Agricultural Meteorology, 1 5 ( 1 9 7 5 ) 97- -111 © Elsevier Scient i f ic Publ i sh ing C o m p a n y , A m s t e r d a m -- P r in ted in The Ne the r l ands

NATURAL CONVECTION HEAT TRANSFER IN COVERED PLANT CANOPIES

M. I Q B A L * and J. A. S T O F F E R S

Institute of Horticultural Engineering, Wageningen (The Netherlands)

(Received May 29, 1974 ; accep ted N o v e m b e r 25, 1974)

A B S T R A C T

Iqbal , M. and Stoffers , J. A., 1975. Natura l c o n v e c t i o n hea t t r ans fe r in covered p lan t canopies . Agric. Meteorol . , 15: 97 - -111 .

A faci l i ty has been des igned and cons t r uc t ed to carry ou t the s tudy of free c o n v e c t i o n f rom leaves in a covered p lan t canopy . The sys tem cons idered is one wi th long parallel rows of canopies w i th f ini te widths , he ights and spacings be tween them. To genera te free convect ive air f low pa t t e rn s inside the canopies , leaves have been s imula t ed by small electr ic bulbs . These " b u l b c a n o p i e s " were c o n s t r u c t e d in such a way so as to p rov ide variable ( leaf surface a r ea ) / (vo lume) ratios. The electr ical sys tem was designed to provide var ious levels of energy inpu t at d i f fe ren t he ights of the canopies which would co r r e spond to desired solar f lux a b s o r p t i o n at these heights .

Convect ive hea t t r ans fe r coef f ic ien ts (in t e rms of Nussel t n u m b e r s ) are p resen ted for a leaf area dens i ty of a p p r o x i m a t e l y 90 cm 2 /1 ,000 cm 3. Hea t - t ransfe r coef f ic ien ts were measu red o n two types of t h i n pla te 8 x 8 cm size, one having b o t h surfaces of coppe r and the o the r having b o t h surfaces covered wi th whi te paper . The first such pla te simula- ted a leaf wi th u n i f o r m surface t e m p e r a t u r e and the second a leaf wi th u n i f o r m surface hea t flux. These plates were f ixed in a vert ical pos i t ion at a b o u t the cen t r e of the canopy . F r o m the expe r imen ta l data , t w o hea t - t r ans fe r cor re la t ions have been p resen ted for t he two plates. The resul ts show t h a t the di f ference b e t w e e n the values o b t a i n e d f rom the two s imula ted leaves is no t greater t h a n a b o u t 15%.

I N T R O D U C T I O N

A leaf subjected to a thermal load exchanges energy with its environment. The modes of exchange are radiation, convection, transpiration and conduc- tion. All these modes are coupled so that after redistribution of energy, the leaf acquires an equilibrium temperature.

In this report, we are interested in the study of convective heat transfer. General theories of convective heat transfer, called boundary-layer theories have been presented in several textbooks such as Schlichting (1967) and Kays (1968). Gates (1962), Rose (1966), Munn (1966) and Lowry (1967)

* On leave f r o m D e p a r t m e n t of Mechanica l Engineer ing, Univers i ty of Bri t ish Columbia , Vancouver , B.C., Canada, 1972.

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have shown how theories of heat transfer originating from the physical sciences can be applied to the energy-exchange processes of plants. However, a s tudy of these transfer processes within a plant canopy is essential for determination and eventual control of plant temperature, evapotranspiration from leaves and photosynthet ic activity. Plant temperature is of key impor- tance. It sets the potential at which the heat-exchange processes are in equi- librium. Leaf temperature influences heat-transfer processes in different ways. Conduction and convection are proportional to the temperature difference between the leaf and its environment. The infrared radiative loss increases with the fourth power of the absolute temperature. The saturation vapour pressure and hence the potential evaporation rate changes exponentially with the temperature. Studies concerned with conditions within plant canopies are relatively few compared to those dealing with the region immediately above them.

Studies of the partitioning of energy within a foliage canopy have been undertaken by Impens (1966), Impens et al. (1967), Landsberg and Thom {1971) and Druillet et al. (1971) among others. However they consider the conditions produced under forced movement of the air. Free convection from isolated individual leaves of a plant in still air has been investigated through Schlieren photography by Gates and Benedict (1963).

DESCRIPTION OF THE PROBLEM

During summer, under strong radiant flux, in most Dutch glasshouses, natural convective cooling of plants is employed by opening windows. At these times, since there is no additional source of energy such as hot-water circulation, solar energy is the only source for the onset of convection currents. The convection patterns in the air are naturally induced by the differences in density arising from differences in temperature between the air and the plants. In these conditions, convective heat transfer between plants and the air takes place only by natural convection, also called free convection.

In the present s tudy we are interested in the free-convection coefficients inside rows of foliage canopies in a glasshouse. It is our purpose to investigate the effects of leaf surface distribution, the ratios of canopy width, height and spacing under various solar irradiation loads. Our ultimate purpose is to space the rows and the distance between plants so as to prevent leaves from attaining lethal temperatures.

General studies of the overall heat balance of glasshouses have been repor- ted by Takakura et al. (1969), Sel~uk (1970, 1971), Seginer and Levav, (1971), and Grac et al. (1969). Walker (1965) and Takakura (1967) have reported studies on the prediction of air temperature in glasshouses. All these studies employ approximate values of various transfer coefficients in order to arrive at the energy balance or the prediction of air temperature in a greenhouse. In these various transfer coefficients, the convective resistances inside the

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canopy are generally much larger than, say, the convective resistance on the outside of the glasshouse or the conductive resistance of the soil. In addition, the convective resistance inside a glasshouse canopy is higher for natural con- vection than when air movement is induced by a fan, blower, or the like. As an accurate determination of this natural convective coefficient is very impor- tant, it is the subject of this report. At the Institute of Horticultural Engineer- ing, Wageningen, the present study is a part of a programme to determine accurate values of various coefficients.

Free or natural convective heat transfer from surfaces with simple geo- metries such as single plates, spheres or cylinders placed in a still ambient fluid has been extensively studied and is well documented in various textbooks. However, in a plant canopy, the leaves and the stems, their surface area dis- tribution and orientation have considerable interaction on the development of airflow patterns and the resulting heat-transfer rates. In addition, the heat-transfer rates are modified by the size and proximity of the rows of

NOTATION

D diameter of a circular plate with same area as tha t o f the copper or paper plates of 8 × 8 cm.

g accelerat ion due to ear th ' s gravity h heat t ransfer coeff ic ient (kcal. m - 2 h -1 ) G m a x i m u m solar insolat ion G~ m a x i m u m solar flux incident on plant canopies G: m a x i m u m of solar flux convec ted by the plants.

(Gr) gqD4 Grashof number (also see Appendix) . v 2 ~,T ' M~ num ber of electric bulbs in series M 2 total number of series in three canopies

qD (Nu) (Ts_Tg)~ , Nusselt number (also see Append ix )

P a tmospher ic pressure P total electrical power dissipated at a layer in plant canopy q heat flux (kcal. m -2 h -1) R total electrical resistance of a layer in plant canopy T absolute t empera tu re of ambien t air Tg mean air t empera tu re in the b o t t o m fif th layer (°C) T s surface t empera tu re of bulb or plate u, v air velocity c o m p o n e n t s in X and Y direct ion V voltage across a layer in canopy X~ one half of the wid th of a canopy X 2 clear dis tance be tween two canopies X, Y co-ordinate di rect ions YI height of a canopy ~ electrical resistance of a single bulb 2, thermal conduc t iv i ty of air at Tg v k inemat ic viscosity of air at Tg p densi ty of air at Tg 3 1

~ , coef f ic ien t o f thermal expans ion of air

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canopies in a glasshouse. In the following section we will describe an experi- ment designed to simulate the foregoing variables so that the results can be expressed in dimensionless quantities. The dynamic dimensionless quantities are, the Nusselt number (Nu) through which the heat-transfer coefficient is expressed and; the Grashof number (Gr) (see Notation). In this instance, the Grashof number expresses the air movement created by the buoyancy forces induced by the temperature differences between the leaves and air, produced by solar flux absorbed in the canopy. From dimensional analysis it can be shown that these two dynamic numbers are related to each other through an expression of the type:

(Nu) = c(Gr)d (1)

where c and d are as yet unknown numbers determined by geometric quanti- ties such as length, area and volume ratios pertaining to the leaves and the rows of canopies. The unknowns c and d also depend on the Grashof number itself, which makes the relation 1 non-linear and in the present case extremely difficult to resolve mathematically.

The aim of the experiments described here is to measure the values of (Nu) and (Gr) and then through regression analysis to obtain the coefficients c and d. It is also our purpose to present the results in such a manner that both the researcher and the designer can use them without great difficulty.

FORMULATION OF THE PROBLEM

Consider the section of a glasshouse as shown in Fig.l , with a number of rows of plant canopies. The separation between the rows is required for plant- ing, harvesting and for any mechanisation involved therein. The height, width and leaf distribution of the canopies depend upon the type of plant, its age and the numbers of plants per unit area of soil.

In the absence of any artificial heating and ventilation, solar energy is the only thermal source establishing temperature levels, temperature distribution and convective patterns in the canopies. Let G be the total maximum solar flux density outside a glasshouse, which in the case, of the Netherlands* is about 660 kcal. m -~ h-1. After reflection and absorption losses of about 30%, let the energy flux density just above a canopy be called G1. About 10% of this G1 is reflected by the canopy and the rest is redistributed by the plant into conduction, transpiration, emission, convection and maintenance of its temperature at a certain level. The convective and evaporative portions are about 40--50% of G1 each. Let the convective portion be called G2. Within the canopy, G2 is absorbed in an exponentially decaying manner as one goes down from the top of the canopy towards the soil (Nilson, 1971). Flow patterns and the temperature distribution inside the plant canopy will

* The present study is very general and not limited to applications on Dutch glasshouses. The value taken is purely to indicate the data used for design calculations.

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g l a s s

G= solar irradiation 660 kcat./m 2 hr or 768 Wat t /m 2

~ o kc~L./~. 2 hr

or 537 Watt/rn 2 ~ j

-rows of plant canopies _~'I

!~'~:~Z~.~!~..F~ 2[.:~'~.~-~v~.~ ?':~ ' : ~ . : :.~:.~ :~.:~̀~:~:k~ :~ :.~ C.~>!~.>'~.:̀ ~q~ ~ K 1 Fig.1. Sec t ion of a glasshouse.

r-expected fLow Lines YI

--Leaves i l l f ~

f- . . . . . . . . [ooooo oOoOo (,oo] ,°o°o°o°° o o ,°o°o 0 0 0 0000 0 0 0 ~/O~!~t[O 0 p 00000( o o o o o oob o, ~°o°oI

, o o o o o o o o~,o~ o) ~, ~, OOD~(~',O0 O0 0"000 O0 01)10001! 0000000000 On n n n nO010000000 O0 O0 00000 O00000C ~ 000000000 0000

_ 2X1 _ ~ x2 _

Fig.2. Coordinate system for the canopies.

partly depend upon the value of G2, its distribution within the canopy and the factors enumerated in the preceding paragraph.

Now let us consider the canopies shown in Fig.2 as if they are in a large space of still ambient air, so that the convective mot ion of the air in the vicinity of the canopies is not affected by the walls of the glasshouse. Con-

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sider the leaves in the canopies as being uniformly distributed. We assume that there are many canopies side by side and their length into the plane of the paper is large. Under the above conditions, the free convective patterns of airflow will resemble that drawn in Fig.2.

In order to obtain a mathematical solution of the problem, we should examine the boundary-layer equations governing the conservation of mass, momentum and energy of the airflow. Assuming that the changes in density are not large and influence only the buoyancy term in the equation of motion; the steady-state two-dimensional conservation equations with invari- ant fluid properties can be written as:

6_u_u + 5v = 0 (2) 5x 6y

( 5v 5u) 5 p + [52v 52v~ P uS-x+Vs-y = 5y tl[-~-~ +5y2} +pg~(Ts--Tg ) (3)

( 6T ~)={62T+~2T] pCp U~x + v \5x2 ~y~ } (4)

For some simple geometries and relatively simple boundary conditions such as a vertical flat plate with a uniform wall temperature, or uniform heat flux, the system of eqs.2 to 4 has been solved by approximate mathematical tech- niques (Sparrow and Gregg, 1956; Ostrach, 1973). In the problem depicted in Fig.2, the velocity at the surface of leaves is zero but the temperature or heat flux are neither uniform nor are they known. Under these conditions, solu- tion of eqs.2--4 is extremely difficult, if at all possible. Therefore, an experi- mental apparatus has been designed to solve eq.1 by simulating the conditions stated above. We now proceed to describe this apparatus.

EXPERIMENTAL APPARATUS

An experiment was designed and apparatus was built to simulate the situa- tions of Figs.1 and 2. It was easy to allow for the flexibility of width, height and spacing of the canopies. However, it was difficult to create conditions within the canopies which closely resemble in-vivo conditions. The main areas of concern here are: leaf shape, leaf surface distribution, leaf orientation, variable heat flux in each leaf and at different layers of the canopy, the result- ing natural convective airflow patterns, and the limit to the maximum tem- perature attained by a leaf.

After considering various possibilities to simulate plant canopies, it appeared most convenient and suitable to use small elliptic shaped electric bulbs of maximum length 99 mm and maximum diameter 35 mm (approximately 90 cm 2 surface area) shown in Fig.3A. The bulbs were held by snap-on clips attached to long thin PVC strips 2 x 10 mm, as shown in Fig.3B. The PVC strips were held on to bars which were themselves supported by columns as shown in the general arrangement, Fig.3C. The structure was designed to pro-

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2Xl0mm. strip

,' tectrtc bu tb A ~

2mrn.x 5rnm, d e e p ~ ~\ / I I / O ~ e "

~'~ /11 ~ . ~ ~,~" ~.hotes~ritLed2.Sc~.~p~rt~ l~<-/ iJ . ~ ~

I T I X ~'Y~_.~ ~ "

63 cm. centre to centre __1 LJ" ~.u- c

F ig .3 . S i m u l a t e d leaves and t h e i r a t t a c h m e n t . A . E l e c t r i c b u l b s used as leaves ( d r a w n fuU scale). B. A t t a c h m e n t of bu lb to PVC strip. C. A t t a c h m e n t of PVC str ips to w o o d e n mem- bers.

vide the capability of changing width and height of canopy in increments of 2.5 cm. Each canopy was built on a wooden plank of its own, so that the distance X2 between canopies could be varied to any desired value. Only three canopies were built and the central one was used as a test section. The whole apparatus was enclosed on four sides by hardboard sheets to avoid ex- ternal wind drafts. Fig.4 shows two photographic views of the canopies.

The arrangement of bulbs was such that each layer at any one height of the three canopies was connected through a Variac to the same power terminal (see Fig.5). This arrangement ensured the same power dissipation in each canopy at a given height. The power distributions at various heights could be varied as desired since each level having its own Variac.

From Fig.5, it is obvious that the total resistance R of the network at any one level will be:

R - Ms (5)

104

Fig.4. Two views of the canopies . A. A diagonal view of t h r ee canopies . B. A f ron t view of one canopy .

105

and hence the to ta l p o w e r fi d iss ipated at t h a t level will be:

/ ~=V-LM-~ (6) M~2

To s imula te condi t ions in a Du tch glasshouse, the spacing b e t w e e n individual bulbs be such tha t the p o w e r P p roduces a m a x i m u m flux dens i ty of 660 kcal. m -2 h -~ a l though , as m e n t i o n e d earlier, the design can be general.

I f z , is the d is tance b e t w e e n the bulbs in series and xa the d is tance b e t w e e n

i I I

M I ~ ~ I ~~~

Fig .5 . Electr ic c o n n e c t i o n o f bu lbs at one layer o f a n y he ight in three c a n o p i e s ( s h o w n in p lant v i ew) . Mj = n u m b e r o f bu lbs in a series; M 2 = to ta l n u m b e r o f series in three canop ies ;

~ = res is tance o f each bulb .

each series, then at any level the f lux dens i ty = ( to ta l wa t t s ) / ( t o t a l area) =

V~M2 1 V ~ .10 4 - - - (Wm -2) (7) M1~2 (z,xgl/I1M2) zlxgVl]~2

where zl and x3 are in cen t imet res . I t was dec ided to use f i f teen bulbs in series a long the lengths of canopies .

Ini t ia l ly , we used five such series of bulbs in each c a n o p y at any one level and five levels were cons t ruc t ed . The initial d is tance z, be tween bulbs in each series was 10 cm, the d is tance x3 be tween each row 10 cm, and the d is tance y2 b e t w e e n each level 10 cm. This resul ted in each bu lb having 1 ,000 cm a space.

Thin a l u m i n u m pape r was glued on to the bulbs to r educe radia t ive losses while the PVC str ip r educed conduc t ive losses.

106

To create airflow conditions approximating an actual plant canopy, the bulbs were mounted in a staggered fashion both in horizontal and vertical directions as shown in Fig.4B.

Two thin flat plates 8 x 8 cm were designed, each to simulate a leaf. One plate consisted of two, simple, white, hard, sheets of paper in between which electric wire was imbedded so as to produce a condit ion of uniform heat flux. The plate had its own power supply. The orientation of the plate could be changed as desired. The second plate was identical to the first except that its surfaces were made of thin copper sheets to produce uniform surface temperature.

The three canopies were supported on 3 x 2 m frames and were enclosed on all four sides by 2 m high hardboard walls as protect ion against wind drafts. Only the top of the frame remained open for the establishment of convection currents.

The temperatures of the electric bulbs and the air were measured by 0.5- mm copper-constantan wire thermocouples. To measure the temperatures of the artificial leaves (8 × 8 cm plates), 0.1-mm wire of the same material was used for thermocouples .

This experimental arrangement permit ted us: (1) to create idealized flow conditions in a simulated plant canopy; (2) to generate heat fluxes in various layers which simulated solar energy absorption in those layers; and (3) to evaluate the sensible heat-transfer coefficients f rom these simulated leaves (by using the two 8 x 8 cm plates).

RESULTS AND DISCUSSIONS

It was ment ioned earlier that the two plates representing artificial leaves could be oriented in any position; hence, we could obtain data for various angles of their orientation. As a first part of this comprehensive study, we shall present in this report data for only vertical orientation of the plates.

The experimental design permits changes in the equivalent leaf density and relative spacing of the canopies. However, the results reported here are for one bulb in a cubic space of 10 × 10 x 10 cm, representing a leaf density of approximately 90 cm 2/1,000 cm 3 . Also the distance from centre to centre between canopies was 85 cm. Each canopy was 50 cm high. The bo t tom layer in each canopy was 10 cm from the floor. Each canopy contained five bulbs per row and per column. All three canopies were fifteen bulbs deep.

Going down in the vertical direction, the percentage power consumed in each row of bulbs was in the following exponential decaying order:

Number of row 1 2 3 4 5

Approximate energy consumption (%) 47 26 14 8 4

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The two artificial leaves (of 8 × 8 cm paper) and the coppe r plates were placed in the th i rd layer of the central canopy . Th ey represen t two idealized condi t ions o f a leaf, un i fo rm surface heat f lux and un i fo rm surface tempera- ture , respect ively.

Heat ing of all layers o f the three canopies crea ted air f low pa t te rns resem- bling though no t ident ical to those of free convec t ion in rows of p lant cano- pies in a glasshouse.

The hea t t ransfer coeff ic ients in terms of the non-dimensional Nusselt num- bers and Grashof numbers are p resen ted in Figs.6 and 7.* Fig.6 is for un i fo rm leaf-surface hea t f lux. Fig.7 is for un i fo rm leaf-surface t empera tu re . Our analy- sis shows tha t the fol lowing corre la t ions give the best fi t to the expe r imen ta l data.

5 O . . . . . . 1 ! ' ' ' I ' ' ' : '

UNIFORM LEAF SLI#~ACE HEAT FLUX

21 ~ ~o

30

o

Z

/ o NU : 1,045GR 0'18/' .... (81

% Lu 15

z

Io L ~ ~ , ~ L J i , i , i L I LJLI,I

0,5 0,6 0.7 02 0,9 10 15 2 3 z, 5 6 '7 8 108

gqD 4 GRASHOF NUNSER GR = -

V2~,T

Fig.& Uniform leaf surface heat flux, Nusselt numbers from a simulated leaf in a row of covered plant canopies.

For paper plate, un i fo rm heat f lux:

(Nu) = 1.045 (Gr) °'1~ (8)

For coppe r plate , un i fo rm surface t empera tu re :

(Nu) = 0 .648 (Gr) °'~14 (9)

Eqs.8 and 9 when r ep lo t t ed on a single graph are shown in Fig.8, which figure shows tha t the lower and higher Grashof numbers have oppos i te effects on the rate of hea t t ransfer as expressed by the Nusselt numbers . In general,

* See Appendix for discussion on the alternative forms of Nusselt and Grashof numbers and their use.

1 0 8

the variation is within about 15% which agrees with the indications of Sparrow and Gregg (1956).

Two qualifying remarks should be made at this stage. Firstly, we have stated that the plate covered with paper produces a uniform surface heat flux. This is only approximately true, since in reality a uniform heat flux can be achieved

5O . . . . . . I 1 l 1 ' I ' ' l '

UNIFORM LEAF SURFACE TEMPERATURE

21,< ~o

Z

m --(91 #

to

z

1 0 , L L , , j J J I , i , I , :

05 "],6 07 0~ 0 3 1 5 15 2 3 , 6 8 9 108

9qD 4 6#ASHOF NUMBE~4 ON - .-

V 2 • T

F i g . 7 . U n i f o r m l e a f s u r f a c e t e m p e r a t u r e , N u s s e l t n u m b e r s from a s i m u l a t e d l e a f in a r o w o f c o v e r e d p l a n t c a n o p i e s

5 0 . . . . . . I [ I I ' I ' I ' J ' I

[ ~ ' UNIFORM LEAF SURFACE T E M P E R A T U R E - - 7

a!~ 30 / NU : 0,6/.8 OR 0,21z. . . . . (9)

;E /' UNIFORM LEAF SURFACE HEAT FLUX

~" - ~ - . - NU = 1,o1.5 GR0,1Bl' . . . . (8}

tn

z

lo J , ~ , L I I 1 L 1 ~ I , I , I , l I 0,5 0,6 07 0,8 (3,9 1.0 1,5 2 3 L 5 6 7 8 9 108

gqD/, GRASNOE N U M B E R GR =

V 2 X T

F i g , 8 . C o m p a r i s o n o f N u s s e l t n u m b e r s f r o m a s i m u l a t e d l e a f in a r o w o f c o v e r e d p l a n t c a n o p i e s .

109

only by having a plate of zero thickness or of zero thermal conductivity. Neither of these two conditions can be met in real life. Therefore, a true com- parison with the theoretical analysis of Sparrow and Gregg (1956) is not possible.

Secondly, theoretical studies such as reported in Sparrow and Gregg (1956) and Ostrach (1973) and other experimental studies of free convection from vertical plates, have assumed that ambient air is still. Also, they have not considered the influences of other bodies near the plates. In the present study, we have deliberately created flow conditions such that the air will strike the plates at some velocity. In addition, we have allowed for the presence of other leaves nearby (the bulbs). For this reason as well, direct comparison with cited literature is not possible.

CONCLUSIONS

Simple correlation expressions for free convective heat-transfer coefficients from leaves in a covered plant canopy have been experimentally obtained. The results indicate that the heat-transfer coefficients differ only by a maximum of about 15% from those for leaves with uniform surface temperature or for leaves with uniform surface heat flux.

ACKNOWLEDGEMENTS

The authors are grateful to the Institute of Horticultural Engineering, Wageningen for providing the facilities and help to carry out this work. The first author is also indebted to the International Agricultural Centre, Wagen- ingen and the National Research Council of Canada for financial assistance; and to the University of British Columbia for granting the leave of absence to carry out this study.

APPENDIX

In this report, we have used the expressions for Nusselt (Nu) and Grashof number (Gr) which are somewhat different than the commonly used forms of these parameters. Therefore, some explanation of this point is desired, which is given below.

The Nusselt number in this analysis is:

(Nu) - qD (Ts--Tg)~ (A-l)

and the Grashof number is:

(Gr) - gqD4 v2~T (A-2)

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The conventional forms of the above expressions are:

hD (Nu)* = ~ (A-3)

and:

(ar)* = l~g(Ts--Tg)D3 (A-4) o 2

In eq.A-3, h is the heat transfer coefficient given by:

h - q (A-5) (Ts - - Tg)

When eq.A-5 is inserted in eq.A-1, it is apparent that one obtains eq.A-3. Therefore, in fact, the expressions A-1 and A-3 are identical

The reasoning for Grashof number is somewhat complicated. Firstly, eq.A-2 is valid only for perfect gases, including air at atmospheric pressure and moderate temperatures, such as encountered in the present study. In such a case, it is known that the thermal expansion of a perfect gas ~ is equal to l/T, where T is its absolute temperature. Therefore 1/T in eq.A-2 replaces f~ in eq.A-4.

The term n T, apparently missing in eq.A-2 is in fact contained in the quanti ty q as expressed by eq.A-5. Therefore, the Grashof number expressed by eq.A-2 does in fact contain the buoyancy parameter ~A T.

The Grashof number expressed by eq.A-2 has been called in the literature (Sparrow and Gregg, 1956) as "modif ied" Grashof number. In discussion of Sparrow and Gregg (1956) it has been shown that the modified Grashof number is equivalent to the products of the conventional Grashof and Nusselt numbers.

Using expressions A-1 and A-2 has the main advantage that in the present study, the heat flux q would be known from estimations. Eqs.8 and 9 or Fig.8 would then give the required unknowns; surface temperature of leaf, Ts, and hence the convective heat transfer coefficient from the leaf, which was the objective of this study.

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