Water boils in a paper cup over an open flame.

a. Ignoring radiative heat transfer, determine a numerical value for the maximum thickness of the bottom of the cup.

Data: Paper burns at \(250^{\circ} \mathrm{C}, \quad h_{o}\) (underside of cup) \(=50 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\)

\[k(\text { paper })=0.5 \mathrm{~W} / \mathrm{mK} \quad h_{i}(\text { inside of cup })=4000 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\]

\[T_{f}(\text { flame temp })=750^{\circ} \mathrm{C} \quad T_{b}(\text { boiling temp})=100^{\circ} \mathrm{C}\]

b. Explain carefully, why a large (rather than small) thickness will promote burning of the cup.

c. By introducing radiative heat transfer from the flame, would you expect a decrease or an increase in the numerical value for the maximum cup bottom thickness? Why?