heat transfer thru irradiated wall 1

[jeremy_rutman]

I've got a simple problem that I'm not able to crack -
A wall of a building has one side in the sun, the other side in an airconditioned area of the building.
The ambient temperature To is known as is the temperature inside the airconditioned area Ti.

I would like to find the equilibrium inner and outer wall temperatures (Twi and Two), and have two equations
q/A - h1(To-Two) = k/w (Two-Twi)
k/w (Two-Twi) = h2(Twi-Ti) [edited]

where q/A is the solar input, h1, h2 are the convective transfer coefficients for vertical wall and k the thermal conductivity of the wall.
THe two equations are not independent and thus I am not able to find the desired Twi,Two; what am I missing?

 

[georgeverghese]

Think there is a minor typo error in the 2nd expression; this should be
k/w (Two-Twi) = h2(Twi-Ti)

Otherwise, expressions are correct.

Make a trial guess for either one of the unknowns. Say you guess Twi and plug this into (2) to get Two-Twi. Plug this into (1) to get Two, and hence Twi. Also do a final check that Qin from the outside, which is the LHS of (1) equals Q into the room through the wall, which is the RHS of (2).
Also do a final check that the values derived for h1 and h2 are consistent with the assumed values for the corresponding surface temps, since h1 and h2 are dependent on surface temp.

 

[IRstuff]

You need to define the interior heat source(s)

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[jeremy_rutman]

If there is in fact only one independent equation and two variables, wouldn't this mean there are infinite possible solutions?

 

[IRstuff]

No, because there are boundary conditions that constrain the problem. You either have no internal sources/sinks, in which case the building is at thermal equilibrium with the outside, or you have sources/sinks (A/C) which drives the temperature internally to whatever the A/C is set to, and the A/C will sink whatever heat comes through the wall to the extent of the cooling capacity of the A/C

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[jeremy_rutman]

Ok I blv the solution is that even though I have two equations of the form
f1(Two)=f2(Two,Twi)
f2(Two,Twi)=f3(Twi)
and therefore
f1(Two)=f3(Twi)
that doesnt mean that I have only one independent equation .
I can still solve f1=f2 for Two in terms of Twi, plug that into f2=f3 and solve for Twi in terms of the constants.