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Filament with lm317 gets hot

At 700 mA, that means there is only about 2.7 V across the LM317. Doesn't it go into dropout when the mains voltage is a bit low?

Edit: this post was based on an earlier power estimate, it doesn't apply anymore.
 
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Coming back to the original question: the TO-220 version of the LM317 has a junction to case thermal resistance of 4 K/W according to the old SGS datasheet I have here. Add 1 K/W for the case to sink thermal resistance, which is a quite conservative estimate, and you have 5 K/W junction to heatsink.

At 3.99 W, the temperature difference between the chip and the heatsink is then 19.95 K, so the junction temperature is 79.95 degrees Celsius. That's well below 125 degrees, so it should be fine.
 
I corrected the power in the lm 317 to 3.99w wich equals 5.7v
5,7V ist a little low. In the datasheed : 6,3V 600mA +- 40mA . Try to use the lm317 in curent mode
1648152236472.jpeg

If you need 600mA use a resistor 2Ohm insted of 4Ohm in the picture
 
TO220 LDO regs are recommended when one wishes low loss low heat. For 6.3V output a 7V transformer with Schottky rectifier diodes is then OK. Or a 6V or 6.3V transformer winding with highish output voltage combined with expensive "ideal" rectifier. With the ideal rectifier things will be OK in many cases but at a price... Make sure to have a large filter cap to keep ripple low like 4700 µF. An LDO that offers the possibility to "slow start" is a nice extra.
 
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The LM317 typically needs about 1.9 V between input and output to stay out of dropout at 700 mA, but it can also be 3 V due to exemplaric spread.

Counting on 3 V, you need to make sure the momentary input voltage remains above the desired output voltage plus 3 V at all times, even when the mains voltage is 10 % below nominal.

What transformer you need for that depends on a lot of things: transformer internal resistance, rectifier diodes, value of the mains filter capacitors. I usually end up using 8 V transformers for 5 V supplies with old-fashioned regulators, so as a rough estimate, I expect 9 V is already about as low as you can go for 6.3 V.
 
It is all up to you but if you will have a custom transformer wound anyway then it would be optimal to have it wound for an LDO. Of course calculate to see if it all works out OK with the regulator you have chosen. LD1084/LT1084 has 1.3V dropout voltage and LT1764A only 0.35V just to show a difference.

If it does not make much of a price difference you could have it wound for let’s say 2 x desired current and a 9V winding with 7V tap. As we are audio people a 12V winding in case you’ll be using 12V filament tubes is maximum flexibility 🙂 The transformer is then reusable in new projects.
 
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7V at rated current would be safer. Better some flexibility than having too low voltage. When calculating too tight and mains voltage one day is too low the regulator might not regulate. Do you know how to calculate this?!

I do it regularly but it all starts to pose challenges. Then transformer A works OK and transformer B does not. Schottky diodes are recommended in any case with small margins just as large filter caps. If you mean a standard 6.3V filament winding well you have to measure under load and verify if it will work. I have 6V 3.4A transformers that work perfectly as they are 6.8V with 2A load in reality.
 
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