Motivation
...
Explicit solution of Proxy model of Pressure Profile in Homogeneous Steady-State Pipe Flow @model in the form of algebraic equation for the fast computation.
Outputs
...
...
Steady-State flow | Quasi-isothermal flow |
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 \rightarrow p(t,l) = p(l) |
---|
|
| LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \frac%7B\partial T%7D%7B\partial t%7D =0 \rightarrow T(t,l) = T(l) |
---|
|
|
Homogenous flow | |
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) |
---|
|
| |
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 |
---|
|
| |
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 |
---|
| \cos \theta \neq 0 |
|
LaTeX Math Block |
---|
| L = \frac{1}{2F\, c^* \rho^*} \cdot (\rho_0^2 - \rho^2)
-+ \frac{2d}{f} \cdot \ln \frac{\rho_0}{\rho} |
| LaTeX Math Block |
---|
| \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 |
| dynamic viscosity as function of fluid temperature and density |
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 |
---|
|
|
|
...
LaTeX Math Block Reference |
---|
|
...
Alternative forms
...
...
...
...
f} \cdot \left[
1 + \frac{ (\ |
|
...
...
...
...
...
/\rho)^{\frac{2}{n-1}} \cdot
\exp \left( |
|
...
\frac{fL/d}{ n-1} \right)}
\right]
|
|
|
LaTeX Math Block |
---|
| 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 |
---|
| n = \frac{f \, L^*}{d}
|
| LaTeX Math Block |
---|
| 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{1+c^* p_0}{1 |
|
...
...
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 |
---|
|
|
|
...
displaystyle \rho_0/\rho = c%5e* \cdot (p_0-p) |
|
| LaTeX Math Inline |
---|
body | \displaystyle \rho_0/\rho = p_0/p |
---|
|
|
Approximations
...
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 | --uriencoded--L%5e* \geq 7,000 \, %7B\rm m%7D |
---|
| |
Alternative
Pressure profile along the pipePressureProfileL{1}Gc^* \rho^* lnG \, \rho^2-F}{G(\rho/\rho_0)^2-1}{1- \exp (2 \, c_0 \, \rho_ |
|
0^2-F}
-0 \, G \, L)}
\right]
=
\frac{2 \, d \, A^2 \, g}{f \, L} \cdot \ |
|
ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G}left [
\Delta Z + ((\rho/\rho_0)^2 -1) \cdot \frac{ \Delta Z}{1 - \exp(2 \, c_0 \, \rho_0 \, g \, \Delta Z)}
\right] |
|
|
1 \costhetaneqG0 | \rho =
\rho_0 \, \exp(c_0 \, \rho_0 \, G \, L |
|
=12Fc^*rho^*, A^2} \cdot \frac{1- \exp(-2 \, c _0 \, \rho_ |
|
0^2 -rho^2)
, L)}{G}}
=\rho_0 \, \exp (с_0 \, \rho_0 \, g \, \Delta Z) \cdot \sqrt{ 1 - \frac{ |
|
2d8}{\pi^2} \cdot \frac{f \, L}{d^5} \cdot q_0^2 \ |
|
ln1 \cos \theta = 0\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} |
|
See also
...