal pipe of known diameter, assuming no slip (i.e. the flow area of the liquid is equivalent to the liquid volumetric flow fraction) determine the height of liquid in the pipe and thus the wetted surface area. This is then used in a fire scenario for blowdown.

I have found more info here

http://mathforum.org/dr.math/faq/faq.circle.segment.html

(Case 15) so I now understand that solving for theta can't be done explicitly and requires a numerical solution.

Using the example excel sheet in that link I was able to construct a single (monster) formula in excel that combined 3 steps of the numerical solution and thus give my height of liquid to sufficient accuracy.

Thanks again!

 

[25362]

A B/D condition seems to me not to be sufficiently (if at all) laminar to warrant such an assumption. Even with an horizontal flow in thelaminar regime the wetted perimeter may gradually drop due to [Δ]P or head differences, thus it appears the result obtained from a volume flow rate would rather be an average. I'd appreciate to have your enlightening comment.

 

[phosty]

You are correct but I am making some simplifying assumptions - immediately prior to blowdown, ESD valves will close. Therefore there will be no flow and the static mix of gas/liq will approximate to the ratio of the volumetric flowrates prior to ESD closure, with the wetted areas calculated as above. There is a plated deck so pool fire heat input via API 521 method is assumed which needs a wetted area (we are including pipework in the assesment).

I appreciate it is somewhat of a simplification - once the blowdown valves open the liquid levels will be all over the place with the turbulent gas flow. In addition any vertical legs would drain back into the horizontal legs. But I need a conservative approach and it was the most reasonable I could come up with?

But that aside, I was puzzled at how to turn the equations around to make them explicit in Theta. I now understand a numerical solution is required and it wasn't just my lack of math skills.