How is it possible to heat a tiny amount (30 ml)[1] of water to a high enough temperature to make a coffee, in less than 10 seconds and possibly instantly?
Most heaters that I know of heat water in no less than 90 seconds (induction heater) or 3 minutes which is way too much for my purposes, and a microwave oven takes me one minute and a half to heat.
[1] A single coffee is between 20 - 30 ml and my coffee machine makes 30 ml for each cup.
Answer
Well, let's do some math:
Assuming 30 mL of water is 30 g, and we want to heat our water from 20 °C to 90 °C, the energy required is: $$\begin{align}E&=C_Pm\Delta K \\ &=\left(4.18 \mathrm{\frac{ J}{gK}}\right)(30\mathrm{\ g})(70\mathrm{\ K})\\ &=8.778\mathrm{\ kJ}\end{align}$$
So how much power do we need to do this in a given time? "Instant" doesn't really mean anything, so let's go with 10 seconds:
$$\begin{align}P&=\frac{E}{t}\\ &=\frac{8778\mathrm{\ J}}{10\mathrm{\ s}}\\ &=877.8 \mathrm{\ W}\end{align}$$
This is not an enormous amount of power, but the trick is that it all has to go into heating the water. A good microwave outputs a fair bit more power than this, but it generally doesn't all get absorbed by such a such a small amount of water in only 10 seconds. Your best bet is probably an electric heating element directly inserted into the liquid, though I don't know if you can get a ~1000 W one small enough to sit in that much water.
As Jon Custer notes, it's not necessary to produce all the heat at once. If you heat some kind of thermal reservoir and flow the liquid past/through it, it reduces the demands on your heat source.
Edit: Also, I just tried this with a 1200 W microwave and it only took 15 seconds. How fast do you really need this coffee?
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