In Oil & Gas operations, it is frequently beneficial to flow fluids (gas, liquid, or both) vertically within an annular space. Some surface facilities are starting to see a late-life benefit from converting large, under-utilized flow lines to dual-service by pulling a second pipe through the large line (and flowing produced water on the inner line and gas in the annulus). There are many other examples in this and other industries, but in every case it is critical to be able to predict performance of the annular flow.
Either the vertical-flow or the horizontal-flow equations can be used to predict performance for annular flow, but both of these families of equations require the conduit inside-diameter. Since the annulus has a very odd surface area (and a large no-flow boundary), you have to be very careful determining an "effective diameter" (i.e., the round-pipe diameter that would perform like the annular space).
There are a couple of different definitions that give you very different answers.
The "Wetted Perimeter" method:
Area = (π/4)*(IDouter2 - ODinner2) WettedPerimeter=π(IDouter + ODinner)
EffectiveDiameter = 4(Area)/WettedPerimeter
This looks reasonable until you do the algebra and see that EffectiveDiameter really equals (IDouter - ODinner) at that point it is well worth discounting.
The "Petroleum Engineering" method (see Petroleum Engineering Handbook, edited by Howard B. Bradely, Third Printing, 1987, published by The Society of Petroleum Engineers, equation 49, page 34-27) uses:
The Petroleum method will give you effective diameters that are about 40% higher than the Wetted Perimeter method.
To use either one, calculate your effective diameter and then just use it in any of the fluid-flow equations that call for ID.
If you have dissimilar materials you'll have to decide on friction factors. An argument can be made to just use the friction factors of the outer pipe (since its surface area is so much larger of a contribution). Another approach is to average the friction factors (either simple average or a weighted average based on surface areas). Neither choice is wrong and either will result in pretty good correlations.
I've found that using the Petroleum method gives me excellent correlation to measured data in vertical flow.
The Wetted Perimeter method is used on the Professional Engineer exam and I've not found it to match measured data very well.