Motivation

One of the key tasks of in Pipe Flow Dynamics is to predict the along-hole temperature distribution during the stationary fluid transport.

In many practical cases the temperature distribution for the stationary fluid flow can be approximated by homogenous fluid flow model.

Pipeline Flow Temperature Model is addressing this problem with account of the varying pipeline trajectory, pipeline schematic and heat transfer with surroundings around pipeline.

Outputs

along-pipe temperature distribution and evolution in time

Inputs

	pipeline trajectory,		fluid density
	pipeline cross-section area		fluid viscosity
	inflow temperature		initial temperature of the surroundings around the pipeline
	inflow pressure		specific heat capacity of the surroundings around pipeline
	inflow rate		thermal conductivity of the surroundings around pipeline
	heat transfer coefficient based on pipeline schematic

Physical Model

Stationary flow

Homogenous flow

Axial symmetry around the pipe

Mathematical Model

Heat transfer in wellbore due to convection and conduction along the wellbore flow

Heat transfer in rocks around the wellbore due to conduction

\rho \, c \, \frac{\partial T}{\partial t} = \frac{d}{dl} \, \bigg( \lambda \, \frac{dT}{dl} \bigg)  - \rho \, c \, v \, \frac{dT}{dl} + \frac{2 \lambda}{\lambda_e} \cdot \frac{r_f}{r_w^2} \cdot U \cdot \left[ T_e(t, l, r_w) - T \right]

\rho_e \, c_e \, \frac{\partial T_e}{\partial t} = \nabla ( \lambda_e \nabla T_e)

T(t=0, l) = T_{e0}(l)

T_e(t=0, l, r) = T_{e0}(l)

T(t, l=0) = T_0(t)

T_e(t, l, r \rightarrow \infty) = T_{e0}(l)

T (t, l=l_{max}) =  T_{e0}(l_{max})

T_e(t, 0, r) = T_s(t)

T_e(t, l_{max}, r) = T_{e0}(l_{max})

Heat exchange between wellbore fluid and rocks around the wellbore

2 \pi \, \lambda_e \, r_w \, \frac{\partial T_e}{\partial r} \, \bigg|_{r=r_w} = 2 \pi \, r_f \, U \cdot \left( T_e \, \bigg|_{r=r_w} - T \right)

(see Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model )

Proxy Models

Linear superposition of Homogenous Pipe Flow Temperature Profile @model and Linear Semiplane Temperature Profile @model