Heat Transfer Rate Calculation for Finned Tube in Commercial Building Heating System

We can approach this problem by using the concept of thermal resistance. The rate of heat transfer to the air in the room can be expressed as:

Q = U A ΔT

where U is the overall heat transfer coefficient, A is the surface area of the finned tube, and ΔT is the temperature difference between the inner surface of the tube and the air in the room. The overall heat transfer coefficient can be expressed as:

U = 1/h + ΣR

where ΣR is the sum of the thermal resistances of the different layers of the system. For this problem, we can consider the following thermal resistances:

1. Thermal resistance of the air boundary layer on the fin surface
2. Thermal resistance of the fin
3. Thermal resistance of the tube wall
4. Thermal resistance of the air boundary layer on the tube surface

We can calculate these thermal resistances using the following equations:

1. Rb = (1/h) (ln(ro/ri)) = 0.00114 K/W
2. Rf = (t/(kA)) (tanh(mL/2t)) = 0.00012 K/W
3. Rt = ln(ro/ri) / (2πkL) = 0.00216 K/W
4. Rb = (1/h) (ln(re/ro)) = 0.00038 K/W

where m = rt/ro, L is the fin length, and re = ro + δ. Note that we assume that the fins are long enough such that their thermal resistance dominates over that of the tube wall.

The total thermal resistance is then:

ΣR = Rb + Rf + Rt + Rb = 0.0048 K/W

The surface area of the finned tube can be expressed as:

A = 2πL (ro + t/2) = 0.335 m2/m

The temperature difference between the inner surface of the tube and the air in the room is:

ΔT = Ts,i - T∞ = 100°C

Putting it all together, we get:

Q = U A ΔT = (1/h + ΣR) A ΔT = (1/25 + 0.0048) (0.335) (100) = 1.17 W/m

Therefore, the rate of heat transfer to the air in the room is 1.17 Watts per meter of finned tube length.