Thermodynamic energy balance

Other than the algebraic mistake and difficulties in properly analyzing the fan, pointed out by @Daniel Hatton.

I would like to add the intent of formulating the problem like this (which is incomplete by design), reproduced from the "Thermodynamics: An Engineering Approach" by Cengel and Boles.

As per the first law of thermodynamics, the energy is conserved as it is converted from one form to another, and so, there is nothing wrong with the conversion of all the electrical energy into kinetic energy of air, for a steady-state system:

$$ dot{Q} - dot{W} = dot{m}_{text{air}} (Delta text{internal energy} + Deltatext{potential energy} + Deltatext{kinetic energy})$$

Now, our ideal case is that there is no heat in or out of our control volume, $dot{Q} = 0$, no change in internal energy of air and no change in potential energy.

That leaves us with: $$ -dot{W} = dot{m}_{text{air}} (Deltatext{kinetic energy}) = frac{1}{2}dot{m}_{text{air}}(v_{text{out}}^2 - v_{text{in}}^2)$$

What if we had an imaginary situation where the inlet flow were completely stagnant $v_{text{in}} = 0$, then as per the first law, all the electrical $20 text{J/s}$ would be converted to kinetic energy of the stagnant inlet flow:

$$- dot{W} = frac{1}{2} dot{m}_{text{air}} v_{text{out}}^2 = - (-20) text{J/s}$$ $$ v_{text{out}} = sqrt{frac{2 * 20}{ 0.25 }} = 12.649 text{m/s}$$

So, the first law has no objection to air velocity reaching 12.649 m/s, but this is the outlet velocity upper bound. Any analysis that obtains a higher velocity violates the first law.

Now as per our first law analysis (and under same assumptions), the following holds:

1. Someone tells you the outlet velocity of this fan is 8 m/s. It could be.
2. Someone tells you the outlet velocity is 13.0 m/s, now that's impossible.

So, the purpose of this problem is just to demonstrate upper bounds enforced by first law of thermodynamics. And the second law has a whole different say!