As a friendly heads up to anyone exploring this space — temperature control is a complex dynamic between many components, chiefly the mixing valve, any local thermostatic dampers in the valve (rarer, but definitely a feature of high end mixers), and the source of hot water itself.
Mixer taps are not simply mixing hot water at temperature H and cold water at temperature C. The temperature H which your hot water source is capable of reaching is highly variable and certainly not constant. Higher flow for example reduces temperature as more cold water cools the heater. Heaters also cut out and oscillate as they overload.
Throughout the year the C temperature may also fluctuate by several degrees in a way that can trigger non linear effects in the hot water generator. (This is all assuming you have a just-in-time hot water source of course — more common in Europe than, say, NYC apartment blocks.)
Study the whole system and be prepared for unexpected non linearity everywhere. Heat loss per foot of pipe is a real thing, as is thermal damping of uninsulated tubing,
to give two more examples.
If you can model the system then next up, you might try your hand at predicting the weather.
Mine doesn't have degrees on it, but better than that, it can be adjusted such that the handle being in the neutral position (vertical in this case) is at whatever temperature you want. I set mine to 39°C which is what most people seem to enjoy. Turn it left gets you as cold as the cold supply. And then you can set a soft limit with a positive detente at some higher temp, which I have set to 42°C of course. So that you don't go overboard when turning right. And above that soft limit, you can also set the maximum temp, which I have set to 45°C. And the valve will shutoff if the temperature still manages to go above that.
This is also the case for electric showers though, where the shower itself contains the heating element and heats its own water, rather than taking arbitrary feeds of hot and cold water.
The shower doesn't need an internal heating element. All you need is an electric control system, which dynamically monitors the temperature coming out of the shower head and constantly adjusts the mix toward a set temperature.
I have a feeling these have existed for a long time in Japan.
You don't even need an electronic thermostat. You have manual thermostatic valves that do the same thing. I believe it has to do with the thermal properties and mechanical interface of a couple pieces of metal. As they expand or contract, it adjusts the mix to maintain the temperature set by the position of the handle.
I use a thermostatic mixer from Grohe (though there are more manufacturers in the recent years, including IKEA), and it handles normally both heating/cooling of the valve itself, and fluctuations of pressure in the pipes (when someone flushes the toilet).
Our hot water supply is off for 2 week per year for maintenance, and I use heater tank (which stores 30 L of hot water, enough to take a generous shower for 1 person). This way, cold water pressurizes both the cold valve, and the tank and the hot valve. I expected some random oscillations, but no, the thermostatic mixer worked just fine.
All this additional complexity, while great for getting an engineer circle jerk going, is mostly irrelevant if you are using the dial for coarse adjustment and dialing it in by feel and if your shower is shorter than the time it takes to make your hot water supply change temp a meaningful amount.
Shower valves were literally two dumb valves for the first 100yr and they worked fine.
Or just active control? Set a desired output water temp, and let a computer constantly fiddle with the valves slightly to keep the output constant. Monitoring the current temp of the input lines would help but not be necessary.
I mostly don't like that every few minutes, I have to turn down the cold water a smidge, as fresher colder water from the ground comes up through the pipe. My shower has two valves for hot and cold separate, so I turn the hot water all the way on, then add just enough cold water to achieve human shower temps at full pressure.
You don't need a computer, or even electricity. Thermostatic shower valves are 100% mechanical, and work beautifully. There are models available that will automatically compensate for both temperature and pressure changes, keeping the output temperature constant.
Interestingly, control valves employed in this way tend to wear out very quickly or else they are slow and expensive so that they can be made more durable. Anybody working on this needs to hit that "good, fast and cheap" triple point.
This is why wax motors are commonly used as valve actuators since the only calibrations are the wax mixture, the mechanical force applicator, and the thermal resistance of various components in contact with each other
Another approach is to go to the other end where you've got the tank running higher than is safe to safely touch (e.g. scalding) put put thermostatic valves on all sinks and showers. This way you can run the water heater at 140° then then use more cold water to reduce it to the proper temperature which is a slightly easier problem than running your water heater at 120° and then cooling it down to 105° which gets to "mostly hot water with cold water for pressure". Running a tank at 140° also inhibits bacterial growth (e.g. legionella)
Not enough for a shower, but for a sink you could look at an under sink water heater ( https://www.homedepot.com/p/Bosch-2-5-Gal-Electric-Point-of-... ) - but you're looking at 6.8 GPH recovery time rather than a 2.5 GPM flow rate. The main advantage there is that you don't have move the hot water from the water heater in the basement all the way to a sink (lots of hot water in the pipe that then gets wasted) and thus less water use to get hot water to the sink and less hot water use overall.
An under the sink tankless looks like https://www.homedepot.com/p/Rheem-Performance-6-kW-1-0-GPM-P... with 0.5 GPM for a 65°F raise (you wouldn't even need to run hot water to that sink - this is what I do for my upstairs bathroom since the water heater is so far away that you need to run the water for a very long time to get the sink up to 'wash with hot water' levels).
Mixer taps are not simply mixing hot water at temperature H and cold water at temperature C. The temperature H which your hot water source is capable of reaching is highly variable and certainly not constant. Higher flow for example reduces temperature as more cold water cools the heater. Heaters also cut out and oscillate as they overload.
Throughout the year the C temperature may also fluctuate by several degrees in a way that can trigger non linear effects in the hot water generator. (This is all assuming you have a just-in-time hot water source of course — more common in Europe than, say, NYC apartment blocks.)
Study the whole system and be prepared for unexpected non linearity everywhere. Heat loss per foot of pipe is a real thing, as is thermal damping of uninsulated tubing, to give two more examples.
If you can model the system then next up, you might try your hand at predicting the weather.