Question

An insulated wall consists of an exterior wood layer of thickness 3.1 cm and an interior layer of foam insulation of thickness 2.3 cm . For the wood, k=0.080W/(m⋅K), and for the foam insulation k= 0.010 W/(m⋅K). The outer surface of the wood is at a constant -15.0 ∘C , and the inner surface of the foam insulation is at a constant 19.0 ∘C .

Calculate the temperature in Celsius where the wood and foam insulation are in contact. T=???

Calculate the heat flow per unit area through the two-layer wall. H=?? W/m^2

Answer 1

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

a = 0.031 m thickness of the wood layer
kw = 0.08 W/mK thermal conductivity of wood
To = - 10°C = 258.15 K outside temperature

b = 0.023 m thickness of styrofoam
ks = 0.01 W/mK thermal conductivity of foam
Ti = 19°C = 292 K inside temperature

Tx = temperature at contact point wood - foam

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

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

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

Tx = ( 0.01*292/0.023 + 0.08*258.15/0.031 ) / ( 0.08/0.031 + 0.01/0.023 ) = 263.03 K

Tx = - 10.12°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 = 0.031 / 0.08 = 0.3875 K/W
Rs thermal resistance of foam layer Rs = b / A*ks = 0.023 / 0.01 = 2.3 K/W
A = 1 square meter

R = 2.6875 K/W
ΔT = 19 - (-15) = 34 K

Φ = 34 / 2.6875 = 12.65 W

Hope this helps :)