AN ELECTRICAL CONDUCTIVITY METHOD FOR DETERMINING THE EFFECTIVE CAPILLARY DIMENSIOXS OF WOOD BY ALFRED J. STAMMI
Four dynamic physical methods for studying capillary structure have been developed and applied a t the Forest Products Laboratory to the determination of the effective dimensions of the fine, continuous, capillary structure of softwoods.2 An electroendosmotic flow method was used for determining the effective continuous capillary cross-section of wood. Data obtained from these measurements were combined with data from hydrostatic flow studies for calculation of the average diameters of the effective openings. Although the electroendosmotic method proved to be of considerable use in determining the average and the limiting values of the various fiber lengths, it is too indirect and inaccurate a method for quantitative capillary crosssection determinations. With wood sections of low permeability, in which the resistance to flow was high, measurements of the velocity of electroendosmosis could not be made with greater accuracy than 2 0 per cent.4 The determination of the increase in electroendosmotic velocity with an increase in capillary cross-section introduced another possible error resulting from the assumption that the increase in capillary cross-section caused by the drilling of fine holes in the sections could be calculated from the cross-section of the bit. The calculated increase in cross-section may have differed from the actual increase by as much as 50 per cent because of the distortion of the holes and the unavoidable tearing and brooming of the fibers. Another approximation, that of estimating the length of the effective capillaries, was necessary in combining the electroendosmosis data with data obtained from hydrostatic flow studies in order to obtain values for the average effective capillary diameters. This estimate could be made with an accuracy of approximately j o per cent. The final calculated diameters were therefore determinable with no greater certainty than a possible error of I O O per cent. This low order of accuracy seems entirely reasonable, however, when it is realized that these openings are in general below microscopic visibility in size. Nevertheless more accurate results were desired for use in connection with a theoretical study of the rate of drying of wood. Because of this need the method described in this paper was developed. Forest Products Laboratory, Forest Service, L-. S. Department of Agriculture maintained a t Madison, Wis., in cooperation Kith the University of Wisconsin. Stamm: Colloid Symposium Monograph, 6, 83 (1928). Stamm: J. Agr. Research, 38, 23 (1929). 0. T. Quimby, working in this Laboratory, found that the inconstancies were due to thermal and polarization effects. The use of non-polarizing electrodes might have given more accurate results, but because of the development of the neu. method this was never tried.
THE EFFECTIVE CAPILLARY DIMENSIONS O F WOOD
313
Structure of Softwoods Electrical conductivity measurements have been used in the past for determining the ratio of the effective capillary length to the effective capillary cross-section of porous material^.^ None of the materials investigated, however, were made up of capillaries of two such distinctly different orders of magnitude as those in wood. Because of this complication a brief consideration of the structure of softwoods is necessary. The fiber cavities that make up the major part of the void volume of softwoods are closed a t both ends, the only communication from fiber cavity to fiber cavity being through the pores in the membranes of the bordered pits. It is the size and the number of these finer openings that control the permeability of wood. Each softwood fiber with an average length of about 0.3 cm. and a diameter of approximately one hundredth of this value has from 30 to 300 pits connecting it with adjoining fibers, and there are from 50,000 to IOO,OOO such fibers in a square centimeter of cross-section.6 Fig. I gives a diagrammatic representation of the capillary path through wood, indicating the manner in which the fiber cavities (shown in black) are connected through the pores of the pit membranes. Method for determining the Ratio of the Effective Capillary Length to the Effective Continuous Capillary Cross-Section The ratio of the effective capillary length to the effective continuous capillary cross-section can be calculated from the electrical resistance of sections of wood, the voids of which are completely filled with a salt-solution, and from the specific resistance of the salt solution in bulk, providing the surface conductivity is made negligible and the conductance of the cell wall is negligible. The surface conductivity for very dilute salt solutions and especially for distilled water may be many times the bulk conductivityof the solution or of the ~ a t e r , * , ~but , ~ ,for ~ *salt * solutions of appreciable concentration this surface conductivity becomes neg!igible in comparison with the bulk conductivity. When the potassium chloride solutions used in this investigation exceeded a concentration of 0.07 mol per liter, the ratio of the specific resistance of the salt solution in bulk to its resistance in the wood structure was found t o be independent of concentration, thus indicating that the surface effects are negligible. Hence measurements were made using salt solutions exceeding this concentration. The specific conductance of dry wood is extremely small, approximately IO+ mho for a cube of wood I cm. on an edge. When water is adsorbed by the cell wall the conductivity increases so as to give a linear relationship Fairbrother and Mastin: J. Chem. Soc., 125, 2319 (1924); Hitchcock: J. Gen. Physiol., 9, 755 (1926); Marshall: J. Soc. Chem. Ind., 46, 373T (1929). For a further description of the capillary structure of wood, see Stamm: Colloid Symposium Monograph, 4, 246 (1926). D. R. Brigas: J. Phvs. Chem., 32, 641 (1928). McBain, Peaker and King: J. Am. Chem: SOC.,51, 3294 (1929); McBain and Peaker: J. Phgs. Chem., 34, 1033 f ~ g j o j .
'
3'4
ALFRED J. STAMM
between the logarithm of the conductivity and the moisture content, up to fiber saturation, a t which point the specific conductance is 3 X IO^ to I x-5 mho.g The water adsorbed in the cell wall at this point shows an increased conductivity over the bulk conductivity of water, because of surface conduction. Salt solutions, however, act differently. The conductivity of such solutions a t the fiber-saturation point, although exceeding that of water, is sufficiently less than the bulk conductivity to make the conductivity of the cell wall in the presence of the salt negligible for the present measurements. This fact may be illustrated by measurements made upon a transverse section of Douglas fir sapwood. The section when completely filled with a potassium chloride solution (0.199 mol per liter and 39.1 ohms specific resistance) had a resistance of 26.0 ohms. The section contained 1.907 gm. of salt solution and 1.869 gm. of water. When the section was dried to approximately the fiber-saturation point, 29.3 per cent water, the resistance was 5,100 ohms. Although the capillary cross section effective for electrical conduction was reduced in proportion to the liquid lost, the concentration of the salt was increased in the same proportion. The electrical resistance should have remained approximately constant if the liquid in the cell wall were fully as conductive as the free liquid in the cell cavities. The increase in resistance, however, is nearly two hundred fold, showing that the salt solution can not be dispersed in the cellwall in a continuous manner. Neglecting the resistance of the cell wall in parallel with the fiber-cavityresistance introduces an error of not more than 0.5 per cent. Further evidence that solutes do not become dispersed in the cell wall in a continuous manner as water does, but on the contrary are confined to the grosser capillary FIG.I spaces, is given in a previous investigation on the effect of Simplified diagramm a t i c representa- solutes upon the apparent density of the wood substance.l0 tion of the capillary The electrical conductivity of a transverse wood path through wood showing the manner section, the capillary structure of which is filled with a salt in which the fiber solution, is thus substantially equal to the sum of the bulk cavities are connected through the conductivities of the solution in all of the individual pores of t h e p i t capillary paths connected in parallel. These capillary membranes. paths in turn are made up of fiber cavities and pores of pit membranes connected in series (Fig. I ) . The part of the electrical resistance for which the fiber cavities are responsible can be calculated from the fractional void volume, V , which in turn can be calculated from the bulk 10
Stamm: Ind. Eng. Chem. Anal. Ed., 1, 94 (1929). Stamm: J. Phys. Chem., 33, 409 (1929).
THE EFFECTIVE CAPILLARY DIMENSIONS O F WOOD
315
density, d, of the wood on a wet volume and dry weight basis, and the density of wood substance, do, which is equal to 1 . 5 2 gm.per cu. cm. Thus,
V
= I
- d/d,
(1)
The fractional void volume obtained in this way includes all void structure. For softwoods free from resin ducts this void volume is made up mostly of fiber cavities, together with the water-filled void structure of the swollen cell wall and the void volume of the ray cells. The ray cell voids, which amount to only I or z per cent of the total, have been neglected to simplify the calculations. The fractional void volume of the cell wall is equal to the product of the moisture content, M, per gram of dry wood a t the fiber-saturation point, and the density of the wood, d. Then the fractional void volume of the fiber cavities per cubic centimeter of wood VJ -- 1’ - M d (2) This void volume is made up of a longitudinal component of I centimeter, and radial and tangential components that approach equality. The void cross-section of the average fiber cavity for transverse sections
Then the electrical resistance of the combined fiber cavities
where R,, is the specific resistance of the salt solution, L the number of unit thicknesses of the section under investigation, and Q the number of unit cross-sections of the specimen. The total resistance, R,, of the salt solution in the pores of the pit membranes will depend upon the fractional cross-section of such pores traversed in parallel and the length of path, that is, the continuous effective capillary cross-section, ym, of the pores expressed as a fraction of the cross-section of the specimen, and the sum, of the thicknesses of the pit membranes traversed in series. Then,
The experimentally measured resistance, R, is equal to the sum of the resistances from equations (4) and (j).
and
for transverse sections.
3 16
ALFRED J. STAMM
For radial sections, where the flow of current is in the tangential direction] equation (7) takes a somewhat simpler form since
Although the radial and the tangential components of the void volume are not exactly equal, they may be considered so for this calculation, since Rfis so small in comparison with R, that a deviation of 50 per cent in R, will cause a n error in Rmof only z per cent.
FIG.2 Cells used in electrical resistance measurements.
The measurements were made upon wood sections that were clamped between the faces of two hard rubber cells and upon cylindrical sections that had been turned in a lathe in a soaked and swollen condition to fit tightly into the ends of the cells (Fig. 2). Several sets of cells and electrodes of different cross-section were used for the measurements. The electrodes, E , were made of heavy platinum wire wound in the form of disk coils and sealed into glass tubes, T,that were filled with mercury for electrical contact. The electrodes were fitted into the cells with rubber stoppers, R. The side tubes, SIserved in filling the cells with the salt solutions. The electrodes were freshly platinized each day before using. The measuring apparatus consisted of a student circular slide-wire bridge, a four-dial resistance box (0.1 to 999.9 ohms), a microphone hummer, telephone receivers, a condenser, a switch, and dry cells connected according to standard conductivity measuring practice. The sections were soaked in distilled water for a t least two weeks. To facilitate the replacement of air by water the soaking was done in a vacuum desiccator to which suction was applied intermittently. Potassium chloride
THE EFFECTIVE CAPILLARY DIMENSIONS OF WOOD
317
was then added. Electrical resistance measurements on the wood sections showed that diffusion was complete and equilibrium of the salt distribution was obtained in less than a week. The specific resistance of the salt solution in equilibrium with the wood sections was determined, using the hard-rubber cells of Fig. 2 clamped together with a ring gasket replacing the wood section. Measurements were made with the electrodes separated by different distances. All measurements were made at 2 joC in a thermostatic air bath. Experimental Results The differences between measurements made with broad transverse sections clamped between the faces of the cells and cylindrical sections cut to fit the cells were first investigated. Current in the broad sections fanned out appreciably beyond the area bounded by the cells. The extent of the spreading of the current proved to be a function of the specific electrical conductivity and of the thickness of the sections and was independent of the cross section of the cell. It is thus possible to correct for this spreading of the current for transverse sections when measurements are made on the same specimen Rrith cells of two different cross sections. Although the extent of the spreading will increase from zero a t each of the surfaces of the section to a maximum at the center, it is simpler for these calculations to consider an average effective extent of spreading X. This must be added to each of the cell radii in order to calculate the effective cross section. Then,
where r, and T ? are the cell radii of two different cells and R1 and Rz are the corresponding resistances. Table I gives the specific resistances calculated in this manner, using three different sizes of cells, as well as the results obtained from measurements made on transverse cylindrical sections cut to fit the cells. The values obtained by the two methods agree quite well. There is a slight tendency, however, for the specific resistance to be slightly less for the cut sections; this will be considered again later in this article. Measurements made in equilibrium with two different concentrations of salt solution are given in Table 11. The results agree within experimental error for the concentrations used. The values of lm/ym for different thicknesses of the transverse sections are plotted in Fig. 3 for all of the Sitka spruce data given in Table I and Table 11, together with data for two other Sitka spruce specimens of different density and two species of cedar. The data in all cases show a linear relationship between /,,/am and the thickness of the section; the graphs when extrapolated to zero thickness pass through the origin. This indicates that the continuous effective capillary cross section, y m , varies but slightly for adjoining sections and that the length of the effective capillaries, I, varies directly with the thickness of the section, thus indicating a rather uniform distribution of
318
ALFRED J. STAMM
TABLE I Comparison of the Specific Electrical Resistance of Transverse Sections of previously Seasoned and Resoaked Sitka Spruce calculated from Measurements made upon Cylindrical Sections cut to fit the Cells and Sections extending beyond the Cell Cross-Sections. Density of the wood (volume green and weight oven-dry)-o.297 gm. per cu. cm. Concentration of potassium chloride solution-0.8 I mol per liter Specific resistance of potassium chloride solution-10.3 I ohms Kind of section
Thickness of section
Extending
0.61j
Cm.
Measured resistance
Extent
Cm. 0.870
Ohms.
,430
1g.00
245
,870
40.80 6.00
Cm. 0.0398 ,0398 ,0399 '0399
.430
20.00
'245 ,870 .430 ,870
46.80 7.75 24. 5 0
,870
10.48 32. I O 14.40 j8,80
Radius of cell
'
I
I
cut Extending cut
,062
,481
,481 2.03I I
2.031
,430 .870 ,430
4.01
IO.
40
of
spreading X
1
,1030
,/
,1030
1 ,1037 '
103 7
'I37 . '37 ,000
.156 ,156 ,000 ,000
Calculated Effective specjfic area of resiscross tance of section section 2.600
Cm.2
Ohms. 16.96
,694
16.92
,255
16.92 16.80 16.87
2.974 ,896 ,382 3.186 1.010
2.378 3.307 1.079 2.378 ,581
16.82 16.70 16.70 16.68 17.07 17.05
16.85 16.82
equally unobstructed capillaries. Such a relationship is rather to be expected, considering the large number of fiber cavities and pit membranes traversed by the current. The ratio Zm/qm per unit thickness for the specimens of Sitka spruce of different density varies directly with the density (Fig. 3 and Table 111).The simplest explanation of this relationship is that the pit membrane thicknesses vary directly with the density, while qm remains practically constant. Values of qm, however, will vary from species to species. Fig. 4 and Table I V show the differences in the ratio lm/q,,, for sapwood and heartwood. The differences in effective capillary cross section are surprisingly small, thus indicating that the large differences in the permeability of sapwood and heartwood are due to another cause. This matter will be considered in more detail later in this article.
THE EFFECTIVE CAPILLARY DIMESSIOKS OF WOOD
3'9
P
FIQ.3 Relationship between the ratio of the effective capillary length to the effective continuous capillary crow section, and the thickness of the section, for spruce and cedar. 0 Sitka spruce, density 0.297 gm. per cu. cm. Sitka spruce, density 0,347gm. per cu. cm. 8 Sitka spruce, density 0.370 gm. per cu. cm. c) Alaska cedar, density 0.442 gm. per cu. cm. o Western red cedar, density 0.290 gm. per cu. ern. x Average fiber length for -4laska cedar y Average fiber length for Sitka spruce and western red cedar
r-
FIQ.4 Relationship between the ratio of the effective capilla length t o the effective continuous capillarv cross section, and the x c k n e a s of the section, for Douglas fir and slash pine. 0 Coast Douglas fir, sapwood 0 C o a t Douglas fir, heartwood o Slash pine, sapwood c) Slash pine, heartwood x Average fiber length for ine y Average fiber length for bouglas fir
320
ALFRED . I . STAYY
TABLE I1 lleasurements made in Equilibrium with Two Different Concentrations of Salt Solution upon previously Seasoned and Resoaked Transverse Sections of Sitka Spruce cut to fit the Cells. Densityof the wood (volumegreenandweight oven-dry)-o.297 gm. per cu. cm.
Thickness of section
Radius of section
Cm.
Cm.
0.751
0,870
,430 ,870
I
481
2.031
Concentration of potassium chloride in mols per liter
Specific resistance of potassium chloride solution
Ohms. 10.31 10.31
Ohms.
0.810 ,810
j8.00
30.4 123.7
Resistance of section
lm!qm
0.196
5.4
. I94 '197 . I87
22.1
,430
I34 ,134
,870
.8IO
10.31
10.4
.870
'
I34
j8,oo
j8.8
,323 '340
'430
'
I34
58.00
240.5
,333
,810
10.31
14.4
,810
10.31 j8.00
58.8 81.3
j8.00
332.0
,870 ,430
'
,870
'
,430
'
j8.00
I34 I34
.484 ,480 492
,491
TABLE I11 Effect of the Density of previously Seasoned and Resoaked Sitka Spruce upon the Ratio of the Effective Capillary Length to the Effective Continuous Capillary Cross Section Density of specimen Gm. Cm.j 0.297
,347 ,370
ldym
for average fiber length
0.091 . I08 ,115
Column Column
2 I
values values
0.306 ,312 ,311
THE EFFECTIVE CAPILLARY DIMENSIONS O F WOOD
321
TABLE IV Comparison of Measurements Made upon Unseasoned Transverse and Radial Sections of the Sapwood and the Heartwood of Slash Pine Sapwood, density (volume green and weight oven-dry)-o.456 gm. per cu. cm. Average number of fibers and rays traversed per mm. in the tangential direction--~ 4 .o Heartwood, density (volume green and weight oven-dry)-0.430 gm. per cu. cm. Average number of fibers and rays transversed per mm. in the tangential direction-33 .o Concentration of potassium chloride solution-0.79 mol per liter Specific resistance of potassium chloride solution-9.84 ohms Part of wood
Section
Thickness Radius of of section
Resistance of section
Zrn/pm per pit memIrnlprn brane traversed in series
Observed Corrected
Cm. Sap
Transverse
I . 940
15.00
.365 .960 .640 1.890 1.395 1.965 1.450 .922 ' 585
10.65 7.38 5.05
0.608 ,453 .292 ,229
58 ' 5 0
,588
43.00 15.30 11.30
.426 '783 .587 ,375 ,267
I
Heart
1.925 I . 400
Sap
Heart
Radial
I . 830
'750 I ,380 '970 I . 680 I . 240 I . 230 ,900
7.20
4.68 59.90 43.80 163.0 68.7 416.0 306.0 158, 5 118.5 391.0 289.0
'0.055
l.070
,771 '575
42.2 17.8
0.0677 ,0698
26.8
.OS71
19.7
20.08~o
,0578 .0743
41.2
30.8 25.2 18.6
,0752
2.0927
.0622 .062 7
_ _
For a travel of half the average fiber lenath: taken from Fig. 2. For the average diameter of t h e fibers a n d rays traversed, corrected for leakage of surface current acroea the sections.
Table IV gives also the data for radial sections of both the sapwood and the heartwood of slash pine. The values for lm/qrnin this case were obtained from equation (9). The number of fibers and ray cells traversed in the tangential direction per unit of distance was determined by direct microscopic
322
ALFRED J. STAMM
measurements upon transverse microtome sections.ll The values of ln,'qnr per membrane traversed in series for the sections turned on a lathe so as to fit the cells vary with the radius of the cell, because of a slight leakage of current over the cylindrical surface of the sections. -4ssuming the thickness of this leakage film to be the same for both cells, the total leakage current will vary directly as the circumference and consequently as the radius of the cell. The leakage per unit of cross section, however, will vary inversely as the radius of the cell. Therefore,
in which R, and Rr are the measured resistances reduced to unit dimensions for the cells with radii r1 and rz, respectively, and C is the leakage conductance for rl. This conductance, in general, amounts to 18 to 2 0 per cent of the total conductance. Presumably there is a similar surface leakage of current with the transverse sections previously considered, but because the resistance of those sections is less than a tenth of the resistance of the radial sections, the error will be less than z per cent. This fact will account for the tendency for the measurements on the cut sections given in Table I to be slightly less than those for the broad sections. No measurements were made on broad radial sections because the necessary correction would be complicated by the longitudinal spreading of the current, which is greater than the spreading in the radial direction on account of the difference in conductivity in these two directions.'? The reason for the deviations between the values for l,,Jq,,, per membrane traversed in series for the transverse and the radial sections is that the values for the transverse sections were calculated by assuming a uniform distribution of the pit membranes along the length of the fiber. Only with such distribution does the current pass on the average through no more than one pit membrane in traversing a distance of half a fiber length. Microscopical observations, however, have shown that the pits are much more numerous near the ends of the fibers, thus actually requiring a greater section thickness than half a fiber length for the current to pass on the average through even one pit membrane. The flow conditions can perhaps be better understood by considering the current that passes through the fiber cavities as being made up of a bundle of threads in parallel, each thread entering a fiber cavity through a single pit membrane opening and leaving through an opening in another pit membrane. A thread may enter a t one pit and leave through the nearest pit or it may enter through a pit close to one end of the fiber and leave through a pit a t the other end. Hence the possible paths for these filament currents vary from a negligibly short length to practically the full fiber length; the actual location of the pits, of course, gives an average length of path greater t>hanthe uniform location assumed. 11 These measurements are further described in "A New Method for determining the Proportion of the Length of a Tracheid that is in Contact with Rays,'' by Stamm, Botanical Gazette, 92, I O I (1931). 1%Stamm: Ind. Eng. Chem., 19, X O Z I(1927).
THE EFFECTIVE CAPILLARY D I M h S S I O N S O F WOOD
323
An approximation of the actual value of ym can be made from the value of 1, y m for transverse sections taken from the graphs for a section whose thickness is equal to that of half the average fiber length, and from the thickness of a single pit membrane, which was found microscopically to range from 2 x 10-5 to 2 x I O - ~ C ~ . it ; would perhaps be better to use ZJq, for radial sections having a thickness of the average fiber and ray diameter. Dividing 2, by l,,,’y, gives a value of ym rangjng from 0.3 X IO-^ to 8.0 X I O + for all of the species studied. The electroendosmosis method gave values of ym ranging from 0.9 X IO+ to 1.7 X I O - ~ . ~ J This agreement is quite satisfactory when the large uncertainty of the effective value of 1, is considered.
Effective Capillary Dimensions The preceding data for Z,,,/q,,, were combined with data from hydrostatic flow studies2 l 3 to calculate the average effective capillary radii. Measurements were also made by the method for overcoming the effect of the surface tension of water in the capillary system to obtain the maximumeffective capillary radii. All of the data are assembled in Table V. The maximum radii range fromabout I t o 6 times the average radii. The radii obtained by overcoming the effect of surface tension decrease with an increase in the thickness of the section. This is to be expected; the most effective path through a number of pit membranes in series will approach the average effective path as the number of pit membranes in series increases, because of the decreasing probability of all the pit membranes in series containing pores of maximum size. For example, measurements made upon radial sections, in which the tangential displacement of water by gas is through the same structure, except that far more pit membranes are traversed in series per unit thickness of the section, gave a value of r for the heartwood of slash pine of 4.0 X IO-^ and for the sapwood of 3 . 7 X IO-^. The number of pit membranes traversed inseriesforthese sections was approximately 40. The data thus show that the radii obtained by the method of overcoming the effect of surface tension approach the values obtained by the electrical conductivity and the hydrostatic flow method. The fact that these two entirely different methods of measuring the size of the effective openings give results of the same order of magnitude provides confirmation of the validity of the methods. The data further show the large differences in the effective capillary dimensions for the sapwood and the heartwood of slash pine. The difference between the capillary dimensions of the sapwood and the heartwood of the Douglas fir specimens is much less. This can be partially attributed to the presence of ring shakes, that is, cracks between the annual rings in the heartwood, which tends to increase the heartwood values. Treatment of Douglas fir with creosote indicates that for this species there is an appreciable penetration through ring shakes. l3
Stamm: Physics, 1, 116 (1931)
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ALFRED J. STAMM
5
THE EFFECTIVE CAPILLARY DIMENSIOKS O F WOOD
325
summary
A method has been developed for determining the ratio of the effective capillary length to the effective continuous capillary cross section by means of electrical resistance measurements of salt solutions filling the wood structure, and the resistance of the solutions in bulk. 2. Measurements made upon sections cut t o fit the electrode cells and sections extending beyond the effective area of the cells agree when a correction is made for the spreading of the current in the oversize sections. 3 . The concentration of the salt solution does not affect the results when it exceeds 0.07 mol per liter. 4. The ratio l,,,/q,,, for a single species varies directly with the density of the wood. 5 . The values of the ratio l,/qm do not differ greatly between the sapwood and the heartwood. 6. The values of the ratio 1,,,!’qm per pit membrane traversed, calculated from measurements made upon tangential and radial sections, agree quite well. 7 . Combining the data for l,,,/q,,, with data obtained from hydrostatic flow studies gives the average radii of the effective capillaries. These values are compared with the maximum effective radii obtained by the method of overcoming the effect of surface tension. The maximum values range from about I to 6 times the average values and approach more nearly the average values when the measurements are made under conditions in which a large number of pit membranes are traversed in series. The effective radii of pit membrane pores are larger for sapwood than for heartwood, the didTerence varying with the species. I.
Madison, U‘is.
