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Pressure profile

Pressure gradient profile

Fluid velocity

Fluid rate

LaTeX Math Block

anchor	PPconst
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p(l) = p_s + \rho_s \, g \, z(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s \, l

LaTeX Math Block

anchor	gradP
alignment	left

\frac{dp}{dl} = \rho_s \, g \cos \theta(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s

LaTeX Math Block

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q(l) =q_s = \rm const

LaTeX Math Block

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u(l) = u_s = \frac{q_s}{A} = \rm const

where

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LaTeX Math Inline

body	\theta(l)

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trajectory inclination

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title	Derivation

Panel

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Incompressible fluid

LaTeX Math Inline

body	\rho(T, p) = \rho_s = \rm const

means that compressibility vanishes

LaTeX Math Inline

body	c(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

anchor	PP

becomes:

LaTeX Math Block

anchor	PP
alignment	left

\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

anchor	gradP

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

anchor	PPconst

.

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