Turbulence Deposition

Transcription

1 Trblene eposition ring trblent flid motions, partiles are transported by the trblene eddies and the Brownian diffsion. Ths, the partile flx is given by T dc J ( ) () dy where C is the average onentration and to Lin et al., T is the trblent eddy diffsivity. Aording T y ( ) 4.5 y < y 5< y < 5 < 0 () where y τ0 y,. () f ρ Assming that the flx is a onstant (or linearly varying) near the wall, dc J. (4) T dy Sbjet to appropriate bondary onditions, the onentration and the flx to the wall may be determined. This approah was first introded by Friedlander and Johnstone (959). Sine then many athors sed a nmber of modified bondary onditions. Here the empirial model desribed by Wood (98) is otlined. Trblent iffsion (Wood, J. Aerosol Siene,, pp 76-8, 98) τ0 For trblent flow near a wall, sing the shear veloity and, a wall f ρ nit may be defined. All qantities may be non-dimensionalized with the aid of and. e.g., the flid mean veloity and the partile relaxation time

2 p p τ ρ d ρ τ ( ) Re f f d, (5) 8ρ 8 ρ where the Reynolds nmber based on shear veloity is defined as, d. The non- dimensional deposition veloity is defined as Re d T J d. (6) dy 0 For very small partiles with τ <, Brownian diffsion beomes signifiant and deposition is affeted by a ombination of Brownian and eddy diffsion. For the Brownian regime, / / s 0.057s (7) 9π where s is the Shmidt nmber. For larger partiles p to τ 0, Eqation (7) mst be modified for the eddy diffsion-impation regime: / s 4.5 τ, (85) 0 The seond term beomes dominant in the eddy-diffsion-impation regimes for < τ <0. In the partile inertia moderated regime, 0. 7 < τ < 00 (86) For τ > 65, it was fond empirially.6 50 ( ). (87) τ τ Figre shows the variation of the nondimensional deposition veloity with nondimensional relaxation time for a trblent flow with shear veloity of 0. m/s. The partiles are assmed to have a density ratio of 000 in this example.

3 Figre. Variation of the nondimensional deposition veloity with nondimensional relaxation time for 0. m/s and S000.

4 eposition on Rogh Walls Wall roghness old signifiantly affet the deposition rate in trblent streams. Wood sggested I B I s, I B 4. for 0.45 < 5, (88) where I s 4.5S / ( φ) [ ln 6 φ φ for 0.45 < 5 tan φ ( φ ) ln 6 φ φ tan φ ] (89) Here, / φ s,.9 / s φ,. s. (90) Eqations (88) and (89) may be approximated as 54 [ 0.45 (0.9 5/ φ) for φ >>, ( > ) (9) 6.] t π )] 7.6 / / [4. 4.5s ( s for >> φ. (9) For 0, eqation (9) redes to (84). and For the highly rogh ase 5 < 0.45 << 0, I s 0 or I B 5. 5ln( ), (9) ln( 0.45 ) 4.8. (94) 4

5 Figre shows the variation of the nondimensional deposition veloity with nondimensional relaxation time for smooth and rogh srfaes as predited by Eqation (89) (Wood s formla.). It is seen that as the roghness oeffiient inreases the deposition rate inreases signifiantly. For a large vale of srfae roghness, the deposition rate beomes roghly a onstant. Figre. Variation of the nondimensional deposition veloity with nondimensional relaxation time for smooth and rogh srfaes as predited by Wood s formla. Using a sblayer model, Fan and Ahmadi developed a semi-empirial model for the deposition rate on smooth and rogh srfaes. Aordingly, d 0.084S 0.4 / d τ g L p ( τ τp g L ( τp L) ( τp 0) / [ 8e ] τ p 0.07 g L( ) 0.07 p otherwise / L) ( τ L ) p if d < 0.4 (95) 5

6 Here nits, d is the partile diameter in wall nits, g g /, and τ p is the partile relaxation time in wall L.08 / Sd. The preditions of Eqation (95) are shown in Figre and are ompared with that of Wood s eqation for smooth wall. It is seen that the model preditions of Fan-Ahmadi are omparable with that of Wood for a smooth srfae. For rogh srfaes the deposition rate inreases sharply as the srfae roghness inreases. Fan Ahmadi (99) showed that the empirial model predition for rogh srfaes are in reasonable agreement with the experimental data. 0.5 Fan-Ahmadi 0 Wood Figre. Variation of the nondimensional deposition veloity with nondimensional relaxation time for smooth and rogh srfaes as predited by Fan-Ahmadi formla. 6

7 Gravitational eposition Partiles deposit on a horizontal srfae de to the ation of gravity. The gravitational deposition veloity is roghly eqal to the terminal veloity. i.e., or t τg (96) τ g, (97) where g g. (98) Figre 4 ompares the gravitational sedimentation veloity as given by Eqation (97) with trblent deposition and thermophoreti deposition for partiles of different sizes nder different flow and thermal onditions. It is seen that the gravitational effet is qite important for partiles larger than a few mirometers, while the thermophoresis effets are signifiant for smaller partiles. In the absene of thermal fores, the gravitational effets is signifiant for partiles larger than 0. µ m. Figre 4. Comparison of deposition veloities de to trblene, gravity and thermophoreti with partile diameter. 7

8 Figre 5 ompares the gravitational deposition veloity with the trblent deposition rate on a smooth wall as predited by the models of Wood (98) and Fan and Ahmadi (99). Here a shear veloity of 0. m/s and partile-to-flid density ratio of S000 are assmed. It seen that the gravitational sedimentation veloity is larger than trblene deposition rate for partile relaxation times greater than in wall nits. For smaller partiles, the Brownian diffsion dominates and the effet of gravity is negligible. Wood Gravity Fan-Ahmadi Figre 5. Variation of the nondimensional deposition veloity with nondimensional relaxation time for smooth and rogh srfaes as predited by Fan-Ahmadi formla. Gaseos Mass Transfer in Trblent Flows Trblene diffsion, sally, dominates the gaseos mass transfer proesses. avies obtained s 0., (99) s 5.04 ln( ) s 0.04 where s is the Shmidt nmber for the speies onsidered. For s >> 5, Eqation (99) redes to 8

9 s 0.. (00) ln( 5.04s ) For rogh srfaes, 0. t ( 0.09 )s ln( 5.04s ) for 0.45 < 5 (0) and 0. ln[(.s 56) / t ] for 5 < 0.45 << 0. (0) Note that when the wall is not absorbing and the onentration on the wall is then the mass flx is C w, Cw J U (C0 Cw ) U C0( ), (0) C where C 0 is the bl onentration. 0 9