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.
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?
Source: http://en.wikipedia.org/wiki/Nucleate_boiling
As can be seen in the above graph, there is a local heat flux maximum for heating water, when the hot surface contacting the water is 30 degrees C above the boiling point, that is 130 degrees C at atmospheric pressure.
So to minimize time to boiling, contact a 130 degree C surface made of high thermal conductivity material (such as silver, copper or aluminum) with the water.
With heat flux over 100,000 Watts/m^2, you just need to make the surface area of contact large enough to acheive any desired heating time.
If you want the practical answer, get an instant hot water dispenser. Practical, HA! Ok, let's do some science!
First, what kind of coffee? At 30 ml I'm guessing you're making espresso. I'm going to go with 92C as the optimal temperature.
Second, I'm going to take your full 10 seconds. The rest of the process of making espresso takes far more time so whether it's 5 or 10 or 15 seconds doesn't make a real difference to how fast you get your morning hit.
Other answers have pointed out the amount of energy needed to bring your 30 ml up from room temperature, but you can make life easier on yourself and do what a lot of "instant boil" systems do and preheat. This allows you to use less energy (for the 10 seconds anyway) and bring the water up to temperature slower. This gives you a wider tolerance between the water being too cold and it being boiled away. If your water is already at 50 C that will only require about 500 W.
To deal with thermal conductivity, a wider surface area in contact with the heating element would serve to heat the water more quickly and more evenly. If the water is too thin, it can overheat before it is removed from the element. Too thick and the bottom may boil before the top reaches temperature.
We need a broad, flat surface that can output at least 500 W and that we can quickly add and remove small amounts of water from. Sounds like you need an electric skillet! A very clean one. Or the precision laboratory version. Buy one that can heat quickly and evenly.
Knowing when to remove the water is a problem. Measuring the temperature of 30 ml of water on a hot electric skillet is problematic. I would suggest trial and error with various skillet settings, times and initial water temperatures. You can do a lot of experimental rounds in 10 seconds. This would also allow you to include the cooling effect of whatever vessel you're transferring it into.
According to Michael Dryden's answer, it takes 877.8W to heat 30ml of water by 70C (from 20C to 90C)in 10s.
My electric shower is about 10 times that power (lets call it 8.778kW for convenience.) I don't like to shower at 90C. I am more than happy if my shower heats the water about half that temperature difference, say 35C (from 5C to 40C). So I would set it to deliver 300ml x 2 = 600ml every 10 seconds. That's 3.6L/minute, so my 5 minute shower uses 18L of water. Of course, if I did choose to raise the temperature by 70C in the shower, I would have to restrict it down to 300ml every 10 seconds, making it a continuous version of your experiment, scaled up by a factor of 10.
The last time I took a shower apart, the element was inside a copper vessel of about 300ml volume. So, if you restricted the flow to 300ml in ten seconds, your residence time in the element vessel would be... ten seconds!
As noted by Schwern, your issue isn't heating it, it's being able to control the temperature accurately. One method that hasn't been mentioned is dunking a calibrated lump of hot metal into it (or pouring it into a hot metal container.) But so far the best method is Michael Dryden's idea of using a microwave.
In industry, batch processes are only used for slow processes like brewing beer. Fast processes (such as those used in oil refining: cracking, reforming, alkylation, etc.) are done continuously. For economic reasons these processes are carried out as fast as possible in the smallest possible equipment, and continuous process is therefore the only practical way of doing them.
If you really need to heat water that rapidly, and the reactants consumed in your experiment are cheap, I suggest you set your experiment up as a continuous process, with a flow of water through it. That way you will be able to control the temperature accurately with an electric heater. A shower will probably be too powerful - try scaling everything down to something like a 25W soldering iron element (make sure it's sealed and 12V DC for safety!)
You can supply your equipment with water from a reservoir at the top and control the flow / residence time with a valve at the outlet at the bottom. If you want to try the same conditions with double the residence time, simply reduce both the water flow and the heater output to 50%.
You may optionally replenish the reservoir with tap water to the point of (near) overflow if you need to mix things into it.
When I did a conversion course from chemistry to chemical engineering, several of the practicals were set up like this. It's definitely the way to go if you want to study fast processes whether they be mixing, reaction kinetics, or whatever.
Regarding Dave's points about critical heat flux, the important thing is to make sure you have sufficient surface area for the volume being heated. I don't see it as a huge issue at your scale though. For 1kW, you need 100cm2 of heating surface for a flux of 100,000W/m2. Many newer kettles provide this (with heating over the whole bottom surface for a neater design) but older models with curly heating elements are above 100,000W/m2, approaching critical heat flux. These electric kettles 'roar' as they come to the boil due to cavitation.
If your reagents are delicate, though, you will need to space your heaters very narrowly, or even use steam injection. I remember one particularly nasty case of electric heaters converting an aqueous polymer solution into volatile aldehydes.
As others have shown the energy requirement is quite reasonable, the problem is delivering it to the water in that timeframe.
I would think a workable approach could be done with a collection of plates with very narrow spaces between them that can hold your 30ml. Make the plates out of something that's not too good a conductor. Feed an extreme current through the plates (very low voltage, though--you'll have a big step-down transformer) and it will get hot quickly. You'll have to figure the energy left in the plates and add it to the amount of power you are dumping through this.
Careful, a second run before the plates have cooled will certainly result in boiling water.
You could try Induction cooktops. 10 seconds for 30 ml should be enough at 700w or more, if you are using a flat and wide vessel (not all utensils are induction compatible).
There are already some good answers on how you could reasonably get a cup of coffee-temperature water within 10 seconds. However, what if you wanted to get it even quicker than that?
If you already have hot water, you won't need to wait for 10 seconds before your cup is ready to go. This may not be the most economical solution to get your morning dose, but if you have a coffee shop and really need to churn out cups during peak times it may be interesting to look into this.
If you don't like the idea of keeping your water hot all the time, you need something to heat it quickly. Of course you could use a huge heating element, but I suppose that it may be even quicker to use something like an explosion. You just need to make sure that no deadly chemicals end up in your cup.
Other answers already indicated that the main problem with using more power, is that it is hard to control the temperature. However, this should be solved with the following steps:
Of course it will never be truely instant, but I imagine that you could get it done in a fraction of a second.
You can easily add $3.9~\mathrm{g}$ steam to $26.1~\mathrm{ml}$ water of $20\mathrm{°C}$ to get $30~\mathrm{ml}$ water of $100\mathrm{°C}$ in less than ten seconds.
We need a wide, level surface that could result at the very least 500 W and that we can quickly include and get rid of small amounts of water from. Sounds like you require an good electric skillet! A very clean one. Or the precision laboratory version. Get one that could heat rapidly and also uniformly.
