# Constant water flow rate?

1. Dec 17, 2009

### blumfeld0

Hi I am a post-doc student in a field having very little to do with physics and math.
currently I am designing an experiement and have (i hope) a relatively simple question.
here is the scenario:

A ziplock bag (or similarly non-elastic reservoir) contains water, but is not maximally filled, so there is no great deal of internal compression on the fluid. A book is placed on top of the bag (the book is large enough to cover the entire surface of the bag, and provides even pressure across the bag's surface) which is lying flat on its side on a hard counter. The bag is now compressed between the mass of the book and the counter surface. If a pin hole is poked through the bag, will the flow rate of the water leaving the bag be constant as the fluid is lost until the book begins to rest (any part of it) on the counter? What equations support your statement?

This question relates the the design of an experiment where the bag set-up is arranged in an attempt to save funds by replacing a peristaltic pump which is normally used to generate constant pressure. The creation of consistent flow rates out of the bag due to constant pressure is necessary. Will the bag system as described above function as a replacement of the more expensive pump?

any help would be greatly appreciated

thank you!

2. Dec 17, 2009

### gmax137

I think it depends upon what you consider 'constant pressure,' - how much variation in pressure is permissible?

If the 'sidewalls' of the bag are relatively short the pressure wouldn't vary much; but it will vary to some extent as the depth of water in the bag decreases, and (more importantly) if the surface area of the bag increases as the water bleeds out and the bag 'flattens.'

3. Dec 18, 2009

### blumfeld0

o if there is an increase in surface area of the bag in contact with the table and book then the rate will change. Even with the weight being fully supported the entire time on the bag and the weight not changing?

Interesting

4. Dec 18, 2009

### BANG!

Ignoring turbulent flow this can be treated using Bernoulli's Eqn. and the Continuity Eqn.

You can use the following link:

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

to see how they are applied to your specific problem.
The main result you need is (e4) from this link:
$$V_{2} = \sqrt{\frac{2}{1-(\frac{A_{2}}{A_{1}})^{2}}(\frac{p_{1}-p_{2}}{\rho}+gh)}$$

In your case you need to use the following values:

g is gravity
h is height from pin hole to top of bag
$$\rho$$ is density of water
$$V_{2}$$ the velocity of the water leaving the pin hole (what you are after)
$$A_{2}$$ the area of the pin hole
$$A_{1}$$ the area of the top of the bag that the book presses on
$$p_{2}$$ the pressure at the pin hole (1atm)
$$p_{1}$$ the pressure at the top of the bag (1atm + (book mass)*g*$$A_{1}$$)

note that since the pin hole $$A_{2}$$ is much smaller the the bag surface area $$A_{1}$$ you can approximate
$$\frac{A_{2}}{A_{1}} = 0$$
This will simplify the eqn some for you, but if you use the approximation then make sure it also applies to the system you will build.

In any case, looking at this equation we see that as the water level (h) drops, then so does the velocity $$V_{2}$$. This may or may not be negligible, you need to plug in values for your real system and decide. But, in general you can diminish this effect by using a heavier book, ie: make the $$p_{1}$$ term outweigh the gh term. Also, as mentioned by gmax137, if $$A_{1}$$ changes then so will $$p_{1}$$ and in turn $$V_{2}$$. Of course, putting the bag between rigid walls would keep it from flattening out and stop this problem.

good luck, and hope this helps.

BANG!

5. Dec 18, 2009

### Staff: Mentor

Yes - flow rate is a function of pressure and as the surface area increases, the pressure decreases.

6. Dec 19, 2009

### blumfeld0

Thank you!! That really helped