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| \biggleft( 1 - \frac{c(p) \, \rho_s^2 \, q_s^2}{A^2} \bigg right) \frac{dp}{dl} = \rho \, g \, \frac{dz}{dl} - \frac{\rho_s^2 \, q_s^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho} |
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| q(l) = \frac{\rho_s \cdot q_0}{\rho} |
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| u(l) = \frac{\rho_s \cdot q_s}{\rho \cdot A} |
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| p(l=0) = p_s |
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| q(l=0) = q_s |
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| \rho(T_s, p_s) = \rho_s |
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-\frac{1}{c} \frac{d}{dl} \left( \frac{1}{\rho} \right) + \frac{\rho_s^2 q_s^2}{2A^2} \frac{d}{dl} \left( \frac{1}{\rho^2} \right) + \frac{\rho_s^2 q_s^2}{2A^2} \frac{f}{d} \left( \frac{1}{\rho^2} \right) - g \frac{dz}{dl} = 0 |
See derivation at
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displaytext | derivation here |
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| anchor | drho |
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| page | Derivation of Pressure Profile in Stationary Quasi-Isothermal Homogenous Pipe Flow @model |
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.It does not give much benefit in computations comparing to
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but it makes an easy path to some proxy models (like Slightly Compressible Fluid and Ideal Gas).
Approximations
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Incompressible pipe flow
with constant viscosity ...
