A falling-head procedure for the measurement of filter media sphericity

Transcription

1 A fallng-hea proceure for the measurement of flter mea sphercty Johannes Haarhoff* an Al Vessal Department of Cvl Engneerng Scence, Unversty of Johannesburg, PO Box 54, Aucklan Park 006, South Afrca Abstract Flter mea sphercty s normally etermne expermentally n a laboratory fltraton column. The pressure rop s measure across a be of known epth whle the fltraton rate s kept constant. The sphercty s then calculate from a theoretcal healoss relatonshp usng the Ergun equaton. Ths paper proposes a metho along smlar lnes, but suggests a much smpler expermental proceure. Instea of havng to mantan a constant flow rate an measurng both the flow rate an the pressure, the column s flle an the water then allowe to ran through the be. The only measurement to be taken s the tme t takes for the water level to rop through a known stance, whch s calle a fallng-hea proceure. The full theoretcal evelopment of the metho s prove, as well as a etale expermental proceure. The practcalty of the metho s emonstrate wth tests performe on a varety of flter mea, an a fully-worke example s presente. Keywors: flter mea, granular fltraton, sphercty, Ergun, fallng-hea test, gran shape Introucton Rap gravty fltraton s the backbone of phase separaton at most water treatment plants n South Afrca. The core of fltraton s a layer of granular flter mea, mostly slca san, whch offers resstance to the flow of water through the mea be urng fltraton, an also expans urng backwash. Flter esgners nee to prect the hea loss through the flter bes, as well as prectng flusaton an expanson of the mea be urng backwash. The esgn moels for hea loss, flusaton an expanson all nclue the sphercty (or rounness) of the mea grans as an mportant varable controllng the behavour of the mea. Numerous efntons were propose to express the egree of rounness of a sol object. A revew by Cerono (1997) conclue that the surface rato sphercty s the efnton most sute an commonly accepte for flter mea. Ths s efne as a rato: surface rato sphercty surface area of a sphere wth equal volume surface area of gran It s qute easy to calculate the surface rato sphercty (smply referre to as sphercty n the remaner of the paper) of a sngle object wth a efne shape. A perfect sphere, for example, wll have a sphercty of 1.00, whle a cube an a typcal sheet of paper wll have spherctes of 0.81 an respectvely. The challenge to the flter esgner, however, s to fn the average sphercty of a flter be whch typcally contans about 3000 mllon grans per cubc metre. Practcally, sphercty can be teste n a number of ways: By comparng a number of grans through a stereoscope an matchng wth a prnte guelne (Far et al., 1968). * To whom all corresponence shoul be aresse ; fax: ; e-mal: jhaarhoff@uj.ac.za Receve 0 October 008; accepte n revse form 10 November 009. Ths metho ha been use on many fferent mea types at the Water Research Group of the Unversty of Johannesburg (UJWRG) an yele results whch were consstently too hgh. By measurng the rate at whch a san gran snks n water, an usng ths value to calbrate the shape factor n a mofe Stokes equaton. Ths test has to be repeate for many fferent grans to get a statstcally robust estmate. Moreover, there s no rect mathematcal lnk between the shape factor from ths test an the surface rato sphercty. By measurng the expanson of a mea sample n a test column at fferent flow rates, an then usng ths expanson to etermne the sphercty from one of the expanson moels (e.g., the moel of Dharmarajah an Cleasby, 1986; or the recently propose moel of Soyer an Akgray, 009). By measurng the hea loss through a mea sample n a test column, an then usng ths hea loss to etermne the sphercty from one of the hea loss moels (e.g., the moel of Ergun, presente n AWWA, 1990; or the moel of Trussell an Chang, 1999). The am of ths paper s to present a smple metho, easly performe wth a mnmum of equpment, to brng the measurement of mea sphercty wthn easy reach of esgn engneers an treatment plant managers. The pont of eparture for ths paper wll be the measurement of hea loss, then obtanng the sphercty from the Ergun equaton. Instea of conuctng the test n a conventonal manner at a constant flow rate, whch requres flow regulaton an the measurement of flow rate as well as healoss, a fallng-hea test wll be evelope whch only requres the measurement of the tme t takes for the water level to rop over a known stance. It shoul be note that the results shoul be entcal, whether the test s performe at a constant flow rate or wth a fallng hea. The avantage of the propose metho s that t s smpler an qucker to perform. The ata analyss, however, s more complcate, so the theory requre wll be fully evelope wth a worke example. ISSN (Prnt) = Water SA Vol. 36 No. 1 January 010 ISSN (On-lne) = Water SA Vol. 36 No. 1 January

2 The Ergun equaton The Ergun equaton reas (AWWA, 1990, p. 465): h V 3 L. g... g... wth: h = healoss through mea be (m) L = epth of mea be (m) V = fltraton rate (m s -1 ) ε = mea be porosty (-) ψ = average surface area sphercty (-) = geometrc gran ameter (m) g = gravtatonal acceleraton (m s - ) ρ = ensty of water (kg m -3 ) μ = ynamc vscosty of water (kg m -1 s -1 ) The attractve feature of the Ergun equaton s that t has both lamnar an turbulent terms. The Reynols number for fltraton s calculate as: Re.. V For a be of flter mea, where the ntersttal spaces cover a broa range n terms of ther sze an tortuosty, there s not a sharp, prectable transton between lamnar an turbulent flow as n the well-known example of ppe flow. Ths s evence by a broa transton range for the Reynols number reporte n the lterature. Typcal rap gravty fltraton rates lea to Reynols numbers whch are n the transton zone. The use of the Ergun equaton, whch automatcally accounts for both lamnar an turbulent flow components, s therefore strongly recommene above some other moels whch only nclue lamnar terms. How oes one fn the geometrc gran ameter of a mxe mea be wth a range of gran szes? Flter san s specfe accorng to ts gran sze strbuton an seve analyses are routnely an easly performe. From such a seve analyss, whch typcally splts the sample nto about 5 to 8 sze fractons, the geometrc mean of the passng an retanng seve szes s calculate for each fracton: geometrc For each fracton, ts fractonal mass contrbuton α can be calculate: fracton through seve above fracton through seve below (4) The Ergun equaton s now apple to each of the fractons n turn, by assumng that each fracton has a epth of α L. Importantly, t s also assume that the gran sphercty of all the fractons are the same an assumpton to be verfe later n the paper. The result s the workng equaton use by esgn engneers for estmatng the healoss through flter mea: h Mea gran ensty seve sze above L. 3. g g... V seve sze below The ensty of the mea grans, requre n the next step for n 1 x. V n 1. L.. V 3 1 () (3) (5) (1) calculatng mea porosty, s measure by pourng a prevously re an weghe sample nto a measurng cylner partally flle wth water. The mass s alreay known, the gran volume s etermne from the volume splacement n the cylner an the ensty s thus rectly calculate. From routne tests one at the UJWRG over about 15 years, typcal values for mea ensty are: For goo qualty clean slca san, the ensty s typcally n the range between 450 an 650 kg m -3. Wth extensve amorphous calcum carbonate eposts on slca san, the ensty coul be as low as 00 kg m -3 A typcal value for flter-grae anthracte s 1400 kg m -3. Mea be porosty The porosty of a ranomly-packe mea be typcally vares between 0.45 an Although ths range may seem to be rather narrow, the Ergun equaton shows that the healoss s strongly epenent on porosty a porosty of 0.45 wll lea to a healoss to 3 tmes hgher than a porosty of Moreover, the porosty of a be s not a constant. A mea be whch s gently settle after be expanson n a test flter, wll compact by as much as 10% after a sngle sharp tap to the se of the flter, whch translates to a sgnfcant reucton n porosty. For the etermnaton of sphercty, however, t s only mportant to know what the porosty s at the tme of the fallng-hea test. It s therefore suggeste that the mea sample s re an weghe before t s transferre to the test column. After the test s performe, the exact mea be epth s measure. The actual n stu porosty for the test s then etermne from: 4. M 1. gran. D. L (6) wth M = mass of mea sample use n test (kg) ρ gran = ensty of mea grans (kg m -3 ) D = ameter of test column (m) Water ensty an vscosty Both the ensty an vscosty of the water can be relably estmate from the water temperature. The followng polynomals were ftte by the authors to values reporte n Le (004) an are wthn 0.00% for ensty an 0.01% for ynamc vscosty: x 10 xt x 10 xt (7) 5.786x 10 xt x 10 xt.3333x 10 xt (8) x 10 xt x 10 xt 1.793x 10 wth ρ = ensty (kg m -3 ) μ = ynamc vscosty (kg m -1 s -1 ) T = water temperature ( C) The propose fallng-hea test A test column s requre for the test, schematcally ncate n Fg. 1. The column has marks one about mm above the overflow level an the other about 100 mm above the overflow level. Each fallng-hea test has parts. In the 1 st part, the test s performe wthout any mea. From the tme t takes for the water level to rop between the marks as t flows through 98 ISSN (Prnt) = Water SA Vol. 36 No. 1 January 010 ISSN (On-lne) = Water SA Vol. 36 No. 1 January 010

3 top mark nepenently etermne for each test: A h B h h. C. t t (1) h 1 h water level n column bottom mark water level above overflow top of overlow ppe mea nlet Fgure 1 Schematc layout of test column A n L. 3. g. 1 n L B. 3 g. 1 The soluton to ths fferental equaton s gven by: (13) (14) t h B A B A. C. 4. h. C. h 4 h1 (15) The above ntegral oes not offer an analytcal soluton an has to be evaluate numercally. For the work reporte further on n ths paper, Smpson s rule was apple wth 10 ntervals to allow a soluton for the sphercty ψ, an the Goal Seek functon of Excel was use to evaluate the sphercty. Verfcaton of the propose proceure 1 the empty column test, the resstance offere by the outlet ppng an the mea support gr can be quantfe. In the n part, the same test s repeate, but ths tme wth the mea n the column. By the mathematcal analyss whch follows, the mea sphercty can be calculate. A etale proceure for performng the test s suggeste n Appenx A. Solvng for the sphercty from the fallng-hea test The water flowng through the test column oes not only have to overcome the resstance of the mea be, but also the resstance offere by the mea support system an outlet ppng of the column. These non-mea losses are turbulent an take the form: h C.V (9) In the case of a fallng hea, where the fltraton rate vares as the water level rops, the fltraton rate s expresse as h/t, leang to a fferental equaton: h h C. (10) t The soluton of ths fferental equaton allows the estmaton of C by usng the tme t takes for the water level to rop from the top mark to the bottom mark: (11). t C h1 h Ths fallng-hea etermnaton of the non-mea losses s superor to the constant-rate metho. The non-mea losses are normally small n relaton to the mea losses an are ffcult to etermne accurately n the constant-rate metho. The tme measurement of the fallng-hea metho s much more accurate. When the column s flle wth mea, the total hea encountere by the water flowng out of the column s the sum of resstance offere by the mea an the outlet ppng. By ang equatons (5) an (10), a 1 st -orer fferental equaton follows, n terms of the constants A, B an C whch are Expermental equpment The tests were conucte n a clear polyethylene tube wth an nse ameter of 67 mm. The mea was supporte on a fne stanless steel mesh wth an approxmate aperture sze of 1. μm, whch n turn reste on a coarser, stff stanless gr wth an approxmate aperture sze of 1.5 mm. The column extene to about 00 mm below the mea support grs an ths uner-floor volume was flle wth glass marbles to equalse the flow patterns n the uner-floor volume. A sngle connecton allowe water n or out of the uner-floor volume. Ths connecton le to both the overflow ppe (whch coul be blocke or opene wth a rubber stopper) an the nlet hose (whch coul be opene or close wth a valve from the muncpal connecton). The overflow ppe was nstalle such that ts top was about 50 mm above the mea support gr, whch etermnes the maxmum mea be epth that can be teste. For all the tests conucte, the actual be epth was between 100 mm an 140 mm. The top an bottom marks n the column, use to etermne the start an stop of the fallng-hea test, respectvely, were mm an 103 mm above the top of the overflow ppe. The fference of mm allowe a relable estmate of the tme taken for the fallng-hea test. As the flow rate ecreases sgnfcantly when the water level n the column approaches the level of the overflow ppe, the bottom mark shoul not be less than about 100 mm above the top of the overflow ppe, to lmt the ran-own tme to a reasonable value. For accurate results, t s necessary to conser the heght of the water above the top of the overflow ppe. Ths can be obtane by rect measurement urng the fallng-hea test. The ameter of the overflow ppe use s 0 mm. For the tests reporte n ths paper, the overflow heght urng the empty be test vare from 5 mm when the water was at the top mark an 5 mm when the water was at the bottom mark. The corresponng values when the column was flle wth mea were 7 mm an mm. Mea samples The proceure was teste on mea samples, lste n Table 1. ISSN (Prnt) = Water SA Vol. 36 No. 1 January 010 ISSN (On-lne) = Water SA Vol. 36 No. 1 January

4 Table 1 Mea samples for expermental work Number Name Source Notes 1 Glass ballotn Commercal As use n general laboratory applcatons. Fne mea Mea suppler Supple after a request for fne mea. 3A 3B 3C 3D 4A 4B 5A 5B Meum mea mm mm mm mm New mea Sample 1 Sample Ol mea Sample 1 Sample Mea suppler Full-scale plant Full-scale plant Supple after a request for meum-sze mea. Fne mea 1 an are from the same suppler an slca epost, but prepare wth fferent nustral screens. Ths sample was separate nto the 4 sze classes before analyss. Mea n use for about yr wth no vsual evence of calcum carbonate eposton. Samples 4A an 4B nepenently rawn from laboratory stockple. Mea n use for about 15 yr wth vsual evence of calcum carbonate eposton. Samples 5A an 5B nepenently rawn from laboratory stockple. #1 # Table Densty an sze analyses for mea samples Number Densty (kg m -3 ) Recovery from seves 10 (μm) 60 (μm) 60 / 10 (-) % % A % A % B % #3B #3D after whch the column constant C was calculate from the average. After each of the sub-samples ha been place n the column, 5 consecutve tests were performe as etale n Appenx A, each wth a slghtly fferent be heght an thus mea porosty. Ths yele a total of 49 nepenent sphercty values (1 test on Sample 5B was unntentonally omtte). To prove a more ntutve grasp of the fferent mea types, Fg. shows a collage of photomcrographs, all at exactly the same magnfcaton. Results #4 Fgure Photomcrographs of mea samples The 5 samples were further subve as shown to yel a total of 10 sub-samples. Seve analyses were performe, 1 test for each sub-sample, an the per cent recovery from the seves was n all cases hgher than 99%. For each sub-sample, the ensty of the mea was measure wth 5 replcates to obtan the average. Each test commence wth an empty be test n trplcate, #5 For the total of 4 nepenent ensty etermnatons, the stanar evaton was 1.0% from the average. The average values for each sample are shown n Table. There was no statstcally sgnfcant fference (α = 5%) between Samples an 3, whch s supporte by the observaton that these samples were rawn an processe from the same geologcal epost. The fference between Samples 4 an 5 was statstcally hghly sgnfcant, showng that the eposton of amorphous calcum carbonate cause a measurable reucton n ensty. From the seve analyses, the 10 th an 60 th sze percentle values were estmate by lnear nterpolaton. The effectve sze ( 10 ) for all the samples was about the same, wth the excepton of Sample. The unformty coeffcent ( 60 / 10 ) for flter mea s normally specfe to be less than 1.4. The calcum carbonate eposton on Sample 5 has a marke an etrmental nfluence on the unformty coeffcent. The tests were performe wth muncpal tap water urng the months of May to July 007, wth the temperature rangng from 13.0 C to 18.5 C. For each test, the ensty an vscosty 100 ISSN (Prnt) = Water SA Vol. 36 No. 1 January 010 ISSN (On-lne) = Water SA Vol. 36 No. 1 January 010

5 Table 3 Sphercty of mea samples Sample # 1 3A 3B 3C 3D 4A 4B 5A 5B Measure sphercty Average Stanar evaton Coeffcent of varaton 1.6% 0.7% 1.6% 1.4% 1.9% 1.% 3.1% 3.6%.5% 3.7% values were calculate from the measure temperature. The column constant C for the column use vare between 39 an 301 s m -1. Ths farly large scatter coul not be relate to any systematc cause, an was presumably the result of the slght blockage of the mea support mesh by small grans remanng after a mea sample was washe out. It s therefore suggeste to conuct the empty be test wth every sample to be teste, t beng a quck an easy preventatve step. The porosty of the mea n the test column was elberately vare for each of the sub-samples by controlle tappng of the column (see the etale proceure n Appenx A). For the slca samples analyse, the porostes all fell wthn a farly narrow ban of 0.46 to 0.53 wth an average of Sample 1 (the glass ballotn) ha a sgnfcantly lower porosty range of 0.37 to 0.4. Ths agrees wth the unversal observaton that ranomly packe roun grans attan a hgher packng ensty (.e. a lower porosty) than grans wth more rregular shapes. The sphercty values, calculate from the prevous results an the proceure propose n ths paper, are shown n Table 3. The ballotn of Sample 1 appear, as s event from Fg., to be perfect spheres wth sphercty of The average measure sphercty turns out to be 0.988, whch s remarkably close. The fference between Samples 4A an 4B (fferent samples rawn from the same stockple) was statstcally nsgnfcant (α = 0.05). These values coul thus be poole to yel an average sphercty of Smlarly, the fference between Samples 5A an 5B turne out to be statstcally nsgnfcant an yele an average of after poolng. Next, the sphercty of the poole Samples 4 an 5 was compare. These values were nee sgnfcantly fferent (α = 0.05). In other wors, the eposton of calcum carbonate have a sgnfcant, albet small, effect of makng the grans less angular. Sample 3A was splt nto 3 fferent sze fractons, 3B, 3C an 3D, to test the earler assumpton of Eq. (4), namely, that all the sze fractons of a sample have equal sphercty. The possblty exsts that the smaller grans wthn a mea sample may be more shar-lke an rregular. The results presente n Table 3 show that ths s not the case. In fact, there s no statstcally sgnfcant fference between the spherctes of Samples 3A, 3B, 3C an 3D (α = 0.05). The assumpton unerlyng Eq. (4) s thus valate. Concluson The objectve of ths paper s to present an alternatve metho for the etermnaton of flter mea, namely to use a fallnghea proceure nstea of the normal constant-rate proceure. The proceure was extensvely teste an refne an a etale test proceure coul be evelope, as shown n Appenx A. The avantage of the smpler proceure s partly off-set by the nee for more complcate ata analyss, fully evelope n the paper. The analyss s llustrate wth an example whch emonstrates that smple spreasheet programmng can eal wth the ata analyss wthout any problem. The test was valate wth the results of near-perfectly sphercal glass ballottn, whch yele an expermental result of 0.988, close to 1 as expecte. The test was senstve enough to scrmnate, wth statstcal sgnfcance (α = 0.05) between the sphercty of clean mea, an the sphercty of the same mea wth some calcum carbonate eposton. The coeffcent of varaton for replcate tests on the same samples (n = 5 n each case) was less than 4%. Acknowlegments The mea samples were obtane through the generous assstance of John Gelenhuys of Ran Water Scentfc Servces an the management of Slca Quartz. References AWWA (AMERICAN WATER WORKS ASSOCIATION) (1990) Water Qualty an Treatment. 4 th en. McGraw-Hll, New York, CERONIO AD an HAARHOFF J (1997) Propertes of South Afrcan slca san use for rap fltraton. Water SA 3 (1) DHARMARAJAH AH an CLEASBY JL (1986) Prectng the expanson of flter mea. J. Am. Water Works Assoc. 78 (1) FAIR GM, GEYER JC an OKUN DA (1968) Water an Wastewater Engneerng, Volume 1. John Wley & Sons, New York. LIDE DR (004) (e.) CRC Hanbook of Chemstry an Physcs. 85 th en. CRC Press, Boca Raton FL, USA SOYER E an AKGIRAY O (009) A new smple equaton for the precton of flter expanson urng backwashng. J. Water Supply: Res. Technol. - AQUA 58 (5) TRUSSELL RR an CHANG M (1999) revew of flow through porous mea as apple to hea loss n water flters. J. Envron. Eng. 15 (11) ISSN (Prnt) = Water SA Vol. 36 No. 1 January 010 ISSN (On-lne) = Water SA Vol. 36 No. 1 January

6 Appenx A Detale test proceure Part A Mea tests 1. Take a representatve sample of the mea to be teste. Wash t gently by han n a 50 μm seve uner runnng water to wash out all lumps an mu balls. Dry overnght at 100 C.. Takng a sample of not more than 300 g, seve t wth a seve shaker through a stack of all avalable seves below an nclung.00 mm. 3. For each fferent seve fracton, calculate the mass fracton α an geometrc ameter an etermne the summaton terms n Eq. (4). 4. Take at least 3 samples of about g each, wegh an a to 500 ml n a ml cylner. (To save mea, f requre, goo results can be obtane by usng half ths mass n a 500 ml cylner.) 5. Calculate the mea ensty for each sample an etermne the average. 6. Retan 500 to 600 g of re mea for the column test escrbe n Part C. Part B Empty column test 1. Set up the empty column an ensure that the column s exactly vertcal usng a post level.. Connect a hose to the nlet an block the overflow ppe wth a rubber stopper. 3. Open the hose an allow the water to rse n the column to the top or at least 50 mm above the top mark. Close the hose. 4. Measure the water temperature n the column wth a thermometer. Calculate the water ensty an vscosty. 5. Measure the stance from the top of the overflow ppe to the top mark (h A ) an the bottom mark (h B ). 6. Remove the stopper from the outlet ppe an measure the overflow epth at the overflow ppe when the water level reaches the top mark (h C ) an the bottom mark (h D ). 7. Calculate h 1 = h A h C an h = h B h D. (The stances h 1 an h are shown n Fg. 1.) 8. Stopper the outlet ppe an fll the column agan by openng the nlet hose. Close the hose when the water level s at least 50 mm above the top mark. 9. Remove the stopper from the overflow ppe. Use a stopwatch to accurately measure the tme t takes for the water to rop from the top to the bottom mark. 10. Repeat Steps 8 an 9 at least 3 tmes to get a relable average for rop-own tme. 11. Measure the nternal ameter of the test column. 1. Calculate the column constant C. Part C Column test wth mea 1. Take a sample of approxmately 500 to 600 g of re mea an wegh.. Pour the mea nto the clean column. 3. Ensure that the column s exactly vertcal usng a post level. 4. Connect a hose to the nlet an block the overflow ppe wth a rubber stopper. 5. Open the hose an slowly fll the column from the bottom, ensurng that no mea s washe over the top. 6. Allow the mea to settle an then gently ncrease the backwash rate to obtan a be expanson of about 50%. Mantan ths rate untl the backwash water s clear. 7. Suenly close the nlet hose an wat for the mea to come to rest. 8. Remove the stopper from the overflow ppe. Measure the overflow epth at the overflow ppe when the water level reaches the top mark (h E ) an the bottom mark (h F ). 9. Calculate h 1 = h A h E an h = h B h F. (The stances h 1 an h are shown n Fg. 1.) 10. Repeat Steps 5, 6 an Remove the stopper from the overflow ppe. Use a stopwatch to accurately measure the tme t takes for the water level to rop from the top to the bottom mark. 1. When no more water s ranng from the overflow ppe, measure the mea epth from the mea support gr to the mea surface. Be careful not to bump or tap the column. 13. Calculate the sphercty. 14. Repeat Steps 10, 11, 1 an 13 at least 3 tmes. The 1 st tme, perform Step 11 mmeately after Step 10. The n tme, just before Step 11, gve the test column a sharp seways tap whch wll cause the mea surface to rop a lttle. The 3 r tme, gve the column taps, etc. (Ths s to ensure that the sphercty s calculate for fferent porostes.) 15. Take the average of the sphercty values etermne n Step 13 an use for further esgn. 10 ISSN (Prnt) = Water SA Vol. 36 No. 1 January 010 ISSN (On-lne) = Water SA Vol. 36 No. 1 January 010

7 Appenx B Example of ata analyss A mea sample s seve, wth the results shown as Columns 1-6 on the left of Table 4. The fractonal mass contrbuton of each fracton (Eq. (4)) s shown n Column 7, the geometrc mean (Eq. (3)) n Column 8 an the summatve twerms of Eq. (5) n Columns 9 an 10.a The ensty of the mea s etermne to be 636 kg m -3 an the water temperature urng the test s 16.0 C. The ensty s thus estmate to be kg m -3 (equaton 7) an the ynamc vscosty to be kg m -1 s -1 (Eq. (8)). The test column has an nternal ameter of m an the top an the bottom marks are 1.1 m an 0.1 m above the lp of the overflow ppe respectvely. Durng the empty be test, the overflow heght over the lp of the overflow ppe s 0.05 m when the water level s at the top mark an m when the water level s at the bottom mark. For the empty be test, therefore, h 1 = m an h = m. Durng the test wth mea, the overflow epths were m an m respectvely, leang to h 1 = m an h = m. Durng the empty be test, the average tme taken for the water level to rop from the top to the bottom mark s.5 s, leang to a column constant C of 38 s m -1 (Eq. (11)). For the mea test, a re mass of g s transferre to the column. The tme taken for the water level to rop from the top to the bottom mark s 54. s an the be epth after ths test s m. From these an earler values, the porosty ε = (Eq. (6)), A = 8.6 (Eq. (13)) an B = 115 (Eq. (14)). Wth the excepton of the sphercty, all the terms n Eq. (15) are now known: 54. or: t K h h f (h, ). h h h By choosng an approxmate value for the sphercty ψ, the tme t can be calculate. Successve approxmatons contnue untl the calculate tme s close enough to the measure tme. If Smpson s rule s apple wth 10 equal heght ntervals (others methos can also be use, of course), then a soluton s offere by: t h1 h K. 10. h a 4h b h c 4h... A convenent way of fnng the best approxmaton for the sphercty ψ s to use the Goal Seek functon n Excel to change ψ untl the measure an estmate tmes are the same. In ths partcular example, the soluton s gven by ψ = j 1 h 4h h k Table 4 Analyss of seve test ata of typcal flter mea Seve (µm) Mass (before) (g) Mass (after) (g) Mass (on) (g) Mass (through) (g) % (through) (%) % α (-) Geometrc mean (m) % % % % % % % % % % % Pan % Sum α / (m -1 ) α / (m - ) ISSN (Prnt) = Water SA Vol. 36 No. 1 January 010 ISSN (On-lne) = Water SA Vol. 36 No. 1 January

8 104 ISSN (Prnt) = Water SA Vol. 36 No. 1 January 010 ISSN (On-lne) = Water SA Vol. 36 No. 1 January 010