# Temperature drop in an insulated pipe over 12 hours

Hello all,

I have been having some problems and would appreciate any help:

I have a length of insulated pipe filled with engine oil at rest, the variables are as follows:

Toil: 40 oC
Tair: 5 oC
h1 (LO): 3600.63 W/m2/K
r1 (pipe inner): 0.1 m
r2 (pipe outer): 0.1095 m
r3 (insulation): 0.1695 m
L (length): 21.5 m
A1 (pipe inner area): 13.50884841 m2
k1 (carbon steel): 50 W/m.K
k2 (mineral wool): 0.0421 W/m.K
h2 (air): 11.6 W/m2/K
A3 (insulation outer area): 22.89749806 m2
Cp (LO): 1928 J/kg.K
Time: 43200 s
Density: 877.8 kg/m3
Volume: 0.68 m3
Mass: 596.904 kg

I am trying to calculate the temperature of the oil in the pipe after a period of 12 hours, assuming that the ambient air temperature remains constant at 5 oC. I have been able to calculate the heat transfer through the pipe and insulation but am having issues calculating the temperature drop over time.

I have derived down to the following equation that gives me the temperature as a function of time: $$T(t)=5+35e^{-\frac{UAt}{mCp}}$$
Where:

U: Overall heat transfer coefficient
A: Surface Area
t: time
m: Mass of oil in pipe
Cp: Specific heat capacity of the oil

I calculated U as 0.3848 using U = 1 / Rt, where Rt is the total thermal resistance of the pipe and insulation.

After plugging in the numbers I get a value of 17.83oC.

If anyone could let me know if I am on the right track and that the formula I have used to calculate the temperature is appropriate that would be great.

Many thanks,

Mark