Motivation
Numerical quadrature solution of Pressure Profile in Homogeneous Steady-State Pipe Flow @model
Outputs
Assumptions
| Steady-State flow | Isothermal or Quasi-isothermal flow |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 \rightarrow p(t,l) = p(l) |
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| | 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(\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
| 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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where
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle j_m = \frac%7B \dot m %7D%7B A%7D= \rm const |
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| mass flux |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \dot m = \frac%7Bdm %7D%7B dt%7D= \rm const |
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| mass flowrate |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle q_0 = \frac%7BdV_0%7D%7Bdt%7D = \frac%7B \dot m %7D%7B \rho_0%7D |
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| Intake volumetric flowrate |
| LaTeX Math Inline |
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| body | \rho_0 = \rho(T_0, p_0) |
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| Intake fluid density |
| LaTeX Math Inline |
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| body | \Delta z(l) = z(l)-z(0) |
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| elevation drop along pipe trajectory |
| LaTeX Math Inline |
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| body | --uriencoded--f(T,\rho) = f(%7B\rm Re%7D(T,\rho), \, \epsilon) |
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| Darcy friction factor |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle %7B\rm Re%7D(T, \rho) = \frac%7Bj_m \cdot d%7D%7B\mu(T, \rho)%7D |
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| Reynolds number in Pipe Flow |
| dynamic viscosity as function of fluid temperature and density |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle c(T,p) = \frac%7B1%7D%7B\rho%7D \left( \frac%7B\partial \rho%7D%7B\partial p%7D \right)_T |
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| fluid compressibility |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D= \rm const |
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| characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe) |
| LaTeX Math Inline |
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| body | G = g \, \cos \theta= \rm const |
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| gravity acceleration along pipe |
| LaTeX Math Inline |
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| body | --uriencoded--F(T, \rho) = j_m%5e2 \cdot f(T,\rho)/(2d) |
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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 Pressure Profile in GFC-Proxy Pipe Flow @model |
See also
