Finding U-value for Wall with Plasterboard Lining and Air Gap

[bobred]

Homework Statement

There is a wall 100 mm thick, find the U-value. Then a plasterboard lining is added of thickness x and a gap between it and the wall of 20 mm

wall thickness [tex]b=0.1[/tex]m, thermal conductivity [tex]\kappa_1=0.5 W m^{-1} K^{-1}[/tex]
plasterboard thickness [tex]x[/tex]m, thermal conductivity [tex]\kappa_2=0.1 W m^{-1} K^{-1}[/tex]
Air gap [tex]g=0.02[/tex]m,
Heat transfer coeff inside [tex]h_{in}=10 W m^{-2} K^{-1}[/tex]
Heat transfer coeff outside [tex]h_{out}=100 W m^{-2} K^{-1}[/tex]
Combined heat transfer coeff air gap [tex]h_{c}=10 W m^{-2} K^{-1}[/tex]
Temp inside [tex]\Theta_1 = 300 K[/tex]
Temp outside [tex]h_{c}= 270 K[/tex]
Cross sectional area [tex]A[/tex]

Homework Equations

For just the wall
rate of heat transfer [tex]q=\frac{UA(\Theta_1-\Theta_2)}{b}[/tex] where
[tex]U=(\frac{1}{h_{in}}+\frac{b}{\kappa_1}+\frac{1}{h_{out}})^{-1}[/tex]

For wall and plasterboard
rate of heat transfer [tex]q=\frac{UA(\Theta_1-\Theta_2)}{b}[/tex] where
[tex]U=(\frac{1}{h_{in}}+\frac{b}{\kappa_1}+\frac{1}{h_{c}}+\frac{x}{\kappa_2}+\frac{1}{h_{out}})^{-1}[/tex]

The Attempt at a Solution

I can work out the U-value for the wall, but I am asked then to find the U-value for the wall after lining, I can't see how to get this without having x.

Any ideas, Thanks

 

[bobred]

Mistake above
[tex]h_{c}= 270 K[/tex] should read [tex]\Theta_{2}= 270 K[/tex] and both [tex]q=\frac{UA(\Theta_1-\Theta_2)}{b}[/tex] should be [tex]q=UA(\Theta_1-\Theta_2)[/tex]

Any ideas?