...
Steady-state 1D inviscid fluid flow | Pipe Flow Mass Conservation |
LaTeX Math Block |
---|
anchor | IY1I1p |
---|
alignment | left |
---|
| \frac{d p}{d l} =
-\rho \, u \, \frac{d u}{d l} + \rho \, g \, \cos \theta + f_{\rm cnt, \, l} |
|
LaTeX Math Block |
---|
anchor | IY1I1jm |
---|
alignment | left |
---|
| j_m(l) = j_m = \rho(l) \cdot u = \rm const |
|
Equation of State (EOS) | Darcy–Weisbach |
LaTeX Math Block |
---|
| \rho = \rho(p, T) |
|
LaTeX Math Block |
---|
anchor | gradPf |
---|
alignment | left |
---|
| f_{\rm cnt, l} = - f \cdot \frac{ \rho \, u^2 \, }{2 d} |
|
where
Substituting
...
and LaTeX Math Block Reference |
---|
|
into LaTeX Math Block Reference |
---|
|
: LaTeX Math Block |
---|
|
\frac{d p}{d l} =
-j_m \cdot \frac{d}{d l} \left( \frac{j_m}{\rho} \right) + \rho \, g \, \cos \theta - f \cdot \frac{ \rho \, }{2 d} \cdot \left( \frac{j_m}{\rho} \right)^2 |
LaTeX Math Block |
---|
|
\frac{d p}{d l} =
j^2_m \cdot \frac{1}{\rho^2} \frac{d \rho}{dl} + \rho \, g \, \cos \theta - \frac{j_m^2}{2 d} \cdot \frac{ |
...
LaTeX Math Block |
---|
|
\frac{d p}{d l} =
j^2_m \cdot \frac{1}{\rho^2} \ |
...
frac{d \rho}{dp} \cdot \frac{ |
...
...
+ \rho \, g \, \cos \theta - \frac |
...
{j_m^2}{2 d} \cdot \frac{f}{\rho} |
LaTeX Math Block |
---|
|
\frac{d p}{d l} =
j^2_m \cdot \frac{1}{\rho} \cdot c \cdot \frac{d p}{dl} + \rho \, g \, \cos \theta - \frac{j_m^2 |
...
...
...
and finally
...
...
j_m = \frac{\rho_0 \cdot q_0}{ A}
\left( 1 - j_m^2 \cdot \frac{c}{\rho} \right ) \frac{dp}{dl} = \rho \, g \, \cos \theta - \frac{j_m^2 }{2 d} \cdot \frac{f}{\rho} |
Alternative forms
...
LaTeX Math Block |
---|
| \left[ \rho -j_m^2 \, c \right] \cdot \frac{d p}{dl} =
\rho^2 \, g \, \cos \theta - \frac{j_m^2 }{2d} \cdot f(\rho) |
|
...
|
LaTeX Math Block |
---|
| \left[ \frac{1}{c} - \frac{j_m^2}{\rho} \right] \cdot \frac{d \rho}{dl} =
\rho^2 \, g \, \cos \theta - \frac{j_m^2 }{2d} \cdot f(\rho) |
|
See Also
...
Petroleum Industry / Upstream / Pipe Flow Simulation / Water Pipe Flow @model / Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model
...