One of the core beliefs of RET is that gravity will cause objects to fall at the same rate regardless of weight. (Given that they are subject to the same gravitational acceleration)

But when you use empirical formulas to calculate terminal velocity, this simply is not the case.

Lets take a perfectly spherical bowling ball, and a perfectly spherical water balloon. We will say that both of their diameters are 8.5in, so that they have the same surface area, and thus have the same wind resistance.

Using an equation derived from the Karamanev Method:

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

where C_{D} is the constant due to the high turbulence of falling: 0.44

You can calculate then terminal velocities of each of the objects.

Lets start with the bowling ball.

0.44 = (4/3) * (32.2ft/s^{2}*0.708ft)/ u_{t}^{2} * (86 lb_{m}/ft^{3}-0.07647 lb_{m}/ft^{3})/ 0.07647 lb_{m}/ft^{3}

You get a terminal velocity of **278.61 ft/s**.

Now lets do it again with the water balloon.

0.44 = (4/3) * (32.2ft/s^{2}*0.708ft)/ u_{t}^{2} * (62.30 lb_{m}/ft^{3}-0.07647 lb_{m}/ft^{3})/ 0.07647 lb_{m}/ft^{3}

The terminal velocity of it is **237.09 ft/s**!

The objects are of the same size and wind resistance, and yet the heavier one falls nearly 50ft/s faster than another!

The gravity hypotenuse claims one thing, but when we use empirical evidence, our findings are very different!

I should come back more often, for all those still wondering at the source of the Karamanev 'Method', I might be able to shed some light. It's some years (18) since I last worked with bioreactors while studying for my MSc in Biotechnology, but here we go.

The Dimitre Karamanev

**Equation** (1996) was published in 1997 in Chemical Engineering Communications (Volume 147, Issue 1) in a paper titled "Equations for calculation of the terminal velocity and drag coefficient of solid spheres and gas bubbles". The terminal velcocity referred to is the terminal

**settling** velocity of bioreactor slurry within the bioreactor environment.

From my notes of the time the variables in the Karamanev equation are:

C

_{D} = Drag coefficient

D = Pipe inside diameter

u

_{t} (that should be greek letter mu) = Fluid absolute viscosity

*p*_{s} (greek rho) = Dry solids density

*p*_{f} = Fluid density

A few terms like g have been thrown in, but they're meaningless as is this equation when applied to the calculating the terminal velocity of a falling object.

Toodles.