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Plaint Physiol. (1967) 42, 651-6l5

Further Studies of the Heat Transfer from a LeafE. T. Linacrel

Commonwealth Scientific and Industrial Research Organization, Irrigation Research Laboratory,Griffith, N.S.W., Australia

Received December 20, 1966.

Summary. The resistance to the diffusion of heat and water vapor external toa leaf, can be derived from measurement of the rate of change of the leaf tempera-tulre, after a sudden alteration of the intensity of irradiation. The theory of themethod has been developed to accommodate the case of a leaf that is freely tran-spiring, exchanging longwave radiation with the environment and with differentinternal resistances on the 2 sides of the leaf. It has been successfully applied tomeasurements on wet blotting paper in the laboratory.

One factor controlling leaf temperatture andtranspiration is the external resistance to the fluxof heat by convection between the leaf surfacesand the ambient air (12). This resistance will betermed rx (sec/cm). In the case of a surface forwhich the heat-transfer coeffic,ient is Xl (cal/cm2sec°C) or hr(cal/cm2min'C), the resistance maybe expressed as follows:

rx = cpp/h - 60cp/hm- 0.018/hm approx. I

where cp is the specific heat of air at constantpressure (cal/g 0°) and p is the density of dry air(g/cm3). In an earlier paper (13), attention waspaid to deriving values of 21i.,, which is the overallheat-transfer coefficient from a leaf regarded ashaving a surface area equial to that of its outline.The following formula was derived:

2hm = 0.69(1/1A)s/(t1/2)m IIwhere (M/A) is the mass of uinit area of the leaf(g/cm2), s is the specific heat of leaf material(cal/g.0C) and tl/2)r is the time taken in minultesfor a leaf's temperatuire to change halfway to thenew equilibrium value, after an abruipt change ofthe incident radiation level.

Equation II is exact only wheni there is anegligible change in the trainspiratioin rate dulringthe temperature change of the leaf. But measuire-ments of MlI/A, s and tn'2 give tunrealistically highvalues of the coefficient with this formula, if theleaf is transpiring freely. I am grateful to Mr.I. C. Mcllroy for pointing this out, in a privatecommulnication. Thus, equation II is useful in thecase of detached succulent leaves (18), for exam-ple, btut needs modification for more general appli-cation.

The assumption that transpiration is uinaffectedby the temperature change has been removed in atreatment of the problem elsewhere (15). Here,another formulla is derived, like equation II, butfreed of the assuimption of uinaltered transpiration

rate, and also accoutnting for changes of the long-wave-radiation flux. In addition, the opportunityis taken to clarify the definition of the externaldiffusion resistance, which could be related eitherto unit area of the leaf's ouitline, or to unit area ofthe surface on both sides of the leaf. Finally,allowance is made for differences between therespective internal resistances on the 2 si(les ofthe leaf.

Theory

Details of the theory are given in Appenidix 1,and here it is proposed simply to describe the basicmodel, assumptions that are made, the outcome ofthe theory, and a few corollaries. This serves aspreliminary to the account of experimental work,described in a later section.

The assumed model of the energy flows aboulta leaf is shown in figture 1. The chief featuires

Qn*

environment -

tctsurface

mesophyll

surface.I

H

environment

a

tenperatur.

rr-T--- T1 *~

x t i H

ris ~ ~

_ _M T s

ri, LE Xi~-------

'x 'r HX<_rr- Hrr

Ta- vavapourpressure

FIG. 1. Model of the energy fluxes and the diffusionresistances about a leaf. Q.. is the incoming net short-wave-radiation flux, H, is the convective heat transfer,and Hr the heat transfer by longwave radia'tion. Thelatent-heat flux LE is used in evaporating the transpiredwater. Suffix 1 is for the upper surface and suffix 2for the lower surface of the leaf.

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are these: i) the vapour-pressure at the evaporatingsturface (ec) is assumed to b)e that of water at theleaf temperature (T, ), ii) the externial resistance(r,) impeding the transfer of heat by convectionfrom the leaf's sturfaces, impedes equlally the diffui-sion of water vapour, iii) (listiiiction is made be-tween the conditions oni the uipper and lowerstirfaces, respectively, iv) ani effective diffusionresistance (r-) is reckoned as governing longwave-radiatioin heat-transfer, between the leaf and itsenvironment; this is considered in Appendix 2.

It is shown in Appen(lix 3 that the longwave-ra(liation flux (I ) is auigmenite(d if the leaf is nottotally einclosed by sturfaces at the ambient tem-peratuire (T.), but has onle sidle exposed to a clearsky-, for example. The increase is practically in-(lepeindent of leaf temperatutre, so the exposuire hasno effect on the validity of a theorv relating theresistanice r. to changes of leaf temperatutre. Thesame applies if the sky is obsctured.

The theoretical treatment of the effect of asuid(len alteration of the shortwave-radiation inten-sity Qns on the leaf temperature T,, shows a de-pendence of the rate of temperatture change, onthe resistance rx. So, conversely, the resistancecan be deduiced from measturemenits of the time fora halfway change of leaf temperatulre (tl/2 sec),using the follow%ing eqtuationis:

r, = 2AcPpt1/ B/0.69 ifs sec/cm III= 2AB sec/cm IV

where xr = A4CPPtII2/O.609 lls sec/cm VB l±r+ rA+ Ar.4r-l-+ri2 + 2rx)

rr 2y(rii + rr) (rO -i r) VIA is the variation of water's vapour pressure withchanige of temperatture (at the ambient tempera-ttire), y is the psychrometric constant, and thecliffuisioni resistances are identified in figuire 1.

Equations I an!d III can be combined to give thefollowing:

2/im 0-0.69 lIfs/.4B(t/2 )T, IIThis is clearly- the same as ecqutation II, except forthe termII B, which dlecreases when r. approaches

zero (as the wiiidspeecl increases) and when riincreases (as stomates close). Equationis III andV7I may also be combiined, anid then the resistanceis given by the real, positive soluttion to the follow-ing equtation:

rA ACI 2 + Fr= G VTIIIwhere C ri-si2 - (y + A)/yP sec/cm IX

F ri, ri2-

(nri + ri2) (2y + A ) /2yP sec2/cm2 XG r-il r Vi2P sec3/cm' XIP= 1/2AN - 1/r, cm/sec XIINY Acppt12/0.69 lls sec/cm V

The psychrometric constant in equatioins IX andX can be reckone(d as O.5 mmHg/° (1.5), so theysimplify as follows:C ra + 7 i2 - (1 + 2A)/P XIIIF r= i ri2 - (ri, + ri2) (1 4- I)/P XIVIt was poinited ouit in the preVioUs paper (13),

that (.MI/A) ofteii has a xalue rouighly about 0.032gm/cm2. Also, over the rainge 0 to 40', the productcpp equals 3.1 to 2.7 X 10-4 cal/cm3.°C, say 3 X10-4 cal/cm3.°C. Furthermore, the specific heatof leaf tissuie may be taken as 0.88 cal/g * (26).So A" in equiatioin V typically has the followingapproximate valuie:

,\ 0.015tw2 sec/cm XVwhere t/2 is the halfw,ay time, measuired in seconds.

The effective resistanice rr has a valuie of about2.1 sec/cm at 200, as shown in Appeindix 2, soequiation XII approximates as follows:

P = 32 t1L- - 0.48 cm/sec XVIThe form of equiation VIII has implications for

the nature of the external resistance rx. If P werenegative, then C and F would be positive and Gnegative, btut then r, wotuld be negative, which isabsuird: so P cannot be negative. It follows fromeqtuation XII that \ muist always be less than 1.05sec/cm, and then equiation XV implies that to/2cainnot exceed 68 seconds, for any leaf of the kindconisidered here. It is noteworthy that the highestvalue listed in table I is 60 seconds: this wasobtained in coinditions of particuilarly low ventila-

Table I. I al/uts oIth/e Hal/f-av Timt( t112 Extracted fromii Pubishld Grap/iv of Lc(af-Tciipcraitirc Variation

Auttlor Plant Place Halfway time

Ctirtis (7) ~~~~~~~~~~~~~~~~~~~~SecCurtis (7) Citrus In a box 60Mellor et al. (16) Tobacco Indoors 50Smith (24) Magnolia 45Casperson (4) Hvydrangea Indoors 40Clumn (5) Lettuce Greenhouse 35Clun3ll (5) Privet Outdoors* 30Mtellor et al. (16) Geranium Indoors 30Ansari and Loomis (1) Tobacco Greenhouse 25Cluiii (5) Fuchsia Outdoors 25Ansari and Loomis (1) Cotton Outdoors** 20Casperson (4) Hydrangea Incdoors v-enitilated 20WVilsoIl (28) Salix artica Outdoors 18* In this case, the leaves were possibly slheltered, since growing juit outside the greenhouse.

The plant may have been in a greenhouse.

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LINACRE-HEAT TRANSFER FROM A LEAF

tion, which implies a relatively high valuie of theresistance r:. In practice, P is small as well aspositive, so C and F in equations IX and X arenormally negative, and G in equation XI is positive.In such a case, there is only 1 positive solution toequationi VIII. It may be found by reiteration,once values have been obtained for the resistancesril and ri2, the temperature (for deducing A), Ml/Aand the time tl/2. An example of the procedure isgiveil in Appendix 4.

For a non-transpiring leaf, the internal resist-ances are very large, and so dividing each term ofequation VIII by the product rni ri2, yields thefollowing equation:

r. = 1/P XVIIThus, rZ is about 1.3 sec/cm if tl/2 is 25 secondsfor example, whatever the temperature of the non-transpiring leaf. However, stomatal closure doesnot make the internal resistance infinite: there isstill some transpiration through the cuticuilar re-

sistance, which has been found to be in the range20 to 80 sec/cm by van Bavel et al (3), Gaastra(8), Shepherd (21), and Slatyer and Bierhiuzen(23). A representative value is 30 sec/cm: thishappens to be the median of values quoted byJ-Iolmgren et al (9). If tl/2 is 25 seconds, Ta is300 and ri, and ri2 are both 30 sec/cm, then r. isfound from equation VIII to be 1.4 sec/cm, whichis higher than the 1.2 sec/cm for a completely non-

transpiring leaf.The amphistomatous leaves mentioned in table

II show minimum values of the internal resistance(ri) of about 0.8 sec/cm. So closure of thestomates may be expected to alter their resistancefrom 0.8 to 30 sec/cm, say, and such a variationappreciably affects the external resistance valuegiven by equation VIII, for a particular halfwaytime. It is clear that the internal resistances mustbe known in making use of equation VIII, and aconvenient instrument for measuring ri in the fieldhas been described by van Bavel et al. (3).

Table II. Pul)1ished Data on the M1inimtum Internal Resistance of Plants (ri), to tile Diffusion of Water Vapourfrom Single Leaves

The data are related to unit area of the surface, w-hich is twice the area of the leaf's outline.

Minimum Method ofAuthor Plant ri determination

Sec/cmBa'nge (2) Zebrina 0.9 iWilliams anid Amer (27) Pelargoniiii; 1.3Gaastra (8) Turnip 1.0 iiZelitch and Waggoner (29) Tobacco 0.2-0.8 iSlatver and Bierhuizen (23) Cotton 1.2 iiiBavel et al. (3) Cottoni 1.2 ivHolmgreni et al. (9) Sunflower 0.7 iiiMilthorpe (17) Wlheat 0.3 iShimshi (22) Maize 0.4 iLee anid Gates (11) Lucerne 0.8 1Sheplherd ('21) Clover 0.6

i. Resistanice vailie- calculated from the number and size of the stomata.ii. Resistanice derived from measurements of rD of a leaf replica, and of the rate of photosynthesis, to determine

(ri + r.).iii. Measured water loss a) from a wvet, filter-paper model of the leaf (for wlhiclh ri is zero), and b) from the leaf.iv. -Made measuremenits with a device which determines the rate of transpiration under standard conditions of hu-

miditv environment.

Table III. MIcasurements of tile External Resistance of a Vertical Piecc of Blotting Paper in Still Air, Indoors.Oai both occasions the ambient temperature was 20°C.

tA12Expt Equationi M/A s Values Mea?n cs -ea E Deduced ra

a 17 0.0124 0.37 sec sec mmHg 10- g/cm2sec sec/cmb 20 0.037 0.79 7.8 6.7 73 7.3 ... ... 3.8c 19 ... ... 30 22 27 27 ... ... 3.8

g/cm2 cal/goC ... *-- 6.9* 19 3.6d 17 0.0130 0.37 7.0 6.4 6.9 6.8 ... ... 2.8

0.035 0.78 19.5 22.0 15.0 18.8~~~~~. . ...

2.6

2.6* In this case, the value is the ambient saturation deficit.

ef

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653

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7.9 30



PLANT PHY'SIOLOGY

To test the halfway-change techliiqtie, measuire-ments were made oni a rectanguilar piece of blottingpaper with an area of 64.5 cm2. It was suspendedby a smaller edge (6.4 cm) on a frame standingon an open weighing machine, wshich was sensitiveto 10 mg. A fine thermocouiple was attached tothe center of the paper, and a 60-watt lamp tusedto irradiate it intermitteintly. The fall of the pa-per's temperattire after removing the lamp wasrecor(led o01 a potentiometric recor(ler with a re-sponse timc of al)out 1 secoli(l. The paper waseither (lry- or wet: in the latter case, the veightchange indlicatedlthe rate of ex-aporation E, whenthe inter-nal resistailce x as zero.

Values of the externil resistaice couild be foundin 3 way-s. Ill the case of Iry- paper, uise was madeof eqilatioll XVII, taking the appropriate valties ofIl/A (0.012 g/cm2) and( s (0.37 cal/g.0C) for

calctilatillg the term P from equlatiolis V an(I XII.When the paper was wet, the internal resistanceri was zero, so the followinig equlation obtains:

r. cpp(c, - ea)/IyLE XVIII1.0 X 100 (es - ea)/E approx. XIX

where es anid( cs are the xapouir pressilres inldicatedin figulre 1, and the term L is the latent heat ofevaporation of water. Also, when ri is zero, eqlua-tion VTIII becomes the following:

r, (I + 2A ) IP XX0.35 Mls

where P - _== -

esp.At 21170 1fs

_0.48

0.48

In this eqtlatioll, the values of (.1/A) and s de-penId on the wAater contenit of the paper, iinferre(dfrom its weight.

Results

Results of 2 sets of measuiremenits are given intable III. The deeuce(I resistalice valtie is seento be little affected either by the wetness of thesurface or 1)) the metho(d of derivation. So theresults suipport the view that the resistances tovapotir ani( heat (liffusioll are inl(lee(l the same,even in the conditions of these measturements, whenthere xvas 11o (liscernilble wind moxement.

The model illustratedl in figiure 1 (liffers fromearlier models (12), as it incorporates the effectivediffusion resistance r,. If this term were ignored(i.e. regardled as infinite), then equiations XVIIandl XX x-vould l)e modifiedl, and yield answersappreciably- less thani does equatioin XIX. Forexample, in place of 3.8, 3.8 and( 3.6 sec/cm showi-in table III, the 2 modified equatioils would leadto 1.4, 2.1 and 3.6 sec/cm. The better agreementof the first triad shows that the model of figuire 1is in(leed more appropriate.

This techniiquie of (ledluciing the externial dliffui-sion-resistanice r. from a measurement of the timefor a halfway chanige of temperatulre, may be un-stlital)le ouitdoors. There, the observed temperatuirechange maybe concealed by fluictuiationis (Ille tow-ind(I gulsts, fluittering of the leaf or irreguilarcloud(ls. However, the difficuilty of irreguilar sunii-shinie could be overcome by shadinig the leaf fromthe suIII, and then intermittently illuminating thelea f by mealns of a lamp. The problem Of xvindguistiness might be dealt with by mnuerouis repeti-tions of the measuiremenit, to obtaini mean valuiesof the half-timiie and(I hellce of the externial re-sistanice.

The technii(qle is uiniqule ini allow-ing measuire-melnts on a leaf whilst it remainis initact and(I inI situi.No assuimptioni is ma(le as to the e(lqivalence of anearby- lblotting-paper alnaloguie, for example. Noris the resistance (lerive(l from somle formuiilla b)asedonl measuiremenlts in a wvindtnilinel, wxhere the wind-spee(d, tilrblulence anid leaf attitiu(le are letter (le-fille(d thani for the leaf in (Iiqestion.

Rapi(d exploration -with a thermocouiple acrossa cottoin leaf reveals temperatilres ranging ov-er1.1° for example, alind Cook et al (6) have reporte(da similar ranige. This meanis that the thermiocouipleregisters onl\- a very- local teniperatiure. and(I itfollows that the halfwva)-change tecihini(que mightbe iise(d for (letermininig the pattern of the externalresistanice over a leaf, provi(le(d ind con(litionisrenlaini steady-.

Also, the half\vay-change techniq(ue cani ie luse(din colijlllctioin with a proce(diure of Imp -ni e:. al.(10), in or(ler to (letermine the transpiration rate ofsigle leaves. His ileaf-ps-chrometer techniq(lue in-v-olves the selecte(d leaf, wvhich trtanspires like thewet-uillb therimiomileter of a psychrometer. matcile(dagainst all adlj'acenit similar leaf, whhich is siitablycoated to prevelnt it from transpiring. Thuis thelatter resembles a psychrometer's (lry-lulb) ther-mometer. Temperatuires of the leaves ( T- andlT.,, respectively-) are compare(d 1y thermocouiplemeasulremelits, to find(I the (liffereice ( T_. - T. ).It maybe (leduiced( from the theory- in Appendix 1that ill steady coll(litionis this (liffer-eilce Is relaite(dto the trallspirillg leaf's rate of water 1u<s< E, asfollows:

7>22 - ~Ts = L1EfJ XXIIwhere J 2c,,p(r, + r.) r XXIII

Heilce a measiuremenit of the temperatiure (liffereice( 7T. ..- T') y-ields the valule of the transpirationrate, if J is knownvi. The factor J cani be calciulate(donice the resistalice rl. is determine(d, and(1L it is clearthat the thierniocouiple attache(d for measiirllig Tcouil(l also be uise(l for (leterminillg r, ly the half-way-change techniquie, on the same leaf. ill thesamiie coll(litions of ra(liatioii and( venItilatioll. Like-wvise, sul(l(lelyly shadiilg the coate(d leaf Nxvmild allowestililatioll of the exterilal resistan1ec for that leaf

654

Experiments Discussion




too, and proviide a check on its equ1ality: it isfutndamenital to the tnse of the leaf-psychrometertechniqute that r is the same for both leaves.

Conclusion

The external resistance of a leaf (ri) can befouind from these eqtuations:

rx." T Cr.2 + Fr, = Gwhere C - r)i + ra2 - (1 + 2A)/P

F = rii ri2 - (ri, + ri2) (1 + A)/PG = r1il ri2/PP = 1/2N - /rA' = AcPptl/2/0.69 Mls

Ii(loor measturements with blotting paper allowderivation of r, from these eqniationis, when thepaper is either wet or dry. The answer proves tobe the same in both cases, and eqtual to the valtneobtained in a third way, by measniring the rate ofwater loss fronm the paper when wet.

The restults indicate the relevance of a model ofenergy fluixes abouit a leaf, showin in figuire 1.

Appendix

Appeiidixl i. Thc Dcrivation of (Itt Expressionfor- the. Resistuiccc ri. The evaporationi of waterfroml a leaf involves the diffnision of water vapotnraway, so the following expression obtains:

E - -D d (c)dxr XXIV

wlhere E is the evaporation rate (g/cm2sec),D is the (liffuisivitv of water vapouir (cm2/sec),an(l c is the concentration (g/cm3) varying in thedirectioni X. Now the term c can be replaced, thnls:

C - pEea/p XXVwhere E is the ratio of the densities of watervapotur al(I (lry air, ea is the partial pressture of thewater vapouir in the air (mmHg), and(l p is theatmospheric pressulre (mmHg). Henice equtationsXXIV anid XXV yield the followinig:

E = DpE(e( - ea)/bp XXVIwhere c. (mmHg) is the vapotur pressuire of theliqutidl suirfaces within the leaf, ea (mmHg) is theambiient vapouir pressture, and b is the effectivethickniess of still air, both within and( ouitside theleaf, throuigh which water-vapour diffusion occurs(cIIm). The thickness b can be expressed in termsof the difftusion resistances ri and r., (sec/cm),respectively iniside and ouitsidle the leaf, as follows:

b = D (ri + r) XXVIIHeince LE = LEp(e - ea)/p(r + r.)

= cpp(e8 - ea)/y(ri + rc) XXVIIIwhere L is the latent heat of evaporation (cal/gm),and y is the psychrometric constant, cpp/LE(mmHg/°C). The external resistance r, refers to each singleside of the leaf and the internal resistances of theupper anid lower surfaces are showni as being dif-

ferent, i.e. rii and ri. Thus, equiation XXVIIIbecomes the following:

LE = LE1 ± LEa = (ea - ea)/K XXIXwhere K = y(rii 4- r.) (ri + rx)/cpp(riL + ri2 + 2r.) mmHg* cm2 * sec/cal XXX

In addition to the diffusion of water vapourfrom the leaf, there is a convection of sensibleheat (He cal/cm2sec) which is proportional tothe temperature difference (T. - Ta), at leastwhen there is stufficient ventilation for forced con-vection. This convective heat flux is a diffusionprocess like the transfer of water vapouir, and soit may be assulmed that it is impeded by effectivelythe same resistance rx. Hence the following equa-tion for the convective heat loss from uinit area ofthe leaf:

Hn1 + H,2 = 2 cpp(Ta - Ta)/rr XXXIEqulation XXXI refers only to convective heat

transfer. However, heat is lost by longwave-radia-tion exchange too. For small temperature differ-ences, this fluix H is also proportional to thetemperatuire difference (Ta - Ta), as shown inAppendix 2. Consequently, it may be treated asthouigh it is aIn extra diffuision of heat through aresistance r. in parallel with the proper externaldiffulsion resistance r,. The result is a loweroverall resistance to heat transfer, and the totalheat flux (H) is given as follows:

H = 2cpp(Ta - Ta) (l/rZ + U/rr)= 2cpp(Tfl - Ta) (rr + rx)/rrrx= J (Ta- Ta) XXXII

where JI= 2cpp(r, + rx)/rirx XXIIINow consider the radiation fluixes about a leaf

onl a plant in(doors, where the enclosuire surfacesare at the ambient air temperature (Ta). Theleaf exchanges longwave radiation (Hr) withnearby sLurfaces, buit also receives shortwave radia-tion in the form of light, a fraction of which isreflecte(l away. WVhen the leaf is suiddenly shadedor exposed to light, only the net shortwave-radia-tion intensity is altered, to a new fixed valuie Q.a.The change of heat stored in the leaf mulst equlalthe difference between the heat receive(l (Q..)and that lost (LEe + H), respectively. Hence thefollowing equation:

(Jlf/A)s d(Ta) - Q.a - (LE + H)dt

Qaa - (MI/A)s d(Ta) = LE + Hdt XXXIII

Eqtlation XXIX can be converted to a relation-ship involving leaf temperature (Ta), in the mannerof Penman (19), as follows:

KLE= e. - ea = (ea - ea,) + (eaa - e.)XXXIV

=(es - ea) + S XXXVwhere eaa is the vapour pressure of a water surfaceat the ambient temperature (Ta), and S is thesaturation deficit, defined as (eaa - ea). Now theterm (ea - eaa) in equation XXXV may be con-sidered, either exactly in terms of the chord to a

655



PLANT PHY-SIOLOGYculrve relatinig water s vapour pressuire to its tem-peratuire, or rouighly in terms of the taangenit to thecurve, signified by A as follows:

C' - as d ( (t-,, )

7T. - 1., ddTZ

XXXVN7IThe approximationi of equiationi XXXNI is reason-able, so long as (T. - T.) is small. For example,the s'ope of the chor(d is 1.25 mmHg/0C, when T, is250 and To. is 200, whereas the tanigenit at 20° is1.08 mmHg/°C. WN ith this approximatioln, equiationsXXXV and(I XXXV'I yieldI the following:

T, - 7'. = KLE/I - SIA XXXVIIIt iS conven:Ient to (lefine a filux Qt as follows:

Q = LE + H =Q0 il/s d(T,)A4 dt XXXVIII

Henice, from e( atilons XXXII, XXXVII andXXXN-IIT:

(9, =1E + J(KLEIA S/A)IE(I +J-AK/A) 1S A XXXIX

From these eclqlations it is poss,ible to derivre theexterlnal resistance r,, as follows. First, equatioinXXXIX is rearranged to obtain an expressioni forLE. Theni this is subtracted from Ot, to yield thesensible-heat fluix H, expressedI free of the termLE. It may be ec(llate(l to the expressioni XXXII,aInl then Ot replace(d by 0,)s - ilIsd(7s), asfollows: A4dt

LU - Ot + JSIA1 + JKAz XL

H =0(, - - Qt JKLz\ -JSIA1 JK/A XLI

==J(T8 - 7,) XXXIIHenice QO (A-t JK) (7'. - 7'a) + S

K XLIIO(.. - Ld(7')

.4dt XLIIIHenice (1(7 ) + VT 21-+1 = 0

dt XLIVwherfle V (A/K + J) .A1-ils sec I XLV

11 A (SIK 0,, - T(A( K-/+ J ) C/seclIs XLVI

Equation XLIV caIn be integrated, provided Vand(I IF are regardedI as constants. This is not quliteso, since the term A varies as leaf temperatuirealters, but the variation tenIs to be lessenedl by thefactors J and K, especially at low temperattures.Henlce the following solultioni:

lit 7 Tf = Vt7'. T7 XLVII

wvhere 7., is the initial value of the leaf tempera-tre(7'.), anId Tf its final v-aluie. The latter equials-IT', i.e. 7T. - (S - 0,,UK)/(A + JK). Inci-

dentally, this last expressioni shows that the leaftemperatuire excee(ls the ambient temperatulre onlyif the provlulct QnxK exceedls the satuiration (lef:c.tS. This is most likely in a suinny but moist climate,with low w\-Ind, or when the ambient temperatuires

are lowv, as cdisculssedI by the aulthor elesewhere ('14,I.')

Equation XLV-II caln be simplifiedl ly in1cor-poratinig the halfway time t11/ (sec ), w-hich is thetime takeni for the leaf temperatuire to fall from7 to T,1 2, (lefine(l thuis'

7,,' 7T.,- 7f

7, 7'J7T TfT1 ,/ 1

-

T

Conibl)illillg e(Illitiois XLVII and Lfollow inlg:

in 2 -- (.69 == 'tIThe equations XXIII, XXX, XLV aii

y-ield( the follow ing:

XLV-III2(T,2 - 7,r) XLIX2

1,Ae- the

LI1(1 LI then

r,. = 2 c,pti I- B ).69 ils

\-here X - l-4C.ptl/2//0.69 -Ils scc,.'cmB = 1 +±r +A (r -i 2 +2--)

IIIIV

K 2y(r2 + ,½) (i2 + r) VI-AlfppeC(di.x' 2. 7he EffectivAt)iffusion Rcsist-(1o1Ce to Lonyugzwcze Radiation r,. The vahie of theresistance r, in eq(utations VI, XII and( XXIII miiaybe derived from (lata on the longwave-radiationtransfer betw eeni suirfaces of a giVen emllissiVitV(a). Froiml onle suirface at an1 absolute temperature7' ',, the longwave-radiation fltux (H -) is gi-en bythe Stefanl equ1ationi:

H, = 1.35 X 10- 12a7'4 cal/ccm'Sec LIIFor v-egetatioIn, the emissivitv for long-wave radia-tioIn is almiost unitv (2))), SO the lnet fluix ( H,)between adjacent sulrfaces, respectively at temlipera-tulres T7i' and T7..ac (OK), is as follow\-s:

H, 1.35 X 102(T42T -K T4a,1.35 X 1()01 (2 7 K TOA)(T, ic +7'.i)(7 7 A 72, , T)

=P' X lJ( ( T. K 7%1i)-/X 10 (7' 7'.) ca/l 24c

LIIIwhere 7'. and Ta are the respective radiauting tir-face temperatures on1 the Cenitigra(le scale. If bothtemperatures are niear 00, then F (cal/Cm112 * sec 0°)is 1.1, anl(l it is 1.5 if the temperatuires are near 300.

Equation LIII clearly resembles equiation XXXI,so the term F has the same natture as cpr,?;-, wherer, is the equiiiv-alent diffuisioni resistanice.

Henice r, = c,p/F LINTw-here c, is the specific heat of air at conistanttemperatuire, andl p is its density at ambient tem-peratuires. Thuis the resistance varies between 2.8sec/cm at 00 and 1.9 sec/cm at 300. If the ranigeof ambient temperatuires lies about 20)', the appro-priate value is 2.1 sec/cni.

Appendi.x 3. The Effect of a Leof being EX-poscd to a Clear Sky. A clear sky transmits long--wave ra(liatioin downwards (Qid), according to anexpression derived from those of Swinbank:

Od., = 1.62 X 10 '2ToK4 - 0.0041 cal cm2secTIN"

656




where T.K is the amlient temperatutre (OK) . Hencethe net tupwards longwave-radiation flux from theexposed side of the leaf is as follows:

QLd = 1.35 X 10-12T48K - 1.62 X10-'2T4.,± 0.0041 cal/cm2 * sec= 1.35 X 1012 (T4oK - T4a.K +[0.0041 - 0.27 X 10 12 T4aK]

= H, + Qa LVIThe fltux H, is defilned in eqtuation LIII. The fluxQ0 has the valuie 0.0026 cal/cm2sec at 0°, and 0.0018cal/cm2sec at 30°, .so it is abouit 0.002 ca'l/cm2secat normal temperatuires. The fact that Qa is prac-tically indepencdent of the leaf temperatulre meansthat it may be lumped wi'th the fixed net short-wave-radiation fluix Q-as in equiation XXXIII, withoutaffectiing the validity of the resuilting equlations.

If the sky radiation is obsculred lby an object ata fixed temperatuire TbK(0K), then the niet long-wNave-radliation flux (Q,i ) is as follows:

AL -- 1.35 x 10nX1(07'4 , - 7'4,) cal/cIm2secLVII

F X 10-4(T8 - 7'b)F X 104(T0 T X 10-4 (Tb - T.)

H, - Ob LVIIIThe flulx Qb is again iid(lepeinlenit of leaf tem-

peratuire. So e(lqlation VTIII is still valid.Appendixr . Calculation of the Externazl Dif-

fuision Resistanitce rx.1. Equiations

r,3 ± Cr2 Fr.- - G VIIIC = ri ± r - (1 + 2A)/P XIIIF = riLr2 -(iri + ri2) (1 + A)/P XIVG rilri2/P XIP 32/t./2 - 0.48 XVI

2. Exanlpletlr2 25 sec; T - 300, i.e. A = 1.8 mmHg/°Cri= 0.5 sec/cm; ri2 = 1.0 sec/cmP = 32/25 - 0.48 = 0.8C = 1.5 - 4.6/0.8 = -4.3F = 0.5 - 1.5 X 2.8/0.8 = -4.8G = 0.5/0.8 = +0.6

rx -4.8rr -,2 -4.3r.2 rx3

5

65.45.2

G-a b c a+b+c(a+b-+c)

-24 25 -107 125 -6 +7-29 36 - 155 216 +32 -31-26 29 -125 157 +6 -5-25 27 -116 140 - 1 +2

Hence r7 = 5.2 sec/cm

AcknowledgmentsI a-1i gratefuil for commenits frcm colleagues, notably

Dr. A. R. G. Lang anid Dr. C. W. Rose.

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