Terminal Velocity and Raindrop Growth

Transcription

1 Terminl Velocity nd Rindrop Growth Terminl velocity for rindrop represents blnce in which weight mss times grvity is equl to drg force. F 3 π3 ρ L g in which is drop rdius, g is grvittionl ccelertion, nd ρ L is the density of liquid wter 000 kg m 3. One formultion for drg is Stokes drg, which cn be clculted from symmetric lminr flow round flling drop, F 6πηv in which η is the bsolute viscosity of the ir nd v is the fll speed. Empiricl studies suggest tht Stokes drg works for droplets up to size of 50 µm. Combining eqns nd yields fllspeed rule for Stokes drg: 3 π3 ρ L g 6πηv v ρ Lg 9η 3 Above 50 µm, Stokes drg fils becuse of incresing cvittion nd then turbulence s higher speeds re ttined by lrger rindrops. A dimensionl nlysis cn show tht b η F k ρv ρv in which k is dimensionless constnt to be determined empiriclly, b is dimensionless exponent, nd ρ in this context is the density of the surrounding ir. The dimensionless group in prentheses is ctully the inverse of the Reynolds number, dimensionless group tht chrcterizes the rtio of inertil forces to viscous forces. In the cse of Stokes drg, eqn reverts to eqn if b nd k 6π. An oversimplified but resonble model of rindrop velocity tht encompsses lrger drop sizes is: v ρ Lg v 9η < 6.5 µm, Stokes drg, b v β 6.5 µm < 65 µm, b 5 / v γ/ 65 µm < 500 µm, b 0 In this expression the empiricl constnts re β 8000 s nd γ 00 m / s, nd these hve subsumed most of the physicl nd empiricl constnts in eqn. The upper limit of 500 µm in the third cse reflects the empiricl observtion tht rindrops growing beyond tht size.5 mm, v 0 m s usully become oblte nd brek up into smller rindrops rther thn continuing to get lrger nd fster. The overprecise sizes for the trnsition rdii re merely to mke the v function continuous but not differentible t the trnsition points. A rindrop growth model must combine velocity rules for v with the formul for growth by collision nd colescence, dt E vw L 6

2 in which E is collection efficiency, w L is the density of suspended liquid wter below the flling drop i.e., the mss per volume of wter in the smll cloud droplets tht re being collected, nd v represents the fllspeed difference between the lrge drop nd the smll cloud droplets. For simplicity, ssume tht the collected droplets re sufficiently smller thn the collecting rindrop tht the fllspeeds of the cloud droplets re comprtively negligible, v v. In wht follows, rindrop will brought from n initil condenstion size 0 to mximum fllspeed size 500 µm, through severl stges. The trnsition from diffusiondominted growth to collision nd colescence will be ssumed to hppen t single rdius rther thn through trnsitionl phse. The remining criticl rdii will be the trnsition points in the fllspeed representtion, 6.5 µm nd 3 65 µm. In the first set of clcultions, we clculte the time elpsed. For the explicit numericl exmple, we will use the mesured vlues T 73 K, S 0.0, w L 0.00 kgm 3, 0 0. µm, nd 0 µm Stge : growing from 0 to by diffusion of wter vpor. [ dt K K De ss ρ L R w T 0 K t t 0 dt 0 Kt t 0 t t t 0 0 K where D is the diffusion coefficient for wter vpor in ir, e s is the sturtion vpor pressure of wter vpor, S is the supersturtion, R w is the gs constnt for wter vpor, nd T is temperture. 7 K De ss ρ L R w T. 0 5 m s 6. P kgm Jkg K K t 0 K 0 5 m 0 7 m s Stge : growing from to by collision nd colescence while flling controlled by Stokes drg. dt ρ L g 9η dt g 8η c t c t c t dt

3 t c 8 c t 9 c g 8η kgm ms kgm 57.3 m s s t c 57.3 m s 0 5 m m 6 s Stge 3: v β. growing from to 3 by collision nd colescence while flling controlled by 3 ln 3 dt v β c t3 [ c dt t c t 3 t t 3 ln c c β 0 3 e c t c β kgm s 000 kgm s t m ln ln c s s m Stge : v γ/. growing from 3 to by collision nd colescence while flling controlled by dt v γ / c 3 / [ t 3 c / 3 dt t 3 / / 3 c 3 t t 3 3 c 3 γ

4 t / / 3 c 3 / 3 + c 3 3 c 3 γ kgm 3 00 m / s 000 kgm m / s t c 3 / / 3 [ m / m / ] m / s 556 s Note tht these equtions cn be used for ny intervl of time nd rdius within given phse, not just for the beginning nd end of phse, so long s both endpoints re within single phse. The remining clcultions bout the growth of wrm rindrop require knowing the distnce fllen. In this, z will be downwrd coordinte leding to clcultion of the distnce fllen within ech phse, z, reltive to the surrouding ir, which is presumbly in n updrft stte most of the time. The most common wy of finding distnce trveled is to integrte velocity with respect to time, but tht does not work well in this cse becuse we hve w lwys s function of drop rdius insted of time. Hence, the integrtion vrible must be trnsformed to : z z i+ z i ti+ t i vtdt i+ i v dt For the Phse, while the drop is growing by diffusion of wter vpor nd experiencing Stokes drg, the resulting integrl is found in question from problem set 7: z ρ Lg 8ηK 0 5 z ρ Lg 8ηK 000 kgm ms [ 0 5 m 0 7 m ] kg m 0.3 m s.6 0 m s For the remining phses while the drop is growing by collision nd colescence, the formuls become independent of the form of v becuse v disppers from the expression for z. Strt with the control on drop size growth tht strts ll of the collision nd colescence phses: dt v Insert the reciprocl of this expression into eqn z v dt v v z 6

5 nd this expression works for ny phse of the collision nd colescence process. z 000 kgm kgm 3 m 0 5 m m z kgm kgm 3 m m 5 m z kgm kgm 3 m m 67 m Completion of rindrop s life history requires clculting its pth reltive to the surrounding ir. Simple ssumptions re tht it begins t cloud bse level, h 0, rises with n updrft whose speed is U, nd flls reltive to tht rising ir. If U is ssumed to be constnt poor ssumption, the rindrop is t height bove ground given by h h 0 + Ut z 7 The equtions bove, long with the erlier growth by diffusion eqution, re incorported in the exmple tble nd spredsheet rindrop.xls. To continue the numericl exmple in the following tble, we use h m nd U 3 ms. The numbers in the tble were copied from rindrop.xls nd hve some lst-digit vritions from those clculted bove, primrily becuse more digits re crried in the spredsheet clcultions, reducing roundoff error. t t z z Ut h µm s s m m m m 5