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- TL;DR Summary
- I'd like to derive an equation for Reynolds number as a function of pressure for a pinhole leak in a pressurized gas line. The line is pressurized to 1 atmosphere and is leaking into vacuum.

Hi all!

This is my first post here, so hopefully I am not in violation of any rules or etiquette. I'm looking to derive an equation for Reynolds number as function of pressure for a pinhole leak in a pressurized gas line. The line is regular air and is pressurized to 1 atmosphere and is leaking into vacuum.

I know that Reynolds number can be defined by the equation

where ##C## is the arbitrary energy constant.

Outside of the line is a vacuum and so ##p=0## and we have

and since energy is conserved, we can equate the two and solve for the fluid flow speed, such that

Substituting this into the equation for Reynolds number, I get

Finally, at STP this site gives ##\mu=1.825\times10^{-5}\ \mbox{kg}\cdot\mbox{m}^{-1}\cdot\mbox{s}^{-1}##, and ##\rho=1.204\ \mbox{kg}\cdot\mbox{m}^{-3}##, and if I just wing a pinhole size with a diameter of 10 microns, then ##L=10\times10^{-6}\ \mbox{m}##, then I can write

So at 1 atmosphere, I get a Reynolds number of ##Re(1\ \mbox{atm})\approx270##. According to the Reynolds number Wikipedia page, the laminar-turbulent transition will occur at a Reynolds number ##Re_x\approx5\times10^5##, where ##x## is the distance from the leak. If I keep the pressure at 1 atmosphere, I find that I need a pinhole diameter of about 18mm to achieve that sort of Reynolds number.

I'm just looking for a rough approximation, and was wondering how confident that I could be in my derivation above. But my real question concerns the suggested Reynolds number for the laminar-turbulent transition boundary. How can I calculate Reynolds number at varying distances away from the leak?

Thanks in advance for any insights!

This is my first post here, so hopefully I am not in violation of any rules or etiquette. I'm looking to derive an equation for Reynolds number as function of pressure for a pinhole leak in a pressurized gas line. The line is regular air and is pressurized to 1 atmosphere and is leaking into vacuum.

I know that Reynolds number can be defined by the equation

##Re=\frac{\rho u L}{\mu}## ,

where:

##\rho## is the density of the air,

##u## is the fluid flow speed,

##L## is the characteristic dimension, and

##\mu## is the dynamic viscosity of the air.

If I constrain the system so that the air does no work and no work is done on the air, and that there is no expansion/compression of the gas, I believe that I can apply Bernoulli's principle to the air and consider it an incompressible fluid. Then, for a rough approximation, I can use Bernoulli's equation for an incompressible fluid##\frac{u^2}{2}+gz+\frac{p}{\rho}=\mbox{constant}## ,

where:

##u## is the fluid flow speed,

##g## is the acceleration due to gravity,

##z## is the distance from the reference plane,

##p## is the pressure at the point of interest, and

##\rho## is the density of the air.

Now, if I place my plane of reference at the pinhole leak, then ##z=0## and the middle term vanishes. It was suggested to me to assume that the gas inside of the line is not moving so that the fluid flow speed in Bernoulli's equation is zero, ##u=0##, so that##C=\frac{p}{\rho}## ,

where ##C## is the arbitrary energy constant.

Outside of the line is a vacuum and so ##p=0## and we have

##C=\frac{u^2}{2}## ,

and since energy is conserved, we can equate the two and solve for the fluid flow speed, such that

##u=\sqrt{2\frac{p}{\rho}}## .

Substituting this into the equation for Reynolds number, I get

##Re=\frac{\rho u L}{\mu}=\frac{\rho}{\mu}\sqrt{2\frac{p}{\rho}} L=\frac{L}{\mu}\sqrt{2\rho p}## .

Finally, at STP this site gives ##\mu=1.825\times10^{-5}\ \mbox{kg}\cdot\mbox{m}^{-1}\cdot\mbox{s}^{-1}##, and ##\rho=1.204\ \mbox{kg}\cdot\mbox{m}^{-3}##, and if I just wing a pinhole size with a diameter of 10 microns, then ##L=10\times10^{-6}\ \mbox{m}##, then I can write

##Re(p)=0.85\sqrt{p}\ (\mbox{s}\cdot\sqrt{\frac{\mbox{m}}{\mbox{kg}}})## .

So at 1 atmosphere, I get a Reynolds number of ##Re(1\ \mbox{atm})\approx270##. According to the Reynolds number Wikipedia page, the laminar-turbulent transition will occur at a Reynolds number ##Re_x\approx5\times10^5##, where ##x## is the distance from the leak. If I keep the pressure at 1 atmosphere, I find that I need a pinhole diameter of about 18mm to achieve that sort of Reynolds number.

I'm just looking for a rough approximation, and was wondering how confident that I could be in my derivation above. But my real question concerns the suggested Reynolds number for the laminar-turbulent transition boundary. How can I calculate Reynolds number at varying distances away from the leak?

Thanks in advance for any insights!