A carpenter builds an exterior house wall with a layer of wood 3.0 cm thick on t

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 2.2 cm thick on the inside wall surface. The wood has a thermal conductivity of 0.080 W/(m*k) and the Styrofoam has a thermal conductivity of 0.010 W/(m*k). The interior surface temperature is 19.0oC, and the extrerior surface temperature is -10.0oC. (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?

PLEASE HELP!! I need step by step so i undestand how to solve the equations. THANK YOU!!!!

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 = 0.01 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 ) = 267 K

Tx = - 6°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 = 0.03 / 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

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