To determine the heat lost through the insulating jacket surrounding the steam pipe, we can use Fourier's law of heat conduction. This law states that the heat transfer rate through a material is proportional to the temperature difference across the material and the area through which the heat is being transferred. Let's break down the problem step by step.
Given Information
- Radius of the steam pipe, r1 = 5 cm = 0.05 m
- Thickness of the insulating jacket, t = 2 cm = 0.02 m
- Outer radius of the jacket, r2 = r1 + t = 0.05 m + 0.02 m = 0.07 m
- Temperature of the steam, T1 = 100 °C
- Temperature at the outer wall of the jacket, T2 = 20 °C
- Thermal conductivity of the insulating material, k = 0.07 W/m-K
Calculating the Heat Transfer Rate
The heat transfer rate (Q) through the cylindrical jacket can be calculated using the formula:
Q = (2 * π * k * (T1 - T2)) / ln(r2/r1)
Step-by-Step Calculation
Now, let's plug in the values into the formula:
- Temperature difference, ΔT = T1 - T2 = 100 °C - 20 °C = 80 °C
- Natural logarithm of the radius ratio, ln(r2/r1) = ln(0.07 m / 0.05 m)
Calculating the logarithm:
- ln(0.07 / 0.05) = ln(1.4) ≈ 0.3365
Now substituting the values into the heat transfer formula:
Q = (2 * π * 0.07 W/m-K * 80 °C) / 0.3365
Calculating the numerator:
- 2 * π * 0.07 * 80 ≈ 35.0 W
Now, dividing by the logarithm:
Q ≈ 35.0 W / 0.3365 ≈ 104.0 W
Heat Loss Over Time
To find the total heat lost through the jacket in one hour, we need to convert the power (in watts) to energy (in joules). Since 1 watt is equal to 1 joule per second, we can calculate the energy lost over one hour (3600 seconds):
Energy lost = Q * time = 104.0 W * 3600 s
Calculating the total energy:
Energy lost ≈ 374,400 J
Final Result
Therefore, the total heat lost through the jacket per meter length in one hour is approximately 374,400 joules.
