*Post by Steve Harris *But take a look at the energy production rate. We can't figure the

first release lasted longer than maybe 5 seconds, so you're running

those 320 kJ out in 5 seconds, for a power of 64 kilowatts.

A lot of the power is produced in an initial spike, followed by a lower

plateau. However, looking at the graph of the plateau, 64 kilowatts

is pretty close to the graph at 1 second elapsed time.

That's

enough to melt the core if you let it keep up. The equalibrium temp is

the one at which the core loses that much in IR radiation from its

surface area of 0.022 m^2. So you need 2.9 megawatts/m^2, and

Stefan-Bolzmann gives you 2676 K for that, way above the melting point

of nickel. With outputs of 10's of kw, this is a meltdown situation at

equilibrium.

Are you sure that radiation is the dominant heat loss mechanism from

the core in that timeframe? These cores were sitting inside

hemispherical hollows in metal reflectors. The thermal contact may not

have been great, but there would be some conduction into that larger

metallic mass, delaying meltdown.

COMMENT:

At 2.9 MW/m^2 and 2676 K you're getting 64 kw out by IR, and to get

more out by conduction your surface total "heat transfer coefficient"

would have to be greater than 2.9 MJ/sec/m^2 divided by 2676 K = about

1084 watts/m^2/degree K of deltaT. This is a pretty high number. A

well-stirred water bath can go up to heat transfer coefficients of

5000 watts/m^2/degree K, but I doubt you could do this with simple

physical contact with most solid materials, and certainly not with a

non-metal. So yes, at these temps, radiation cools better than just

about anything but running water.

Now, this doesn't mean radiation was the dominant mode of cooling

through the temp range of the two prompt-critical Los Alamos

accidents, which I'm guessing was 300 K up to . For one thing, with

only the top of the assembly open, at best, the view factor for heat

radiation leak into the room would have to be less than 1/6th of

theoretical (think of basically only one open side of a cube, in this

case the top). There's no way of easily calculating heat leak across a

small air gap into WC bricksÂ—-- or at least I know of none (my guess

is that IR would quickly heat the surface of the reflector to sphere

surface temp, and then the heat sinking limit would be diffusivity of

a large amount of tungsten carbide from limited 0.02 m^2 contact

point, which I'm not going to be masochistic enough to attempt. And

finally the temp reached in the real accident was a lot lower if you

figure 320 kJ into an 800 J/K heat capacity Pu-239 sphere = +400 C. If

it got up to only 300 K room temp + 400 K for heating = 700 K, now

your max IR loss for an uncovered sphere is only 300 watts. And a

sixth of that is 50 watts. So yes, no doubt at *that* point, free air

convection and conduction through the bottom of the sphere into the

cradle is winning. But the temp is still going up fast.

The point is the sphere is generating 64 kw = almost 3 MW/m^2 and you

have to get rid of that somehow. Even with a stirred water bath and

its 5000 watt/K/m^2 heat transfer coefficient, you're going to get

calculated surface temp deltas of 2.9MW/5000 = 580 K. Which when

added to liquid water temp of at least 273 K gives you a minimum

sphere surface temp of 853 K, which of course is 580 C. Plutonium

melts around that temp, so if it's going to melt even in a running

water bath at ice temp, it's CERTAINLY going to melt at equilibrium

power output with the much worse conduction/convection conditions in

the Daghlian and Slotin accidents. Even if you subtract IR radiation

losses at 850 K (650 watts with viewfactor of 1/6 = 100 watts), you

still need to get rid of 2.8 MW, which gives a delta of 560 K between

the sphere and the "best" conductive bath you can imagine (which in

this case we didn't have), and that's still too close to Pu melting

point. So in the real world, it would have soon melted.

It's interesting what a little physics can tell you, eh?

*Post by Steve Harris **Post by Steven Sharp*In another posting I mentioned a Soviet accident that did reach

thermal equilibrium, lasting 6.5 days before being disassembled.

This involved a uranium core with copper reflector, stabilizing at

around 865 C and 480 watts. Those figures certainly support your

prediction of eventual meltdown for 10s of kw in a smaller core.<<

COMMENT:

Yes, indeed, and those number, imply a fairly insulated sphere, with

IR cooling very much surpressed. If we figure a critical mass U-235

uranium sphere has to be at least 20 kg, then the surface area is 0.05

m^2 or so. Theoretical max IR cooling at 865 C = 1138 k and this area

is 4754 watts, which is 10 times what you say was actually getting

out. If the sphere is enclosed sphere in a copper reflector, then the

whole thing heats up, and you have an even bigger surface area

somewhere at 1138 K trying to cool by radiation. You shouldn't be able

to get that hot. The only way to do this is a big temp gap between

sphere and reflector. The implied "heat transfer coefficient" for such

a sphere and whatever bath is cooling it, is 9.6 kW/m^2 divided by 845

C delta or so = 11 watts/m^2/K. That's lousyÂ—- about what you'd

expect from free air convection. So there must have been a bad air gap

between sphere and reflector, insulating it. If you had good thermal

contact between sphere and a piece of copper, then you'd see only a

small sphere-refelector temp gap, and equilibrium temp would be

limited by the radiation+convection contact resistance of this copper

reflector and the cool space around it, at equilibrium, and things

couldn't have gotten this hot. BTW, this implies the temp above has to

be the core temp-Â—it can't be the copper reflector temp unless they

have somehow insulated the reflector, too.

*Post by Steve Harris *I figured the total heat capacity of the sphere at around 800 J/K, so

at a power output of 64 kw you're getting a temp increase of 80 C/sec,

which is only +400 C over the course of the event, in worst case. Not

*quite* enough to melt the plutonium, but getting very close (within

another few seconds). I figure only 2 or 3 times the total energy of

this though WOULD have melted the sphere, nickel can and all, which

means that in both cases the prompt attentions of the operators

probably did prevent meltdowns.

The nickel canning was only 5 mils thick, so it wouldn't have

provided

*Post by Steve Harris *much structural support.

COMMENT:

Interesting. So no doubt meltdowns were very narrowly averted by just

seconds, in both cases.

*Post by Steve Harris **Post by Steven Sharp*As an aside, Daghlian may have made the situation worse in the short

term by moving back in, since his body would have provided additional

neutron reflection and moderation.<<

COMMENT:

Yeah, but if he had run away, look what this would have become. Melted

plutonium running like mercury through the gaps in the WC bricks

pilled around the assembly, and then in globs and splatters,

mercury-style, all over the floor. It least it would have killed the

reaction. Thank goodness OSHA, the EPA and Hazmat didn't exist. Can

you imagine...?

If Slotin historically used Daglian's intact core, as you say, then

that means if Daghlian had run like hell instead of pilling out the

brick, he would (perhaps) have saved not only himself, but later

Slotin too. It would have been such a mess after Daghlian with all

those plutonium splatters all over hell that probably nobody else

would have done that ever again.

Sharp

*Post by Steve Harris *If you are interested in reading the Los Alamos report yourself, I

can send it to you. It is a 4M PDF file.<

Yes, mail it to the "ix" address above. Thanks!!

Steve Harris