How Does Heron's Ancient Steam Engine Calculate Rotational Speed?

[Arcarius]

Homework Statement

Heron of Alexandria invented the steam jet engine in the first century A.D. One of his many inventions, the one shown below was invented for amusement but employs many concepts not again used until the 18th century.
A caldron with water in it was heated by fire and the steam generated was fed up and into a hollow spherical container with two spouts on each side. The exiting steam would spin the container at high speeds. We want to estimate the highest rotational speed using the little facts we have about this ingenious device developed almost two thousand years before it was rediscovered as the steam engine.
The spherical container has a radius of 0.2 m and mass of 10 kg. The two spouts can be considered massless but extend an additional 0.1 m above the surface of the container. The container is hollow and do not consider the moment of inertia of the steam contained inside.

a. What is the moment of inertia of this container?
b. If the pressure inside of the container reaches 1.5 atmospheres, what is the angular acceleration of the container? Take the area of the spouts to be circles of radius 0.010 m. Remember that outside pressure is 1 atm.
c. How much energy is stored by the steam inside the container?
d. If this energy were somehow completely converted into rotational kinetic energy, what would be the final angular velocity of the container?

Homework Equations

I = (2/3)MR^2 for spheres.
P = F/A
T = F * Perp Distance
T = Ia
E = 1/2Iw^2

The Attempt at a Solution

a. I = (2/3)MR^2 because the problem assumes that the two spouts are massless. This gives me 0.27 kg * m^2

b. P = F/A
F = (1.5 atm - 1.0 atm)(pi(0.010m)^2)
Convert 0.5 atm to Pascals --> 50,662.5 Pa.
F = 15.9 N

Net Torque = 2(15.9N)(0.30m) = 9.5 N * m = Ia
Angular Acceleration = 9.5 N *m / 0.27 kg *m^2 = 35 rad/s^2

c. This is where I ran into trouble. I don't know how I could calculate this. I only know E = 1/2(Iw^2) but I feel like this is only considering kinetic energy, and doesn't help me with the heat energy and such of the steam.

d. I can simply plug in the answer for c into the Energy equation to solve for d. However, I am stuck on c.


Any help is greatly appreciated!

 

[CWatters]

c) They only want to know how much energy is stored in the steam pressure. (eg ignore the thermal energy).

Energy = Δpressure x volume

d) Apply conservation of energy. They say to ignore the moment of inertia of the steam/water.

Sorry for all the edits to this reply.

 

[Arcarius]

c) They only want to know how much energy is stored in the steam pressure. (eg ignore the thermal energy).

Energy = Δpressure x volume

d) Apply conservation of energy. They say to ignore the moment of inertia of the steam/water.

Sorry for all the edits to this reply.

Thanks for the response!

So, for c I got E = (50,662.5PA)(4/3 * pi (0.20m)^3) = 1700 J

And for D:
1700 J = 0.5Iw^2
w = sqrt(3400J / I)
I = 0.27 kg * m^2
W = 110 rad/s