Thanks for the reference ione, I haven't seen that equation in a while. I just finished reading several of the papers that Google pointed me to and one thing that jumped out at me is the lengths involved. All of the papers show two cases, first a "winter" case with a gas temperature of 130F and a ground temperature of 62F. The second is a "summer" case with a gas temperature of 150F and a ground temperature of 80F.
The way I approach this kind of problem is with a "Thermal Entry Length" which is defined as "the distance required for the dT between the fluid and the surroundings to be reduced to 20% of its original value". After the thermal entry length, heat transfer is too slow to be a practical consideration. So in the first case we reach the thermal entry length at 75.6F and in the second cast it is at 93.6F. What I found interesting is that you reach these values at around 30 miles and 50 miles respectively. These numbers are far longer than I've ever observed in the field.
I have frequently compared temperatures at meter stations on wellheads to the temperature of gas entering compressor stations less than 5 miles away. Two things I've noticed is that temperature at the compressor station is very constant winter to summer and it is constant regardless of the mix of inlet temperatures (turning large high-temperature sources on or off does not seem to affect station inlet temperature).
Far be it from me to suggest that my observations in any way trump a graduate student's manipulation of "fourth order ODE's", but I just can't buy either the lengths involved or the shape of their curves.
For the OP's situation, I would call a 60F inlet and a 45F ground temperature the same value (3% difference in absolute terms) and get on with the rest of the evaluation without a temperature profile. I do a lot of pipeline modeling and the model I use has some robust temperature modeling modules. In gas lines, turning them on or off has never made any difference at all to the recommendation that I would make about the model results. If I'm concerned about hydrate prediction then I have to turn the temperature modeling on and I have been able to explain a couple of line freezes that way, but if I model a line heater (in field gas saturated with water vapor), I get hydrate formation in the same place as the model predicted it without the heater.
David
