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There are few popular practical approximations based on assumption of constant friction factor and  linear density-pressure equation of state.


Approximations

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LaTeX Math Block
anchor
c0
2
alignmentleft
\
left( \frac{dp}{dl} \right)_f = 
Delta p(L)=- \frac{
j_m^2}{
2 d
\rho_0} 
\cdot \frac{f_0 \, L}{2 \, d }


LaTeX Math Inline
bodyf(l)= f_0 = \rm const

LaTeX Math Inline
body\rho(l)=\rho_0

}

=

\rm

const


LaTeX Math Block
anchor2
alignmentleft
\Delta p (L) =- \frac{\rho_0}{c^*} \cdot  \left[
1 - \sqrt{  1 - j_m^2 \cdot \frac{c^* \rho^*}{\rho_
0
0^2} 
\cdot \frac{f_0
\, 
L}{
2 \, 
d
} 
}}
\right]


LaTeX Math Inline
bodyf(l)= f_0 = \rm const

LaTeX Math Inline
body--uriencoded--\rho(l)=\

rho_0= \rm const

rho%5e* \cdot (1 + c%5e* \, p)

LaTeX Math Inline
body--uriencoded--c%5e* \, p \ll 1


LaTeX Math Block
anchor
c0
2
alignmentleft
\
left( \frac{dp}{dl} \right)_f = - \frac{ j_m^2}{2 d} 
Delta p (L) =- p_0 \cdot \left[ 1- \sqrt{ 1 -\frac{j_m^}{\rho_0 \, p_0} \cdot \frac{f_
0}{\rho(p)} 
o L}{d}} \right]


LaTeX Math Inline
bodyf(l)= f_0 = \rm const

LaTeX Math Inline
body--uriencoded--\displaystyle \rho(l)= \frac%7B\rho_0%7D%7Bp_0%7D \cdot p



LaTeX Math Block
anchor2
alignmentleft
\Delta p (L) =- \frac{j_m^2}{\rho_0} \cdot \frac{f_0}{2 \, d} \cdot 
\frac{ 1- \exp \left( - c^* \rho^* G \, L \right)}{c^* \rho^* G}



LaTeX Math Inline
bodyf(l)= f_0 = \rm const

LaTeX Math Inline
body--uriencoded--\rho(l)=\rho%5e* \cdot (1 + c%5e* \, p)


See also

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Physics / Fluid Dynamics / Pipe Flow Dynamics / Pipe Flow Simulation / Pressure Profile in Homogeneous Quasi-Isothermal Steady-State Pipe Flow @model

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