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

where

l

pipe length 

j_m = \dot m \, / A = \rho_0 \, q_0 \, / A

mass flux

\dot m (l) = \dot m = \rm const

mass flowrate 

q_0

intake flowrate 

\rho_0

intake fluid density

d

pipe diameter

A = 0.25 \, \pi \, d^2

pipe cross-section area

f= f({\rm Re}, \epsilon)

Darcy friction factor

\epsilon

inner pipe walls roughness

\displaystyle {\rm Re} = \frac{j_m \, d}{\mu}

Reynolds number 

\mu(T, p)

dynamic viscosity as function of fluid temperature T and pressure p


The accurate calculations require
solving of a self-consistent equation of Pressure Profile in Homogeneous Quasi-Isothermal Steady-State Pipe Flow @model.

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


Approximations



(2) \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

f(l)= f_0 = \rm const

\rho(l)=\rho_0= \rm const

Incompressible fluid
(3) \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]

f(l)= f_0 = \rm const

\rho(l)=\rho^* \cdot (1 + c^* \, p)

c^* \, p \ll 1

Slightly compressible fluid
(4) \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]

f(l)= f_0 = \rm const

\displaystyle \rho(l)= \frac{\rho_0}{p_0} \cdot p


Ideal gas
(5) \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}


f(l)= f_0 = \rm const

\rho(l)=\rho_0 \cdot \exp (c^* \rho^* G \, l)

Gravity dominated fluid density distribution


See also


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

Darcy friction factor ] [ Darcy friction factor @model ] [ Reynolds number in Pipe Flow ] 

Fluid friction with pipeline walls ][ Darcy friction factor in water producing/injecting wells @model ]


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