A composite wall, having unit length normal to the plane of paper, is insulated at the top and bottom as shown in the figure. It is comprised of four different materials $$A, B, C$$ and $$D.$$
Dimensions are $$$\eqalign{ & {H_A} = {H_D} = 3cm, \cr & {H_B} = {H_C} = 1.5cm, \cr & {L_1} = {L_3} = 0.05m,\,\,\,\,\,\,\,\,\,\,{L_2} = 0.1m \cr} $$$

Thermal conductivity of the materials are $$$\eqalign{ & {K_A} = {K_D} = 50\,\,\,W/mK, \cr & {K_B} = 10\,\,W/mK,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{K_C} = 1\,\,W/mK \cr} $$$
The fluid temps and $$HTC$$(see fig) are $$$\eqalign{ & {T_1} = {200^ \circ }C,\,\,\,\,{h_1} = 50\,\,W/{m^2}K, \cr & T{}_2 = {25^ \circ }C,\,\,\,\,{h_2} = 10\,\,W/{m^2}K, \cr} $$$

Assuming one dimensional heat transfer condition. Determine the rate of heat transfer through the wall.

A gas filled tube has $$2$$ $$mm$$ inside diameter and $$25$$ $$cm$$ length. The gas is heated by an electrical wire of diameter $$50$$ microns ($$0.05$$ $$mm$$) located along the axis of the tube. Current and voltage drop across the heating element are $$0.5$$ amps and $$4$$ volts, respectively. If the measured wire and inside tube wall temperatures are $${175^ \circ }C$$ and $${150^ \circ }C$$ respectively, find the thermal conductivity of the gas filling the tube.

An electric hot plate is maintained at a temperature of $${350^ \circ }C,$$ and is used to keep a solution boiling at $${95^ \circ }C.$$ The solution is contained in cast iron vessel of wall thickness $$25mm,$$ which is enameled inside to a thickness of $$0.8mm$$. The heat transfer coefficient for the boiling solution is $$5.5$$ $$kW/{m^2}K$$ and the thermal conducti-vities of cast iron and enamel are $$50$$ and $$1.05W/mK,$$ respectively. Calculate the $$OHTC$$ and the rate of heat transfer per unit area.

Steam is flowing through an insulated steel pipe shown in the fig, is losing heat to the surroundings. The details are as follows

Inner radius of steel pipe $$=50mm,$$
Outer radius of the steel pipe $$=57mm,$$
Outer radius of insulation $$=157 mm,$$
Thermal conductivity of steel $$=43$$ $$W/mK$$
Thermal conductivity of insulating material $$=0.1$$ $$W/mK$$
Heat transfer coefficient on steam side $$ = 570W/{m^2}K$$
Heat transfer coefficient on air side $$ = 12W/{m^2}K,$$
Temperature of steam $$ = {500^ \circ }C$$
Temperature of surroundings $$ = {30^ \circ }C$$

Calculate the heat loss per meter length of pipe and temp of the outer surface of the insulation.