A carpenter builds an outer house wall with a layer of wood 3.0 cm thick on the

Question

A carpenter builds an outer house wall with a layer of wood 3.0 cm thick on the outside and a layer of styrofoam insulation 2.2 cm thick as the inside wall surface. The wood has k = 0.080 W/m middot K and the styrofoam has k = 0.010 W/m middot K. The interior surface temperature is 19.0 degree C, and the exterior surface temperature is -10.0 degree C. a) What is the temperature at the plane where the wood meets the styrofoam ? b) What is the rate of heat flow per square meter through this wall ?

Explanation / Answer

a = 0.03 m thickness of the wood layer
kw = 0.08 W/mK thermal conductivity of wood
To = - 10°C = 263 K outside temperature

b = 0.022 m thickness of styrofoam
ks = 0.010 W/mK thermal conductivity of styrofoam
Ti = 19°C = 292 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*292/0.022 + 0.08*263/0.03 ) / ( 0.08/0.03 + 0.01/0.022 )

Tx = 267 K

Tx = - 6°C = ...............Ans(a)

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 = 0.03 / 0.08 = 0.375 K/W
Rs thermal resistance of styro layer Rs = b / A*ks = 0.022 / 0.01 = 2.2 K/W
A = 1 square meter

R = 2.575 K/W
T = 19 - (-10) = 29 K

= 29 / 2.575 = 11.26 W ................Ans(b)

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