A cubical container full of hot water at a temperature of 90 °C is completely lagged with an insulating material of thermal conductivity 6.4 X 10-² Wm-¹K-1. The edges of the container are 1.0 m long and thickness of the lagging is 1.0 cm. Estimate the rate of flow of heat through the lagging if the external temperature of the lagging is 40 °C. Discuss qualitatively how your results will be affected if the thickness of the lagging is increased considerably, if the temperature of the surrounding air is 18 °C. (7)
To estimate the rate of flow of heat through the lagging, we can use Fourier's Law of Heat Conduction, which states that the rate of heat flow (Q) through a material is proportional to the temperature gradient (ΔT) across the material and inversely proportional to the thickness (L) and thermal conductivity (k) of the material.
Mathematically, this can be expressed as:
Q = (k * A * ΔT) / L
where:
Q is the heat flow rate,
k is the thermal conductivity of the lagging material,
A is the surface area of the lagging,
ΔT is the temperature difference across the lagging,
L is the thickness of the lagging.
First, let's calculate the surface area of the lagging. Since it is a cube, the surface area can be computed as:
A = 6 * (edge length)^2
Given that the edge length is 1.0 m, the surface area is:
A = 6 * (1.0 m)^2 = 6.0 m^2
Next, let's calculate the temperature difference across the lagging. This can be found by subtracting the external temperature (40 °C) from the internal temperature (90 °C):
ΔT = 90 °C - 40 °C = 50 °C
Now, we can plug in the values into Fourier's Law to find the heat flow rate:
Q = (k * A * ΔT) / L
Given that the thermal conductivity of the lagging material (k) is 6.4 × 10⁻² Wm⁻¹K⁻¹ and the thickness (L) is 1.0 cm = 0.01 m, we can substitute these values:
Q = (6.4 × 10⁻² Wm⁻¹K⁻¹ * 6.0 m² * 50 °C) / 0.01 m = 1.92 W
Therefore, the estimated rate of flow of heat through the lagging is approximately 1.92 Watts.
Now, let's discuss how the results will be affected if the thickness of the lagging is increased considerably and if the temperature of the surrounding air is 18 °C.
If the thickness of the lagging is increased considerably, the rate of heat flow will decrease. This is because a larger thickness (L) in the denominator of Fourier's Law will result in a smaller Q.
Additionally, if the temperature of the surrounding air is lower (e.g., 18 °C), the temperature difference (ΔT) will be greater, resulting in a higher rate of heat flow (Q) according to Fourier's Law.
Therefore, increasing the thickness of the lagging will decrease the rate of heat flow, while decreasing the external temperature will increase the rate of heat flow.