Let's start with Pressure Profile in Homogeneous Steady-State Pipe Flow @model:
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| \left[\rho(p) - j_m^2 \cdot c(p) \right] \frac{dp}{dl} = \rho^2(p) \, g \, \cos \theta(l) - \frac{ j_m^2 }{2 d} \cdot f(p) |
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| p(l=0) = p_0 |
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and assume constant pipe inclination:
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\theta(l) = \theta = \rm const |
Let's define constant number:
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G = g \cdot \cos \theta = \rm const |
and rewrite the equation
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as: LaTeX Math Block |
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\frac{\left[\rho(p) - j_m^2 \cdot c(p) \right] \, dp}{\rho^2(p) \, G - \frac{ j_m^2 }{2 d} \cdot f(p)} = dl
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The integration of the left side of
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with the boundary condition LaTeX Math Block Reference |
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leads to: LaTeX Math Block |
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anchor | PressureProfile |
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alignment | left |
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L =\int_{p_0}^{p} \frac{ \rho(p) - j_m^2 \, c(p) }{G \, \rho^2(p) - F(\rho(p))} \, dp
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where
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body | --uriencoded--\displaystyle F(\rho) = \frac%7B j_m%5e2 %7D%7B2 d%7D \cdot f(\rho) |
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This can be further re-written as:
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L = \int_{p_0}^{p} \frac{ \rho \, dp}{G \, \rho^2 - F(\rho)}
- j_m^2 \cdot \int_{\rho_0}^{\rho} \frac{1}{\rho} \, \frac{d \rho}{G \, \rho^2 - F(\rho)}
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or
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L =\int_{\rho_0}^{\rho} \frac{ 1/c(\rho) - j_m^2/\rho }{G \, \rho^2 - F(\rho)} \, d\rho
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See also
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Physics / Fluid Dynamics / Pipe Flow Dynamics / Pipe Flow Simulation / Pressure Profile in Homogeneous Steady-State Pipe Flow @model / Pressure Profile in LG-Proxy Pipe Flow @model