A carpenter builds an exterior house wall with a layer of wood 3.2 cm thick on t

Question

A carpenter builds an exterior house wall with a layer of wood 3.2 cm thick on the outside and a layer of Styrofoam insulation 2.3 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 20.0 C , and the exterior surface temperature is -15.0 C .

Part A

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

Express your answer using two significant figures.

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Part B

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

Express your answer using two significant figures.

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A carpenter builds an exterior house wall with a layer of wood 3.2 cm thick on the outside and a layer of Styrofoam insulation 2.3 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 20.0 C , and the exterior surface temperature is -15.0 C .

Part A

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

Express your answer using two significant figures.

T =   C  

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Part B

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

Express your answer using two significant figures.

H =   W/m2  

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Explanation / Answer

a = 0.032 m thickness of the wood layer
kw = 0.08 W/mK thermal conductivity of wood
To = - 15°C = 258 K outside temperature

b = 0.023 m thickness of styrofoam
ks = 0.01 W/mK thermal conductivity of styrofoam
Ti = 20°C = 293 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 = ( ks Ti/b + kw To/a )/ (kw/a + ks/b)
Tx = ( 0.01*293/0.023 + 0.08*258/0.032 ) / ( 0.08/0.032 + 0.01/0.023 ) = 263.2 K
Tx = - 9.8 °C

The rate of heat flow per square meter is :
H = T / R ; where R is total thermal resistance of the wall R = Rw + Rs

Rw thermal resistance of wood layer Rw = a / A*kw = 0.032 / 0.08 = 0.4 K/W
Rs thermal resistance of styro layer Rs = b / A*ks = 0.023 / 0.01 = 2.3 K/W
A = 1 m^2

R = 2.7 K/W
T = 20 - (-15) = 35 K

H = 35/ 2.7
H = 12.96 W/m^2

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