Final storage containing spent HTR fuel elements

[Lucky mkhonza]

can anyone help me with the following problem: I have been asked to estimate the rise in temperature in the salt in a final storage containing spent HTR-fuel elements in a time span of 100 years. Given the following data: Average heat flux 10 W/m; Heat capacity of salt 0.9 kJ/kg.K; density of salt 2100 kg/m3; distance between storage vessels in salt S=30m.

What I know is that the following relation might in solving this problem (not sure if is correct)
$$\int_{\tau_1}^{\infty}{P_D(t)}dt = \tau.S^2.\rho.C.\Delta T$$

Thank you in advance

 

[Astronuc]

Are you assuming an infinite array of storage containers?

One can then assuming a square cell of salt with symmetrical BC (e.g. q" = 0), with a cylindrical storage container (heat source).

 

[Lucky mkhonza]

Astronuc "Are you assuming an infinite array of storage containers?"

The problem does not state whether we have an infinite array of storage containers or certain number of containers. storage containers are in cylindrical shape.

 

[Astronuc]

Yes, I was proposing one approach to obtain the boundary conditions - and it would represent the worst case for an individual container.

Think about it. If it is one container in a 'very large' salt deposit, then one has a single heat source an infinite heat sink. That's fairly easy to deal with. One has a cylindrical heat source, with a heat flux in an infinite medium of some temperature. There will be radial temperature profile in the salt around the container.

Back to the equation:
$$\int_{\tau_1}^{\infty}{P_D(t)}dt = \tau.S^2.\rho.C.\Delta T$$

The integral of power over time is the energy produced during that time, in this case from time of insertion to infinity. The terms on the right hand side do not produce the units of energy.

$\tau$ = time, s
$S^2$ = area, m²
$\rho$ = density, kg/m³
$C$ = specific heat, J/kg-K
$\Delta T$ = temperature, K

the product on the right hand side has units of J-s/m.

So let's look at the integral. It represents some amount of energy produced over a long period of time, and since the source is radioactive, it will be a decaying exponential, more or less. (In reality, the thermal source could be increasing for a while because the decay of Pu-241 to Am-241 to U-237 into Np-237, and both Am241 and Np237)

Anyway, one can take that energy and put it into the surrounding salt. Then there is the relationship between energy (enthalpy), specific heat and temperature.


On the other hand, part of problem statement is "distance between storage vessels in salt S=30m" - so consider the case where there are 'many' similar containers. Then one container, assuming a square lattice is surrounded by 8 containers, and so on, or rather a large array of containers. The surrounding containers are also raising the temperature of the salt formation, and the salt surrounding the containers heats up.

One could assuming 'tall' (or stack of) containers, and simply make this a one dimensional radial heat conduction problem. There does not appear to be any height dimension given, so one could assume one meter of container heats one meter of salt. There also does not seem to be a radial dimension of the container.

Perhaps one is to assume one dimensional Cartesian coordinates, i.e. the problem involves a semi-infinite slab, 30 m thick with a heat flux at the surface, which is decreasing in time. If the heat flux is constant, then the amount of heat continually increases and the temperature would continually increase - to infinity - which is certainly not realistic.