Motivation
One of the key challenges 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 the matter around pipeline.
| Inputs | Outputs |
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pipeline trajectory | LaTeX Math Inline |
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| body | {\bf r} = {\bf r}(l) = \{ x(l), \, y(l), \, z(l) \} |
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| along-pipe temperature distribution and evolution in time |
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inflow temperature , inflow pressure , inflow rate |
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initial temperature of the medium around the pipeline |
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Assumptions
Equations
| LaTeX Math Block |
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| \rho \, c \, \frac{\partial T}{\partial t} = \frac{d}{dl} \, \bigg( \lambda \, \frac{dT}{dl} \bigg) - \rho \, c \, v \, \frac{dT}{dl} |
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| LaTeX Math Block |
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| \rho_e \, c_e \, \frac{\partial T_e}{\partial t} = \nabla ( \lambda_e \nabla T_e) |
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| LaTeX Math Block |
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| T(t=0, l) = T_{e0}(l) |
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| LaTeX Math Block |
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| T_e(t=0, l, r) = T_{e0}(l) |
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| LaTeX Math Block |
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| T(t, l=0) = T_0(t) |
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| LaTeX Math Block |
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| T_e(t, l, r \rightarrow \infty) = T_{e0}(l) |
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| LaTeX Math Block |
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| 2 \pi \, \lambda_e \, r_w \, \frac{\partial T_e}{\partial r} \, \bigg|_{r=r_w} = 2 \pi \, r_f \, U \, \bigg( T_e \, \bigg|_{r=r_w} - T \bigg) |
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(see Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model )
Approximations
References
https://en.wikipedia.org/wiki/Darcy_friction_factor_formulae
https://neutrium.net/fluid_flow/pressure-loss-in-pipe/
