How much ash enters the engines?

I looked for some calculations like this on the net – but found none.

Please comment with corrections if these are wrong.

Maximum concentrations of ash (mainly finely shattered volcanic glasses)are of the order of 300 µg/m³.

So how much ash does the engine turbines encounter? We have to calculate the volume of air passing through the engines.

A few facts:
In the standard atmosphere, at 11,000 m the pressure is approx. 1/5 of standard pressure (1 atm and the absolute temperature is approximately 4/5 of standard temperature.
At standard temperature and pressure, 1 mole of a gas occupies a volume of approx. 25 l. So at 11,000 m 1 mole of a gas occupies four times as much volume: approx. 100 l. Thus each m³ of the standard atmosphere at 11,000 m contains 10 moles of gas. Approximately 20% of this is oxygen – so we have 2 moles of oxygen (~ 60 g) per cubic metre, and 300 µg of ash.

Thus this air in these dense parts of the plume carries ~ 5 µg of ash per 1 g of oxygen.

We approximate the combustion of aviation fuel as
(CH₂)_2n + (O₂)_3n => (CO₂)_2n + (H₂O)_2n

So each 28 g of fuel requires ~ 96 g of oxygen for combustion.

In addition, only about 25% of the oxygen entering the engine is used in combustion – so for every 1 g of fuel burned, the engine must take in ~ 14 g of oxygen, which brings with it 70 µg of ash. Thus, in burning 1 Kg of fuel we bring 70 mg of ash into the engine, and for 1 tonne of fuel, 70 g of ash.

An Airbus burns around 2 tonnes of fuel per hour. So 70 g of ash is an upper limit for a half-hour encounter with a plume containing 300 µg of ash per m³.

This may not seem much. But imagine taking a can of epoxy to gum up the works of a delicate jet engine – 70 g of epoxy, a yoghurt carton-full, seems like enough to do plenty of damage to a couple of engines. Molten glass is probably more effective.