...

LaTeX Math Inline

body	--uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0

LaTeX Math Inline

body	--uriencoded--\displaystyle \frac%7B\partial T%7D%7B\partial t%7D =0 \rightarrow T(t,l) = T(l)

Homogenous flow

Constant cross-section pipe area

LaTeX Math Inline

body	A

along hole

LaTeX Math Inline

body	--uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial \tau_x%7D =\frac%7B\partial p%7D%7B\partial \tau_y%7D =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l)

LaTeX Math Inline

body	A(l) = A = \rm const

Constant inclination

Constant friction along hole

LaTeX Math Inline

body	--uriencoded--\displaystyle \theta(l) = \theta = %7B\rm const%7D \rightarrow \cos \theta = \frac%7Bdz%7D%7Bdl%7D = %7B\rm const%7D

LaTeX Math Inline

body	f(l) = f = \rm const

Linear density

LaTeX Math Inline

body	--uriencoded--\rho = \rho%5e* \cdot ( 1 + c%5e* \cdot p)

which leads to

LaTeX Math Inline

body	--uriencoded--\displaystyle c(p) = \frac%7Bc%5e%7D%7B1 + c%5e \cdot p %7D

and

LaTeX Math Inline

body	--uriencoded--c%5e* \, \rho%5e* = c_0 \, \rho_0

Equations

...

Pressure profile along the pipe

LaTeX Math Block

anchor	PressureProfile
alignment	left

L = \frac{1}{2 \, G \, c^*  \rho^*}  \cdot \ln \frac{G \, \rho^2-F}{G \, \rho_0^2-F}
-\frac{d}{f} \cdot \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G}

LaTeX Math Block

anchor	1
alignment	left

 \cos \theta \neq 0

LaTeX Math Block

anchor	G0
alignment	left

L = \frac{1}{2F\, c^* \rho^*} \cdot (\rho_0^2 - \rho^2)
 -+ \frac{2d}{f} \cdot \ln \frac{\rho_0}{\rho}

LaTeX Math Block

anchor	1
alignment	left

 \cos \theta = 0

...

LaTeX Math Inline

body	--uriencoded--\displaystyle j_m = \frac%7B \dot m %7D%7B A%7D

mass flux

LaTeX Math Inline

body	--uriencoded--\displaystyle \dot m = \frac%7Bdm %7D%7B dt%7D

mass flowrate

LaTeX Math Inline

body	--uriencoded--\displaystyle q_0 = \frac%7BdV_0%7D%7Bdt%7D = \frac%7B \dot m %7D%7B \rho_0%7D

Intake volumetric flowrate

LaTeX Math Inline

body	\rho_0 = \rho(T_0, p_0)

Intake fluid density

LaTeX Math Inline

body	\Delta z(l) = z(l)-z(0)

elevation drop along pipe trajectory

LaTeX Math Inline

body	--uriencoded--f = f(%7B\rm Re%7D(T,\rho), \, \epsilon) = \rm const

Darcy friction factor

LaTeX Math Inline

body	--uriencoded--\displaystyle %7B\rm Re%7D(T,\rho) =\frac%7Bj_m \cdot d%7D%7B\mu(T,\rho)%7D

Reynolds number in Pipe Flow

LaTeX Math Inline

body	\mu(T,\rho)

dynamic viscosity as function of fluid temperature

LaTeX Math Inline

body	T

and density

LaTeX Math Inline

body	\rho

LaTeX Math Inline

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)

LaTeX Math Inline

body	G = g \, \cos \theta = \Delta Z/L = \rm const

gravity acceleration along pipe

LaTeX Math Inline

body	--uriencoded--\Delta Z = Z_%7Bout%7D - Z_%7Bin%7D

altitude drop in downwards direction (positive if descending)

LaTeX Math Inline

body	--uriencoded--F = j_m%5e2 \cdot f/(2d) = F(l) = \rm const

Expand

title	Derivation

Panel

borderColor	wheat
bgColor	mintcream
borderWidth	7

See Derivation of Pressure Profile in GF-Proxy Pipe Flow @model

Alternative forms

...

\cos \theta \neq

Volumetric Flowrate in inclined pipe:

LaTeX Math Inline

body	\cos \theta \neq 0

LaTeX Math Block

anchor	q_0G
alignment	left

q_0^2 = \frac{2 d A^2 G}{f} \cdot \left[ 

1 + \frac{ (\rho/\rho_0)^2 -1}{1- (\rho_0/\rho)^{\frac{2}{n-1}} \cdot 
\exp \left( \frac{fL/d}{ n-1}  \right)}
\right], \quad n = \frac{f}{2 \, d \, G \, c^* \, \rho^*}

LaTeX Math Block

anchor	1
alignment	left

Volumetric Flowrate in horizontal pipe:

LaTeX Math Inline

body	\cos \theta =

0

LaTeX Math Block

anchor	q0_G0
alignment	left

q_0^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{1 - (\rho/\rho_0)^2}{2 \ln (\rho_0/\rho) + fL/d}

where

LaTeX Math Block

anchor	n
alignment	left

n = \frac{f \, L^*}{d}

LaTeX Math Block

anchor

1

L*
alignment	left

 \cos \theta = 0

where

LaTeX Math Inlinebody--uriencoded--\displaystyle

L^* = \frac{1}{2 \, G \, c^* \, \rho^*} = \frac{1}{2 \, G \, c_0 \, \rho_0}

LaTeX Math Block

anchor	rho_rho0
alignment	left

\rho_0/\rho = \

frac%7B1

frac{1+

c%5e

c^* p_

0%7D%7B1+c%5e* p%7D

0}{1+c^* p}

with the following asymptotes:

Low compressible fluids:

LaTeX Math Inline

body	--uriencoded--c%5e* p \ll 1, \, \, c%5e* p_0 \ll 1

High compressible fluids:

LaTeX Math Inline

body	--uriencoded--c%5e* p \gg 1, \, \, c%5e* p_0 \gg 1

LaTeX Math Inline

body	--uriencoded--\displaystyle \rho_0/\rho = c%5e* \cdot (p_0-p)

LaTeX Math Inline

body	\displaystyle \rho_0/\rho = p_0/p

Approximations

...

Pressure Profile in GC-proxy static fluid column @model

LaTeX Math Inline

body	n \geq 1

which is equivalent to

LaTeX Math Inline

body	--uriencoded--L%5e* \geq d

and holds true for the most of practical tube diameters, as the lowest practical values of

LaTeX Math Inline

body	--uriencoded--L%5e* \geq d

are

LaTeX Math Inline

body

\dot m = 0, \, q_0 = 0 (no flow)

--uriencoded--L%5e* \geq 7,000 \, %7B\rm m%7D

LaTeX Math Block

anchor

static

q0_G
alignment	left

p(L)

q_0^2 = 
\frac{

-

2 \, d \, A^2 \, G}{f} \cdot \left [
1 + \frac{(

1+c^* \, p_0) \cdot \exp(c^* \rho^* G \, L)}{c^*} Inclined pipes

\rho/\rho_0)^2-1}{1- \exp (2 \, c_0 \, \rho_0 \, G \, L)}
 \right]
=
\frac{2 \, d \, A^2  \, g}{f \, L} \cdot \left [ 
\Delta Z + ((\rho/\rho_0)^2 -1) \cdot  \frac{ \Delta Z}{1 - \exp(2 \, c_0 \, \rho_0 \, g \,  \Delta Z)}
\right]

LaTeX Math Block

anchor	q0_G
alignment	left

\dot m = \rho_0 \, q_0

LaTeX Math Block

anchor	static
alignment	left

p(

\rho =
\rho_0 \, \exp(c_0 \, \rho_0 \, G \, L)

=

\, \

frac

sqrt{

-

+ (1+c^*

- \frac{f \,

p_0)

q_0^2}{2 \, d \, A^2} \cdot \

exp(c^* \rho^*

frac{1- \exp(-2 \, c _0 \, \rho_0 \, G \, L)}{

c^*}

G}} 
=\rho_0 \, \exp (с_0 \, \rho_0 \, g \, \Delta Z) \cdot \sqrt{ 1 - \frac{

j_m^2

8}{\

rho_0

pi^2} \cdot   \frac{f \, L}{d^5} \cdot q_0^2 \cdot \frac{1 - \exp(- 2 \, c_0 \, \rho_0 \, g \, \Delta Z) } {

2

 g \, \Delta

d} \cdot \frac{ 1- \exp \left( - c^* \rho^* G \, L \right)}{c^* \rho^* G}

Z}}

LaTeX Math Block

anchor	1
alignment	left

p(L) = p_0 + \frac{\rho/\rho_0 -1}{c_0}

Pressure Profile in GC-proxy static fluid column @model

LaTeX Math Block

anchor	static
alignment	left

\rho = \rho_0 \, \exp (c_0 \, \rho_0 \, g \, \Delta Z)

LaTeX Math Block

anchor	static
alignment	left

p(L) =  p_0 + \frac{\exp (c_0 \, \rho_0 \, g \, \Delta Z) -1}{c_0}

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