The critical radius of insulation for asbestors with k = 0.125 W/mK surrounding a pipe exposed to room air with h = 2.5 W/m2K is
50 mm
Critical Radius of Insulation Calculation
The concept of critical radius of insulation is important in heat transfer, especially when dealing with cylindrical pipes or spherical shells. It represents the radius at which adding more insulation to a pipe or sphere actually increases the rate of heat transfer, rather than decreasing it. This occurs because while insulation reduces heat transfer by conduction, it simultaneously increases the surface area available for convection and radiation to the surroundings. The critical radius is the point where these two opposing effects balance out.
Understanding Critical Radius
For a cylindrical pipe, the heat transfer from the pipe to the surroundings involves both conduction through the insulation layer and convection from the outer surface of the insulation to the ambient air. As the thickness of the insulation increases, the conductive resistance increases, but the convective resistance decreases due to the larger surface area. The critical radius is the outer radius of the insulation at which the total thermal resistance is minimum, leading to maximum heat transfer.
Formula for Critical Radius
The critical radius (\(r_c\)) for a cylindrical pipe with a given insulation material and surrounding fluid is determined by the ratio of the thermal conductivity of the insulation (\(k\)) to the convective heat transfer coefficient (\(h\)) of the surrounding medium. The formula is given by:
\[r_c = \frac{k}{h}\]
Where:
- \(r_c\) is the critical radius of insulation (in meters).
- \(k\) is the thermal conductivity of the insulation material (in W/mK).
- \(h\) is the convective heat transfer coefficient of the surrounding air (in W/m\(^2\)K).
Given Data
Let's list the given values from the problem statement:
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Thermal conductivity of asbestos (insulation) | \(k\) | 0.125 | W/mK |
| Convective heat transfer coefficient of room air | \(h\) | 2.5 | W/m\(^2\)K |
Step-by-Step Calculation
Now, we will substitute the given values into the critical radius formula:
\[r_c = \frac{k}{h}\]
Substitute the values:
\[r_c = \frac{0.125 \text{ W/mK}}{2.5 \text{ W/m}^2\text{K}}\]
Perform the division:
\[r_c = 0.05 \text{ m}\]
The question asks for the answer in millimeters (mm). To convert meters to millimeters, we multiply by 1000:
\[r_c = 0.05 \text{ m} \times 1000 \text{ mm/m}\]
\[r_c = 50 \text{ mm}\]
Final Answer
The critical radius of insulation for asbestos under the given conditions is 50 mm.
Important Questions from Critical Thickness of Insulation
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