Calculation of Log-Mean Heat Transfer Area for Insulated Pipe

Given:
- Length of pipe (L) = 2.5 m
- Inner diameter (ID) = 50 mm
- Outer diameter (OD) = 56 mm

Assumptions:
- The pipe is insulated at both ends.
- Heat transfer is steady and one-dimensional.
- The thermal conductivity of the insulation is much lower than that of the pipe material.
- The heat transfer coefficient is constant along the length of the pipe.

Solution:

Step 1: Calculation of the inner and outer surface areas of the pipe

- Inner surface area (Ai) = π x ID x L = 0.196 m²
- Outer surface area (Ao) = π x OD x L = 0.219 m²

Step 2: Calculation of the log-mean temperature difference (ΔTlm)

- For an insulated pipe, the log-mean temperature difference can be calculated using the following equation:
ΔTlm = (ΔT1 - ΔT2) / ln(ΔT1 / ΔT2)
where,
ΔT1 = T1 - T2 (temperature difference between the hot fluid and the pipe wall at the inlet)
ΔT2 = T3 - T4 (temperature difference between the pipe wall at the outlet and the cold fluid)
T1 = Inlet temperature of the hot fluid
T2 = Wall temperature of the pipe at the inlet
T3 = Wall temperature of the pipe at the outlet
T4 = Outlet temperature of the cold fluid

- For simplicity, assume that the pipe is at a uniform temperature along its length and that the insulation is perfect. Therefore, T2 = T3 and ΔT1 = ΔT2 = (T1 - T4).
- Let's assume the inner and outer wall temperatures of the pipe are T1 and T4, respectively. The insulation prevents heat transfer from the outer surface of the pipe to the surroundings. Therefore, T4 = Tsurroundings.
- Let's assume that the hot fluid enters the pipe at a temperature of 100°C and the cold fluid leaves the pipe at a temperature of 20°C. Therefore, T1 = 100°C and T4 = 20°C.

ΔTlm = (100 - 20) / ln(100 / 20) = 53.7°C

Step 3: Calculation of the log-mean heat transfer area (Am)

- For an insulated pipe, the log-mean heat transfer area can be calculated using the following equation:
Am = (Ai x Ao)^(1/2)

Am = (0.196 x 0.219)^(1/2) = 0.207 m²

Conclusion: The log-mean heat transfer area of the insulated pipe is 0.207 m². However, this is not one of the answer options provided. It is possible that the answer options are rounded or that there is an error in the question. If we assume that the correct answer is between the given options, then the closest option is 0.41 - 0.42 m².