Page 43 – A Moment for Calculation

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The engine is driven by a pressure oscillation. But, the pressure is created from heat.

It is very easy to measure the pressure inside the engine. Sensors are very quick acting and
many readings can be taken per revolution. Temperature sensors by comparison, are very slow responders. They are only good for average temperature values.

So we can measure the pressure accurately and quickly. And pressure is distributed throughout the engine very rapidly. Not so with temperature.

Can we use the pressure to calculate the temperature with the pv/T = nR formula?

Yes, for certain moments of the cycle.

In the beta engine, when the displacer is at top dead centre, all the gas has to be between
the bottom of the displacer and above the piston. And cold.
Likewise, there is a moment when the displacer and the piston “kiss”.
All the air is above the displacer. .And hot.

Looking at the diagram of the sinusoidal motion curves of the bottom of the displacer and the
top of the piston, we can calculate the gas volumes at these two moments.
The vertical gap for cold gas at displacer TDC is 2.4. For the “kiss”, the hot gap is 1.7.
If we use a cross-sectional area of 1, then the volumes are in the ratio, hot to cold of 1.7 : 2.4

Now we use the fact that the engine has a constant amount of gas in it – at all times.
So, the term nR, the mass of gas is the same whether the gas is hot and pressurised or
cold and contracted.
…………………………………..p^hot v^hot / T^hot = p^cold v^cold / T^cold

but we know the volumes …... p^hot 1.7 / T^hot = p^cold 2.4 / T^cold

……………………and ………..2.4 T^hot / T^cold = 1. 7 p^hot / p^cold

From measurements of a 90mm atmospheric beta, the pv work diagram shows that
the pressure varies from 24 psi down to 8 psi (absolute)

Substituting…………..2.4 T^hot / T^cold = 1. 7 x 24 / 8

and the ratio of hot to cold temperature becomes ……. (1. 7 x 24) / (8 x 2.4) = 2.13

We can guess as to the temperature of the cold end. The cooling water does not boil.
It is less than 100C (373Kelvin). It is also above ambient say 20C (293K)
Lets settle for somewhere in the middle - 333K

Using the 2.13 factor, the hot end gas temp could be 333 x 2.13 = 709K
Or 436^oC

What’s the point of all this?

We have a measurement method to estimate the temperature of the hot gas.
We can measure – and then manage.

We can assess changes to the burner, or the hot cap finning.