Interesting paper, as I had no idea that Karamanev's work had been extended out of the bioreactor. I consider myself duly enlightened and chastised.

However, I note that the equation in the paper and the one you have posted are not identical.

yours is:

C_{D} = (4/3)*(g*D/u_{t}^{2})*(*p*_{s}-*p*_{f})/*p*_{f}

the paper states:

v_{t} = ((4/3)*(g*d/C_{D})*(*p*_{p}-*p*_{a})/*p*_{a})^0.5

Note that I'm not querying the differing variable names, rather the different form of the two equations. They give quite different results, although the outcome remains the same.

Toodles

His equation has just been solved in terms of drag coefficient instead of velocity.

Take his equation and raise both sides to the 2nd power.

v_{t} = ((4/3)*(g*d/C_{D})*(*p*_{p}-*p*_{a})/*p*_{a})^0.5

=

v_{t}^{2} = ((4/3)*(g*d/C_{D})*(*p*_{p}-*p*_{a})/*p*_{a})

Now we want to isolate the drag coefficent, so multiply both sides by C_{D}

v_{t}^{2}*C_{D} = ((4/3)*(g*d/)*(*p*_{p}-*p*_{a})/*p*_{a})

Divide over velocity.

C_{D} = (4/3)*(g*D/v_{t}^{2})*(*p*_{s}-*p*_{f})/*p*_{f}

It is now in terms of drag coefficient, and the same equation that I had used.

Where does the form you used come from, as I would be interested in reading the paper?

His equations are actually standard in most recent fluid mechanic books. I will see if I can find a version online, but that may be difficult since these are usually books you buy for uni. It would defeat the purpose if they were free online.

I found my text book from a few years ago and scanned relevant page instead. It isn't the best image, but it was the best I could get with my scanner and a thick book. If you look at the bottom, it even references the journal you mentioned.

On the second image it should the same equation in the previous paper I had linked. As I said, mine is just in terms of drag coefficient, but you can also consider on the first page when it has (3/4)C

_{D}*Re that the Reynolds number is just dissolved into the right side of the equation.

Fluid Mechanics for Chemical Engineers by James O. Wikles