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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^{o}C

Tair: 5^{o}C

h_{1}(LO): 3600.63 W/m^{2}/K

r_{1}(pipe inner): 0.1 m

r_{2}(pipe outer): 0.1095 m

r_{3}(insulation): 0.1695 m

L (length): 21.5 m

A_{1}(pipe inner area): 13.50884841 m^{2}

k_{1}(carbon steel): 50 W/m.K

k_{2}(mineral wool): 0.0421 W/m.K

h_{2}(air): 11.6 W/m2/K

A_{3}(insulation outer area): 22.89749806 m^{2}

Cp (LO): 1928 J/kg.K

Time: 43200 s

Density: 877.8 kg/m^{3}

Volume: 0.68 m^{3}

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^{o}C. 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 17.83^{o}C.

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

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# Temperature drop in an insulated pipe over 12 hours

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