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\neq
 Volumetric Flowrate in inclined pipe


LaTeX Math Block
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 \cos \theta \neq 0



LaTeX Math Block
anchorq_0G
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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^*}




Volumetric Flowrate in horizontal pipe


LaTeX Math Block
anchor1
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 \cos \theta 
= 0



LaTeX Math Block
anchorq0_G0
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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}
LaTeX Math Block
anchor1
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 \cos \theta = 0




where

LaTeX Math Inline
body--uriencoded--\displaystyle \rho_0/\rho = \frac%7B1+c%5e* p_0%7D%7B1+c%5e* p%7D

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Pressure Profile in GC-proxy static fluid column @model

LaTeX Math Inline
body\dot m = 0, \, q_0 = 0
 (no flow)


LaTeX Math Block
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p(L) = \frac{-1 + (1+c^* \, p_0) \cdot \exp(c^*  \rho^*  G \, L)}{c^*}


 Inclined pipespipe



LaTeX Math Block
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\dotrho m = A\rho_0 \cdot, \sqrt{2 \,exp (c^* \rho^* (p_0 - p)}\, G \, L) \cdot \sqrt{ 1 - \frac{f}{12d} + 0.5\cdot \frac{j_m^2}{G \, c^* (p_0 + p)}{\rho_0^2} \cdot ( 1 - \exp(-2 \ln, \frac{1+c^*p_0}{1 + c^* p} + \frac{f \, L}{d}}} \rho^* \, G \, L))}


LaTeX Math Block
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\rho = \rho_0 \,p(L) = \frac{1}{c^*} \cdot \left[ 
-1 + (1+c^* p_0) \cdot \exp (c^* \rho^* \, G \, L) \cdot \sqrt{  1 - \frac{f}{2d}  \cdot \frac{j_m^2}{G \, \rho_0^2} \cdot \big( 1 - \exp (-2 \, c^* \rho^* \, G \, L) \big) }
\right]


 Horizontal pipe


LaTeX Math Block
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\dot m = A \cdot \sqrt{2 \, \rho^* (p_0 - p)} \cdot \sqrt{ \frac{1 + 0.5 \, c^* (p_0 + p)}{2 \ln \frac{1+c^*p_0}{1 + c^* p} + \frac{f \, L}{d}}}



LaTeX Math Block
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p(L) = \frac{1}{c^*} \cdot \left[ 
-1 + (1+c^* p_0) \cdot \exp (c^* \rho^* \, G \, L) \cdot \sqrt{  1 - \frac{f}{2d}  \cdot \frac{j_m^2}{G \rho_0^2} \cdot \big(1 - \exp (-2 \, c^* \rho^* \, G \, L) \big) }
\right]


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