A black dwarf is a hypothetical star in our universe that do not exists now but will exist in future after a white dwarf star will cooled down. Black dwarfs takes quadrillions of years to form. At less than 14 billion years old, the universe is too young to have any existing black dwarf.
Our sun is a main-sequence star. A main-sequence star lacks the mass necessary to explode into supernova to become a black hole or a neutron star. Instead, it will become a white dwarf, which is a kind of dead star that has burned through all of its fuel, meaning it’s no longer capable of continuing the fusion of hydrogen into helium that makes a star to glow and produce heat. In about 8 billion years our sun will become a white dwarf & over trillions of years it will no longer emit any light. But the white dwarf will remain hot for other trillions of years. After, long time all the heat will radiated away & our sun will become a black dwarf. The estimated time for our sun to cool enough to become a black dwarf is 1015 (1 quadrillion) years.
The Possible Fate of Earth
After the first week without the Sun’s heat, the Earth’s surface temperature would drop to 0°C (32°F). After a year, it would go down even more – dropping to a freezing cold -100°C (-150°F). By that time, all our oceans would be covered in ice. Despite that, the only warm place on the planet that you’d have even a slight chance for survival would be near the Earth’s geothermal vents at the bottom of one of these oceans.
Temperatures on Earth will hit -100°C only if Earth survived the expansion of Sun when it will become a red giant in about 5 billion years from now. But this is not possible when Sun will become a red giant, it will expand beyond the orbit of Earth, engulfing up to asteroid belt. So, there is no chance that Earth will survive this expansion.