Is 'heat has only sources, no sinks' an equivalent formulation of the second law? I was thinking about it and in a sense heat is an interesting quantity that can only be created and not destroyed. In comparison with the first law this means that ultimately for a closed system all internal energy will be converted into heat. Does this make any sense? How would you write this in continuity equation form?
$$ \frac{d\rho}{dt} + \nabla \cdot\vec{\jmath} = \sigma, $$ where $$ \sigma \geq 0.$$
Would a good analogy be that in comparison with charges, imagine if all charge is conserved, and charge can only be destroyed and not created (so the inverse for heat). This means that the total net charge will stay the same but the total different charge (or potential) will go down in time. Then there would be a 'second law' for charge it seems because ultimately any system would end up with only as neutral as possible charge.
 A: 
In comparison with the first law this means that ultimately for a
closed system all internal energy will be converted into heat. In comparison with the first law this means that ultimately for a closed system all internal energy will be converted into heat. Does this make any sense?.

No. A system does not "contain" or "store" heat. Heat is energy transfer between a system and its surroundings due solely to temperature difference.

How would you write this in continuity equation form?
$$ \frac{d\rho}{dt} + \nabla \cdot\vec{\jmath} = \sigma, $$ where $$
> \sigma \geq 0.$$

Since you have not defined the terms in your equation, I can't comment on it.

Would a good analogy be that in comparison with charges, imagine if
all charge is conserved, and charge can only be destroyed and not
created (so the inverse for heat).

No it would not be a good analogy. Heat is not a "conserved" quantity like charge.

Then there would be a 'second law' for charge it seems because
ultimately any system would end up with only as neutral as possible
charge.

Sorry, I have no idea what you are talking about.
