Climbing to hold?

[Nikker]

I know this question is probably very stupid since I didn't find any mentioning of this on the forum but here it goes:

So let's say you just performed a missed approach at destination aerodrome because runway is blocked( or whatever reason,not so important). You are the only arriving traffic at this airport for the foreseeable future. ATC tells you expected delay ~1h from now and asks intentions. You have ~2 hours of extra fuel because its tankering sector so naturally you request to hold.

Now the question is how high do you climb? ATC doesn't care.
Do you climb the altitude of IAF? Or higher? How much higher if so?

My instinct tells me that there's no reason to climb above altitude at which you commence the next approach,because amount of fuel you spend to climb those few extra thousand feet will kill all the savings. At the same time it is reasonable to assume that the higher we go the less fuel flow we are going to have and therefore more time in a hold is available before diversion. Is there a point when it becomes reasonable to climb higher? How to calculate?

Hopefully my question is clear.
Thanks in advance

[AerocatS2A]

There’s very little difference in holding fuel flow with altitude. Being high is good for going somewhere.

[Luc Lion]

At minimal power speed, the consumed power (and thus, to some extent, the fuel consumption) increases with decreasing density.
Thus climbing brings a double penalty.
The only benefit of altitude is the increase in engine efficiency,
... for power needs that also increase.
The savings brought by altitude is over a distance, not over time.

Vmp = [ (4/3) . (W/S)² . (1/rho²) . (1/CD) . (1/(Pi.e.AR)) ]^1/4
Or Vmp² = k . (W/rho)
And Pmp = Dmp . Vmp ~= [(½ . CD . rho . Vmp² . S) + (1/(Pi.e.AR) . W² . 1/(½ . rho . Vmp² . S)] . Vmp
Obviously, Dmp, drag at minimal power speed, is constant through density variations as Vmp² is inversely proportional to density.
Pmp = k’ . (W/rho)
Pmp increases with decreasing densities
(with Vmp=minimal power speed, W=weight, S=wing area, rho=density, CD=parasitic drag coefficient, 1/(Pi.e.AR)=induced drag coefficient, Dmp=drag at minimal power speed, Pmp=power at minimal power speed, k and k' indicate a constant value)

Note: What I wrote above is only valid for airplanes whose endurance is linked to power consumption (ie: piston and turboprop planes).
For jet airplanes, the endurance is related to thrust generation with max endurance matching the minimal drag condition.
As the thrust-specific fuel consumption (TFSC) is slightly decreasing with altitude, there is a small benefit in holding at a higher altitude if you don't have to consume more fuel to climb there. I tried the NASA jet engine simulator at https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/enginesimu/ and found that the change in TFSC, when keeping a constant IAS, more or less matches the thermodynamic efficiency variations derived from the inlet temperature and the turbine exit temperature.

[Capt Fathom]

Why not use the Climb, Descent and Holding charts and run the calculations for some different altitudes?

[AerocatS2A]

You can get a feel for the answer looking at just the holding tables. An A320 ceo at max landing weight burns about 2400 kg/hr at 1500’ vs 2200 kg/hr at FL250. A saving of a about 9%. You would easily burn through that climbing to altitude. OP might like to look at the tables for their own type of course.

[hans brinker]

With the difference in fuel burn per hour between holding low and high not significant I would make the decision based on the likelihood of the diversion. Think you are going somewhere else? Might as well start at cruising altitude. Think you will get in? Might as well be at approach altitude.