Step by Step just as you asked.
now be happy
The formula that IRstuff provided is a good start.
Q= S x Km x (T1-T2)
S= 2piL/Ln(4zD)
S = shape factor of pipe
L = Length of pipe
z = depth of pipe
D = diameter of pipe
All have to be in Meters
In your case
L= 457.2m
z = 1m
D = .0762m
which then
S = -2417.87 <-- don't worry about the negative heat loss is negative.
Q = S x Km x (T1-T2)
Here is where we make a whole of assumptions
Km is the heat transfer coefficient of the soil, from workign with one of our customers in the power plant business the varies tremendously especially if the soil is wet. I am gonig to use 0.9 as the formula recommends but I have seen in vary from 0.6 to 1.2, usually this is empirically determined based off field measurements
Next is the assumptions on temperature. I am going ot assume that the ground is air temperature. This is a conservative assumption generally speaking if you bury a hot pipe in the ground during winter you may see the snow melt where the pipe is buried. That means the ground is actually warmer then air temp. From a practical standpoint this is not good, but from an equation stand point it actually will yield less heat transfer because the delta between T1 and T2 is less. all other things being equal, which is another assumption entirely.
Anyway. That gives you this
Q= -2418 x 0.9 x (60-5)
Q= -120,000 watts
That is assuming the pipe outer sking temp is 60 degrees, that is also assuming the ground temp is air temp which is 5 degrees.
So how does the insulation play in? Well I ran a simple thermal simulation using foamed insulation in my heat transfer software and found that you'll pick up roughly 10 degrees in outer pipe surface temperature. Again assuming the insulation doesn't degrade in some manner. So instead of 55 degree delta your looking at 45. So that 10 degrees buys you roughly the following.
Q = -2418 x 0.9 x (50-5)
Q = -97,929 watts
How much is 22kW watts worth to you?
Keep in mind, look at the variability of your soil in the equation and also the temp delta, we are assuming best case justification for the insulation.
As a design reference, steam pipes for the power industry run a pipe in a pipe with mineral wool insulation in between. granted they are at 500 deg steam.
Hopefully I have been of assistance
StrykerTECH Engineering Staff
Milwaukee, WI