Document v1BgG8eRyN88jJJo5EE4JyjK8

crsity of j vestigat* f \* I recall ed to wt Whatj jy* in the | Reynolds qjjj best tfcefc dock t <r, bet tl fftfftert*19 authors (i, 4, 6, 7,11) have | recognized the importance of por- I in filtration. At long ago as 1937, (11) developed equations relating filtration resistance to the porosity Nation throughout a fixed bed of . ST 0722787 Process Equipment . .. *8 family 81 Filtratiei ^ p, a liquid passing through a solid changing and that die porosity was constant While Tour's procedure ' gound, some of his equations relat**porositjr to pressure were not tested, i *f, work was not published. Ruth i. in the period 1932-37, also devd| Jd a procedure for relating the filtra- ^Vvariahles in a solid of variable`porCarman (J) focused attention on faafi equation involving the rate of u a function of the porosity and variables. Several other inrestiga* i >10 considered Koxeny's equation but i ^ not used it in the best manner in ikrttiofl equations. In this paper the fee*? equation is used at a basis for fcrtloping *** differential equations of drsrioa These equations are used in lection with fixed beds of solids, coot rate filtration of a solid with and out precoat, and constant pressure itiou with and without a precoat | nows tevetvad <* nUrailM [a the filtering of solids from suspen^ or in any process in which a liquid I gras through the interstices of a solid, fie flow is in the direction of decreasing ptssare. The solids are supported on a <Mcn, cloth, or. other solid bed known the septum or filter medium. In ordi ary filtration practice, the solids closest to d* septum are packed more densely I 4 the rest of the cake. The porosity . a minimum at the point of contact I toween the cake and the septum and is i onzonum at the surface where the apd enters. Since the filtration resis| :we depends upon the porosity, the re- once steadily increases as the liquid | a*** through the cake if the solids are 'mpmsibte. In order to treat qtuntita` tdy the filtration problem, it is neces<7 le have relationships between the it of flow, pressure on the liquid, and j .* porosity in addition to the various I ^meters. The information necessary j * u understanding of the methods used* this paper is summarized 14, follows; Mumerass! Meihefa far Canagfint flats end Cenag&mt Prosura Emigration issod on BCosony's Loot F. M. Tiller iamar Steto Cnlbge of Tnchnetegy, icnnmnnt. Tea-- Based upon the Kozeny low relating the rote of flow to the porosity, a method has been developed for determining (a) the pressure drop vs. the depth and the flow rate vs. thg applied pressure In a fixed bad of solids, (b) tha pressure vs. time relationship in constant rote filtration, and (e) the volume vs. time relationship in constant pressure filtration, it Is assumed that the.jpnrosity is solely a function of the pressure an the solids, thus eliminating solids such px certain naturally occur ring days In which application uf pressure causes only a slaw approach to the equilibrium porosity. Experimental porosity vs. pressure dota are presented for such materials as koolm, calcium carbonate, carbon block (suchar), dlafomocoovt earth (hyfloX asbestos, and mixtures of these materials. As a satisfactory approximation, the per cent voids can be related to the pressure by a power function with powers ranging from 0.01 to 0.05 for applied pressures up to 100 Ibr/sqJn. J. Thn Keeney eqnnrinn at preeentef by Carman h wand at n bnde lar ralaUsg Ora rale of Sew in die prnunre gradient an ike tpariBe wifaca af dm mOd, d H Other tnwi raloHwg the. Saw rate in 1 Perarity vs. Prougro m SnBdg In the (told el salt KthawIn, `the par can! enidi w eoM radn In ftmyinatly detarmlhed at e faction of tho. appSed bod. Vhe apparatvt swplaytd castfrb ecMatioSy nf and the UqeM within the ha IMd b free * Sew the parent platet, the este* of. BgsM | hi et the 1 leave the voids (Kg. I), rowBing Met tfoa. af the --Ws. As the 1 neatrel 1 the eppAod bad. When ba eadra lead b berao by the taBdo at nifTrafd I* Kgnra 2 the attaal premora ea the wBdt b gbaa by pressure on solids =* be placed and -Matted to vertical loading. In *e almplnti type of Iodine, the temple U plaeod btfiraen paraiu platoo and kept mterotod wM mater mhlln dm lead It oppibd. Al the ntiawt the lead b added, the tiro* b applied te both applied pressure/(l --) (1) where y is the equilibrium value of the - per cent voids by volume or porosity and (I'-s) is equal to the average true area of the solids per square foot. MuSm f porokUr o | 12) ara pmeatod m 1 Wlh Mfcal O-WBy nhnrf la the f I U sever fvadlas, In * | *%./*** I **** pnmoro om salUi 1 hil g* iliirfpifar el m prafclaw In I * Iran kady saalyili nf the mW and m the Bgnid 1 1 McltnnaBy thrangk FronkM. Tiller, who hod received tha AJ.Ch.fi. Junior Mombar Aword In 1950*was again a wiruwr ot the 1952 Clovglond meatirig, where tfira paper tamed the best prosonfgd paper award. A fi.Ch.fi. qf,the University of LovbviQo and a Ph.D. of the University of Cincinnati, ho taught two yoars at Cincinnati, nlno years at Vanderbilt University, and a short time at the ' Institute do Oloos In RIo de Janeiro. He Is now director of englneerintf ofiomtsr State College of Technology, fieoumont, Tml O. N. 9 Chemical Engfaaarlite Pregrai. Pag* 467- lp lp FJg. 1. Coaiproislsw process m nmsolfdo--otor. la this type of test, the liquid is not confined, and the neutral stress must ap proach aero. The rate of flow of the water from the sample depends upon the permeability, and the time required to reach equilibrium ranges from a few minutes for the materials used in this investigation to 24 hr. for soils classified as days. In addition to the primary equilibrium which is reached, there is a secondary creep effect which may con tinue more or less indefinitely. In another procedure known as triaxial compression testing (JO) , the sample is placed in a water-tight cover ing in which the liquid pressure Pm or neutral stress, can be controlled. After adjusting the neutral stress, an inde pendent external load is added which results in compression of the solids as indicated in Figure 3. The neutral stress acts equally in all directions and does not affect the porosity; thus the external load over and above the neutral stress (P -- P9) is responsible for compres sion and decrease of die voids. Actually in triaxial testing, the solid is not con fined laterally and changes in dimensions occur along all three axes. In filtration, the neutral stress in the consolidometer and triaxiai soil compres sion tests is equivalent to the liquid pressure in the interstices of the cake. In the static tests the neutral stress is constant throughout the solid; where**, in filtration the liquid pressure decrease in the direction of the flow of the filtrate. In this investigation it is assumed that the liquid pressure does not affect the porosity. Then the results obtained in the simple consolidometer can be meg for predicting relationships between por osity and pressure on the solids hi * filter cake. Ruth (8) developed an ingenious per. meability-compression apparatus in which filiations could be carried out while the pressure on the solids was con trolled independently. Loads were added in a manner similar to that used in the consolidometer except that a perforated plate confined the solid at both top and bottom. After loading the solids, liquid was introduced separately and allowed to flow through the solids under rdativefy small heads. With this procedure the porosity of the solids could be con trolled at an essentially constant value while the filtration was performed. EXPERIMENTAL DETERMINATION Of POROSffT VS. APWED LOAD Tbe porosity i deteralaod la coomIdewater cMfbtfftf of a brow cap Ea wkldt 9 mot wild could be cowprsnod bp a seftebfc loading procedure. Tba cep wot -fjrrrluitulL 5 aqJn. In area, aad tba i*Ud daptb was taMaly oboot 1 M. Tba mot taUd railed mm a porees (dole ibroegfc wbkb walstere ceald flew m t> was sqweesed out of the solid. la Most of fto runs a lew sysfoai was eaed sa that tho on the redd was toor Hum Hie weight employed. An Amos dial served for Meowing the sHA^^j of tbe taUd. Tbe proetdere canilued af tbe fodoebsi 1. Tbe total depth was feead before the aa^s was placed In dn seaiandeaeler en top f porows staaa. <> 2. After the mot seOd was placed hi Hw a^tf ST 0722788 i <l to elepse before tba first reodtag was recorded. X Weights war# added aad 10 tile. eflowed betweea readlegs 4. After this precedere, tbe aebturo ceatewt f the solid was determined, aad perodder were colcnloted bared upoa the readings aad tbe Rad porosity. .J The final per cent voids were calculated from the per cent moisture of the cafcq at the end of the tests in the following maimer: Mg. 2. loft, IqelObrieM (two In cemprosrioa * cO - void vol. void roL + solid voL m/62.4 " x/6Z4 + (l-ej/p, (2) Fig. X Abate, WaHbeHen of bod batwooa soflds owd liquid. ,(pr.) (p-r-) + (i -- ) 0) where ia the density of the solids and Pqgo 468. Chemical Engineering Progross September, 1953 used i and i the i > &A * tb* *P-8T- Data and sample r tof^ioni are shown in Table 1. deprocedure outlined was applied to ^etomaceous earth (hyflo), kaolin, *^yated carbon (suchar), calcium carmixtures of kaolin and hyflo, suchar and hyflo, and as- final moisture content was Th* final porosity is calculated '\a Equation (2) .using a value of 260 ^the specific gravity of kaolin. Thus (2.6) (0.2586) = 0.476 *(B) (0.2586) -07414 of porosity at 11:30 , .uctioo in thickness = 0.4542 -04192 - 0.1350 in. *skc thickness -- initial value - 0.1350 , 1.018 - 0.135 = 0.883 in. I volume = (area) (thickness) (16.4 cc./.in.) ** voL in cc. = (4.988) 6.4) (thickness) (81.7) (0.883) *711 cc Totsl volume of solids remains constant j ^ can be obtained from the final por- *ity. Vohune of solids (1 - .476) (627) a 326 cc. porosity at 11:30 (72.1 - 32.8)/72.1 s 0.545 Ifnpirieal equation as found from Fig ure 5 TeMo li--Doto omd Swnplo Cekefodoas for CemetMecwtor Tests Poredty rv Ns. JO Sybrtewcoi kaefin ftnei Mototars tmmlomi Area wt cake 4.9SS tq.to. 21UK TIMI tnllfal 10*30 10.40 11,00 11,10 11i20 11,30 11,40 1140 12>00 12,10 11,20 12,30 12,40 1240 U10 1,10 AMES MAI HEADING IN. 0 4542 04240 040tf 04790 04490 0.32*3 04192 04097 04942 04742 0.2439 0.2337 04394 04279 04191 04T1S 0.2030 A9PUO LB./SQ.IN. 04S 1.42 3.04 3.1S 4.17 4S 12.92 19.94 27.00 33.99 47.93 41JS 73.S2 0942 10347 CAKE THICKNESS IK 141S 0943 0.913 0192 OBS3 0*73 0437 0.139 0427 0419 0.103 0491 04*2 0773 0441 VOL CC *34 77J) 744 724 72.1 714 700 414 474 44.9 434 444 434 434 424 poeosiTY 49S 474 440 449 443 440 431 421 414 409 499 .493 .444 .441 474 lo the first stages of the investigation, Ae effect of both time and pressure on dr porosity were determined. In Figure | l the porosity of crushed limestone is | (totted against the time at different ap* i (bid pressures. It can be seen that about isua. were required to approach appar ent equilibrium closely. With natural dan and similar materials, the time re* pared to reach an apparent equilibrium Much longer. In some runs thickness and porosity 'nenninatioQl were made as- the pres ere was reduced, Le^ after completing *e addition of loads on the sample in ae consolidometer, the loads were re ared and Ames dial readings recorded, little increase in porosity was noted, ad the conclusion was reached that the oeipression process was essentially irnnnible. Carman (2) also indicated fiat bis data pointed to the irreversibility 4 the compression praces s with most krfids. Is Figure 5 some of the porosity vs. Ptwsre data are plotted logarithmically. A* a fair approximation in the ranges *Ttttigated, Figure 5 indicates that a ^ function of the form (4) PRC88URS, P.8.1, n*. a hmAy . (wm. ** *, No. Chemicol Engineering Progress Page 469 ST 0722789 f.' may be employed for representing th data where p, is in pounds force/sq.ii In Figure 6 data of Lindquist and You (7), Ruth (8), and Walas (12) plotte in a similar manner appear to foliothe same relation. Values of , and as found from the graphs, are given i Table 2. Terzaghi and Peck (10) have used semilogarithmic plot for representing linear relation of void ratio to app|je pressure in the range of 10-100 kg. sq.cm. (142-1420 Ib./sq.in.) for sand. / the higher pressure, the sand grair begin to crush. For some days the son logarithmic relation yielded linear rei; tions up to applied pressures as high ; 1000 kg./sq.cm. In terms of the va> iables used in this investigation. t> Terzaghi relation is given by SUtSTANCC Cnisbod Ilamtono Caktua carbonate Cold-- carbonate Cofciw wKato Ferric oildo Moanociow carbonate IOOW M* OH boob 75% hyb 25% kaolin 50% Me 30% keeb 25% M* 73% bob 0% M 100% boob 66J% Mo 354% teebar 0% byb 100% tocher Atberieo Aleainoa axUo CeMo Copper oxide Carbon, nerit A f- - J37 = Cl In p, + (J Toblo ls Literature Cited ftirtfc (4) Waia* (12) Walai (12) Waist (12) Welat (12) UodawUt end YewU (7) Undqeid wd Toole (7) iMdpwlit and Towle (7} Undewbt and Toole (7) CAKE 049 0.77 045 044 041 0.90 0.M 0.15 041 040 049 0.00 0.19 0.91 040 044 042 OJO < 0.015 0.034 0.034 0.010 0.037 0.011 ' 0.014 0.020 0.032 0444 0445 0.013 04091 0417 0.0047 0413 0.031 0.021 PRECOAT OCVIATION IN t 0.009 0.002 0.005 0.002 0.02 0.003 041 0.02 042 0410 0406 0400 0.002 0.000 0.03 0.03 044 0.05 MEDIUM While this expression is adequate in th high pressure range, it is not satisfy tory at low pressures which are impor tant in filtration. In this paper the ana lytical form of the pressure-porosit relationship is not of importance excej as it may be used for interpolation. Hov ever, in the succeeding paper (Part II where an analytical equation is deve! oped for the filtration variables, the fon of the equation is vital to the mathc matical work. Grace (5) has indicated that th power function approximation (4) fo the porosity is not satisfactory above 10 lb./$q.in. in accord with Terzaghi an Peck. Therefore, at ail times the us of Equation (4) should be examine with care. Equation (4) is not satisfactory as f (Ib./sq.in.) approaches zero (atmoc pheric pressure) since an infinite vala of is predicted. Actually, after th pressure is dropped below about 0.1 lb./ sq.in. gauge pressure, the porosity tends to approach a limiting value. As a gen eral guide, Equation (4) should not be used below 0.1 ib./sq.in. for compres sible materials. Little error will be in volved if the porosities below approxi mately 0.1 Ib./sq.in. are assumed con stant. w Rf. 7. IHwproflon W wpmMUHy mUi In ftfcof wo. Rrouocot M UqeM Md kDdi Darfag WHiitlea A schematic diagram of the Sow of : liquid through a bed formed from a com pressible material is illustrated in Figun 7. The particles are shown in a maxme which indicates that the porosity is de creased as the liquid passes through thi solids and approaches the septum it accord with common knowledge. lx Figure 8 the liquid is shown as it passe frictionally along the particles in th filter bed. The drag on each partich Pago 470 Chemical Engineering Progress September,  forci}! 'll in ().! ar toil of are j have \ pres to to \ 10-10 fori sand * *y*t 1 lia i 1 of tigati )T equate^ not i hi per t sure-] taace < lation.*! er (P on is i ItM the i ed ion rya cn CM CM I oo h from a? id in 1 in a f -osity i thn septs wlet" as its Ides ich jjj to the drag on the previous paryid consequently, the net compres- t pressure builds up as the filter med* U approached, thus accounting for ^decreased porosity. | w the entrance to the cake, both the | tf^$ore on the liquid Pm and the pres- | on the solids are equal to the applied tltJsuit F.t However, the applied presJ^does not represent the net compres- <rt pressure on the solids. Grace (5) I pi Roth (9) have assumed that there [ pxnt contact between the particles as pasted in Figure 9 which illustrates I ^ conditions at the surface. The presL^e P at the surface of the solids is | (gaiterbalanced by a pressure P + dP, I ltd the net increase in compressive presre on the first layer of solids is I i-dP,)- Thus, as the differential presI drop is negligible for die surface r of the cake, the net compressive n^sure P$ at the surface is essentially ttfo regardless of the applied pressure. tV build-up of compressive pressure ftfoufbout the cake will be in accord vilfa the relation The pressure on the solids can be eliminated from (7) to give a direct relationship between the liquid pressure and the porosity. The elimination of P, can be accomplished either with the ex perimental porosity vs. pressure data or by means of Equation (4). If Equation (4) is used, there results (in terms of lb./sq.in.) p-t.= or solving for = t.(p-^.)- (8) (9) The slope of the vs. p9 curve is given by rf = -c*.(P-Ps) --1 (10) The variation of with p0 throughout a cake for various filtration pressures for the kaolin of Table 1 is illustrated in Figure 10. Equation (10) indicates that the slope of the curves depends only on the porosity and not pm and this is borne out by the curves in Figure 10. Even at the lowest pressure, the porosity if, * -dPm (6) bkh is equivalent to \r,+P* = P (7) I Equation (6) is predicated on the basis : point contact and would have to be t sodified if there were area contact beI -veen particles. Equation (7) is equiva| rst to writing a force balance stating Iret at surface = force on solids + force on liquid [ It should be noted that pressures of ^nation f7) are computed on the basis J -t the total cross section and not upon IU area of the voids or solids. When the liquid enters the cake P = |7,, and from (7) it can be seen that I * 0. As decreases, Pt increases; I nd if the liquid leaves at atmospheric I * ttro gauge pressure, P, will increase | vP. 'WW fattart P, P,, end P. wiO bo owd far la lh./q.fLf well lofton P* 4 p. vitt be Med who* h mbcr/1 ** *>. No. 9 Chemical Engineering Progress varies from 51.5 to 57.5%, which accord ing to Koceny`s law would give a two fold change in filtration resistance. Sev eral authors (7, 6, 72) have determined porosities under flow conditions without investigating the variations in voids throughout the bed. Since the porosity does vary, sampling procedures should be such that a true average value of If consistent units, i.e., pound mass, poundai, foot, second; or slug, pound force, foot, second, are used, k has the value of 5?r However, if inconsistent units are used with the pressure being pounds force/sq.ft and ft being lb.m./ (ft)(sec.), k has approximately the value of 5.0. In this paper, P, is used in lb.f./sqit and k is taken as 5.0. Carman differential equation in P, and x. Meth ods of this paper will be devoted to th graphical integrations of the equation: In assuming Equation (4) to be vtlu the assumption is made that the porosit depends only on the pressure on th solids. This is equivalent to assunrin that the equilibrium porosity is attaine immediately and is contrary to the fac* moisture content is determined. At low investigated Equation (16) for values of as illustrated in Figure 4. When th pressures the porosity may change quite S0 ranging from 1,800 to 600,000 sq.ft/ pressure is built up slowly as in ind^ rapidly with materials like asbestos and cu.ft. of solid. trial constant rate filtrations, it is pro) kaolin, and low heads are no guarantee For round or square particles and for able that the time effect on consolidate of a negligible variation in porosity. cylindrical particles with length equal to is unimportant However, if the rate diameter, has the value of 6/D. For quite high as exemplified by many laboi Keioajr Equation The Kozeny equation (3) relating rate of flow through a bed of solids can be derived from the following modified form of the PoiseviUc equation. cylindrical particles with length twice the diameter, St = 6/D. For irregular par ticles, Dallavalle (2) reports that in general the surface area is given by a constant divided by an average diameter. For rounded particles the constant was atory investigations, the time effect pro! ably cannot be neglected. In constar pressure filtration, the initial depositic of solids at a high rate may represer a case in which the time effect is impotant Both the nature of the solids ar dP. *'-dT l>7u (ID 6.1; for subrounded particles, 6.4; for sharp particles, 7.0; and for angular par ticles, 73. the total filtration time will determir the importance of the transitory peric in which the porosity is a function c where k is a dimensionless constant, u the lineal velocity in ft/sec., ft the vis cosity in lb.m./(fL)(sec); Re the hy draulic radius, Pa the pressure in Ib.f./ sq.ft, gj*,, is the pressure m lbl./sqit, and x is the distance. The hydraulic radius is given by Ruth (6, 7) has indicated that the Kozeny equation should be modified by replacing by (r--r,) where q is i constant representing a portion of the voids unavailable for flow, termed the dead volume. Grace (5) has indicated that Rath's dead volume hypothesis is not valid and that in general Kozeny's time as well as pressure. Suppose a liquid flows into a rectanf ular bed of fixed thickness (Fig. ; under an applied pressure P, while at tl opposite side the liquid leaves at pressui Pi which is necessary to overcome tl resistance of -the septum. At the poii where x * 0, the liquid pressure is i equation is inadequate. Grace also has and at the other side where x L. tl ( flow area \/ length of path \ ^ wetted perimeter /\ length of path / shown that the value of k in Equation (16) is not constant and has investigated its variation with porosity. No attempt value of Pm is Pv The Kozeny equation may be rea ranged for integration to give void volume surface of solids (12) Since the value /(l --) represents the ratio of the void volume to the volume of the solids, the void volume is. given by void volume =------(volume of solids) (13) Substituting (13) in (12) yields has been made in this paper to evaluate either the validity of Ruth's modification or Grace's experimental data, and calcu lations have been made directly with the unmodified Kozeny equation. The meth ods fundamentally would be the same regardless of the functional nature of the right-hand side of Equation (16). Plow Thiweeb a Pfxod Sod The fundamental problem involved in the flow of a liquid through a fixed bed 4 O-O* a; Reversing the limits on the right-har side to eliminate the negative sign ar integrating the left-hand side yields dP. (W 9* ________(volume of solids) 8 (1 -- t) (surface area of solids) of solids revolves about the calculation of the pressure as a function of the dis tance. The use of Equations (7) and where the right-hand side must 1 evaluated numerically. The value / (16) in conjunction with either Equa represents the pressure at which tl ~ (14) tion (4) or actual porosity vs. solid liquid leaves the solid at x * L, and eo The surface area of the solids per cubic foot of solids (exclusive of voids) is termed the specific surface, St which can pressure data permits the calculation of Pa vs. x. Variables involved may be summarized as follows: responds to the pressure required overcome the resistance of the septa If the upper limits in (77) are x and i be substituted in Equation (14) to yield . _1_ Ra 1 -i S, (IS) EQUATION DESCUrnON VAUASIES Koiony Sow rote..................................................... P-- , a Solid and Dqvlrf procure* ............................ .... Pm, P, CONSTANTS q, k, ft, S, P EQ. MC (I < The true velocity u in the interstices of Pom**? m solid pnmvn .......... t, P, r, i 1 the solids is given by the superficial ve locity f cuit/(see.)(sqit) or ft/see. There are four variables Pm , and corresponding to some arbitrary point divided by the per cent voids, i.e., x from which it is desired to eliminate the bed of solids, integration produces * B q/o. Making substitutions in the modified Poiseuille equation leads to dPm _____ *_ (1 ix ~ g, * (16) P, and . Theoretically, P, should be eliminated between (4) and (7) to give a relation between < and Pm similar to (9). The porosity should then be elim inated between (9) and (1$) to give a W* - fP * am (I! ST0722792 Poge 472 Chemical Engineering Progress September, 19 ' '* by myjSI mecfle, 1. In , itial i may r*} Igt the i will < ansitory^ [ i pot iqvare Inch) (Mr t^von IkK) ..A'-"' 1 ,4p. i* 0 too 0.575 14)52 10 90 0435 07)5 20 10 0J2O 0.510 20 70 0412 0454 40 50 0405 042* 50 40 0.499 0.495 60 40 0.494 0.47! 70 20 0.4*9 0.440 00 20 0.405 0.430 90 !0 0.40! 0413 100 0 0.475 0492 0 t41 14.15 2072 25.17 3177 35.10 40.59 4540 4940 5242 0 0.155 0.271 0.209 0490 0.599 0.676 0.755 0.045 0.915 1400 30| -- * * bJ ,u/^ K UJ IMITI SLOP r / J (&*3*< fu !ty .a: ' . : v 20 40 60 APPLIED- PRESSUR^J?^ Fig. 11. Uft* Variation gf prawn in o Mar ^o. Fig. 12. Abo*o, Rota f Row w. oppKod piawiro. I then* rbitraiy J idoo | iPiroling (18) by (19) gives ~P ydP, T = JJfr~ I ydP, (20> Jhcre y * */(l -- )* Equation (20) I -atet that the value of Pa for a gives J i^lied pressure P depends only on xfL I nd not on q, p, or 5r __ < Is order to solve Equations (18)-(2Q) ' is oecessary to relate to Pa as was Imioosly done in Figure 10. With this 1 agination it is possible to obtain the litbe Of the Kozeny function */ J'l-c)8 for each P, and numerically to |e|rate Equations (18)-(20). tXAMFtl 1. Swppoto HeId k to how |Ski bod of Dm boon* to which a w |* plotted la Ftgwro 10. CobvkritoM for | ft* p at i/l rofottodup of tOO lb/ |*W or* km l Tobb 3. * pita of tba dot* ! Tobb 3 to oo oppttad if 100 Ib/ipk. It ibra la Ftgwra 11 l^f Mi cvrvoa for 25, 50, oad 75 I hfc okvcm d* tot daport groofty tool I **ibt Raw, there k doOoito cwrvatvro. Om tbe 100 lb./aq.la. Cwrvo, tho ibpo at x/L * 141 ti obota 2.4 rime* tbo slope f i/i 0. Qorto af Flaw vs. Applied Pnsiuro Of more interest than the pressure drop throughout a fixed bed if the varia tion of the rate of flow with the applied pressure. Consider a rectangular solid of constant thickness L in which the liquid flows in one surface and out the opposite. Suppose the total masj of dry solid is ter lb./$q.ft. If < represents the per cent voids at distance x below the surface and the true density is then the mass of solids per square foot in distance dr is given by dto* (density)(volume of solids) " Pt(l-*)dr (21) where (1 -- *)dx represents the true vol ume of the solids. Integrating from zero to L yields t)dx (22) The variable of integration in (22) can be changed from x to Pa with the aid of the Kozeny equation (16). The dif ferential dx is obtained from (16) and substituted into (22). The limits of in tegration are changed from L to Pi and O to P. Thus combining (16) and (22) and changing the limits of integration to avoid a negative sign leads to dPa (23) kpwSf jp (l-) The value of this integral can be found graphically in a manner similar to that used in Example 1. A plot of a relative rate kptvS^/PtSt ** ? P lor the solid of Example 1 is shown in Figure 12. It can be seen that the de crease In the porosity with increasing pressure has a pronounced effect on the fiow rate. Calculations show that the average resistance defined as F/$ at 100 lb./sq.in. is 1.5 times the resistance at 5 lb./sq.in. Tvr* ttDde lo torfoc If the liquid is flowing through two solids in series as would occur with two layers of graded sand or a precoat in filtration, Equation (23) must be modi fied. Suppose that the pressure drops V 49. No. 9 Chemical Engineering Progress Page,.473 from P to Px in the first solid and from tally, it is desired to obtain relationships <1 Pi to P2 in the second solid, where P% between the volume filtered, pressure, is now the pressure required to overcome and time. the septum resistance. Equations similar If the filtration is carried out at con to (23) can be set up for each solid sub* stant rate, the mass of material deposited ject to the condition that the flow rate per square foot w and the applied pres is the same through each. Thus sure P which is the upper limit of the 9 kpwtSx* (-*!> integrals in Equations (23) and (?') will vary together. As ter increases in Equation (23) or (24), the value of P must also increase if q remains constant If v is the volume of filtrate per square When does not vary too widely, Equ*. tioo (34) may be replaced by m= 1+ P* (35) ffcPt kpuifSi* dP, (24) where the subscripts 1 and 2 are used to denote the values of the parameters t, p, vo, and St for the two solids. foot, m the average ratio of the mass of wet to dry cake, and s the per cent solids in the slurry, then w = spv/(\ -- mj) = spq9/(l -- ms) (29) where 9 is the time and v -- q6. In the which is usually sufficiently accurate if an average value of for liquid pressure* between Px and P is used. For the case of constant porosity, tegration of (31) gives the usual simple form of the constant rate equation. Thus Theoretically, Pi can be eliminated from these two equations to give q as a func tion of the islet and outlet pressures P and Pf. If both solids were incompres sible and i and were constants, inte gration of (24) would produce _ ft i* g. ti>A (l-i) <P-Pi) = _?*__!*!_ (Pi-P,) A' ('-,) (25) Elimination of Px yields S.jP-Pt) H (,! +W*i) (26) where case of a dilute slurry where (1 -- ms) is approximately unity/ (29) reduces to XO = spC = spq$ (30) Combining (29) with (23) gives l -- ms w. (l-) dPs P\ (31) This equation represents the desired solution for constant rate filtration in which 9 is given as a function of the upper limit P. The lower limit Px is constant only when the septum resistance does not change, e.g., when the solids are laid down directly on something like a wire screen. If a precoat is involved, qH 1 -- MS 9* (36) where , the filtration resistance, 1* given by _ *(1-)S, (37) Equation (36) would only be valid for an incompressible solid in which the voids were independent of the pressure. SXAMne t. Suppew mOd fcmfcg tfco ma poratify chuiortnrhtki or Hw booth In 11 to faa Ritorod te a and wtt MpKglblo ophQ retbtanco at a cmPMt roto W OJ gd/lahj (iqAJ. FwAar twppoto that tho ihoy fa Afa end uwitoht 04% mW* by wolght. Olko poraoMton hovo the nbw the changing pressure P would cause the Pl*l and _ *(l-sW " -------W------ (27) (28) precoat resistance to increase, and Px would not be constant. If Px is small, slight changes would have little effect on the resnlts. If s is not small enough so that ms may be neglected, m must be obtained p =s S2A p, 1244 lb S, = 2,000,000 teA/cuA. f = mh. = 4/SO ,, Vim S3 0.0QS, m wBI ba mmB, end (1 -- The quantities t and oj are the re sistances per pound mass per square foot of solid. If the porosities are not constant, it is necessary to perform die integrals in (24) by numerical methods and adjust the value of Px so that the by an integration. Thus mass wet cake mass dry cake ' (32) nay ba taboo a* volty. Chwiibn the Me ban por iwm * commoot mm pw q mho of (0J)/<toK7.4t> - 040111 aJk/ (mcXmA). Sufaufrwdnp ifco vrivaa la (31) ytakh (Q.005X424X0.000473)t3)(4)00^{OJ01 13ft4Wf <1*44X32.0 equality is realised. CoMUHl Soto nftrttioi In the previous sections the analysis has been simplified in that the thickness and the mass of cake have not changed with time. In a filtration process, the thickness changes with time; and the equations must be modified. Fundamen (33) Substituting for dx as obtained from Kozeny's equation yields In (31) P h In pound* por tqimr* foa< ato lag It poandt por tqvaro inch bodt to GO * pt Pm * J --4 0 8475 100 0-44S 10 MM 0 0431 30 0430 so 0493 30 04)3 70 0475 ToMo 0 40 0405 0499 10 50 0440 0441 40 0494 40 0433 70 0459 30 0435 50 20 0422 90 0401 10 0413 "V 04to . Page 474 Chemical Engineering Progross September, 1W#. ft if* CO In terms of *A and t as defined by Equations (27) and (28), Equation (40) may be transformed into q = 1 ~ ms ft(p ~ ^l) - ft(*N - pt) Spq$ Oj OgfOt (41) Elimination of F, and substitution of pRmq for g^Pj yields geP-li{Rm-i-atat)q = | (42) The solution of Equation (39) in the most general case requires the numerical determination of Px and 9 for each ap plied pressure P. The determination of the exact solution it rather tedious unless the simplifying assumption is made that the porosity of the precoat is constant. If Pi is not large, the variation of Pt and] through the precoat wilt be small. The porosity wiQ be at its minimum value at the aide where the liquid exits. As a satisfactory approximation the value of <| may be taken as constant at its minimum value and can be calculated on the basis of the maximum value of Pr At the side where the filtrate leaves the precoat and ft = ft p, is at a maxi mum and is given tty P*=P~Pt (43) The value of s corresponding to this value of ft may be calculated by using Equation (4) to #*cb mppUtd pfwin p, Uto latognd aw# to ndaatod wwkafly to (tog lb* Ifaao, la i iiW* Mm Bolndartati Wwitratod far f - 100 Ito/^Ja. The integral corresponding to Equa tion (31) is found by calculating the nea under the curve of */(! -- ) plotted against pm The graphs of */ H-) vs. p,, for applied pressures ranging from 5-100 Ib./qtin. are demonrrafed in Figure 13. The area under the : cane for 100 tb./sq.in. is found to be 3539 Ib./sqjn. or 3800 lb-/sq.ft. The ime required to reach 100 lb./sq.tn. is ; obtained from Equation (39) and is ! f (1.85)(26.15) = 48.3 min. InFig*e 14, the applied pressure is plotted ; sgtinst the time. The curve for Extnple 2 is typical of the type encountered . h the constant rate filtration of a com1 Possible material btoNet Qto PtHreMea wftb a Proton! Equation (24) for two fixed beds of "Wo in series can be modified to give fie equations for constant rate filtration a precoat The mass of solids per square foot te of the precoat will remain constant while Wi will change with the time in accord with Equation (29). Solving for ta in (29) and substituting the result in (24) in the place of tv. gives L kuspqOSf dP, 9efii kftWfSt1 'Pi dP, (39) The value of Pt win equal i*Rmq/gc if the septum resistance does not change and the flow is viscous, while Px will in crease as the filtration process continues. For incompressible solids (39) becomes on integration kfuj _ (1 -mj)^__ j (P-Pi) a. infs? (l -i) WlV (i - ) (P, - p,) (40) = <.(P-P*)~e (44) or it may be obtained directly from the experimental data. Since * is being assumed constant, the value of Px may be determined from Equation (41), thus Px a Pt + Q -H)V. to*1 = + ft (45) Values of a and a, will change during the course of the filtration. As the filtra tion pressure P builds up, the value of s will decrease and j will increase. With the values of P and /*, fixed. 9 is found to be (1 - mr) g<fix 9 liipq* kSx* dP, <l-i) Pi (46) EXAMPlf 3. Swppaw a liiwy praaoat af 1.4 fWtoJt. af a tttar aid wbb rito faflawfeg baf ctofifMa k laid daw* paa a uptow witI* nogUaiUo rotitoaacn 23 St a 1300300 qA./ca.ft. . -- ass Fvrtbar wpptn tba! tba watorial bwapfe 2 (i bafaq Mtorad aador t*a fallawtoa caatfMaat l i i i > i *1 49, No. 9 Chamlcol Engl J< Page 475 whldi arc n-->ploi wirtt tb*o of 4m proriovt p^r. TO S. 3 2400400 KjJt./cu.ft. I 3 0009 1-w = 14 (pprMlHwMy) 1 a 04 eol./a(n.Aiq.ft.) # 3 (2.4 pi = 0 #* 3 min. On dtoa^Mg to Ib./tqJn. end nUnutoc, Equation (39) btwoi .00103 / .' ~X n-* dp* 3 0400197--!!------f (l-*i> M7) aharq pikh pound* par quart Inch and q b * BquaHawi (49) and (44) bocowo p, - <o4oo672)(5kiqmxo4oii 1 -*i (3i2Xt3X^KI44) ^ - 5.4*-!^ * (4*) to reach a given pressure will be smaller with the added resistance of the precoat. In order to calculate the value of the integral in- Equation (46), it is 6rst necessary to obtain the values of px for different values of the applied pressure p. Since Pt is zero in this eximple p4 will equal p at the outlet, and a can be calculated from the relation f 3 0.85 P$~9M* ~ 0.85 p-s*w. With 3, Pi can be obtained from Equation (4) u illus trated in Table 5. Examination of the first two columns in Table 5 reveals the fact that p is less than px. Since p is the applied pressure, it must be greater than or equal to px. At p * 1.46 lh./sq.iiu, both p and p% are equal, and the time is zero. There fore 1.46 Ib./sqJn. represents the initial pressure necessary to overcome the re* sistacce of the precoat In Figure 14 the variation of px with time is shown. When the total pressure had increased to 100 Ib./*q.ia, px had changed to only 2.37 Ib./sq.in. Actually with a smaller and more reasonable quantity of precoat the values of px would have been smaller. The applied pressure for the filtration of the solid with a precoat is plotted against the time in Figure 14 and can be compared with the same filtration without a precoat Although px is actually a variable, tittle error would have ensued from having assumed px constant and equal to about 2j0 Ib./sq.in. the integral P will remain constaaj while dv/di and Px will both vary. T*t cases present themselves depending upoc whether the septum resistance is cotj. slant or not In the simpler case of coq. slant septum resistance, Px may be re placed by (51) where Pm is the septum resistance am viscous flow through the septum is jj. sumed. If is constant, on integratioc Equation (50) becomes dv 9t(l-ur) Hi 1 p-px _ 1 _ dv (52) where a is defined by Equation (33) Integration of (52) leads to the usna parabolic relation of v to 9 expressed u apjL tp gc(\-ms)~2 9 = P$ (*> If is not constant, a graphical inte gration must be carried out to obtain tb> v vs. 9 relationship. It is possible a solve for the volume in Equation (50] to obtain c f" - u 3 dv/it /, ..(I-.) OP. The value of ? in minutes correspond* uig to the applied pressure P in pounds per square inch as givea by (49) can be obtained by graphical integration of iV(* -- 4i) between px and p. Since the solid being filtered is the same as that used in Example 2, the value of iV(l -- l) will be the same for equal values of pm and the curves of Figure 13 may be employed for obtaining the values of the integrals. In Example 2, where there was no precoat and neg ligible septum resistance, the integration was carried out with the lower variable of integration in (49) equal to zero. In this example, the lower limit of integra tion will not be tero, and the numerical value of the integral in this example will be smaller for a given applied pressure p than in the previous example. This amounts to saying that die time required CaDoloat Prusere flitwittoo (54) If the applied pressure P remains con stant, the rate of filtration will decrease as the cake thickness builds up. Equa tion (25) can he adapted to a constant pressure filtration if qd is replaced by t1 and q by dv/d$. Thus xpft kSj* dp vdv (1 -- ms) d9 Cd9 where C represents the combined value of the constants on the left-hand side of the equation. Conditions for solving (50) will differ from those of constant rate filtration in that the upper limit of Since P is fixed, the value of the Integra will depend upon the lower limit o* dv/d$. Thus, the integral is a functia of dv/di. It is possible to choose valne of dv/di and then calculate v. The vain of 9 then can be obtained by a secon . numerical integration as illustrated g the following example. *.J EXAM91C 4. Tho ipluttow al Cquatioo (3Si wilt be carried out for tho mm toUd d fc Eweplw 2 ood X Suppotc to uppllod prwuri !| 100 fe./c*fck, toe product pJU/p. = (10400) (144) or ttmptf 10400 wton dto prcowwi 1* ia poaodc par tqoaro loch and C 3 1.1S fa ardor la cafwe (tot prabfee, wW valwac el dr/d# will ba cbocon, ton Osina t and Hto &mAi ea (54). Tha taUM rata of flb* Hon h tuck Htot aO Hm proctor* drop a ocm p. cppBod procure c* vo - * pv f ' - in 045 44? 1.39 B (1 - A) Jpt r.mhu 14 0.149 4.04 \JO 1.44 0.149 3.90 1.44 0 0 ToMo J 9 0.129 333 1JC 10 0.131 3.09 143 137 9.14 244 MS 29 o.ati 212 24)1 7J1 14.3 90 0J02 >41 2.17 1444 164 75 0497 240 244 2047 374 too 0J93 2.24 247 4! 2544J 474 Pago 476 Chemical Engineering Progress September, 195 -P, /?* a graphic out to G It is i EqnatiS and = p = 100 lb./M|Ja. Tig* Tfclo 0 100 . : 041 ft/H ?0400 {IS) I jjH to* ra*3 Caftoipawdi to 4J I V f b (1w la sliNrtM mi p. to ptt qvor Inch (54) fat--tat J/MOO 10400 4** 1 ~* 0^1 * (T44)(60)(1J5K10^). Cvkuto- I fa afaltfntng r va, dr/d# from (54) arc ' ^ in TaUa 4. TW 100 Ik/iqJ. wr*s for .01 fr./mW 0.4 h 100 1.^ 0 r r (*u.) 0 0 .0095 044 95 2.013 0.359 0.4 409 047 90 177 0.498 U 4075 0*45 75 141 1.(41 34 405 040 50 I4.M 4.94 1X4 .0025 0.15 25 20.77 13.(3 57.4 | . | -- *) P* 1* Rf*f* D be mod for ' ^trAn Hm ktogra! of Equadoo (54). plots of df'Jdv vt pu well u lines | < ^psunt ft are shown in Figures 15 ' Zi 16. It can he seen in Figure 15 | 2tjie lines of dl'/rfev*.* are virtually [ ^[ight, a fact that has frequently been I Zgcostrated experimentally. In ana- I pied form the straight lines in Figure i gay be represented by **-Lv + t *K (57) | ,hicb integrates into 1$ a o* 4- Kbv (58) Elation (58) is simply the equation of I ( ell~known filtration parabola. Thus I fa graphical procedure of Example 4 ' fa i compressible solid eonfinns the fact j dot constant pressure filtration data can k accurately represented by a parabola. Acre is, however, a deviation from the I piabolic form at the start of the filtra| ms where v is zero. Is Figure 16 the line for 10 Ib./sq.in. | a shown on an enlarged scale in order tfustrate the deviation from linearity in the dft/dv relationship at the begin ning of the filtration. The line which is indicated as coming from Figure 15 represents the straight line through the points for times greater than 20 min. It is apparent that for the points up to 20 min. a line of smaller slope could be constructed. Since the slope is smaller at die beginning of the filtration, the re sistance must increase as the filtration proceeds. An average slope up to 20 min. is 25.5 nun./(ft)c; whereas, the final limiting value of the slope is 2551. The value of 25.9 corresponds to a filtra tion resistance of 4.16(10u) ft/lb. mass. The initial slope and filtration resis tance cannot be calculated with certainty as can be seen from Figure 17. In this figure, the Kozeny function */(l --) is plotted against the liquid pressure for an applied pressure of 10 lb./sq.in. The cross-hatched area under the curve be tween 6.0 and 10 lb./sq.in. represents the value of the integral in Equation (54). In this region the liquid pressure falls from 10 to 6.0 tb./sq.in. in the cake and from 6.0 to zero across the filter medium. The area under the curve can not be determined with certainty in the portion of the curve between 8J8 and 10 lb./sq.in. since an extrapolation of the experimental data is necessary as in dicated by the dotted extension of the curve. The porosity data were taken from Table 1 with an initial value of ft * M2(*. =* 10- 1.42 8.581b./sq. in.) and an initial porosity of 0.574 cor responding to V(1 -- ) * 0.444. Ex trapolation to f, =0 leads to a value of */(l -- i) = 0.508 for which s 0.592. In this region of low compressive pres sure, the solids may be quite plastic; and it is difficult to obtain the porosity-pres sure relations with accuracy. Therefore, there is an initial region of uncertainty in evaluation of the volume-time rela tionship. As time progresses, the rela tive effect of this region becomes smaller. The double-hatched region from 8.58 to 10 lb./iq.m. has an area of 0.045 while the total area from zero to 10 Ib./sqin. is 3.86 lb./sq.in. Thus the uncertainty becomes quite small as the volume of filtrate increases. The initial filtration resistance can be calculated from Equation (37) using the initial value of e*/(l -- *), tiro* _ *(1-W _ (S)(4)(10) ** (124.8X0.508) = 3.16(1011)ft./lb.m. (59) The average filtration resistance at any time can be written as _ *5.. .----f----------------------------------- Septemb ** 4, No. ? (60) The ultimate resistance approached as dv/dQ becomes aero ia _ *v (61) (1-.) iP. Chemical Engineering Progress Page 477 For the 10 lb./sq.in. run just considered. The right-hand side of Equation (63) is the maximum pressure is applied to th. Equation (61) becomes a constant for a given applied pressure precoat Initially then . ,, (5)(4><10)(1Q) P (124.8) (3.86) P. Thus integration of (63) yields an equation of the form kfi dv ~g7~df = 4.16 (I0")fL/lb.mass (62) This is equal to the value previously found from drawing a tangent through the dtf/dv vs. p line. From these calcu lations it can be seen that the filtration resistance increases approximately 30% above its initial value. Nevertheless, the ultimate value is approached quite rap idly, the average resistance at 20 minutes being within 1.5% of the maximum value. It should be recalled that in the initial stage of the filtration, equilibrium consolidation may not be reached. Since the porosity decreases with time, the initial porosities would be larger than the equilibrium values as reported in Table 1, thus leading to smaller initial resistances than those which have been calculated. From this standpoint, the filtration resistance would be expected to increase more than 30% above its initial value. Constant Ptrasaaro PiltiwHoo vritfe KoeHgiblo Soptvia BosfstaMO The previous method may be simpli fied somewhat if Rm is taken as zero. If Rm is zero, the value of Px representing the lower limit of Equations (50), (54), and (56) will be zero. Thus Equation (50) becomes *-JW (64) where the if is identical with the con stant in Equation (58). From (64) it can be seen that v and 9 are related by a perfect parabola in contrast to the prev ious case with a finite septum resistance. Again (64) is subject to the limitation of the time effect on consolidation. The filtration resistance found from (63) is not only constant but is also equal to the limiting value given in Equation (61). Thus, as the filter time is increased, the resistance of the medium has a lessening effect on the filtration resistance. Caotlaa* Prowwro PIKretko with Proteat The equation for constant rate filtra tion with a precoat can easily be modi fied for a constant pressure filtration. In Equation (39), q is replaced by dv/d$ and q& by v, thus ip jh 9, *9 = --?-- f * dr "A1 I, , ..(!-,) * () To start the solution, a value of Avfty must be* found which wiU satisfy tfjtj equality. Smaller values of dv/ds chosen, thereby fixing the left-hand of (65) and the value of Pv In die ne*t step Fj which replaces P in (66) ^ determined. With Px known, the inte. gral of the center member of Equation (65) may be calculated along with values of v. After v has been found as a function of dv/d$, the values of $ may be determined by graphical Integra, tion. J Achnewledgmeot This work was done in connection with a research project sponsored by .the Niagara Filter Corporation, Buffalo, N. Y. The author wishes to express lfe appreciation to E. A. Ulrich of that Corporation for his cooperation. The fundamental ideas underlying this paper rest in part on the work of the late R. S. Tour of the University of Cincinnati. B. F. Ruth and H. P. Grace have given a number <4 helpful dv suggestions. The author wishes to ex press his appreciation also to J. 14. With the applied pressure P fixed and Dallavalle of the Georgia Institute of 1 -- tm gJP% * the problem rests in Technology for comments on the Rozcsy spfk kS,* (1-0 finding values of Px for various values law. Tom Pickel of Humble Oil Co. and of v. At the start of the filtration proc Leo Goldschiag helped obtain the por (63) ess, the value of Px is equal to P, and osity data. CcrO> " CM CM a go : u 'pOQK ation, $ to ex& UlriehTjl pend eu on the i the Ud lUth ; mber erf A(J--d; 4. fair, G. ML, and U P. Hatch, J. Am. Water Host of the work on sands, days, and simi Work* Aieet, 28, 1591 (1733). lar materials, has been done by men in the *qA. 0 fctortopS so Hto grtff of d4/dr vt v, reciprocal of Ikg labial ralo of Idrafloa* toe. or *in./ft. 0 oxponoat occvrrtog to tqvatfoa (4) 0 con***"* dofiood bp fqvottoa (50) 0 contoot In Equofloa (5) 0 particle diameter, ft. a doflood bp Jqwttec (23), kpwSf/ f,g. cM.ft./(wc)(tb.f.) b convorMon factor fro* U>.f. t IbL, nuworicol oqwivaUnt of the don* dard accoloratiaa of grovftp, 111 JbL/lbJ. 0 proportionality canOon* in the Kazoo? oq^ approximately oqwol to 5J0 0 fwtoo tfcc reciprocal of the dope of df/rfr n. v, toe. or ioin./qA a coho thichnoM, ft. * ratio of M0 of eat to tea* of dry csko s appHod proMoro, IM./rqJn. c applied prorwre, lb.f./sqJo. K* a pro*ore at Squid oxlt fro* roBd, fc.f./q.ft. s prOMtire of Kquirf oxft fro* tofld, 5. Groce. H. tH Personal cowawnication (1792). 4. HoAng, f. H, and P. J. Lockhart, Cho*. Cap. Progrow, 47, 1 (1751) 7. Undqokt, C O- and J. J. Yoelo, RHraHo* Syntposiv* of the Am. Ind. Cho*. fapn* Cleveland (Paeonbor, 1952). 8. tvth, t. fH iod. fog. Chon., 98. 544 0944). 9. Ruth, I. 9,, Portenoi connwokotien (1952). 10. Tenoghl, It, end ft. t Pock, "Soil Mochan* ks In Engfaworiog Proctko," John Wiloy, New York (1948). 11. Tewr, ft. 1, M5. Thesis of Robert Stewart, University of Cincinnati (1937). 12. Wolat, 5. Mw Trane. Am. fad. Che*, fngr**, 42, 783 (1744). W. H. Elliott, (National Lead Co., Fairhaven, N. J.): In using the Koaeny equa tion we assume that we have streamlined flow. Is it possible ever to form a bed of solids or a filter cake in which the flow of fluid through the cake is not streamlined and the equation is not applicable? soil mechanics field. As to whether sand is compressible or not depends largely upon partide size. Many fine sands are compres sible. In fact the differentiation between day and sand is primarily based on partide size. W. O. Webber (Humble Oil & Refining Co., Baytown, Tex.) : In these derivations porosity is assumed to be a single valued function of pressure; that is, the time re quired to reach equilibrium is assumed to be quite short. In other words, one can see that even with a cake, possessing a signifi cant variation of porosity with pressure, as the cake is laying itself down on the bed and approaching its equilibrium porosity, an amount of time would be required that is significant with respect to the amount of filtrate flowing. Presumably this factor would increase the effective porosity of the cake. Have you considered tune lag and eliminated it as being an insignificant item ? 9. M. Tillers No, we certainly haven't eliminated it as an insignificant item. You'll find that there is a general assumption here that the equilibrium porosity is obtained in Jb.f./rq.iA. * prouvro on Dqvid, Ihf/iqft, * proMoro oa BquW, ib.f./tqin. s pronwro on mRc^ lb.f./tq.ft. s proMoro on teUdt, Ib.f./tqjn. : rate of filtration, ev.ft./(qA){*oc.) or superficial velocity ft./toc J rosbteMO of fW mptwm, f/ft r hydrouik rorfWs, ft. 1 per cent wlidl io dwrry LSwfc s terfoco oroo/cu.ft. of retid notorial tq.fc./cA = overage velocity of filtrate in veldt of filter coho, ft./cot s reknoo of fihrate/iq.ft^ coA/tq.ft. t total not of cafco, IbJO./iqJ*. trance to coho, ft. ` velwo of */{1 -- if i per cont aoictere P. M. TJMon That is quite possible. In the equations which I showed, I factored out the viscosity term as 1/p, indicating that the velocity was inversely proportional to the viscosity; if we had turbulent flow, the vis cosity would be much less important in cal culating the resistance which would be pro portional to about 02 power of the viscosity. We would find also that the velocity would be proportional to the square root of the pressure drop. There is no reason that we couldn't modify the equations for turbulent Aiwaywevat Have you ever found any data showing application of the Kozeny equation for highly viscous materials passing through porous beds--materials having a viscosity of 1,000 poises with high pressure drop? stantaneously, sod that is not true. Figure 4 shows the variation of porosity with time. For materials like caldum carbonate, filter aids, etc., you'll find h takes about ten min utes to reach an initial equilibrium after which a gradual creep continues for quite some time. With materials like natural clays, it may take 24 hr. to reach the initial equilibrium with the creep going on contin ually. Actually, the equations presented at the present time, represent the first approxi mation, and we can hope that the next set of equations win include the time effect. There are not many data on the time effect It is important practically inasmuch as the porosity is at a maximum when the bed is laid down and the resistance is at a mini mum. In the initial states, you may have a smaller resistance than yon will have after the equilibrium porosity has been reached. : filtration reslvtoncet, ft./lbJO. : fibrotlon recktanoo during coa prusswro fUtrotion, ft./Tbj*. u a dood vokrwo * por cont voldc at one pound per square inch, constant bi Eq. (4) p = deadly of Dqvld, (h./cu.N. -Kpt = tree density of eottdf, lb./coJl X * vhceefty, tkje,/(ft.X0 4 a Hm nf filtration, soc. 1 = Hew of filtration, win. tesetoro cited Cerann, t. C, Trane, fed. Chow. fog**. Omdon). 14, 148 (1938). [ l Mmh. J. M. "Mia--H*V M oi. P- 49, Pttnoo Pek Carp* Now York . 0*48). [ 1 Oolavefto, J. M- Porwnal cawmonkntioo 11951). f. M. Tt&ori That represents quite a diffi cult problem. In the first place, many of these liquids with so high a viscosity, par ticularly like Du Pont viscose, may behave as non-Newtonian liquids. The laws gov erning the flow rates will not follow the ordinary Koseny cquatioa. That is a prob lem which I don't think has been completely solved. There are many other problems which are side products of the filtration of highly viscous materials. It is difficult to tell when the filtrate has been properly clarified and to evaluate filter aid character istics. . Awewyeasi Would you expect sand to compress under-pressures of possibly 3,000 or 5,000 lh./qJa? f. M. TUtoi Yes, you certainly would In fact, you'd probably have crushing of the particles with pressures of that magnitude. AiMaymoM] In the operation of a Schriver plate and frame-type press or a Sweetland leaf-type press, would your remarks about controlling the pressure of the liquid indi cate that longer operation of the filter with out cleaning the leaves or plate or more rapid filtration could be obtained by keeping a relatively high back pressure on the liquid leaving the unit? f. M. TtlUn The compressibility of the solids is dependent upon the pressure drop of the liquid through the solid. If there is point contact between particles, it will not make any difference ai far as the compres tive pressures are concerned, whether there is a pressure drop from 10 lb. to 0 lb. or 100 lb. to 90 lb. through the take. The resistance should essentially be the same. TrwtbS at Aj.ChJ. Itrtg-Jt/tS bmimm! miU*, OUvutead, OM. ptwnl 1,1 <*, No. 9 Choplcctl Engineering Progron Pago 479