When liquids have to be transported at relatively high temperatures, e.g. viscous crudes/products, temperatures and cooling rates can be determined using the following formulae:

where:

T_{L} = average temperature in cross-section of pipeline at distance L, K

T_{s} = soil temperature at pipeline depth, K

T_{inlet} = temperature at pipeline inlet (L = 0), K

U_{eff} = Effective Heat Transfer Coefficient, W/m^{2}-K

d = pipe internal diameter, m

L = pipeline distance at which T_{L} is to be measured, m

m = mass flow rate of liquid, kg/s

C_{p} =Specific Heat Capacity of pipeline contents, J/kg-K

**Calculation of U _{eff}**

**U _{convective}**

For turbulent flow (Re > 4000)

For Laminar flow (Re < 2300)

where:

U_{convective} = Convective heat transfer coefficient, W/m^{2}-K

λ_{liq} = liquid thermal conductivity, W/m-K (for crude oil, λ_{liq} = 0.13 W/m-K)

u = average liquid velocity, m/s

ρ = liquid density, kg/m^{3}

d = pipe internal diameter, m

C_{p} = Specific Heat Capacity of pipeline contents, J/kg-K

ν = liquid kinematic viscosity, m^{2}/s

g = acceleration due to gravity = 9.81 m/s^{2}

α = thermal expansion coefficient, 1/K (for crude oil, α = 8*10^{-4} K-1)

µ_{b} = bulk liquid viscosity, Pa.s

μ_{w} = liquid viscosity at wall temperature, Pa.s

T_{b} = bulk temperature, K

T_{w} = inside wall temperature, K

**Notes**:

*1. The inside wall temp. (T _{w}) is generally assumed 2-3 degrees lower than the bulk temp. (T_{b}) in laminar flow*

*2. The bulk liquid viscosity (μ _{b}) and the liquid viscosity at wall temperature (μ_{w}) can be assumed same for turbulent flow thus reducing the term (μ_{b} / μ_{w}) to 1*

**Specific Heat Capacity of Crude Oil**

where:

C_{p} = Specific heat capacity at temperature T, J/kg-K

T = Temperature, ⁰C

ρ = crude oil density at temperature T, kg/m^{3}

**U _{wall}**

where:

U_{wall} = steel wall heat transfer coefficient, W/m^{2}-K

λ_{steel} = thermal conductivity of steel, W/m-K (for Carbon Steel, λsteel = 52 W/m-K)

d_{o} = outside diameter of the steel pipe, m

d = pipe internal diameter, m

t_{wall} = wall thickness, m

**U _{coating }(Note: External Coating could be a polyolefin liner or insulation)**

where:

U_{coating} = coating heat transfer coefficient, W/m^{2}-K

d_{o} = outside diameter of the steel pipe, m

λ_{coating} = thermal conductivity of coating or insulation, W/m-K

t_{coating} = coating thickness, m

**U _{environment} (For Buried pipes the environment is soil)**

where:

U_{environment} = heat transfer coefficient to the environment (soil), W/m^{2}-K

λ_{soil} = thermal conductivity of soil, W/m-K

z = burial depth of the pipe up to the pipe axis, m

I have developed a spreadsheet for temperature drop in a buried crude oil pipeline based on the above methodology. For a set of conditions that I have used and for a pipeline inlet crude oil temperature of 55⁰C, the temperature drops to 48.5⁰C at a distance of 50 km for a 16 inch NPS pipeline based on the above method. The pipeline is also externally coated.