Heat Loss by Convection to Air

Determine the rate of heat loss from a 100 mm diameter steam pipe placed horizontally in ambient air at 30^oC. The length of the pipe is 4 m and wall temperature, T_w = 170^oC.
Use the following empirical expression:
Nu=0.53 (Gr x Pr)^1/4
Properties of air at 100^oC are as following
β =1/373 K^-1; ν = 23.13 x 10 ^-6 m² /sec
k= 0.0325 W/m.K
Pr = 0.7
Calculations:

Gr = gβD³(T_w - T_∞)ρ²/μ²

Where β = 1/T_f ; T_f = 0.5 (T_w + T_∞) = 0.5 x (170 + 30) = 100^oC = 373 K

ν= μ/ρ

Substituting for the values,

Gr = 9.812 x (1/373) x 0.1³ x (170 - 30) / (23.13 x 10^-6)² = 6.884 x 10⁶

Nu = 0.53 x (6.884 x 10⁶ x 0.7)^1/4 = 24.832

Nu = hD/k = 24.832

h = 24.832 x 0.0325 / 0.1 = 8.07 W/m².^oC

Rate of heat loss Q = hA(T_w - T_∞) = h 2πRL (T_w - T_∞)

= 8.07 x 2 x 3.142 x 0.05 x 4 x (170 - 30) = 1419.8 W