Question

u UUUUL) 26. A gas bubble of diameter d rises with velocity V in a liquid of density p and viscosity u. (a) Determine the dimensionless groups that include the effects of all the significant variables, in such a form that the liquid viscosity appears in only one group. Note that the driving force for the bubble motion is buoy- ancy, which is equal to the weight of the displaced fluid. (b) You want to know how fast a 5 mm diameter air bubble will rise in a liquid with a viscosity of 20 cP and a density of 0.85 g/cm'. You want to simulate this system in the laboratory using water (u = 1 CP, p= 1 g/cm) and air bubbles. What size air bubble should you use? (C) You perform the experiment, and measure the velocity of the air bubble in water (Vm). What is the ratio of the velocity of the 5 mm bubble in the field liquid (V) to that in the lab (Vm)? Va muiet nodiet the nerformance of a large industrial mixer under various

Chemical Engineering Fluid Mechanics. Ron Darby

Answer 1

} 26 (2) The variables involved are d, V, 3, h&g (acc. due to ' gravity). Formulate the dimension matrin as, follows: ... 1 d. vg ig M oö i To: Mathie A = 0 0 107 Eli í -3 - TID.-10 - 12 : Loto12 . The dimension-less groups lie in the null-space of mattin A (ie given by the null-rectors of A) - O

To find the null space of x x= [ ] such that a consider a vector Ax=0 where O-fo NUL rool | | | -3 - 1 1 0 -1 0-1-2 K2 23x1 → Hz + nu = 0 az=-nu a Mit42+(-3) mg - Mut us Co > > ^, = 313+14 –M2 - M5 SEBA -Xa - 14 -2x =o 2 = x. --X, -2x2 = 3a) 2a la & a → X = -3Hy + My + My +245 - Ms = xs - My → @ X. - 15-247 al A4-2xc| Nu ĪTĪ - It us -2 -Oos- dimensionless These null-vectors ni & 12 12 give us the groups: I = d'right'g the T2= d'v-2 po kg = dg. the dimensionless groups are: – svo & dg sud

(6) Mi=20cp; Si= 0.85.9 /em? ;.d,= 5mm M2 = lep ;92 = 1g/em² ; dz=? Fes similitude: This form the 41,4,9. - dago . Sividi 12 Heft da=dy Sedantis 3/2 0.47516464521 de (smatraons) A = 6:4751644452) da ~ 0.609 mm i diametes of the bubble to 2120 - I be used in the lab. 5mm ~ 2.866 0.609 mm