You need to specify what is the color of the LED.
Each one have different drop voltages.
you need 8*3.1V = 24.8V for 8 leds in series
and to limit the current you can use 100 ohm 1/4 W resistor (need 0.02 A * 100 ohm = 2 V)
you can Adjust the LM317 o/p at 24.8+2 = 26.8 V by usin 240 ohm between Vout & Adj , and Pot between Adj & GND
you need 30 V Dc as i/p the adjust o/p using pot
then connect 100 ohm between Vout & LEDs.
i need to use 9Volt battery to make circuit simpler and do i need heat sink for LM317?
You might be aware that the brightness will decrease a few with the battery discharge, even considering that the alkaline batteries have a soft slew rate on its voltage during discharge.
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