The rate of particle settling in a water column, representing a lake, pond, or any other water impoundment, can be described by Stokes' Law:
(9.13) where
t = time (sec)
d = particle diameter (m)
h = distance for particle to fall, meter
n = viscosity of water at given temperature (kg ms-', 0.1 at standard temperature)
g = gravitational constant (9.8 kg ms-2)
Ps = density of the solid (kg m-3, for clays, approximately 2650) P f = density of the fluid (kg m -3)
Equation 9.13 shows that nonphysically and nonchemically interacting particles of a given density and relatively large diameter settle faster than particles with similar composition but relatively smaller size. Additionally, particles with a given diameter and a relatively large density settle faster than particles with the same diameter but a relatively smaller density. The rate of settling (Sr) in meters per hour (m s-') can be described by
(9.14) where g(Ps - Pf)/18n is a constant denoted as Co. Therefore, the rate of colloid settling can be expressed by
(9.15) Equation 9.15 points out that a small increase in particle diameter due to colloid agglomeration has a large impact on the rate of particle settling (Sr).
Apparent particle size or particle diameter of colloids in natural bodies of water (e.g., lakes, ponds, and human-made water impoundments) varies depending on the type of interactions between the colloidal particles (physical versus chemical or outer-sphere, versus inner-sphere colloid to colloid interactions) and type of colloids participating in these interactions (e.g., inorganic versus organic colloids). Colloid to colloid interactions result in particle agglomeration or colloid flocculation. Therefore, the greater the degree of flocculation is, the greater the rate of settling. The rate of colloid flocculation or agglomeration depends on the frequency of successful colli- sions between colloids. Successful collisions depend on the force by which colloids collide with proper orientation. There are two types of colloid collision forces in a liquid medium (e.g., water). One force is due to Brownian (thermal) effects producing perikinetic agglomeration and a second force, which exceeds that of Brownian motion,
384 SOIL COLLOIDS AND WATER-SUSPENDED SOLIDS
is due to velocity gradients and is referred to as orthokinetic agglomeration (Stumm and Morgan, 1970).
Perikinetic agglomeration applies to a monodisperse suspension and can be repre- sented by a second-order rate law:
-dp/dt = k p2.
P (9.16)
or
(9.17) where p = number of colloid particles at any time t, Po = initial number of particles, and kp a conditional second-order rate constant. Orthokinetic agglomeration under a constant particle velocity applies to larger colloid particles and can be described by first-order kinetics:
-dp/dt= kop (9.18)
or
(9.19) where p = number of colloid particles at any time t, Po = initial number of particles, and ko is a conditional first-order rate constant (Stumm and Morgan, 1970). In nature, it is difficult to distinguish perikinetic from orthokinetic agglomeration because both modes take place simultaneously. Thus, the overall agglomeration rate is the sum of orthokinetic and perikinetic agglomeration:
(9.20) Agglomeration rate constants depend on many factors, including chemical makeup of colloids, size of colloids, surface charge of colloids, and solution concentration and ionic composition. For this reason, particle settling in a water column is difficult to predict. However, it is easy to produce experimental data describing the settling of a particular colloidal system under a given set of experimental conditions. Settling includes the overall rate of agglomeration as well as particle movement by gravity deeper in the water profile. Such systems are referred to as polydispersed systems.
Figures 9 . 16a-c show that the settling behavior of suspended particles, as expected, is dependent upon ionic strength (EC). The settling rate at the highest ionic strength is initially very rapid because of high collision frequency (Oster et al., 1980) and then it declines. The majority of the suspended particles at the highest ionic strength settle out within a 60-rnin period. However, at the lowest EC values shown, an insignificant quantity has settled out, even after 7 hr. Also, the water EC in these systems is well within drinking standards (tap water may have an EC of up to 0.700 dS m-I or mmhos cm-I ).
9.3 FLOCCULATION AND SETTLING RATES 385
~
01
E c:
II>
"0
o
VI
"0
'"
"0
c: IV
a.
V>
:J
VI
400~--~--~--~--~--~~--~--'----r--~---'
360 320 280 240 200 160 120 80 40
Sample 20
• •
28}4mhas em-I
•
•
o~--~--~--~--~--~~--~--~--~--~--~
0.00 32.5 65.0 97.5 130.0 162.5 195.0 227.5 260.0292.5 325.0 Time (Minutes)
Figure 9.16a. Particle settling characteristics as a function of time at three levels of suspension electrical conductivity (EC) (from Evangelou, 1990, with permission).
380 342
'7 304
...J
01 266
E c::
fI) 228
...,
"0 190
VI ..., 152 ..., Q)
c::
Q) 114
0. .. :>
VI 76
O~--~--~----~--~--~----~ o 4 5 6 __ - L 7 __ ~
8 Time(Hours)
Figure 9.16b. Particle settling characteristics as a function of time at three levels of suspension electrical conductivity (EC) (from Evangelou, 1990, with permission).
386 SOIL COLLOIDS AND WATER-SUSPENDED SOLIDS 450
400 •
-, 200 fLmhos em-I •
...I 350
C'
E 300
<: 280 fLmhos em-I
II>
:!2 250 0
(/) 200
"'C
'"
"'C
150
., c Q. V) •
:::>
100
(/)
50 0
0 51 102 153 204 255 306 357 408 459 510 Time (Minutes)
Figure 9.16c. Particle settling characteristics as a function of time at three levels of suspension electrical conductivity (EC) (from Evangelou, 1990, with permission).
The data in Figures 9.17 and 9.18 demonstrate the kinetics of settling characteristics of colloids with mixed mineralogy. By comparing data in Figures 9.l7a and band 9.l8a and b, it can be seen that as SAR approaches zero with a total salt concentration of about 4 mmolc L-1, Ca and Mg have approximately the same influence on the kinetics of settling of the clay colloids. Within a 2-hr settling period, the suspended solids for both systems approach 35 mg L -I (Fig. 9.18a and b). However, at a SAR of approximately 24 (mmol L-I)1I2 after an 8-hr settling period (Fig. 9.18a and b), the Na-Ca system maintains approximately 300 mg L-1 of suspended solids, while the Na-Mg maintains approximately 140 mg L -I suspended solids.
These findings suggest that the particular colloids in the Na-Ca system are more dispersive than the Na-Mg system. This conflicts with some previous studies (Yadav and Girdhar, 1980) which showed that some soils have Na-Mg states that are more dispersive than the Na-Ca states. However, the soils employed by Yadav and Girdhar (1980) contained appreciable amounts of expanding minerals (smectites), whereas the material in Figures 9.17 and 9.18 is predominantly kaolinitic. Kaolin-type clay minerals exhibit stronger basic behavior than smectites and have a tendency to adsorb Mg2+ (a stronger acid) with higher affinity than Ca2+ (weak acid). This could explain the apparent low dispersivity of the Mg-clay system studied (Hesterberg and Page, 1990). Similar trends are shown in Figure 9.17c and d. A comparison of data shown in Figure 9.17 a and c and 9.17b and d suggests that an increase in total salt concentra- tion from 4 to 8 mmolc L -I greatly increases the settling rate of the suspended solids.
Note also that after 6 hr of settling time at SAR of 23 (mmol L-I)112 and total salt
w 00 -.l
ôXI ,r--,,-- 3IlO
,...
~31!0
~ ~ 2IlO
'"
o 2ôl
~ 3D~il
fil 1111
~ 12111 a
::J a:! L9
'" r\
ôlh,
Lo-..o
o U-L'C.o=:::::"'"
o 12 24 311
o SIR - 0.1
'" SIR = 4.7 II 1
D SIR - 8.7 ~!
• SAR - 16.4 !
+ SAR = 23.1 J.
... No-Co 1-
"'_____ I = 4n1ro11l
"'-'"
6 ' " ,
~ III n ~ ~ 1~12II1~1~ I~I~
TIr-'E(Hours)
.-~
'" SAR - 5.4
~ 2IlO<;> '"
~:fI "'\ I I .
~11Il~ ,,~.
w 12II~ '" ~
D SAR = 8.3
• SAR - 14.9 +SAR=21.7
No-Co
I - B nrrolll
~ a:!~ "'~ ~
ôloS "'-f: C
o ~O---'---'---'~'-'-'-",--",---L---L--""""""'"
o ~ ~ 311 ~ III n ~ ~ I~ 12II1~ I~ I~ I~
TIME (Hours)
4001) '"
~:~ ~ .'f
~ 2IlO 0
~ 2ôl~
"'3D
o 1
~* "
~ IlIljg
~ 12II~\ ",'
'"
..
oSffi -:l
'" Sffi - 3.9
D Sffi - 8.2
• SAR - 13.2
+ SAR - 18.9
No-Mg
I - 4 nrroi/L
:~ ol \0'0 ' ----... O>----'-~-~~~--'-~-~~...J b
o 12 24 311 ~ III n ~ ~ 1~ 1211 1~ 1~ I~ 1~
TIME (Hours)
4OOJ>.:
~:"'\ ~2IlO + D : SAR SAR - 15.7 - 8 5 •
+
8 2ôl~ ~ 8
fil la:!
@3D \
: \ ~+ ~ = : ~ã6D
~ ':~ , d .~
~ vO!!, ~ 0 _ D " - - - - D
i ' - u
ôl!l. "'~"'--"'=' ,---=" ' 1211 I~ 1~ I~
NO_Mg
I 1
I = B nrroi/L ,~ n ~ ~ I~ -
0
0 12 24 311 ~ III TIr-'E (Hours)
!
1~
Figure 9.17. Settling behavior kinetics of clay particles of Na-Ca and Na-Mg systems at various SAR values and a constant pH of 7,2 (from Evangelou and Karathanasis, 1991, with permission),
w
00 00
ED X-~~~O~NO_CO IL
3!Ifi " 0 I _ 4 rmol
3J) " 6
'"' ,
~: ;"""~
gJ 2-46 SfR _ 0.1 ' ....
~ 0 _ 24.1 ' .... ,
,.! 210 II. SfR 27 5 .... ' ....
liS SfR _ • _ .... ,
@ no 0 ' • • • • • • •
2'~ ~ i
~ ,m ~. QH'~
Ul 70 ãs .... 9 ... : ... .... 9. . .... , 9 ... ,
31
o a 2
~: ~\ :~
Ul2IID \
82-46 I
<I
d ':t.
Ul 210 II. 0 D
fijl75 \ 0",-
• TII'E 10 12 (Hours) 14 18 18 2D 22 ~
No-Co
I = 8 rmollL
o SfR - 0.1 '" SAR - 25.3 o SfR - 48.9
~ 141 Q \ " ' "
~ Uli ~ ~'" 9 C
70 Q... "_"'~
~ g .... g ... :~~ ~ ~~. ~.~ ~.~ ~ gi--:--._-.. -~-:--.-~-. _-.. -:--._-. :~
o 2 <I 8 8 la 12 W 16 18 2D 22 ~
TII'E (Hours)
e1ro---~
386 o~c ~ u
""'!BJ '\6""-
~315 \ ~""
... 211) \ 0
el ' "-
... 241 , ""
No-Mg
I - 4 rmollL
a210 Ul .~ , g~
176 ' ...
~14I 'a- .... , ...
a SfR - 0.1
II. SfR - 23.3 o SfR - 31.3
! I: \ b """" '" ~
ãã0 - ... ~
~ uãããgãã .. 9 ... oããã .. ãããã ... 0 ... ~ .. ~ .. ~.~ -~-.-b
o 2 4 8 8 III 12 1<1 18 18 2D 22 24
TIME (Hours)
e1rl---,
31169
'"' ",'
...J 310 _0
" I I
gằ3115 '.n
~ II.
gJ2IID \
"-"245 ,0
~210 \ \
fijl75 'I>
2141 A
W \ 0
III 105 ",0'-...
No-Mg
I - 8 rmollL
o SfR - 0.1
II. SfR - 26.3
o SAR - 48.9
d
ffi 70 ~ "', 0...0
... ~ '-"'-_ l/---_, __ ~o .
31 '0-.... - - - _ _ t! 0
o g ... ãããgãã .. ;ããã .. ããã_;-:-ã-::u..-:-::;:-:-ã-:-,-:-ã.,.ã';-;-:a
o 2 4 8 8 10 12 1<1 18 18 2D 22 ~
THE (Hours)
Figure 9.18. Settling behavior kinetics of clay particles of Na-Ca and Na-Mg systems at various SAR values and a constant pH of 5.2 (from Evangelou and Karathanasis, 1991, with permission).