Gravity Separation Droplet Settling Laws 2

[Nosey]

GPSA 11th Edition (SI Units) - Figure 7.4, gives an "Intermediate Law" for the gravity settling of a sphere in a still fluid. This apparently is valid for droplets between 100 and 1000 microns and a Reynolds number of between 2.0 and 500. As a result it lies between the effective ranges of good old Stoke's Law and Newton's Law.

I cannot find any reference to this "Intermediate Law" anywhere else i.e. Perry's, Campbell's.

The relationship given is:
V_t = [3.5*(g^0.71)(d^1.14)(D_L-D_g)^0.71] / [D*(g^0.29)*(viscosity^0.43)]

Where V_t = terminal velocity (m/s), g = acceleration due to gravity (m/s²), d = particle diamter (m)
D_L = Liquid density (kg/m³), D_g = Gas density (kg/m³)

I believe that the formula (as it is given in the GPSA) is incorrect (though I have been known to be wrong in the past) for the following reasons.

As you can see the gravity term (g) appears on both the top and the bottom of the relaton. If the exponential laws were applied this would reduce to:

g^0.71 / g^0.29 = g^0.42

One fact that makes me suspicious is that 0.71 + 0.29 = 1.00 and that is just too neat to be right, or is it?

Additionally the density term on the bottom is not defined as any phase, is it gas, liquid or an average?

Am I correct in this, or is the GPSA the bible?

I'd appreciate you thoughts or comments on this.

 

[25362]

To Nosey, after checking with other books I concluded you are right: it appears there is a typing error in the GPSA formula: instead of D*(g^0.29) in the denominator, remove the * and you'll get the correct formulation: (D_g^0.29).
It happens in the best of families.

 

[Nosey]

Ahhh, it all makes sense now. Having stared at it for so long I couldn't see that permutation.

Thanks

 

[25362]

Nosey, there is still a discrepancy in the numerical factor 3.5; the formula I got has 0.153 instead.

A worked out example on a vertical spray drier shows as follows:

Hot (at 120^oC), humid air viscosity: 0.000021 N s/m².
Its density: 0.88 kg/m³.
Droplet max diameter: 1.3mm = 0.0013m.
Droplet density: 930 kg/m³.

The terminal velocity obtained: 5.4 m/s.

Which gives a Re = (0.88)(5.4)(0.0013)/0.000021 = 294, within the stipulated range 2<Re<500.

3.5, as a factor, leads to an illogically high terminal velocity and the Re comes out of range.
Is it possible there is another typing error or that the units used to measure viscosities differ ? Could you re-check ? Good luck.

 

[Nosey]

Hi 25362,

You were right about the viscosity units. GPSA (chapter 7) uses milli Pa.s so your viscosity is 0.021 mPa.s

This results in a terminal velocity of 6.35 m/s (close to your value of 5.4 m/s).

The Reynolds no. then becomes 346, again within range of applicability.

Where did you source the 0.153 factor from?

 

[25362]

The book is 24 years old, titled: Fluid Flow and Heat Transfer by Aksel L. Lydersen (John Wiley and Sons). Lydersen was then a professor of Chemical Engineering at the University of Trondheim, Norway. The formula appears on p.129, formula (6.8). The factor is 0.072^1/1.4= 0.153. The worked out example 6.1 appears on p.167 +.

 

[Nosey]

I e-mailed GPSA with the concern relating to the error in the formulae and they replied (through JM Campbell & Co.) as follows:

looking at your attachment, you've made a mistake with the
equation (Intermediate Law terminal velocity). In the denominator, the &quot;g&quot;
is not gravity but the subscript for the density symbol (rho) thus depicting
the gas phase density. Maybe not as clear as it could be. I think this
clarification addresses both of your concerns.

Out of curiosity, I went thru the derivation of each equation (Stoke's,
Intermediate, & Newton's) and they are all correct. The only slight error I
found was that I calculate the value of the constant at the front of the
Intermediate Law equation to be 3.49 instead of the 3.54 in the data book.
This difference is a result of roundoff of the exponents which affects the
value of the equation when it is simplified into the form shown in the book.
For what its worth, the exponents to 3 decimal places are 0.714, 1.142,
0.286 and 0.428. These are close to the book values but the round off error
is magnified when you start multiplying exponents.

A good discussion on the derivation of these equations is included in McCabe
and Smith, Unit Operations of Chemical Engineering, 3'rd Edn, pp. 153 - 156.

Regards

Nosey