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Steady-State flowIsothermal or Quasi-isothermal flow

LaTeX Math Inline
body--uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 \rightarrow p(t,l) = p(l)

LaTeX Math Inline
body--uriencoded--\displaystyle \frac%7B\partial T%7D%7B\partial t%7D =0 \rightarrow T(t,l) = T(l)

Homogenous flow

Constant cross-section pipe area

LaTeX Math Inline
bodyA
along hole

LaTeX Math Inline
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)

LaTeX Math Inline
bodyA(l) = A = \rm const

Incompressible fluid 

LaTeX Math Inline
body\rho(T, p)=\rho_0 = \rm const
 → 
LaTeX Math Inline
body\mu(T, \rho) =\mu_0 = \rm const


Equations

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Pressure profilePressure gradient profile


LaTeX Math Block
anchorPPconst
alignmentleft
p(l) = p_0 + \rho_0 \, g \, \Delta z(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 \, l



LaTeX Math Block
anchorgradP
alignmentleft
\frac{dp}{dl} = \rho_0 \, g \cos \theta(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0


Mass FluxMass Flowrate


LaTeX Math Block
anchorMassFlux
alignmentleft
j_m = \rho_0 \cdot \sqrt{\frac{2 \, d}{f_0 \, l }} \cdot \sqrt{g \, \Delta z(l) + (p_0 - p)/ \rho_0}



LaTeX Math Block
anchorMassFlowrate
alignmentleft
\dot m = j_m \cdot A = \rho_0 \cdot A \cdot \sqrt{\frac{2 \, d}{f_0 \, l }} \cdot \sqrt{g \, \Delta z(l) + (p_0 - p)/ \rho_s}


 Volumetric Flowrate

Intake Fluid velocity


LaTeX Math Block
anchorPPconst
alignmentleft
q_0 = \dot m / \rho_0 = A \cdot \sqrt{\frac{2 \, d }{ f_0 \, l }} \cdot \sqrt{  g \, \Delta z(l) + (p_0 - p)/ \rho_s }



LaTeX Math Block
anchorPPconst
alignmentleft
u_0 = j_m/ \rho_0 =q_0 / A = \sqrt{\frac{2 \, d }{ f_0 \, l }} \cdot \sqrt{  g \, \Delta z(l) + (p_0 - p)/ \rho_s }


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