# Quick Hagen "ladder" calc



## vauxhallmark (27 Aug 2008)

I recently used a glass diffuser and wasn't happy with the noise of the tiny bubbles fizzing out of it, so went back to using an old Hagen "ladder", from their yeast based kit (but with a bottle of CO2).

I have suspected that it was very efficient for a while, because of the size of the bubbles that reach the top, but I couldn't remember the formula for the volume of a sphere (shame - it has been (ahem) a few years though..).

If you've never used one, you probably know what they look like (big and ugly), but if you've never seen one in action, this is what happens:

A bubble of CO2 enters at the bottome of the ladder. It's about 8mm in diameter. 

It travels left and right up the ladder, shrinking as it goes (and the CO2 dissolves). As it shrinks it travels more slowly, so at the bottom you have a few big bubbles, and at the top, loads of bubbles which have shrunk to between 1.5 and 2mm.

A 2mm bubble doesn't sound that much smaller than an 8mm one. Even a 1.5mm bubble doesn't sound _that_ much smaller than an 8mm one. But I knew that the formula for formula for volume of a sphere (eg bubble) included a power of 3, and would help me calculate how much CO2 was dissolving in the aquarium water and how much was being wasted.

Here's what I found (thanks Wikipedia !)

The volume of a sphere is 4/3 x Pi x r^3 (4/3 x Pi x r x r x r) - I'm trying to say 4/3 Pi r cubed, but I don't know how to type it.

I can't be sure whether the bubbles that finally reach the top of the ladder are 1.5 or 2 mm in diameter - let's say they're 2mm.

Therefore the volume of a bubble of CO2 entering the ladder is 4/3 x Pi x 4mm^3 = *268mm cubic mm*.

The volume of a 2mm bubble at the top of the ladder is  4/3 x Pi x 1mm^3 = 4*.19 cubic mm*.

That means 4.19 cubic mm is 1.56% of 268 cubic mm. so *98.44%* of CO2 has been dissolved in the aquarium water - result!

if the bubbles are in fact 1.5mm in diameter when they reach the top that would mean that *99.34%* of CO2 had dissolved in the tank water.

Obviously the choice of diffusers is wide and personal these days - I'm just posting these observations so that if people have a set up that could accommodate this kind of diffuser (probably either a grow out tank where looks aren't too important, or a tank that allows you to fulfil the opposing criteria of hiding a big ugly diffuser, and making sure that it receives enough current to circulate the dissolved CO2 round that tank) this humble bit of plastic could be a good way to get the most out of your CO2 bottle.

M


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## Ed Seeley (27 Aug 2008)

What you also have to remember is that the gas produced by DIY CO2 may also contain tiny amounts some other substances (certainly some water vapour) and, as the CO2 in the bubbles is going into solution as the bubbles shrink, there will also be some other gases coming out of solution into the bubble.  This means it's possibly even more efficient than your calculation!


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## vauxhallmark (28 Aug 2008)

Ed Seeley said:
			
		

> What you also have to remember is that the gas produced by DIY CO2 may also contain tiny amounts some other substances (certainly some water vapour) and, as the CO2 in the bubbles is going into solution as the bubbles shrink, there will also be some other gases coming out of solution into the bubble.  This means it's possibly even more efficient than your calculation!



Yes, that would be the case - however, I'm using a pressurised bottle of CO2 on it (in a grow out tank).


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## a1Matt (28 Aug 2008)

Nice work Mark, thanks.

I am using a reactor rather than a diffuser, but as this also takes large bubbles (apprx 8mm) and churns them until they are small (apprx 1-2mm) I figure that your calculations are just as relevant for me.

I am amazed and pretty pleased that 98% (or more) of the CO2 is dissolved    I would never have guessed that it was that efficient.


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## vauxhallmark (28 Aug 2008)

a1Matt said:
			
		

> Nice work Mark, thanks.
> 
> I am using a reactor rather than a diffuser, but as this also takes large bubbles (apprx 8mm) and churns them until they are small (apprx 1-2mm) I figure that your calculations are just as relevant for me.
> 
> I am amazed and pretty pleased that 98% (or more) of the CO2 is dissolved    I would never have guessed that it was that efficient.



The figures would only be the same if one big bubble got churned up to one tiny one - if you chop one big bubble into (say) 100 small ones you'd need to know how many "waste bubbles" came from each big bubble. 

I'm sure all the diffusers on the market these days are pretty efficient!


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## aaronnorth (28 Aug 2008)

wow 98% is a lot!


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## a1Matt (28 Aug 2008)

vauxhallmark said:
			
		

> The figures would only be the same if one big bubble got churned up to one tiny one - if you chop one big bubble into (say) 100 small ones you'd need to know how many "waste bubbles" came from each big bubble.
> 
> I'm sure all the diffusers on the market these days are pretty efficient!



That makes a lot sense, thanks for clarifying.  (I'm still more than pleased with going the reactor route, probably just not quite 98% pleased   )


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