A furnace wall is to be constructed of brick having standard dimensions 9 by 4.5 by 3

Question:

A furnace wall is to be constructed of brick having standard dimensions 9 by 4.5 by 3 in. Two kinds of material are available. One has a maximum usable temperature of 1900?F and a thermal conductivity of 1 Btu/(h ft?F), and the other has a maximum temperature limit of 1600?F and a thermal conductivity of 0.5 Btu/(h ft?F). The bricks cost the same and can be laid in any manner, but we wish to design the most economical wall for a furnace with a temperature on the hot side of 1900?F and on the cold side of 400?F. If the maximum amount of heat transfer permissible is 300 Btu/h for each square foot of area, determine the most economical arrangements for the available bricks.GIVENFurnace wall made of 9 × 4.5 × 3 inch bricks of two typesType 1 bricks Maximum useful temperature (T1,max) = 1900?FThermal conductivity (k1) = 1.0 Btu/(h ft?F)Type 2 bricks Maximum useful temperature (T2,max) = 1600?FThermal conductivity (k2) = 0.5 Btu/(h ft?F)Bricks cost the sameWall hot side (Thot) = 1900?F and cold side (Tcold) = 400?FMaximum heat transfer permissible (qmax/A) = 300 Btu/(h ft2)ASSUMPTIONSOne dimensional, steady state heat transfer conditionsConstant thermal conductivitiesThe contact resistance between the bricks isnegligible