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An energy balance model for estimating leaf cuvette environments
EnvironmentalandExperimentalBotany,Vol. 24, No. 4, pp. 321-330, 1984.
0098-8472/84 $3.00 + 0.00 © 1984 Pergamon Press Ltd.
Printed in Great Britain.
AN E N E R G Y BALANCE M O D E L F O R E S T I M A T I N G LEAF C U V E T T E E N V I R O N M E N T S B. D. AMIRO,* T. J. GILLESPIE and G. W. THURTELL
Department of Land Resource Science, University of Guelph, Guelph, Ontario, Canada, N1G 2W1
(Received 9 December 1983; accepted in revisedform 14 May 1984) AMIRO B. D., GILLESPIET. J. and THURTELL G. W. An energy balance model for estimating leafcuvette environments. ENVIRONMENTALAND EXPEmMENTAL BOTANY 24, 321--330, 1984.--A model for predicting the environment in a leaf cuvette is described. This model requires an estimation of energy input into the system and some knowledge of the physical characteristics of the cuvette. These inputs are used to predict leaf and air temperature, ambient vapor pressure, transpiration and other energy-related parameters within a cuvette. This allows the researcher to pre-select appropriate conditions to achieve a desired cuvette environment. This also allows appropriate cuvette design before the start of an experiment. Data are presented from the model to reveal the influence of various inputs on leaf and air temperature and transpiration rate. The complexity of the plant-environment system in response to specific variables is observed.
INTRODUCTION
CUVETTES or leaf c h a m b e r s are often used for studies of gas exchange by a plant. (9'1z) However, these closed environments m a y cause excessive leaf h e a t i n g (13) or r a p i d t r a n s p i r a t i o n leading to w a t e r stress. M e a n s for reducing cuvette t e m p e r a tures m a y include the circulation of cooling w a t e r a r o u n d the cuvette (a4) or the installation of a Peltier-type cooler. (1°) H o w e v e r , it is felt that such a p p a r a t u s e s m a y be a v o i d e d if the energy inputs to the cuvette are suitably m a n i p u l a t e d . Cuvette e n v i r o n m e n t s can be m e a s u r e d d u r i n g an e x p e r i m e n t b u t it is desirable to be able to predict the e n v i r o n m e n t d u r i n g the p l a n n i n g stages of a project. This p a p e r presents a model that enables a researcher to predict leaf a n d air temperatures, a m b i e n t v a p o r pressure, transpiration rates a n d other e n e r g y - r e l a t e d variables
from an estimation of the energy i n p u t to the cuvette system a n d a knowledge of leaf diffusive conductance. This allows changes of specific inputs to the system in o r d e r to select a desired cuvette environment. T h e m o d e l is also seen as an instructive tool to describe the complexities of interactions between plants a n d their environment. METHODS
The model T h e m o d e l assumes a knowledge of i n p u t energy. Shortwave r a d i a t i o n (KJ,), longwave r a d i a t i o n (L J,), v a p o r pressure entering the cuvette (el) , and air t e m p e r a t u r e s entering ( T i) and outside of the cuvette ( Toc ) must be known. These variables can be measured or estimated for a given set of conditions. T h e model then simultaneously
* Present address: Environmental Research Branch, Whiteshell Nuclear Research Establishment, Pinawa, Manitoba, ROE 1L0. This research was supported by Environment Canada. 321
322
B . D . AMIRO, T. J. GILLESPIE and G. W. T H U R T E L L
solves the energy balances for the cuvette wall, the leaf, and the air inside the cuvette. All energies are expressed in units of W / m 2 and temperatures in degrees Kelvin (K). The cuvette is free-standing and is of any geometry. The Stefan-Boltzmann equation is used to account for longwave emissions (Q) by a black body surface
0..= o-T 4,
(1)
where a is the Stefan-Boltzmann constant ( - ~ 5 . 6 7 x 10 - s W/mZ/K 4) and T is the surface temperature. It is assumed that no shortwave radiation is absorbed by the cuvette wall. T h e energy budget tbr the wall is then expressed as a balance between the net longwave radiation and the convective heat transfer away from each wall surface so that 5[L~ + ( 1 - 5 ) (1 -fl)L~.
= pC, ho(Tw- too) +pcd,,(Tw- T.),
(2)
where e is the longwave absorptivity (or emissivity) of the cuvette wall (5 = 1 if opaque) and fl is a radiation view factor for the cuvette wall. Iffl = 1, the cuvette wall cannot see any other part of the cuvette so that longwave exchange for the wall is restricted to the leaf and L~, (first and third terms in Eq. 2). I f fl < 1, the wall also receives some longwave energy which has been either transmitted or emitted by the opposite cuvette wall (second and fourth terms in Eq. 2). TL, Tw and Ta are the temperatures of the leaf, cuvette wall, and air inside the cuvette, respectively. T h e leaf is assumed to be isothermal and to have an emissivity of 1 in these calculations (Eq. 2, third term). Typical leaf emissivities are usually between 0.94 and 1.0 ~5~ and the error in neglecting this difference is proportional to the temperature difference between the two exchanging surfaces. Most of the temperature differences in this system are < 1 0 K and the reflection error for an emissivity of 0.95 is about 0.5 K for this difference. Multiple reflections make this error even smaller, and since it is m u c h less than other constraints in the analysis, it can be ignored. Reflections by the cuvette wall are also ignored but the transmission characteristics are included to encompass a variety of cuvette wall materials. This model,
then, will only consider materials that are highly transparent in the wavelengths represented by K~. This is a workable assumption for glass and polyethylene but, if necessary, a shortwave absorbance term could be included in the analysis. T h e net longwave radiation is balanced by convective heat transfer in Eq. 2. This is represented as products of temperature gradients with heat transfer coefficients for the inside (h/, m/s) and outside (ho) of the cuvette wall and with p@ (J/m3/K1), the volumetric heat capacity of air. Equation 2 assumes that the heat conductance of the cuvette wall is infinite so that there is no temperature gradient across the wall. This can be safely assumed for m a n y construction materials since the b o u n d a r y layer conductances (ho and hi) are much smaller than the heat conductance through the wall itself. Since the wall is assumed isothermal, L,~ is taken as a mean value incident from all directions. A similar energy budget for the leaf is written where the net radiation is balanced by the sensible and latent heat transfer such that
2Mw(ghw = pCphL(TL- T,)+ - ~ - \ g - ~ / ( % - e a ) ,
(3)
where K~ is the total shortwave radiation incident on the leaf and is absorbed according to the fraction cc. The terms containing 5 depend on the cuvette wall properties so that ire = 1, the wall is opaque and all longwave exchange is between the leaf and the wall. T h e radiation balance accounts for both sides of the leaf. The convective heat transfer is again represented as the product of a temperature gradient and a heat transfer coefficient h L. This transfer coefficient is for the whole leaf so that the two surfaces are included. The latent heat transfer is proportional to a vapor pressure gradient ( % - - % Pa) from the inside of the leaf to the air inside the euvette. It is also dependent on the leaf diffusive conductance (g, m/s) and the transfer coefficient for water vapor across the b o u n d a r y layer (hw). This last parameter is related to hL by the ratio of the diffusion coefficients for heat and water vapor to the 2/3 power, (3) so that h w ~ 1.08 h L. Then h,, and g are added in parallel. The saturated vapor pressure of the leaf, eL, is uniquely related to leaf
ESTIMATING LEAF CUVETTE ENVIRONMENTS temperature. 2 is the latent heat of vaporization (J/g), M w is the g r a m molecular weight of water (g/tool), R is the universal gas constant (J/mol/K) and T is the absolute temperature of the system. An isobaric condition is assumed as is perfect mixing, which implies no gradients exist in the cuvette for Ta and e,. An energy balance for the air in the cuvette can also be written. If it is assumed that there is no net radiative exchange for the air, the energy gained by convection from the leaf and the cuvette wall is balanced by the heat flux to the air stream flowing through the cuvette:
pc. ~ h, ( T~- Ta)+pC~hL(~-- L) F = oCp~-[L(TO-- 7~),
(4)
where F is the air flow rate through the cuvette (m3/s) and T o is the air temperature leaving the cuvette. Again if perfect mixing is assumed, To = T,. A~ and A L are the areas of the cuvette wall and leaf (one side), respectively. T h e ratio Aw/A L is necessary to put h i o n a similar area basis to hL. These heat fluxes are then expressed as energy per unit leaf area. T h e latent heat flux from the leaf is similarly balanced by the energy leaving the cuvette as water vapor so that
323
experimentally or calculated through empirical relationships. T h e latter approach is suggested for ho where the geometry and size of the cuvette and the flow characteristics of the air stream in which the cuvette is immersed are known. This transfer coefficient can then be calculated through dimensional analysis. (11) The transfer coefficient for the leaf can be experimentally determined by placing a mock leaf in the cuvette. This mock leaf can be a section of saturated filter paper of similar size and geometry to a typical leaf. (6) Equation 5 can then be used to determine h w with g ~ oo. Measurements of the temperature of the evaporating surface, the flow rate and the vapor pressure entering and leaving the cuvette must be made to satisfy Eq. 5. Alternatively, a mock leaf can consist of a heating pad and the transfer coefficient for heat (hL) calculated as described by Amiro et al. (1) The transfer coefficient of the inside of the cuvette wall (hi) can also be experimentally determined. For the purposes of this model, the transfer coefficient was chosen as equal to ALhL/A w. This puts hi on a suitable area basis but ignores the difference in b o u n d a r y layer characteristics caused by the difference in geometry between the leaf and cuvette wall. However, it is felt that it is a reasonable first approximation since it considers that both surfaces are subjected to similar flow characteristics inside the cuvette.
An analytical solution \ g ~ - ~ / ( e L --ea) = ~
\~--~L/ (eo- el).
(5) As in the case of air temperature, the vapor pressure leaving the cuvette, eo, is equal to e, if no gradients exist. All other terms are defined previously. T o summarize, the known energy inputs are K~, L+, Ti, ei and TOc. T h e remainder of the variables is either estimated from some knowledge of the system or calculated through the above equations. T h e system defines A~, At. and F. T h e construction materials of the cuvette determine whereas fl is a function of cuvette geometry and the quantity of leaf in the cuvette. Constants are ~r, R, Cp and M~, while 2 and p can be chosen since they are relatively weak functions of temperature. T h e transfer coefficients must be determined
T h e unknown variables in this analysis are now restricted to TL, Tw, Ta, el., e, and g. This last variable is determined by the plant and must remain as an independent variable in the analysis. Since eL is a non-linear function of TL, only four unknowns remain in Eqs 2-5 and the system appears to be solvable. However, both TL and Tw also appear as fourth power functions and TL is exponentially related to eL, which effectively adds three extra unknowns. The system is still solvable by numerical means through successive iterations, but it is desirable to find an analytical solution if possible. Several approximations are made in this paper to solve the cuvette system analytically. T h e fourth power functions of temperature can be approximated and expressed in terms of known temperatures. For example,
crT[ = ~rT~ + a ( T [ - - T~) ~_ ~rT~ + d Q
(6)
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B.D. AMIRO, T. J. GILLESPIE and G. W. THURTELL
can be written where a difference in temperatures to the fourth power is expressed in terms of a radiation difference (dQ). Equation 1 can be differentiated so that a linear approximation to dQis found: dQ= daT 4 = a dT 4 = 4~T 3 dT -___4 ~ T ~ ( T L - T i ) .
(7)
Combining Eqs 6 and 7:
oT~ _~ ~T,a+~T,3(TL- 7-,)= ~,T,~ T~- 3~T,4 (8) This approximation is very good for small differences between TL and Ti. For example, if TL - - 2 9 5 and T i = 2 9 0 , the error in o'T~ is approximately 0.7 W / m 2 in 429 W/m 2 or approximately 0.2O/o. This would give an incorrect TL of about 0.12 K. A similar approximation is also done for T~, where 3
~ T ~ ~- 4aTo~Tw-3aT~o ~.
X = F + ALh L + Awhl
,~ [ 1 7 - 2 6 9 ( T - 273.2)1 e~=610.yoexpL~--~9 j.
(10)
By differentiating Eq. 10, the slope (S) of the saturated vapor pressure curve is obtained:
x ( 2 - e ( 1 - f l ) ) + p C v h o T o c + P@hiFTi X
pCvAw(hi)Z X D = \RTi/\g+hJ
~-~.~
/"
Making use of Eqs 14-17, the four terms in the equation for TL are
8 ea T3oc Terml =~K~+--Y
g
x
(II)
g
eL = (%--esi) +e~i,
(19)
(pCphLAwh'')
Term3=pCphL(l_~)
(12)
where e~i is the saturated vapor pressure of the incoming air that is uniquely related to T i through Eq. 10. For small differences in temperature, and hence vapor pressure, the slope of the saturated vapor pressure curve (S) can be used at the known temperature T i so that
et,--qi = S ( T L - 7~.
(18)
/
--DF(esl-ei--STi). The saturated vapor pressure at leaf temperature is expressed as a gradient plus a known value:
(16)
ALghw F+-g + h~,
-2(1-e)L~+6a(eT4o~ - T~).
~ 1 7 . 2 6 9 ( T - 273.2) 1
(15)
Z = PCp(ho+h,) + 4eaTo3c(2-e( 1 -/~))
610 78~ -(17-'269) (237'3) l " L ( T - 3 5 . 9 ) 2 _]
x exPL.
(14)
Y = eLi,[1 + (1 --e) (1 --fl)] -3eflaT~*+3eaT4oc
(9)
The saturated vapor pressure (es in Pa) can be related by an empirical formula to temperature ( T i n K)(17):
S=
lations, an iteration is made to select S at the mean temperature of TL and Ti. The vapor pressure of the air in the cuvette can then be calculated through Eq. 5 as a function of eL and hence S and T L. These linear approximations now make the system solvable. The model is then used to calculate TL, Tw, 77,, eL and % The first step is to solve for TL by simultaneously solving Eqs 2-13. The resulting equation for TL contains four terms, within which the following factors appear :
(13)
This approximation is very good for small differences in TL - Ti. In the actual model calcu-
Term 4 = FSD + 8a
x[T3i
eTa°c( pCvhiALhL +4efl~T/3)l.
Z ',
(21)
x
The solution for TL is then Term 1 + Term 2 TL = T e r m 3 + Term 4"
(22)
ESTIMATING LEAF CUVETTE ENVIRONMENTS After obtaining
Y as can
TL, Tw can
be solved :
TL(pC~xALhL t-4eflcrT~)
(23)
Ta : 77,= (AwhiTwwALhLTL-k-FTi)(1).
(24)
T h e saturated vapor pressure of the leaf (eL) can be found through Eq. 10 and e, can be solved by
e,={ ghw %+Fei~ 1 \g+hw ALJ ghw + F g+h~ AL
(25)
It is difficult to evaluate conceptually each of the terms in Eq. 22 but an overall evaluation can be carried out by testing the model with a variety of inputs.
RESULTS AND DISCUSSION T h e model predicts the temperature characteristics for any leafcuvette design. As an example of the model predictions, the cuvette described by AMIRO et al.m will be used as a test subject. This cuvette is a spherical glass flask of 1 1 volume with no independent stirring mechanism. Since the flask is glass, and opaque to longwave radiation, e = 1. fl is chosen as 0.5, A L = 100 c m 2, ho = 5 cm/s, e = 0 . 7 , and T / = Toc=20°C. L~, is chosen as 450 W / m 2 which corresponds to surrounding surfaces that are perfect emitters at a temperature of 25°C. This would be characteristic of a growth c h a m b e r environment where the lights and c h a m b e r wall temperatures average this value. The other variables are changed in this demonstration to depict different conditions. Since the system is multivariate, a single graphical representation of conditions is not possible. Therefore, a set of conditions (which are termed 'default' conditions) are chosen and only one variable is allowed to change from these conditions during each test. These default conditions are F = 5 x l 0 -5 m3/s, h L = 0 . 0 2 m/s, K ~ = 240 W / m z and e; = 700 Pa. For m a n y physiological studies, TL and T, are
325
variables that are of interest. Leaf temperature is used to assess thermal stress and the rate of metabolic processes (photosynthesis and respiration). Air temperature is of special interest if TL - T ~ is large. Leaf temperatures measured with thermocouples then become uncertain because the thermocouple bead is actually measuring some intermediate temperature. ~2~ Another parameter of interest is the transpiration rate, which m a y give some indication of potential plant water stress. Hence, TL, T, and transpiration rate are plotted as functions of leaf diffusive conductance (g). As previously mentioned, this independent variable is determined by the plant and is the governing factor in partitioning energy between. sensible heat flux and latent heat flux. T h e stomatal mechanism, therefore, determines the cuvette microenvironment but it should be noted that the stomatal mechanism is influenced by humidity so that a feedback situation is likely. ~ 5~ Figure la illustrates the change in TL and T, with g while F, h z and ei are held at default values and K~ assumes two different radiation loads. T h e lower radiation load (K~ = 240 W / m 2) represents a case similar to a typical growth chamber. As the stomates open, the leaf temperature cools, as does the cuvette air temperature. With conditions as described, TL becomes 4 K warmer than the air entering the cuvette. However, ifa high radiation load is used (K~. = 840 W/m2), the leaf temperature becomes 16 K w a r m e r than Ti, even at high values of g. This condition is similar to a full sunlight radiation load with 700 W/m 2 arriving from above and a surface albedo of 20% below the cuvette. In this situation, air temperatures also become high. High leaf temperatures at the higher radiation load increase transpiration rates (Fig. lb). Even a t g = 0.8 cm/s, the air inside the cuvette does not become saturated for the described conditions. However, a determination of Tw reveals that condensation occurs on the cuvette walls for the higher radiation load a t g > 0.25 cm/s (marked by C in Fig. lb). This is because the cuvette walls are cooler than the dewpoint temperature in the cuvette. T h e model does not include the latent heat release when condensation is occurring on the walls since it is assumed that this condition is undesirable for physiological studies. Hence, the data beyond this condensation point will not reveal the true temperature and
326
B . D . A M I R O , T. J. G I L L E S P I E and G. W. T H U R T E L L
t r a n s p i r a t i o n rate for the leaf b u t this m o d e l clearly indicates w h e n such a s i t u a t i o n will arise, T h e flow rate t h r o u g h the c u v e t t e is v a r i e d in Fig. lc while o t h e r variables are held at default values to illustrate t h a t b o t h the leaf a n d air t e m p e r a t u r e c a n be r e d u c e d b y i n c r e a s i n g the v o l u m e of air passing t h r o u g h the system.
A l t h o u g h this increased flow cools the leaf, it also increases the t r a n s p i r a t i o n rate (Fig. ld). I f the flow rate is too low, c o n d e n s a t i o n will occur o n the walls with i n c r e a s i n g g as m a r k e d b y C in Fig. ld. T h i s illustrates t h a t a c o m p r o m i s e m u s t be r e a c h e d u s i n g the flow rate so t h a t there will be n o c o n d e n s a t i o n , the leaf will r e m a i n s u i t a b l y cool,
KI varied (W m -2) 40
a
~
400
35
~
b
300
30
~'~.~. ~"-~,...
g 200 "~
K~ = 840
25
~- 100 ~ " ~
200
.
K~ = 2 4 0
0.4
018
O0
0.4 g (em s -1)
g (cm s -1)
30
Flow rate (F) varied 200
c
0.8
m3 S"! x 10.`5 ) d
~150 'E
g
~I00
25
1.7 50
""
2o 0
'
0.4 g (cm s -1)
F
17
'8
O.
0
o14
o18
g (cm s -1)
FIO. l. Model predictions of TL, Ta and transpiration. Symbols listed in Appendix. Except where varied, conditions are K,[ = 240 W/m 2, L~, = 450 W/m 2, Ti = Toc = 20°C, F = 5 × 10 - s m3/s, hL = 2 cm/s, h 0 = 5 cm/s, AL = 100 cm 2, Aw = 300 cm 2, ~ = 1, fl = 0.5, ei = 0.7 kPa, and ~ = 0.7. The point marked by C indicates start of condensation on cuvette wall. (a) TL and T= vs g for K J, varied ( TL - - ; T= - - ) . (b) Transpiration vs g for/(~ varied. (c) TL and T= vs g for F varied ( TL ; T~- -). (d) Transpiration vs g for F varied.
ESTIMATING LEAF CUVETTE ENVIRONMENTS
327
h L varied (cm s -1) 30
200
150 E
50
I 20 |
0
hL=5 !
04 g (cm s 1)
J
0.8 9
014 (cm s 1)
018
e i varied (k Pa) 30 I c
200- d
~
. 150 'E
~_~ 25
g
100
S • =
20 I 0
I 04 g (cm s -1)
J 08
0 0
I
I
04 g (cm s 1)
0.8
FIG. 2. Model predictions of TL, 7-, and transpiration. Default conditions same as for Fig. 1. (a) TL and T. vs g for hL varied (TL - - ; To ). (b) Transpiration vs g for h L varied. (c) TL and T. vs g for e i varied ( TL - - ; T. - - ) . (d) Transpiration vs g for e i varied.
and the transpiration rate will not be excessive. In addition to these constraints, the flow rate determines the relative change in gas c o n c e n t r a t i o n entering a n d leaving the cuvette. Sufficient differences must exist to be measurable for such gases as H 2 0 and C O / d u r i n g transpiration a n d net photosynthesis studies. Flow rate is an easily adjusted variable a n d the model indicates a suitable rate to be selected. T h e transfer coefficient for the leaf to both heat and water vapor is varied in Fig. 2a. As the
transfer coefficient increases, the leaf cools as heat loss is facilitated. Since the model uses this p a r a m e t e r as a measure of turbulence within the cuvette, the transfer coefficient tbr the inside of the cuvette wall also increases. This allows increased heat transfer out of the cuvette through the wall a n d therefore cools the air inside the cuvette. T h e increase in h L does not have a d r a m a t i c effect on transpiration. This is due to the fact that cooling the leaf and reducing e L is countered by an increase in the transfer of water vapor away from
328
B . D . AMIRO, T. J. GILLESPIE and G. W. T H U R T E L L
the leaf. I n practice, the change in hL can be induced by a change in cuvette geometry, by an increase in flow rate a n d turbulence, or by the i n t r o d u c t i o n of an i n d e p e n d e n t mixing system such as a fan. If this last option is adopted, the heat generated by the fan must be incorporated into the model. It should be noted that a change in hL will change the b o u n d a r y layer c o n d u c t a n c e for other gases a n d d e p e n d i n g on the m a g n i t u d e of g, photosynthesis m a y be affected. (4) Methods of cooling the system include introduction of cool or dry air into the cuvette. I n p u t
air cooling is a practical solution and works by increasing the sensible heat flux away from the leaf. Plants n o r m a l l y cool by p a r t i t i o n i n g most of the heat loss through transpiration, so a modification of the vapor pressure entering the cuvette is a n o t h e r available method of control. This siiuation is illustrated in Fig. 2c where moist air (e i = 1700 Pa) and relatively dry air (e i = 700 Pa) are introduced. This i n p u t variable has no influence when the stomates are closed since no energy is lost in evaporation. As g increases, a lower e i yields reduced leaf a n d air temperatures.
I00
~
8C
4
3
S.. 2
4O
AIR
I
0
2o~ RH
15 ~
~2o
:>
0
25
b
014
J
0 8
0
) 5 ~>
0,4
0
08
g (cm s -1)
g (cm s 1) 30
2°°Fc
d
150[ c~
'E
~ I00 50
I
0
04 g {cm s 1)
J
08
20 0
0 14
Oi8
g (cm s l)
FIo. 3. Model predictions for default conditions indicated in Fig. 1. (a) eL and e, vs g. (b) Relative humidity and vapor pressure deficit (V.P.D.) vs g. (c) Energy partitioning of net radiation (Q*) to sensible heat flux (QH) and latent heat flux (Q~) for the leaf with a change in g. (d) Influence ofcuvette longwave transparency on leaf temperature (s = 0 = transparent, s = 1 = opaque, e = 0.5 = 50% transparent).
E S T I M A T I N G LEAF CUVETTE ENVIRONMENTS This lower el causes an increased g r a d i e n t for e v a p o r a t i o n so t h a t t r a n s p i r a t i o n is e n h a n c e d (Fig. 2d). A l t h o u g h a higher ei m a y be desirable from this point of view, an ei = 1700 Pa will induce condensation on the cuvette walls at g > 0.57 cm/s. This b a l a n c e must then be considered to m a k e the system operable. T h e m o d e l gives i n f o r m a t i o n concerning other parameters. F i g u r e 3a illustrates the c h a n g e in both eL and e, with leaf diffusive c o n d u c t a n c e for the default conditions cited. H u m i d i t y in the cuvette can also be considered in terms of relative h u m i d i t y a n d v a p o r pressure deficit as d e p i c t e d in Fig. 3b. T h e p a r t i t i o n i n g of energy is presented in Fig. 3c. As the stomates open, the leaf cools a n d increases the net radiation. Similarly, the o p e n i n g increases t r a n s p i r a t i o n a n d reduces sensible h e a t loss. These energy balances can be d e t e r m i n e d for a n y selected set of conditions. T h e foregoing analysis describes a glass cuvette that is o p a q u e to longwave r a d i a t i o n (~ = 1). T h e model is designed to be used also with other cuvette materials that m a y be totally or p a r t i a l l y t r a n s p a r e n t to this radiation. F i g u r e 3d illustrates leaf t e m p e r a t u r e d a t a for such materials t h a t m a y include very thin polyethylene (e "~ 0). Since the effective t e m p e r a t u r e of the surroundings is greater than the t e m p e r a t u r e of the cuvette walls in this case, a t r a n s p a r e n t wall causes the leaf to warm. This e x a m p l e is realistic tbr a cuvette placed in a g r o w t h c h a m b e r u n d e r hot lights. Since L{ is lower than 450 W / m 2 u n d e r field conditions, a t r a n s p a r e n t wall would give lower leaf t e m p e r a t u r e s t h a n an o p a q u e one in this situation. T h e m o d e l therefore allows the researcher to select construction materials for a given e n v i r o n m e n t . E a c h of Figs 1 3 illustrates an i m p o r t a n t p h y s i o l o g i c a l - e n v i r o n m e n t a l principle. As g increases, all curves become increasingly less sensitive to changes in the s t o m a t a l mechanism. Therefore at high values of g, the p l a n t loses its c a p a b i l i t y to control the cuvette environment. T h e m a g n i t u d e of this change in slope of TL or T, with g is d e p e n d e n t on both the leaf b o u n d a r y layer c o n d u c t a n c e and g. This is illustrated in Fig. 2a. L e a f diffusive c o n d u c t a n c e is controlled by the p l a n t and will v a r y between species, tS) with
329
e n v i r o n m e n t °5) and with p l a n t w a t e r status. (7) T h e researcher usually has some knowledge of the relative m a g n i t u d e of this value for a p a r t i c u l a r experiment. This model allows n u m e r i c a l exp e r i m e n t a t i o n over any range in g to indicate w h e t h e r some change in i n p u t variables is necessa r y to o b t a i n a desired cuvette environment. This m o d e l l i n g a p p r o a c h quantifies inputs in a known w a y a n d complements empirical comparisons of cuvette environments that have been done elsewhere .~16~ A p i e - e x p e r i m e n t a l consideration of the model can save the researcher m u c h time that would otherwise be used in e x p e r i m e n t a l l y c h a n g i n g inputs to arrive at a satisfactory situation.
REFERENCES I. AMIROB. D., GILLESPIET.J. and THURTELL G. W. (1984) Injury response ofPhaseolus vulgaris to ozone flux density. Atmos. Env. 18, 1207 1215. 2. BEADLEC. L., STEVENSONK. R. and THURTELL G. W. (1973) Leaf temperature measurement and control in a gas-exchange cuvette. Can. J. Plant Sci. 53, 407-412. 3. BraD R. B., STEWARDW. E. and LIOnTFOOT E. N. (1960) Transport phenomena. John Wiley, New York, 780 pp. 4. GRACEJ. (1981) Some effects of wind on plants. Pages 31-56 in J. GRACE, E. D. FORD and P. G. JARVIS, eds. Plants and their atmospheric environment. Blackwell Scientific Publications, Oxford. 5. Inso S. B., JACKSON R. D., EHRLER W. L. and MITCHELL S. Z. (1969) A method for determination of infrared emittance of leaves. Ecolocv 50, 899-902. 6. JARVlS P. G. ( 197 I) The estimation of resistances to carbon dioxide transfer. Pages 566 631 in Z. SESTAK, J. CATSKY and P. G. JARVIS, eds. Plant photosynthetic production, manual of methods. Junk, Hague. 7. KANEMASU E. T. and TANNER C. B. /,1969) Stomatal diffusion resistance of snap beans. I. Influence of leaf water potential. Plant Physiol. 44, 1547 1552. 8. KORNER C., SCHEEL J. A. and BAUER H. (1979) Maximum leaf diffusive conductance in vascular plants. Photosynthetica 13, 45-82. 9. LAYZELL D. B., PATE J. S., ATRINS C. A. and CANVIN D. T. (1981) Partitioning of carbon and nitrogen and the nutrition of root and shoot apex in a nodulated legume. Plant Physiol. 67, 3(~36. 10. LEGGE A. H., SAVAGED. J. and WALKER R. B.
330
11. 12.
13, 14.
15.
16.
17.
B . D . AMIRO, T. J. GILLESPIE and G. W. T H U R T E L L (1979) A portable gas-exchange leaf chamber. Pages 16-11-16-24 in W. W. HECK, S. V. KrtUPA and S. N. LINZON,eds. Methodologyfor the assessment . of air pollution effects on vegetation. A i r Pollution Control Association, Pittsburgh, PA. McADAMSW. H. (1954) Heat transmission, 3rd edn. McGraw Hill, New York. RADINJ. W. and ACKERSONR. C. (1981) Water relations of cotton plants under nitrogen deficiency. Plant Physiol. 67, 115-119. RITcmE G. A. (1969) Cuvette temperatures and transpiration rates. Ecology 50, 667 670. SHARKEYT. D., IMAI K., FARQUHARG. D. and COWAN I. P. (1982) A direct confirmation of the standard method of estimating intercellular partial pressure of CO z. Plant Physiol. 69, 657-659. SHERIFFD. W. (1979) Stomatal aperture and the sensing of the environment by guard cells. Plant, Cell Envir. 2, 15-22. SXEBERTB., KOCHW. and ELLERB. M. (1980) Leaf temperatures in a gas exchange chamber and in the open air. 07. exp. Bot. 31,.863-871. TETENS O. (1930) lJber einige meteorologisebe Begriffe. Z. Geophys. 6, 297-309.
APPENDIX AL Aw Cj, e. ei
leaf area (m 2) cuvette wall area (m 2) specific heat of air (J/g/K) cuvette air vapor pressure (Pa) vapor pressure entering cuvette (Pa)
eL
e, esi
F g hi
saturated vapor pressure at T L (Pa) saturated air vapor pressure (Pa) saturated vapor pressure at T/(Pa) flow rate through cuvette (m3/s) leaf diffusive conductance (m/s) heat transfer coefficient of inside cuvette wall
(m/s) hL ho h~
*q L$ M~
o.. Q* Q~ Q,, R S T
L T~ L L~ T~ O~
2 P
heat transfer coefficient of leaf (m/s) heat transfer coefficient of outside cuvette wall (m/s) water vapor transfer coefficient for leaf (m/s) total incident shortwave radiation (W/m 2) mean incident longwave radiation (W/m z) gram molecular weight of water (18 g/mole) longwave radiation (W/m 2) net radiation on leaf (W/m 2) latent heat flux of evaporation for leaf (W/m 2) sensible heat flux for leaf (W/m 2) universal'gas constant (8.31 J/mol/K) slope of saturated vapor pressure curve (Pa/K) temperature (K) cuvette air temperature (K) air temperature entering cuvette (K) leaf temperature (K) air temperature exiting cuvette (K) air temperature outside ofcuvette (K) cuvette wall temperature (K) shortwave absorptivity of leaf radiation view factor ofcuvette wall longwave absorptivity of cuvette wall latent heat of vaporization of water (J/g) density of air (g/m 3) Stefan-Boltzmann constant (.5.67 x 10 -8 W/mi/K '*)
0098-8472/84 $3.00 + 0.00 © 1984 Pergamon Press Ltd.
Printed in Great Britain.
AN E N E R G Y BALANCE M O D E L F O R E S T I M A T I N G LEAF C U V E T T E E N V I R O N M E N T S B. D. AMIRO,* T. J. GILLESPIE and G. W. THURTELL
Department of Land Resource Science, University of Guelph, Guelph, Ontario, Canada, N1G 2W1
(Received 9 December 1983; accepted in revisedform 14 May 1984) AMIRO B. D., GILLESPIET. J. and THURTELL G. W. An energy balance model for estimating leafcuvette environments. ENVIRONMENTALAND EXPEmMENTAL BOTANY 24, 321--330, 1984.--A model for predicting the environment in a leaf cuvette is described. This model requires an estimation of energy input into the system and some knowledge of the physical characteristics of the cuvette. These inputs are used to predict leaf and air temperature, ambient vapor pressure, transpiration and other energy-related parameters within a cuvette. This allows the researcher to pre-select appropriate conditions to achieve a desired cuvette environment. This also allows appropriate cuvette design before the start of an experiment. Data are presented from the model to reveal the influence of various inputs on leaf and air temperature and transpiration rate. The complexity of the plant-environment system in response to specific variables is observed.
INTRODUCTION
CUVETTES or leaf c h a m b e r s are often used for studies of gas exchange by a plant. (9'1z) However, these closed environments m a y cause excessive leaf h e a t i n g (13) or r a p i d t r a n s p i r a t i o n leading to w a t e r stress. M e a n s for reducing cuvette t e m p e r a tures m a y include the circulation of cooling w a t e r a r o u n d the cuvette (a4) or the installation of a Peltier-type cooler. (1°) H o w e v e r , it is felt that such a p p a r a t u s e s m a y be a v o i d e d if the energy inputs to the cuvette are suitably m a n i p u l a t e d . Cuvette e n v i r o n m e n t s can be m e a s u r e d d u r i n g an e x p e r i m e n t b u t it is desirable to be able to predict the e n v i r o n m e n t d u r i n g the p l a n n i n g stages of a project. This p a p e r presents a model that enables a researcher to predict leaf a n d air temperatures, a m b i e n t v a p o r pressure, transpiration rates a n d other e n e r g y - r e l a t e d variables
from an estimation of the energy i n p u t to the cuvette system a n d a knowledge of leaf diffusive conductance. This allows changes of specific inputs to the system in o r d e r to select a desired cuvette environment. T h e m o d e l is also seen as an instructive tool to describe the complexities of interactions between plants a n d their environment. METHODS
The model T h e m o d e l assumes a knowledge of i n p u t energy. Shortwave r a d i a t i o n (KJ,), longwave r a d i a t i o n (L J,), v a p o r pressure entering the cuvette (el) , and air t e m p e r a t u r e s entering ( T i) and outside of the cuvette ( Toc ) must be known. These variables can be measured or estimated for a given set of conditions. T h e model then simultaneously
* Present address: Environmental Research Branch, Whiteshell Nuclear Research Establishment, Pinawa, Manitoba, ROE 1L0. This research was supported by Environment Canada. 321
322
B . D . AMIRO, T. J. GILLESPIE and G. W. T H U R T E L L
solves the energy balances for the cuvette wall, the leaf, and the air inside the cuvette. All energies are expressed in units of W / m 2 and temperatures in degrees Kelvin (K). The cuvette is free-standing and is of any geometry. The Stefan-Boltzmann equation is used to account for longwave emissions (Q) by a black body surface
0..= o-T 4,
(1)
where a is the Stefan-Boltzmann constant ( - ~ 5 . 6 7 x 10 - s W/mZ/K 4) and T is the surface temperature. It is assumed that no shortwave radiation is absorbed by the cuvette wall. T h e energy budget tbr the wall is then expressed as a balance between the net longwave radiation and the convective heat transfer away from each wall surface so that 5[L~ + ( 1 - 5 ) (1 -fl)L~.
= pC, ho(Tw- too) +pcd,,(Tw- T.),
(2)
where e is the longwave absorptivity (or emissivity) of the cuvette wall (5 = 1 if opaque) and fl is a radiation view factor for the cuvette wall. Iffl = 1, the cuvette wall cannot see any other part of the cuvette so that longwave exchange for the wall is restricted to the leaf and L~, (first and third terms in Eq. 2). I f fl < 1, the wall also receives some longwave energy which has been either transmitted or emitted by the opposite cuvette wall (second and fourth terms in Eq. 2). TL, Tw and Ta are the temperatures of the leaf, cuvette wall, and air inside the cuvette, respectively. T h e leaf is assumed to be isothermal and to have an emissivity of 1 in these calculations (Eq. 2, third term). Typical leaf emissivities are usually between 0.94 and 1.0 ~5~ and the error in neglecting this difference is proportional to the temperature difference between the two exchanging surfaces. Most of the temperature differences in this system are < 1 0 K and the reflection error for an emissivity of 0.95 is about 0.5 K for this difference. Multiple reflections make this error even smaller, and since it is m u c h less than other constraints in the analysis, it can be ignored. Reflections by the cuvette wall are also ignored but the transmission characteristics are included to encompass a variety of cuvette wall materials. This model,
then, will only consider materials that are highly transparent in the wavelengths represented by K~. This is a workable assumption for glass and polyethylene but, if necessary, a shortwave absorbance term could be included in the analysis. T h e net longwave radiation is balanced by convective heat transfer in Eq. 2. This is represented as products of temperature gradients with heat transfer coefficients for the inside (h/, m/s) and outside (ho) of the cuvette wall and with p@ (J/m3/K1), the volumetric heat capacity of air. Equation 2 assumes that the heat conductance of the cuvette wall is infinite so that there is no temperature gradient across the wall. This can be safely assumed for m a n y construction materials since the b o u n d a r y layer conductances (ho and hi) are much smaller than the heat conductance through the wall itself. Since the wall is assumed isothermal, L,~ is taken as a mean value incident from all directions. A similar energy budget for the leaf is written where the net radiation is balanced by the sensible and latent heat transfer such that
2Mw(ghw = pCphL(TL- T,)+ - ~ - \ g - ~ / ( % - e a ) ,
(3)
where K~ is the total shortwave radiation incident on the leaf and is absorbed according to the fraction cc. The terms containing 5 depend on the cuvette wall properties so that ire = 1, the wall is opaque and all longwave exchange is between the leaf and the wall. T h e radiation balance accounts for both sides of the leaf. The convective heat transfer is again represented as the product of a temperature gradient and a heat transfer coefficient h L. This transfer coefficient is for the whole leaf so that the two surfaces are included. The latent heat transfer is proportional to a vapor pressure gradient ( % - - % Pa) from the inside of the leaf to the air inside the euvette. It is also dependent on the leaf diffusive conductance (g, m/s) and the transfer coefficient for water vapor across the b o u n d a r y layer (hw). This last parameter is related to hL by the ratio of the diffusion coefficients for heat and water vapor to the 2/3 power, (3) so that h w ~ 1.08 h L. Then h,, and g are added in parallel. The saturated vapor pressure of the leaf, eL, is uniquely related to leaf
ESTIMATING LEAF CUVETTE ENVIRONMENTS temperature. 2 is the latent heat of vaporization (J/g), M w is the g r a m molecular weight of water (g/tool), R is the universal gas constant (J/mol/K) and T is the absolute temperature of the system. An isobaric condition is assumed as is perfect mixing, which implies no gradients exist in the cuvette for Ta and e,. An energy balance for the air in the cuvette can also be written. If it is assumed that there is no net radiative exchange for the air, the energy gained by convection from the leaf and the cuvette wall is balanced by the heat flux to the air stream flowing through the cuvette:
pc. ~ h, ( T~- Ta)+pC~hL(~-- L) F = oCp~-[L(TO-- 7~),
(4)
where F is the air flow rate through the cuvette (m3/s) and T o is the air temperature leaving the cuvette. Again if perfect mixing is assumed, To = T,. A~ and A L are the areas of the cuvette wall and leaf (one side), respectively. T h e ratio Aw/A L is necessary to put h i o n a similar area basis to hL. These heat fluxes are then expressed as energy per unit leaf area. T h e latent heat flux from the leaf is similarly balanced by the energy leaving the cuvette as water vapor so that
323
experimentally or calculated through empirical relationships. T h e latter approach is suggested for ho where the geometry and size of the cuvette and the flow characteristics of the air stream in which the cuvette is immersed are known. This transfer coefficient can then be calculated through dimensional analysis. (11) The transfer coefficient for the leaf can be experimentally determined by placing a mock leaf in the cuvette. This mock leaf can be a section of saturated filter paper of similar size and geometry to a typical leaf. (6) Equation 5 can then be used to determine h w with g ~ oo. Measurements of the temperature of the evaporating surface, the flow rate and the vapor pressure entering and leaving the cuvette must be made to satisfy Eq. 5. Alternatively, a mock leaf can consist of a heating pad and the transfer coefficient for heat (hL) calculated as described by Amiro et al. (1) The transfer coefficient of the inside of the cuvette wall (hi) can also be experimentally determined. For the purposes of this model, the transfer coefficient was chosen as equal to ALhL/A w. This puts hi on a suitable area basis but ignores the difference in b o u n d a r y layer characteristics caused by the difference in geometry between the leaf and cuvette wall. However, it is felt that it is a reasonable first approximation since it considers that both surfaces are subjected to similar flow characteristics inside the cuvette.
An analytical solution \ g ~ - ~ / ( e L --ea) = ~
\~--~L/ (eo- el).
(5) As in the case of air temperature, the vapor pressure leaving the cuvette, eo, is equal to e, if no gradients exist. All other terms are defined previously. T o summarize, the known energy inputs are K~, L+, Ti, ei and TOc. T h e remainder of the variables is either estimated from some knowledge of the system or calculated through the above equations. T h e system defines A~, At. and F. T h e construction materials of the cuvette determine whereas fl is a function of cuvette geometry and the quantity of leaf in the cuvette. Constants are ~r, R, Cp and M~, while 2 and p can be chosen since they are relatively weak functions of temperature. T h e transfer coefficients must be determined
T h e unknown variables in this analysis are now restricted to TL, Tw, Ta, el., e, and g. This last variable is determined by the plant and must remain as an independent variable in the analysis. Since eL is a non-linear function of TL, only four unknowns remain in Eqs 2-5 and the system appears to be solvable. However, both TL and Tw also appear as fourth power functions and TL is exponentially related to eL, which effectively adds three extra unknowns. The system is still solvable by numerical means through successive iterations, but it is desirable to find an analytical solution if possible. Several approximations are made in this paper to solve the cuvette system analytically. T h e fourth power functions of temperature can be approximated and expressed in terms of known temperatures. For example,
crT[ = ~rT~ + a ( T [ - - T~) ~_ ~rT~ + d Q
(6)
324
B.D. AMIRO, T. J. GILLESPIE and G. W. THURTELL
can be written where a difference in temperatures to the fourth power is expressed in terms of a radiation difference (dQ). Equation 1 can be differentiated so that a linear approximation to dQis found: dQ= daT 4 = a dT 4 = 4~T 3 dT -___4 ~ T ~ ( T L - T i ) .
(7)
Combining Eqs 6 and 7:
oT~ _~ ~T,a+~T,3(TL- 7-,)= ~,T,~ T~- 3~T,4 (8) This approximation is very good for small differences between TL and Ti. For example, if TL - - 2 9 5 and T i = 2 9 0 , the error in o'T~ is approximately 0.7 W / m 2 in 429 W/m 2 or approximately 0.2O/o. This would give an incorrect TL of about 0.12 K. A similar approximation is also done for T~, where 3
~ T ~ ~- 4aTo~Tw-3aT~o ~.
X = F + ALh L + Awhl
,~ [ 1 7 - 2 6 9 ( T - 273.2)1 e~=610.yoexpL~--~9 j.
(10)
By differentiating Eq. 10, the slope (S) of the saturated vapor pressure curve is obtained:
x ( 2 - e ( 1 - f l ) ) + p C v h o T o c + P@hiFTi X
pCvAw(hi)Z X D = \RTi/\g+hJ
~-~.~
/"
Making use of Eqs 14-17, the four terms in the equation for TL are
8 ea T3oc Terml =~K~+--Y
g
x
(II)
g
eL = (%--esi) +e~i,
(19)
(pCphLAwh'')
Term3=pCphL(l_~)
(12)
where e~i is the saturated vapor pressure of the incoming air that is uniquely related to T i through Eq. 10. For small differences in temperature, and hence vapor pressure, the slope of the saturated vapor pressure curve (S) can be used at the known temperature T i so that
et,--qi = S ( T L - 7~.
(18)
/
--DF(esl-ei--STi). The saturated vapor pressure at leaf temperature is expressed as a gradient plus a known value:
(16)
ALghw F+-g + h~,
-2(1-e)L~+6a(eT4o~ - T~).
~ 1 7 . 2 6 9 ( T - 273.2) 1
(15)
Z = PCp(ho+h,) + 4eaTo3c(2-e( 1 -/~))
610 78~ -(17-'269) (237'3) l " L ( T - 3 5 . 9 ) 2 _]
x exPL.
(14)
Y = eLi,[1 + (1 --e) (1 --fl)] -3eflaT~*+3eaT4oc
(9)
The saturated vapor pressure (es in Pa) can be related by an empirical formula to temperature ( T i n K)(17):
S=
lations, an iteration is made to select S at the mean temperature of TL and Ti. The vapor pressure of the air in the cuvette can then be calculated through Eq. 5 as a function of eL and hence S and T L. These linear approximations now make the system solvable. The model is then used to calculate TL, Tw, 77,, eL and % The first step is to solve for TL by simultaneously solving Eqs 2-13. The resulting equation for TL contains four terms, within which the following factors appear :
(13)
This approximation is very good for small differences in TL - Ti. In the actual model calcu-
Term 4 = FSD + 8a
x[T3i
eTa°c( pCvhiALhL +4efl~T/3)l.
Z ',
(21)
x
The solution for TL is then Term 1 + Term 2 TL = T e r m 3 + Term 4"
(22)
ESTIMATING LEAF CUVETTE ENVIRONMENTS After obtaining
Y as can
TL, Tw can
be solved :
TL(pC~xALhL t-4eflcrT~)
(23)
Ta : 77,= (AwhiTwwALhLTL-k-FTi)(1).
(24)
T h e saturated vapor pressure of the leaf (eL) can be found through Eq. 10 and e, can be solved by
e,={ ghw %+Fei~ 1 \g+hw ALJ ghw + F g+h~ AL
(25)
It is difficult to evaluate conceptually each of the terms in Eq. 22 but an overall evaluation can be carried out by testing the model with a variety of inputs.
RESULTS AND DISCUSSION T h e model predicts the temperature characteristics for any leafcuvette design. As an example of the model predictions, the cuvette described by AMIRO et al.m will be used as a test subject. This cuvette is a spherical glass flask of 1 1 volume with no independent stirring mechanism. Since the flask is glass, and opaque to longwave radiation, e = 1. fl is chosen as 0.5, A L = 100 c m 2, ho = 5 cm/s, e = 0 . 7 , and T / = Toc=20°C. L~, is chosen as 450 W / m 2 which corresponds to surrounding surfaces that are perfect emitters at a temperature of 25°C. This would be characteristic of a growth c h a m b e r environment where the lights and c h a m b e r wall temperatures average this value. The other variables are changed in this demonstration to depict different conditions. Since the system is multivariate, a single graphical representation of conditions is not possible. Therefore, a set of conditions (which are termed 'default' conditions) are chosen and only one variable is allowed to change from these conditions during each test. These default conditions are F = 5 x l 0 -5 m3/s, h L = 0 . 0 2 m/s, K ~ = 240 W / m z and e; = 700 Pa. For m a n y physiological studies, TL and T, are
325
variables that are of interest. Leaf temperature is used to assess thermal stress and the rate of metabolic processes (photosynthesis and respiration). Air temperature is of special interest if TL - T ~ is large. Leaf temperatures measured with thermocouples then become uncertain because the thermocouple bead is actually measuring some intermediate temperature. ~2~ Another parameter of interest is the transpiration rate, which m a y give some indication of potential plant water stress. Hence, TL, T, and transpiration rate are plotted as functions of leaf diffusive conductance (g). As previously mentioned, this independent variable is determined by the plant and is the governing factor in partitioning energy between. sensible heat flux and latent heat flux. T h e stomatal mechanism, therefore, determines the cuvette microenvironment but it should be noted that the stomatal mechanism is influenced by humidity so that a feedback situation is likely. ~ 5~ Figure la illustrates the change in TL and T, with g while F, h z and ei are held at default values and K~ assumes two different radiation loads. T h e lower radiation load (K~ = 240 W / m 2) represents a case similar to a typical growth chamber. As the stomates open, the leaf temperature cools, as does the cuvette air temperature. With conditions as described, TL becomes 4 K warmer than the air entering the cuvette. However, ifa high radiation load is used (K~. = 840 W/m2), the leaf temperature becomes 16 K w a r m e r than Ti, even at high values of g. This condition is similar to a full sunlight radiation load with 700 W/m 2 arriving from above and a surface albedo of 20% below the cuvette. In this situation, air temperatures also become high. High leaf temperatures at the higher radiation load increase transpiration rates (Fig. lb). Even a t g = 0.8 cm/s, the air inside the cuvette does not become saturated for the described conditions. However, a determination of Tw reveals that condensation occurs on the cuvette walls for the higher radiation load a t g > 0.25 cm/s (marked by C in Fig. lb). This is because the cuvette walls are cooler than the dewpoint temperature in the cuvette. T h e model does not include the latent heat release when condensation is occurring on the walls since it is assumed that this condition is undesirable for physiological studies. Hence, the data beyond this condensation point will not reveal the true temperature and
326
B . D . A M I R O , T. J. G I L L E S P I E and G. W. T H U R T E L L
t r a n s p i r a t i o n rate for the leaf b u t this m o d e l clearly indicates w h e n such a s i t u a t i o n will arise, T h e flow rate t h r o u g h the c u v e t t e is v a r i e d in Fig. lc while o t h e r variables are held at default values to illustrate t h a t b o t h the leaf a n d air t e m p e r a t u r e c a n be r e d u c e d b y i n c r e a s i n g the v o l u m e of air passing t h r o u g h the system.
A l t h o u g h this increased flow cools the leaf, it also increases the t r a n s p i r a t i o n rate (Fig. ld). I f the flow rate is too low, c o n d e n s a t i o n will occur o n the walls with i n c r e a s i n g g as m a r k e d b y C in Fig. ld. T h i s illustrates t h a t a c o m p r o m i s e m u s t be r e a c h e d u s i n g the flow rate so t h a t there will be n o c o n d e n s a t i o n , the leaf will r e m a i n s u i t a b l y cool,
KI varied (W m -2) 40
a
~
400
35
~
b
300
30
~'~.~. ~"-~,...
g 200 "~
K~ = 840
25
~- 100 ~ " ~
200
.
K~ = 2 4 0
0.4
018
O0
0.4 g (em s -1)
g (cm s -1)
30
Flow rate (F) varied 200
c
0.8
m3 S"! x 10.`5 ) d
~150 'E
g
~I00
25
1.7 50
""
2o 0
'
0.4 g (cm s -1)
F
17
'8
O.
0
o14
o18
g (cm s -1)
FIO. l. Model predictions of TL, Ta and transpiration. Symbols listed in Appendix. Except where varied, conditions are K,[ = 240 W/m 2, L~, = 450 W/m 2, Ti = Toc = 20°C, F = 5 × 10 - s m3/s, hL = 2 cm/s, h 0 = 5 cm/s, AL = 100 cm 2, Aw = 300 cm 2, ~ = 1, fl = 0.5, ei = 0.7 kPa, and ~ = 0.7. The point marked by C indicates start of condensation on cuvette wall. (a) TL and T= vs g for K J, varied ( TL - - ; T= - - ) . (b) Transpiration vs g for/(~ varied. (c) TL and T= vs g for F varied ( TL ; T~- -). (d) Transpiration vs g for F varied.
ESTIMATING LEAF CUVETTE ENVIRONMENTS
327
h L varied (cm s -1) 30
200
150 E
50
I 20 |
0
hL=5 !
04 g (cm s 1)
J
0.8 9
014 (cm s 1)
018
e i varied (k Pa) 30 I c
200- d
~
. 150 'E
~_~ 25
g
100
S • =
20 I 0
I 04 g (cm s -1)
J 08
0 0
I
I
04 g (cm s 1)
0.8
FIG. 2. Model predictions of TL, 7-, and transpiration. Default conditions same as for Fig. 1. (a) TL and T. vs g for hL varied (TL - - ; To ). (b) Transpiration vs g for h L varied. (c) TL and T. vs g for e i varied ( TL - - ; T. - - ) . (d) Transpiration vs g for e i varied.
and the transpiration rate will not be excessive. In addition to these constraints, the flow rate determines the relative change in gas c o n c e n t r a t i o n entering a n d leaving the cuvette. Sufficient differences must exist to be measurable for such gases as H 2 0 and C O / d u r i n g transpiration a n d net photosynthesis studies. Flow rate is an easily adjusted variable a n d the model indicates a suitable rate to be selected. T h e transfer coefficient for the leaf to both heat and water vapor is varied in Fig. 2a. As the
transfer coefficient increases, the leaf cools as heat loss is facilitated. Since the model uses this p a r a m e t e r as a measure of turbulence within the cuvette, the transfer coefficient tbr the inside of the cuvette wall also increases. This allows increased heat transfer out of the cuvette through the wall a n d therefore cools the air inside the cuvette. T h e increase in h L does not have a d r a m a t i c effect on transpiration. This is due to the fact that cooling the leaf and reducing e L is countered by an increase in the transfer of water vapor away from
328
B . D . AMIRO, T. J. GILLESPIE and G. W. T H U R T E L L
the leaf. I n practice, the change in hL can be induced by a change in cuvette geometry, by an increase in flow rate a n d turbulence, or by the i n t r o d u c t i o n of an i n d e p e n d e n t mixing system such as a fan. If this last option is adopted, the heat generated by the fan must be incorporated into the model. It should be noted that a change in hL will change the b o u n d a r y layer c o n d u c t a n c e for other gases a n d d e p e n d i n g on the m a g n i t u d e of g, photosynthesis m a y be affected. (4) Methods of cooling the system include introduction of cool or dry air into the cuvette. I n p u t
air cooling is a practical solution and works by increasing the sensible heat flux away from the leaf. Plants n o r m a l l y cool by p a r t i t i o n i n g most of the heat loss through transpiration, so a modification of the vapor pressure entering the cuvette is a n o t h e r available method of control. This siiuation is illustrated in Fig. 2c where moist air (e i = 1700 Pa) and relatively dry air (e i = 700 Pa) are introduced. This i n p u t variable has no influence when the stomates are closed since no energy is lost in evaporation. As g increases, a lower e i yields reduced leaf a n d air temperatures.
I00
~
8C
4
3
S.. 2
4O
AIR
I
0
2o~ RH
15 ~
~2o
:>
0
25
b
014
J
0 8
0
) 5 ~>
0,4
0
08
g (cm s -1)
g (cm s 1) 30
2°°Fc
d
150[ c~
'E
~ I00 50
I
0
04 g {cm s 1)
J
08
20 0
0 14
Oi8
g (cm s l)
FIo. 3. Model predictions for default conditions indicated in Fig. 1. (a) eL and e, vs g. (b) Relative humidity and vapor pressure deficit (V.P.D.) vs g. (c) Energy partitioning of net radiation (Q*) to sensible heat flux (QH) and latent heat flux (Q~) for the leaf with a change in g. (d) Influence ofcuvette longwave transparency on leaf temperature (s = 0 = transparent, s = 1 = opaque, e = 0.5 = 50% transparent).
E S T I M A T I N G LEAF CUVETTE ENVIRONMENTS This lower el causes an increased g r a d i e n t for e v a p o r a t i o n so t h a t t r a n s p i r a t i o n is e n h a n c e d (Fig. 2d). A l t h o u g h a higher ei m a y be desirable from this point of view, an ei = 1700 Pa will induce condensation on the cuvette walls at g > 0.57 cm/s. This b a l a n c e must then be considered to m a k e the system operable. T h e m o d e l gives i n f o r m a t i o n concerning other parameters. F i g u r e 3a illustrates the c h a n g e in both eL and e, with leaf diffusive c o n d u c t a n c e for the default conditions cited. H u m i d i t y in the cuvette can also be considered in terms of relative h u m i d i t y a n d v a p o r pressure deficit as d e p i c t e d in Fig. 3b. T h e p a r t i t i o n i n g of energy is presented in Fig. 3c. As the stomates open, the leaf cools a n d increases the net radiation. Similarly, the o p e n i n g increases t r a n s p i r a t i o n a n d reduces sensible h e a t loss. These energy balances can be d e t e r m i n e d for a n y selected set of conditions. T h e foregoing analysis describes a glass cuvette that is o p a q u e to longwave r a d i a t i o n (~ = 1). T h e model is designed to be used also with other cuvette materials that m a y be totally or p a r t i a l l y t r a n s p a r e n t to this radiation. F i g u r e 3d illustrates leaf t e m p e r a t u r e d a t a for such materials t h a t m a y include very thin polyethylene (e "~ 0). Since the effective t e m p e r a t u r e of the surroundings is greater than the t e m p e r a t u r e of the cuvette walls in this case, a t r a n s p a r e n t wall causes the leaf to warm. This e x a m p l e is realistic tbr a cuvette placed in a g r o w t h c h a m b e r u n d e r hot lights. Since L{ is lower than 450 W / m 2 u n d e r field conditions, a t r a n s p a r e n t wall would give lower leaf t e m p e r a t u r e s t h a n an o p a q u e one in this situation. T h e m o d e l therefore allows the researcher to select construction materials for a given e n v i r o n m e n t . E a c h of Figs 1 3 illustrates an i m p o r t a n t p h y s i o l o g i c a l - e n v i r o n m e n t a l principle. As g increases, all curves become increasingly less sensitive to changes in the s t o m a t a l mechanism. Therefore at high values of g, the p l a n t loses its c a p a b i l i t y to control the cuvette environment. T h e m a g n i t u d e of this change in slope of TL or T, with g is d e p e n d e n t on both the leaf b o u n d a r y layer c o n d u c t a n c e and g. This is illustrated in Fig. 2a. L e a f diffusive c o n d u c t a n c e is controlled by the p l a n t and will v a r y between species, tS) with
329
e n v i r o n m e n t °5) and with p l a n t w a t e r status. (7) T h e researcher usually has some knowledge of the relative m a g n i t u d e of this value for a p a r t i c u l a r experiment. This model allows n u m e r i c a l exp e r i m e n t a t i o n over any range in g to indicate w h e t h e r some change in i n p u t variables is necessa r y to o b t a i n a desired cuvette environment. This m o d e l l i n g a p p r o a c h quantifies inputs in a known w a y a n d complements empirical comparisons of cuvette environments that have been done elsewhere .~16~ A p i e - e x p e r i m e n t a l consideration of the model can save the researcher m u c h time that would otherwise be used in e x p e r i m e n t a l l y c h a n g i n g inputs to arrive at a satisfactory situation.
REFERENCES I. AMIROB. D., GILLESPIET.J. and THURTELL G. W. (1984) Injury response ofPhaseolus vulgaris to ozone flux density. Atmos. Env. 18, 1207 1215. 2. BEADLEC. L., STEVENSONK. R. and THURTELL G. W. (1973) Leaf temperature measurement and control in a gas-exchange cuvette. Can. J. Plant Sci. 53, 407-412. 3. BraD R. B., STEWARDW. E. and LIOnTFOOT E. N. (1960) Transport phenomena. John Wiley, New York, 780 pp. 4. GRACEJ. (1981) Some effects of wind on plants. Pages 31-56 in J. GRACE, E. D. FORD and P. G. JARVIS, eds. Plants and their atmospheric environment. Blackwell Scientific Publications, Oxford. 5. Inso S. B., JACKSON R. D., EHRLER W. L. and MITCHELL S. Z. (1969) A method for determination of infrared emittance of leaves. Ecolocv 50, 899-902. 6. JARVlS P. G. ( 197 I) The estimation of resistances to carbon dioxide transfer. Pages 566 631 in Z. SESTAK, J. CATSKY and P. G. JARVIS, eds. Plant photosynthetic production, manual of methods. Junk, Hague. 7. KANEMASU E. T. and TANNER C. B. /,1969) Stomatal diffusion resistance of snap beans. I. Influence of leaf water potential. Plant Physiol. 44, 1547 1552. 8. KORNER C., SCHEEL J. A. and BAUER H. (1979) Maximum leaf diffusive conductance in vascular plants. Photosynthetica 13, 45-82. 9. LAYZELL D. B., PATE J. S., ATRINS C. A. and CANVIN D. T. (1981) Partitioning of carbon and nitrogen and the nutrition of root and shoot apex in a nodulated legume. Plant Physiol. 67, 3(~36. 10. LEGGE A. H., SAVAGED. J. and WALKER R. B.
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11. 12.
13, 14.
15.
16.
17.
B . D . AMIRO, T. J. GILLESPIE and G. W. T H U R T E L L (1979) A portable gas-exchange leaf chamber. Pages 16-11-16-24 in W. W. HECK, S. V. KrtUPA and S. N. LINZON,eds. Methodologyfor the assessment . of air pollution effects on vegetation. A i r Pollution Control Association, Pittsburgh, PA. McADAMSW. H. (1954) Heat transmission, 3rd edn. McGraw Hill, New York. RADINJ. W. and ACKERSONR. C. (1981) Water relations of cotton plants under nitrogen deficiency. Plant Physiol. 67, 115-119. RITcmE G. A. (1969) Cuvette temperatures and transpiration rates. Ecology 50, 667 670. SHARKEYT. D., IMAI K., FARQUHARG. D. and COWAN I. P. (1982) A direct confirmation of the standard method of estimating intercellular partial pressure of CO z. Plant Physiol. 69, 657-659. SHERIFFD. W. (1979) Stomatal aperture and the sensing of the environment by guard cells. Plant, Cell Envir. 2, 15-22. SXEBERTB., KOCHW. and ELLERB. M. (1980) Leaf temperatures in a gas exchange chamber and in the open air. 07. exp. Bot. 31,.863-871. TETENS O. (1930) lJber einige meteorologisebe Begriffe. Z. Geophys. 6, 297-309.
APPENDIX AL Aw Cj, e. ei
leaf area (m 2) cuvette wall area (m 2) specific heat of air (J/g/K) cuvette air vapor pressure (Pa) vapor pressure entering cuvette (Pa)
eL
e, esi
F g hi
saturated vapor pressure at T L (Pa) saturated air vapor pressure (Pa) saturated vapor pressure at T/(Pa) flow rate through cuvette (m3/s) leaf diffusive conductance (m/s) heat transfer coefficient of inside cuvette wall
(m/s) hL ho h~
*q L$ M~
o.. Q* Q~ Q,, R S T
L T~ L L~ T~ O~
2 P
heat transfer coefficient of leaf (m/s) heat transfer coefficient of outside cuvette wall (m/s) water vapor transfer coefficient for leaf (m/s) total incident shortwave radiation (W/m 2) mean incident longwave radiation (W/m z) gram molecular weight of water (18 g/mole) longwave radiation (W/m 2) net radiation on leaf (W/m 2) latent heat flux of evaporation for leaf (W/m 2) sensible heat flux for leaf (W/m 2) universal'gas constant (8.31 J/mol/K) slope of saturated vapor pressure curve (Pa/K) temperature (K) cuvette air temperature (K) air temperature entering cuvette (K) leaf temperature (K) air temperature exiting cuvette (K) air temperature outside ofcuvette (K) cuvette wall temperature (K) shortwave absorptivity of leaf radiation view factor ofcuvette wall longwave absorptivity of cuvette wall latent heat of vaporization of water (J/g) density of air (g/m 3) Stefan-Boltzmann constant (.5.67 x 10 -8 W/mi/K '*)