In order to illustrate the main variables of interest in the sodium fire representation, the following model is worth discussion. The model com­prises heat balance equations for the pool, flame, and the room, respectively, and mass balance equations for the sodium and the oxygen content of the room. The model is not a spatially distributed one; therefore the heat transfer from the room to ambient temperatures outside is necessarily crude. Nevertheless it does give reasonable results for large pools.

The pool loses heat to the vault walls at temperature Tv but gains heat from the flame at temperature Tt. Figure 4.26 shows the assumed configura-

+ See Hines et al. (33) and Humphreys (34).

Sodium pool Pool wall heat

mass Mp losses h4 (Tp-Tv)

Fig. 4.26. Model for calculation of containment pressures following a sodium pool fire.

tion of the pool lying in a pit at the bottom of the room with a flame above it. Heat transfer rates are shown between the various components. Thus the pool heat balance equation is

MpCp dTpjdt = h3(Tt — Гр) — Л4(ГР — Tv) (4.41)

The flame heat balance equation includes no heat capacity term, but it does include a heat production term based on the rate of burning of the sodium mass

— 6H dM^/dt = hi(Tf — Гг) — ht(Tt — Гр) (4.42)

The room heat balance includes heat received from the flame and that given up to the ambient temperatures T& outside the walls.

MA dTJdt = — Лх(Гг — rr) — h2(Tt — Г.) (4.43)

And finally the mass balance for oxygen depends directly on how much sodium is used; that is,

dM0Jdt = A dM^Jdt (4.44)

The mass of sodium burned depends on the concentration of oxygen, and it is proportional to the square root of the absolute temperature as shown.

This is an experimental correlation.

dMNJdt = —AcM0% (7V + 273.2)1/2 (4.45)

Finally the pressure in the room, according to the ideal gas law, is

Pt = k{Tt + 273.2) (4.46)

In the preceding equations, 8H is the heat of combustion (4850 Btu/lb), and c is a coefficient to make the units correct in the burning equation. The heat of combustion is based on an initial burning rate of 5 lb/hr-ft2 and has a value of 0.17-10~10 in cgs units. The heat transfer coefficients are all of the order of 10~4A’ cal/cm2-sec-°C, where A’ is the surface area of interest.

This model does not allow for the effect of a throat above the fire or for blanketing of the fire by the oxide the fire produces. It does not include time lags to account for nonimmediate mixing of the heat in the room or the pool, although this could be included by using a spatially distributed model. It naturally does not have enough nodes for more than illustrative accuracy. Nevertheless, very similar models are used in the sodium fire codes (see Appendix).

Figure 4.27 shows typical results that are obtained from such a model. After the fire is initiated by allowing contact between sodium and air, the temperature, and therefore the pressure in the room, rises rapidly to a peak where there is a balance between the heat input and that lost through the walls of the room. After this peak, due to lack of oxygen, the temperatures

slowly diminish. The time scale is long, the maximum pressure of 20 psia not being attained in this case for 5 hr. Also shown is a parametric case in which the room volume was doubled.

The pressures resulting from the sodium fire may in certain cases set the design pressures for the containment building (34) if nothing worse than this fire could be envisaged. The sodium and its combustion products may of course be slightly radioactive and the smoke is caustic.

The burning rate in the above model was based on a proportionality with the concentration of oxygen and the square root of the absolute temperature, as predicted by experimental work and theory. The square root accounts for the relative velocity of the sodium and oxygen molecules. If the reaction takes place in the pool, then the expression of Eq. (4.45) can be used, but if the reaction actually takes place in the flame, then the concentration of sodium molecules also ought to be included in the expression.

burning rate = c’AM0tM-si(Tt + 273.2)1’2 (4.47)

Some codes, instead of assuming a semiempirical burning rate of this kind, actually calculate the chemical balance at each point in time from the re­action between available sodium and oxygen molecules. This approach is more useful in spray calculations.