thermodynamics - How large would the steam explosion at Chernobyl have been?

So the second episode of the HBO series began to cover the risk of a steam explosion that led to them sending three divers into the water below the reactor to drain the tanks.

This occurred after the initial explosion that destroyed the reactor, and after the fire in the core had been put out. But at this point the decay heat and remaining fission reaction kept the core at more than 1200°C, causing it to melt through the concrete floors below the reactor.

And below the reactor were water tanks which contained 7,000 cubic meters of water (according to the TV show. If anyone has a real figure, I'd love to hear). When the lava of the melted core hit it, it would cause an enormous steam explosion.

Finally, my question: About how large would this explosion have been? The character in the show says "2-4 megatons" (of TNT equivalent, I assume). I'm pretty sure this is absurd and impossible. But real estimates are hard to come by. Other sources vary wildly, some repeating the "megatons" idea, and others saying it would've "level[ed] 200 square kilometers". This still seems crazy.

tl;dr:

I know a lot of it hinges on unknowns and the dynamics of the structures and materials involved, so I can simplify it to a constrained physics question:

Assuming 7,000 cubic meters of water instantly flashes to steam, how much potential energy is momentarily stored in that volume of steam occupying the same volume as the water did?

I don't know what to assume the temperature of the steam is. There were hundreds of tons of core material at temperatures near 1200°C, so worst case scenario you could assume all the steam becomes that temperature as the materials mix. Best case scenario, I guess we could assume normal atmospheric boiling point (100°C)?

Answer

In my view the water isn't really the thing to focus on here. The real energy reservoir was the partially-melted core; the water wasn't dangerous because it held energy, but rather because it had the potential to act as a heat engine and convert the thermal energy in the core into work. We can therefore calculate the maximum work which could conceivably be extracted from the hot core (using exergy) and use this as an upper bound on the amount of energy that could be released in a steam explosion. The exergy calculation will tell us how much energy an ideal (reversible) process could extract from the core, and we know from the Second Law of Thermodynamics that any real process (such as the steam explosion) must extract less.

Calculation

Using exergy, the upper bound on the amount of work which could be extracted from the hot core is

\begin{align} W_\text{max,out} &= X_1 - X_2 \\ &= m(u_1 - u_2 -T_0(s_1-s_2)+P_0(v_1-v_2)) \end{align} If we assume that the core material is an incompressible solid with essentially constant density, then \begin{align} W_\text{max,out} &= m(c (T_1 - T_2) -T_0 c \ln(T_1/T_2)) \end{align} where $T_0$ is the temperature of the surroundings, $T_2$ is the temperature after energy extraction is complete, and $T_1$ is the initial temperature. At this point you just need to choose reasonable values for the key parameters, which is not necessarily easy. I used:

• $T_1 = 2800\,^\circ\text{C}$ based on properties of corium


• $T_2 = T_0$ as an upper bound (the most energy is extracted when the system comes to the temperature of the surroundings)


• $T_0 = 25\,^\circ\text{C}$ based on SATP


• $c = 300\,\text{J/(kg.K)}$ based on properties of UO$_2$


• $m = 1000\,\text{tonnes}$ based on the text in your question.

This gives me $W_\text{max,out} = 6.23 \times 10^{11}\,\text{J}$ or 149 tonnes of TNT equivalent. This is several orders of magnitude lower than the "megatons" estimate provided in your question, but does agree with your gut response that "megatons" seems unreasonably high. A sanity check is useful to confirm that my result is reasonable...

Sanity Check

With the numbers I used, the system weights 1 kiloton and its energy is purely thermal. If we considered instead 1 kiloton of TNT at SATP, the energy stored in the system would be purely chemical. Chemical energy reservoirs are generally more energy-dense than thermal energy reservoirs, so we'd expect the kiloton of TNT to hold far more energy than the kiloton of hot core material. This suggests that the kiloton of hot core material should hold far less than 1 kiloton of TNT equivalent, which agrees with your intuition and my calculation.

Limitations

One factor which could increase the maximum available work would be the fact that the core was partially melted. My calculation neglected any change in internal energy or entropy associated with the core solidifying as it was brought down to ambient conditions; in reality the phase change would increase the maximum available work. The other source of uncertainty in my answer is the mass of the core; this could probably be deduced much more precisely from technical documents. A final factor that I did not consider is chemical reactions: if the interaction of corium, water, and fresh air (brought in by an initial physical steam explosion) could trigger spontaneous chemical reactions, then the energy available could be significantly higher.

Conclusion

Although addressing the limitations above would likely change the final upper bound, I doubt that doing so could change the bound by the factor of ten thousand required to give a maximum available work in the megaton range. It is also important to remember that, even if accounting for these factors increased the upper bound by a few orders of magnitude, this calculation still gives only an upper bound on the explosive work; the real energy extracted in a steam explosion would likely be much lower. I am therefore fairly confident that the megaton energy estimate is absurd, as your intuition suggested.