Shear-induced flocculation: The evolution of floc structure and the shape of the size distribution at steady state

~

Wat. Re5. V01. 30, N0. 5, pp. 1049-1056, 1996

Per9am0n

0043-1354(95)00253-7

C0pyr19ht ~. 1996 E15ev1er5c1ence Ltd Pr1nted 1n 6reat 8r1ta1n. A11 r19ht5 re5erved 0043-1354/96 $15.00 + 0.00

5HEAR-1NDUCED F L 0 C C U L A 7 1 0 N : 7 H E E V 0 L U 7 1 0 N 0 F F L 0 C 5 7 R U C 7 U R E AND 7 H E 5HAPE 0 F 7 H E 512E D157R18U710N A7 57EADY 57A7E PA7R1CK 7. 5P1CER and 5071R15 E. PRA751N15* Department 0f Chem1ca1 En91neer1n9,ML 171, Un1ver51ty0f C1nc1nnat1, C1nc1nnat1, 0H 45221-0171, U.5.A. (F1r5t rece11~ed June 1995; accepted 1n rev15edf0rm 0ct06er 1995)

A65tract--7he f10ccu1at10n0f p01y5tyrenepart1c1e5 1n a 5t1rred tank wa5 5tud1ed at var10u5 5hear rate5 (63-129 5-~)and a1Um1num5u1fate, A1.,(504)316H•,0, f10ccu1antc0ncentrat10n5.7he c0mpet1t10n6etween c0a9u1at10n and fra9mentat10n dur1n9 5hear-1nduced f10cCU1at10ndeterm1ned the e4u1116r1um0r 5teady 5tate part1c1e (f10c) 5tructure and 512e d15tr16ut10n. 7he ev01ut10n 0f the f10c 5tructure w1th t1me wa5 m0n1t0red 6y 1ma9e ana1y5150f d191t12ed f10c 1ma9e5. 7he avera9e f10c 5tructure 6ecame 1e550pen 0r 1rre9u1ar a5 the f10c 512e d15tr16ut10n atta1ned 5teady 5tate a5 a re5u1t 0f 5hear-1nduced 6reaka9e/ re5tructur1n9. At h19h a1um (f10ccu1ant)c0ncentrat10n5,the 5teady 5tate f10c 512e d15tr16ut10nappeared t0 6e 5e1f-pre5erv1n9w1th re5pect t0 5hear rate. 1n c0ntra5t, at 10werf10ccu1antc0ncentrat10n5,the 5teady 5tate f10C 512e d15tr16ut10n narr0Wed w1th 1ncrea51n95hear rate a5 the 1ar9e ta11 0f the d15tr16ut10nwa5 pu5hed t0 5ma11erpart1c1e 512e56y 5hear-1nducedfra9mentat10n.C0pyr19ht • 1996 E15ev1er5c1ence Ltd. Ke),~ w0rd5--5e1f-51m11ar1ty, 5hear-1nduced f10cCu1at10n,a1um1num5u1fate, part1c1e 512ed15tr16ut10n5,f10c 5tructure, fracta1

1N7R0DUC710N 7he rem0va1 0f part1c1e5 fr0m a 114u1d 5u5pen510n 15 0ne 0f the 6a51c type5 0f 5eparat10n5 1n 60th dr1nk1n9 water and wa5te water treatment. 7he part1c1e5 are fre4uent1y 1n the m1cr0n 0r 5u6-m1cr0n 512eran9e and are d1ff1cu1t t0 rem0ve 6y c0nvent10na1 mean5 11ke 5ed1mentat10n 0r f11trat10n. F10ccu1at10n 15 0ften u5ed t0 1ncrea5e the avera9e part1c1e (f10c) 512eand enhance 5011d5 rem0va1. 5evera1 pha5e5 0f f10c 9r0wth 0ccur dur1n9 f10ccu1at10n. 1n1t1a11y, part1c1e (f10c) 9r0wth 15 d0m1nant, part1c1e5 c0m61ne 6y c0a9u1at10n and the1r 512e 1ncrea5e5 rap1d1y. A5 f10ccu1at10n c0nt1nue5, the f10c5 f0rm 1ar9e, p0r0u5 and 0pen 5tructure5 that are m0re 5u5cept161e t0 fra9mentat10n 6y f1u1d 5hear (7am60, 1991). A5 a re5u1t, the f1na1 f10c 512e d15tr16ut10n 15 the 6a1ance (5teady 5tate) 6etween part1c1e 9r0wth and 6reaka9e (Re1ch and V01d, 1959; Parker et a1., 1972; Lu and 5p1e1man, 1985; 01e5, 1992; 5p1cer, 1995). Furtherm0re, the f10c 5tructure 15 1mp0rtant 51nce 1t determ1ne5 f10c 512e and den51ty and, thu5, w1111nf1uence the 5011d5 rem0va1 eff1c1ency dur1n9 5ed1mentat10n. 7he 5tructure 0f the f10c5 at 5teady 5tate a150 depend5 0n the pr0ce55 c0nd1t10n5. V1rtua11y n0 exper1menta1 5tud1e5 0f the dynam1c ev01ut10n 0f the f10c 5tructure have 6een perf0rmed. M05t 5tud1e5 have f0cu5ed 0n the determ1nat10n 0f

*Auth0r t0 wh0m a11 c0rre5p0ndence 5h0u1d 6e addre55ed.

f10C 5trUCtUre at 5teady 5tate 0r 0n 5amp1e5 C011eCted at a C0n5tant re51dence t1me (1.e. K11mpe1 and H099, 1986; 50nnta9 and Ru55e1, 1986; L1 and 6anC2arC2yk, 1990; C1ark and F10ra, 1991; J1an9 and L09an, 1991; L09an and K11p5, 1995). Re1Ch and V01d (1959) 5tUd1ed the effect 0f 5hear rate 0n a 5U5pen510n 0f ferr1C hydr0X1de and f0Und that the f1na1 avera9e f10C 512e wa5 the re5U1t 0f a 5teady 5tate 6etWeen 6reaka9e and 9r0Wth. 7am60 and Watana6e (1979a) the0ret1Ca11y and exper1menta11y 5tud1ed the tur6u1ent f10ccu1at10n 0f ka011n u51n9 a1um1num 5u1fate a5 a f10ccu1ant. 7hey f0und that the 5teady 5tate d15tr16ut10n 0f 5ed1mentat10n rate5 f0r the 5u5pen510n c011ap5ed 0nt0 a 51n91e curve when n0rma112ed 6y tw0 var1a61e5.7hey 1nferred that the 5teady 5tate f10c 512e d15tr16ut10n wa5 5e1f-pre5erv1n9 6ut d1d n0t 0ffer d1rect exper1menta1 pr00f. M0re recent1y, 01e5 (1992) m0n1t0red the avera9e f10c 512e at 5teady 5tate a5 a funct10n 0f 5hear rate f0r c0a9u1at10n 0f p01y5tyrene part1c1e5 u51n9 NaC1 a5 a de5ta61112er (f10ccu1ant). He 065erved a rap1d 1n1t1a1 f10c 9r0wth rate that 510wed a5 f10c 6reaka9e 6e9an t0 0ccur. A 1eve11n90ff 0f the f10c 512e5at 5teady 5tate wa5 065erved and attr16uted t0 an e4u1116r1um 6etween f10c 9r0wth and 6reaka9e. 1ncrea51n9 the am0unt 0f f10c 6reaka9e 1ncrea5ed the f10c den51ty 6ut decrea5ed the avera9e f10c 512e (01e5, 1992). 7h15 paper addre55e5 the atta1nment 0f 5teady 5tate 6y a f10ccu1at1n9 5u5pen510n a5 a funct10n 0f pr0ce55 c0nd1t10n5. Empha515 15 p1aced 0n the ev01ut10n 0f

1049

1050

Patr1ck 7. 5p1cer and 50t1r15 E. Prat51n15

the f10c 5tructure a n d the pr0pert1e5 0 f the 5teady 5tate f10c 512e d15tr16ut10n. Mea5urement5 0f f10c 512e 6y 1ma9e ana1y515 are u5ed t0 eva1uate the deve10pment 0 f the 5teady 5tate f10c 5tructure a n d the f0rm 0 f the 5teady 5tate 512e d15tr16ut10n. C0nd1t10n5 t h a t 1ead t0 a5ympt0t1c 0r 5e1f-pre5erv1n9 5teady 5tate 512e d15tr16ut10n5 w1th re5pect t0 5hear rate a n d f10ccu1ant c0ncentrat10n are 1dent1f1ed. 7h15 15 part1cu1ar1y u5efu1 1n under5tand1n9 the de9ree 0f c0ntr01 that can 6e exerted 0ver f10c f0rmat10n. EXPER1MEN7AL Apparatu5 and mater1a15

A11 exper1ment5 were c0nducted 1n a c105ed, 2.8 1, 6aff1ed, 5t1rred tank 0f 5tandard c0nf19urat10n (H011and and Chapman, 1966) U51n9a 51x-61ade, d15k-m0unted (RU5ht0n) 1mpe11er. F19ure 1 5h0w5 the re1at1ve d1men510n5 0f the emp10yed 1mpe11er-tank. 7he tur6u1ent 5hear rate w1th1n the 5t1rred tank wa5 CharaCter12ed 6y the avera9e ve10c1ty 9rad1ent 0r 5hear rate, 6 (Camp and 5te1n, 1943) 6 =

(1)

where v 15 the k1nemat1c v15c051ty 0f the 5u5pend1n9 f1u1d (here, water) and ~ 15 the avera9e tur6u1ent ener9y d1551pat10n rate (60dfrey et a1., 1989)

where P0 15 the 1mpe11er p0wer num6er, N 15 the 1mpe11er 5peed, V 15 the 5t1rred tank v01ume, and D 15 the 1mpe11er d1ameter. 7he P015 06ta1ned u51n9 the 5tandard p0wer curve f0r the emp10yed 51x-61ade 1mpe11er (p. 78, F19.4-4 1n H011and and Chapman, 1966). A1th0u9h the f10w c0nd1t10n5 w1th1n a 5t1rred tank are n0n-h0m09ene0u5 (Cutter, 1966; 5pr0w, 1967; K0nn0 et a1., 1983), E4uat10n5 1 and 2 are u5ed t0 pr0v1de a 6a515 f0r c0mpar150n w1th prev10u5 w0rk. 7he 1mpe11er r0tat10na1 ve10c1ty wa5 mea5ured w1th an 0pt1ca1 tach0meter (0n0 50kk1 H7-4100). 7he K01m090r0v m1cr05ca1e, ~, character121n9 the 1en9th 5ca1e 0f the ener9y d1551pat1n9 tur6u1ent edd1e5 15 (5p1e1man, 1978)

8aff1ed 5t1rred 7ank 1.5 cm

1.2 cm

15 cm

v

5 cm

A

15 cm

F19. 1. 5chemat1c 0f the emp10yed 5t1rred tank.

M0n0d15per5e, 5pher1ca1, p01y5tyrene part1c1e5 w1th a d1ameter 0f 0.87 #m (6y 7ran5m15510n E1ectr0n M1cr05c0py) and a den51ty 0f 1.05 9/cm 3 were f10ccu1ated 1n d15t111ed water (that had 6een 6uffered w1th NaHC03) at 5011d5 v01ume fract10n5 0f 4~ = 8.3 x 10 -5 and 2.1 x 10 -5. P01y5tyrene 15 an 1dea112edm0de1 part1c1e 6ecau5e 0f 1t5 10w den51ty and 5pher1ca1 5hape. 7he p01y5tyrene part1c1e5 had 6een prepared u51n9 the meth0d 0f De 80er (1987) and c0ncentrated 6y vacuum evap0rat10n t0 m1n1m12e the v01ume 1njected 1nt0 the 5t1rred tank 5u5pen510n. 7he f10ccu1ant wa5 a1um1num 5u1fate hydrate 0r a1um, A12(504)3 16H20 (A1dr1ch, 98%). 7he 5u5pen510n wa5 6uffered w1th 50d1um hydr09en car60nate (NaHC03) (A1dr1ch, 99%) at a c0ncentrat10n 0f 1 mM and the pH wa5 ma1nta1ned at 7 + 0.2 6y the add1t10n 0f 0.1 N HC1 0r N a 0 H 1n a11 exper1ment5.

Pr0cedure and mea5urement5

A 5ma11 v01ume (1-2 cm 3) 0f the p01y5tyrene 5u5pen510n wa5 1njected 1nt0 the water v01ume 1n the 5t1rred tank and m1xed 6ef0re f10ccu1at10n wa5 1n1t1ated. 7he part1c1e5 were 5u5pended and 5t1rred f0r 5 m1n at a h19h 5hear rate, 6 = 540 5-~ (500 r.p.m.), t0 6reak up any 1n1t1a11ypre5ent a9910merate5. 7h15 pr0cedure wa5 checked u51n9 a c0ntr01 exper1ment 1n wh1ch n0 f10ccu1ant wa5 added and the 1nd1v1dua1 pr1mary part1c1e5 were 065erved t0 6e 5ta61e and rema1n un-f10ccu1ated. U51n9 a hyp0derm1c need1e, a mea5ured d05e 0f the ac1d1c 5t0ck f10ccu1ant 501ut10n (2 9 a1um per 11ter) wa5 then added at the t1p 0f the 1mpe11er and m1xed w1th the 5u5pen510n f0r 1 m1n at 6 = 540 5-L 7he f10ccu1ant d05e wa5 expre55ed a5 t0ta1 m11119ram5 0f A12(504)3 16H20 per 11ter 0f 5u5pen510n. F0110w1n9 the rap1d m1x1n9, f10ccu1at10n wa5 then carr1ed 0ut at a c0n5tant 5hear rate. A 1 cm 3 5amp1e wa5 w1thdrawn per10d1ca11y f0r 1ma9e ana1y515. 7h15 5amp1e v01ume wa5 f0und 5uff1c1ent t0 pr0v1de repr0duc161e 512e d15tr16ut10n mea5urement5. A m1n1mum 0f 500 part1c1e5 were c0unted dur1n9 the determ1nat10n 0f each 512e d15tr16ut10n. M1cr05c0p1c ana1y515 wa5 carr1ed 0ut u51n9 an 0pt1ca1 m1cr05c0pe (N1k0n, La60ph0t) c0nnected t0 a CC7V camera (H1tach1-Den5h1). 5amp1e5 were taken 6y 1n5ert1n9 the w1de end 0f a 0.5 cm 1.d. p1pette (61665 and K0nwar, 1982) 6e10w the 5u5pen510n 5urface, c0ver1n9 the end, and w1thdraw1n9 carefu11y. 7he 1ma9e ana1y515 0f the f10c5 wa5 carr1ed 0ut 6y p1ac1n9 a dr0p 0n a f1at m1cr05c0pe 511de w1th0ut a c0ver 511p and v1de0tap1n9 the c0rre5p0nd1n9 1ma9e. 7he v1de0 wa5 then d191t12ed and ana1y2ed w1th 6106a1 La6 1ma9e v. 2.0 50ftware. 7he re501ut10n 0f the m1cr05c0pe def1ne5 the 10wer detect10n 11m1t 0f the 1ma9e ana1y515 techn14ue. Under 400X ma9n1f1cat10n a 1 #m f10c appear5 t0 6e 0.4 mm 1n 1en9th. 7he 10wer detect10n 11m1t 0f the 1ma9e ana1y515 50ftware, h0wever, 15 0n1y a60ut 1 mm ( ~ 2 . 5 / J m f10c at 400X). At 100X and 40X ma9n1f1cat10n5, the 10wer detect10n 11m1t5 were 10 and 25/~m, re5pect1ve1y. 70 m1n1m12e the 1nf1uence 0f the 10wer detect10n 11m1t, 1ma9e ana1y515 data were u5ed 0n1y when m0re than 95% 6y c0unt (0r 9reater) 0f the v15ua11y 065erved part1c1e5 c0u1d 6e re501ved 6y the 1ma9e ana1y515 50ftware. 7he max1mum 1en9th (512e), per1meter, and cr055 5ect10na1 area 0f the f10c5 were 06ta1ned 6y 1ma9e ana1y515, a5 the v1ewed f10c5 had 5ett1ed t0 the1r m05t 5ta61e c0nf19urat10n. 7he 1ma9e ana1y515 50ftware 1dent1f1e5 an area 0f the 1ma9e a5 a part1c1e 1f the 9rey va1ue5 0f the c0nt19u0u5 p1xe15 exceed a u5er def1ned thre5h01d vah1e. 8ef0re ana1y515 the 50ftware wa5 ca116rated u51n9 a 511de marked at kn0wn 1nterva15. An 1ma9e 0f the 511de wa5 d191t12ed and the num6er 0f p1xe15 6etween tw0 mark5 c0rre5p0nd1n9 t0 10 #m wa5 rec0rded 0n the 5creen.

F10c 5tructure and 512ed15tr16ut10n 150 A1Um C0nCentrat10n (m9/11te0 -0-

5hear Rate, 6 = 63 5"1

32 32 (6=95 5-1)

-0--

10.7 4.3

100

Y

0f 0

A

40

2•0

60

71me (m1n)

F19.2. Ev01ut10n 0f the avera9e f10c 1en9th, L,, a5 a funct10n 0f t1me f0r the f10ccu1at10n 0f p01y5tyrene at var10u5 a1um c0ncentrat10n5 (c=4.3-32 m9/1) and 6=63 5 ~ (0pen 5ym6015) and 6 = 95 5 ~(5011dd1am0nd5). After atta1nment 0f 5teady 5tate, L. n0 10n9er chan9e5 w1th t1me.

R E 5 U L 7 5 AND D 1 5 C U 5 5 1 0 N

7he atta1nment 0f 5teady 5tate

F10c 512e d15tr16ut10n5 were 06ta1ned 6y 1ma9e ana1y515 0f v1de0taped, d191t12ed 1ma9e5 0f the 5u5pen510n 5amp1e5. A11 data p01nt5 repre5ent the re5u1t5 0f 2-3 rep11cate exper1ment5 w1th a repr0duc16111ty err0r 1e55 than 5%. 7he emp10yed a1um c0ncentrat10n5 were ch05en t0 repre5ent the three c1a551ca1 re91me5 0f part1c1e de5ta61112at10n, ad50rpt10n de5ta61112at10n: 4.3 m9/1; 5weep f10ccu1at10n: 32 m9/1; and the 1ntermed1ate re91me: 10.7 m9/1 (Am1rtharajah and M1115, 1982). 7he5e c0ncentrat10n5 are near the 10w end (10-100 m9/1) 0f pract1ca1 app11cat10n5 0f f10ccu1at10n (Am1rtharajah and M1115, 1982) 6ut a150 a110w c0mpar150n w1th the trend5 pred1cted 6y the prec1p1tat10n-char9e neutra112at10n the0ry (Dente1 and 6055ett, 1988). A11 5u65e4uent d15cu5510n 0f a1um de5ta61112at10n effect5 15 6a5ed 0n th15 the0ry. 7he emp10yed 5hear rate5 (63, 95, 129 5-L; 116, 152, 184 r.p.m.) were ch05en t0 e11m1nate the p05516111ty 0f 519n1f1cant 5ed1mentat10n 0f the 1ar9er f10c5. A1th0u9h the 6 va1ue5 are re1at1ve1y h19h when c0mpared t0 th05e 0f a pract1ca1 f10ccu1at10n 5y5tem (10--40 5-~), they a110w the 065ervat10n 0f 60th part1c1e c0a9u1at10n and fra9mentat10n phen0mena that are re5p0n5161e f0r the atta1nment 0f the 5teady 5tate 512e d15tr16ut10n. A 5u5pen510n wa5 c0n51dered t0 have atta1ned 5teady 5tate when the f10c 512e d15tr16ut10n n0 10n9er chan9ed w1th t1me. F19ure 2 5h0w5 the ev01ut10n 0f the ar1thmet1c (num6er) avera9e f10c 1en9th, L,, at 6 = 63 5-~ (0pen 5ym6015) f0r three a1um c0ncentrat10n5. Depend1n9 0n the a1um c0ncentrat10n, after 50me t1me [a60ut 25m1n f0r 6 = 6 3 5 -~ and c = 10.7 m9/1 a1um (F19.2)] the f10c 512e d15tr16ut10n

1051

rema1n5 c0n5tant and ha5 reached 5teady 5tate. 1ncrea5ed a1um c0ncentrat10n pr0duce5 5tr0n9er and 1ar9er f10c5 and re5u1t51n a fa5ter atta1nment 0f 5teady 5tate. Furtherm0re, 1ncrea51n9 the 5hear rate (F19. 2: 5011d d1am0nd5) decrea5e5 the 5teady 5tate f10c 512e a5 a re5u1t 0f 1ncrea5ed f10c 6reaka9e. F19ure 3 5ummar12e5 the avera9e 5teady 5tate f10c 1en9th, L,~ 06ta1ned 6y 1ma9e ana1y515 a5 a funct10n 0f 5hear rate ( 6 = 6 3 , 95, 129 5-t) and a1um c0ncentrat10n (c = 4.3, 10.7, and 32 m9/1). 7he data 0f 01e5 (1992) are a150 5h0wn f0r the 5hear-1nduced f10ccu1at10n 0f 2.17 ~m p01y5tyrene part1c1e5 w1th NaC1 at ~6 = 10 5. 7he L.~ depend5 0n the re1at1ve 519n1f1cance 0f 5hear-1nduced 9r0wth and 6reaka9e, each 0f wh1ch 15 1n turn a funct10n 0f the app11ed 5hear rate and a1um c0ncentrat10n. 1n F19. 3, 1ncrea51n9 5hear 1ncrea5e5 the rate 0f f10c 6reaka9e, re5u1t1n9 1n a 5ma11er avera9e f10c 512e at 5teady 5tate. At the ear1y 5ta9e5 0f f10ccu1at10n, 5hear 1ncrea5e5 part1c1e c0111510n5 and 9r0wth. A5 the part1c1e5 9r0w, 6reaka9e 6ec0me5 m0re 519n1f1cant a5 the 512e ran9e 0f f10c5 5u5cept161e t0 6reaka9e 6y tur6u1ent edd1e5 1ncrea5e5 (Ku5ter5, 1991). 7h15 15 c0n515tent w1th Parker et a1. (1972), 7am60 and Watana6e (1979a), and 01e5 (1992) wh0 065erved that the avera9e f10c 512e decrea5ed w1th 1ncrea51n9 5hear rate. F19ure 3 a150 5h0w5 the K01m090r0v m1cr05ca1e 0f tur6u1ence, 4, f0r the emp10yed ran9e 0f 5hear rate5. 7he var1at10n 0f 4 w1th 6 f0110w5 the var1at10n 0f L, w1th 5hear f0r c = 32 m9/1 and 10.7 m9/1 a5 we11 a5 the data 0f 01e5 (1992). Reca111n9 that the 5ma11e5t eddy 512e 15 0f the 0rder 0f 4, we 065erve that the 5hear f1e1d determ1ne5 the 512e character15t1c5 0f the re5u1t1n9 f10c5 at h19h f10ccu1ant c0ncentrat10n5. 7he 1ncrea51n9 L, w1th a1um c0ncentrat10n c1ear1y 1nd1cate5 the 5tren9then1n9 0f the 11nka9e 6etween the

150 32 m9/11ter

E" 125

0

10.7 m9111ter

A

4.3 m9/11ter

-- - - 01e5, 1992 100

~

7~

~ A

0 50

2~ 75

100

0 A 125

150

5hear Rate, 6 (5"1)

F19. 3. 7he avera9e 5teady 5tate f10c 1en9th, L,,., a5 a funct10n 0f 5hear rate and a1um c0ncentrat10n. A150 p10tted are the f10ccu1at10n re5u1t5 0f 01e5 (1992) and the ca1cu1ated va1ue5 0f the K01m090r0v m1cr05ca1e, 4.7he data 0f 01e5 (1992) are 1n 900d a9reement w1th the re5u1t5 0f th15 5tudy. 7he var1at10n 0f r/and L,,. w1th 6 are 1n 900d a9reement.

1052

Patr1ck 7. 5p1cer and 50t1r15 E. Prat51n15 1000

//I Dpf = 1 /

,~

2

/ j./5~0~, = -55•,

1

1'0

100

1000

F10c Per1meter(pm)

F19. 4. Determ1nat10n 0f the fracta1 d1men510n, 0pf, fr0m 1ma9e ana1y515 mea5urement5 0f f10c cr055-5ect10na1 area and per1meter at ff = 2.1 x 10 -5. 7he tw0 11m1t1n9ca5e5 0f Dpr are a150 5h0wn f0r c0mpar150n 0f the re1at1ve f10c 5tructure. pr1mary part1c1e5 0f the f10e. Depend1n9 0n the m0de 0f de5ta61112at10n (Dente1 and 6055ett, 1988), 1ncrea51n9 the f10ccu1ant c0ncentrat10n w111 e1ther 1ncrea5e the avera9e part1c1e 512e 0r the part1c1e num6er c0ncentrat10n. E1ther re5u1t w111 1ncrea5e the f10e 9r0wth rate un1e55 re5ta61112at10n 6y char9e rever5a1 0ccur5 (Dente1 and 6055ett, 1988). 1ncrea5ed a1um c0ncentrat10n 1ncrea5e5 the f10e 9r0wth rate and 5tren9th 0f the 11nka9e 6etween the pr1mary part1c1e5 1n the f10e (6y 1ncrea51n9 the A1(0H)3 5urface c0vera9e), mak1n9 1t m0re re515tant t0 6reaka9e. A5 a re5u1t, the f10e 512e at 5teady 5tate 1ncrea5e5 w1th 1ncrea51n9 a1um c0ncentrat10n. 7h15 15 1n a9reement w1th Franc015 (1988) wh0 f0und an 1ncrea5e 1n 60th the f10e 9r0wth rate and the c0111510n eff1c1ency w1th 1ncrea5ed a1um c0ncentrat10n dur1n9 ka011n f10ccu1at10n.

the f10e 5urface m0rph0109y 51nce Dpf var1e5 6etween 2 (a 11ne) and 1 (the pr0jected area 0f a 5011d 5phere, a c1rc1e). Lar9e va1ue5 0f Dpr theref0re repre5ent 0pen and 1rre9u1ar f10e 5tructure5. F19ure 4 5h0w5 the5e 11m1t1n9 ca5e5 a5 we11 a5 the data 06ta1ned after 120 m1n 0f f10ccu1at10n at 6 = 6 3 5-t and c = 2.1 m9/ 1. F0r th15 ca5e, re9re5510n ana1y515 91ve5 Dpf = 1.19. F19ure 5 5h0w5 the ev01ut10n 0f Dpf and the c0rre5p0nd1n9 L, at the5e c0nd1t10n5. 7he var10u5 ve10c1ty 9rad1ent5 w1th1n a 5t1rred tank pr0duce a d15tr16ut10n 0f f10e 5tructure5 a5 a re5u1t 0f 51mu1tane0u5 5hear-1nduced 9r0wth, 6reaka9e, and re5tructur1n9. 7he err0r 6ar5 1n F19. 5 1nd1cate the ran9e 0f Dpf va1ue5 1n the 5amp1e. 7he5e va1ue5 were determ1ned fr0m the 5tandard err0r (95% c0nf1dence 1nterva1) 0f the 510pe f1t t0 the area v5 per1meter data. F19ure 5 5h0w5 that Dpr 1ncrea5e5 fr0m 1t5 1n1t1a1 va1ue (D0f ---- 1) and reache5 a max1mum va1ue 0f Dpf = 1.29 a5 the f10e5 9r0w and 6ec0me m0re p0r0u5. When 6reaka9e 6e91n5 t0 d0m1nate, Dpf decrea5e5 5119ht1y unt11 1t reache5 a 5teady 5tate va1ue 0f a60ut 00f = 1.19. 7h15 15 1n a9reement w1th 7am60 and Watana6e (1979a) and 01e5 (1992) wh0 5u99e5ted that a decrea5ed f10e p0r051ty (a5 1nd1cated here 6y the decrea51n9 fracta1 d1men510n) re5u1t5 fr0m the pr0duct10n 0f 5tr0n9, den5e fra9ment5 6y preferent1a1 6reaka9e 0f the f10e5 at weak p01nt5. L1 and 6anc2arc2yk (1991) rep0rted Dp~= 1.13-1.22 f0r act1vated 51ud9e f10c5. 1n F19. 5, the 5pread 0f the Dpr narr0w5 a5 the f10e 512e reache5 5teady 5tate. 7h15 may re5u1t fr0m re5tructur1n9 and/0r c0mpact10n 0f the f10e5 6y 5hear f0rce5 (Ju111en and Meak1n, 1989). Franc015 (1987) 065erved a 51m11ar c0mpact10n 0f f10c5 at extended f10ccu1at10n t1me5. F19ure 6 5h0w5 the mea5ured 5teady 5tate per1meter-6a5ed fracta1 d1men510n, Dpr~ at a11 emp10yed a1um c0ncentrat10n5 and 5hear rate5. Err0r 6ar5 repre5ent the var1at10n 0f the mea5ured va1ue 0f

10

7he ev01ut10n 0f f10c 5tructure 7he c0ncept5 0f fracta1 9e0metry (Mande16r0t, 1983) pr0v1de a mathemat1ca1 framew0rk f0r de5cr1pt10n 0f the 5tructure 0f 1rre9u1ar f10e5. A character15t1c d1men510n 0f the f10e, 1n th15 ca5e the per1meter (P), can 6e re1ated t0 the pr0jected area 0f the f10e (A) 6y (Mande16r0t et a1., 1984) A 0c P ~p,

6=635 "1 c = 2.1 m9111tera1um 2

E3

~

~

~

-8 1.75-

~ 1.5-

(4)

-c 1.25-

where Dpr 15 the per1meter-6a5ed fracta1 d1men510n 0f the f10e. 7h15 e4uat10n wa5 u5ed 6y L1 and 6anc2arc2yk (1991) t0 character12e very th1n 511ce5 0f act1vated 51ud9e f10e5. Pr0v1ded that there are 5evera1 pr1mary part1c1e5 1n each f10e, the avera9e per1meter-6a5ed fracta1 d1men510n, Dpf, 0f the f10e5 can 6e e5t1mated fr0m E4uat10n 4 u51n9 a 109-109 p10t 0f the f10e per1meter and cr055 5ect10na1 area (F19.4). 7he Dpf 15 re1ated t0

2

1

11

0

60

120

0 180

7~e (m1n)

F19. 5. Ev01ut10n 0f the L. and fracta1 d1men510n, Dpr, f0r 6 = 6 3 5-~ and c = 2 . 1 m9/1 at ~=2.1 × 10 -~. After 5uf11c1ent1y 10n9 t1me5, 60th L, and Dpf appr0ach an a5ympt0t1c, 5teady 5tate va1ue.

F10c 5tructure and 512e d15tr16ut10n 1.34 61

1.2

=,

•-

1.1

A1um C0ncentrat10n(m9 / 11ter) 32 0 10.7 A 4.3 60

90

120

150

5hear Rate, 6 (5-1)

F19. 6. 7he 5teady 5tate fracta1 d1men510n, Dpt~, a5 a funct10n 0f 5hear rate at three a1um c0ncentrat10n5 at = 8.3 x 10 -5. 1ncrea5ed a1um c0ncentrat10n pr0duce5 m0re 0pen f10c 5tructure5 at c0n5tant 5hear rate. 7he effect 0f 1ncrea51n9 5hear 15 m0re dramat1c at h19h a1um c0ncentrat10n5.

1053

1ma9e ana1y2er. At the5e C0nd1t10n5, 5hear ha5 n0 effect 0n the 5hape 0f the n0rma112ed 5teady 5tate 512e d15tr16Ut10n, mak1n9 1t 5e1f-pre5erV1n9 W1th re5pect t0 5hear. 7he 5e1f-pre5erV1n9 nature 0f the 5teady 5tate f10C 512e d15tr16Ut10n 51mp11f1e5 f10C 512e character12at10n. 1f the 5teady 5tate 512e d15tr16ut10n 0f a f10ccu1at1n9 5u5pen510n 15 5e1f-pre5erv1n9, 0ne need5 0n1y t0 e5t1mate 0ne var1a61e 0f the d15tr16ut10n, e.9. the avera9e d1ameter, and then the ent1re d15tr16ut10n can 6e 06ta1ned. F19ure 8 5h0w5 (a) the d1men510na1 and (6) the n0rma112ed 5teady 5tate f10c 1en9th d15tr16ut10n5 06ta1ned at an a1um c0ncentrat10n 0f 10.7 m9/1. A5 1n F19. 7(a), the 5teady 5tate d15tr16ut10n5 1n F19. 8(a) narr0w w1th 1ncrea51n9 5hear a5 fra9mentat10n 1ncrea5e5. U p 0 n n0rma112at10n, the5e d15tr16ut10n5 d0 n0t c011ap5e 0nt0 a 51n91e 11ne a5 1n F19. 7(6) 6ut a5 the 5hear rate 1ncrea5e5 they 6ec0me narr0wer

(a)

0.31 32 m9/11tera1um 5hear Rate, 6 (5"1)

--.p- 129

Dpf~ 6etween rep11cate exper1ment5.7he Dr,r~, wa5 n0t affected 519n1f1cant1y 6y f10ccu1ant c0ncentrat10n at the tw0 10wer a1um c0ncentrat10n5 (10.7 and 4.3 m9/1) at a11 5hear rate5. H0wever, at a h19h a1um c0ncentrat10n, c = 32 m9/1, the D~r~ wa5 1ar9er (1nd1cat1n9 m0re 0pen 5tructure5) than at the tw0 10wer a1um c0ncentrat10n5. At h19her a1um c0ncentrat10n5 and rather 10w 5hear rate5, the 1ncrea5ed 11nka9e 5tren9th w1th1n the f10c 5tructure may 1ncrea5e re515tance t0 6reaka9e 0r re5tructur1n9 re5u1t1n9 1n m0re 0pen f10c 5tructure5 at 5teady 5tate. 7h15 15 c0n515tent w1th 7 a m 6 0 and Watana6e (1979a) wh0 f0und 1ncrea51n91y 0pen f10c 5tructure5 a5 the rat10 0f a1um c0ncentrat10n t0 5u5pended 5011d5 wa5 1ncrea5ed. 1ncrea51n9, h0wever, the 5hear rate ( 6 = 129 5-~) at h19h f10ccu1ant c0ncentrat10n5, ( c = 32 m9/1) decrea5ed the Dp~ (F19. 6). A5 the 5hear rate 1ncrea5e5, the 1ncrea5ed a m 0 u n t 0f 6reaka9e/re9r0wth and re5tructur1n9 pr0duce5 m0re c0mpact f10c 5tructure5 (7am60 and Watana6e, 19796). At h19h 5hear rate5, there 15 v1rtua11y n0 d1fference 1n the f10c 5tructure re9ard1e55 0f f10ccu1ant c0nd1t10n5.

Effect 0f pr0ce55 var1a61e5 0n the 5teady 5tate f10c 512e d15tr16ut10n

.•

0.2

0

~60

•

95

~

63

30~

260•

,60

F10eLen9th,L (0m)

(6)

0.3-

32 m9/11tera1um 5hear Rate, 6 (5"1)

+

129

•

95

<3, 63

0.2 • + <> +

41.

+

0.1

,3 0

%

+C,

+ 0

7he effect 0f 5hear rate 0n the 5teady 5tate f10c 512e (1en9th) d15tr16ut10n 15 5h0wn 1n F19. 7(a) at c = 32 m9/1 and ~6=8.3 x 10 -5. C1ear1y, 1ncrea51n9 5hear reduce5 the 1ar9e ta11 0f the d15tr16ut10n and decrea5e5 the 0vera11 f10c 512e, 5h1ft1n9 the ent1re 512e d15tr16ut10n 1nt0 the 5ma11er 512e5. When, h0wever, the f10c 1en9th, L, 15 5ca1ed w1th 1t5 avera9e, L., a11 data c011ap5e 0nt0 a 51n91e curve [F19. 7(6)]• 50me 5catter 15 065erved 1n the 10wer 512e ran9e5 that may 6e attr16uted t0 the 10wer 512e 11m1t 0f the

+•

•

+

,3,

D111en510n1e55F10CLen9th (L1 Ln )

F19. 7. (a) 5teady 5tate f10c 1en9th d15tr16ut10n5a5 a funct10n 0f 5hear rate f0r an a1um c0ncentrat10n 0f 32 m9/1 at ~6 = 8.3 x 10 -5 (Nr = 2774 at 6 = 129 5-~, 6573 at 95 5-t, 3181 at 63 5-~). 1ncrea5ed 5hear narr0w5 the f10c 512e d15tr16ut10n 6y decrea51n9 the 0vera11 f10c 512e. (6) N0rma112ed f0rm 0f the 5teady 5tate f10c 1en9t6 d15tr16ut10n. 7he curve5 c011ap5e 0nt0 a 51n91e11ne and are 5e1f-51m11ar w1th re5pect t0 5hear.

1054

(a)

Patr1ck 7.5p1cer and 50t1r15 E. Prat51n15 0.310.7 m9111tera1um

cr9 = exp

5hear Rate, 6 (5-1 )

~ 2n~(1nL - 1nL9)-• ~ ~ - [

(5)

---p-- 129

where N7 15 the t0ta1 num6er 0f f10c5 1n the d15tr16ut10n and L9 15 the 9e0metr1c avera9e f10c (max1mum) 1en9th (H1nd5, 1982)

0.2-

exp[- Y~n11nL1 L~ ~"

L9 =

2

(6)

01

100

0

200

300

F10c Len9th, L (pm)

(6)

7he tr9 1nd1cate5 the w1dth 0f the 512e d15tr16ut10n, where cr8 = 1 c0rre5p0nd5 t0 a m0n0d15per5e d15tr16ut10n wh11e 1ncrea51n9 tr~ > 1 1nd1cate5 1ncrea51n91y 6r0ad d15tr16ut10n5. At the tw0 10wer a1um c0ncentrat10n5 (4.3 and 10.7 m9/1), 1ncrea51n9 5hear rate narr0w5 the 512e d15tr16ut10n 6y tr1mm1n9 the 1ar9e ta11 0f the f10c 512e d15tr16ut10n wh11e the 5ma11 ta11 rema1n5 v1rtua11y unchan9ed a5 the 5ma11e5t

0.3-

10.7 m9111tera1um

0.3

(a)

4.3 m9111tera1um

5hear Rate, 6 (5-1)

5hear Rate, 6 (5-1) +

+ • 0

0.2-

0

0

--.f-- 129

129 A

95

•

95 , 63

63

0.2

+0 E

0 0.1-

0

0.1 +

01-

0 +0

0,~

04•

A 0

2~5

0

50

=

131rnen510n1e55F10cLen9th ( L / L n ) F10c Len9th• (pm)

F19.8. (a) 5teady 5tate f10c 1en9th d15tr16ut10n5a5 a funct10n 0f 5hear rate f0r an a1um c0ncentrat10n 0f 10.7 m9/1 at ~6=8.3 x 10 5 ( A f 7 = 2 8 1 3 a t 6 = 1 2 9 5 -~, 1820 at 95 5 ~, 3215 at 63 5-~). (6) N0rma112ed f0rm 0f the 5teady 5tate d15tr16ut10n. Decrea51n9 a1um c0ncentrat10n 1ncrea5e5 the effect 0f 5hear-1nduced 6reaka9e 0n the f10c 512ed15tr16ut10n and 1ncrea5e5 the dev1at10n fr0m 5e1f-51m11ar1ty 6y narr0w1n9 the n0rma112ed 512e d15tr16ut10n. [F19. 8(6)]. 7h15 effect 15 m05t pr0n0unced at the 10we5t a1um c0ncentrat10n, 4.3 m9/1 [F19. 9(a) and (6)] where the 11nka9e 6etween pr1mary part1c1e5 15 the weake5t. 1n F19. 9(a), 1ncrea51n9 5hear-1nduced fra9mentat10n pr0duce5 a 5ec0nd m0de at the 5ma11e5t f10c 512e5. 7he n0rma112ed f0rm 0f the5e d15tr16ut10n5 dev1ate5 fr0m 5e1f-51m11ar1ty a5 a re5u1t 0f the dramat1c 5hear effect5 [F19. 9(6)]. 7he 9e0metr1c 5tandard dev1at10n at 5teady 5tate, a5~, f0r each 0f the f10c 512e d15tr16ut10n5 1n F19. 7(a), F19. 8(a), and F19. 9(a) 15 5h0wn 1n F19. 10 a5 a funct10n 0f 5hear rate f0r a11 three a1um c0ncentrat10n5. 7he 6ar5 1nd1cate the var1at10n 0f the mea5ured a9~ f0r rep11cate exper1ment5. 7he tr~ wa5 ca1cu1ated u51n9 (H1nd5, 1982)

(6)

0.3

4.3 m9/11tera1um 5hear Rate, 6 (5"1) 4-

0.2

~.~A

4-

129

•

95

A

63

~x +~x 0.1

•

• A

4-

A+

A

•+ 0 •

0

1

2

3

A 4

•

1

D1men510n1e55F10cLen9th (L 1 Ln )

F19.9. (a) 5teady 5tate f10c 1en9th d15tr16ut10n5a5 a funct10n 0f 5hear rate f0r an a1um c0ncentrat10n 0f 4.3 m9/1 at = 8.3 x 10-5(N~ = 503at 6 = 129 5- t, 592 at 95 5-~, 791 at 63 5-t). (6) 1ncrea5ed f10c 6reaka9e at th15 10w a1um c0ncentrat10n re5u1t5 1n the f0rmat10n 0f a 5ec0nd m0de 1n the d15tr16ut10n.

F10c 5tructure and 512ed15tr16ut10n 2.3-

A1um C0ncentrat10n,(m9/11ter)

5

A ~

4.3 10.7

2,1

.8

1.9

N 1.7

1.560

90

120

150

5hear Rate, 6 (5-1)

F19. 10. 6e0metr1c 5tandard dev1at10n, a5~, 0f the 5teady 5tate f10c 512e d15tr16ut10n5 a5 a funct10n 0f a1um c0ncentrat10n and 5hear rate. 7he va1ue 0f a9~ decrea5e5 w1th 1ncrea51n95hear rate a5 fra9mentat10n 6ec0me5 m0re d0m1nant. 1ncrea51n9 a1um c0ncentrat10n at a c0n5tant 5hear rate a150 decrea5e5 the va1ue 0f a9~. At h19h a1um c0ncentrat10n5, the a~ rema1n5 e55ent1a11y c0n5tant (~9~ ~ 1.7) w1th1nexper1menta1 var1at10n. part1c1e 512e 15 that 0f the pr1mary part1c1e. H19her 5hear rate5 m05t1y reduce the upper end 0f the d15tr16ut10n, re5u1t1n9 1n a narr0wer 0vera11 d15tr16ut10n. 7he cr~ a150 decrea5e5 w1th 1ncrea51n9 a1um c0ncentrat10n. At the h19he5t a1um c0ncentrat10n, f0r wh1ch the 512ed15tr16ut10n5d15p1ay 5e1f-51m11ar1ty,the cr9~ 5h0w5 n0 trend w1th 5hear and rema1n5 c0n5tant. 5e1f-51m11ar1ty 15 apparent1y exh161ted 0n1y at exce55 f10ccu1ant c0ncentrat10n5, pr06a61y a5 a re5u1t 0f the effect5 0f the 1ar9er am0unt 0f A1(0H)3 prec1p1tate pre5ent (1ncrea5ed re515tance t0 6reaka9e and 1ncrea5ed f10c 9r0wth rate). 7he 5teady 5tate f10c 512e d15tr16ut10n 15 the re5u1t 0f the 6a1ance 6etween c0a9u1at10n and fra9mentat10n. 7he re1at1ve 5tren9th 0f each phen0men0n determ1ne5 the ev01ut10n and 5pec1f1c va1ue 0f the 5teady 5tate f10c 512e d15tr16ut10n. At 10w a1um c0ncentrat10n5, the 11nka9e 6etween the pr1mary part1c1e5 1n the f10c 15 weak and the f10c5 are m0re 11ke1y t0 6reak. A5 a re5u1t, fra9mentat10n 9reat1y affect5 the 1ar9e ta11 0f the d15tr16ut10n and prevent5 the advancement 0f the f10c 512ed15tr16ut10n fr0m the 10wer 512e ran9e. 8y 1ncrea51n9 the a1um c0ncentrat10n, the f10c 5tren9th 15 1ncrea5ed, 50 5hear-1nduced c0a9u1at10n and fra9mentat10n 6a1ance each 0ther far away fr0m the 1n1t1a1 part1c1e 512e. 7he re5u1t1n9 5teady 5tate f10c 512e d15tr16ut10n 15 fu11y deve10ped and 6ec0me5 5e1f-pre5erv1n9 w1th re5pect t0 5hear. C0NCLU510N5 An exper1menta1 5tudy 0f the effect 0f 5hear rate and f10ccu1ant c0ncentrat10n dur1n9 f10ccu1at10n 0f p01y5tyrene part1c1e5 wa5 carr1ed 0ut. After 5uff1-

1055

c1ent1y 10n9 t1me5, the f10c 512e d15tr16ut10n reached an e4u1116r1um 0r 5teady 5tate 512e d15tr16ut10n fr0m the c0mpet1t10n 6etween C0a9u1at10n and fra9mentat10n. 1ncrea5ed 5hear decrea5ed the avera9e 5teady 5tate f10c 512e a5 determ1ned 6y 1ma9e ana1y515. 1ncrea51n9 the f10ccu1ant c0ncentrat10n 1ncrea5ed the f10c 9r0wth rate and the 5teady 5tate f10c 512e wh11e 1ncrea5ed 5hear a1way5 reduced f10c 512e at 5teady 5tate. 7he dynam1c ev01ut10n 0f f10c 5tructure wa5 m0n1t0red 6y 1ma9e ana1y515 0f d191t12ed f10c 1ma9e5. 7he per1meter-6a5ed fracta1 d1men510n, Dpf, 0f the f10c5 wa5 f0und t0 1n1t1a11y 1ncrea5e (m0re 0pen 5tructure) and then 5119ht1y decrea5e (m0re c0mpact 5tructure) and reach a 5teady 5tate a5 a re5u1t 0f 5hear-1nduced 6reaka9e and/0r re5tructur1n9. 5hear1nduced re5tructur1n9 a150 narr0wed the ran9e 0f f10c 5tructure5 a5 5teady 5tate wa5 atta1ned. 7he n0rma112ed 5teady 5tate f10c 512e d15tr16ut10n5 appear t0 6e 5e1f-51m11arw1th re5pect t0 5hear at h19h f10ccu1ant c0ncentrat10n5 where the f10c 512e d15tr16ut10n 15 fu11y deve10ped. At 10wer f10ccu1ant c0ncentrat10n5, 1ncrea51n9 f1u1d 5hear appeared t0 narr0w the 5teady 5tate f10c 512e d15tr16ut10n5 a5 1ncrea5ed f10c 6reaka9e 11m1ted f10c 9r0wth and narr0wed the f10c 512e d15tr16ut10n.

Ackn0w1ed9ement5--We w15h t0 ackn0w1ed9e D0u9 80w1-

1n9 and the Mater1a155c1enceDepartment 0f the Un1ver51ty 0f C1nc1nnat1f0r pr0v1d1n9 acce55 t0 the 1ma9e ana1y515 e4u1pment. We ackn0w1ed9e 5t1mu1at1n9d15cu5510n5w1th Dr Kar1 Ku5ter5 (current1yw1th 5he11011 C0.) at the ear1y 5ta9e5 0f th15 pr0ject. 5upp0rt 6y the Nat10na1 5c1ence F0undat10n, 6rant C75-8957042, 15 9ratefu11y ackn0w1ed9ed. We thank the j0urna1 referee5 f0r he1pfu15u99e5t10n5 dur1n9 the rev1ew 0f the manu5cr1pt.

REFERENCE5

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