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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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| LaTeX Math Block |
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| n = \frac{f \, L^*}{d}
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| | LaTeX Math Block |
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| L^* = \frac{1}{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+c^* p_0}{1+c^* p}
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which is equivalent to | LaTeX Math Inline |
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| body | --uriencoded--L%5e* \geq 1d |
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and holds true for the most of practical tube diameters (< 1 m ), 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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| LaTeX Math Block |
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| q_0^2 =
\frac{2 \, d \, A^2 \, G}{c^* \rho^*f} \cdot \frac{left [
1 -+ \frac{(\rho/\rho_0)^2-1}{1- \cdotexp (2 \exp, c_0 \left( -L/ L^* \right)}{2 \ln (\rho_0/\rho) + fL/d \cdot (1- \exp \left( - L/ L^* \right))/(L/L^*)}, \rho_0 \, G \, L)}
\right]
=
\frac{2 \, d \, A^2 \, g}{f \, L} \cdot \left [
\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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| LaTeX Math Block |
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| \dot m = \rho_0 \, q_0 | | LaTeX Math Block |
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| \dot m^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{\rho_0^2 - \rho^2 \cdot \exp \left( -L/ L^* \right)}{2 \ln (\rho_0/\rho) + fL/d \cdot (1- \exp \left( - L/ L^* \right))/(L/L^*)} |
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| LaTeX Math Block |
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| anchor | static |
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| alignment | left |
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| \rho =
\rho_0 \, \exp (2(c_0 \, \rho_0 \, G \, L/L^*) \cdot, \sqrt{ 1 - \frac{f}{2d \, q_0^2}{2 \, d \, A^2} \cdot \frac{j_m^2}{G1- \exp(-2 \, c _0 \, \rho_0^2}0 \cdot, ( 1 - \exp(-L/L^*)} | | LaTeX Math Block |
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| p(L) = \frac{1}{c^*} \cdot \left[
-1 + (1+c^* p_0) \cdot \exp (2 \, L/L^*G \, L)}{G}}
=\rho_0 \, \exp (с_0 \, \rho_0 \, g \, \Delta Z) \cdot \sqrt{ 1 - \frac{f8}{2d\pi^2} \cdot \frac{j_m^2f \, L}{Gd^5} \rhocdot q_0^2} \cdot \left(frac{1 - \exp(- 2 \, c_0 \, \rho_0 \, g \, \Delta (-L/L^*) \right) }
\right]Z) } { g \, \Delta Z}} |
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| LaTeX Math Block |
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| p(L) = p_0 + \frac{\rho/\rho_0 -1}{c_0} |
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| Pressure Profile in GC-proxy static fluid column @model |
| LaTeX Math Block |
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| anchor | static |
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| alignment | left |
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| \rho = \rho_0 \, \exp (L/L^*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{-1 +\exp (1+c^*c_0 \, p\rho_0) \, g \cdot, \exp(L/L^*)}{c^*} Delta Z) -1}{c_0} |
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See also
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