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Analytical model of temperature build up step-response in a Homogenous Stationary Pipe Flow with account for the heat exchange with surroundings
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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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| G_e = \frac{dT_e}{dl} |
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| t_D(t) = \frac{a_e \, t}{r_w^2} |
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anchor | RelaxationRamey |
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alignment | left |
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| R(t) = \frac{q_s\dot m \, c_p}{2 \pi \, a\lambda_e} \,cdot \left( T_D(t) + \frac{\lambda_e}{r_f \, U} \right) |
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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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