A study was undertaken to determine if the Klinkenberg equation for gas flow through porous media could be applied to a wide variety of wood species. When the results proved negative, a second study was initiated to determine if the cause of the nonconformities could be traced to turbulence. The combined experiments indicated that neither molecular slippage, as described by the Klinkenberg equation, nor turbulence can adequately explain the reduction in apparent permeability with increasing mean pressure

Choong, E. T.

Wiley, A. T.

English

Wood and Fiber Science (E-Journal)

SOME ASPECTS OF NON-DARCY BEHAVIOR OF GAS FLOW IN WOOD' A. T .  lliiley' and E.  T .  Choong School of Fo~c l \ t~y  and Wildlife Slanagement, Louisiana State Univcnity, Baton Rouge, LA 70803 (Rccvived 12 Noveml)rl 1974) ABSTRACT A stlldy \\,as nndcrtakrn to drtrrmine if the Klinkenberg equation for gas flow through p o r o ~ ~ s  media conld be applied to a wide variety of wood species. When the results proved nrgative, a second stndy was initiated to determine if the cause of the nonconforrnitirs conld 1 ) ~  traced to turblilence. The combined experinic,nts indicated that neither molecular slippage, as described by  tllc Klinkenberg equation, nor turbnlence can adrquatrly explain the redl~ction in apparent perlneability with increasing mean pressme. Atltlitiol~nl ~C!/IL'OIY~.T: IIard\voods, soft\voods, gas pemreahility, tt~rblllcnt flolv, moleculal flow. INTRODUCTION THEORY A large number of studies on the perme- Darcy's Law for gases expressed the ability of wood to fluid have been made apparent pcr~~~eabil i ty K:, as follows: using Darcy's law. However, Darcy did his work for flow through sand, a porous QaW', K = -  medium that would be expected to have a AF(AP) a different porcL structure than does wood; 3 therefore deviations from Darcy's law are Darcy [cm ( f l u i d  )-c~/sec-atm-cm(wood)l (1 ) cxpccted in wood. In the flow of gases, have been proposed by Adzumi "here Qa (cnlvscc) is the flow rate at and Klinkenberg, and used in wood (Pfalz- Pressure, P:) = 1 AP ncr 1950; Resch alld Ecklund 1964; Corn- (at") is the p'.'%~r'" drop (P,  - P,) be- stock 1968; Sebastian et al. 1965; Siau tween Pressure - PI and down- 1971) to account for molecular slippage stream pressure P,, P is the mean pressure bccausc of decreasing apparent permca- ( PI + P, ) 12, p. ( cp ) is thc viscosity of the bility as nlean pressure increases. permeating gas, A (cm2) is the cross- This study was originally undertaken to sectional area of the sample, and L (cm)  determine, if the Klinkenbcrg equation is the sample length. could be applied to a wide variety of wood Klinkenberg (1941) derived the follow- spccies. \T7hen the results showed consider- ing expression for materials involving mo- al)lc deviations, another study was under- lecular flow: taken to dctcrmine if those deviations coultl bc, attributccl to nonlamir~ar or tur- Ka = ~ ~ ( 1  + b/P) ( 2 )  bulcnt flow. Sincc more is known about the flolY of fluids through rock, it was "here Ks is the true permeability, b is a dcemcd rcllcvant to corllparc its gaseous material constant equal to ( 4 c ~ F ) / r ,  is flow behavior with that in wood. the mean-free-path of gas molecules, c is a constant assurlled equal to 1.0, and r is the t r -  Lllis rcscarcl~ \vas partially supported from radius of the R~lals providcd by thc, Mc1ntia.-Stmnis Coopera- ~~~~~i~~~ 2 illdicates the following flow tivr Research Act. present address is T~~~~ ~ ~ , ~ ~ ~ t  ~ ~ ~ ) d , , ~ t  ~ ~ 1 ) -characteristics, which Klinkcnbcrg verified oratory, L~lfkin, TS 75901. with rock and glass filters: n 7 0 0 ~  ANII F I B I ~  298 WINTER 1975, V. 6 ( 4 )  NON-IIARCY GAS FLOW IN WOO11 I\- ) w o o d  ( 1 1  1.4 4.-. r o c k  - . > U 1.2 n d A - + ~ - - P . ~ . - ~ - . ~  r o c k  0.8 RECl  P R O C A L  O F  M E A N  P R E S S U R E  ( I / F ,  0 t m - I )  0.2 l2~r; .  1. Typical plot of nitrogen gas permeability as a fl~nctiorr of ~.eciprocal mean pressure for scvcral woonl and rock snmplrs. 1 ( 1 ) white elm, ( 2 )  white pine, ( 3 ) fir.] C - 1. Thc apparent gas permeability is a l i ~ ~ c a r  function ot the reciprocal of mean pressure. At high pressure, the pcnneability becomcs a character of thc material and not of the perincat- ing fluid. 2. 'The apparent gas pcrlneability is not tlepcllclent on thr pressure difference. -3. 'Tho constant 11 is expected to be small for high pcrnleability samples, sinec it is inversely proportional to the radius of the capillary. 4. The b-factor is directly proportional to the mean-free-path of the perme- ating gas molecules. - ,-. ,. - - - wood (3)  -- randomly occurring eddies that dissipate fluid motion as heat and increase the re- sistei~ce to flow ( Carman 1956). The Reynolds number ( R , )  is a dimen- sionlcss valuc given by: ~vherc d is the density and v thc scalar velocity of the fluid. The Rcynolds num- ber is an exprcrsion of dynamic similitude, which is to say that flow systenls having the same nunlber will appear roughly the same. For a straight circular tube, the tlansition to full turbulence is abrupt and occurs at a Reynolds number of a ~ ~ r o x i -  Both Eqs. 1 and 2 apply only when flow lnatcly 2200 ( scheideggcr 1957). 1n;rved is laminar, i.e. in smooth streamlines. Tur- tubes, however, the transition to turbulence l)ulc~lcc, is nonlaminar f lo~v consisting of is gradual, 1,c.ginning as a rc,gular circula- 300 A. T. WILEY AND E. T. CIIOONG tory motion, perpendicular to the axis of the tube. During the transition period, a lcss than proportional increase in flow with pressurc drop occurs because of increasing eddy motion. Porous mcdia, having tortu- ous flow paths, would be expected to behave more as curved tubes than as straight onvs (Carman 1956). In wood, there, arc rliany flow channels that are not straight, p,irticularly in the softwoods; thcrc,forc, thc' Reynolds' number at which nonlaminar flow ensues can be expcctcd to 1)e low, ancI if the permeating fluid is c*omprcl\sible, a less than proportional in- crcasc in flow with mean pressure should 1)e o1)scrved bccausc, ot the changing tl~xnsity f'ictor in Eq. 3. l?XPERIhlENTAL PROCEDURE Experiment I In orc1c.r to tcst the validity of the Klink- oil,c~rg c\c!uation, scvcral species of lvood, with two replic~~tions pcr species, were sc~lccted '15 follows: white elm', soft maple, red oak, sycat-norc, hickory, sweetgum, whit(, pint,, fii, Douglas-fir, and baldcy- press 111 addition, four sandstone rock samples wcrc also usctl. Each of thcse sampl(,s n7as subjected to pressurc drops of 5 1 ant1 10.2 cin IIg, respectively, at an upstrcarn prcssurc ranging from 10 to 155 em ITg ah01 c, atmospheric. Nitrogcn gas was used on all samples, ancl helium gas on cigllt \vood s ~ ~ i l ~ p l ~ ~  and two rock sam- p1c.s Floin thcsc measurements, linear rc- grcssion analyses werc madc to determine thc \lop? ot thc relationship between K, and 1/T, for thc two prcssmc drops and I luitl media Experiment ZZ The rc%sults of the above cxperiment showcd iilinost no confornlity to character- "'I 'l~c ltc.?;nolds number presented here is to sllo\v, in a general way, 11ow different flow paranlc.trrs could affect the degree of turl)ulcnce of a flo\x. system. ' Botanical scientific names are omittcd here sirlcc the \voods were: selected for convenience to pl.ovidr. a \vide range of pernteal~ility values and  rot for spc.cirs rcxprc-sentation. istics 2, 3, and 4 of the Klinkcnbcrg equa- tion; thercforc,, anothcr experiment was undcrtakcn to determine if the dcviations could be traccd to nonlaminar flow. The following wood species wcre used, also with two replications per species: honey- locust, soft maple, hard maplc, Douglas-fir, hcmlock, and southcrn pine, as well as the four aforementioned rock samples. Sincc molecular slippage is a function of the nlcan pressure, the mean pressure was maintained constant at 40 cm Hg abovc atmospheric, and only the pressure drop was varied from 0.147 to betwecn 3.6 and 70 cm Hg. The upper limit of the pressurc drops was dependent upon the specimen pcrmcability, since largc pressure drops could not be maintained across highly permeable specimens at the specified mean pressures. A water manometer was used when thc pressurc drop was belo\v 4 cm Fig. The apparent permeability was plotted against pressurc drop for each specimen. Since thc flow appcared to be laminar at 0.37 cm Hg pressure drop for all specimens, Klinkenbcrg extrapolations ( to infinite pressurc) were carried out at  this pressurc drop as in Exprrirnent I using both heliunl and nitrogen Extrapolations wcre also collducted at 5 cm IrTg prcssurc drop for cornpsi 1-'  son. Thc apparatus used for permeability i~l~asurcment was similar to the one de- scribed by Choong et al. (1974). Thc upstrc-am pressure was adjusted by means of a pressure regulator, and the down- stream prcssurc by a metering valve. Cas was prcvented from flowing arouncl thc spccimen by lining the holder with a prcs- surized rubber sleeve, which held the sam- ple tightly so that no flow could occur around the edges. The rate of gas flow was n~casurecl by determining the time for a film of soap inside a buret to travcl bet\veen two marks, delineating a known volume. Diftcrent burcts were used, so that the measured volume ranged from 1 to 1000 cc. The size of the test sa~nple wu5 X-inch i n  cliamctcr ancl about %-inch NON-DAHCY (:AS FLOW IN WOOD 30 1 100 - - - W O O D  - - o H e l i u m  Nitrogen PRESSURE D R O P  ( AP,  cm Hg ) Flc;. 2. Typical plot of tlrc logarithm of apparent gas permeal)ility as a fimction of thc logarithm of 1>1.1,~~111. (~  drop for several wood samples iisirrg l~el i i in~ :111cl ilitrogc11. 1 ( 1 ) I I ~ I I C ' ~ ~ O C I I S ~ ,  ( 2 )  soft ~rlaple, ( :3) hard maple, ( 4 )  11crnlock.I i l l  lcngth. Flo\v was in thc longitudinal tlircction. All thc sanlples wcrc storccl in- side a small desiccator containing P,O,; thu5 thdr  nloisture contcnt5 were ncar zcro pcrccnt. The fluids used were nitrogen ( h  = 9.29 x 10 "cm and p = 1.78 x lo-' poise) ;illd heliunl (A = 27.45 x 10 Qm, and ,L = 1.94 x 10.' poise). RESULTS Typical plots of K, vs. 1 / P  obtaincd from Lt:x~x~i~n(~nt I with nitrogc.11 arc shown in Fig. 1 for both rock and wood samples. In rock, t h ~  slope is horizontal, showing no apparent etfcct of pressure drop over the range studied. On the other hand, in wood thcb plotr generally are lower and with somcwhat steeper slopes at 10 cm than at 5 cm IIg pressure drop. The effect of pressure diop became morc pronounced in the higher perme'ibility range. For woods with permeability less than 0.2 Darcy, the cffcct of pre5sure drop was negligible. The results of Experiment I1 arc reprc- A. T. WII.EY AND E. T. CHOONG o Hel ium R O C K  Nitrogen 0.1 I  I  I 1  1 1 1 1 1  I  I  I I  I 1  1 1 1  I  I  I 1  l l l l  0.1 I 10 100 PRESSURE DROP ( a P ,  cm H g )  FIG. 3. 'Typical plot of the logaritll~n of apparent gas pcrlncability as a fnnction of the loaaritl~nl of p~.t~ssrn.c drop for three rock s:i~~lples using heliun~ a n d  nitrogi.n. scmtetl by the relationship bctwecn pcrme- ability and pressurc drop. Figurc 2 shows typical plots of log K,, vs. log AP for representative wood samples with nitrogen and helium, whereas Fig. 3 shows similar plots for the rock samples. In most cases, tho curves appeared horizontal up to son~c critical value of AP, then began to drop 5harpIy with an increasc in P due, pre- sumably, to turbulence. This critical point is a function of permeability, type of po- rous mcclia, and fluid rncdium. For thc salnc cornparativc, perineability value, the critical point occurrc~d at  a much higher AP value in rock than in wood; &ap- parently no turbulence (i.e. no critical point within the range pf pressures used) can be assunled to occur at permeabilities below 1 Darcy for rock and 0.2 Darcy for \vood since thc d o t  remained horizontal L throughout the entire measured range of AP. \Vhen the permeating fluid was he- lium. thc rock sam~les  exhibited no critical point, while the wood samples show critical points only at permeabilities higher than about 1 Darcy. NON-DAHCY GAS FLOW IN WOOD 303 TABLE 1. Vul~ies f o r  varior~s ratios o f  h-factors in u:ood Ratio Expected Value Observed Value Reference bOxygen/b~i r bOxygen/b~ir  btiel i u r n / b ~ i  trogen b ~ e l  iu ln /b~ i  trogen 1.083 Pfalzner  (1950)  1.171 Resch and Ecklund (1964)  1.51-2.35 Cornstock (1968) 0.00-2.32 Present  study IIISCUSSION OF RESULTS The more pronounced cffcct of pressure drop on apparcmt permeability of wood, as compared with rock may be traced to the structure of the porous media. I t  has been shown by Meyer (1971) that completely 1)locked flow paths within short distances of cach other can be expected to take place in wood. \\'here paths are not completely l)lockcd, only one bordered pit between two tracheids could bc espccted to remain opt'11. This tortuosity and restriction to olily one or a very fc\v small flow channels pclr parallel path greatly restrict flo\v through wood and probably account for the dcviations from permeability as pre- dictccl by the equations. Also, in view of the, lack of siinilarity in flow paths between soft\voods a i d  hardwoods, other factors contril)uting to this cffect should bc con- sitlcrccl. For instance, in capillary flow, a cvrtain arnomlt of inertial resistance duc to velocity changes 011 eutrancc and exit of fluid from the capillary is creatcd ( Car- lnan 1956). The amount of inertial re- sistanccb increases not only with a reduction in capillary length, but also with an in- crcasc in fluid density and velocity, and thcrcforc, could result in decreasiilg appar- c11t pc:rnleability with increasing pressure (hop and mcwl prtsssurc. In softtvoods, whcrcb most of thc rc~sistancc to flow takcs placc i l l  vcLry short capillaries, namc>ly thc pits, incrtial resistance may bc pronounced. In hard\voods, particularly ring-porous, a large, variation in pore sizes rcsults in a high percent of flow volume taking place through a small fraction of thc speciincn cross scction. I11 rock, the, porc size is Figures2 and 3 show that the pcrmea- bility to ht:lium was less subject to the effect of pressure drop than the perinca- bility to nitrogcn. Thc reason is due to thc density factor of E q  3, for nitrogen is seven tirnes more dense than helium. The 1)-factor for nitrogen (by,), which was obtained froin Eq. 2, was generally below 0.4 for permeabilities ranging from 0.1 to 10 Darcies. Only two sainples show a high b-factor (i.c. over 1.0).  These sam- ples have Iocv pernieability values of about 0.1 Darcy. Contrary to the prediction of the Klinkenberg equation, there was no relationship between K, and by,, but this may have been due to largc sanlplc vari- ation since permeability is a function of both pore size and pore density. The mean-free-path of a helium rnoleculc is 2.95 times greater than that of nitrogen (West 1971); thus, if the Klinkenberg cquation is applieablc to wood and its slope bK, is \vholly due to molecular slip- page, then the ratio b I f ( % / b N 2  should be equal to 2.95. On the other hand, if a sigllificant portion of the slope is due to molecular slippage and the rest to 11011- laminar flow, the ratio should tend toward 2.95 with dccreasiilg permeability as well as pressure drop. Analyscs of the rcsults show no correlation between this ratio and pcnneability. Thc ratio was less than 2.95 in all cases. Also, thrrc was no tendency for the ratio to dpcreascx with an increase in pressure drop. The extrapolations car- ried out at 0.37 cnl IIg pressure drop intli- cated less tendency to conform to the Klinkenberg equation than at any othcr prcssurc drop uscd. The values of the hrr,./bs., ratio may l ~ e  comparecl \\7itl1 data ol~taincd in other 304 A. T. WlLEY AND E. T. CHOONC studies. Pfalzner (1950) compared the flow of oxygen and air through an unidenti- ficd species of wood. Resch and Ecklund (1964) compared the flow of oxygen and nitrogen in redwood. Both used the Ad- zumi approach, which does not generate b-factors, but the values may be readily calculated from the data. Comstock (1968) uscd the Klinkenberg approach to correct for molecular slippage on hclium and ni- trogen in eastern hemlock. The results are suminarized in Table 1. The observed ratios of Pfalzner and of Resch and Ecklund are in close agreement with the expected values, probably because of the similarity in density, viscosity, and mean-free-path of the gases they studied. Cornstock's data showed a smaller range of variation in the ratio, as compared with this study, which may be due to smaller sample variation since he studied only one species. There was no tendency for the bHe/bN2 ratios to tend toward 2.95 as the pressure drop was decreased; therefore it &not be concluded that the slope of the K, vs. 1/P rcIation was due to a combination of mo- lecular slippage and turbulence; nor can it be concluded that turbulence was the only factor causing the slope, as this would have given rise to the ratios being less than one since nitrogen is more subject to turbulence than heliurn. I t  appears that in wood a third factor that cannot be explained in terms of inolecular slippage or-by turbu- lcncc, as normally conceived, may also affect the flow of gases in wood. CONCLUSION From this study, it can be concluded that the flow of gas in wood is different from that in rock. The gas permeability of wood is greatly affected by pressure drop at a high range of pressures and permeabilities. Molecular slippage alone cannot account for this deviation from Darcy's law; effect of turbulence and other factors must also be taken into con- sideration. REFERENCES CARMAN, P. L. 1956. Flow of gases throngh porous media. Rutterworths Sci. Publ. Lon- don. 182 pp. CIIOONG, E. T., F. 0. TESORO, AND F. G. MAN- WILLER. 1974. Permeability of twenty-two small diameter hardwoods growing on south- ern pine sites. Wood Fiber 6(1):91-101. C o l r s ~ o c ~ ,  L. (>. 1968. Physical and structural aspects of the longitudinal permeability of wood. Ph.D. dissertation, State Univ. New York, Coll. Forestry, Syracuse. 285 pp. KLISKENBERG, L. J. 1941. The permeability of porous media to liquids and gases. Am. Pet. Inst. Prod. Prac. 200-215. MEYER, R. W. 1971. Influence of pit aspiration on earlywood permeability of Ilouglas-fir. Wood Fiber 2 ( 4 )  3328-339. PFALZNER, P. M. 1950. On the flow of gases and water vapor through wood. Can., J. Res. 28A:389-410. RESCH, H., AND R. A. ECKLUND. 1964. Permea- bility of wood-exemplified by measurements on redwood. For. Prod. J. 14(5):199-206. SCHEIDEGGER, A. E. 1957. The physics of flow through porous media. McMillin Co., N.Y. 236 pp. SERASTIAN, L. P., W. A.  CAT^, JR., AND C. SKAAR. 1965. Relationship of gas phase permeability to ultrastrncture of white spruce wood. For. Prod. J. 15(9 ) :393-404. SIAU, J. F. 1971. Flow in wood. Syracuse Univ. Press, Syracuse, N.Y. 131 pp. WEST, R. C. (Ed. ) 1971. Handbook of physics and chemistry. 52nd ed. Chem. Rubber Co., Cleveland, 0. 

Some Aspects of Non-Darcy Behavior of Gas Flow In Wood

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Some Aspects of Non-Darcy Behavior of Gas Flow In Wood

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