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/m²-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/m²-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/m³

d = pipe internal diameter, m

C_p = Specific Heat Capacity of pipeline contents, J/kg-K

ν = liquid kinematic viscosity, m²/s

g = acceleration due to gravity = 9.81 m/s²

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/m³

U_wall

where:

U_wall = steel wall heat transfer coefficient, W/m²-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/m²-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/m²-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.