Plant Physiol. (1974) 54, 177-181

Boundary Layers of Air Adjacent to CylindersESTIMATION OF EFFECTIVE THICKNESS AND MEASUREMENTS ON PLANT MATERIAL1

Received for publication January 29, 1974 and in revised form April 2, 1974

PARK S. NOBEL2Department of Environmental Biology, Research School of Biological Sciences, The Australian NationalUniversity, Canberra, A.C.T., 2601, Australia

ABSTRACT

Using existing heat transfer data, a relatively simple expres-sion was developed for estimating the effective thickness of theboundary layer of air surrounding cylinders. For windvelocities from 10 to 1000 cm/second, the calculated boundary-layer thickness agreed with that determined for water vapordiffusion from a moistened cylindrical surface 2 cm in diam-eter. It correctly predicted the resistance for water vapor move-ment across the boundary layers adjacent to the (cylindrical)inflorescence stems of Xanthorrhoea australis R. Br. andScirpus validus Vahl and the leaves of Allium cepa L. Theboundary-layer thickness decreased as the turbulence intensityincreased. For a turbulence intensity representative of fieldconditions (0.5) and for vwindd between 200 and 30,000 cm2/second (where V,ind is the mean wind velocity and d is thecylinder diameter), the effective boundary-layer thickness incentimeters was equal to 0.58 N/(/,/ wnd

separate from the surface, adverse velocity gradients develop,vortices are shed, and so the air movement there depends in acomplicated way on vwind, d, and other parameters (3, 7, 13,16, 17, 29). Here we will consider that a hypothetical or effectiveboundary layer of uniform thickness surrounds a cylinder. Thishighly simplified approach sacrifices a detailed description; e.g.,the complicated dependency of heat transfer on angular positionaround a cylinder (16, 29) is not incorporated. However, rela-tively simple relations are obtained for estimating the boundary-layer thickness controlling heat and gas fluxes from cylindricalplant surfaces.

THEORYWe will define a thermal boundary layer of effective thickness

a such that the temperature goes from that at the surface of thecylinder (T,) to that of the ambient air (Ta) across this hypo-thetical distance. Assuming that there are no heat sources withinthe boundary layer, and for the case of cylindrical symmetry(and no change along the axis of the cylinder, which is assumedto be infinitely long), the heat conducted per unit area and perunit time in a radial direction to or away from the surface canbe represented as follows:

Boundary-layer theory (3, 17, 29) has been applied to thestudy of the energy and gas fluxes for leaves, which are generallytreated as flat plates (5, 8, 21, 24, 28). However, many plantparts such as stems, branches, and certain leaves (e.g., onion)are actually cylindrical. Considerable photosynthesis and pre-sumably water loss as well occurs for the (cylindrical) branchesand stems of Cercidium floriduni Benth (1), Populus treinuloidesMichx. (30), and other plants (1, 22). Moreover, stems andbranches often represent 30 to 50%^ of the above ground surfacearea of trees and shrubs (1, 34). The present objective was todevelop expressions, as simple as possible, describing the thick-ness of the boundary layers adjacent to cylinders and to comparethe resulting predictions of boundary-layer resistance withmeasurements on water vapor diffusion from cylindrical plantmaterial.We will consider a cylinder of diameter d whose axis is normal

to the free-stream wind velocity (Vwind). On approximately theupwind half of the cylinder, there exists a laminar boundarylayer which is analogous to the boundary layer adjacent to flatplates and which can be theoretically analyzed (13, 15, 16, 29).On the rear portion of a cylinder, the boundary layer tends to

I This research was made possible by a fellowship from the JohnSimon Guggenheim Memorial Foundation.

2 Permanent address: Department of Biology, University of Cali-fomia, Los Angeles, Calif. 90024.

k (T8 Ta)JH

rlnr + a (1)r

where JH is the average heat flux at the surface of a cylinder ofradius r, and k is the thermal conductivity coefficient (13, 16).Heat transfer from objects is also described by JH = h(T, - Ta),where h is the conventional heat transfer coefficient averagedover the cylindrical surface (8, 13, 16). We can insert this expres-sion for JH into equation 1, cancel T8 - Ta from each side, andrearrange to give

r+6 k 2r hr Nusselt number (2)

where we have incorporated the convenient aerodynamic param-eter known as the Nusselt number (hdk,- d being the cylinderdiameter).Heat transfer experiments generally involve determining

relationships between the Nusselt number and another dimen-sionless parameter known as the Reynolds number, vwindd/v,where v is the kinematic viscosity (0.150 cm2/sec at 20 C fordry air, ref. 13). Hilpert (10) found that the Nusselt number forcylinders equaled a times the Reynolds number raised to thepower m, where a and m depended on the Reynolds number(as the Reynolds number increased from 1 to 250,000, a wentfrom 0.891 to 0.0239 and ni from 0.330 to 0.805). Upon solving

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Plant Physiol. Vol. 54, 1974

equation 2 for a and using Hilpert's relationship, we obtain

r exP )a (Reynolds number)

Although equation 3 is relatively simple, a and in depend on

both d and i,v ilil. Hence, various exponential relations and

power series were considered in an attempt to provide an approxi-

mate but more convenient formula for the effective boundary-

layer thickness. The following relationship was found to agree

within ±5%c with Hilpert's data for 1 cm2 sec < v'vindd < 30,000

cm2/sec:

b(crn) = 0.74 d(cm) [exp I"/N/(C_'V'sec)d(cm) - 1] (4a)

where the subscripts indicate the units. The exponent in equation

4a can be expanded in a power series, which converges rapidly

for wi"'id > 200 cm2 Jsec and leads to

6(crn) = 0.74/d(c n)/V(%cinllsec) (4b)

Equation 4 applies at 20 C; to correct within 1I c from 0 C to

40 C, a

can be multiplied by T/293 (T in degrees Kelvin). A

more important problem is turbulence intensity (equation 4

applies for a turbulence intensity of 0.009), which we will con-

sider in detail below.As a working hypothesis, we will assume that the concentration

of water vapor drops from c', at the cylindrical surface to the

ambient air value (c',) over the same boundary-layer thickness 5

as for heat transfer. The flux of water vapor (J1,,) can be de-

scribed by Fick's first law, which in the case of cylindrical sym-

metry (6, 24) isDucv-cuv ACt

JV=- rb (5)ln-±r

where Jwv is measured at the cylinder surface,D.,v is the diffusion

coefficient of water vapor (0.25cm2/sec at 20 C, ref. 24), and

Rbl is the resistance of the boundary layer to the movement of

water vapor.

MATERIALS AND METHODS

A cylindrical sleeve of wet filter paper (Whatman No. 2)

provided a convenient source of water vapor. The paper, which

had an axial length of 10.0 cm and a 2.00 cm diameter, was

centrally placed around a cotton-filled Perspex tube 30 cm long,

with a thin slit on one side. The filter paper was inserted into the

slit and kept wet by its contact with the moistened, absorbent

cotton. The tube was supported at both ends in a conventionally

arranged open-circuit wind tunnel with a 4:1 contraction ratio, a

downstream axial fan, downstream louvered vanes to control

wind speed, and a test section that was 30 cm wide, 35 cm tall,

and 100 cm long. The air stream was normal to the axis of the

cylinder, end effects were neglected (32), and it was assumed that

the boundary layers had the same profile at any cross section

along the central third of the tube where the filter paper was

mounted.The plant material used should ideally meet the following

criteria: (a) circular cross section, i.e., no ridges or other surface

irregularities; (b) constant diameter for 10 cm along the axis;

(c) a high rate of water loss; and (d) constancy of stomatal

aperture during measurement. Thirty-cm lengths of the inflores-

cence stems of a water plant (Scirpus i'alidus Vahl) and the native

grass-tree (Xanthorrhoea australis R. Br. subsp. australis) as

well as the leaf of onion (Al/lint cepa L.) were selected. The

pith of the inflorescence stems was partially removed and re-

placed with water to provide essentially 100%- relative humidity

in the cylinders, while the hollow onion leaves were filled withmoistened cotton for the same reason. Before placing the plantmaterial in the wind tunnel, all but the middle 10 cm werewrapped with adhesive tape and the ends were blocked to limitwater loss to the central region. The diameter of the central 10cm varied from 0.84 to 0.87 cm for Scirpus, from 2.00 to 2.09cm for Alliont, and from 4.11 to 4.13 cm for Xainthorrhoea.Measurement of the resistance of the plant material with adiffusive resistance porometer indicated that the stomatal aper-ture did not change significantly during the period in the windtunnel. Experiments were also performed using needles of Pintisradiatai Don, although the above criteria were not met. A newgroup of 30 needles averaging 10.7 cm in length were mountedin the wind tunnel for each condition.The temperature of the filter paper or plant material was

continuously monitored using calibrated thermistors and thermo-couples placed just under the surface. T, was the average ofsurface temperatures determined at 600 intervals around thecylinder at 1 cm in from the end of a test section and at 2-cmintervals along the cylinder at 120° from the frontal stagnationpoint (all values generally were within ±0.3 C of T8). In practiceT8 was usually obtained 1 cm in and 120° from the front. Thewater vapor content of the air was measured with a CambridgeSystems EG and G Model 880 dew point hygrometer. Weightsof the objects before and after a 10 to 30 min period in the windtunnel were determined with a Mettler HIOTw semimicro ana-lytical balance. The CO2 concentration in the ambient air was330,ul, 1 as determined with a Hartmann and Braun URAS 2infrared gas analyzer.The resistance to water vapor loss from the tissue was meas-

ured with a conventional diffusive resistance porometer (12, 14).A foam pad with a cylindrical trough and a rectangular aperture(0.50 cm wide by 1.00 cm along the axis) was placed over the hu-midity sensor to ensure that only water vapor which emanatedfrom the cylindrical surface was detected. Calibration was donein the manner of Kanemasu et al. (12), but using perforated cy-lindrical surfaces of approximately the same radius of curvatureas the plant material. For a calibration surface with a diameterof 2.00 cm and a calculated resistance of 5.0 sec, cm, the elapsedtime for a porometer measurement (14) was only 5 % lowerthan that for a planar calibration plate of the same resistance.The wind velocity was routinely measured with the test object

in placeusing a hot-wire anemometer. The "probe" was thecoiled tungsten filament removed from a flashlight bulb (12 v,0.1 amp, 19 ohms when covered and approximately 14 ohms whenplaced in a wind of 500 cm, sec) and mounted on a thin rod inthe air stream. The probe was incorporated into a conventionalbridge circuit, and the instrument was calibrated in the windtunnelusing a Pitot-static tube and a Fuess inclined micro-manometer to determine the wind velocity. A DISA 55A25 hot-wire probe, a 55D05 constant temperature anemometer circuit,and a 55D15 linearizer were also used to measure pI'lld, whichwas constant within ±41'< along the evaporation surface.

Turbulence intensity is defined as '\"v'f w1i,d( where v' isthe instantaneous fluctuation from the mean wind velocity, Vv"i'ld(2, 15, 19, 29, 31). Thus \/i"' represents the standard deviationof the wind speed from its mean (2). To create a turbulence in-tensity of 0.01 at the position of the test object, a grid of 0.3-cmdiameter wires spaced 1.5 cm apart was placed 70 cmupwind ofit. Higherturbulence intensities were created by using rectangularparallelepipeds (5 cm wide, 25 cm tall, and 12 cm along the winddirection).

RESULTS

When the wind velocity was increased 100-fold, the rate ofweight loss from a moistened cylindrical surface increased 10-

178 NOBEL

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BOUNDARY LAYERS ADJACENT TO CYLINDERS

Table I. Calculationi of the Resistalice for Water Vapor Loss anidEffective Bounidary-Layer Tlhicktiess Adjacenit to a Cylinzdrical

Piece of Wet Filter PaperThe cylinder diameter was 2.00 cm and the axial length of the

wet filter paper was 10.0 cm, giving a surface area for water evapo-ration of 62.8 cm2. The turbulence intensity was 0.01, Ta was 20.1 i0.3 C, D0, was set equal to 0.25 cm2 sec, and r was 1.00 cm. Theboundary-layer resistance equals Acwlt,'.'J,., which is (r'D5r) In-(r + 6,'r) by equation 5.

.,id a a bl WAeighti bl" ,Ticnd c Loss Over J.,, Ru v

20min

ci?z/sec g/cm3' C pg/cm3 g/can' g pg/cm2- sec sec,,/cm cm

10 9.81 18.91 16.22 6.41 0.3662 4.86 1.32 0.392100 9.80 19.11 16.41 6.61 1.1982 15.9 0.416 0.110

1000 9.78 20.1' 17.40 7.62 4.2880 56.9 0.134 0.0342

30 100 300Wind velocity (cm/sec'

1000

FIG. 1. Comparison of the effective boundary-layer thicknessdetermined from heat transfer data as represented by equation 4a( ~) and that calculated from water vapor loss from a wet cy-lindrical surface 2 cm in diameter (0). The turbulence intensity inthe wind tunnel was 0.01.

fold (Table I). The wet surface of the filter paper was assumedto be saturated with water vapor and so c8V was determined bythe surface temperature (T.), while c'. was measured directly.By invoking equation 5 where J.,, is the rate of weight loss perunit surface area, both R"V and a could be calculated. For the2-cm diameter cylinder at the low wind velocity of 10 cm,/sec,the effective boundary-layer thickness was 0.392 cm, which de-creased to 0.0342 cm for the high wind velocity of 1000 cm, sec(Table I).

Equation 4a can represent (within + 5%) the effective bound-ary-layer thickness based on heat transfer data. Let us now com-pare such an evaluation of a with that calculated from equation5 for the diffusion of water vapor from a wet cylindrical surface.As Figure 1 indicates, the two methods of calculating the effectiveboundary-layer thickness are equivalent within experimentalerror, suggesting that equation 4 can be used to estimate 6 forboth heat and mass transfer for cylinders.The resistance of the stomates. the cuticle, and the intercellular

air space (24) affects water vpaor diffusion from the cylindricalplant parts used. We will let R""l represent the resistance ofthe plant plus that of the boundary layer. The total drop in watervapor concentration includes that across the intercellular airspaces, the stomatal pores, and the boundary layer. Since dis-tilled water was placed within the plant material studied (exceptfor pine needles), the internal water vapor concentration (cPv)was the saturation value at the temperature of the plant (Tp).Thus R 1,1a equals (cP,V - c v)/'JW, where the water vaporflux is that at the surface of the cylinder.

Table II summarizes data obtained at various wind velocitiesfor cylindrical plant parts. When v,-ifln was increased from 20 to200 cm sec, Rt°al decreased in each case. To see whether thedecrease in total resistance could be accounted for by a decreasein boundary layer resistance, RI'V was calculated using theboundary-layer thickness determined from equation 4a and therelation implicit in equation 5, viz., R".,' = (r/Dw.v) In (r +b/r). D,v was set equal to 0.25 cm2, sec, the value appropriatefor 20 C (the temperature of the plant tissue did not vary appre-ciably, e.g., Tp for onion was 19.4 C at 20 cm/sec and 19.6 Cat 200 cm/sec, while Ta was 20.9 -±- 0.3 C). The numbers in thesecond to last column of Table II were obtained by subtractingthe calculated value of R"C'V from the measured value for Rttal.As can be seen, the resistance of a particular plant was substan-tially constant for the various wind velocities. This conclusionwas supported by measurements with the porometer, whichindicated that the stomatal aperture did not vary significantlyduring the course of the experiments. More importantly, thevalue of the resistance of the plant parts determined by theporometer (after adaptations for cylindrical geometry and ap-propriate calibration) agreed well with the values determinedin the wind tunnel (comparison of the last two columns, TableII).The measurements described above were performed using a

wind tunnel with a turbulence intensity of 0.01, while plants underfield conditions generally experience a turbulence intensity of0.3 to 0.8 (2). Higher levels of turbulence intensity were thereforeintroduced into the wind tunnel and the effective boundary-layerthickness for water vapor diffusion determined. Figure 2 indicates

Table II. Resistanzces for Water Vapor Diffusioln from CylintdricalPlanzt Parts

Rtotl was determined from the rate of weight loss and RbWV waweemndrmh aeo egh osadR wascalculated from equations 4a and 5 (their difference is given in thesecond to last column). Data in the last column were obtainedusing a diffusive resistance porometer calibrated for the diameterindicated; results are presented as the average + SE for 10 to 12measurements in each case.

Plant

Xanithorrhoeaaiustralis

Alliuim cepa

Scirplus validuts

Piniius radiatca

Dia- Rind otalmeter

Cmt cmz,'sec seccmn

4.12 20 8.450 7.9100 7.8200 7.4

2.05 20 4.3150 4.03100 3.73200 3.65

0.86 20 4.7650 4.56100 4.44200 4.34

0.10 20 15.4200 15.3

bl

sec/cmn

1.310.840.590.420.890.570.410.290.580.370.270.190.180.06

total blRuv - RWV

sec/c,n

7.1 7.02 it 0.057.17.27.03.42 3.33 -- 0.053.463.323.364.18 4.31 4t 0.074.194.174.1515.215.2

Plant Physiol. Vol. 54, 1974

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Plant Physiol. Vol. 54,r 1974

K

0o50 cm/sec

J _

K-

therefore recommended for estimating the effective boundary-layer thickness surrounding cylindrical plant parts under naturalconditions:

6(cm) = 0.58 d(cm) [exp 1/'X/¾'cXhjsec)dCcn) 11,

1 cm2'sec < vwindd < 30,000 cn2j/sec

'(cm) = 0.58Xe

200 cm2. sec < vin,ld < 30 000 cm2/sec

(6a)

(6b)

The form of the relation for the effective boundary-layer thick-ness adjacent to a cylinder (equation 6b) is quite similar to anapproximate one for flat leaves tinder field conditions:

0-

-0 o 200 cm,'secD_ >~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(7)

0 05 where '(ciii) is the distance in cm across the leaf in the directionof the wind (5, 18, 21, 24, 26, 27). The same general conclusionshold in the two cases, e.g., the boundary layer will be thinner forsmaller objects or at higher wind velocities. Also, the resistanceof the boundary layer of air is usually less than that of the plant

0 00 - part which it surrounds. We note further that the surface area0 0 0.2 0.4 06 per unit volume of a cylinder is inversely proportional to the

Turbulence intensity dimensionless radius, and so thin cylinders such as pine needles are bettersuited for photosynthesis on a tissue volume basis than are large2. Effect of turbulence intensity on the effective boundary- cylinders, and also they will have a lower boundary-layer re-hickness adjacent to a wet cylindrical surface 2 cm in diame- sistance. Large cylindrical plant parts generally serve i*wind velocities of 50 and 200 cm/sec. ssac.Lreclnra pntatsgealyeveimportantroles besides photosynthesis, such as support, conduction, ors the turbulence intensity was increased from about 0.005 storage.the effective boundary-layer thickness adjacent to a 2- The present approach can describe heat (equation 1), water

ameter cylinder went from 0.169 cm down to 0.116 cm vapor (equation 5), carbon dioxide, and oxygen fllxes across1,winii Of 50 cm,'sec and a similar fractional decrease oc- the air boundary layer adjacent to cylinders. We can readilyfor a wind velocity of 200 cm, sec. incorporate specific resistances encountered within a plant. Ifboundary-layer thickness calculated from equation 4a Acj is the concentration drop of species J across some component

sented by 50.009, since the turbulence intensity for Hilpert's and Jj is the flux at the cylinder surface, then the resistance ofveraged 0.009, refs. 15, 29) was compared with that ob- that component, expressed per unit area of the surface, is Ac J.at a turbulence intensity of 0.5 (30.5), a turbulence intensity The tissue resistances of Xinthorr/lhoea, Alliumn, and Scirpus forentative for many plants under natural conditions (2). water vapor movement quoted in Table 11 are actually fairly lowllowing values of bo.5 /bo.0oo9 were obtained for a wet cylin- compared with other cylindrical plant parts that were checkedsurface 2 cm in diameter: 0.81 at a wind velocity of 20 with the porometer. For Xantliorrloea the base of the inflorescy 0.78 at 50 cm, sec, 0.80 at 100 cm sec, 0.75 at 200 cm, cence stem where the cuticle was not fully developed was used,id 0.74 at 400 cm/sec. Thus bo.5 6(9oog averaged 0.78, which and so cuticular transpiraition was appreciable. Stomates occupyAts that the factor 0.74 in equation 4 is too high for field a large fraction of the surface area of onion (20), which helpsations, e.g., 0.78 x 0.74 or 0.58 would be more appro-

Table Ilf. Effective Boundary Lajyer Thickn-ess for Variou.s WilladVelocities aniid CYliilner Diacmeters

DISCUSSION Values in cm were determined with equation 3 usinig Hilpert's

effective thickness of the boundary layer of air surround- data (upper numbers) compared with thickness calculated froml"I-rlr;"+l -aim

m r Nha, ;cbt;t frr%r" ths,,(mSequation 4a (lower numbers).ing cy1iHinZrica pianit miateriai canl oe estimiateci uirom Ine amienuitwind velocity and the cylinder diameter, paying due regard tothe effect of turbulence intensity. Figure 2 shows that the effectiveboundary-layer thickness decreased substantially when the tur-bulence intensity was increased up to about 0.1, in agreementwith previous findings for heat (4, 15, 33) and mass (4, 19) trans-fer from cylindrical surfaces. As the turbulence intensity wasfurther increased up to 0.7, the effective boundary-layer thicknessdecreased more gradually, i.e., became less- sensitive to theactual magnitude of the turbulence intensity in the range ofvalues occurring in the field. Plants under natural conditionsoften experience a turbulence intensity near 0.5 (2), a value whichreduced the boundary-layer thickness an average of 22- fromthe value predicted by equation 4a. Although the effect of aturbulence intensity of 0.5 appeared to increase somewhat astne wind velocity was raised from 20 to 400 cm/sec (for a 2-cmdiarieter cylinder), for convenience we will simply reduce thefactor 0.74 in equation 4 by 22%. The following relations are

d cm)lvin(l~~~~~~~~~~~~~d, n

V(cm sec) --100 32 1 O 3.2 1.0 0.32 0.10 0.032

1000 0.13 0.072 0.041 0.0231 0.013 0.0081 0.00480.13 0.075 0.042 0.0241 0.013 0.00781 0.0046

320 0.40 0.22 0.13 0.072 0.042 0.026 0.015 0.00950.42 0.24 0.13 0.075 0.043 0.025 0.015 0.0087

100 0.72' 0.41 0.23 0.13 0.081 0.048 0.030 0.0190.75 0.42 0.24 0.13 0.078 0.046 0.028 0.018

32 1.3 0.72 0.42 0.26 0.15 0.095' 0.058 0.0371.3 0.751 0.43 0.25 0.15 0.087' 0.056 0.040

10 2.3 1.3 0.81 0.48 0.30 0.19 0.122.4 1.3 0.78 0.46 0.28 0.18 0.13

3.2 4.2 2.6 1.5 0.95 0.58 0.374.3 2.5 1.5 0.87 0.56 0.40

0.20

-E 0.15--~

a)

-c

o

0

FIG.layer t]ter for

that asto 0.7cm ditfor acurredThe

(represdata atainedrepres(The fcdricalcm, se(sec, ansuggesapplicupriate.

The

180 NOBEL

cc

-

"N' indb(cill) -`= 0.4-\/] .'c III) /V(cm/sec)

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BOUNDARY LAYERS ADJACENT TO CYLINDERS

account for its low resistance. Scirpus grows in water and so itcan usually withstand a substantial rate of water loss.Many simplifying assumptions have been incorporated into

the above analysis. For instance, heat and gas fluxes by natural(free) convection have been ignored. Natural convection is moreimportant at low wind speeds, large diameters, and large tem-perature differences between the plant tissue and the ambientair (9, 16, 23, 25). For a temperature difference of 5 C (moder-ately large) and a wind velocity of 10 cm/sec (low), forcedconvection exceeds natural convection for diameters up to 100cm for horizontal cylinders (16, 25). For vertical cylinders up to300 cm tall and a AT of 5 C, forced convection dominates freeconvection for wind velocities greater than 20 cm/sec (16, 23,25). When the ambient wind velocity is not normal to the cylinderaxis, the effective boundary-layer thickness depends on cylinderlength (29); as an approximate guide we can let vwind be thevelocity component normal to the cylinder axis. For moderatelytapering plant parts and for noncylindrical cross sections (butstill smooth surfaces, ref. 11) we can let d be the mean diameteras a useful approximation. The above equations are inaccuratewithin a few diameters distance from the end of a cylinder (32).These conditions must be kept in mind when using equation 6to estimate effective boundary-layer thicknesses adjacent tocylindrical plant material.

Acknowledgments-The author gratefully acknowledges the outstanding co-operation of many individuals in the Department of Environmental Biology,Resealch School of Biological Sciences, The Australian National University andthe F. C. Pye Laboratory, Division of Environmental Mechanics, CSIRO, espe-cially 'Mr. J. J. Finnigan,Mr. P. J. MIcGruddy, and Dr. W. G. Allaway.

APPENDIX

Table III compares the effective thickness of the boundarylayer of air adjacent to cylinders calculated from Hilpert's data(10), as represented by equation 3 (upper numbers), with 6 de-termined from the approximate but simpler-to-use formulationdeveloped in this paper, equation 4a. As can be seen, the twoexpressions for 6 agree to within about + 5 %- for 1 cm2/sec <vwind ,)d(d.) < 32,000 cm2,sec. The calculations are for aturbulence intensity of 0.01; to adjust to a turbulence intensitymore appropriate for field conditions (0.5), the numbers shouldbe reduced by 22%-. Table III indicates that the effective bound-ary layer could vary from approximately 0.005 cm up to over 4cm in thickness for the wide range of wind velocities and cylinderdiameters occurring for plants.

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181Plant Physiol. Vol. 54, 1974

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