Calculate the R-value of the stack of materials

[ninaw21]

Homework Statement

Calculate the R-value of the stack of materials whose total thickness is made up of the
individual thicknesses:
material 1; k = 0.123 W/m/K, thickness = 0.103 m
material 2: k = 0.234 W/m/K, thickness = 0.092 m
material 3: k = 0.345 W/m/K, thickness = 0.081 mHence calculate the heat flow per unit area through such a stack with has temperatures of
20 °C and -5 °C on opposite sides of the stack

Homework Equations

R = ΔT/Q_A, Where Q_Ais heat flux

The Attempt at a Solution

I've no idea how to do this problem! Please help!

 

[I like Serena]

Welcome to PF, ninaw21!

In equilibrium the heat flow through material 1 must be equal to the flow through material 2, which in turn must be equal to the flow in material 3.

If you introduce 2 variables representing the temperature between materials 1 and 2, respectively materials 2 and 3, you can set up a system of equations that you can solve.

Do you know how to do that?

 

[ninaw21]

Thank you! I know that they're equal but I don't know the variables to use..

 

[I like Serena]

Well, what can you come up with?
Which symbols can you think of?

 

[ninaw21]

q = specific heat x m x Δt,

where q is heat flow, m is mass in grams, and Δt is the temperature change. ??

 

[I like Serena]

Hmm, that is the formula that relates absorbed heat to change in temperature.
I'm afraid that is not the formula to use here.

Do you have a formula that relates the R-value to the thermal conductivity k?

Actually, to find the R-value of the stack, you can simply add the R-values of the 3 materials.

 

[ninaw21]

Is this the formula that is used then :

deltaQ/deltat = kAdeltaT/d, where: deltaQ = heat flow, deltat = time, k = thermal conductivity, deltaT = temp, and d = distance ??
(Thanks for all the help!)

 

[I like Serena]

That's closer.

So you have:
$$R = {\Delta T \over {dQ \over dt} / A}$$
and
$${dQ \over dt} = {k A \Delta T \over d}$$
where d is distance the heat travels, or in other words, the thickness of the material,
and where A is the surface of the material.

Note that the heat flux $Q_A$ that you had in your opening post, is actually the heat flow per unit area and per unit of time.Can you deduce what the R-value will be of each material?

 

[I like Serena]

I'll make it simpler.
The relation between R and k is: $R = {d \over k}$.

 

[ninaw21]

Thank you so much! :)

 

[I like Serena]

So... do you have your answer now?

 

[ninaw21]

Yes! :)

 

[I like Serena]

Good!