Calculation of collision efficiency factor by trajectory ... - Springer Link

KSCEJournal of Civil Engmlerln9 Vol 2, No

1/ M a ~ h 1998

I EnvwenmentalEr~neen~ [

pp 91--95

Calculation of Collision Efficiency Factor by Trajectory Analysis in Dissolved Air Flotation By Mooyoung Han*, Seok Dockko** and Chunghyun Park***

Abstract

In order to calculate the collision efficiency factor o f bubble and particle ( a ~ ) , trajectory analysis is performed. The formulation and equations are adopted f r o m the trajectory analysis d e v e l o p e d to calculate the collision e f f i c i e n c y factor in differential sedimentation (a~). The most recently d e v e l o p e d h y d r o d y n a m i c equation and interparticle force based o n D L V O t h e o r y are included. T h e effect o f e a c h governing parameter is calculated. Although the inclusion o f electric repulsion as a function of surface charge o f particle mad ionic strength o f suspension is possible, only hydrodyrmmic and attraction forces are considered in this research to simulate the case w h e n the bubble-particle system is destabilized. T h e collision efficiency factor o f bubble and p a r t i c l e ( a ~ ) is found to be functions o f bubble diameter, particle diameter, H a m a k e r constant, and particle density. T h e result o f a t~ shows sitrfilar trend with published a~, and experimental results, a ~ increases as bubble size decreases a t constant size ratio, as particle size ratio approaches unity and as H a m a k e r constant increases. T h e r e is only a slight effect o f particle density o n a bpK e l / w o r d s " DAF, ct bv, trajectory analysis, imerparticle f o r c e

1. Introduction Dissolved air flotation is a process that uses fine rising bubbles (diameter 10 I20/~, average 40am) to remove particles in water. D A F is found most effective in treating algae and l o w density flocs that a r e hard to r e m o v e in a sedimentation process. Flotation performance is depen-

d e n t o n r a w water quality, degree o f pretreatment, bubble size, and bubble v o l u m e concentration. Addition o f chemicals and provision o f flocculation time is n e c e s s a r y for particle destab'flization and f o r m a t i o n o f larger floe. Although there are a lot o f e x p e r i m e n tal researches in D A F , only a f e w att e m p t s have b e e n m a d e to study theoret-

* Associate Professor, Dept of Civil Eng., Kyunghee Univ., Kytmgkt-Do, Korea ** Post Doc. Industrial halson Research Institute, Kyunghee Univ., Kyungkl-Do, Korea. *** Professor, Dept. of Civil Eng., Seoul National Unlv, Seoul, Korea The rnanusc~aot for this paper was submitted for review on October 11, 1997 Vol. 2, No 1/March t998

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By Mooyotmg Hart, Seok Dockko and Chunghyun Park.

ically the movement and collision behavior o f bubbles and/or particles in a DAF system. Existing kinetic models used a single collector collision efficiency V developed for "water filtration, which accounts for the collisions between particles and a single collector(Malley Edzwald, 1991; Yao, et al. 1971). It is defined as: (particle-bubble collision rate) V= (particle-bubble approach rate) (1) for three transport mechanisms can be formulated, and the total efficiency is defined as the sum o f the three. In this model, hydrodynamic and interparticle forces are not included. Instead, the term a h, is used to include the attachment or adhesion efficiency which accounts for the effect o f all tmknown factors including chemistry, stability and hydrodyrmmics. Here, ~ bp is a calibration parameter which can be only determined from experiment. Okada et at. (1990) and LaFrance(1994) calculated the trajectory o f bubble and particle including the hydrodynamic and interparticle forces and estimated flotation efficiency. They used Brenners hydrodynamic equation and DLVO theory for the interparticle forces. They did not show their result systematically which include the collision efficiency o f bubble and particle o f different size and density range. Recently, Hart & Lawler(1991) did trajectory analysis to study the behavior o f two settling spheres and calculated the collision efficiency factors for differential sedimentation(aos). They used the most recent and accurate hydrodyluamic equation developed by Jeffrey and Onishi (1984). And then, they showed that hydrodynamic and interparticle forces can be mathematically incorporated in the cat-

culation o f collision efficiency factors for other collision mechanisms(Hart & Lawler, 1992). In that paper, they extended the analysis o f g os and showed that the effect o f all independent parameters besides the size ratio can be accounted for by a single nondimensional parameter Ng. From that results, aDS for destabilized condition can be obtained from N g - a DS relationship without calculating the trajectory. In this research, the collision efficiency factor in DAF is defined and calculated from trajectory analysis. The sensitivity o f a bp on each governing parameter is determined. The ramification o f this research is discussed.

2. Method Traditionally, collision o f a rising bubble and a falling particle is assumed to occur when the vertical projectiles o f the two overlap, because each particle is assumed to move rectilinearly without considering the hydrodynamic and interparticle forces. The collision frequency function expresses the rate at which such collision occurs, wSth the effect o f bubble concentration extracted. In fact, however, when two unequalsized particles move in a low Reynolds number region with a smaller particle falling above a larger rising bubble, each particle follows a curved path due to the influence o f the other. The exact relative trajectory o f the small particle with respect to the bubble is determined by curvilinear theory. Even when the vertical projections o f the two spheres overlap, some trajectories are open and others are cIosed. Collision depends on the initial horizontal separation distance between the centers o f the two spheres, as shown in Fig. 1. B y changing the initial x coordinate at a y coordinate far enough - 92 -

KSCE Journal of Civil Engineering

Calculation of coR~ston efficiency factor by tralectory analysis in dzssolved air flotatton

above that the effect o f e a c h other is negligible, the critical distance (Xc) which separates the o p e n and d o s e d trajectory is obtained. CnncalTmje~ry ~_,D X or

2 ~C

center-to-center separation distance(s) and size ratio o f particle to bubble(~t ). r and 0 are the location in polar coordinates, V is particle velocity, D is diffusion coefficient, k is B o l l z m a n n ' s constant, T is the absolute temperature and g) is interparticle force. This interparticle force, as defined in D L V O theory, is a fimction o f the size ratio( A ), the radius o f larger(A1) and smaller particle(or bubble) size(A2) and separation distance(s), Vs12 is the relative v e l o c i t y o f bubble and particle.

abe= (a,+ aj) z OPEN

CLOSED

Fig. 1. Definition of collision efficiency factor ( a b,,)

T h e collision efficiency factor o f bubb•e and p a r t i c l e ( a ~ , ) is defined as the ratio o f the circular area determined by the critical horizontal separation distance (Xc) to the circular area m a d e by sum o f the radii o f two spheres. F r o m this deftnition, ~bp, in rectilinear theory, is alw a y s unity; stated another way, a bw in curvilinear theory, represents a correction to the collision f r e q u e n c y calculated f r o m rectilinear theory. W h e n two unequal-sized spheres move, the relative trajectory o f the small sphere with respect to the large sphere can be obtained b y numerically integrating the first-order differential F_x/. (2) with different initial values. dr V, Y'-- dO - S v ~ -

M(s,~t) sin O V ~

(2) where L, G, M are the h y d r o d y n a m i c functions w h i c h are determined f r o m the Vol 2, No 1/March 1998

(3)

V ~ = 2g( p ~-aa~- p ,~.~)A~a2/9/~

(4)

V m = V~I- V~

(5)

In this work, a nondimensional vertical coordinate o f sr = 2 0 was used for the initial point. T h e capture cross section is the circular area defined by a radius o f the critical initial horizontal distance(Xc). F o r numerical convenience, collision was defined in this research w h e n the centerto-center distance was less than a specific value (s < 2.0001). In this case, the t w o spheres are assumed to f o r m a doublet. T h e numerical integration was m a d e b y using I M S L Library UseI's Manual subroutine D V E R K , w h i c h is based on the R u n g e - K u t t a - V e m e r fifth and sixth-order method. P r o g r a m s w e r e written in F O R T R A N 77. 3. Result

Dr2 - cos OL(s,A) Va2 - ~ G(s,,0 v ~glz s

Vsl = 2g( p ~bblr - P ,,,~t~)A12/9/t

and

Discussion

T h e standard values o f variables used in this research are as follows. T h e diameter o f bubble is a s s u m e d 40pro, and varied f r o m 20 to 200tan. T h e size ratio is calculated for 0.5 and varied f r o m 0.1 to 0.9. T h e H a r n a k e r constant is taken as 10kT (3.54 x10-14erg), arid varied f r o m 1 - 93 -

By Mooyoung Hart, Seok Dock;ko and Chunghyun Park

to 20 kT. Particle density is 1.1g/cm 3, and varied f r o m 1.1 to 2.65g/era ~, T h e effect o f e a c h force in the calculation o f a ~v is s h o w n in Fig. 2. W h e n no forces are included, no correction is needed in the a ~, so the a bp is always equal to 1 (Line 1 in the figure). W h e n both hydrodyrmmics and v a n der Waals attraction are included, a ~ b e c o m e s less lO +__

-._

_-

~

= +--

_-

u

it

2 gYD~AT~ 0 01

i

Dd;,ml

:

:

:

0B

07

0a

-+-0+

0 O01 0+000"I 0

01

02

03

04

05

051

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Fig. 2.

ab.

:

Effect of bubble and particle size

than 1 and varies fi:om 0.001 to 0.02 depending on the size o f bubble and particle (Curve 2). ~ ~, b e c o m e s a function o f bubble size, particle size, size ratio. G~p increases as the particle size decreases, and bubble-particle size ratio approaches unity. W h e n the repulsion force is included for the case o f the bubble surface charge is -60mV, the particle surface charge; - 2 0 m V and suspension ionic strength; 0.001M, ~r~ becomes v e r y l o w and is not dependent o n the size o f bubble and particle(Curve 3).

The effect o f particle density ( f r o m 1.1 g/cm 3 for algae to 2 . 6 5 g / c m 3 for sand) to a ~v is shown in Fig. 3 for the case noted in the figure, a bp is affected b y the size o f bubbles. Inclusion o f electric repulsion force lowers the value o f a ~ . H o w e v e r , there is n o significant difference due to the density o f particle. This suggests that the collision efficiency can be considered the same within the density range studied. This shows the advantage o f D A F process over sedimentation process, especially for l o w density particles. Fig. 4 shows the sensitivity o f both H a m a k e r constant(A) and particle size ratio at the conditions n o t e d in the figure. T h e effect o f H a m a k e r constant was determined at a fixed bubble diameter o f 40 /_~ and particle o f 1.1g/cm3( ,t =0.5). a ~ increases as H a m a k e r constant increases and size ratio increases. 1

I;

0oI e ,r, =

~ 001

0r 01

O~

03

34

05

~

C:$

s

~

S~zr rat,.. ;~-DJI~.

Fig. 4 .

ab,

:

Effect of Hamaker constant

Similar analysis can be m a d e for the two rising bubbles to calculate the col-

1 [pz--O 0~11?gjem] A=IQZT, z - n d n , < s HyI)+ATT =~

1

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P~"~K'I 17Went~ A= I 0~.T

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-

I'tV b+s~1"11" ~ 2 " _-

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of

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=

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i

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0 _-

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PplWem 31

particle

O

density

01

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for

different bubble size

OS

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Fig. 5. ebb : Effect of bubble size

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KSCE

Journal of Qvi[

Engineering

Calculation of colllszon e~ciency factor by trajectory analyszs m dzssolved mr flotation

Iision efficiency factor o f bubble and b u b b l e ( a ~ O b y m o d i f y i n g equation(4). T h e results are s h o w n as in Fig.5. ~r ~b increases as b u b b l e size decreases at c o n stant size ratio, as particle size ratio a p p r o a c h e s unity. T h e r e is no significant difference b e t w e e n the values o f a bp and a bb within the r a n g e o f calculation.

4. Conclusion T h e collision efficiency factor o f b u b ble and particle ( a up) in D A F is calculated f r o m trajectory analysis using the h y d r o d y n a m i c a n d interparticle forces. The sensitivity o f a bp o n e a c h g o v e r n i n g p a r a m e t e r is calculated and discussed. T h e result s h o w s similar trend w i t h published aDS and e x p e r i m e n t a l results, a ~ increases as b u b b l e size decreases at c o n stant size ratio, as particle size ratio a p p r o a c h e s unity and H a m a k e r constant increases. T h e r e is o n l y a slight effect o f particle density in ~ . T h e collision efficiency factor o f b u b b l e and b u b b l e ( a bb) is similar to the v a l u e o f a by o n the r a n g e studied in this research. T h e curvilinear collision efficiencies calculated in this research can b e s u m m a rized systematically and applied to the m o d e l i n g o f dissolved air flotation p r o c ess. Inclusion o f electric repulsion force in the trajectory analysis will give results for

Vol. 2, No. I/ March 1998

the c a s e o f stable system. E x p e "rtmental verification o f this m o d e l is needed.

5. References 1 Han, M Y. and Lawler, D. F. (1991) Interactions of Two Settling Spheres: Setthng Rates and Collision Efficiencxes, Jour Hydraulic Eng, ASCE, 117, No.10, 1269 1289. 2. Hart, M. Y. and Lawler, D F. (1992). The (Relative) Insignificance o f G in Flocr Jour. AWWA, 79. No 10, 79 - 91. 3. Jeffrey, D. J. and OnJshi, Y. (1984). Calculation of the Resistance and Mobility Functions for Two Unequal Rigid Spheres in Low Reynolds Number Flow, Jour Flrad Mech, 139, 261 - 290. 4. LaFrance. P. (1994). Masters Thesis, University of Connecticut. 5. 1VLally, J. P., Extzwald, J K.(1991) Concepts for dissolved air flotation treatment o f drinking waters. Water SRT-Aqua, 40, No.l, 7-17. 6. Okada, K , Akagi. Y., Kogure, M., Yosbaoka, N (1990). Analysis o f particle trajectories of small particles m flotation when the particles and bubbles are both charged. Can J Chem. Eng, 68, 614-62t 7. Yao, K. M., Hablbian, ~ T., O'Melia C. R. (1971). Water and Wastewater Filtration. Concepts and Applications. Environ. Sci Technol, 5, No. 11, 1105-1t12.

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