An insulated wall consists of an exterior wood layer of thickness 3.1 cm and an

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/(mK), and for the foam insulation k= 0.010 W/(mK). 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

Explanation / Answer

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 :

= T / 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
T = 19 - (-15) = 34 K

= 34 / 2.6875 = 12.65 W

Hope this helps :)

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