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

$\begin{array}{l}T(t, l)\end{array}$

Along-hole Temperature Profile

where

$\begin{array}{l}t\end{array}$	Flowing duration
$\begin{array}{l}l\end{array}$	Length along pipe

Inputs

$\begin{array}{l}T_s\end{array}$	Intake temperature	$\begin{array}{l}T_e(l)\end{array}$	Background temperature of the surroundings
$\begin{array}{l}\dot m\end{array}$	Mass flowrate	$\begin{array}{l}U\end{array}$	Heat Transfer Coefficient (HTC) between pipe fluid and surroundings
$\begin{array}{l}a_e\end{array}$	Thermal Diffusivity of the surroundings	$\begin{array}{l}r_f\end{array}$	Flowing pipe radius
$\begin{array}{l}\lambda_e\end{array}$	Thermal Conductivity of the surroundings	$\begin{array}{l}r_w\end{array}$	Wellbore radius

Equations

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

(2)	$\begin{array}{l}\displaystyle G_e = \frac{dT_e}{dl}\end{array}$

(3)	$\begin{array}{l}\displaystyle t_D(t) = \frac{a_e \, t}{r_w^2}\end{array}$

(4)	$\begin{array}{l}\displaystyle R(t) = \frac{\dot m \, c_p}{2 \pi \, \lambda_e} \cdot \left( T_D(t) + \frac{\lambda_e}{r_f \, U} \right)\end{array}$

(5)	$\begin{array}{l}\displaystyle T_D(t) = \ln \Big[ e^{-0.2 \, t_D} + (1.5 - 0.3719 \, e^{-t_D}) \, \sqrt{t_D} \Big]\end{array}$

Assumptions

Intake Flowrate is constant in time	Intake Temperature is constant in time
$\begin{array}{l}q_s(t) = q_s = {\rm const}\end{array}$	$\begin{array}{l}T_s(t) = T_s = \rm const\end{array}$
Thermal diffusivity of the surroundings is constant along-hole	Thermal Conductivity of the surroundings is constant along-hole
$\begin{array}{l}a_e(l) = a_e = \rm const\end{array}$	$\begin{array}{l}\lambda_e(l) = \lambda_e = \rm const\end{array}$
Flowing pipe radius is constant along-hole	Wellbore radius is constant along-hole
$\begin{array}{l}r_f(l) = r_f = \rm const\end{array}$	$\begin{array}{l}r_w(l) = r_w = \rm const\end{array}$
Heat Transfer Coefficient (HTC) between pipe fluid and surroundings is constant along-hole
$\begin{array}{l}U(l) = U = \rm const\end{array}$

References