A hydrogen bomb may be approximated by a fireball at a temperature of 7200 K according to a report published in 1950 by the Atomic Energy Commission. (a) Calculate the total rate of radiant-energy emission in watts, assuming that the gas radiates as a blackbody and has a diameter of 1.5 km, (b) If the surrounding atmosphere absorbs radiation below 0.3 determine the per cent of the total radiation emitted by the bomb that is absorbed by the atmosphere, (c) Calculate the rate of irradiation on a 1 m2 area of the wall of a house 40 km from the center of the blast if the blast occurs at an altitude of 16 km and the wall faces in the direction of the blast, (d) Estimate the total amount of radiation absorbed

Question

assuming that the blast lasts approximately 10 sec and that the wall is covered by a coat of red paint, (e) If the wall were made of oak whose flammability limit is estimated to be 650 K and that had a thickness of 1 cm, determine whether or not the wood would catch on fire. Justify your answer by an engineering analysis stating carefully all assumptions.



GIVEN

- A hydrogen bomb fireball

- Fireball temperature (T1) = 7200 K

- Surrounding atmosphere absorbs radiation below 0.3



- The blast occurs at an altitude (H) of 16 km = 16,000 m

FIND

(a) The total rate of radiant-energy emission in watts (qr)

(b) The per cent of the total radiation absorbed by te atmosphere

(c) The rate of irradiation on a 1 m2 area of the wall of a house 40 km (40,000 m) from the center of the blast and facing the blast (G2)

(d) Total amount of radiation absorbed if the blast lasts 10 seconds and the wall is covered with red paint

(e) If the walls are oak with a flammability limit of 650 K and a thickness (s) of 1 cm, will the wood catch fire?

ASSUMPTIONS

- The gas radiates as a blackbody

- Diameter of the fireball (D) = 1.5 km

- The air and surrounding temperature (T?) = 10°C

- The surroundings behave as a blackbody enclosure

- The heat transfer from the oak walls to its surroundings during the 10 seconds of irradiation can be neglected

- The house wall is initially at the surroundings temperature

SKETCH



PROPERTIES AND CONSTANTS

the Stephan-Boltzmann constant



the emissivity of red paint at short wavelengths



the emissivity of red paint at long wavelengths



Specific heat



Thermal conductivity



Density



Thermal diffusivity

Answer 1

(a) The total rate of radiation emission is the blackbody emissive power times the area



(b) For

The fraction of energy absorbed is the fraction, e of the total radiation below 0.3 ?m which can be read directly



(c) The distance between the house and the fireball center (L) is



The energy calculated in part (a) will spread evenly in all directions from the fireball. Therefore, the flux at the distance L = q1/AsL where AsL is the surface area of a sphere of radius L



However, the atmosphere will absorb 9.4% of this energy.



The angle between this flux and the (normal to the) wall surface, ?, is given by



Therefore, the irradiation on the wall is



(d) By Kirchoff’s law, the absorptivity

Energy absorbed=
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(e) The radiative heat transfer coefficient (hr) is given by



Since Ts << Tf, the heat transfer coefficient will not vary much as the Tf changes. To estimate hr, let Tf = 500 K



The Biot number for the wall is



Therefore, the internal thermal resistance of the oak is significant and the chart solution

will be used to estimate the surface temperature of the oak after 10 seconds: (L = s/2 =

(0.01 m)/2 = 0.005 m) The Fourier number is





(The center of the oak is still at the initial temperature after 10 s).

(b) for x/L = 1.0



where T(L,t) = the surface temperature of the wall after 10 seconds of exposure to the radiation.