Two key values required to calculate pipeline temperature gradients are rarely known with certainty: the temperature and the thermal conductivity of the surrounding material. However, given reasonable estimates of these values, the equations described below allow calculation of flowing pipeline temperature changes to within 10%. This is adequate for most design purposes (e.g, process considerations and anti-corrosion coating design). For systems where more precision is needed (e.g,. inventory calculation for loss control monitoring), temperature should be measured at regular intervals along the length of the pipeline and the equations "tuned" as operating experience is gained.

Although the method is suited to hand calculation, the following equations are more typically used as the basis for computer calculations where the pipeline is divided into a large number of segments. This approach can be extended to cover transients and to interface with pressure loss routines to provide iterative solutions for gas pipeline flows.

For a simple hollow cylinder, the conduction heat transfer coefficient based on diameter D is

where is D_{1} and D_{2} are the outer and inner diameter, respectively, and λ_{1} is the thermal conductivity of the material. The coefficient relating to the pipe surroundings (based on the outside diameter) has been found to be adequately predicted for engineering purposes by:

where h is the depth of cover to the pipe axis, D is the outside diameter of the wrapped pipe and λ_{n} the thermal conductivity of the surroundings.

The Overall Heat Transfer Coefficient for the pipe and any coatings or insulation is obtained by adding the inverse coefficients for each layer (based on the same diameter D):

The heat flow ratio, a, can then be calculated:

The pipeline axial temperature profile can then be expressed as:

where T_{2} is the temperature at a distance L from the inlet and T_{1} is the inlet temperature. T_{a} is the asymptotic temperature or the temperature, which the fluid approaches as it flows along the line. This is often wrongly assumed to be equal to the ground temperature where in fact it can be above or below ground temperature depending on the balance between the cooling effect as the product expands with falling pressure and heat gained through friction. These effects are represented through the Joule-Thomson Coefficient, μ_{JT} and, correcting for elevation change, Δz

At typical pipeline conditions, light hydrocarbon gases can have Joule-Thomson coefficients, μ_{JT}, of the order of 0.4°C/bar resulting in asymptotic temperatures below ground temperature. For hydrocarbon oils the coefficient becomes negative with frictional heating effects outweighing cooling due to expansion. A coefficient of —0.04°C/bar would be typical for a heavy oil.

Typical thermal conductivities:

Material | λλ(W/mK) |

Steel | 45 |

Epoxy coating | 0.21 |

Concrete | 1.4 |

Soil (landlines) | 0.9 -- 1.7 |

Soil (subsea—water saturated) | 2.7 |

More complex systems involving multiple lines, trace heating etc. are addressed in King (1980). Jee (1992) deals with multiple pipelines within an outer carrier pipe and Maddox and Erbar (1982) provide an approach for pipelines with two-phase flow.

#### REFERENCES

Jee, T. P. (1992) *Assessing the Thermal Behaviour of Flowline Bundles*, IIR conference on Optimizing Design and Construction of Flowline Bundles, April.

King, G. G. (1980) *Geothermal Design of Buried Pipelines*, ASME Energy Conference and Exhibition, New Orleans.

Maddox, R. N. and Erbar, J. H. (1982) *Fluid Flow of Gas Conditioning and Processing*, Ch. 16, 3, Campbell Petroleum Series.