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LaTeX Math Inline
bodyT_0

Inlet temperature 

LaTeX Math Inline
bodyT(l)

Along-pipe temperature profile 

LaTeX Math Inline
bodyp_0

Inlet pressure 

LaTeX Math Inline
body\rho(T, p)

Fluid density 

LaTeX Math Inline
bodyq_0

Inlet flowrate 

LaTeX Math Inline
body\mu(T, p)

Fluid Dynamic fluid viscosity 

LaTeX Math Inline
bodyz(l)

Pipeline trajectory TVDss

LaTeX Math Inline
bodyA

Pipe cross-section area  
LaTeX Math Inline
body\theta (l)


Pipeline trajectory inclination,

LaTeX Math Inline
body--uriencoded--\displaystyle \cos \theta (l) = \frac%7Bdz%7D%7Bdl%7D

LaTeX Math Inline
body\epsilon

Inner pipe wall roughness

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LaTeX Math Inline
body--uriencoded--\displaystyle j_m =\frac%7B \rho_0 \, q_0%7D%7BA%7D

mass flux

LaTeX Math Inline
bodyq_0 = q(l=0)

fluid flow rate at pipe intake

LaTeX Math Inline
body\rho_0 = \rho(T_0, p_0)

fluid density at intake temperature and pressure

LaTeX Math Inline
bodyс(p)

Fluid Compressibility

LaTeX Math Inline
body--uriencoded--f(%7B\rm Re%7D, \, \epsilon)

Darcy friction factor

LaTeX Math Inline
body--uriencoded--\displaystyle %7B\rm Re%7D = \frac%7Bu \cdot d%7D%7B\nu%7D = \frac%7Bj_m \cdot d%7D%7B\mu(T, p)%7D

Reynolds number in Pipe Flow

LaTeX Math Inline
body--uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D

characteristic linear dimension of the pipe

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

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Reynolds number at intake

LaTeX Math Inline
body--uriencoded--f_s 0 = f(%7B\rm Re%7D_s0, \, \epsilon)

Darcy friction factor at intake inlet point

LaTeX Math Inline
body--uriencoded--\displaystyle %7B\rm Re%7D_s 0= \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7B4 \rho_s q_s%7D%7B\pi d%7D \frac%7B1%7D%7B\mu_s%7D

frac%7Bj_m \, d%7D%7B\mu_0%7D

Reynolds number at inlet point

LaTeX Math Inline
body\mu_0 = \mu(T_0, p_0)

Dynamic Fluid Viscosity at inlet point


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titleDerivation


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Incompressible fluid 

LaTeX Math Inline
body\rho(T, p) = \rho_s = \rm const
 means that compressibility vanishes 
LaTeX Math Inline
bodyc(p) = 0
 and fluid velocity is going to be constant along the pipeline trajectory 
LaTeX Math Inline
body--uriencoded--u(l) = u_s = \frac%7Bq_s%7D%7BA%7D = \rm const
.

For the constant viscosity 

LaTeX Math Inline
body\mu(T, p) = \mu_s = \rm const
 along the pipeline trajectory the Reynolds number 
LaTeX Math Inline
body--uriencoded--\displaystyle %7B\rm Re%7D = \frac%7B4 \rho_s q_s%7D%7B\pi d%7D \frac%7B1%7D%7B\mu_s%7D = \rm const
 and Darcy friction factor 
LaTeX Math Inline
body--uriencoded--f(%7B\rm Re%7D, \, \epsilon) = f_s = \rm const
 are going to be constant along the pipeline trajectory.

Equation 

LaTeX Math Block Reference
anchorPP
 becomes:

LaTeX Math Block
anchorPP
alignmentleft
\frac{dp}{dl} = \rho_s \, g \, \frac{dz}{dl}  - \frac{\rho_s \, q_s^2 }{2 A^2 d} f_s

which leads to 

LaTeX Math Block Reference
anchorgradP
 after substituting 
LaTeX Math Inline
body--uriencoded--\displaystyle \cos \theta(l) = \frac%7Bdz(l)%7D%7Bdl%7D
  and can be explicitly integrated leading to 
LaTeX Math Block Reference
anchorPPconst
.


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