Question

(a) What thickness of cork would have the same R-factor as a 1.0 -cm thick stagnant air pocket? (b) What thickness of tin would be required for the same R-factor?

Short answer

Answer: (a) The thickness of cork required to have the same R-factor as a 1.0 cm thick stagnant air pocket is 1.667 cm. (b) The thickness of tin required to have the same R-factor as a 1.0 cm thick stagnant air pocket is 2750 cm.

Step-by-step solution

Find the R-factor of the stagnant air pocket

To find the R-factor (thermal resistance) of the air pocket, we need the thickness (d) and thermal conductivity (k) of the air. The thickness of the air pocket is given as 1.0 cm. The thermal conductivity (k) of stagnant air is approximately 0.024 W/mK. Conversion of thickness from cm to m: \(d = 1.0 cm = 0.01 m\) Now, let's calculate the R-factor of the air pocket using the formula R = d / k. \(R_{air} = \frac{0.01 m}{0.024\:W/mK} = 0.4167\:m^2K/W\)

Calculate the thickness of cork needed

To find the thickness of cork required to have the same R-factor as the air pocket, we need the thermal conductivity (k) of cork. The thermal conductivity of cork is approximately 0.040 W/mK. Now using the formula R = d / k, we can find the thickness (d) of cork needed as: \(d_{cork} = R_{air}\times k_{cork} = 0.4167m^2K/W \times 0.040 W/mK = 0.01667 m\) Convert the thickness of cork to cm: \(d_{cork} = 0.01667 m = 1.667 cm\)

Calculate the thickness of tin needed

To find the thickness of tin required to have the same R-factor as the air pocket, we need the thermal conductivity (k) of tin. The thermal conductivity of tin is approximately 66.0 W/mK. Now using the formula R = d / k, we can find the thickness (d) of tin needed as: \(d_{tin} = R_{air}\times k_{tin} = 0.4167m^2K/W \times 66.0 W/mK = 27.50 m\) Convert the thickness of tin to cm: \(d_{tin} = 27.50 m = 2750 cm\)

Results:

(a) The thickness of cork required to have the same R-factor as a 1.0 cm thick stagnant air pocket is 1.667 cm. (b) The thickness of tin required to have the same R-factor as a 1.0 cm thick stagnant air pocket is 2750 cm.