Planta (Berl.) 125, 281--287 (1975) 9 by Springer-Verlag 1975

The Thermal Conductivity of Leaves Robert L. Hays Department of Biology, San Diego State University, San Diego, California 92182, USA Received 17 December 1974; accepted 28 May 1975

Summary. Thermal conductivities of fresh leaves, both unmodified and infiltrated with water, were measured. Samples were placed between silver plates of known and differing temperatures, and the time required to boil off a constant volume of liquid was measured. The species used are evergreens: Eucalyptus globulus Labitl. (sclerophyllous) with isolateral leaf symmetry; and Peperomia obtusi/olia A. Dietr. (succulent), Citrus limon Burm. f. (mesophyllous), Arbutus menziessii Pursh. (sclcrophyllous), and Heteromeles arbuti]olia M. Roam. (sclerophyllous), all with bilateral leaf symmetry. Mean values found were in the range of 0.268 to 0.573 W/re. ~ for fresh leaves, and 0.540 to 0.548 W/re. ~ for leaves infiltrated with water. An analysis of errors in the technique indicated that these values may be somewhat low. These results are several times higher than previously reported values. It is concluded that ordinary mesophytic and xerophytic leaves will not develop large gradients in temperature between the surfaces. Introduction

I n studies of gas and energy exchange between plant and environment, leaves are commonly assumed to be isothermal. Slatyer (1971) has shown t h a t small errors in the measurement of leaf temperature can result in large errors in calculated resistances to gas diffusion. Deviations from isothermality might produce similar errors. Is it safe to assume isothermality between the surfaces ? Reports exist of temperature differences between the surfaces (e.g. Gale et al., 1970) but such reports are subject to question, since there are formidable difficulties in making these measurements (Perrier, 1971). The existence of such a temperature difference, however, can be argued on theoretical grounds. If the fluxes of radiation impinging on the surfaces differ greatly in magnitude, and if radiation is absorbed most strongly near the surface, then the two sides should differ in temperature. The magnitude of this difference should depend upon the distribution of the absorption of radiation inside the leaf, and upon the thermal conductivity of the leaf between the surfaces. Measurement of the thermal conductivity of fresh leaves is technically very difficult. Perrier (1968) calculated a thermal conductivity of 0.15 W / m . ~ from measurements of the temperatures of the surfaces, and the various fluxes of energy into and out of leaves in air. His report, however, contains few details. The difficulties involved in making adequate measurements of surface temperature leaves the reliability of this report in doubt. Turrell and Austin (1969) reported con. ductivities of fresh leaves (in the transverse direction) of between 0.11 and 0.13 W/re. ~ The technique used, however, suffers from several shortcomings.

First, the tissue samples were necessarily left in the apparatus for a long time (i-2 h, Turrell et al., 1967). Secondly, they were placed under pressure (810 or 1620g, unknown area; Turrell and Austin, 1969). If they were compressed 6

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R.L. Hays

s i g n i f i c a n t l y , t h e i r t h e r m a l c o n d u c t i v i t i e s w o u l d b e a l t e r e d . Also, if t h e l e a f t h i c k n e s s w e r e m e a s u r e d w h e n i t was n o t u n d e r c o m p r e s s i o n , e r r o r w o u l d r e s u l t f r o m u n d e r e s t i m a t i n g t h e t e m p e r a t u r e g r a d i e n t across t h e leaf. F i n a l l y a n d p r o b a b l y m o s t i m p o r t a n t l y , since T u r r e l l a n d A u s t i n m a d e t h e i r m e a s u r e m e n t s , i t h a s b e e n r e c o g n i z e d (Fried, 1969) t h a t t h e r e o f t e n is a s u b s t a n t i a l r e s i s t a n c e t o h e a t c o n d u c t i o n b e t w e e n t w o o b j e c t s i n c o n t a c t . T h i s c o u l d p r o d u c e errors of p e r h a p s m o r e t h a n 100% i n t h e m e a s u r e m e n t s b y T u r r e l l a n d A u s t i n . T h i s p a p e r r e p o r t s m e a s u r e m e n t s of t h e r m a l c o n d u c t i v i t i e s u s i n g a t e c h n i q u e w h i c h a v o i d s s o m e of t h e p r o b l e m s m e n t i o n e d a b o v e .

Methods The method chosen to measure the thermal conductivity of leaf material was that described by Schroder (1963). The sample or calibration disc rests between two silver plates. Below the lower one is a reservoir of boiling liquid. Vapor is directed over the lower plate where it can condense, transferring the latent heat of condensation to the plate. This maintains the plate at the boiling point of this liquid. Surplus vapor is condensed in a condensor and returns via a vapor trap. The upper plate acts as the bottom of a second reservoir containing a liquid having a lower boiling point than the one in the lower reservoir. Heat traveling through the sample heats the upper plate to the boiling point of the second liquid. Additional heat added is carried away by vaporizing the upper liquid. This vapor is condensed and its quantity measured in a graduated container. After a quantity has condensed so that the system can reach equilibrium, the time necessary to boil off a constant quantity of the upper liquid is measured. This, of course, measures the rate of heat flux through the sample. The last two variables necessary to calculate the thermal conductivity are determined by measuring the dimensions of the sample. Calibration is accomplished by using discs of known thermal resistances to produce a linear relationship with time. An apparatus was constructed utilizing a simplification of Schroder's design. My apparatus differs from his in that it lacks the complicated heat guard system of evacuated and silvered envelopes, and the boiling liquid heat barrier. These modifications could be made because of the small temperature difference between the boiling point of the " c o l d " liquid and the ambient room temperature. Instead of the spring system used by Schroder to maintain a constant known pressure on the sample, the upper portion of my apparatus was affixed to a sliding pipe on a ring stand rod. The weight of this portion furnished the necessary pressure (about 475 g/cm~). The liquids chosen were Freon 11 (B.P. 23.8 ~ and methylene chloride (B.P. 41.6~ Dow Coming {Midland, Mich., USA) DC-704 diffusion pump oil was used to reduce the contact resistance between the sample and the apparatus. Calibration standards were discs of Pyrex 7740 glass of various thickness and 18 mm in diameter. Leaf samples of that diameter were punched out with a cork borer from areas of the leaf with the smallest veins to minimize variations in thickness. Samples were taken close together from comparable tissue, and from a single leaf whenever possible. Leaves from five species were used to sample a range of morphological types. To reduce the temperature on the " h o t " side of the leaf, one of the calibration discs was placed between the leaf and the apparatus on that side. This also facilitated positioning and alignment of the sample. To reduce infiltration of the leaf sample by the oil during the experiment, the cut edges were sealed by rolling the disc across a plate coated with Lubriseal (Arthur H. Thomas Co., Philadelphia, Pa., USA). Infiltration was usually insufficient to change the obvious reflective properties of the sample. Attempts to weigh the uptake of oil, if any, were unsuccessful, presumably because of water loss from the sample. Some leaf types did absorb oil somewhat through the epidermis, but no superior technique was found. Only data for leaf samples having less than about 5 % of the area obviously infiltrated by oil are reported below. The thickness of the samples was measured with a micrometer in the middle of the sample and at four places equally spaced around the edge both before and after measuring thermal resistance. The mean of the measurements after removal from the apparatus was used. Timing

The Thermal Conductivity of Leaves

283

was by means of a step-watch. Samples were typically in the apparatus for less than 5 rain. Calibration used a minimum of three points with each of two discs. The expected resistance of a calibration disc was calculated by: Rdisc - -

4 Zdisc 2 ~D disc ]Cdisc

(I)

where Zdisc is the thickness and Ddisc the diameter of the disc, and kdisc, the thermal conductivity, is 1.11 W/re. ~ (Powell et al., 1966). Calibration was quite repeatable. (Maximum coefficient of variation for a calibration disc was 2.3%.) The additional contact resistance present when measuring a leaf sample (between it and the glass disc) was estimated by placing two glass discs in series and measuring the additional resistance (0.432 ~ C/W). The thermal conductivity of the leaf (klf) was calculated by use of the formula: Z 1f

k l f ~ Alf Rlf

~D~lf(ZR

-

4 Zlf -~disc-- ~cont) -

(2)

where Z l f is the sample thickness (after removal from the apparatus), Alf is the area of the sample disc with diameter Dlf , Rlf is the resistance of the leaf to thermal conduction, 2:R is the total resistance of the leaf and disc in the apparatus from the calibration, Rdisc is the resistance of the glass disc in series with the leaf sample, and Rcont is the resistance of the additional contact between the glass disc and the leaf sample. To estimate the heterogeneity of the samples, they were weighed as well as individual thicknesses measured. To measure the percent air space, leaf samples were vacuum infiltrated with distilled water, and the air space determined by weighing.

Results The measured values of the thermal conductivities of leaves of the various species are given in Table 1. Arbutus menziesii Pursh., Citrus limon Burro. f., and Heteromeles arbuti/olia M. Roam. have typical biracial n o n - K r a n z dicotyledonous a n a t o m y . There is considerable variation between these species in the t h e r m a l c o n d u c t i v i t y values. Eucalyptus globulus Labill. leaves have isolateral s y m m e t r y . The thermal conductivity, however, is similar to t h a t of the species with biracial leaves. This indicates t h a t the s p o n g y mesophyll does not function as an effective thermal insulator. Peperomia obtusi]olia A. Dietr. is characterized b y thick, succulent leaves. This succulence is due to a thick hypodermis-like layer. Air space occupies only 8 % of the volume, as measured b y infiltration. There is little t o r t u o s i t y to pathways for heat conduction in the liquid phase between the surfaces. B o t h unmodified and water-infiltrated samples exhibit thermal eonductivities a r o u n d 0.54 W / m - ~ This value represents the best available estimate of the thermal c o n d u c t i v i t y of the liquid phase of leaf tissue and is slightly lower t h a n t h a t of pure water (Table 1). This value m a y be too high for Arbutus menziesii and Citrus limon since infiltrated leaves of these species are in the same range, while their percent air space is considerably larger. I t would be expected t h a t the higher c o n d u c t i v i t y of water would have to be counter-balanced b y a lower c o n d u c t i v i t y for the liquid material in these species. A sample of the measurements of t h e r m a l conduetivities was subjected to an u n c e r t a i n t y analysis using the m e t h o d of Moffat (1970). Details of these calculations are given in the Appendix. The means of the final calculated uncertainties are given in Table 1. These values represent the 95 % confidence intervals a r o u n d a given d a t a point for the true value, based u p o n expected r a n d o m variation

284

R . L . Hays

Table 1. Thermal eonduetivities of fresh leaves, dry air and water (a leaves with raised veins; b values from Weast, 1968) Material

1~o. of Mean thermal Standard Uncertainty deterconductivity error minations (W/mK) (Qk) %

Citrus limon a Sun leaf No. 1 Sun leaf 1~o. 2 Sun leaf No. 2 (water infiltrated)

Airspace %

2 2 3

0.37 0.41 0.54

0.01 0.05 0.13

0.08 0.08 0.11

20 17 19

Heteromeles arbuti/olia a Shade leaf 3

0.27

0.02

0.03

11

3 3 3 3

0.36 0.38 0.38 0.55

0.02 0.02 0.02 0.08

0.05 0.05 0.05 0.10

13 12 t3 15

29 29

3 1

0.56 0.55

0.01

0.05 0.05

9 9

8 8

2 4 3

0.32 0.36 0.31

0.01 0.02 0.01

0.05 0.04 0.03

15 11 10

Arbutus menziesii Young sun leaf No. Young sun leaf No. Young sun leaf No. Young sun leaf No. (water infiltrated)

1 2 3 3

Peperomia obtusi/olia Shade leaf No. 1 Shade leaf No. 1 (water infiltrated) Eucalyptus globulus Juvenile leaf Young adult leaf Old adult leaf Dry Air b Water b

30 30

0.026 0.59

i n h e r e n t in t h e technique. I t is i m p o r t a n t to n o t e t h a t t h e confidence l i m i t s given r e p r e s e n t r e a s o n a b l e m i n i m u m values, since some sources of error could n o t be a n a l y z e d . T h e v a r i a b i l i t y in t h e m e a s u r e d s a m p l e s is g r e a t e r t h a n t h e e x p e c t e d rep e a t a b i l i t y ( a b o u t 6-12 % of k ; see A p p e n d i x ) . T h e v a r i a b i l i t y in leaf tissue, as e v i d e n c e d b y disc weights (Table 2), u n d o u b t e d l y p r o d u c e s some v a r i a t i o n in t h e r m a l c o n d u c t i v i t y , b u t a regression of t h e r m a l c o n d u c t i v i t y on disc weight/disc t h i c k n e s s (as a m e a s u r e of d e n s i t y ) was n o t significant.

Potential Errors T h r e e classes of p o t e n t i a l errors, which were n o t a n a l y z e d , were a p p a r e n t d u r i n g t h e course of t h e m e a s u r e m e n t s . Firstly, since t h e t h i c k n e s s of t h e s a m p l e s was g e n e r a l l y small, a s m a l l error in m e a s u r i n g i t w o u l d p r o d u c e a large error in/elf. Compression of leaf discs f r o m t h e weight of t h e a p p a r a t u s , i m p r o p e r a l i g n m e n t , or v a r i a b i l i t y in tissue thickness could cause this error. A s m a l l b u t significant decrease in t h e thickness of t h e s a m p l e s a f t e r m e a s u r e m e n t was f o u n d (Wilcoxon

285

The Thermal Conductivity of Leaves Table 2. Weight variation and compression of leaf discs used for thermal ductivity measurements Species

con-

Mean weight (rag)

Variance (mg)

Mean thickness (~m)

Decrease from initial thickness (%)

59.8 87.9 87.9

0.07 0.02 0.50

279 380 374

2 1 2

64.3

0.08

246

8

61.3

0.01

278

6

75.4 90.4 81.8

0.02 0.01 0.02

370 398

2 3

397.1

7.19

1611

2

Eucalyptus globulus Juvenile leaf Young adult leaf Old adult leaf

Citrus limon Sun leaf

Heteromeles arbuti]olia Shade leaf

Arbutus menziesii Sun leaf No. 1 Sun leaf No. 2 Sun leaf 1~o. 3

Peperomia obtusi/olia

signed rank test p ~ 0 . 0 1 ; Table 2). Attempts to measure the compression of discs under similar loading were unsuccessful. There were, however, no obvious signs (such as tissue damage) of tissue compression. To minimize this error, the thickness used was that of the sample after it was removed from the apparatus. If compression of a sample reduced the thickness below the value used in the callcnlations, the result would be to over-estimate/~lf. Since DC-704 has a thermal resistance of about twice that of leaf tissue, veins on the leaf which held the apparatus farther apart than the rest of the tissue would result in under-estimation of /~lf. Leaves with veins which protruded are indicated in Table 1. I t seems unlikely that within a leaf there would be enough variation in this attribute to account for much of the scatter. Improper alignment would also cause /elf to be under-estimated. This also seems unlikely to cause much of the scatter.

Secondly, potential over-estimation of/elf would occur if All were effectively increased by excess DC-704. Gross excesses were not observed and it is thought that this was not a significant problem. Finally, it was difficult to prevent entrapment of air bubbles in the DC-704 during assembly of the apparatus with tissue present. Since air is a poor conductor of heat (Table 1), and is thought to be responsible, to a great extent, for the contact resistance phenomenon, bubbles would produce substantial errors resulting in the under-estimation of klf. I t is not clear whether or not these sources of error, taken together, have resulted in systematic bias. The fact that measurements on leaves infiltrated with water are close to that of pure water argues against a substantial systematic error in the technique.

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R . L . Hays

Discussion T h e values of t h e t h e r m a l c o n d u c t i v i t i e s of fresh leaves m e a s u r e d a r e subs t a n t i a l l y higher t h a n t h o s e r e p o r t e d b y Turrell a n d A u s t i n (1969) or P e r r i e r (1968). M y v a l u e s for Citrus limon are a b o u t 3 t i m e s T u r r e l l a n d A u s t i n ' s values (K---0.122 W / m . ~ standard deviations0.016 W/m.~ for t h i s species. This disa g r e e m e n t m a y , of course, be due to t h e v a r i a b i l i t y in Citrus leaves, or error in technique. To help d e t e r m i n e t h e t r u e value, it is useful to e s t i m a t e l i m i t s t o t h e t h e r m a l c o n d u c t i v i t i e s of Citrus leaves. A s s u m e t h a t t h e leaf consists of o n l y d r y air a n d w a t e r (30% air space b y v a c u u m i n f i l t r a t i o n of water) a n d t h a t h e a t is t r a n s f e r r e d inside t h e leaf o n l y b y conduction. U s i n g values of k for w a t e r a n d a i r f r o m T a b l e 1, l i m i t i n g values of klf are 0.079 W / r e . ~ for a p a r a d e r m a l l a y e r of a i r in series w i t h one of w a t e r a n d 0.418 W / r e . ~ for t r a n s v e r s e columns of a i r a n d w a t e r in parallel. These e s t i m a t e s are p r o b a b l y low since t h e y ignore h e a t t r a n s f e r b y i n f r a r e d r a d i a t i o n , a n d b y t h e e v a p o r a t i o n diffusion a n d c o n d e n s a t i o n of water. T h e a c t u a l o r g a n i z a t i o n of cells inside t h e leaf involves m a n y t r a n s v e r s e connections. This i n d i c a t e s t h a t t h e real t h e r m a l c o n d u c t i v i t y is on t h e higher side. T h e m e a s u r e d high t h e r m a l c o n d u c t i v i t i e s of leaves i m p l y t h a t t r a n s v e r s e t h e r m a l g r a d i e n t s a r e small. F o r m o s t purposes, t h e y m a y be a s s u m e d to be i s o t h e r m a l in t h e t r a n s v e r s e d i r e c t i o n ( b u t n o t in t h e p a r a d e r m a l p l a n e ; R a s e h k e , 1956). P u b l i s h e d r e p o r t s (e.g. Gale et al., 1970) of significant t e m p e r a t u r e differences b e t w e e n t h e surfaces are l i k e l y to be d u e to e x p e r i m e n t a l artifacts.

Appendix:

Thermal Conductivity Uncertainity Analysis The equation used to calculate the thermal conductivity of the leaf (Eqn. 1) includes five variables. A suitable analytical method for determining the uncertainty associated with the calculation of 27R -- R4isc using a regression line was not found. Consequently, a Monte Carlo analysis was performed on the computer. This analysis simultaneously and randomly (using normal distributions of appropriate means and standard deviations to give the expected variation of each parameter) varied the resistance of the calibration discs and the elapsed times of the calibration and leaf sample runs. After each randomization, a calibration linear regression line and the predicted 27R -- ~disc were calculated. After 100 repetitions, the standard deviation of the X R - Rdise values was calculated. Since 27R and Rdisc are not independent, this leaves four independent variables which can be combined to give the uncertainty in klf by:

[[ ~]Clf

1- / c q ] c l f ~2_~_(_ (9]Vlf \21112 [t Qz, )\2+ ~|\ -~-Dlf @DII] \ ~(~''-- Rdisc) e(ZR--Rd,s ))J

where @xis the uncertainty in x at a given odds, and the odds are the same for all the variables (Kline and McClinteck as quoted in Moffa% 1970). For an estimate of the expected repeatability, the resistances of the calibration discs and the diameters of the samples were taken to be non-varying, i.e., @= 0. The partial derivatives were evaluated by use of the approximation: ~]Zlf

klf(X -t-/I x) -- k1f(x)

~x

Am

'

where A x was 0.001. Values for the terms used are as follows (odds ~ 95% confidence limits); Dlf was taken as the diameter of the glass disc on which the sample rested. This is somewhat low, since the samples were slightly larger than that. The approximation is good since all samples were punched with the same punch, and the error due to a poor alignment of a sample on the disc

The Thermal Conductivity of Leaves

287

would have been larger than that associated with a somewhat larger area available for heat transfer, oDlf was taken as twice, and 0Zlf as equal to the smallest division of the micrometer scale (0D]f = 5.0 ~zm for D l f = 1.7920 cm, o Z l f = 2.0 • 10-6 tzm). Errors due to this parameter are discussed above. For the Monte Carlo analysis, the following values were used. The thermal conductivity of Pyrex 7740 was taken as 1.11 W/m.~ and ~ was taken as 0.054 W / m . ~ (Powell et al., 1966). Since the discs were quite flat, the uncertainty in the thickness was taken to be 1.3 • 10-6 tzm. The uncertainty in the elapsed time of the measurements was taken as 0.5 s. This work was supported by National Science Foundation Grant GB21441 to Dr. H. A. Mooney. The Ames Research Center of the National Aeronautics and Space Administration built the thermal conductivity apparatus. Their Mr. John Kirkpatriek supplied valuable technical advice. Drs. Mooney and P. H. Zedler offered valuable criticism of the manuscript

References Fried, E. : Thermal conduction contribution to heat transfer at contacts. In: Thermal conductivity, vol. 2, p. 253-275, Tye, R., ed. New York: Acad. Press 1969 Gale, J., Manes, A., Poljakoff-Mayber, A.: A rapidly equilibrating thermocouple contact thermometer for measurement of leaf-surface temperatures. Ecology 51, 521-525 (1970) Moffat, R.: Uncertainty analysis. Stanford, Calif., U S A : Thermosci. Measur. Ctr., Dep. of Mech. Eng., Stanford Univ. 1970 Perrier, A. : Contribution ~ l'6tude des 6changes thermiques en biologie v6g~tale. Rev. g~n. thermique (Paris) 79-80, 721-740 (1968) Perrier, A. : Leaf Temperature Measurement. In: Plant photosynthetic production, p. 632-671, Sestak, A., Catsky, J., Jarvis, P., eds. The Hague: J u n k 1971 Powell, R., tto, C., Liley, P. : Thermal conductivity of selected materials. National Standard References Data Set. NBS-8. Washington, D. C. : U.S. Nat. Bur. Standards 1966 Raschke, K. : Uber die physikalischen Beziehungen zwischen W~rmeiibergangzahl, Strahlungsaustausch, Temperatur und Transpiration eines Blattes. Planta (Berl.) 48, 200-238 (1956) Schroder, J. : Apparatus for determining the thermal conductivity of solids in the temperature ranges from 20 to 200 ~ C. Rev. sci. Instr. 84, 615-621 (1963) Slatyer, R. : Effect of errors in measuring leaf temperature and ambient gas concentration on calculated resistances to C02 and water vapor exchanges in plant leaves. Plant Physiol. 47, 269-274 (1971) Turrell, J., Austin, S. : Thermal conductivity and mass in stems, leaves, and fruit in relation to frost resistance. Proc. 1st Internat. Citrus Syrup., Riverside, Calif., USA, March 1968, vol. 2, p. 601-608, Riverside, Calif. U S A : Dept. of Soil and Plant Nutr., Univ. of Calif. 1969 Turrell, J., Austin, S., McNee, D., Park, W. : Thermal conductivity of functional citrus tree wood. Plant Physiol. 42, 1025-1034 (1969) Weast, R., ed. : Handbook of Chemistry and Physics, 49th edn. Cleveland, O., USA: Chem. Rubber Co. 1968