Motivation
One of the key challenges in Pipe Flow Dynamics is to predict the along-hole temperature distribution during the stationary fluid transport.
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| along-pipe temperature distribution and evolution in time |
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Assumptions
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Equations
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| \rho \, c \, \frac{\partial T}{\partial t} = \frac{d}{dl} \, \bigg( \lambda \, \frac{dT}{dl} \bigg) - \left( \sum_{\alpha} \rho_{\alpha} \, c_{\alpha} \, v_{\alpha} \right) \, \frac{dT}{dl} + \frac{2 \lambda}{\lambda_e} \cdot \frac{r_f}{r_w^2} \cdot U \cdot \left[ T_e(t, l, r_w) - T \right] |
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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_{e0}(l) |
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| T_e(t=0, l, r) = T_{e0}(l) |
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| T(t, l=0) = T_0(t) |
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| T_e(t, l, r \rightarrow \infty) = T_{e0}(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 \cdot \left( T_e \, \bigg|_{r=r_w} - T \right) |
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[ Homogenous Pipe Flow Temperature Profile @model ][ Pipe Flow Temperature Analytical Ramey @model ]
References
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