Calculate time it takes non=flowing water to freeze in ductile iron pipe?

Question

16" unlined ductile iron pipe, ID = 17.1", OD = 17.4"
Starting water temp = 50 °F
Ambient air temp = 7 °F
Therm cond of ductile iron pipe, k = 36 W/m K
Heat loss, Q = -kA(T2-T1)
A = surface area of pipe per meter of pipe = PI*(.5*OD)^2 = 0.15 sq m / m of pipe
Q = -36*0.15*(283-260) = -132 Watts/meter of pipe

Now calculate the time it takes to cool the mass of water in 1 m of pipe from starting temp
Of 50 °F to 32 °F

Time to cool = Q / heat capacity of water / mass of water per m of pipe / (273 K – 283 K)
Where
Heat cap of water = 4200 J/kg per degree K
Mass of water in 1 m of pipe = 148 kg per m of pipe
Freezing point of water = 32°F = 273 K
= 13 hours

Now, determine the time to freeze water at 32°F

Time = Q / latent heat of fusion for water / mass of water in pipe per meter
Where
Latent heat of fusion of water = 334,000 J/kg
= 104 hours
Total time to cool water from 50°F to freeze it is 13 hr +104 hr = 117 hr, this seems too high

I found my mistake. I was calculating cross-section area instead of surface area of the pipe (cylinder). I found my mistake and am getting much more realistic values. Of course I do neglect the insulation properties of the ice layer as it forms from the pipe wall inward, but I've got my worst case answer and it supports what we want to achieve.

Thanks for your help, though! You ceretainly convinced me to go back and triple check my work and that's how i found my math error.

Answer 1

Does seem high, but you're dealing with an
exothermic process and a low 'Delta T`.
'Trust the math, check it, but trust it.`