Severe Accidents would result with corium temperatures in the range [59] 3500 to 5000 K, for which Planck’s Black-body radiation spectra
[218,219] W(l) in Figure 5.6 has an effective waveband of 0 < l < 5 mm. Similar to neutrons, the probability of an interaction

13

between an EM wave and the atoms or molecules in a medium is found to increase with its density and thickness. Accordingly, it is reasonable to postulate that the absorption of thermal radiation in a steam film depends on the product of its pressure P and thickness L. Though Doppler broadening of absorption bands also occurs with increasing temperature, the decrease in density at constant PL can be expected to predominate. With these principles Hottel [219] successfully correlates seven different

X-ray studies [16] might have prompted the original conjecture.

independent data sets for the total emissivity (= absorptivity) of steam as a function of temperature by a family of curves indexed by constant values of PL. For example, the total absorptivity of a 100 mm thick steam film at 1 MPa is derived from his graph at T(K)as

a = -3.2 x 10—5(T — 1944)) + 0.055 (5.18)

More generally, Hottel’s correlation and data in Ref. [233] establish a largely insignificant absorption of thermal radiation in the steam film around an evolving coarse mixture. The total emissivity of a water surface is 0.96 [218], so it acts as a perfect absorber for present purposes. Over the waveband 4.58-6.47 mm the total emissivity of solid or liquid urania measures as 0.82 [220], and so it can be regarded as a gray emitter. On this basis the thermal radiation flux entering the interfacial liquid in Figure 5.4 is[84]

frad = гмs(T4MB — TLg) (5.19)

where

гм — Emissivity of UO2 = 0.82

TMB; TLB — Interfacial temperatures of melt and liquid (water), respectively

s — Stefan—Boltzmann constant = 5.67 x 10-8 W/m2 — K Because

TLB ~ TMB=10

then to a close approximation

frad = гмsT4MB = гм j W(l)dl (5.20)

0

where

W(l) — Planck’s radiation intensity function (in Figure 5.6)

The absorbed thermal radiation flux fab in a slab of incremental thickness Sz is generally specified by

A(1, Sz) — spectral absorption factor = 1 — exp (—a^Sz) al — spectral absorption coefficient

Measured values [221,222] of a(l) for water at 1 bar and 20 °C are graphed in Figure 5.7. As a result of Doppler broadening, increases in temperature of over 20-60 °C decrease the resonant amplitude at 2.95 mm by around 25% but pressure has little effect [223] because water is relatively incompressible. Table 5.4 provides values of the absorption factor A(1, Sz)for a range of wavelengths and water thick­ness (mm). In the waveband 0 to 2.5 mm, less than 5% of a radiation flux

„ 10°

T

e

Figure 5.7 The Absorption Coefficient of Water at 20 °C and 0.1 MPa

Table 5.4 Absorption Factor A(1,8z) as a Function of Wavelength and 8z in the Range 1-500 mm at Around 20 °C l(mm) A(1,1) A(1,5) A(1,10) A(1,50) A(1,500) 1.0 3.2E-5 1.6E-4 3.2E-4 1.6E-3 1.5E-2 1.5 2.5E-3 1.2E-2 2.5E-2 1.2E-1 7.1E-1 2.0 9.0E-3 4.4E-2 8.2E-2 3.6E-1 1.0E0 2.5 1.0E-2 4.9E-2 9.5E-2 1.0E0 1.0E0

is absorbed over the first 5 mm from an interface. For corium at 3500 K over 87% of the radiated power lies in this waveband [218] with an even larger fraction at higher temperatures [59]. Though the total radiant flux at 3500 K evaluates from equation (5.20) as 7MW/m2, it follows that less than 0.3MW/m2 can be absorbed in this crucial interfacial region. By comparison, Table 5.2 reveals an average molec­ular conductivity for steam that is well in excess of 200 mW/m — K, so the corresponding heat flux across a typical 100 mm film with a temperature difference of at least 3000 K is

fconduction0200 x 10—3 x (3000/10—4) = 6 MW/m2

Furthermore, over an incremental time 8t heat absorbed at a plane surface approximately diffuses the distance [224]

d ~ V4 a8t (5.22)

For water and a typical steam film destabilization period of 20 ms

awater = 0.16 x 10—6 m2/s so d = 3.6 mm (5.23)

Consequently the radiant heat absorbed by water beyond 5 mm can hardly influence film stability. It is therefore concluded that molecular conduction across a steam film is the effective stabilizing heat — transfer process. Several published simulations [198,225,226] of steam film destabilization omit the effect of a wavelength-dependent absorption coefficient on the radiant energy absorbed by the sur­rounding water.

Monatomic and symmetric molecules, like those of sodium vapor, undergo neither vibrational nor rotational transitions. Also symmetric molecules have no electrical dipole moment, so they can neither significantly absorb nor emit radiation by vibration or rotational bands. It follows that gases of such molecules are transparent to thermal radiation at low to moderate temperatures [218]. At high temperatures, however, these gas molecules can radiate or absorb appreciably by electron bound to bound or bound-free transitions. However, the asymmetric molecules of steam have all these degrees of freedom yet still absorb negligible amounts of thermal radiation. A fortiori, the same must be true for sodium vapor. For a good electrical conductor, like liquid sodium, the effective total absorption length lab can be estimated from the skin-depth equation [227] of EM wave theory

where

kGB — thermal conductivity of interfacial sodium vapor

@T

@x GB