A brass pipe that is 10 cm indiameter and has a wall thickness of 0.26 cm carries steam at `100^(@)C` through a vat of circulating water at `20^(@)C`. How much heat is lost per meter of pipe in 1s?

Answer

To solve the problem of how much heat is lost per meter of a brass pipe carrying steam at 100°C through a vat of circulating water at 20°C, we will use the formula for heat conduction in cylindrical coordinates.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Internal diameter of the pipe, \(d = 10 \, \text{cm} = 0.1 \, \text{m}\)
- Radius of the inner wall, \(r_1 = \frac{d}{2} = 0.05 \, \text{m}\)
- Wall thickness, \(t = 0.26 \, \text{cm} = 0.0026 \, \text{m}\)
- Outer radius, \(r_2 = r_1 + t = 0.05 + 0.0026 = 0.0526 \, \text{m}\)
- Temperature of steam, \(T_1 = 100 \, \text{°C}\)
- Temperature of circulating water, \(T_2 = 20 \, \text{°C}\)
- Thermal conductivity of brass, \(k = 109 \, \text{W/m·K}\)

2. **Calculate the Temperature Gradient**:
- The temperature difference across the pipe wall is \(\Delta T = T_1 - T_2 = 100 - 20 = 80 \, \text{°C}\).

3. **Use the Heat Transfer Formula**:
The heat transfer per unit time per unit length through a cylindrical wall is given by:
\[ \frac{Q}{T} = \frac{2 \pi k (T_1 - T_2)}{\ln\left(\frac{r_2}{r_1}\right)} \]
where:
- \(Q\) is the heat lost,
- \(T\) is the time,
- \(k\) is the thermal conductivity,
- \(r_1\) and \(r_2\) are the inner and outer radii, respectively.

4. **Substitute the Values**:
\[ \frac{Q}{T} = \frac{2 \pi (109) (80)}{\ln\left(\frac{0.0526}{0.05}\right)} \]

5. **Calculate the Natural Logarithm**:
\[ \ln\left(\frac{0.0526}{0.05}\right) = \ln(1.052) \approx 0.05069 \]

6. **Calculate the Heat Transfer Rate**:
\[ \frac{Q}{T} = \frac{2 \pi (109) (80)}{0.05069} \]
- Calculate \(2 \pi (109) (80)\):
\[ 2 \pi (109) (80) \approx 2 \times 3.1416 \times 109 \times 80 \approx 54789.504 \, \text{W} \]
- Now divide by \(0.05069\):
\[ \frac{Q}{T} \approx \frac{54789.504}{0.05069} \approx 1080874.0 \, \text{W/m} \]

7. **Convert to Megajoules**:
- Since \(1 \, \text{W} = 1 \, \text{J/s}\), the heat lost per meter of pipe in 1 second is:
\[ Q \approx 1080874.0 \, \text{J/m} \approx 1.08 \, \text{MJ/m} \]

### Final Answer:
The heat lost per meter of pipe in 1 second is approximately **1.08 MJ/m**.