Quintiere $9\mathrm{.}5)$ A thick polystyrene wainscotting covers the wall of a room up to $1$ m from the

floor. It is ignited over a $0\mathrm{.}2$ m and begins to spread. Assume that the resulting smoke layer in

the room does not descend below $1$ m and no mixing occurs between the smoke layer and the

lower air.

Polystyrene properties:

Density, r $=40$ kg$/$m$3.$

Specific heat, cp $=1\mathrm{.}5$ J$/$gK$.$

Conductivity, k $=0\mathrm{.}4$ W$/$mK$.$

Vaporization and ignition temperature, Tv $=400$ C$.$

Heat of combustion, Dhc $=39$ kJ$/$g$.$

Stoichiometric oxygen to fuel ratio, r $=3$ g$/$g$.$

Heat of gasification, L $=1\mathrm{.}8$ kJ$/$g$.$

Other information:

Initial temperature, T$=20$ C$.$

Ambient oxygen mass fraction, Yox,$=0\mathrm{.}233.$

Specific heat of air, cp $=1$ J$/$gK$.$

a$.$ The fire spreads to the top of the wainscotting, but does not spread laterally. What is the

burning $($pyrolysis$)$ rate of the polystyrene over this region assuming a steady state. The

vertical natural convection heat transfer coefficient, h $=15$W$/$m$2$K$.$

b$.$ Later the fire spreads laterally to $2$ m and the surface experiences a heat flux due to

radiation from the heated room of $1\mathrm{.}5$W$/$cm$2.$ Using the small B$-$number approximation, account for all radiation effects and calculate the mass loss rate per unit area of

polystyrene.