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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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| 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
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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} |
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| \rho_e \, c_e \, \frac{\partial T_e}{\partial t} = \nabla ( \lambda_e \nabla T_e) |
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| T(t=0, l) = T_g(l) |
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| T_e(t=0, l, r) = T_g(l) |
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| T(t, l=0) = T_0(t) |
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| T_e(t, l, r \rightarrow \infty) = T_g(l) |
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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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| T_g(l) = T_{0e} + \int_{z_0}^{z(l)} G_T(z) dz = T_{0e} + \int_{l_0}^l G_T(z(l)) \cos \theta dl |
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| G_T(z(l)) = \frac{j_e}{\lambda_e(l)} |
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where
(see Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model )
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