I. The Permeability of a Bed of Smooth Spherical Particles. II. Non-Steady Flow of Gas through a Porous Wall

Citation

Green, Leon, Jr. (1950) I. The Permeability of a Bed of Smooth Spherical Particles. II. Non-Steady Flow of Gas through a Porous Wall. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/p2m3-ce79. https://resolver.caltech.edu/CaltechTHESIS:08132025-225352011

Abstract

Part I

The resistance of a bed of smooth spherical particles to gas flow in both Darcy and quasi-turbulent regimes was investigated experimentally in order to check the accuracy of two conflicting resistance equations widely reported in the literature. Variations in specific surface and porosity of the bed independent of one another were realized by the use of spheres of different sizes systematically packed in different configurations. The experimental results are correlated on the basis of a general pressure drop equation defining two length parameters which are necessary for complete characterization of an arbitrary porous structure. These results are used to extend the range of validity of the more accurate of the two conflicting sphere bed resistance expressions. A general definition of the Reynold number of a fluid flowing through a porous structure of arbitrary complexity is suggested, and the special conditions which permit a sphere bed to be characterized by a single length parameter are discussed.

Part II

The results of a calculation of the time required for establishment of steady flow of gas through a porous wall following the sudden application of a pressure difference across the wall are presented in dimensionless form. The calculation was made by using an electrical analogy to satisfy the non-linear partial differential equation governing the pressure distribution in the wall.

Item Type:	Thesis (Dissertation (Ph.D.))
Subject Keywords:	(Mechanical Engineering and Aeronautics)
Degree Grantor:	California Institute of Technology
Division:	Engineering and Applied Science
Major Option:	Mechanical Engineering
Minor Option:	Aeronautics
Thesis Availability:	Public (worldwide access)
Research Advisor(s):	Duwez, Pol E.
Thesis Committee:	Unknown, Unknown
Defense Date:	1949
Record Number:	CaltechTHESIS:08132025-225352011
Persistent URL:	https://resolver.caltech.edu/CaltechTHESIS:08132025-225352011
DOI:	10.7907/p2m3-ce79
Default Usage Policy:	No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:	17626
Collection:	CaltechTHESIS
Deposited By:	Benjamin Perez
Deposited On:	14 Aug 2025 18:10
Last Modified:	14 Aug 2025 18:10

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Thesis by Leon Greon• Jr. Fer the Dsg:roo 0£ Doc-co~ of Pl,J.losophy Ck11ifo1Yda Institute of Technology Pa.sooona., Cali£o:mia 1949 i t he m.r'chol" wia1bJ s to OX[Jl"OBS his doep appx-ooin'c:ton for ·bho edvloo Md 00.n::rta.;1-t; 0ncru..~er:ent gi~'l Ju.a by Proi'casor Pol !)u.lmz fl[Jl;1 in bo-;;h 'Ga'te flru•oy and qti!ll!li...tm:bulen·c Teg-li:oo waa iwre1:.rt:lgatod e.1r:p•.,ri,';l.-.m:l;ally in m:•dor t o che ck tho acourooy of two oonflicrt,ing 0 res-is-i';einoo oqum:Uona w_\c1cly i~oported in. 'bhe literntu.i"eo \fadatiooa :ln apoolf"'lc ou?'f\-\00 o.ncl porosity of ·ch0 bod irulopon.dC!.1·~ of ano an... o-chor WGro roalized by tho u.se oi' sp~10.res of d:1.ff'et--c.m t slzes syster.Ja:tically packed in dif'fol"ant con.t:igµz-9.tions. Tho o:iqx,r:b:1::n1tai. i-esu1:cs B.i'O oorx-ole!ood on tho baaio of n gono:-al p,~oasul"o drop oqu.atlon dofitting t"WO long·th pru:·ru,nto~s which az.•e :noooasru:"'J for complete ohall•aotoi"ize:tion af 11).![l a..""bita•ru.;7 ).)(~O'ilS otruc-tuse.. Thooo roeulte Vn"o usGd to ~00t"'ld the ro::ige of validity of tho m.oro O.CO'Jr~oo of' tho ·~70 coof'.U.c-tinr; spheiX:i l)ed I."0Erl.sta.noo 0:icpt'CH'ltdcrna. J. 60:aorlll def'initio.n 0£ ·tho TIO:/n.olds nu; 1iJO:r of a fluid flowing 0 tho s .t;: ocial eond:H;;ioi1.e vkdch porr.:'.l.i t a. sphoro b0d t o be chs.1..uct0 l'::i.~ed b;sr a single length para.m·tci:· are cl:i.aoutrned. Port II The rom.tl'ta of n oaloulo:tioo of t he tin1; roq1l11;•0,1 i"'or: es-ta'l.,lish= !!10nt cit: u'tO'O.C(:! .r:laur of eno -cl.rough e. porous ;;,roJ.1 fo1lmt.i:n.g the sudden a i;91:loation of: mo px·osmu·o d.i:Cf'o ronoo aoross tho i11;1.U :l!'Cl prosontod i n dir'Bns:lonleao f o:.."fLi;. 'l'ho eo.lcu.lttl.iion was ):'IMC by usinr; an el0ot1:-ioal aaa.lc~ ·to r; i1t i ~f"J th,(:) non-linear pai-tia.1 dii'.!'0rontial oquatio.,.'1. iii Pago I Introduction n l 4 7 9 16 16 B. D:taoussion of the I:qua·Mon 22 22 B. Inoort Dosign 24 26 V Procedure and f:esuH;s 27 27 Quad1~:U-e P,quation 51 iv ConcluaiG.is 36 !Jom:molature 37 Tables 41 I Il The Eleetr-loaJ. 1\no.loe;y ll! The Prooodu..re of SoluM.on 6'7 Hosults t\c'1d Cc~:iolusiooc 68 'lO Tables Figuroa '11 V PART I Page 41 1. Iloo-Sto&'iy Pressm"€l Distribution. in a Porous Vfe.11 71 vi Foodoroff 2. Idont:U:ico.tion of' t ho ?io...lee of: Sphere PMk:1.ng 44 s. Uluatrntion of l.o3lii.n.g Proood".u-@ 45 4. Top ,r1ew of Colu."ln Packed, wlth o.125- in~h Spheroe 6. View sh<ffl'l."'lg Datnils or Bed :Support Pla:-te Spheres 50 51 52 53 61 55 vii 14. C~a.:dsoo cf the Viaeous rtes1.sta.'loo 0oe£:f"1 ciant M a Fu.i-iction (>f Porof.l:1:cy with the ?radio-too Lela·tioosl">J.ps of Kotony an.d Balch1mte£'£...Foodo:rot'f 15. 2:'ho Vru."'ln.tion of tho Visoou.s a..".lCl ln0lrl;;ial I~sisto.."11.00 w·lth tho Speo:tfie Ul.U'i'noo of' the Bed 5'! at Constant Porosity 16. f'roaau.&-e• Squlll.I'o GI"ti,H.ont vs . ,-id.gh·c i?l::w, Hate fm.~ Spheres in C:1-ibio Pucking 17. '£he Entio of th0 Iri.ertil1.l I'.ssiir.;anc0 Cooi'f'lcient; to tho ViscouG Eesis-tnz"100 Ooof':flclont as a 59 Ii'wlction of the Spoc:i..:r-le 8u1•faoo oi' the Bod. 16,. Pl"essure- fiqmu·e Gi•oi!.iont vs. \JeigM; Flow F:.o:co (Data fr-om Othor i~porin::rnts) U1. l. • 2a s. (50 FI"'lC"cian Iractor vs . I;.e;/nolda Zluxiboi· fo'E' Va.1:--.i.<Yl.lB Sphere Beds . 61 Typical ItJah \:;lomnt foX> Dif'f'eronoo Y;quation 72 'l'ypioal Ifodc of an f.:l.ee,-tr:lci'!.l tlotwork 72 Ana.1. <,gmw Hle otrl ool nctvmrk Si'l:ti sf-yin~ the Giii'.!:n 73 D-1i'foronoo .P.quntioo 4. Dir,i ,msioulesa l?roseuro v.s . Diirons:i.onless 'l'iY"I(•~ a"'G Vt'l.dOtl.S 6. Positions, N a:> 2 Uimi:m.s:lonlom:i Presstiro ·vs. Uim:maionloss Tir:l;} Vaidous ?osit:'l.o:n.a., n~ 4 t,t 7fj viii Vru:i i,us Positions, i! aa 10 79 80 D1 I Appliocl Pressur0 to I.-:1:ttial Pressure PART I I ni'l'RODUCTIOiJ A t,X;)thod ot• eoping with the hiGl1 ro.toel of hoa.t fa•ru1.sf'er oncoun.. tex-ed in t ...¾.e i'ie 1d of je-t propulsion oonsista of making the parts to be cool/4~d of a porous mator:Lal and forcing ·the cc,oli:ng flu.i.d ;chroogh t he por0s in o. dirootion opposite to that of the heat flmir. !OClthod has oooon:a kncmn a.a 11 SV':ec;i.t eool1n.g 11 •. This (1). Perhapo t he nost mtl tablo l1J8tl100 for fabricating porous seotions fo-:r SVJ\9at oool.i ng a.pplioationo involves t ho uoo 0£ powd0E" ne·t.allurgy techniques . (2). The principle of the method 0£ preparing porous metals f or s-uch uao consists of adding ·to tho matnl p,mder a pmvdored substance that dooomposea into a gaseous ph.a0e at'i a 'temporature oolmr t he sinterin.g temporatu.re of the m-t:ml. The two powders are intina.tely l:ni:i,:0d.o prossed in a die in 01•doi- to .for.n e. eam.pao-'c, then sintered at high temperature . fr11.en the n<m•ril3tallic pOllider dacompoi10s, am¢;y apaces ru-e le.ft in the :tntorior of. the oomp-aet. S1noo the gasoa f.',-enerated by the daoomposition of: ·cha poroaity-i'orminp; material ?'Jl.1.ot esco.pe. small oha.r.nels a.re f'orm3d in the eor,lJ?ao-l; nnd tho pores are t h0rofo,"0 il1·001·connoctod. Althoi.1g-,h Som@ ot -these chn.."mela imy b0 cloaod during ·the sintering of tho m ·ca.l pm1deX", enough of them remain open to :i.•endo:t tho r.:w rtal penooable . In dosir;.ning n. e,:reat cooled part, it ie imperative to a.sau.r© a certnin rate of flow of ooolai"1t l..U'!d,Jr given condi t:i. ona of pressu.:re drop. Such di;Hdgn requires knowledge oi: the pennoo.b:U:l'by of! t he 100tal of whioh the p!ll"l:i is to be r:m.do ,. The oomploxit:r of the metal sti"l.lcrtuli'0 1 of oours0 11 prc~cludes 1,1.."ly wi.a.lytioe.J. approach to the problem of prod:lcJ~ing ·t..'10 pcni::ie:ab:U.:l-t.y. !n -the cou:i.·se o:e an o:x:peri• rxmto.J_ stud.y ot t ho f'lm7 0£ gas t .hrour,.h r.1:1.fi'ererrt po t'\1U. S ioot a.1 speeiroo,.7.0 ocmduoted a·b tho Jot Propulsion. l.~bore.tor-.t ( 2) _. it ,oms found -that for n given r.iatal prep$..red undeT a fixed set or conditiorto or oompao"l.;ion e..'1.d sintering., the pe1-meo.bility oooffic:tent (dof-lncd on t l10 hasia ot Darcy•a lmi1) is e. 'l.U'lique f.tmotion of t he r:l(\)tal porosity. .A more genora.l oorrela-cion of the me9.sured pornenbility values obtained from different meta.ls could not be realized, hrn°re,.,"er. Tho permco..bil:tty problem in t he ca.so of porous t1etnls of t he type desor-l'bed above is 00:r.1 ,plioated by the .fact t hat -tho mathod of mo.nu.f'o.oture produces oo.~ p-oro-sity which does not 0ffe3et po?'l'neability. '.fuo.t ia., not all of' t he pores or ohe.nnele are neoesaa:ril;? oontinuoua t h roughout the nstal speoimen; sonu my be r'dead ends n. As tho porosit:r of t he trr:>tal is inei--easedi the proportion of continuous or con:::w cted pc)res nloo inerea,ses. and so ·che pern,.eabilH,";/ :ts doubly sonsi tive to porosit-y ohangoe, often varying as rapidly as t i:,& ·tenth p01.1ro1q oi" 'che porosity. Ano-bher omr.plication ls i.'1t:i·oducod by tha fact t h!l\t Darey ' s law, whioh def"lnes the permeability ooe.ft.lcient, is valid only for viscous la..-nina.l' f'l <m. The 111.,1.;_ ting rnto o.1' £lor1 v.t whi oh the fl ov1 coaaea to be truly ,tiaeoua in na:ture CW'l be e.xpressod in the ooa.e of a porous imdiun.., o.s well na £01· pipes, by a. critionl Raynolds nw.uber. L"l t he case of porous iootala propnrod b;y• t ho toohnique d®scr:tbed above., h<ffl'9vor, it :Ls not possiblo by inap0otion to define the che.rt1.cteristic long·bl.1 ootering into t he Reynolds nu:nber expression. The t11ro complicating factors r:e n.tioned s.bove made tt desirable 5 the 1)(:lhavior of: t\ medium possessing both oontinuous po:roEJ:tt:r and a ;;,,--ell dofin0d charr.oto:dstic J.cmr,th, a n10diur.1 which mir,ht serve aa a basis for conparison . Th.o ~d.mplest such r,~rsten, a..'1d ono ,·1h:t ch .hru:l boon n;.i1i;,lec-'cec1 to considerable ~era.t iny by r. large !l'.X':lbc!' oi' invosti ... ea.tors., vro.s a bod of smooth spherl.oo.1 particles . :-::r.:rom.nnt:1.-0:i of tho litorature, hoWl~VBi~, roveaJ.ed inoonplete tmdorsta."1clinr, of t)ven this s;y-Gtom.. It v.-as .f.ou.~d t hat the gemeral topic of gi"armln:r beds ':,-as the subject of oonsiderablo con·trO"-rersz,; prims.r:i.l:r c01:cGr-t.d.11g tho effoot of bed porosity upon the reaiste..-ioo i,o i'1ow. T-1:0 major schools of t h•'.J',1sht, on thiu m,1bject eu·c extant, eaeh w:i:t;h its ovrn porosity .i'unot:Lan en.tpported by larr.;a a;'"loo.nts of ru:p-Dr:.i.m.o~tnl data. !n viaw of· thia sitnu-c:i..on, it wns f.elt t he.t f.t1rther 02,:pcrin;)ntal wor1: upon eph,:n•e bods ·wae dosirnble'~ A. The Pipe Flow l,nalog;;r The physical pr-lnoiplee r;ovorn:lng the f'l cnr of fluids th:rour,1 porotut ~:'ll:>dia tmro i'irnt inwstigatod b:;r Th1.rey (3) in J.G50. His exporimante on tho flew ,,f vm:ter through SR"lds established that th® rato of flow of e. liquid tbrruGh a u.,-1 tt o.ren or porottS material io d.5. root,ly ::>ropo1~~tono.l to the µressure drop par unit length M: the mater.ial. 'fhis stateniont, Jmown o.s DnrO";•rn law, ,raa ®baeqv.0nt1.y Shcmi1 to hold f.or the i'lOW of gUGS fil.R Woll 8.8 liquids, and was t'0tmd to be 8trictly valid only in th{'J rogit:l1l of flow in which pr@ssu.1•0 loasae oootll' as a result of v:l.se0t\S tangential shearing stressea arising in la."!tl.na.l" "creeping" motion of t-he fltd.do Th® similtu'ity of: Darcy's lav.r t.o Poiseuillo •a law fo't" tho l!l.minar flow of a. visoous flu5,d th1'0ti.gh a ca,p:Ul~ tubs lod t'1an.y wbsoquont invaa-tigatoi..s to base t heir wol'k upon tho aeeu..>npt:i.an thci.t n gra."llllar bod is analogous too. group of' ea.piH~rloe parallel to the direetion o:f .f.10\"'-.. F~r aui3h a hypo"~hetieal. bed the porosity {def'inod a.s the ratio of void voluroo to total volun-e) of' a la.yor no!'!t!.al to the d:.treotion of flow will be equal to tho porosity, € , of the bod as n whole. Binoe, ro1· ea.oh au.oh la:v~r, the fre.e.ttont\lJ. free volums vd.11 be equal to the fractional free nrea, eontinv.ity eonsideraticms re• quire thnt t ho apparent bulk velooity, u. , of the fluid oo leas thn.Yl t he aotual velooit~l in the pores, '-le , w'!"..ich VJO'J.ld oo '-'/£ • Thia ast.ru1:"lption was originally rn.&clo by Dupuit (4) :ln. 1863 and equations 'baaed upon it are still in use. 1\. numbo:i.· o.r attompta w:er·e rw.de to 5 equatiO:..'".IB ·to beds of hypothetical coru."i~u"a.tia.-,,, but none vJUS mthod of plotting b--3 dimnsionlesw groups . tho case of pi~ flow, for which a unique correlation :ts obtained by number; u8 lfJ eftoc·tiw intat"Stltial fluid velocity ~ • fluid densi t"-/ /A-- e fluid visoooit~r• h1 order to Gvaluat.e the L"'lto1.~st1tia.l velocity and the h~l'().l"$.!liC ! .P C:f e. cross soot!® area x len,ratlt = t:ree voluas.:.. ~al volu,J.w lll""" wetted pe1.~m:atcr lengbh -...-etted area • total vo ltirooi A p Thie roproeentat:1·an ·was o.-i~s.lly proposed by ~&1ly (6) in 1927 o.nd later worbd out independently by Fa.u- and I-Iatc.i.. (4) ·ariting t..lw tact that in the vieoau.a regime ( f T \ Ue z. ) (L lte ,,,,« A ) = p was uaad to derive e.n equa:bion for the pressure gradient a.ci-osa e. lo.,yer of a porous rnooium 6$ o. f'Unotion of th® poiroei ty. and speeii"lo surtaoo of the bed and the bulk velooi ty of flow tl:i;rough it which is generally known a.a Y>Q-zany• a equation: (l) whera k is a dimanoionless eonato.nt equal to o.boot 5.0. Thie equo.tion has boen applied by Car.man (7 ,8) to '!;he !!Warurel'll0'!:rt in .riltration theory. A aim:l.lar equation latsr giwn b'-j' Fnit• and Hateh (6, 9, 10 1 11) is 'l a. Th@ Dlmeneional Approach l'.t would appear mdent t hat the idea.J.:laed Duptdt model upon 15) Who, arguing trorn. dimensioo8.l 0011sids.r oti.ons, point out thM; tor a genorallzed poroo.o YOOdium (ao opposed to t..1'1.0 Du:guit rr-.odel) the fraotione.l f'rG0 oross aoot:i01i.al area is not necesaartly equ&l to the f'raotional void voluiOO., Md pr.opoae ths:t ·the inte?oM.tial void velocity should be not u /t , but re.th.er u ( £.1-/ 1 • such .6 F L Bakhmteff and F'3odorot£ do not attompt t o cri.ti.oize ·the structun of th.e Fa.ir-na·c.oh equatio.'1 ( the ideaUzed 'baais for imioh. should prove inadequat;o for flows 1:io:,rond the viscous rog:i.!!I.;)) 1 bu.t merely point out th.at the two equations disagree a.s ·co the efi'eo-t 8 SUJT!?01"tors of the F..oseny or Fro.r•Hateh theory will :()'-Jint out in ~oonect:ion that i t has ooen ®tper:tm.:mtally "<-"1'0!'ified ovoT a w-1do a. Repreoorrca.tion of. i)hG state of Flow c011s-ideration o.£ the flow reglme °b'J including the ezqxmont n a.a a o:t. a rate of! flaw oh~-Wr"lted by a Reynoldi1J nu.m'ber (ba.sed on particle di~~) of tho ot'der of J.. T'.ne existenee of ctJndi t i ~ of tt'tW turbulent r.iixing withL"l the interstiooa or tJw bed at su.eh a low RoJ,,nolde nu..'1lbsl" is highlJ• improbable• al:1d ~torimoos to ~"he state of .fl~, eho.raotori.med b~· n 7 l as n~ooesarily 11t-i.Wbulez.1t 1} (a.,-,, ad.jectiw too tr,oquently en~tered 1n -t•;he literature on the subjeot) mair be miel~ading. * ~ f i' W".d Feodoroff, on -the other ri.and, :rnaJ.-..e a precise dis·tinotian in t his CG':U10etion by pointing out that o. more correct: orua.lo~ irould be no·t:. to Po!seuille flow in pipea 1 .f01r which inertia :aovemont of a flui.d a:round a sphere., in whioh oo.se ·tr.e equations are tion, valid 1.ntb.L:"'l oortai.'1 lbdt&, that th.a inertia terms containing unalogu, tho D'!lr<r.f :re~ oorre-eponds to the realm ot ve.lid.1-ty of: the Stokes ~'?Proximation in whioh t he inertia. forces a.."'e negligible !'lQ matt<:Jr whether ·the flow is t u:r-bulent or laminar., the oontTil:ution dOC'J.nant un•cil .f'lnally t"he q~i-tul'"bule11t regime is reachad. The above ana.l~'tlis o.1' the mGl:1.w.1.i~m o£ rosiatance of a gra"!Ular bod to fluid flow. was e.ppai"ently .first nnd0 b;y· Forcheim$t>. ,;i Considaring A ft.wther disoussian of this matter is given in S@ction III B. ll the ~,.; that the visorus forces a~ proportional to the first powe~ of the velooity and the inert.ial forces are propo~cionnl to ·too ~ : : . au.-+ bu. z. (5) L (19) o.pparontly have oo-en f'.i:t to ignore .Fo-rcl1eirrm:ts quadratic equation and to correlate> their data. in tol"'n.5 of the .friction te.ctoi' "f'Orsus Heynolds .ri.u.'Tiber l!"(')pi-eoontat:ton o-o..."lvent.:t,")nnl :L."1 pipe fl~v . problems. Althoogh 11rJ.ch a r.wthod nay oo aatisi"aetc4j• £or use with bed$ 0£ a:nooti-1 splwre-11.k" particles~ pac1:L7'gs of mor~ compliooted s.he.pes agaii1 introd.1.100 t ho problc!n of' evnlul!-ting t he He~molds number, and tiruo :lnvolWi eon.s iderable difficulty . In c.n o:t'forb to based upon a,_"1 oqu~tion oi' t he Fo1"oheiroot" t~"l)ea not only oooauso the 12 ·to the flmv- of a. fluid is influonood t:r.r tho size 0£ tto tl.i.oo into 13 '.l.'1;0 o f'i"oct 1'1-ds nppro:d:nate or:i.pit~.tool coi~roction was not t.ho-<1ght ou£ficiontly accurate ~~or the preoon:I;; invostigatirn.1., hmvove1. 0 , sinoo ·tho nature of ·the wall ef'i'oct: nny bo 1t"t:trie.bJ.o, dopmd.ing :.1po."'l tho flo;rr 1~of,irJo obt;ab:u.11r, v-r:i.thin. the bod (20 , 22) • In the Da.1."~/ rer,iro the on0:rr_;y- loaoes al:'o duo to t ho vineoua Erl;z,oaoos in tho .f1.u1d and so ohould be upon which ·: .: ho str<.-nioos u.cr~o for' t his oano of vioco:..1.0 flow, it '1.'l'OU:ld v.ppear that t ho o.o·tua.l particle d.:totriblrticm oho;.tl be of secondary inpo-r'co.noo, -'c hat t.he roeiJ.stru.100 of t ':;_o heel should bo pz,opor-'donal to tho tmmJ. ex-Jor:cd wr00., a.--id t hat tho correct oonponse.·tioo f oz, wall effect ohould b0 to a-.-id the ontix-o wall ar-0a. ln t he case not 1.:mff':i..01ont 0 sinoo it ic O'vident thnt t he distl.·ibution of tho por'cioles in t ho bed in.f'luonoos t ho tortuosity of t he lY.;1d, whio.11 will !:.nvo groat offuot upon tho inertia f'orooo which pt"eclon:lnate at h:'i.g-)-1 :cat00 of' ±'loo,. From -ch@ e.1)0VO oons:idorut:i.ons :lt wus felt -t.Jiut i n t l\0 prooont i n·v,Jtrtignt:i.an a G'..1.i'fio:i.ontly ©.ccur&.t0 corrooti on f0:.• tho C'.dd;'J. tic:nal :triotion o~fi'o:t'"t!od. b:'l t;he con/G~...i.ner- wall cou ld ot\sily be nooe . It was WO'ulcl p~eol n dc f'.l:'z\;/ "".Btl:l od. ,':Jf. l'.'O.t:i.onal oor\ tm1.sntioa 11 and that a :::oire f l'l~eJ.c'cical 1:1.no .-:,f fl.J}iYt·oo.c>. ,1001.d bo an S1.ttm,'1[.rt to r.emle~ th:i.ei ·t.Y'J.:>e of wa.E orfuct no 0H.p_:ible lry poold.nr:; the bod 1n a_ t1odo conpntible 15 16 " h_ 1... c_..n,,_<;+ . (6 ) cJ.. L / - ~ viscos:i:'c:r of i'ln:td J ~ e. lonr;"ch eJ iaractorizi.'1 6 thG nizc ei t.he r o:C tho gr ains oli" = (7) euf.'f.icien.tly !~:igh value s o~ Eoynolds nu:.ibor JI hQl 10vor-, are ohru"r.o1 ~ d. L C-O'Y. s+ . ( 8) ~ +- (9) &.L Th@ -uv;o OOf~fi'ieionts oL and (3 detined 'b~, t;h:tra oqunti,m ru--e :l.ndopsndo..'1.'t of ·t ho roe.hru:1:lcal pr:-0031-ties of t,h0 fJ;uitl. 0 }3:lnoo they (lo) (11) 18 and GO ,::::. a (13) / G- 2. L as =o (14) (15) (17) n =t l and n e:, 2 ; +- F-1--\ : eE L (21) {22) 20 exporionood by the fluid particles dopcmds upon t;ho et.'.f"v-"'ll\'ruro of the Feodoroi'f' :t :Lvi. u,tc:: ce.oo /3 would represent tho cun-o.ture 0£ t ho bed • pa:r·ticleo~ or by analo{;-;t -to oantJ:"¥-..,i.ctian ernl c,::pt"..i:!Gion lossos dUD to 21 c5.i:x1.t l y 11-'.tch vaJ.uos o:2 I:.o:.1.10J.ds nu::J."i:tJ:."_, p::-omu~li1y$ a t!"uly turb1v1lon't roi;.Lru :::us-'.;; 0v·ont1.mny 'bo roached, in whiob tl:0 &'osir;rta.nc0 is 1od;:J)d i."'1 a."tresf:.'08 0.1":ls.1.ne, froc1 t1°anot0r of momntun ca r:. r.1ac!'oocopic sea.le • :Jinco ·t;hooo fa7o cf;;;ucto ru"o 'bcr'c;h propol'•·b5. on.al to t.ho LJquo.re of the l:fl.rt thoy '.7.t~Y ncvcl"f.-;h0l000 tx) 0,"<1,t)od t ;ogothor U:':idm.~ ·tho collcc-'dvo title o:2 ro;_ .,.no!"1:;:lal rotd.artmi.00:1 • 22 A. fhe &]there Bed Steel ball b e ~ sal"V8d ae the pm:-tlolea tor the bGu- atudied 1n the pJ:esont i.nwatigaticm. Bearings ot 0.125, o.250,. o.soo, and o.87&--inoh diarxrlier wore used. Vane.ti~ in pol"O&lty were obta!ned by pa.eking too b<Jds* aoc~g to ~tw 001ltlgurations shown dia~ matioally in Figure 2. Cubic a.nd haegona.l mode$ ot packing pl'G'dded 'b1w ma:rlFum and t'Jinimum liml.to. Ns:p$otiWly, of the bed pol'Osi°by obtainable with sphorit.Je.l ptu"tielas. Since it was- ood.Nd to haw s~ intermediate porositie.a 1 e random pooldng was 1.noluded• even thou.eh, it noc,e~~ly _in:,rolved IJOln'.) wall ei't$eta• In Oddition, a. mixs.d m.ode oonoisting of altornato le.yore ot cubic and hexo.~oru~l paoking 'l11aS used. Although the n<X1~orm poro$ity distriwticn ot this ,_,atra.titiodn packing could be ~»oted t-o atfeot the n8UUr8d values ot ol and (3 be u n ~ . , it vm.s hoped tha;t; the i-e.tio o£ the G'$ mi!tit The experl.rroot~ data later tended to justify thie Since lt was essential to obeai-ve the bed in order to prevent r..istilkes during pa.oking, the fl°'" tube, 28 :1.nohe.s 1n lenef.'uh, was fabricated from l•1noh lucite shoot, wlth o.o3-!l1oh Meopreno shoot aeoti~ &.6 incl-we, wide sinplitied tho ivall insert problom by permitting mt integnil nu.l'!lOOr of spheres in each le..jro?' in order to ~r ree.oane ~von lr...or• anly the o.25o.-tnoh dio.t~tor ~ e •re pacl~d. 1n eaoh ot tho nb<m modes. 23 The r1Ythod of: s:n.r..,ernat:ioaJ.ly pa.eJrJ.ne the sph01-ea into the -'"ube c..-an bo soon f.l.•om Firr,u:re 3. By Ot.tpportJ.ng the bed upon tho top of a <t;l.J';'oaded rod n.nd tho -cube upon a disk -::;:;;~avoninc on thia rod.. tlw uuoo could oo l!"aised around tho bod• t hus penlittL'1G e.11 the loading operations to oo oonductod at tho top of' tho tube. A top v:low of the arrny of 0.125-1.noh spho:t.'OS dtu..lng t he loo.ding proced-..11·0 is ci vr-.in in F.'igurn 4 . Onoo paol::nd., the bods wore i•it~dl~· 0011.a-tndned 1~r end soc-tions cmployinp; 16 car~ v.rlro screen sugpo:rtoo 'b~,· a porfo- z-nu:;d mta.l pln-te. Tho dct~ti.ls of t '.:10 plo.·to end am-eon. can be ooon iu F'.l rµ..""e 5. 1:17.,-s-corna.tic paoklngo. [~inco incr·tin.:l. e2£oo·:~s dopend upon t ::c cul....m ... -'.;;..u:o of tLo a-troainl:i.ncsp :i.:c was desired tJ,o.t; "Ghc wo.n :i.nool....GO not half spllo:c00 0 ·i:iu.t :1.n ~~!C -L;t:m vm:rat. ca.sos (-t,he o . n'7[j... and 0.500..iri.ch spl,.G;~e beds) o. reasanablo coD./}ro., .d.St; 1re.c l"eaehod by t: ,o uco ot insorte 0£ 0.875-inch bearings with ® inoort of i.;his typo. Fo1· tho amallor spllel"Os. n.o hali'-ro-..u'ld tiaterla.1 waa available, and eo the :i.nse1~s were r:-n(kJ !Tom rou.ncl roo. For these t\10 OS.OOB 11 hat·.rever, the vrall ofi'ect io of reduced importanoo d.w to the largo ratio of tube width to ba.11 diar~tei" • c. i'b:as-J.ring Equiprrsnt '.!he oxporlimn~s wore I)Dl'foi'l:oo -ali·t h nitrogan gas e.e the \ll'OJ."ld.ng fluid. 'l.'m 9roooduro oon s:lsted o~ deternin:tng the premrt.1..""'0 a..l'.l.d temperature at both ends of t h0 granulaz- bod as a function of tha rate oi' fl~r thr®r)i th.G bed. Sir100 difi'Gt"Onee ~ o n the inlet a.· K1 o-u.tlet presSUl"os viel;'O often small. the inle-'c: and outlet proomuros were 1:l )asured with Uk)raur.y- 1:is.n0r:1Jtors., v1hile t ho pro$1:i'.U·e drop across t he bed ires :mruru_red ivith e. \m1001;.• i:nn.o;~aer or one of tv10 drn--t't g86(>S» dope:1ding upon its roo.gni-:..-udo. 1'he msnomter rr-Jstem usod for t heoo n.eam.tror»nts is ahown in Figu:ro ?• 1'he gas te,{lpGl'G:l:.'Ul"OS v10ro i:nasured with r-nretU-J thermomte1•·s • • Tho :r.eaaurom:mts of go.a tla1 rate woro made -.vi th tho 0quipn1:1nt usod a-l'. thG Jot Pt·opul aion Labora·ool'jl £or routine p0rr1eabili-t:y testi."lC of porous ~tel spaoiT.10ns. 'this equiprrPntj) ehO'JJn in PigJtre 8., utilized four Fisher-Porter :V1lowra.tors in parallel ·to mo.sure g;o.s .flow ratos up to 100 cu i't,A:tl.n. The gas prosmu"e v~ controlled by ·thrae Grove ro_gule.tors, also 1n pru~an01. Por the sphere bed r1easul"Gn1.mts. tho flort WllS diwrliad t h.z-0t1~1 the packad oolu.!in and back :L:.to tho 1'"1.ovJTe.t.01:" eyston.o P.l(';Ure 9 shcn.".8 all t he oqu.ipmont aseompled td th the !'1""1 tuoo soo·cions in position is shown in. F"lgur.o 10. Tho 1.mcomroirtional location of the upstroo.n prosBw.·o i;ap was selected 0000000 the ~1nw::uo prosaure pick.-ed up at thnt rGgian was found to oo leas thW'l tho:c at 0.063 inch) woro originally locutod ct'l t:;o oylind:dce.1 p ort;,i.onei of genoro:te-..1 by tho r:lnr; OU;?;_;ort.lnr, the f'olt dii'iueinc diak introoucod upa--troa.n tap had to re Dmrcd to ·the loord:;i(.>11 descnbod a.1,ovo. only fox tl,o djm.w.'1.io I>T0SSt.iro p:lckod up and foi- tl"!.o preerr,.1re drop -ch.rouf.p ·cho supporting sc!'oons, and not to includo tho pross-,:i.,."'Q d1top due to f'l"icticn along tho tuoo wall. Conoequontly a apeo:tal ca1i- brati an tuoo of sheet bn1sa wns rooa vt.1. th n length just suff'io:1.ent to gi w a l/4~inch oopuation bot-woon tho supporti.7.G soroons whon t '.10 0m:1 00ctlcm.s r:-ero .ft'.stonod :i.n ple.oo. 'i'b.0 p1."0ssui·0 drop s.cro9s to correct the road.in.gs ob't,c.inod vri. t;~:: the paoknd tube~ Th1s cali• 1:cl)·er¾"'O tenporo::::.uro V,"$$ a1x1-.1t niderod neglir;t blo. i. ,_. two ·-y,z, aon-t, 8":ld this 0~1~or is ca:1- Al•;_;hoafji. tl·,o cno ~oo; :11)02.~at11r0 vrt:i.s thua O.}?]_)!' o:cl, ... tho di.f.f'el-onoo bct.:vrcon tho :Jqum""Oo oi: tLo 1nle't m.d outlcrt absoluto pr•ocodlu-o. oould bo calculatc--d dh"Getly. ?ho wlueo 20 a.'1.d the 0:1.:pori1Xlln-tal po:'Lnts ia 10013 so.-J.:rls:rac·toi+:1 :tn th,) 00.00 of t.!10 two largest h\':I.J.l s-lzes t!inn i.,(ll~ tl10 sr.i..a11ot> i:,a11So attr:11:n'tetl to unco:-."'tain:'0;:,-- in the values 0£ c,(_ Tho poor rt. ..; is obtained w-lth the J.ru.•goi· sphoroe as a Z>OS'.1.lt of scatt.01· L"1 1·0ading very small pl"'GaGu.ree ( of' the om ~n· of o . 001 inch of vmtcr) on. th0 lau:1 rongo dro.ft gage . giva"l by those spho!'OS i;1 the hi[;11or po::·o'Oity packi.."lgs would nc1G oo reLi.ablett snd the.t t.he tootm w:i:ch vm~r:lns potos5.-t:r would have 'lio be l.::>Gri'onmd wi'th ·0h0 s-..,siallor sr>hores. Ori t ho otJ:~!' hand,, it had boon found that the tt:l.sk of s~_rs·oomat:1.oally po.okinr; tl:.o 0.125 ..in.oh 'bearings in O.:."ly node 1.Jtl.ei• t;l,a.n the a-table ho.:{a.r;r.: inal arre.y vro.s iraposs:i.ble. Gonooquontlya Anothar cmolusion t hn"c oon b0 d ~ .from Figw:-e 11 is t hat the r-at::"i.o of t,.ibe tv:1.dth to rn:>I:ero tl:lrui:o-tor in tl:e oaao of' the t\i.1'0 largest ball sizos is too sr.w.11 to o.n,:iw 0linination of t he wall of:l'.'ocr'c in t ho :1!1.m:...t i o.l ~egi:.n ., evon wlth t he uoo of inserts to fill t ho voids a.t t 110 w-aH . 1~~.is ?ac.rt is 0vldon00d by tho o.bnorno.lly wide oopai~at:lo:ti of t,ho tlvo lo,,..-:01~ curvoe :b tl:e slop:<::) 2 region . The apaoinr; of t ho m.u·ves r;i'li-en b:;- t -lKl tv:o S?118.llm: sphct"oe., on tho other hai."ld.; i.a t i.-.o.t pz·0dio·ood ir,{ the thoor:/ • Uup;_:x,rt \;m.s t17UG r;ivon -to tho belief r.:d.nox- i ;::iporta.noo in. i.~h 0 00.00 of t~':G o.260-inch sphet'0a cw.Jing ti'J th0 lm·c;e :ratio of i,··uoo....to.. bru.1 aise . 29 in f'igar:0 12 o (3 It rn;r ho noted that t ho e:q;,erl7;xmtal cr:.1.t"V'C ci~ 'tr; ind:i.\.-'1.dually w0ro hi cJ1e:r than norncd as a result of non.,.it.:i.forn por·oi:tlty d.tstr:i.bution. In contrast to °U!G oohnvlor ed.' oc:: and f , tho viscous cooi'rtc:i.ont.s of Fig,H.'B 13 vrlt,h tho crn1tlictinc r0s:lota.noo to prodiat; t ho rowul.t;a of 11:"t.gure 13., v:hio11 ind,:loo.tos t;I:;at t,ho :11:i."" --- -- - t ho A"eoulte of Plc,1ro 11 could not be naod 'to i.nvesti(;a-'W t !:0 W1"1° i1w.ticelJ.y ol:'l.m:.tnaws t ho m1ow11 po:ror.:ity distr:trut:t.::m at tho boundru:ics included in I"ir,ure 15 as f\mctione of s0 • AJ.t..11.ouch the agreeront 11it'h tho predicted behavlor is less convincing th..,an vmo the case VJ'"ith the visco-..is coo.f'ficiont, a diroot linear relationship betrean /3 and ·t.ho specific surfaoo seems apparent. o. 1:!llrtansion of the Ko.Reny Equation to tho Inertial Rogiroo Although ·the data. obisained in tho pl('escnt investigation wore not nur:isrou.s enour;.li to juotify the detor;Jina:tion of the conatrurc required for e.n indopondmrc empirioe.l oquation for prossuro drop, the rolatianehip 1-,etweon the pertinent variablea desoribod a.b(YV® seetned convincing cncugh to permit at least o.n extension of the Kozon:? equation to tb:1 regino ot' iner-tial or quasi-tu.rbulcm'c r0Giste.n.oo b:l ·the addH:.icm. of a quadratic torn • From b"'ic,i1,.0 13 it was soon that the ratio f-, /ex. ima approrlnµitoly indep0~1dont of the bad porosity,. a. reB'v.lt not prooicrted by o:l.thor of tho Y ·:quationa (21) be expootod to vary inversel? vr.lt;h t.hc specifio su.rftlce of tho !)Ode ,:hon tho data o1:ttoined i'z-on -the O.l25•inoh and 0.250-inch boe.z.·int'tl (the ,:,mrt reliable 1.X)oo.uso of tho negligible wo.11 effect) were plotted in f"'lgure 17 11 it was found that the best fit 1.-vas obtt-t lnod b;,." the linonr t"0lat;ion If the KozmrJ 0qua-tion is uccopted as valid L"'l the visoo>.1a ro?,iiOOt this evaluation of the s."3.tio /3 lot.. iue-e tho value of the ompil"?ioal dirensionlcas consta'"lt of ·tllc quadra:bic resiatanoo 1-;erm,.=i e.nd the 32 complete oqmit:i.on expreso:1.ng the prossuro drop a.cross o. bed of snootb spherical p~~1"tiolet: as r. i'u.11c-cio-r;. ot' -the flu.id proport.ios, the frcrc1 Tu11o flow ol nnturm.1 gas -thl"our)1 lead shot. Equation (23),, Corroctloa tor tho Considering tho w1oort1:'..inties in the si.x:,oific surtaoo 33 the inortia i"oroos to tho viscous ±'o:rceo, it sooma logitima-Go (at least in thG 0000 of lar.n.nar flow) ·to asour.-:o that this ratio io the (24) nur:lbor cx--~ression ie f / o ( rosist...<:1J;1.oo coc)f'f:lcionts. C £ , t he ro.:tio of th0 im~rt:i.al a..vid viscoua r.:quation (17) can aloo bo used to tlo.t"lno a a :z6T6J.. = 2 a 4- ia Dru.·cy t..egim l8 2 Re. +- 2. (2G) gra.ii(mt exp:z-ession of ·t.110 Forcheir,xn. . tm>e~ My de.t-e. i"Jttloh .f'lt au.ah a qtiad:ra.-c,ie equation "i'rlll i'it ftiua:'-cion. (2G). As f!l'l ruca.':lplo s01ro ·c:,1?ioal data obtained f'r0t1. the sphere, bed o,q,e,rlimnto ro-o presontod in Fi r,.1re 19. " it i7e..y also be so.:'i.d that 'vhore io little value in a. nnthod of oorrolation which r0quir0s faro chaz-acrv0rist:lc lenr~-.;h pm:-nrl$tors ( ..c 11..nd f ) vft.ten it ia wo11 known that data on fl(:,w thl"OUf:;11 sphoro bods can oo cm.·rolo,tod •.rith oi;1.ly on.o length pa;rru:a:Jtoi.'.' • tho sphi.3re dia•noter. Such a ota'cer~z::;; , howevor., onphnsiz,-;s tho r.s.in point of the dovelopr:ien-b prooontod abO"'ve: 1.n gonero.l a porou.a ??aa'Ccl~ial can only oo conplotol~'l charactotizod uy both oi' the long;t:h pw.e.r:nters d.ofiuod b;i tho two--·t.e:rm qu&<.lro:tic rosista.noo oqua-tion.. 'l.:h:1.e t;71.Je of r0pi•ooonta:cion is t.he only one -.1hicL can be nppliod to comple% eansolldated 1:iatG:&'iala such as poi~ou.s r:mn1s 3 s:l.nco it etin eorrelo:t;o tho bohavlo!r of spocimons w"hooo pt'OG&l!."O ff.'.'OOiont vs o flaw rate cm-voe 1:nter:;oet oi· ci--oos ono ru10'bhor . SuCL'1 bohavlot· :i.o of-1"...en on• 001. m:tered in porou.a ~tala, sinco the in.01.'tia.1 resistance ooofficiout is ap ptU'Ontl~i noro sensitiw to ointc:dng oond:l:t:lons thun is tho viscous cocf'fic ient . Int0rooc ting curveo of' -this t :_,1Je haw also boon obtained in tosta upon consol:ldatotl sands (2e). bad ia a special. case in wh:lcl! oorrolatloo m t ,ho haaio of ono lonit..;h 35 CX f .__._ _s- D --'/er are directly pi'oportiono.1 to the sphero Tho con<litiona noooasa.r-.1 for the val:i.dity of ·(;he assumption lea.ding 'to r4u0.tian ( 24) aro not obv"loue o.t the present tit"~. poi·ous ;:l3tl:ll 1 -;:;);10 oxistenoo of stationary vortices o.nd eddios ie not unlil:oly. UndeT such oondit:tous the onergy diaa5.pa:'..;ion would still which turbulent oondi tio:::.0 c.lnoat oortainly obtain o.-'c hir)1 z-atoa of flow. Conaequoutl:-.,- it; a.pporu·s f;h a-t; tho V'al:ldi ty 0£ E\quatioo (24) does not depend upon a oo,-,,,10a:ln at;o:t.o of flow~ but 1:equires only e. 30 rrr:.:o by a siz~t1lo quru1.ra·i~ic 0quat ian . .:>'twh e..n equat;1a.'1 doi"in.es two loncth pN."'OJ:l::lto2.·s cho..r.a.ctcr:i.otic of t he porous :ri;.ructu.t'O. 'I.'.bo pru."eJJffto:r b:t wrtl.ch ·b:11<0 linct.'\r tor'~ is r.cltiplio.d (di,~:m ai ons [ L - 2. J ) in t.h0 inverse of -the p0:i."l:;1 0rlbilit--;:r cooft'icient dof.'l.,_71.0d bJ Do.l'C"J 'e lrcU'l, and is coJ.lod the avisccr.ls rosistanoo oooff•:lC'..i0:nt 11 • The !TIUltipl ier of ..cho quadi"atic ·(;o!"r!l. (dine n sions [ L - i ] ) is ce.lled t he ninc1...t:tal rasi ste.."loo coof"i'ic:l0nt 11 • 2. T.he influenoo of porosity upon the rosiste.noo of o. bed of S,".looth aphe:i:~oal pe.1.""ticle.s to fluid i'li:n.r in the Daro--.7 ireginB ie oxpreaaed 1110zc-e acO't1rt1.tel:/ by t he Kozeny {or F'air-Ha.tch ) 0quG1.tian t;hM 'by 't!:10 oqua-tion of Bak'lu:xrtei'f and Feodoroi"'f • e:n.191-eosions, 'G hc ln!'luenoo o:t.' porosity in the inertial or quasi ... turbulent rec;-J.iro wns found to be approxi rl1ltely tl-:-.o aar:ie e.a :l n the 4,. The in..i'luen.oo of t 11c spo c:i.. i':lo surface of: tlie bed upan the linmir lllr.d que.dra'c:io rosinta.nce t0rrm :ls pr0dicted correctly lr,1 -tho equations of Fair-=-1i"o.tch imd Bal.drnnto£f...Feodt.>ro:2:f'. Thie spoc:lru s:\mct:1.onal dopendonoo of ·::.ho 'viscous a.w:l inor-.;lal r;.•esia-tm,n oo coof... f i cio;rto upon the spccif'l c S'J.r:Cuce is responsi'ble for t he faat tha:b ephero bed data can be ·,oonu1atet1 wlth the uoo or only one length e A constant b A eonstent .Po~toet gas consta"l"t, li?'lcti.on factor (dirensionleaa) F A function f A :r-,moti on._ G /bif)1t flor1 ra·oo/u,i"'l.it ru"oa [ :1t..1i-3 g Aooolea.-ation d:.l0 te (7t\V:i.t:;i· [ k A ocucrtant ra D}ui'V't\.lont hydre.ulio :rnd:lus p .Prom:ru20 S Su.:r£a.co aroa of porous material por u."li·c total volua:.i of r1aterial J L'..r-'"2 ] [ L l [ L -1 J f:,'urf~oo ru:-ea of' pol"ous nntor-iaJ. por unit solid volu.'1):) of' r.ntorial [_ L -, j .i\bscluto tenpcrattu-o u Pluid voloci·t:i [ ~l ] 38 E. ·;"'- KOl"OSity (dii:X3nai o.nloas) D-.Yns;:n:lo viscosity [ ,.,.L-l , - - • , J r·\ f xn.u.:ta. densi t:-,r [ M L- 1 ] (j Spoo:li'ic 1rroir)1t [ M L - 2.. T - L 'L Shearing i:rl;roem [ M L - 1 T J - -z.. ] 1 3. Thru:•c:;•; T\e Pe C·•• 0 Lcs F'ontv~'lr!OS Publiqooe de la Villo do Dijcm' Vi ct;oi~ Delr1m1t , I?ru.;---ls, 1856. 4. Du.pu.it 0 A. ,J. ,: . J., 0 Dcu.des 'lheoZ"etiques <3t P!'atiquos Li01.ivcr:1::mt des Eau..<tn D 1003. Fair, Ge ;,'I. ~.1d Ho.toh• Le Po, 1 aw.? le 1 ~\:ID.d$.:"'.0V."Cal I< e.cto1."s Gov'el"r'J...Vlg 'tho S-t;1•eQTnlino Plow1 of rfntol." 'hl1.ratigh Sand 11 , J. l\J¾ Viator Wo1..lm Assoc. 25, 1551 (1933). Ca.."'1:uin, P• Go, ii.F'lu:td. Flem thra1.:1.p;h Gr9.11Ular. Bods O , 'l'r • Inst o Cho I:nss., 15, lf)O (1937). Cn.:rna."1, .P o Ooj 11 '.l'I':0 Dotorni.-iatio:.1 or ·the Specif-lo s-Lurf'aoo of PovruoI' 11, :r Journ. Soeo Gh0:m. lru'lust. 57, 225 (10313)0 rracoh, L. P. 4 Diaouasion of paper 11Flmv thrcmr7>1 Graaulru:· l'bd.i.n 11 J. ;\;?P. _:.b ch • ., Trl'l.\'1.C /lJfr'.E1 00 1> A-86 ( l 938) • m,.toh, Lo P e~ "Plow of Jllulcls tI--irongh Gi"ru:ra.hi.r 1'.f ato!'io.ls: Fi'.!.. t?e.ticm.., r~::pano:lon, t>..:'"1tl i:ilnd.01"0d Gettlin,/\ '£roz1.s o /\.m,. Cloopl1~·a . Ura.ion, po 036., 1D43. Ba..':dm:ote.ff0 Do Ao 1 3i.1d 11 Foodoroi'f', !J. V., Flor1 thro---1.gh Gr.ant'iloz 1 I.bdiu 1, ,;• Ap-9 0 :.bch ., Tro..'1.s. AS, TJ, 59 1 i\.•97 (1037). Bak.h1::otoi'f tJ Bo A. ar:il F'oodoa-o:s-.'f, '.1 o 'Vos 11 Flow t:::.1."'cmr:;h Granular.~ 1oili.a[! ~ \?:roe. o.f Fi~~h Il1·c. Cong. i'o:r i\ppl. ;be!: •$ 1937. 11 . i/ ., <lia01.1:;m5.an of l?l½."X>ll~ ,J. Ar,p o ; bch., 'rTe.tl.Go 1\SIIB 11 Gm·rlDn, P. C. 8 nP.~tnda.\m,nta:i. J;">z•ino:l pl es o£ lnc1:ustri eJ. I•'i l tr~ti o.~s 11 Trnns. Ind . of Ch. !::h go., 16, 168~ {1938). 22. P.os0, E. ,::., 1nm. of ?bah . 11:r;rs., Ap~'.}• rcch. ;;)roo., 153 0 1<2:1 (194,C) . 23. ruskntt i::., 11 Plow ,of Eon.oc:enoous I''.L..i e.s 'l'hrcrt:r;h Poro:as :odio, 11 ., 7.1c(ll"m1-r11:n, 1D37. 24. " Chiltor.1 end Ooll:ri.1m, 1'78 (19~Sl) • 25. L!!tpplG, c . ;;., Oort:l3ntary on a paper b-.1 Brownell and i:>:e.tz., (,'hem. llig,. Pf'og. 1 4,3; G37 (1947) • Trana . Am::i:r. Ins-',.;. Chon. 1, 27 (lfJ31 ) . / r~o·s., 2G, TABLE' I Geometric Pro~rties of Sphere Beds Bed Number 1 2 3 4 5 6 7 8 9 Packing Type Sphere Diameter A A B C D A D A D d 0.125 0.250 0.250 0.250 0.250 0.500 0.500 0.875 0.875 Bed Length (in.) 17.1 24.0 23.9 24.0 16.5 24.2 24.0 24.1 24.5 137 28 1/16 round 96 95.8 96 66 48,4 48 27.6 28 14 1/8 round 14 14 14 7 7 round none none hi{f none 4 4 5/8 half none round 210 295 294 295 203 297 295 296 149 4.1 203 11.7 184 106 6.9 178 0 0 189 24.7 154 0 172 157 37 .4 0 (in.3) 153 215 191 178 106 214 154 210 157 Insert Volume Packing Volwre (percent) 2.7 5.4 J.6 0 0 11.6 0 17.9 0 Bed Porosity (percent) 27.1 27.0 35.2 39,7 47,8 2ELO 47.8 29 ..2 47.8 (in . 3) 1.3 3.3 2.0 - 0 4.2 0 9.2 0 (percent) 0.8 1.5 1.0 - 0 2.0 0 4.4 0 239 336 335 336 231 160 336 171 264 374 240 0 0 0 239 7170 7434 4880 5254 4410 4650 4260 4260 2540 2540 264 2272 2536 343 0 1850 1850 1180 1419 1078 1078 Ifominal Surface Area of Packing with Ideal Packing (in.2) at Walls 7440 5240 4620 - 2540 2620 1850 1500 1078 Difference in Packing Surface due to Inserts (~rcent) -0.1 +o.3 +o.6 - 0 -3.2 0 -5.4 0 7673 5590 4985 4596 2771 2696 2186 1590 1421 3.4 6.7 5.2 0 0 9.8 0 15.0 0 (percent) 3.1 6.0 7.2 7.3 8.3 5-9 15. /4, 10.8 24.1 Bed 3urface Area Packing Volume 50.0 26.0 26.1 25.8 26.1 12.6 14..2 7.59 9.05 L/d D/d L (in.) (D = tube width, in.) I Description of Insert Rod Material Bed Volume (in.3) Sphere Volume Insert Volume (in.3) (in . J) Total Packing Volume :Y.issing .Sphere Volume - Insert Volume ,.. . Sphere Volume - Insert Volume ;:11ss1ng Packing Volume (in. 2 ) Exposed Surface Area of Tube Exposed Surface Area of Inserts Surface Area of Spheres (in. 2 ) Surface Area of Packing (in. 2 ) (in. 2 ) Total Bed Surface (Tube Included) Surface Area of Inserts Bed .Surface Area Surface Area of Tube 13ed Surface Area Specific Surface (in.-1) • so = (in/-) (percent) 1/8 round 301 14 . 2 7.59 <: .05 35 .2 39. 7 47 . J 28.C 47 .8 29.? 47 , 3 5 6 '7 3 0I / 4 2 3 R ,1. ..')_ (, u t 26.l c'.). " ~ r 26.0 26 .1 50 . 0 27.1 27.0 1 (in . - s0 1) • ' tr. ) ( 1\ Porosity NurrJ:)e r J ed II - ·- 63 63 70 78 56 59 55 (OF) l o,: ,,, SuF1mary of .Sphe r e Bed Data T.\BLE .•. /3 ~ '1 43 42 33 ..-,~ 35 63 65 30 46 ( {_, - ' ) oC.. - 5.46x.103 l .. 66x104 2 .25x10o3 7.3lx.l 5. 20xlot 3.2sx10I ~. 19x10 ('lf!. . . -2) 2 2 .69xl0-'5 .-49 7. 06x.l01 9.94 l l.64x101 6.nx10 1 l.28xl0 ✓-: 4.2lxl0~ 2.G?xlO: r5> , ( i.n. - ... \; ·- 4 . c,.5~ .-,:J. n-- 3 / 2.?lxlo-3 2 ') 41 x1. ' c-3 ~·; • -3 '-l- •"- 5xl0 2. 53x10-J 2 .4 6X ,.1.<..1,.._- -'< l.23xlo- 3 (in.) /3/cx. l'-l ~ Lf3 ,.-... (,0 .._,, t.-i § 3000 Ko -eny or Fa · r - Hatch Fi ction, •rl +J r( e) """ 35s(1-t:) 3 C. 2 (.) ~ ~ 2600 •rl (/) 0 ~ 0 0.. Q) ,c: +J 2200 t+-i 0 f1) (l) - Bakhme eff- Feodo off FuncL on, ;j r-1 ~ 1 800 I r( e )-=..640; c..-4/3 1 - - - - - ---1 37 38 39 40 41 Porosity (J ercent) Fig . 1 Graphs of the conflicting porosity f unction3 Bokhr:et~ff and Feodoroff, and by Koz.eny or Fair and Hatch. E) ven by 0 IJj iv ,.., p_, r.i :::, c+ ,-.,. H ) !-'· 0 ~ 0 (__) ~ "' 1--'• 0 c-t· ,-J • 0 :::i s:? ~ ::l t=i () p, 0 ill " H) (1) ()..1 CL 0 a::i 0 '".! v ;:i· (1) 0 >< )l·, c-q g 0" ::S' w '°X. r:; .... 1. 0 r!:'.'. ~•· x (() :j Ill f-' (/) f-'• d c i· I--'• 0 :::i ,.) '"f-·' (0 ~ 0 Q.. (v c,ry 0 ~ ~.) Cf.1 '0 ;:i· (C "") (l) :-a C,., ,l.) 0 () ;,;- ;:., w. ~ ·: ~ :l 1~fQ '~-'•"' c+ () ;::J (' ::;I '1 ,) (J} 8 ::1 ci-(\.: c t- 0 ~j hh Ft ➔ . 3 ..L II tt rt-rC<.-h~ o-f La4-d,;,, 7 P~ ce&t.c.-'e.. lf8 Vie~ s,ow,~1 l,,Jtf-4 I ~te o .<f'7S--,:... . ()~ .../-(;(serf- ~tere~ f,/r·e ,I L/'1 :>~ ~ ~! :::I -~• 0 f~ t:: c:, 0 r: ...1 .. Fti . ff Er.-<r;,u-.e1,t_f- o.f- ~ Ka«-h~e ?or--uu s l>erl,4,(eq_l,,/,-ii, t11e-1a:. /.r. lc:-rtr ... ,, A..rs-Fu-.,1~ a.CA.A J o~ Flrw t=lo-w 77:,,,e,/ Me.Qr-t.<.n"f. 1 Vvl«~""'<<:fer- E'/'"" ' P"""e 4 -I- P<::<~~1,, .;) U) f:: g Cl I I Qi .a i tr; (I) C $.. (.) 0 ,--J o.. I nl I E-i : C 0 ~- t 0 •ri {[) '-; :::i ½,.➔ _',;'. ~-·i U) •r l ., -l C..l C '. 3: .µ 0 r-¾ r ·i (l) !;:., ~1.i __ . i_ __ ___i_ ! ! --- !___ _ !__ I I , - - --1----1-+~-~+--H--+-+-+-H+-H - --1 I -4 IO -2 10 1 li~ilZ~ :f •· ~l~:'= '====:====:=:::::=:::::====":r:_.:··=i=·~:•~-:~=f=f-.,-T:.,.,l,---L--I-.1it!ttJ,Jf • : . F - ~ ~ · - - ~-~~4~it1 t -- j -'= · ~ , - ~ -~Tip ~iric· is;rr~-1~·-;;-:f~~o±"o.:f in:1)· :----~ 5 ___ ...j..:..:__;____:_.:.j.....:__;__.;::::.:_+.:__;__::: ·--::.;:_··:.+ - _;_;__:_--'--+-~-"t~---:i~--:-1-· .7 ;· -- . ... ... - . - -- . t·· -- --------- ---t-· -- -· -------- ------ - --------- --- ----r~----. • . . .. ·1 ___ _ __ __ _ ___ _! _ _ • l . • .. 1 3 ----- - - - ---l-~r---,-----'-1-----+-, ---'----1--~~--+___:....:..+--t----__-+i-_ - - - - t -_---,~------ --- ---- 2 - '. c· -- T ----+--· ·_··-+!---+---·--=□=:,I~_-_·_: : •: • : ·• I • ---'-,---+--•.-----;-----+--L-._ -+----t1, < ; .• . :.~ : ::: :.~ ---------- • --·- I . . . ·- - -· . : . r -- - . . .. . . - - -- . . ·- - --·- - - . --. . - .-----. ]._ _ -- 1· - 1 . --- - -~: :. . · I- ----t---. ;_:~~-J-_~.~~- -- - - ~- • . • .-- . - - •. .- - -_ - . ! : . .. ! • . .. . . -· . -·-.. · . . .. - - . ' .. :• Ti ·_ I :. l i: i J . . -. .... _I .- :_~ r -------- --- -i - ·__ :_1 ' ' . • ,: . I ·---- ----!---- ' 1 • · .c: - ---·- . 4__ ·_ I i_ - - - - - ·- 5__ 1 • • I • I --· -------- _J ____ --6__ :L . --+------- - -- - J ---~ i I [ . I - -· - - • I • --1 -- · - -- I' I ·1 . --1 j ----,---_ ---1------, ----i I -T · ·· i 10 ____ _,..~_,__.__ _ _ _ _~ - - -- -~ - 9 __ _ g _ __ ------ +· ·_ ... · - ----: ------ -.....~-------~-..--- 4 ___ 3 __ _ ::> --- - --- -- - - l - - - + - - •1 . - ·- L --··-·- _• .. - ; . - - --•---- - ! - ! i - - --; --- - ! i ·- _______J ! i i' • --- • -·r - --• •-- ➔---~---,-..- --• - I - b - -l-------1 ; ! -· . l i •. - -· - •-· ; • . • I I. 1 I ! i i i i r. ! i : i, -- - - ---+-------- --I L ! I . .. -7·-···· .... -- ! :.lI T I : ' ····· ·-· ·· - .• ··· ·i' ··-" ·- · ····--•· I Fig. 15 The Variation of the Viscous and Inertial Resistance with the Specific Surface of the Bed at Constant Porosity J0 -1:+-"'--!';----- '1 - - -- · - -=-- ··-· - · -- - - - ·-;-··- - ·--·- .. ·--·=- ------ -- 9 - __;__ ·--· -'--··---- -.. ~ - i i I S- • I -· l - , _: · ~'~: L __ . I • I i !- l- - - ' 1 I ·1 .; ·; I: ' - ·-L - ·-·· -: ;6 iI iI ! . I i j . J i - - ~--- - . ---1-- ..J. ___ - - ... - ···+-·· I I • l , , I {.Q½!,.t_cL<>bt,~_;trn~.cJ.fr<?m_ b~_ds . Qf _oi 1?5~in ~ 1 and 0.250-in. spheres) I __ J._ I i 1- 3 •• ·- - --- - -- - . -- - · -· ··· ·· - I ; -- --+-·- ·- -·-! . . i I I 1Q z - I ·---- --- - ·- -I .. I -- { -· 0 ! i I Specific 3urface •of Be d ( ill ,L..l) '. .!· ~ I ! Jo The i"tat :i o of the I nertial ne s i st:rnce r;oef fj_('. i.ent t, o the Visc ous Resist ancP Coefficient as ::.i. Function of t he ::.5pec ific J urfac e of t he :.~d. 80 I _j_l • ·•-+--,. ' l l • I I I I T I, I l --t-· ._l ____ _ __ __ •~ - I I I ! :i -t---- -~- :·::~=-:--=~~~=~~:-----\-1----J---j-++-! . 'I 10 -3 -- - ' . ·---: -----t--l--l- ..-+-+-+ I -2 10 -I 10 l 10 10 2 1o3 ! , :J 2 3 4 - N ·t t , ±=~ r • : I l I 4 1 1 1 56 7 fJ9 1 2 3 1 4 1 I 55 7 89 1 2 1 3 1 4 I 1 , 56789 i 1 j , __ _i ____ ~1____L ~-!__~ __;_ -- --- - _•i ___ ---- -!- __, .1~__ __;_:_J_i-j..-LtiJ::d • 1 1 r-_ _-----~---±-.. ~_ , 1 +.J 3 ' 1 I ' : : , 1 1 : l . • i Ii • r-:-t - _ ---·---+--_+-- l I j I ' -:·· ' 'I • ';] • ·1 I L 1 . •• : 1: ! 1· 1 • ; , . ! ! ; . l - , .. ·· : - 1·· --· ., - - 1 1 . I l I 1 ! - --+--- -+:-:- :-+--,_- --:o - :· · _ ,. . . ., , u-An. :•::>P, c17v ,'f) ' : ~ r •h....Lng ]" • I - -- -- 1-- ·- '-·-t ·I o ·o/-"O _-·1rr--· -.s·p•!..,_:rr ·"'- _:""1'~ i,--.e-yr,;_-m_- _ - ! ' ' • : •• ..: . ' ' I i • !i 1 : ' '.,. 1· ....' ! / l i, . ,, ' i l j I' . . I. . . .. I • ' - I . l'- --~ +- : l I i --·!·, ' .. '1 ; I ! _, I ! i i J Ii j I . '. : I i ! !. , I .· - -~i -- I, .. ..; -~I ' . ; f tJr11 m. j _,! i 1 10-1 10-2 , - ' ' I ____ L ! ;- i i i. l ! r l L ~-- L . ..l L _ •.. : : I , 1 . ! • 1 •· · . /-- f ' • ' • : i ' . , ' i I 11 i . J\ , I f , ! ·I - • I!_ - I ·1· -- -1 . r','-n : 1 I i ; t· l 1 • i t ' '1 • r ' ' 1 ' 1.. • • • •• 10 •. ••• I : I --1 I I I I I ' i:·· •i- ·1 I 4 , •• 1 !· I i i~i- I I ·I 1.0'.'.' !-"L..L J... !_ If 1 ' 1:' i •I-~(L:i' .! I I;!~, _ +-1- l • - ·•• -~!---- lI -- -1--· I --,---··· :1 I i 11· -, l . I ,- I ' 1 -: l 1-,1.+++· -~ t • . -, ~ - .. ! '. • 1 I' •' I ! ; ..-IC':,:_ , ..., ! .. I ,. ; • ' ·: . --- - ·•-· ' . • 1 l (I_:.!.i.!j ----- --: -- ..:. . .+--:;- . 1J;.. , ' l 1' i : LL• , i ! ! : ! I 1 - - .. - --- --, -.1.- L'ft4; .. - ' -xso'"'-l!.J I I I ;- : I .j. Friction Factor vs. Reynol ds Number Gra ph of Sphere Bed Da te;. Fig. 19 ; ; : ---- - ,-------- -- ,- '-- . ,· J -- - -- ~ ! •• • ! -· ;I •· i , L.......L.___..~-----'--------L _(i "'r'/'"' c,: I. . . , i ·1- --; •,..,,, Jo a ' ' ~- . : l--1:---~H: --------+-·--t----~---l---~,: + -H+-------- I -2 - ! ---~ • ·· !Cr• ..iI.- ·~ I : : • ! :i"'ll. : I ., . . '. 'i ·' "j " " : L-1- 1-r- : . : : :... i •• -- ·, ' • • • I-- l?4- i ·: . o. 2'.:>0--:1.n~ :s~eire$ , : pa~tt:1:ng -d -- ' +'· LtJ~~----'--L_~--~--~~~-l.l1~tJ 2"' _:__; :.J ~ ~:~?j i-14P .o:f~o~i~:.lsr~;r~~;,~1Jki~fut~ i .,: I . . : • --~---+-·--t1 ------ -- -,---- --- I t=c : ; :t.:i1=~ .• :·~·1~~1T~H.~·.··~·.~i-.",j TffUf~1 I : 2 -~--~-!- ~--- ---------- : ___ 6 7 891 ~-~-~1-~~~-L:~i~ ~=~~~L~~ -~~~--1~{-IffI~~=~-~=l~-:~i=l~~LiJ~r- ~~ -~:c::~=~~i~fiff11~L~~-~ _;-_J~~~J=--1~ f-t::1;B l , ___L ! 5 (, 7 89 1 2 3 4 5 1 - ~- --=-: --:-- ~ t ---- -- --- t - ----i---- ____ 4 ---i------i-~:-i--_j-::H·-h --- --~----:·+-~+~---•-+·-~l---H 0- t-1 - f ••• ' r 1 9 ,- 2 I -- -- ---- - ' : 1-·----- ---,-- 1 3~ •1 ~ t~~-~=-1- =_: ~ r=:=:~--: __--__J--_ 2 l ~ '-- A poss:i.'ble ~ppl.ication of mvoa:-t cooling :bwolvoe oo.ncll:'ciona (1) I pu (2) o(_ E 0 Conhini~1.r~ E(F,mtion s (2) o~:~d (:::,) g:l·vDB cl ( _f_ o .z f ' / d 2J, L ) dX L + C ?_e_ = ® ~t (4) 63 ::: (5) which lo t ho t'Ll..f'.i'orontial equa1;ion. governing tho prom:n.1z-e distl•i.,. (fi) sub ject to the £oJ. l ~ ·::l.nc bmli."1.darz,- conditiono: p( O .J -t- t ) = l\J f>o -= t-, f>:, y,., p-=N~o-i(-t.'> -:=. f' -i ( t 1 tltlokri.ess . 0 - .P X p(R,t) -=- r ~ l X ) \::>e 0 ) =:: o 64 l-\ ~ ;:::. Po X s =- -.£ 't ~o -:::. fo1...r ...e. z.. -t Since &1u0.tion (5) booomas or (_ 'J .And since (6) wholtO (_ ' .I r)==-1 o ) = I This non. 1:1.neru." parabolic cqnat:1on is not oolvo.bl e b~1 S"c&.ndard ane.ly-'"Q.ool rnthod.s, but rrll!!Prleal solutions are obtai..11.a.bl0 if the ~"'1!1('; th® dif'ferenoa oqut\-'\j:ton wl th ru:1 oloctr-lcal analo~.r aur,gostod b;r '• A dif'teronoo oquat:lon equit'111le.nt to r:;qu.a.tion (G) con bo vn:-.ttton (7) Nultipl;i.dng 'both aides by 6 s"' one eets {O) bo OOI'O: (9) ::: 0 Thus, if' vm oot (10) equal i~rlcally to the bovin.dru."°:)• valoos o.i' the p1~ssure 1.~tio y, Prom E<-rtui.tion (10) i t con be soon that in 01"der to matisfy Ix1ue.t1ou (8) truree oi.' tho fol.l.l"' :i.npede.noo olor!l9nts join:bs each node of the 01eetrical n0•l;wo1."'k 1:1.urh oo oppoa.i.te in s:le,i to t.he fourth. This requii-ennnt was r~t t)y the ua0 of o.n al ternatin.g ba.mdf:\r'"J space n:--id timo. Th@ n.an-linoar nat"'llro of the pt"obleJ:t, h~vor, is na.'11.if'ested in ·tho bo1.~zon:bal i~?Oda.noos, r1hich o.r0 soon to be func·tions of the pressure re:'do obtaining at each point :i.n question. 'l'his dopandonoo roqu:lrod a.n iterative pi"ooesis of solution, in til'hioh the init:tal values of' the hodzoo:'cal ir:rpoda."loos vro1"'0 detorr'.d.nod f'rc::1 a x,ourj1 ootims.te oi: y M a f"<li.1c-tion of a for varl.cr..xs valuos o£ Z. • Three typical bm.L-idar~r ve.lne ·in•oblamo of the t~J"jp doscrl bed abavo wore a :,lvad ,sith tho une of tho BloC'1.iric Analog Computet" by Pl"'oi'ossor ;,·a1ts and tho ,-n-ltai·. Probler:3£1 with i.."'lOl"eCl.BinE dogl"Oee of ncm-linoa:d.ty wore studiod by consider-lag oe.oos vtho:ro N (t,ho ratio of t ho O.l?{)U0d "botw..da.r.i' proecuro to tho ird ti9.l pz-esrui·e) asm.JJX'>d tho -..ra.luoe .,f 2., 4 6 and 10. Fron li'ig,1tn:--e 3 H; ma;/ oo coon t he.t t ho tir:l:3 varinblo cruJ.d m,t . approach in:rinl ty- in tho static ai,'1.alor;., s:L.100 only e finite numlxn.. of tire ineren::mta itJG:ro avoilo.ble. 'fhis rat'ldc :lt ooooGsru-y to choose the sine of th0 t;ir-d"i:I :1J1·ooi""ii'<ll rn.ioh t l:.at steady Gtat0 conditions woro t"Oachod 01· clooo1;;,,• arprozlmo.ted i:.u."toi: eir:;ht tir:o 1:ncrornntm i'olr' '1..-.ho ah:r.--up·b teri-r,-J..na:ti~ of tho net,1ork in -t~10 tirn cli::--ection af'oor oteoo;,r stato ha.a ooon r0achcr.l has e. noglir;:i.ble efi'oot upon the solution for rs,:1al le1~ -values oi~ ·cir;}) o l t 1·taa i'crmltl by ox,_:ioit'lmon't;ati o..VJ. that a Vt.1.luo of -tho dimnsionloon i:10:rerxmt LI T. of: al.>out 0.010 t\'t\S lru:-g;o onoo..c;h to [yh/'e o.ppro:d.rcJLYcol;;, stoad.-y ot::1.to conditions 1n tho Cru30 of 0 68 the lowest bou..r1d~J pres:m.1re l":a.tio. Thim interval ?JM halved in the oa.se o£ l'7 e.l 10. The 001-iverr~,,moo of the :J.toratio:ri proeooe dooCl"ibed abovo was fou.nd to bo m1.r.-ptioingl;7 rnpld. 4 For t he 0000 o.r U ,:,;, 2, tlie rnasv.1"'8d pot..ontie.ls OO'.'.:'l"Oopontlir.,fi t;o t he d:i..rwnsionlosa proSS'"ul"O y were £cru.nd to have ic\ppi-o~·ied to w;l tbin ()D.O l.;Xll"CGnt cf their i'lna.1 value n:rte:ir only ane .i tet·o.-tion. For 'bh0 moi"e noo-lineru• cases (N e 4 0 10) 0 a. larger nu.rnlJ0l~ or iterations \-ma i~equirod f or ooovm.~genoo. The Z'tlpidity of tho pi•oooss, of oot.1z-so. depends upon the noouraO"J or the inH;ia.1 ©S'timatie of' tho prossv.ro disti~ibution. IV f!.BSULTS AllD CCllCLUHIONS 'Z•xD f'.lno.1 ~luoe or :'1 which -v10ro obt.U-1. ned in. eo.ch or the tJ:1roo 0!1oos doac.;:-lhed abo'\1--e a.ro s--.;:.:-:narized in To.1.110 r. Wheri. theao clt).te. v10re plotted 'to giv~ pressure d.:lstrlbut:i.on ourws., it ui-ns disoovered ths-c ste~ s-00:00 oanditions hcv..1 not alwev·s boon approooh0d closely e11c'f..1g!"'. to cl :t:,tl.nevt@ completely tho ~.n:fluenoo oi' th0 netwo:.~k tet"'rrd.nntion in the ti1w d1root:1 on upon the aohrtions for tbe lui:,goat masured va.lu.as of: tima . It ·was fuJ.t the:'c a repoti-tioo of tl.'!O solution would not bo naoossary~ -the:t the sli ght scatter of the pointm vr.1. 'th time resulting ft•oi:. tho abz"Up-h te:-t:.1.ina:i;ioo. coD.ld be ror,1ovod equo.11;/ well 1:,;r smoo·t;h:1.IiC the ov..rva. V0.1U0G of :r ·vo.lues of o, as sh\J'J'Ti1 in F:lgure0 4 11 5 11 and 60 For this pu.rpooo, the gh~n Te.blG I v10re l)lotted WX'GUS 'l. fot~ ircwloua 'i'he proosm-o d:l.st;rl.,. 69 to ~1uation (2) t he f'10t'1 rato is propo~tioo.a.l to the pt...3Ssure ... squru.-e grooiont. In orc.ror to do,.cornrl.no t hi a gradient the date of li'igt,U'oa 4, 5, and G were used to caloule.to tho OU!"ii'OS of -;;2 Wl'C"'IJ.S " proaentod 6ra9hioally detoirn.:lned fl•o,11 t he so cur""vOa 0 are plo·ttoo. wirsua 7: in. prossuro to ono sid0 of tho vrall io s0on to doereaao as tho E'ntio of the uppliod prossur0 to 'cho initinl pi:."Osnur.G is ino:r0ased. Sinoo a fi. vo.,.fold inc:roaoo in t:.l,c pressure ratio eff'octed or1ly about o. 40 ·10 a A oo:nstant g Aooolero;tio.n d:,e to grnvl ty I f::'lootrlc CUfi'ent i A subaor.·ipt L A 1,.mgch vru:ia.ble [ L] 1 Tllicl::nc.HH3 of t>ro.11 [ L 1l A IT'rJ..'11tJ0X' n A. subsct"ipt S A dir10n.aionlees loo&th variable u Fluid wloci·by [ r..~ • l l x A long-th ValB:'i.13.ble l [ L [ LT -2 J l Z. A dineaeionleiss prossuro .. Glootzsioe.1 :tnpedimoo ( r:iQ• 2 L2 x• 1 J ot. V'.i.scous rend stance ooci'i':iolent [ 1.,•2 J (3 !nort.;:tv.1 rcs:tstanoo cc~fflcient [ L • l l :.? D,,.ma';',ic v.1scoe1.ty [ :rr:1 i•1 J J F'lu :i.d d ;:in.s:i. 'ty [ 7.J_,-3 8poe-:1£'1c i,VBi c:;.7:t [ !!rr.• 2 T• 2 l ~lectrioal potontinl T -2 1 J 2. 000 20000 2. 000 2. 000 • 2. 000 2. 000 2.000 2. 000 2. 000 0 00169 .0322 o•'.½91 0025('~ 00688 o-0591 0050'7 . 0423 00333 9.30 8 0 26 6099 o. ,J 3 7.35 7. 56 7. 6G 7. 7D 5. 53 6. 46 8. 52 8. 57 Jg 1. 00 3. 78 Ce.se s . 14 5o4G 7 0 0G 7. 73 1. 00 3 . 41 3. 43 3.3'1 s.oo 2. 59 2. 79 2. 92 s . 02 3. 08 3. 22 3.Sl 2~83 loGH Z.27 2. 30 8 .2G 10..0 10.0 10. 0 10. 0 10. 0 10. 0 10. 0 10. 0 10. 0 0 . 00645 00169 1. 00 8. 41 4 . 00 4. 00 oll35 2a 1. 00 Case 1. 243 1 . 430 1. 536 l . 59'2 1. 631 1. 654 1. 663 1. 680 1. 433 1. Gl'l 1. 685 1. 726 1. 754 1 . 76'7 1. 775 1. 792 a.21 4:oOO 00966 li 1 . 0()0 C'.a.se 3/o l o307 1 . 377 1. c-01 l . 423 1. 43D 2. 35 2.46 G. J3 c . 02 0. 11 G.35 G. 13 5oBG S. fJG l oOO 2. 17 2. 53 3. 03 4.69 Go24 5.53 5. 79 5. 92 1. 13 1. 00 2. 00 2. 00 2 ol'7 ?'J 2 ov- 2. 53 Z. 67 2.'73 2 . 81. 1. 00 1. 3G 1 . 10 1. 00 l . '76 2. lS 2. 37 4.82 !J:.47 4 . 66 4. 15 2. 57 s.oo 1 . 43 1. 00 1. 00 1.S6 1. 61 1. 79 1. 00 1. 00 1. 00 1. 00 1. 12 1. 220 1.252 1. 232 1. 293 1 . 333 3. 28 3 . 40 s . 12 1. im 2.33 2 .as 1. 00 1. 00 1. 00 1. 5G l o61 1. 50 1. 41 1.:w 1. os 1. 14: 1.00 1.00 1 . 006 1. 074 1. 112 l . 130 1 . 147 1. 153 1ol 63 1. 08'1 1. 1G1 1 . 266 l . 166 1.000 1.002 ?/e, 1. 000 1. 015 3/4 1.0t'.l:8 1. 000 5/8 1. 2G 4 1 . 00 '1lJ IJ .,. 10 n 1. 501 1. 532 1. 546 lo564 l . S92 1. 463 1. 000 1. 110 1. 287 N:-:.:2 lfi Dino110ionleoo Dista.11.00 ,, S 1. 000 1/4 0. 59 8.91 9. DG 9 . 1 '1 l oO'J 'l oGO 4. 00 4 . 00 . 0644 o0-313 .1288 1 . 00 3. 07 S o4S 3. 55 5.,Gl 5o67 3o'70 3. 72 s . 7s 4:. 00 4 . 00 4 o00 4 . 00 l . 870 l.884 1. 000 1 . 000 1. 842 1. 864 1. 000 l o6G2 1. 798 1/8 o01G9 . 0322 . 0491 0 . 1280 . 1135 00813 o09G6 . 0044 0 Dil'X!ns:i.oPJ..oss T:1.r.::a D TABI.E I Ifo.n--ri·ceady I-"r--.3ssur0 D:i.st:r-l b ution in o. Porous tiall t'or T.hreo Difi'm"a:nt P,0.tioo of Applied Pr-~om.-ire tc I nit:i.G.1 ih." essui~e. Ce.lculo.:'coo by Eloctr-ioo.1 An.a.logy· 1. 00 1. 00 1. 000 1. 000 1. 000 1 . 0<X) 1. 000 1.000 1 . 000 1.000 1. 000 l 1.00 1. 00 1. 00 1 . 00 1. 00 1. 00 1. 00 1. 00 1 . 00 1. 00 1. 00 1. 00 1.00 1. 00 1.00 1 . 00 ~ ~..;.1 1--J 0 0 ~n-1,vn 'j >, rn I ; 0 0 Fig. 1 Typical :.:esh Slement for Difference i~quation 0 0 r h-lJ n-, 0 0 73 t 1 I T I I T' -_, - I Fig. 3 Analogous Electrical Network Satisfying the Given Difference Equation 1 I i ·- -+ i --r -- 1 r . - -- 1.1 1.0 ------ - --+' - -l. 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I '• I• ' I ' I • I ; · 1, I • I 1- ! i il;r ;\F:•r y i.I i ~-i:! #it'h 'T l ·., ir ;~~t6~6! r· ~Li-=FP-Hi r - 1 I I i :,'••,., ,J: [:; "}~IT ;m.± H1t;:° I r 1 1 l~l 1 I I I .;,. :'.J(fff ~llt, ,..,. ,.,.• 1 !\]; . ., -~~", .,. I j l -~t- ·I i ·1-1 ,7 i , n , ., , , , 1 111• 1-1_:_~_,_1..u I I w I '! ' ,·:•' • ,il+E Hr~1t i : , , 1 1.11 1..1-j. L, .,. ,_! ~- . •···'• ,,·- , ,-;-,-r ' ·~ -, ' 'j f, '' u ., 1 : :IF :·· ,:'•f~~ : H-h-i- ~~Jr~ :bf lni' ::ii :F!~e~ i i l r ,-; ,.•.• ·rT:-: ·: ·r-rr-l- i~~ 1 Fn :·.:'---;1··:l f1J:t _l;_;__:_;_ ,_ i-+ \ J-H- · 1 ,--,- ,, ,· :-;,· 1·1--: -:- 1 i - -, , 1-rn,Trr:: fll'.=; rr o:1·, x, .,· ,1: :: 1'T' 1,::. 1,1 , i:. 1 1 , , i , - J-·t-i -T·• 1t~_ :\r· t =tiJ=t"{~+t: I I .... ....