Question

The thermal conductivity of a certain concrete is 0.80 W/m • K and the thermal conductivity of a certain wood is 0.10 W/m • K. How thick would a solid concrete wall have to be in order to have the same rate of heat flow through it as an 8.0-cm thick wall made of solid wood? Both walls have the same surface area and the same temperature difference across their faces.

Answer 1

A concrete wall would need to be 1.0 cm thick to have the same rate of heat flow as an 8.0-cm thick wooden wall, considering that the concrete has a thermal conductivity of 0.80 W/m·K and the wood has a conductivity of 0.10 W/m·K.

Step-by-step explanation:

The question asks us to determine the thickness of a concrete wall needed to have the same rate of heat flow as an 8.0-cm thick wooden wall, assuming equal surface areas and temperature differences across both materials. The thermal conductivity of concrete is given as 0.80 W/m·K and that of wood is 0.10 W/m·K.

The rate of heat flow through a material can be described by Fourier's law of heat conduction, which is Q = (k × A × ΔT) / d, where Q is the rate of heat flow, k is the thermal conductivity, A is the area, ΔT is the temperature difference, and d is the thickness of the material.

To solve for the thickness of the concrete wall (d_concrete), we set the equation of heat flow for the wood equal to that of the concrete since the Q, A, and ΔT are the same for both materials:

Q_wood = Q_concrete
(k_wood × A × ΔT) / d_wood = (k_concrete × A × ΔT) / d_concrete

Simplifying and solving for d_concrete gives us:

d_concrete = d_wood × (k_wood / k_concrete)

Using the given values:

d_concrete = 8.0 cm × (0.10 W/m·K / 0.80 W/m·K) = 1.0 cm

Therefore, a solid concrete wall would need to be 1.0 cm thick to have the same rate of heat flow as an 8.0-cm thick wooden wall.

Answer 2

The solid concrete wall would need to be 0.064 meters (or 6.4 centimeters) thick to have the same rate of heat flow as the 8.0 cm thick wall made of solid wood.

Step-by-step explanation:

To find the thickness of the concrete wall required to achieve the same rate of heat flow as the wood wall, we can use the formula for heat conduction:

Q = (k * A * ΔT) / d

Where:

Q is the heat flow rate

k is the thermal conductivity

A is the surface area

ΔT is the temperature difference

d is the thickness of the wall

For the wood wall:

Q_wood = (k_wood * A * ΔT) / d_wood

For the concrete wall:

Q_concrete = (k_concrete * A * ΔT) / d_concrete

We want Q_wood = Q_concrete. Since A and ΔT are the same for both walls, we can set the two equations equal to each other:

(k_wood * A * ΔT) / d_wood = (k_concrete * A * ΔT) / d_concrete

Canceling out common terms:

(k_wood / d_wood) = (k_concrete / d_concrete)

Rearranging the equation to solve for d_concrete:

d_concrete = (k_concrete / k_wood) * d_wood

Substituting the given values:

d_concrete = (0.80 W/m • K / 0.10 W/m • K) * 8.0 cm

Simplifying:

d_concrete = 8 * 0.80 cm

Converting cm to meters:

d_concrete = 6.4 cm = 0.064 m

Therefore, the solid concrete wall would need to be 0.064 meters (or 6.4 centimeters) thick to have the same rate of heat flow as the 8.0 cm thick wall made of solid wood.