Heron's Fountain as a Shower - Seeking advice on the Physics

[some bloke]

TL;DR Summary

I want to try and make a Heron's Fountain as a shower for a camper, and I am looking for advice on how far apart the tanks need to be!

Hi all!

I have had a sudden inspiration that I could use the Heron's Fountain to make a shower for a campervan, which wouldn't need overhead water tanks (unstable) or electrical pumping!

The concept of the heron's fountain is on Wikipedia:

Using the diagram above, A would be the shower tray where you stand, B would be beside the tray rather than below it, P3 would be the showerhead (approx 2m above the tray) and C would be put outside, about 450mm below tank B.
Here's the layout I'm thinking of using:

The idea being that then all you need to do is prime it ( a foot pump or something will start it off) and then the fountain is self-sustaining until it runs out of water in tank B. Tank B can be heated, and then tank C is already outside and can be easily emptied responsibly (it'll have wheels!).

So, to the physics!

If I were to assume that I want it to flow at 0.125L per second (7.5L per minute, approx. 2 gallons per min), how much of a height difference will I need between the tanks?

I am anticipating that the important difference will be between the upper level of water in thank B and the showerhead, as this is how high the water needs to be raised to reach it, and excess pressure from here will define the pressure of the shower. It takes 2.85psi to raise water 2m, so anything over, say, 3psi will be the pressure of the shower, does that seem right?

Basically what I need to know is whether this has any hope of working before I buy some water containers and muck around with it in the garden to see how well I can get it to work!

Thanking you all in advance for any assistance you can offer!

 

[anorlunda]

Is your shower heated? You will have to heat the entire tank B water before starting, but only a small fraction of the water goes out the shower. Most of it goes to tank C.

You can find a technical analysis here.
https://arxiv.org/ftp/physics/papers/0310/0310039.pdf

But your question is incomplete. It is not only how high the fountain is, but the flow rate. An adequate shower cannot have a drip of one drop per second.

The paper says that the velocity of the stream is:

Note the qualifier about friction and compressibility. Thus, the diameter of the tubes will play a role. For a DIY, achieving 50% of that maximum velocity might be good.

So, you must specify how much flow is adequate for a shower. I suggest 1 gallon per minute (~ 0.06 L/s). When I use my sun shower, 1.5 gallons in 90 seconds is about right for a shower. Sun shower in the picture, but not me. Personally, I like the sun shower better than the Heron's Fountain because of KISS.

 

[some bloke]

Is your shower heated? You will have to heat the entire tank B water before starting, but only a small fraction of the water goes out the shower. Most of it goes to tank C.

You can find a technical analysis here.
https://arxiv.org/ftp/physics/papers/0310/0310039.pdf

But your question is incomplete. It is not only how high the fountain is, but the flow rate. An adequate shower cannot have a drip of one drop per second.

The paper says that the velocity of the stream is:
View attachment 286257
Note the qualifier about friction and compressibility. Thus, the diameter of the tubes will play a role. For a DIY, achieving 50% of that maximum velocity might be good.

So, you must specify how much flow is adequate for a shower. I suggest 1 gallon per minute (~ 0.06 L/s). When I use my sun shower, 1.5 gallons in 90 seconds is about right for a shower.Sun shower in the picture, but not me. Personally, I like the sun shower better than the Heron's Fountain because of KISS.

View attachment 286258

Thankyou for the response!

Regarding the heating, yes I would need to heat tank B (possibly via log burner or solar panels) before it could be used as a warm shower. I'm not sure what you mean when you said "only a small fraction of the water goes out the shower. Most of it goes to tank C." The only way the water can get to tank C is by coming out of the shower, isn't it?

Estimating (worst case for distances) h1 = 2m from tank B level to shower (basically empty) and h2 = 0.5m from tray to outside container (keep outside container wide and flat!), then we get a velocity of 5.4ms-1, which seems high! I'm not sure how to convert this to a flow rate, either!

 

[anorlunda]

The only way the water can get to tank C is by coming out of the shower, isn't it?

You're right. I stand corrected.

I'm not sure how to convert this to a flow rate, either!

Estimating (worst case for distances) h1 = 2m from tank B level to shower (basically empty) and h2 = 0.5m from tray to outside container

Look at how H1 and H2 are defined in that paper. They are from shower head to the height of the water surface in tanks B and C.

If you know the tube diameter then calculate cross sectional area. Area times velocity has units of volume flow rate, maybe in cubic cm per second. Convert that to liters per second.

 

[jack action]

Just to illustrate the model presented by @anorlunda for your shower, assuming the following:

Then $h_2 = H_a - H_c$ and $h_1 = H_h - H_b$ and:
$$\Delta P = \rho g (H_a - H_c - (H_h - H_b))$$
Or:
$$\Delta P = \rho g (H_b - H_c - (H_h - H_a))$$
Or:
$$\Delta P = \rho g (H_b - H_c - h_s)$$
Where $h_s$ is your shower height (approx. 2m).

You can see that to obtain a positive pressure at the showerhead, then:
$$H_b - H_c \gt h_s$$
So assuming your tank C is a very flat tray on the ground ($H_c \approx 0$), your tank B level $H_b$ must be at least at 2m from the ground for the system to work.

In the end - unless your camper is really high off the ground! - I'm not sure the system will be that different from an overhead tank (but much more complicated).