1 Gravty Dranage Pror to ake Fltraton Sott A. Wells and Gregory K. Savage Department of vl Engneerng Portland State Unversty Portland, Oregon 97207-0751 Voe (503) 725-4276 Fax (503) 725-4298 ttp://www.e.pdx.edu/~wellss e-mal: sott@eas.pdx.edu Abstrat Durng te ntal stages of a Buner funnel or spef resstane test, gravty dranage ours pror to applaton of te pressure dfferental. Some allow tme for a small ake to form by gravty dranage. Fltrate data from te gravty dranage perod an be used to determne onsttutve propertes of te ake under a ydrostat pressure gradent. Te onsttutve propertes tat defne te struture of te ake nlude te permeablty and porosty as funtons of te appled stress. Equatons governng te dranage rate durng a gravty fltraton experment assumng a onstant and a non-onstant average ake permeablty and ake porosty were developed. Numeral solutons were sown predtng te gravty dranage rate gven known onsttutve relatonsps. Also, a proedure was sown llustratng ow onsttutve relatonsps ould be determned usng gravty dranage data.
2 Introduton Durng te ntal stages of te Buner Funnel test, gravty dranage ours pror to applaton of te pressure dfferental. Some researers reommend allowng tme for a tn ake to form by gravty pror to applaton of te pressure dfferental (Veslnd, 1979). rstensen and Dk (1985a, 1985b) and Wells and Dk (1988) evaluated te mpat of allowng a gravty dranage perod (or ake formaton perod) on omputed values of spef resstane. Ts resear note explores te teory and applaton of gravty fltraton data pror to applaton of te pressure dfferental n a Buner funnel test. Governng Equatons: onstant Average ake Permeablty and Porosty For flow troug a seres of ake layers of varyng permeablty, Greenkorn (1982) sowed tat Dary s Law ould be wrtten as were Α p Α p kα p Q (1) * µ Σr L L L µ µ Σ k Q:flow rate troug ake Α :ross - setonal p :appled pressure dfferent al aross entre ake * r :nverse permeabl ty of layer L :lengt of layer µ :dynam vsosty L :total ake lengt ΣL area of fltraton k :average permeabl ty of ake ell For gravty fltraton, te appled pressure dfferental and te ake lengt s a funton of tme, su tat Dary s Law an be wrtten as Q kα p t kα m L t m rg t L t (2) were :dstane from top of ake to free water surfae Usng mass onservaton, te relatonsp between L(t) and (t) an be determned as follows: For gravty dranage, te rate of fltrate produton s equal to te ange of volume of te ntal slurry (assumng no solds are lost troug te flter medum),.e., Q - Α (3) t
3 were :ange t :ange n egt n tme of slurry Te ntal mass of solds n te slurry s Mass ntal Α (4) were : ntal :ntal egt of slurry onentrat on of slurry : volume of slurry added to fltraton ell If te ake s formed at an average onentraton of, ten te ntal mass s equal to te mass of solds n te slurry (at onentraton ) and te mass of solds n te ake (at onentraton ), su tat Smplfyng and solvng for L, Α Α + L Α (5) L (6) Also, note tat an be desrbed by te porosty of te ake, ε, as s ρ 1 (7) were ρ s : sold densty Also, te ntal onentraton of suspended solds an be determned usng te ntal porosty, ε, as s ( ) ρ 1 (8) Ten substtutng Eqs. 3 and 6 nto Eq. 2 and smplfyng, d dt kρg µ ( ) (9) Usng an ntal ondton tat at t 0, te soluton to Eq. 9 assumng k and an mplt equaton of te form are not a f(t) s
4 exp αt exp 1 (10) were k ρg α µ kρg (1 ) µ (1 ) Hene, f data of (t) are determned from expermental results, te value of k and an be determned assumng tey are onstant as a f(t). Note tat Eq. 10 an be rearranged as (11) ln t α +1 (12) Tere s also a mnmum value of mn or onversely a maxmum value of L. Tese an be determned from a mass balane as n Eq. 6 su as or L max [ ] mn (13) mn 1 + (14) Te relatonsp between L and s sown n Fgure 1. L L L max L slope mn
5 Fgure 1. Relatonsp between ake lengt L and egt of slurry n fltraton vessel. Governng Equatons: Non-onstant Average ake Permeablty and Porosty Te average ake permeablty and porosty (k and ) an be assumed to be funtons of tme by assumng approprate onsttutve relatonsps. For example, te permeablty an be desrbed as a funton of porosty,.e., k f( ), and te relatonsp between pressure dfferental and ake porosty an be defned usng te defnton of te average ake oeffent of volume ompressblty, m v, su tat were m v - ρ σ' p : pure water pressure σ ':effetve stress p total p (15) Any funtonal relatonsps for k and m v an be osen, but for matematal onvenene and aordng to data from Wells (1990a), te followng forms were osen: ( b ) k a exp (16) m exp d (17) v were a [L 2 ], b[-], [ML -1 T -2 ], d[-]: empral onstants. Substtutng Eq. 17 nto Eq. 15, ntegratng, usng ( ) p ρ g, and solvng for, 1 ln d [ dρg ( ) + exp (- d )] (18) Ten, rewrtng Equaton 8 by substtutng Eq. 7 and Eq. 16, d dt ( b ) ρg( ) µ ( ) kρg a exp 1 µ ( ) ρ s (19) Ten substtutng Eqs. 18 and 8 nto Eq. 19, we ave te fnal dfferental equaton for, d dt aρg exp b d ln { dρg( ) + exp ( d )} 1+ ln{ dρg( ) + exp ( d )} µρ (1 ) s ( ) 1 d (20)
Ts non-lnear ordnary dfferental equaton an be solved numerally. 6
7 Numeral Results Eqs. 12 and 20 were solved for a gven set of onsttutve relatonsps for kaoln lay sown n Table 1. Eq. 12 was solved mpltly for as a f(t). Eq. 20 was solved usng a Runge-Kutta numeral tenque. Fgure 2 sows te resultng egt of flud n a olumn or Buner funnel and Fgure 3 sows te expeted fltrate volume as a funton of tme for bot Eq. 12 and Eq. 20. Table 1. onsttutve parameters for kaoln lay (Wells, 1990a). Parameter values for Eqs. 16 Value Unts and 17 a 2.04E-20 m 2 b 28.9-2.0E-16 kg/m/s 2 d 28.9-35 onstant ake porosty and k Nononstant ake porosty and k 30 25 20 15 10 5 0 0 100 200 300 400 500 600 mn4.2 m Tme, s Fgure 2. Predtons of slurry egt from Eq. 12 and 20 for onsttutve parameters from Table 1, ε 0.7, A150 m 2, ρ s 1.5 g/m 3, 30 m, ε 0.95, 75 kg/m 3, µ0.01 g/m 2 /s, ρ0.9982 g/m 3.
8 4500 onstant ake porosty and k Nononstant ake porosty and k 4000 3500 3000 2500 2000 1500 1000 500 0 0 100 200 300 400 500 600 Tme, s Fgure 3. Predtons of fltrate volume from Eq. 12 and 20 for onsttutve parameters from Table 1, ε 0.7, A150 m 2, ρ s 1.5 g/m 3, 30 m, ε 0.95, 75 kg/m 3, µ0.01 g/m 2 /s, ρ0.9982 g/m 3. Determnaton of onsttutve Propertes from Gravty Dranage Data Te nverse problem s to determne te permeablty and porosty of te ake gven te dranage rate. Eqs. 16 and 17 ntrodued 4 empral oeffents neessary to spefy te onsttutve propertes. Oter onsttutve relatonsps also usually requre a mnmum of 4 parameters (Vorobjov et al., 1993). Te tenque would be smlar to one developed by Wells (1990b) were a non-lnear, least-squares urve fttng proedure was used to determne tose oeffent values wt te mnmum error. Summary Te rate of fltrate produton data from a gravty fltraton experment an be used to determne onsttutve propertes of te slurry. Tese propertes nlude ake porosty and ake permeablty as a funton of pressure dfferental. Equatons determnng te dranage rate over tme were presented for te bot onstant and non-onstant average ake porosty and permeablty. Te onstant average ake permeablty and porosty model predted a slower rate of fltrate produton tan te non-onstant porosty and permeablty model. Tese models an be used to determne slurry propertes by usng
laboratory fltrate data to alulate te relatonsp between permeablty and porosty and porosty and effetve stress. Slurry propertes allow researers to ompare slurry aratersts and to develop omplex models of te dewaterng beavor. Referenes rstensen, G. L. and Dk, R. I. (1985a) Spef Resstane Measurements: Metods and Proedures," Journal of Envr. Engr., ASE, 111(3), 258-271. rstensen, G. L. and Dk, R. I. (1985b) Spef Resstane Measurements: Nonparabol Data," Journal of Envr. Engr., ASE, 111(3), 243-257. Greenkorn, R.A. (1982) Flow Penomena n Porous Meda, Marel Deker, New York. Veslnd, P. A. (1979) Treatment and Dsposal of Wastewater Sludges, Ann Arbor Sene, Ann Arbor, MI. Vorobjov, E. I.; Ankeev, J. V.; and Samolyotov, V. M. (1993) Dynams of fltraton and expresson: new metods for ombned analyss and alulaton of te proess wt due aount of te ake onsoldaton dynams and te flter medum ompressblty, emal Engneerng and Proessng, Vol 32, 45-51. Wells, S. A. and Dk, R. I. (1988) Synrotron Radaton Evaluaton of Gravty Sedmentaton Effets Pror to Dewaterng," Proeedngs ASE-SE Natonal onferene on Envronmental Engneerng, Vanouver, Brts olumba, 845-852. Wells, S. A. (1990a) Modelng and Analyss of ompressble ake Fltraton, P.D. Dssertaton, Dept. of vl and Envronmental Engneerng, ornell Unversty, Itaa, N.Y. Wells, S. A. (1990b) "Determnaton of Sludge Propertes for Modelng ompressble ake Fltraton from Spef Resstane Tests," Proeedngs A.S..E. Natonal Envronmental Engneerng onferene, Wasngton, D.., 125-131. 9