A carpenter builds an exterior house wall with a layer of wood 3.1 cm thick on t

Question

A carpenter builds an exterior house wall with a layer of wood 3.1 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/(mK), and the Styrofoam has k= 0.010 W/(mK). The interior surface temperature is 19.0 C , and the exterior surface temperature is -13.0 C .

Part A

What is the temperature at the plane where the wood meets the Styrofoam?

Express your answer using two significant figures.

Part B

What is the rate of heat flow per square meter through this wall?

Express your answer using two significant figures.

Explanation / Answer

SOLUTION: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 = - 13°C = 260 K outside temperature

b = 0.02 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*292/0.02 + 0.08*260/0.031 ) / ( 0.08/0.031 + 0.01/0.020 ) = 816.97/3.08=265.25 K

Tx = - 7.75°C

PART B

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 = 0.387 K/W
Rs thermal resistance of styro layer Rs = b / A*ks = 0.02 / 0.01 = 2 K/W
A = 1 square meter

R = 2.387 K/W
T = 19 - (-13) = 32K

= 32 / 2.387 = 13.40 W ANSWER

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