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.

\left(   \frac{dp}{dl} \right)_f = - \frac{ j_m^2}{2 d}  \cdot \frac{f(l)}{\rho(l)}

where

	pipe length
	mass flux
	mass flowrate
	pipe diameter
	pipe cross-section area
	Darcy friction factor
	inner pipe walls roughness
	Reynolds number
	dynamic viscosity as function of fluid temperature and pressure

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

Approximations

\Delta p(L)=- \frac{j_m^2}{\rho_0} \cdot \frac{f_0 \, L}{2 \, d }

Incompressible fluid

\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]

Slightly compressible fluid

\Delta p (L) =- p_0 \cdot \left[ 1- \sqrt{ 

1 - \frac{j_m^2}{\rho_0 \, p_0} \cdot \frac{f_o L}{d} 

} \right]

Ideal gas

\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}

Gravity dominated density distribution

See also