Question

A carpenter builds an exterior house wall with a layer of wood 3.0 cm thick on the outside and a layer of Styrofoam insulation 2.0 cm thick on the inside wall surface. The wood has k=0.080W/(m⋅K), and the Styrofoam has k= 0.010 W/(m⋅K). The interior surface temperature is 20.0 ∘C , and the exterior surface temperature is -12.0 ∘C What is the rate of heat flow per square meter through this wall?

Answer 1

We don't consider here heat transfer for the air .

a = 0.03 m thickness of the wood layer
kw = 0.08 W/mK thermal conductivity of wood
To = - 12°C = 261 K outside temperature

b = 0.020 m thickness of styrofoam
ks = 0.01 W/mK thermal conductivity of styrofoam
Ti = 20°C = 293 K inside temperature

Tx = temperature at contact point wood - styrofoam

The rate of heat transfer must be the same through the wood and styrofoam layers :

ks ( Ti - Tx ) / b = kw ( Tx - To ) / a

Tx ( kw/a + ks/b) = ks Ti/b + kw To/a

Tx = ( 0.01*293/0.020 + 0.08*261/0.03 ) / ( 0.08/0.03 + 0.01/0.020 ) = 266 K

Tx = - 7°C

The rate of heat flow per square meter is :

Φ = ΔT / R ; where R is total thermal resistance of the wall R = Rw + Rs

Rw thermal resistance of wood layer Rw = a / A*kw = 0.03 / 0.08 = 0.375 K/W
Rs thermal resistance of styro layer Rs = b / A*ks = 0.020 / 0.01 = 2.0K/W
A = 1 square meter

R = 2.375 K/W
ΔT = 20 - (-12) = 32 K

Φ = 32/ 2.375 = 13.47 W