How can I eliminate Tw in the equation for heat loss from a house?

[Donna14]

Homework Statement

I'm struggling to understand question 1.C
How do I incorporate T_w and later eliminate this?

Homework Equations

-k*A*(ΔT/ΔX)

The Attempt at a Solution

For 1.a I used this equation: -k*A*((T_inside-T_outside)/thickness wall) I thought to put T inside first as they ask for the inward heat flow and that will be negative (heat loss)...

For 1.c I'm really stuck...

 

[mfb]

You get two heat flow equations, one for the styrofoam and one for the bricks. Using both together, you can elimitate T_w.

 

[Donna14]

Thanks for your response.


J_s=-k_s*A*((Tinside-T_w)/thickness styrofoam)
J_b=-k_b*A*((Tw-Toutside)/thickness brick wall)

How do I now eliminate Tw?

 

[mfb]

With the standard ways to manipulate equations.

Why did you use two different J?

 

[Donna14]

I thought that I had two different J as they both represent a different number...
Im really not sure how to continue from here... I am sorry... Can you give me a hint?

 

[Chestermiller]

I thought that I had two different J as they both represent a different number...
Im really not sure how to continue from here... I am sorry... Can you give me a hint?

J is the same for both because the heat goes through both walls in succession. The walls are in series.

Chet

 

[Donna14]

J is the same for both because the heat goes through both walls in succession. The walls are in series.

Chet

But won't the flow through styrofoam be different from the flow through brick as they have a different k? And one Goes from inside to the brick wall and the other from the brick wall to outside.
I understand that there is also a 'total' flow from into outside.

And how do I combine these 2 equations. I might be very dumb, but really trying to see it!

 

[Chestermiller]

But won't the flow through styrofoam be different from the flow through brick as they have a different k? And one Goes from inside to the brick wall and the other from the brick wall to outside.
I understand that there is also a 'total' flow from into outside.

And how do I combine these 2 equations. I might be very dumb, but really trying to see it!

This is very much analogous to an electric circuit with two unequal resistors in series. The overall voltage drop is analogous to the overall temperature difference, and the current is analogous to the heat flow. The current flow through each of the resistors is the same. The temperature at the interface between the styrofoam and the brick is analogous to the voltage in the wire between the two resistors. So:

I = ΔV₁/R₁

and

I = ΔV₂/R₂

ΔV=ΔV₁+ΔV₂=I (R₁ + R₂)

I = ΔV/(R₁ + R₂)

Chet

 

[Donna14]

This is very much analogous to an electric circuit with two unequal resistors in series. The overall voltage drop is analogous to the overall temperature difference, and the current is analogous to the heat flow. The current flow through each of the resistors is the same. The temperature at the interface between the styrofoam and the brick is analogous to the voltage in the wire between the two resistors. So:

I = ΔV₁/R₁

and

I = ΔV₂/R₂

ΔV=ΔV₁+ΔV₂=I (R₁ + R₂)

I = ΔV/(R₁ + R₂)

Chet

I really appreciate your effort!
I understand now that J is the same, but still don't know what to do with the to equations... Sorry!

 

[Chestermiller]

I really appreciate your effort!
I understand now that J is the same, but still don't know what to do with the to equations... Sorry!

Do the same thing we did here with the two voltage drops. Solve for the two temperature differences, and then add them to eliminate T_w.

Chet

 

[Donna14]

Do the same thing we did here with the two voltage drops. Solve for the two temperature differences, and then add them to eliminate T_w.

Chet

I got it!
Thank you so much!