Help me with the leakage rate of a crack in a pressure pipe.

[zax]

I am thinking about the following situation. In a stagnation water loop, the pressure is building up. A crack occurs due to the expansion of the water. What will be the solution approach to determine the flow rate of the leakage?
Please give some hints or references. Thank you.

 

[stewartcs]

Bernoulli.

 

[zax]

stewartcs, thank you for your reply!
I have another question here. If I use Bernoulli's equation, the pressure at upstream must be constant. What if the pressure is decreasing with time since the water is flowing out?
Thank you for any approaches.

 

[stewartcs]

Can you describe your specific setup a little more? Is this a vertical, or horizontal pipe? Once pressurzied, is it isolated prior to rupture? Etc...

 

[zax]

Sorry for lack of information.
The leak occurs in the heating coil in a water storage tank of a solar hot water system. The glycol/water mixture in the collector loop expands due to the stop of the pump.
Under the high pressure, we assume there is a crack in the heating coil. So I am thinking how to determine the flow rate of glycol/water mixture flowing into the storage tank.
Thank you.

 

[stewartcs]

It looks like the pump is just circulating the fluid from the panel to the water heater. If so, the only high pressure I would think exists due to sudden pump stoppage would be a transient pressure spike (water hammer). This may or may not cause the coil to rupture (I would think not though). The pressure wave will dissipate naturally, and very quickly, due to the associated flow losses. After that, it seems to me that the only pressure in that coil line would be hydrostatic pressure from the vertical fluid column (from the panel down to the hypothetical rupture point).

If so, then the velocity (and thus flow rate) may be approximated with Bernoulli's equation (see the link below). The flow rate is then of course Q = v*A. You'll have to estimate the cross-sectional area (A) of the rupture point. Since you have a glycol/water mixture, you'll also have to figure out the mixture's density and use that instead of pure water or glycol. However, if the glycol content is really low, it probably won't matter that much and you can just assume the fluid density is that of pure water.

http://www.engineeringtoolbox.com/bernouilli-equation-d_183.html

This approach will probably give you a decent approximation. The only thing to be wary of is that the diameter of the top of the tank (in the link example) is assumed to be significantly greater than the outlet. I think this is a reasonable assumption in your case since you are talking about a small ruptured area in the coil and presumably a relatively large diameter coil.

Hope this is of some use to you!

 

[zax]

Thank you for your answer.
The glycol/water mixture is circulated by the pump from the collector to the storage tank where the heat transfer occurs. The glycol/water mixture carries the heat from the solar collector to the cold water in the tank. The high pressure is not due to water hammer but from stagnation. Because the collector continues absorbing heat from the sunlight after the pump stops. Since the flow of glycol/water mixture stops, the pressure builds up.

The question is that if the heating coil ruptures, what will be the flow rate of mixture through the small ruptured area? There is water in the tank and the tank is connected to the city water main. The pressure difference will be the only thing driving the flow. And the pressure difference can be assumed as a constant.
The question became complicated.

Thank you for any thoughts.

 

[stewartcs]

The high pressure is not due to water hammer but from stagnation. Because the collector continues absorbing heat from the sunlight after the pump stops. Since the flow of glycol/water mixture stops, the pressure builds up.

As far as I know, stagnant water is just still water and doesn't cause a pressure increase. The heating of the water by the collector will cause a pressure rise due to the increase in temperature. I guess your point is that since there is no "heat exchanging" going on due to the water not flowing, the pressure rises too high and ruptures the pipe. So why not use a relief valve to prevent the rupture?

The question is that if the heating coil ruptures, what will be the flow rate of mixture through the small ruptured area? There is water in the tank and the tank is connected to the city water main. The pressure difference will be the only thing driving the flow. And the pressure difference can be assumed as a constant.

The method I gave you before should still work (use equation e4 from the link). P1 will be the added pressure in the coil due to the increase in temperature; P2 will be the city water main pressure in the storage tank. Keep it simple by using water as both fluids.