Mass Transport. Laboratory 3

Transcription

1 Laboratory 3 HWR 431/

2 Itroducto: I the prevous laboratory we derved a expresso descrbg the flow of groudwater through a porous medum kow as Darcy's Law: Q = dh KA dl where Q = volumetrc dscharge [L 3 ] K = hydraulc coductvty [L/t] A = cross sectoal area [L 2 ] dh = gradet of the hydraulc head [-] dl Ths expresso may also be wrtte terms of the specfc dscharge (q [L/t]): q = dh K dl Recall from the prevous laboratory that we used a sol colum to determe the hydraulc coductvty of a sol. We also determed the average lear pore water velocty (v) by dvdg the specfc dscharge (q) by the effectve porosty ( e ) of the sad. v = K e dh dl The lear pore water velocty s the average rate at whch water molecules travel through the porous meda. I ths laboratory we wll expad our study of flow through saturated porous meda to clude the trasport of dssolved solutes. We wll exame the processes cotrollg mass trasport ad derve a goverg equato. I the laboratory, we wll attempt to quatfy the effects of dfferet solute propertes o the rates of mass trasport usg a colum expermet. 3-2

3 Backgroud Processes cotrollg the movemet of solutes through a porous medum: Dffuso: Ths process causes the solute to move from areas of hgher cocetrato to areas of lower cocetrato. As log as the cocetrato gradet s mataed, dffuso wll cotue. Ths process s descrbed mathematcally by Fck's frst law. It says that the mass of flud that dffuses a ut tme s proportoal to the cocetrato gradet: F = D d ( d dx) where F = mass flux of solute per ut area per ut tme [ML -2 t -1 ] D d = dffuso coeffcet [L 2 /T] = solute cocetrato [M/L 3 ) d dx = cocetrato gradet [M/L 3 /L] Fck's secod law gves a related expresso for the case whch the cocetrato gradets are chagg wth tme: 2 2 t = Dd x Because the path traveled by dvdual partcles s loger porous meda tha water, the dffuso coeffcet (D d ) must be replaced by a effectve dffuso coeffcet (D*) whch accouts for the tortuosty of the porous meda. 3-3

4 D * =D d ω where ω = coeffcet of tortuosty, geerally rages from 0.5 to 0.001, determed expermetally The soluto to the dfferetal equato above was determed (rak, 1956) for the approprate boudary codtos: where x ( x, t) = 0erfc 2( D * t) = the cocetrato at a dstace x from the source at tme t sce dffuso bega [ML -3 ] = the orgal cocetrato [ML -3 ] erfc = the complemetary error fucto [-] Advecto: Ths s the process by whch dssolved solds are trasported alog wth the groudwater flow. The fgure below llustrates what purely advectve trasport of a cotuous solute source. Note that the solute frot s etrely vertcal because o mxg occurs. 3-4

5 Fgure1: Dagram of the purely advectve trasport of a cotuous solute source at tme t=0, t=1, t=2 The amout of solute trasported depeds o solute cocetrato ad the volumetrc flow rate. The mass flux due to advecto may be descrbed by the followg relatoshp: where: F x = oe-dmesoal mass flux [ML -2 t -1 ] F x = q x q x = specfc dscharge [Lt -1 ] = cocetrato [ML -3 ] Dsperso: Groud water travels at veloctes that are both faster ad slower tha the average lear pore water velocty. Ths meas that groudwater wll trasport solute both faster ad slower tha the average velocty. Ths results mxg of the solute frot called mechacal dsperso. 3-5

6 Fgure 2: omparso of the solute frot wth advecto ad advecto+dsperso There are prmarly two types of mechacal dsperso: logtudal ad trasverse dsperso. Logtudal dsperso refers to mxg that occurs the drecto of the flow. Trasverse dsperso s mxg that occurs parallel to the advacg frot ad s ormal to the drecto of flow. There are three prmary pore-scale causes of dsperso: (A) Frst, flud wll always move faster through the ceter of a pore tha alog the sdes. Ths s because the velocty profle across a pore s a parabolod of revoluto wth zero velocty at the wall ad a maxmum velocty at the ceter gve by: v max = 2 r 4µ dh dl Fgure 3: Varatos velocty across a sgle ore wll cause dsperso of the solute frot. 3-6

7 (B) Secod, sce the volume of water passg through a gve pore s coserved, varatos the radus of the pore cause varatos the relatve velocty. Fgure 4: Varablty pore dameter alog a gve flow path wll cause varatos velocty alog the path, resultg dsperso () Fally, the flud wll take dfferet paths through the porous meda, causg some flud elemets to travel loger dstaces tha other flud elemet. Ths causes the local velocty to vary both drecto ad magtude. Fgure 5: Dfferet path legths may cause dsperso Schedegger (1954) showed that mechacal dsperso s learly related to the magtude of the lear pore water velocty (v) ad troduced the term dspersvty (α) to relate the two. D m = α v 3-7

8 where s a emprcal costat roughly equal to 1.0. It s frequetly assumed that the mechacal dspersve flux may be descrbed by a "fcka" type curve. The dspersve mass flux ca therefore be descrbed: F = D m ( d dx) Hydrodyamc Dsperso: I atural systems t s dffcult to separate the mass trasport by dffuso from trasport by advecto. Therefore, the two are combed to form a parameter called the hydrodyamc dsperso coeffcet. D D L T = α = α L T v v + + D * D * We ca substtute ths expresso for hydrodyamc dsperso to the dspersve mass flux equato to obtaa ew expresso for the mass flux. The equato for the mass flux becomes: F = m + ( D D *)( d dx) Relatve Importace of Dffuso ad Dsperso: The hydrodyamc dsperso s the sum of the mechacal dsperso ad the dffuso terms. The relatve mportace of these quattes ca be evaluated usg the Peclet umber calculated as: where: P L = Peclet umber [-] P = L v xd D d v x = average lear pore water velocty [Lt -1 ] d = dameter of gras [L] 3-8

9 D d = molecular dffuso coeffcet [L 2 t -1 ] The Peclet umber s commoly plotted agast the rato D L /D d or the rato D T /D d. The fgure below shows the rage of values at whch dffuso domates ad at whch dsperso domates. Dervato of the advecto-dsperso equato We ca ow use these expressos for the mass flux due to advecto ad dsperso to derve the advecto-dsperso equato. If we cosder the flow through a cotrol volume, we ca wrte that the egatve dvergece of the mass flux equals the chage mass storage. I other words, what goes to the cotrol volume must equal the amout whch comes out + the amout stored. F x = t Now, the mass flux to the volume (F ) has two compoets, oe s related to the advectve mass flux (F A )ad oe s related to the dspersve mass flux (F D ): 3-9

10 F = F + F A D If we substty=ute our expressos for the advectve flux ad the dspersve flux to ths equato, we obta: F = q D h j x j We also kow that the chage the flux through the cotrol volume s equal to the chage cocetrato the cotrol volume F x = t If we take the dervatve of the above expresso for F (wth respect to x) ad set equal to the chage cocetrato wth tme we obta (rearragg): + t x x x j ( q ) D = 0 j If we assume the porosty () s costat both tme ad space, we ca dvde through by the porosty () to obta the advecto-dsperso equato + v { t { x storage advectve D = 0 j x x j dspersve Sorpto ad Retardato of Solutes: I ths laboratory we wll be examg solutes that travel wth the groudwater flow (coservatve tracers) ad solutes that are retarded as a result of sorpto to the aqufer meda. For the later case we wll eed to troduce a sorpto term to the advecto-dsperso relato. Sorpto refers to the removal of a solute from a soluto through assocato wth a sold surface ether by electrostatc or chemcal forces. There are three basc types of sorpto: 3-10

11 (A) Specfc Sorpto (chemsorpto): caused by chemcal teractos, usually occurs at specfc stes o the sold due to a covalet bod. (B) No-specfc Sorpto: solutes attach to solds va Va der Waals or electrostatc forces () Exchage adsorpto: os of oe substace are sorbed ad replace os of aother substace Oce sorbed oto the sold meda, solutes are o loger aqueous soluto ad caot be trasported by advecto ad dsperso. Usually, the solute wll sorb to the sold utl the cocetrato sorbed s equlbrum wth the cocetrato the lqud phase. Ths relatoshp betwee the amout soluto ad the amout sorbed s called parttog. We ca clude ths our advecto-dsperso relato by addg a sorpto term: t + v x 2 D 2 x + ρ b S t 123 sorpto where: ρ b = bulk desty of the aqufer = porosty S = amout of solute sorbed per ut mass of aqufer Ofte laboratory expermets are coducted to determe the value for S at a partcular temperature. The relatoshp betwee the quatty of sorbg materal per ut mass of sold ad the cocetrato of solute soluto s called the adsorpto sotherm. May types of adsorpto sotherms have bee developed. The smplest (lear adsorpto) sotherm assumes a lear relato betwee the sorbed mass per ut mass of sold (S) ad the solute cocetrato soluto (). The slope of ths relato s called the dstrbuto coeffcet (K d ). S = K d 3-11

12 3-12 Fgure 6: A lear adsorpto sotherm If we substtute ths to the advecto-dsperso relato for S: = + + t K x D x v t d ρ b ad rearrage, we obta: = + + x D x v t ρ b K d The term + ρ b K d 1 s called the retardato factor (R). Sce R s costat we ca multply through by 1/R to get: = + x R D x R v t Sorpto sotherms may be lear or olear. Oe example of a o-lear adsorpto sotherm s the Freudlch sotherm:

13 S = K log S = log K + log We ca therefore wrte the sorpto term as: ρ b S t = ρb K t = ρb KN N 1 t ad R = 1+ ρbkn N 1 Laboratory Determatos of Retardato (R), Peclet umber (P) ad Dspersvty (D). I ths laboratory, we wll be usg sol colum expermets to obta estmates of values of retardato coeffcet, Peclet umber ad dspersvty. A cotuous source of tracer of kow cocetrato s jected to the sol colum at a costat rate. Small fracto effluet samples are collected at the colum outlet ad sample cocetratos are measured. Whe relatve cocetrato (% of orgal cocetrato) s plotted agast ether tme, volume of effluet or the umber of pore volumes eluted, we obta a characterstc break-through curve lke the oe llustrated below. 3-13

14 Fgure 7: A typcal break through curve Note: The stadard way to costruct ths dagram uses pore volumes. The umber of pore volumes (T) may be calculated by dvdg the amout of water leached through the colum (V) by the lqud capacty of the colum(v 0 ). T = V = vt V L 0 Usg the break-through curve, a umber of parameters may be estmated farly easly. It should be oted however, that a more rgorous procedure should be employed for accurate estmate of these parameters. Such procedures are beyod the scope of ths class. However, the estmates descrbed below should provde a bass for uderstadg how the system works. Usg the approprate boudary codtos we wll assume a soluto of the form: c( x, t) 1 Rx vt erfc 1/ 2 2( DRt) = 2 Do t worry about how ths equato was derved, ths s a expresso whch s frequetly used to descrbe these types of dsplacemet expermets ad should be suffcet for our eeds. Usg ths equato we may obta aother equato for relatve cocetrato (c e ). c e ( T) = 1 2 erfc 1/ 2 [ ( P 4RT) ( R T) ] where c e = c / c 0 Estmatg the retardato coeffcet (R): Sce we kow that erfc(0) = 1, we ca use ths equato to see that c e (R) = 0.5. Ths meas that we ca estmate a value for R by locatg the value of T where c e = 0.5. Estmatg the Peclet umber (P): If we dfferetate the equato for c e (T), wth respect to T, we obta the followg expresso: 3-14

15 P = πr ST where S T s the slope of the curve at the pot where T = R. If we estmate a value for S T we ca the calculate a value for the Peclet umber. Estmatg the Dspersvty (D): Oce we have estmated values for the Peclet umber, we ca use the formula for the Peclet umber to calculate a value for the dsperso coeffcet (D) P = vl D 3-15

16 Laboratory Assgmet: Ths laboratory wll use two groudwater tracers to make laboratory colum estmates of mass trasport a saturated porous medum. We wll use l- as a coservatve tracer. Sce l does ot readly sorb to colum materals ts movemet through the colum should be due to advecto ad dsperso oly. A orgac dye wll be used as a o-coservatve tracer. Because orgac dyes have a strog tedecy to sorb to orgac matter, the movemet of the dye through the colum should be retarded as t sorbs to the charcoal the sol matrx. I ths lab, a soluto of kow l ad orgac dye cocetratos wll be appled to the laboratory colum. The soluto wll be appled at a costat rate ad wth a costat hydraulc head mataed. Small volume fractos of effluet wll be collected at the colum outlet ad ther l ad dye cotets measured usg a o sestve electrode ad a spectrophotometer. From the breakthrough curve produced, estmates wll be made of the retardato factor, peclet umber ad dsperso coeffcet. Whe performg laboratory colum expermets o solute trasport, the results are ofte reported terms of pore volumes of flud that are eluted. Oe pore volume (T) s the cross sectoal area of the colum (A)tmes the legth (L) of the colum tmes the porosty of the meda (): T = AL Rememberg that the total dscharge over a tme (t) s just qt = vat pore water velocty), we ca calculate the total umber of pore volumes eluted: Ths s a dmesoless umber related to tme. vat T tot = = AL vt L (q = specfc dscharge, v = lear 3-16

17 Laboratory procedure: 1) albrate the strumets as descrbed the above procedures 2) Prepare calbrato curves for the tracer soluto dlutos ragg from 0 to 100 % of the pure tracer (If you do t kow how to do ths see the TA BEFORE the lab 3) Label 15 to 20 plastc cups. These wll be used for collecto of water passg through the colum 4) Lower the water the colum utl t cocdes wth the surface of the upper flter pack. Do NOT let the water level drop below ths pot. 5) Add the tracer soluto to the top of the colum utl t s approxmately 5cm above the top of the colum. Mark ths ad mata ths level of tracer soluto the colum throughout the expermet. 6) Ope the clamp ad beg drag the flud from the colum 70 ml cremets to umbered beakers. Try to keep the flow cotuous ad mmze the tme the outlet hose s clamped betwee cremets. 7) After collectg 20 samples, measure the coductvty ad trasmttace of each. 8) Flush the colum wth 2-3 volumes of water 9) alculate the volume of the followg before leavg the lab a) Upper/lower flter pack b) Sad sample c) Dra tube 3-17

18 Measuremet of Solute ocetratos Durg the laboratory, the solute cocetratos of the eluted soluto must be measured as a fucto of tme to determe the dspersve propertes of the porous meda. There are may aalytcal techques whch may be used to measure solute cocetratos. We have chose two methods whch should provde reasoably smple ad accurate measures of cocetrato suffcet for ths laboratory. I practce, t may be ecessary to explore more accurate methods of measuremet. Nal: For the measuremet of Nal, we have chose to use a coductvty meter. I water, Nal separates to the os Na+ ad l- ad the cocetrato of Nal the soluto may be related to the soluto s electrcal coductvty. Dye: The cocetrato of orgac dye a soluto may be measured optcally over a wde rage of dlutos. Therefore, we wll use a spectrophotometer to measure the percet trasmttace of the soluto. Beer s Law may be used to relate ths to the cocetrato of the soluto. Usg the oductvty Meter: The coductvty meter uses a electrode to measure how well a soluto coducts electrcty. The mportat thgs to remember whe usg a coductvty meter to measure cocetratos are: a) always calbrate the strumet wth kow solutos beforehad. b) Measure the backgroud coductvty of the MQ water c) Rse the electrode ad shake off all excess water before usg aother sample d) Make readgs µmhos/cm e) ostruct a calbrato curve (smlar to the oe produced for the dye) relatg coductvty to cocetrato Usg the Spectrophotometer The use of optcal laboratory methods detemg soluto cocetratos s based upo Beer s law. 1 I 0 log = log T I where: T = Trasmttace = A = ab I 0 = testy of lght trasmtted througha referece soluto laboratory blak I = testy of lght trasmtted through the test soluto 3-18

19 A = Absorbace a = Absorptvty costat b = lght path legth of the cell = solute cocetrato The spectrophotometer s a strumet capable of provdg a measure of the amout of lght passg through a soluto. It uses a tugste lamp to sed a arrow wavelegth bad wdth (20 m) through the soluto. All lght ot absorbed s measured by the phototube the spectrophotometer as % trasmttace. However, the spectrophotometer must be adjusted to the correct wavelegth order to sure that t adheres to Beer s law. Sce Beer s law says that absorptvty (a) s costat, the wavelegth chose should be oe where values of trasmsso are farly costat wth dfferet wavelegths. Ths occurs at the rego of maxmum absorptvty (a) ad occurs as a flat secto o a graph of absorptvty vs wavelegth. Procedure for usg the spectrophotometer 1) Tur o spectrophotometer ad allow t to warm up for 20 m. 2) albrate spectrophotometer a) Isert opaque lght block to cuvette holder ad adjust AMPLIFIER cotrol kob to 0%T b) Place blak cuvette cotag MQ water to cuvette holder ad set AMPLIFIER cotrol kob to 100%T 3) Set wavelegth (expected wavelegth approxmately 400) a) Perform a umber of measuremets of trasmsso o a cuvette cotag tracer soluto, each tme varyg the wavelegth by approxmately 5 m tervals aroud 400m b) Each tme wavelegth s chaged, check calbrato (step 2) c) Plot absorpto versus wavelegth to fd the best wavelegth (flattest part of the curve) 4) Prepare a calbrato curve for the tracer soluto. a) Prepare several stadard solutos of varyg cocetratos by dlutg the prmary soluto wth MQ water (remember that a soluto may be prepared by 1V1 = 2V2, see TA f you do t remember how ths works) b) Measure the trasmttace of each c) alculate absorbace of each d) Plot absorbace versus cocetrato (Beer s law says ths should be lear) 5) Now you are ready to beg measurg samples 3-19

20 Laboratory Report: Ths laboratory report should be wrtte the form of a scetfc paper. It should clude a abstract, troducto, procedures/methodology, results ad a dscusso secto. Iclude the followg your lab: calbrato curves produced for the spectrophotometer ad o electrode all measuremets made (dmesos of the colum, lower ad upper flter packs, dra tubes) table of data ( terms of eluted volumes, tme) 2 breakthrough curves for the two samples dscuss shapes of breakthrough curves ad factors whch cotrol the shape calculate the hydraulc coductvty ad porosty of the colum calculate the retardato factor (R) for each sample calculate the peclet umber (P) for each sample (what mechasm cotrols solute movemet more, dsperso of dffuso?) calculate the dsperso coeffcet try chagg the 3 values R,P,D. How do they chage the shape of the curve? What doe ths tell you? 3-20