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Assumptions
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Stationary Isothermal or Homogenous flow | Constant cross-section pipe area conditionsA | along hole LaTeX Math Inline |
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body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 |
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\rightarrow p(t,l) = p(l) | LaTeX Math Inline |
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body | --uriencoded--\displaystyle \frac%7B\partial T%7D%7B\partial t%7D =0 \rightarrow T(t,l) = T(l) |
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Homogenous flow | |
LaTeX Math Inline |
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body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial \tau_x%7D =\frac%7B\partial p%7D%7B\partial \tau_y%7D =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l) |
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Constant inclination |
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LaTeX Math Inline |
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body | --uriencoded--\displaystyle \theta(l) = \theta = %7B\rm const%7D \rightarrow \cos \theta = \frac%7Bdz%7D%7Bdl%7D = %7B\rm const%7D |
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Equations
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Pressure profile along the pipe |
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LaTeX Math Block |
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anchor | PressureProfile |
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alignment | left |
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| L = L(p) = \int_{p_0}^{p} \frac{ \rho(p) - j_m^2 \, c(p) }{G \, \rho^2(p) - F(\rho(p))} \, dp
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Density form |
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LaTeX Math Block |
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anchor | PressureProfile |
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alignment | left |
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| L = L(\rho) =\int_{\rho_0}^{\rho} \frac{ 1/c(\rho) - j_m^2/\rho }{G \, \rho^2 - F(\rho)} \, d\rho
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Pressure-Density form |
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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 \, 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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This form is useful for derivation of Pressure Profile in GF-Proxy Pipe Flow @model and |
Approximations
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See also