Figure 4.13 shows the state of the system in its initial state of compression at point A. It has an internal energy of a Btu/lb. If it now undergoes an isentropic expansion all the way to 1 atm, then it finally arrives at a state with an internal energy of у Btu/lb.

Thus, in expanding, work has been done by the core on the surrounding material. From the first law of thermodynamics

work done dW = —dU = (a — y) (4.26)

This work is available to do damage to the structure around the core. The rest of the energy (y Btu/lb) remains in the fuel material to be released on a somewhat slower basis to the surrounding sodium through heat conduction. This residual heat forms a large proportion of the total energy release and following transfer to the sodium, which may vaporize, further damage can be done.

The calculation of the exact isentropic expansion process can be performed if generalized tables are available. However, this is not so for uranium oxide.

Hicks and Menzies derived an approximate analytical method (9):

work = U,— U,-U, In (UJUj) (4.27)

The available work is the change in internal energy minus the unavailable work during the irreversible change from state 1 to state 2.

work = U, — 17,— unavailable work during

the energy generation phase (4.28)

= 17,-17,- — S,) (4.29)

= U,-U,-T, ldQ! T (4.30)

Assuming that dQ = cv dT for a constant volume process, but that for a liquid cv is approximately cp and then using an average value of cp, Eq. (4.34) becomes:

work = U, — U, — cpT, ln(7’2/7’1) (4.31)

This expression is similar (17) to the Hicks and Menzies version in [Eq. (4.27)] but it does not assume that cpT, = U,. Actually U, = cp(T,— Tm) where Tm is the base temperature chosen as the fuel-melting temperature.