Motivation
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Explicit solution of Proxy model of Pressure Profile in Homogeneous Steady-State Pipe Flow @model in the form of algebraic equation for the fast computation.
Outputs
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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 inclination | Constant friction along hole |
| 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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which leads to | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle c(p) = \frac%7Bc%5e*%7D%7B1 + c%5e* \cdot p %7D |
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and | LaTeX Math Inline |
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| body | --uriencoded--c%5e* \, \rho%5e* = c_0 \, \rho_0 |
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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 = \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 |
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| 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 |
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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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| mass flux |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \dot m = \frac%7Bdm %7D%7B dt%7D |
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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 = f(%7B\rm Re%7D(T,\rho), \, \epsilon) = \rm const |
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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 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 = \Delta Z/L = \rm const |
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| gravity acceleration along pipe |
| LaTeX Math Inline |
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| body | --uriencoded--\Delta Z = Z_%7Bout%7D - Z_%7Bin%7D |
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| altitude drop in downwards direction (positive if descending) |
| 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 |
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_0)^{\frac{2}{n-1}} \cdot
\exp \left( \frac{fL/d}{ n-1} \right)}
\right]
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| LaTeX Math Block |
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\cos \theta \neq |
| LaTeX Math Block |
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| q_0^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{1 - (\rho |
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_0-1}{2 \ln (\rho_0/\rho) + fL/d} |
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where
The equation
| LaTeX Math Block Reference |
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for horizontal pipelines can be re-written explicitly in terms of pressure:...
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2 \, G \, c^* \, \rho^*} = \frac{1}{2 \, G \, c_0 \, \rho_0}
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| | LaTeX Math Block |
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| anchor | rho_rho0 |
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| alignment | left |
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| \rho_0/\rho = \frac{1 |
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with the following asymptotes:
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/\ |
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| 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
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which is equivalent to | LaTeX Math Inline |
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| body | --uriencoded--L%5e* \geq d |
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and holds true for the most of practical tube diameters, as the lowest practical values of | LaTeX Math Inline |
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| body | --uriencoded--L%5e* \geq d |
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are | LaTeX Math Inline |
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| body | --uriencoded--L%5e* \geq 7,000 \, %7B\rm m%7D |
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Alternative
Pressure profile along the pipePressureProfileL1}{Gc^* \rho^* lnG \, \rho^2-F}{G(\rho/\rho_0)^2-1}{1- \exp (2 \, c_0 \, \rho_ |
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0^2-F}
-0 \, G \, L)}
\right]
=
\frac{2 \, d \, A^2 \, g}{f \, L} \cdot \left [
\ |
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ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G}Delta Z + ((\rho/\rho_0)^2 -1) \cdot \frac{ \Delta Z}{1 - \exp(2 \, c_0 \, \rho_0 \, g \, \Delta Z)}
\right] |
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1 \costhetaneqG0 | \rho =
\rho_0 \, \exp(c_0 \, \rho_0 \, G \, L |
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=12Fc^*rho^*, A^2} \cdot \frac{1- \exp(-2 \, c _0 \, \rho_ |
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0^2 -rho^2)
, L)}{G}}
=\rho_0 \, \exp (с_0 \, \rho_0 \, g \, \Delta Z) \cdot \sqrt{ 1 - \frac{ |
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2d}{f}8}{\pi^2} \cdot \frac{f \, L}{d^5} \cdot q_0^2 \ |
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ln1 \cos \theta = 0\rho = \rho_0 \, \exp (c_0 \, \rho_0 \, g \, \Delta Z) |
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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) = p_0 + \frac{\exp (c_0 \, \rho_0 \, g \, \Delta Z) -1}{c_0} |
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See also
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