I've been wanting to design my own water carbonation device for a while. For reference, I've been looking at devices like the Sodastream, which injects pure CO$_2$ into water to dissolve it with up to seven injections per bottle. The more injections, the higher the CO$_2$ contents. I have not actually built anything yet; for now I am just trying to understand how it works.
I'm assuming that the device keeps the pressure constant, but the CO$_2$ doesn't dissolve all at once and needs multiple injections to get to the right carbonation level.
Assumption: the device opens the valve for a set amount of time with a known pressure, which releases a certain volume of CO$_2$ into the liquid.
I would assume that injecting the CO$_2$ creates a lot of small bubbles which increases the effective contacting surface area, leading to faster mass transfer.
Through calculation and use of Henry's law + an approximation for the solubility constant I've found a temperature and pressure range for my desired carbonation levels: temperature 15 °C and pressure 2 bar CO$_2$ for a desired 4 g/L CO$_2$ (perhaps with variable pressure and/or temperature to control the carbonation levels).
I have a few questions related to this:
How can I apply my desired pressure and carbonation level to a direct injection method?
How long would I need to inject, and how many times? Is there a relationship with injection pressure(preferably a formula if available)
If the desired carbonation level isn't achieved after a single injection, how far would the injection pressure need to be raised?
Is there any theoretical description of this process available?
Is direct injection of CO$_2$ an efficient method of carbonisation, or are there better/cheaper/more efficient ways to dissolve CO$_2$ in water as fast as possible?
Edit: I have a background in analytical chemistry. Any engineering knowledge is self-taught. I never received (much) formal education on subjects like fluids/gases/physics, so I hope the information I've provided is applicable to some degree.
(If you require any further information please let me know.)
(originally posted on SE Physics, they recommended me to ask here)