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\left[\rho(p) -  j_m^2 \cdot c(p)   \right]  \frac{dp}{dl} = \rho^2(p) \, g \, \cos \theta(l)  - \frac{ j_m^2 }{2 d} \cdot  f(p)



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p(l=0) = p_0




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u(l) = \frac{j_m}{\rho(l)}



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q(l) =A \cdot u(l)


where

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body--uriencoded--\displaystyle j_m =\frac%7B \rho_0 \, q_0%7D%7BA%7D= \rm const

mass flux

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bodyq_0 = q(l=0)

Fluid flowrate at inlet point (

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)

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body\rho_0 = \rho(T_0, p_0)

Fluid density at inlet point (

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bodyl=0
)

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body\rho(l) = \rho(T(l), p(l))

Fluid density at any point 

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bodyl

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body--uriencoded--\displaystyle с(p) = \frac%7B1%7D%7B\rho%7D \left( \frac%7B\partial \rho%7D%7B\partial p%7D \right)_T

Fluid Compressibility

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body--uriencoded--f(T,

p

\rho) = f(%7B\rm Re%7D(T,

p

\rho), \, \epsilon)

Darcy friction factor

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body--uriencoded--\displaystyle %7B\rm Re%7D(T,

p

\rho) = \frac%7Bj_m \cdot d%7D%7B\mu(T,

p

\rho)%7D

Reynolds number in Pipe Flow

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body\mu(T,\rho)

dynamic viscosity as function of fluid temperature 

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bodyT
 and density 
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body\rho

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body--uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D= \rm const

Characteristic linear dimension of the pipe

(or exactly a pipe diameter in case of a circular pipe)



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titleDerivation


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See Derivation of Pressure Profile in Steady-State Homogeneous Pipe Flow @model.


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