Question About Drying Theory

..............another homework question....

Barring losses to different leakage paths, then yes, the total heat supplied depends only on the starting and final states, independent of path. How could it be otherwise? It would violate conservation of energy.

That said, leakage paths are often significant, and there may be practical reasons to prefer one path over another. One also has to be careful in defining the initial and final states. In your paths a) and b), for example, you only mention water, steam and air. However, in the potato drying problem you're trying to work, there's also potato material to consider. Path a) leaves the potato at 24 C throughout the process, and only heats the water vapor at the end. Path b) heats the water -- and the potato with it -- to 71 C, leaving it there. So the initial and final states are not, in fact, identical.

As a side note, the definition of "efficiency" in this type of calculation is problematic. From a rigorous thermodynamic perspective, the efficiency for any drying process will be close to zero; no work is produced and almost no change in potential energy. There is a small change in potential energy from the heat of hydration of the potato mass, but it's tiny compared to the energy used to dehydrate the potato in any realistic system. Nor is its value specified, or easily determined from theory.

Since the rigorous thermodynamic definition is useless for practical engineering, a looser utilitarian definition is used. The minimum energy for an idealized but quasi-realistic process is determined. In your potato drying example, it would be the latent energy of vaporization, at 24 C, of 2.2 kg of water per 100 kg of potato. That energy is then compared to the actual energy consumed in the process under evaluation to derive what might be termed the "functional efficiency" for the process. But note that there's an implicit assumption that the heat of vaporization is always unrecoverable.

That's not necessarily a valid assumption. One can imagine a vacuum drying system in which water vapor is pumped from the drying chamber, boosted to a slightly higher pressure, and condensed on a surface that transfers heat back into the drying chamber. The transferred heat balances the cooling from the evaporation of water in the drying chamber. This is a heat pumping process that could theoretically yield a functional efficiency of several hundred percent, compared to the non-condensing reference model.

Hope that helps. It's a good thing I'm retired, because that was a $50 answer if I were still billing my hours.

Thank you very much. That is an amazing answer.

If this is a small family operation air drying would be most cost efficient.

But, drying time is important in any commercial operation. If throughput is to be maximized, then heat is dictated. So, cost of production in total is most important.

I don't have an opinion about the formulas.