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
Inputs
Equations
| LaTeX Math Block |
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| anchor | Tf_Ramey |
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| alignment | left |
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| 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)} |
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| LaTeX Math Block |
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| G_e = \frac{dT_e}{dl} |
| | LaTeX Math Block |
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| t_D(t) = \frac{a_e \, t}{r_w^2} |
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| LaTeX Math Block |
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| anchor | RelaxationRamey |
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| alignment | left |
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| R(t) = \frac{q_s}{2 \pi \, a_e} \, \left( T_D(t) + \frac{\lambda_e}{r_f \, U} \right) |
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| LaTeX Math Block |
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| T_D(t) = \ln \Big[ e^{-0.2 \, t_D} + (1.5 - 0.3719 \, e^{-t_D}) \, \sqrt{t_D} \Big] |
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Assumptions
Intake Flowrate is constant in time | Intake Temperature is constant in time |
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| LaTeX Math Inline |
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| body | --uriencoded--q_s(t) = q_s = %7B\rm const%7D |
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| | LaTeX Math Inline |
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| body | T_s(t) = T_s = \rm const |
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Thermal diffusivity of the surroundings is constant along-hole | Thermal Conductivity of the surroundings is constant along-hole |
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| LaTeX Math Inline |
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| body | a_e(l) = a_e = \rm const |
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| | LaTeX Math Inline |
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| body | \lambda_e(l) = \lambda_e = \rm const |
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| 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 |
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| | LaTeX Math Inline |
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| body | r_w(l) = r_w = \rm const |
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| Heat Transfer Coefficient (HTC) between pipe fluid and surroundings is constant along-hole |
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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
