Motivation

Analytical model of temperature step-response in a Homogenous Stationary Pipe Flow with account for the heat exchange with surroundings

.

Model equally works for wellbore flow, ground, on-ground and bottom-water pipelines.

Outputs

Along-hole Temperature Profile

where

	Flowing duration
	Length along pipe

Inputs

	Intake temperature		Background temperature of the surroundings
	Mass flowrate		Heat Transfer Coefficient (HTC) between pipe fluid and surroundings
	Thermal Diffusivity of the surroundings		Flowing pipe radius
	Thermal Conductivity of the surroundings		Wellbore radius

Equations

T(t, l) = T_e(l) - R(t) \, G_e(l)  +  \Big[ T_s - T_e(0) + R(t) \, G_e(l) \Big]  \cdot e^{ - l/R(t)}

G_e = \frac{dT_e}{dl}

t_D(t) = \frac{a_e \, t}{r_w^2}

R(t) = \frac{\dot m \, c_p}{2 \pi \, \lambda_e} \cdot \left( T_D(t) + \frac{\lambda_e}{r_f \, U} \right)

T_D(t) = \ln \Big[ e^{-0.2 \, t_D} + (1.5 - 0.3719 \, e^{-t_D}) \, \sqrt{t_D} \Big]

Assumptions

Intake Flowrate is constant in time	Intake Temperature is constant in time

Thermal diffusivity of the surroundings is constant along-hole	Thermal Conductivity of the surroundings is constant along-hole

Flowing pipe radius is constant along-hole	Wellbore radius is constant along-hole

Heat Transfer Coefficient (HTC) between pipe fluid and surroundings is constant along-hole

References