The pressure drop in pipe flow due to fluid friction with pipe walls depends on mass flux density and friction factor distribution along the pipe.

(1)	$\begin{array}{l}\displaystyle \left( \frac{dp}{dl} \right)_f = - \frac{ j_m^2}{2 d} \cdot \frac{f(l)}{\rho(l)}\end{array}$

where

$\begin{array}{l}l\end{array}$	pipe length
$\begin{array}{l}j_m = \dot m \, / A = \rho_0 \, q_0 \, / A\end{array}$	mass flux
$\begin{array}{l}\dot m (l) = \dot m = \rm const\end{array}$	mass flowrate
$\begin{array}{l}q_0\end{array}$	intake flowrate
$\begin{array}{l}\rho_0\end{array}$	intake fluid density
$\begin{array}{l}d\end{array}$	pipe diameter
$\begin{array}{l}A = 0.25 \, \pi \, d^2\end{array}$	pipe cross-section area
$\begin{array}{l}f= f({\rm Re}, \epsilon)\end{array}$	Darcy friction factor
$\begin{array}{l}\epsilon\end{array}$	inner pipe walls roughness
$\begin{array}{l}\displaystyle {\rm Re} = \frac{j_m \, d}{\mu}\end{array}$	Reynolds number
$\begin{array}{l}\mu(T, p)\end{array}$	dynamic viscosity as function of fluid temperature $\begin{array}{l}T\end{array}$ and pressure $\begin{array}{l}p\end{array}$

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

Approximations

(2)	$\begin{array}{l}\displaystyle \Delta p(L)=- \frac{j_m^2}{\rho_0} \cdot \frac{f_0 \, L}{2 \, d } = - \frac{8}{\pi^2} \frac{\rho_0 \, q_0^2}{d^5} \, f_0 \, L\end{array}$

$\begin{array}{l}f(l)= f_0 = \rm const\end{array}$

$\begin{array}{l}\rho(l)=\rho_0= \rm const\end{array}$

(3)	$\begin{array}{l}\displaystyle \Delta p (L) =- \frac{\rho_0}{c^} \cdot \left[ 1 - \sqrt{ 1 - j_m^2 \cdot \frac{c^ \rho^*}{\rho_0^2} \cdot \frac{f_0 L}{d}} \right]\end{array}$

$\begin{array}{l}f(l)= f_0 = \rm const\end{array}$

$\begin{array}{l}\rho(l)=\rho^* \cdot (1 + c^* \, p)\end{array}$

$\begin{array}{l}c^* \, p \ll 1\end{array}$

(4)	$\begin{array}{l}\displaystyle \Delta p (L) =- p_0 \cdot \left[ 1- \sqrt{ 1 - \frac{j_m^2}{\rho_0 \, p_0} \cdot \frac{f_0 L}{d} } \right]\end{array}$

$\begin{array}{l}f(l)= f_0 = \rm const\end{array}$

$\begin{array}{l}\displaystyle \rho(l)= \frac{\rho_0}{p_0} \cdot p\end{array}$

(5)	$\begin{array}{l}\displaystyle \Delta p (L) =- \frac{j_m^2}{\rho_0} \cdot \frac{f_0}{2 \, d} \cdot \frac{ 1- \exp \left( - c^* \rho^* G \, L \right)}{c^* \rho^* G}\end{array}$

$\begin{array}{l}f(l)= f_0 = \rm const\end{array}$

$\begin{array}{l}\rho(l)=\rho_0 \cdot \exp (c^* \rho^* G \, l)\end{array}$

Gravity dominated fluid density distribution

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