A tubular furnace must be designed so that it can treat materials at an inner surface temperature of \(1200^{\circ} \mathrm{C}\) yet have its outer surface be at a temperature no more than \(45^{\circ} \mathrm{C}\). The inner layer of the furnace is made of a ceramic with thermal conductivity of \(0.5 \mathrm{~W} / \mathrm{mK}\). Around that we have a layer of high temperature insulation with a thermal conductivity of \(0.038 \mathrm{~W} / \mathrm{mK}\). Finally, the outer surface is stainless steel with a thermal conductivity of \(15 \mathrm{~W} / \mathrm{mK}\). The external environment is held at \(20^{\circ} \mathrm{C}\) and air movement about the furnace gives a heat transfer coefficient of \(10 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\). We can buy materials in any thickness we want but, the inner radius must be \(3 \mathrm{~cm}\) and the outer radius no more than \(35 \mathrm{~cm}\) :

a. Design a furnace that meets the requirements.

b. If the insulation can take a temperature of no more than \(1125^{\circ} \mathrm{C}\), how thin a wall can be used to meet the requirements?

c. Given the constraints above and in part (b), what wall will yield minimum heat flow?