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 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

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»,