he thermal conductivity of concrete is 0.80 W/m-C° and the thermal conductivity of wood is 0.10 W/m-C°. How thick would a solid concrete wall have to be in order to have the same rate of flow through it as an 8.0 cm thick wall made of solid wood? (Assume both walls have the same surface area.)
To determine the thickness of the solid concrete wall needed to have the same rate of flow as the 8.0 cm thick wall made of solid wood, we can use the equation for heat conduction:
Q = (kA ΔT) / L
Where:
Q is the rate of heat flow (same for both walls)
k is the thermal conductivity of the material (0.10 W/m-C° for wood and 0.80 W/m-C° for concrete)
A is the surface area of the wall (assumed to be the same)
ΔT is the temperature difference across the wall (assumed to be the same)
L is the thickness of the wall
We can rearrange this equation to solve for the thickness of the concrete wall:
L(concrete) = (k(wood) x A x ΔT) / Q
Let's assume the temperature difference is ΔT = 1, and the rate of heat flow is Q = 1. Now we can calculate the thickness:
L(concrete) = (0.10 W/m-C° x A x 1) / 1
L(concrete) = 0.10 W/m-C° x A
Therefore, the concrete wall would need to be 0.10 times the thickness of the wood wall to have the same rate of flow.
L(concrete) = (0.10) x (8.0 cm)
L(concrete) = 0.8 cm
So, the solid concrete wall would need to be 0.8 cm thick in order to have the same rate of flow as an 8.0 cm thick wall made of solid wood.
To find the thickness of the solid concrete wall, we can set up a proportion using the known thermal conductivities and the desired rate of flow:
Let x be the thickness of the concrete wall (in cm).
The rate of flow is directly proportional to the thickness and thermal conductivity (assuming the surface area is the same). Therefore, we can write the equation:
8.0 cm / 0.10 W/m-C° = x cm / 0.80 W/m-C°
Now we can solve for x:
8.0 cm / 0.10 W/m-C° = x cm / 0.80 W/m-C°
Cross multiplying:
0.80 W/m-C° * 8.0 cm = x cm * 0.10 W/m-C°
6.4 W/cm = 0.10 W/cm * x cm
Dividing both sides by 0.10 W/cm:
6.4 W/cm / 0.10 W/cm = x cm
64 = x cm
Therefore, the solid concrete wall would need to be 64 cm thick to have the same rate of flow as an 8.0 cm thick wall made of solid wood.