Example

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1 Chapter Example sold slab of 5.5 wt% agar gel at 78 o K s.6 mm thk and ontans a unform onentraton of urea of. kmol/m 3. Dffuson s only n the dreton through two parallel flat surfaes.6 mm apart. The slab s suddenly mmersed n pure turbulent water so that the surfae resstane an be assumed to be neglgble;.e, the onvetve oeffent h m s very large. The dffusvty of urea n the agar s m /s. ) Calulate the onentraton at the mdpont of the slab (5.8 mm from the surfae) and.54 mm from the surfae after hr. ) If the thkness of the slab s halved, what would be the mdpont onentraton n hr? (Ref: Transport Proesses and Separaton Proess by C.J. Geankopls, Prente Hall, 4 th Edton, 3) Soluton ) Calulate the onentraton at the mdpont of the slab (5.8 mm from the surfae) and.54 mm from the surfae after hr. The onentraton at any pont s gven by the approxmate soluton os( x Dt ) where C exp(- F o ) and F o.6 mm > 5.8 mm m Dt F o (4.7 )(36) (5.8 ) >. t the mdpont of the slab x > x x m hm K ' D >.578 and C.73 os( x ) C exp(- Fo).73 exp( ) K ' K ',,,.5 Sne, >,.5 > (.5)(.).5 kmol/m 3. t x.54 mm > x x.5 -

2 C exp(- F o )os( x ).73 exp( )os(.578.5), (.5)os(π/4).77 (.77)(.).77 kmol/m 3. ) If the thkness of the slab s halved, what would be the mdpont onentraton n hr? Dt F o (.54 ) (4.7 )(36) os( x ) C exp(- Fo).73 exp( ),. (.)(.). kmol/m 3. Example slab of whte pne, 5 m thk, has a mosture ontent of wt% at the start of the dryng proess. The equlbrum mosture ontent s 5 wt% for the humdty ondtons n the dryng ar. The ends and edges are overed wth mosture-resstane oatng to prevent evaporaton. The dffusvty of water through pne s. - m /s. ) If the two large surfaes are exposed to the dryng ar and the resstane to mass transfer outsde the pne s neglgble, estmate the tme requred to redue the mosture ontent of the slab at the enter to wt% and the tme to redue the (average) water ontent to wt%. ) Repeat part () f the external mass transfer resstane s equal to the nternal mass transfer resstane. 3) Repeat part () f only one large surfae s exposed to the dryng ar whle the other surfae s also overed wth mosture resstane oatng. (Ref. Fundamentals of Momentum, Heat, and Mass Transfer by Welty, Wks, and Wlson, 4 th Edton, ) Soluton ) If the two large surfaes are exposed to the dryng ar and the resstane to mass transfer outsde the pne s neglgble, estmate the tme requred to redue the mosture ontent of the slab at the enter to wt%. The dmensonless onentraton an be expressed n terms of mass fraton,, / / / /, ω + > ω ω -

3 , , / / / /, m hm K ' D >.578 and C.73 Solvng for F o gves.64 os( x ) C exp(- Fo).73 exp(.578 F o ) Dt F o.5 Sne 5 m >.5 m t (.5)(.5) se. Estmate the tme to redue the (average) water ontent to wt%. sn( ) (, ) (, ) Solvng for gves M t sn( ) (.736).578 sn(.578).73 exp(.578 F o ) Solvng for F o gves Dt F o.48 t (.48)(.5).55 5 se. -3

4 ) Repeat part () f the external mass transfer resstane s equal to the nternal mass transfer resstane. m hm K ' D >.863 and C. Solvng for F o gves.64 os( x ) C exp(- Fo). exp(.863 F o ) Dt F o.75 t (.75)(.5). 6 se. Estmate the tme to redue the (average) water ontent to wt%. sn( ) (, ) (, ) Solvng for gves M t sn( ) (.736).863 sn(.863). exp(.863 F o ) Solvng for F o gves Dt F o.64 t (.64)(.5).5 6 se. -4

5 3) Repeat part () f only one large surfae s exposed to the dryng ar whle the other surfae s also overed wth mosture resstane oatng. m hm K ' D >.578 and C.73 Sne mass transfer ours only on one sde we have.5 m and x s zero at the oatng surfae. t the enter of the slab x.5 m and x.5 Solvng for F o gves.64 os( x ) C exp(- Fo) os( x ) exp(.578 F o ) os(.578.5) Dt F o.45 t (.45)(.5).6 5 se. Estmate the tme to redue the (average) water ontent to wt%. sn( ) (, ) (, ) Solvng for gves M t sn( ) (.736).578 sn(.578).73 exp(.578 F o ) Solvng for F o gves Dt F o.48 t (.48)(.5). 5 se. -5

6 Example pure water stream s to be hlornated by ontatng t wth pure hlorne gas n a ounter urrent spray tower. The spray tower s operated at 5.5 o C and. atm. The spray droplets are spheral wth a dameter of. -4 m and the tower heght s.5 m. The onentraton of hlorne on the surfae of the droplet an be expressed n terms of the hlorne pressure as,s (kmol/m 3 ).5 P (atm). The dffuson oeffent of hlorne n water at ths ondton s 4. - m /s. The resdene tme of the water droplets n the olumn an be determned from h.5gt, where h s the tower heght and g s the aeleraton of gravty,.8 m /s. ) Determne the hlorne onentraton at the enter of the droplet at the olumn ext. ) Determne the moles of hlorne transferred to a droplet at the olumn ext. (Ref: Transport Proesses and Separaton Proess by C.J. Geankopls, Prente Hall, 4 th Edton, 3) Soluton ) Determne the hlorne onentraton at the enter of the droplet at the olumn ext. The approxmate onentraton for a spheral system s gven by, r t r, C exp(- F o ) sn( r ) There s no external resstane to mass transfer: m hmro K ' D > π and C. The ontat or dffuson tme for hlorne wthn the water droplet s determned from t.5 hg se Sne pure water enters the tower,,. The equlbrum relaton s,s (kmol/m 3 ).5 P (atm). Therefore (.5)(.).6 kmol/m 3 Dt F o r o (4. )(.553) ( ) 4.673, C exp(- F o ). exp(-π.673) (.6)( +.34).47 kmol/m 3-6

7 ) Determne the moles of hlorne transferred to a droplet at the olumn ext [sn( ) os( )] [sn(π) π os (π)] 3 π π The maxmum moles that an be transferred to a water drop s 4 3 πr o 4 3 π (-4 ) 3 (.6).53-3 kmol Moles of hlorne transferred to a droplet at the olumn ext s then kmol Example wet ylnder of agar gel at 78 o K ontanng a unform onentraton of urea of. kmol/m 3 has a dameter of 3.48 mm long wth flat parallel ends. The dffusvty s m /s. Calulate the onentraton at the mdpont of the ylnder after hr for the followng ases f the ylnder s suddenly mmersed n turbulent pure water. ) For radal dffuson only. ) Dffuson ours radally and axally. (Ref: Transport Proesses and Separaton Proess by C.J. Geankopls, Prente Hall, 4 th Edton, 3) Soluton ) Calulate the onentraton at the mdpont of the ylnder after hr for radal dffuson only. For radal dffuson only, the soluton of an nfnte ylnder s used, J ( r ) t the enter of the ylnder, r > J () C exp(- F o ) The ylnder s mmersed n turbulent pure water so that. m hmro K ' D >.448 and C.6-7

8 Dt F o r o (4.7 )(36,) (.54).73, C exp(- F o ).6 exp( ).33 nfnte ylnder.33, (.33)(.).33 kmol/m 3 ) Calulate the onentraton at the mdpont of the ylnder after hr f dffuson ours radally and axally. For a slab wth thkness 38. mm,.5 m P(x, t) os( x ) slab t the enter, x, C exp(- F o ) slab m hm K ' D >.578 and C.73 Dt F o (4.7 )(36,) (.5).468 C exp(- slab F o ).73 exp( ).4 fnte ylnder, nfnte (.33)(.4).34 slab ylnder,.34 The onentraton at the mdpont of the fnte ylnder s then.34, (.34)(.).34-4 kmol/m 3-8