Motivation
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Proxy model of Pressure Profile in Homogeneous Steady-State Pipe Flow @model in the form of algebraic equation for fast computation.
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Inputs
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Assumptions
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| Steady-State flow | Quasi-isothermal flow |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 |
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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(t, \tau_x,\tau_y,l) = p(l) |
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| Constant inclinationConstant friction along hole | Linear density |
| 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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| | Linear density | | LaTeX Math Inline |
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| body | --uriencoded--\rho = \rho%5e* \cdot ( 1 + c%5e* \cdot p) |
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Equations
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| LaTeX Math Block |
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| anchor | PressureProfile |
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| alignment | left |
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L = \frac{1}{2 \, G \, c^* \rho^*} \cdot \ln \frac{G \, \rho^2-F}{G \, \rho_0^2-F}
-\frac{d}{f} \cdot \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G} |
| LaTeX Math Block |
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\cos \theta \neq 0 |
| LaTeX Math Block |
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L = \frac{1}{2F\, c^* \rho^*} \cdot (\rho_0^2 - \rho^2)
- \frac{2d}{f} \cdot \ln \frac{\rho_0}{\rho} |
| LaTeX Math Block |
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\cos \theta = 0 |
where
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| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle j_m = \frac%7B \dot m %7D%7B A%7D |
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| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \dot m = \frac%7Bdm %7D%7B dt%7D |
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| 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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| LaTeX Math Inline |
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| body | \rho_0 = \rho(T_0, p_0) |
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| LaTeX Math Inline |
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| body | \Delta z(l) = z(l)-z(0) |
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| LaTeX Math Inline |
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| body | --uriencoded--f = f(%7B\rm Re%7D(T,\rho), \, \epsilon) = \rm const |
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| 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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dynamic viscosity as function of fluid temperature
and density ...
| 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)
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| LaTeX Math Inline |
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| body | G = g \, \cos \theta = \rm const |
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| LaTeX Math Inline |
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| body | --uriencoded--F = j_m%5e2 \cdot f/(2d) = F(l) = \rm const |
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Alternative forms
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| LaTeX Math Block |
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q_0^2 = \frac{2 d A^2 G}{f} \cdot \left[
1 + \frac{ (\rho/\rho_0)^2 -1}{1- (\rho_0/\rho)^{\frac{2}{n-1}} \cdot
\exp \left( \frac{fL/d}{ n-1} \right)}
\right], \quad n = \frac{f}{2 \, d \, G \, c^* \, \rho^*}
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| LaTeX Math Block |
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\cos \theta \neq 0 |
| LaTeX Math Block |
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q_0^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{1 - (\rho/\rho_0)^2}{2 \ln (\rho_0/\rho) + fL/d} |
Equation
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| LaTeX Math Block |
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\cos \theta = 0 |
where
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| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \rho_0/\rho = \frac%7B1+c%5e* p_0%7D%7B1+c%5e* p%7D |
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with the following asymptotes:
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Low compressible fluids:
| LaTeX Math Inline |
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| body | --uriencoded--c%5e* p \ll 1, \, \, c%5e* p_0 \ll 1 |
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High compressible fluids:
| LaTeX Math Inline |
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| body | --uriencoded--c%5e* p \gg 1, \, \, c%5e* p_0 \gg 1 |
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| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \rho_0/\rho = c%5e* \cdot (p_0-p) |
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| LaTeX Math Inline |
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| body | \displaystyle \rho_0/\rho = p_0/p |
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Approximations
Pressure profile in static fluid column, no flow: | LaTeX Math Inline |
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| body | \dot m = 0, \, q_0 = 0 |
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| LaTeX Math Block |
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| anchor | static |
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| alignment | left |
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| p(L) = \frac{1}{c^*} \cdot \left[ -1 + (1+c^* \, p_0) \cdot \exp(c^* \rho^* G \, L) \right] |
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| Expand |
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| Panel |
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| borderColor | wheat |
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| bgColor | mintcream |
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| borderWidth | 7 |
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See also
References
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| Panel |
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| bgColor | papayawhip |
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| title | ARAX |
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