Motivation
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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.
In many practical cases the along-hole temperature distribution during the stationary fluid flow can be approximated by homogenous fluid flow model.
Inputs & Outputs
Outputs
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| along-pipe temperature distribution and evolution in time |
Inputs
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| Inputs | Outputs |
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pipeline trajectory | LaTeX Math Inline |
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| body | --uriencoded--%7B\bf r%7D(l) |
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{ r} = {\bf r}{ | %7B x(l), \, y(l), \, z(l) \ |
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}along-pipe temperature (t lpipeline cross-section area distribution and evolution in timefluid density rho and \mu(T, p) | inflow temperature 0t, inflow pressure | initial temperature of the medium around the pipeline |
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, inflow rate q_0 | initial temperature | LaTeX Math Inline |
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body | T_g the specific heat capacityc_p(l) | , thermal conductivity of heat transfer coefficient based
Assumptions
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Equations
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| 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} + \frac{U}{r_w} \cdot \big( T_e(t, l, r_w) - T \big) |
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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 \ |
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,biggleft( T_e \, \bigg|_{r=r_w} - T \ |
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bigg(see Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model )
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Approximations
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See also
References
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https://en.wikipedia.org/wiki/Darcy_friction_factor_formulae
https://neutrium.net/fluid_flow/pressure-loss-in-pipe/
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