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| \bigg( 1 - \frac{c(p) \, \rho_0^2 \, q_0^2}{A^2} \bigg ) \frac{dp}{dl} = \rho(p) \, g \, \frac{dz}{dl} - \frac{\rho_0^2 \, q_0^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho(p)} |
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| u(l) = \frac{\rho_0 \cdot q_0}{\rho(p) \cdot A} |
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| q(l) = \frac{\rho_0 \cdot q_0}{\rho(p)} |
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where
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| body | --uriencoded--f(%7B\rm Re%7D, \, \epsilon) |
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| Darcy friction factor |
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| body | --uriencoded--\displaystyle %7B\rm Re%7D = \frac%7Bu \cdot d%7D%7B\nu%7D |
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| Reynolds number |
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| body | --uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D |
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| characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe) |
See Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model.
Approximations
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Incompressible pipe flow
with constant friction ...
| %7B\rm Re%7D, \, \epsilon) = f_0 |
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| Pressure profile | Pressure gradient profile | Fluid velocity | Fluid rate |
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| p(l) = p_0 + \rho \, g \, z(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 \, l |
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| \frac{dp}{dl} = \rho \, g \cos \theta(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 |
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| u(l) = \frac{q_0}{A} |
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| q(l) =q_0 = \rm const |
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