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
...
where
| T_b(l) | Background temperature
Inputs
q flowrateabThermal diffusivity of | Background temperature of the surroundings |
|
r_f | | | Thermal Conductivity of the surroundingsr_w | Wellbore radius | | LaTeX Math Inline |
|---|
body | (l) where
| l | wellbore trajectory Measured Depth with reference to Earth's surface (l=0 | )Assumptions
| Constant rate | Constant intake temperature |
|---|
--uriencoded--q_s(t) = q_s = %7B\rm const%7DT_s(t) = T_s = \rm const
Equations
| LaTeX Math Block |
|---|
| anchor | Tf_Ramey |
|---|
| alignment | left |
|---|
| T(t, l) = T_be(l) - R(t) \, G_be(l) + \left(Big[ T_s - T_be(0) + R(t) \, G_be(l) \right)Big] \,cdot e^{ - l/R(t)} |
|
| LaTeX Math Block |
|---|
| G_be = \frac{dT_be}{dl} |
| | LaTeX Math Block |
|---|
| t_D(t) = \frac{a_e \, t}{r_w^2} |
|
| LaTeX Math Block |
|---|
| anchor | RelaxationRamey |
|---|
| alignment | left |
|---|
| R(t) = \frac{q_s\dot m \, c_p}{2 \pi \, a\lambda_e} \,cdot \left( T_D(t) + \frac{\lambda_e}{r_f \, U} \right) |
|
| LaTeX Math Block |
|---|
| T_D(t) = \ln \ |
| bigBig[ e^{-0.2 \, t_D} + (1.5 - 0.3719 \, e^{-t_D}) \, \sqrt{t_D} \ |
| big | | LaTeX Math Block |
|---|
|
t_D(t) = \frac{a_e \, t}{r_w^2} |
Assumptions
Intake Flowrate is constant in time | Intake Temperature is constant in time |
|---|
| LaTeX Math Inline |
|---|
| body | --uriencoded--q_s(t) = q_s = %7B\rm const%7D |
|---|
|
| | LaTeX Math Inline |
|---|
| body | T_s(t) = T_s = \rm const |
|---|
|
|
Thermal diffusivity of the surroundings is constant along-hole | Thermal Conductivity of the surroundings is constant along-hole |
|---|
| LaTeX Math Inline |
|---|
| body | a_e(l) = a_e = \rm const |
|---|
|
| | LaTeX Math Inline |
|---|
| body | \lambda_e(l) = \lambda_e = \rm const |
|---|
|
|
| Flowing pipe radius is constant along-hole | Wellbore radius is constant along-hole |
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| LaTeX Math Inline |
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| body | r_f(l) = r_f = \rm const |
|---|
|
| | LaTeX Math Inline |
|---|
| body | r_w(l) = r_w = \rm const |
|---|
|
|
| Heat Transfer Coefficient (HTC) between pipe fluid and surroundings is constant along-hole |
|---|
|
See Also
...
Physics / Fluid Dynamics / Pipe Flow Dynamics / Pipe Flow Simulation / Temperature Profile in Pipe Flow @model / Temperature Profile in Homogenous Pipe Flow @model
References
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