I hope this a question that can be answered and isn't too vague. I'm also only after a very rough answer.
I'm adding a small amount (~500mL) of sugar solution into a carboy with a much larger volume of water (~20L).
I don't particularly want to stir it in very much because there is a lot of sediment that has already settled out.
I'm wondering roughly how long it would take for the sugar water to diffuse roughly evenly through the larger volume?
Best Answer
You can get a rough idea from Fick's law (keeping to 1D for simplicity):
$$J = -D \frac{\partial \phi}{\partial x}$$
$J$ is the diffusion flow rate, $D$ is the diffusion coefficient (about $10^{-9}m^2s^{-1}$ for water and $\phi$ is the concentration in $moles.m^{-3}$. let's assume a 1 molar sugar solution (i.e. $10^3$ moles.$m^{-3}$). When you first add the sugar solution the concentration gradient is very high, so lets assume the mixing is partly under way and the sugar has diffused 10cm. That means $\partial \phi$ is $10^3$ and $\partial x$ = 0.1.
$$J = -10^{-9} \frac{10^3}{0.1} = -10^{-5}moles.m^2.s^{-1}$$
Suppose your blob of sugar solution is a freely floating sphere of volume 500mL containing 0.5 moles of sugar, then it's radius is about $0.05m$ and hence the surface area is $0.03m^2$. The flow rate out of the drop is therefore:
$$flow rate = 10^{-5} \times 0.03 = 3 \times 10^{-7} moles.s^{-1}$$
You can't just divide the amount of your sugar (0.5 moles) by the flow rate to get the time to disperse all the sugar, because the concentration gradient is changing all the time. I've just chosen what seems to me to be a reasonable average concentration gradient. However doing the division will give you a rough idea of the timescales, and it comes out at around 1.6 million seconds.
Unless you're very patient I would stir your carboy.