Broken pipe
Broken pipe is set of approximations and formulas of a long pipe filled with Methane while one of its tips is sadenly open.
Some qualitative estimates are expected to appear at the consideration.
It is very draft; the text below is expected to have many misprints and, perhaps, even more serious errors.
Notations
\( R=\mathrm{Re}=uD/\nu \) , Reynolds number
\(u\) is flow speed. For example, Meter/Second
\(D\) is linear size (for example, diameter of pipe). Say, one Meter.
\(v\) is kinematic viscosity. Usually it is measured in Meter^2/Second
\( \lambda\) is turbulent friction coefficient.
Kinematic viscosity
In the first approach, density is proportional to pressure, and the dynamic viscodity does not depend on pressure. The kimematic viscosity is inversely proportional to the pressure.
\( \rho = \rho_0 P/P_0 \)
\( \nu = \nu_0 P_0/P \)
The engineeringtoolbox [1] suggests values for visvosity. At temperature \(T=300\) Kelvin, values are follpging:
\( \begin{array}{rc} P,{\rm bar} & \nu, 10^{-6} {\rm m^2/s} \\ 1 & 17.28\\ 10 & 1.720\\ 100 & 0.1829\\ \end{array} \)
In such a way,
\( \displaystyle \nu \approx 18\times 10^{-6} \mathrm{\frac{Meter^2}{Second}} \frac{P_0}{P} \approx 18\times 10^{-6} \mathrm{\frac{Meter^2}{Second}} \frac{\rho_0}{\rho} \)
where
\( P_0 = 101325 \ \mathrm{ Pa} = 101325 \mathrm{ \frac{kg}{Second^2 \ Meter} }~\) is atmospheric pressire
and
\( \rho_0=0.657 \mathrm{\ kg/Meter^3} ~\) is density of methane at armospheric pressure,
The values above refer to temperature of 300 Kelvin. The efficient thermal exchange with the walls of the pipe is assumed; so, the temperature is supposed to remain near of 300 Kelvin; only the first minutes the expansion is rather adiabatic than isothermic.
Loss of pressure
The loss of pressure is described with equations
\( \displaystyle P'= \lambda \rho \frac{u^2}{2D} \)
\( \dot P= P u'\)
here prime differentiates with respect to length, and dot differenciates with respecto to time.
Then, by Blasius (see Reynolds number)
\( \displaystyle \lambda=\left(100 R\right)^{-1/4} = \left(100 u D / \nu \right)^{-1/4} = \left(100 \frac{u D}{\nu_0} \frac{\rho_0}{\rho} \right)^{-1/4} \)
The basic pipe equations be expressed as follows:
\( \displaystyle \rho'= \frac{\rho \rho_0}{P_0}\lambda \frac{u^2}{2D} \)
\( \dot \rho= \rho u'\)
Warning
The formlas above are combined from various sources and are not yet verified; perpahse, there are many misprints and even more serious errors there.
References
- ↑ https://www.engineeringtoolbox.com/methane-dynamic-kinematic-viscosity-temperature-pressure-d_2068.html Methane - Dynamic and Kinematic Viscosity vs. Temperature and Pressure Online calculator, figures and tables showing dynamic and kinematic viscosity of methane, CH4, at varying temperature and pressure - Imperial and SI Units. .. For 26.9 degree celsius, [m2/s*10-6] : 1.720
http://thermalinfo.ru/svojstva-gazov/gazy-raznye/dinamicheskaya-vyazkost-gazov-i-parov Динамическая вязкость газов и паров.
https://ru.wikipedia.org/wiki/Формула_Дарси_—_Вейсбаха
Keywords
«Darcy-Weisbach», «Dynamic viscosity», «Kinematic viscosity», «Methane», «Reynolds number»,
