III)Deformation

To see how much drops and bubbles deform from its spherical form while rising in a fluid, the ratio of dynamic pressure $P_{dyn}$ to that of the surface tension stresses \frac{\sigma}{2d} is used. The surface tension forces will drive a drop or bubble to a spherical shape while the dynamic forces, due to the non-symmetri of them will drive it to a non spherical shpae. This dimesnionless ratio is called the weber number $W_e$

$W_e=\frac{\rho_fw^2d}{\sigma}$

where $\sigma$ represents the surface tension and is expressed in $\frac{N}{m}$. for values of $W_e<<1$ the particles are approximately of a spherical form, for values of $W_e=1$ there is a moderate deviation, while for $W_e>>1$ there are large deviations. However sicne the velocity is to be known a priori to know how much a drop or bubble will deform, the dimensionless BONd number is used. It represents the ratio of the effective gravitational forces to tension forces & is expressed as follows

$B_O=\frac{gd^2(\rho_p\rho_f)}{\rho_{f}^{2}\sigma^3}$

To relate the bond, Weber and reynolds number in a satisfying way, the Mortion number is introduced. This number has no phsical interpretation but is convenient and can be calculated easily. It is defined as 

$M_O=\frac{g\mu_{f}^{4}(\rho_p-\rho_f))}{\rho_{f}^{2}\sigma^3}$ 

Which yields

‚Äč$\frac{W_{e_{terminal}}^{2}}{R_{e_{terminal}}^{4}}=\frac{M_O}{B_O}$

And make it possible to study the deformation by keeping $M_O$ constant and varying $B_O$ or viceversa. Also solving it further :

$W_{e_{terminal}}=\frac{4B_O}{3C_{D_{terminal}}}$

Though the dimesnionless parameters $B_O$ and $M_O$ seem convenient, they are only to be used for the case of terminal velocity $u_{max}$

 

 

Transient Velocity for non spherical particles

Non-spherical particles can have spherical shapes or completely random and strange shapes that from now on are referred as non-spherical shapes. For axisymmetric shapes Gabitto and Tsouris finf it convient to use the aspect ratio,E , defined as the ratio of length projected on the axis of symmetry to the maximum diameter normal to the axis.

Wadell also proposed the degree of sphericity defined as 

$Ø=\frac{A_V}{A_p}$

where $A_V$ is the surface of a sphere of the same volume as the non-spherical particle. This sphericity term is considered the best for isometrically shaped particles. Using this parameter, Haider and Levenspiel Presented a $C_D$ vs $R_e$ correlation. Which is as follows

$C_D=\frac{24}{R_e}[1+0.1118(R_eK_1K_2)^0.65657]+\frac{0.4305}{1+\frac{3305}{R_eK_1K_2}}$

Where the used $C_D$ and $R_e$ are based on the equal volume sphere diameter called volumetric diameter. $K_1$ and $K_2$ are shape factors applicable in sphericity for solids of spherical shape for non-shperical shpaes $K_1$ and $K_2$ are functions of the sphericity and particle orientation.

Chien, re-analysing pre-existing data in the petroleum engineering & processing literature proposed the following expression for the drag coeffiecient 

$C_D=\frac{30}{R_e}+67.289exp(-5.03Ø)$

The later expression can be used to find a transient velocity profile. This together with the transient velocity method yields.

$\lambda_{non}=\frac{\lambda}{n_1}u_{max_{non}}=\frac{u_{max}}{n_1}$

$n_1=\frac{[1+0.1118(R_eK_1K_2)^0.65657]+\frac{0.4305}{1+\frac{3305}{R_eK_1K_2}}}{K_1K_2}$

For Drag coefficient by Ganser we have :

$\frac{du}{dt}=n_2-n_3u-n4u^2$

For the drag coefficient by chen, which by calculating the constans n_2,n_3 and n_4 would give a perfect fit.