11.5 Influence of Energy Components on Leaf Temperature

It is useful to evaluate the major components separately of the leaf energy budget in order to get a feeling for their values and influence. If a non-transpiring leaf is in a vacuum, its temperature is determined solely by radiative exchange and its energy budget is \[\begin{equation} Q_a = \varepsilon \sigma [T_l + 273]^4 \tag{11.5} \end{equation}\]

Let \(\varepsilon = 0.96\).

#radiation only
sigma= 5.67 * 10^{-8} #Stefan-Boltzmann's constant (W m^-2K^-4)
#Q_a is total amount of radiation absorbed in W m^-2

f <- function(T_l, Q_a=800) epsilon *sigma*(T_l + 273)^4 - Q_a
uniroot(f, interval=c(-100, 100))$root
## [1] 75.18409
f <- function(T_l, Q_a=600) epsilon *sigma*(T_l + 273)^4 - Q_a
uniroot(f, interval=c(-100, 100))$root
## [1] 51.02181
f <- function(T_l, Q_a=400) epsilon *sigma*(T_l + 273)^4 - Q_a
uniroot(f, interval=c(-100, 100))$root
## [1] 19.78675

As calculated above, if \(Q_a = 800 W m^{-2}\), \(T_l = 75.2 ^\circ\) C; \(600 W m^{-2}\), \(T_l = 51.1 ^\circ C\); and \(Q_a = 400 W m^{-2}, T_l = 19.8 ^\circ C\). A leaf temperature of 75°C is very warm indeed and no plants as we know them would survive such a temperature. Even at 51°C, most plants would sustain serious heat damage. However, plants live in air, a fluid, which flows about the plant and takes away excess heat by convection. Just as a plant loses heat by radiation, it must lose heat by convection as well, and this it cannot avoid. We shall now determine the extent to which a plant leaf may have its temperature influenced by convection.

If the non-transpiring leaf is placed in air, the leaf temperature is significantly affected by the air temperature. If the ratio \(V/D\) is large, convection becomes very significant and the leaf temperature is tightly coupled to the air temperature. Large \(V/D\) suggests high wind speeds or small leaf dimension. If \(V/D\) is small, the leaf temperature is less strongly affected by the air temperature, but still it is significantly influenced. The energy budget is given by \[\begin{equation} Q_a = \varepsilon \sigma [T_l + 273]^4 + k_1 \bigg( \frac{V}{D} \bigg)^{1/2} [T_l - T_a] \tag{11.6} \end{equation}\]

If the air temperature is \(30°C\), one can determine the exact numerical relationship between \(Q_a\) and \(T_l\) for any given ratio \(V/D\). For example, we will determine values of \(T_l\) for \(Qa\) = 800, 600, and 400 W m-2 for \(V/D\) = 1.0, 10.0, and 100 below. These values are listed in Table 11.1.

#radiation and convection
sigma= 5.67 * 10^{-8} #Stefan-Boltzmann's constant (W m^-2K^-4)
k1=9.14 #an experimentally determined coefficient 
        #(J m^-2 s^-0.5 °C^-1)

#Q_a is total amount of radiation absorbed in W m^-2
#estimate root with radiation and convection
rc_root= function(VDratio1, Q_a1){
  f <- function(T_l, VDratio=VDratio1, Q_a=Q_a1, T_a=30) epsilon *
    sigma * (T_l + 273)^4 + k1 * (VDratio)^0.5 * (T_l - T_a) - Q_a
uniroot(f, interval=c(-100, 100))$root

rc_mat= rbind( sapply( c(1,10,100), FUN=rc_root, Q_a1=800), 
               sapply( c(1,10,100), FUN=rc_root, Q_a1=600), 
               sapply( c(1,10,100), FUN=rc_root, Q_a1=400) )
colnames(rc_mat)=c('V/D=1', 'V/D=10', 'V/D=100') 
##         V/D=1   V/D=10  V/D=100
## [1,] 51.49638 39.67761 33.49723
## [2,] 39.12380 34.02487 31.44818
## [3,] 26.10093 28.31561 29.39652
Table 11.1: Leaf temperature.
V/D=1 V/D=10 V/D=100
51.49638 39.67761 33.49723
39.12380 34.02487 31.44818
26.10093 28.31561 29.39652

Table 11.2. Leaf temperature in °C for various amounts of absorbed radiation exchanged by radiation only, by radiation and convection, and by radiation, convection, and transpiration. Values of variables used are \(Ta=30°C\), r.h.=50%, and \(r_l\) = 100 s m-1. The addition of convective exchange to the energy budget will decrease the difference between leaf temperature and air temperature compared to the leaf temperature computed when only radiation is included. As the ratio \(V/D\) increases, the magnitude of the difference between leaf temperature and air temperature decreases. Note that if absorbed radiation levels are above black body value (for \(Q_a\) = 800 and 600 W m-2 because here black body level is \(Q_a=\sigma T_a= 5.67\times 10^{-8} \times (273+30)^4 = 478 W m^{-2}\). \(T_l-T_a\) is positive and if \(Q_a\) is below the black body value \(T_l\) -\(T_a\) is negative. If transpiration is included, it will reduce all leaf temperatures from the values computed when only radiation and convection transfer are included.

\(Q_a\) (W m-2) Radiation only Radiation and convection \(V/D\) (s-1) Radiation, convection, and transpiration \(V/D\) (s-1)
1.0 10.0 100.0 1.0 10.0 100.0
800 75.2 61.5 39.7 33.5 36.8 31.8 30.3
600 51.1 39.1 34.0 31.2 30.4 28.5 28.7
400 19.8 26.1 28.3 29.4 23.0 24.9 27.1

For \(V/D = 1.0\), simply placing the non-transpiring leaf in air of temperature 30°C dropped the leaf temperature from 75.2 to 61.5°C at \(Q_a = 800 W m^{-2}\), a decrease of 13.7°C. Clearly, convection may play a very strong role in influencing leaf temperature. If \(V/D = 10.0\), the decrease is, of course, even more dramatic. At \(Q_a = 800 W m^{-2}\) and \(T_a\) = 30°C, \(T_l\) = 39.7°C, there is a decrease of 35.5°C from the pure radiation regime. If \(V/D = 100\) and \(Q_a = 800 W m^{-2}\), the leaf temperature is 33.5°C with convection, a change of 41.7°C.

We tend to forget the significance of convection in the ordinary world but when we make comparisons of this kind to show what would happen to leaf temperature without convective cooling, the influence of convection becomes very clear. At lower amounts of absorbed radiation, the influence of convection is less strong, but still important until the leaf temperature is equal to the air temperature. Then, when the quantity of absorbed radiation is very low, which will occur for an exposed leaf on a clear night, the leaf temperature is below the air temperature and convection will deliver heat to the leaf, thereby, warming it. This is seen in Table 11.1 for \(Q_a = 400 W m^{-2}\); when, without convection, the non-transpiring leaf had a temperature of 19.8°C, and the in air at 30°C, the leaf temperature increased to 26.1, 28.3, and 29.4°C, respectively, at \(V/D\) = 1.0, 10.0, and 100.0.

The influence of transpiration is now readily seen by including the transpiration term in the full energy budget expression as given in Equation (11.4). Now one must decide on a value for the internal resistance of a leaf and the relative humidity of the air. Let \(r_l = 100 s m^{-1}\) and \(r.h. = 50\)%. It is more difficult to solve Equation (11.4) then it is to solve Equations (11.5) or (11.6). Graphical methods are possible although cumbersome. It is far easier to use a calculator or a computer and this is now done. The values are listed in Table 11.1.

At \(Q_a = 800 W m^{-2}\), \(T_l\) = 36.8, 31.8, or 30.3°C for \(V/D\) = 1.0, 10.0, and 100.0, respectively. Hence, transpiration has reduced the leaf temperature by 24.7, 7.9, and 3.2°C for the three ratios of \(V/D\), as shown by comparing the values listed in columns 6, 7, and 8 with those in columns 3, 4, and 5. These temperature reductions caused by transpiration are quite striking and would be significant in terms of photosynthetic and respiration rates. At lower levels of absorbed radiation, the influence of transpiration if reduced, since there is less energy available for the evaporation of water. Transpiration always produces leaf temperatures lower than the case of radiation and convection only.