Quantitative Estimation of Leaf Heat Transfer Coefficients by Active Thermography at Varying Boundary Layer Conditions

Quantifying heat and mass exchanges processes of plant leaves is crucial for detailed understanding of dynamic plant-environment interactions. The two main components of these processes, convective heat transfer, and transpiration, are inevitably coupled as both processes are restricted by the leaf boundary layer. To measure leaf heat capacity and leaf heat transfer coefficient, we thoroughly tested and applied an active thermography method that uses a transient heat pulse to compute τ, the time constant of leaf cooling after release of the pulse. We validated our approach in the laboratory on intact leaves of spring barley (Hordeum vulgare) and common bean (Phaseolus vulgaris), and measured τ-changes at different boundary layer conditions.By modeling the leaf heat transfer coefficient with dimensionless numbers, we could demonstrate that τ improves our ability to close the energy budget of plant leaves and that modeling of transpiration requires considerations of convection. Applying our approach to thermal images we obtained spatio-temporal maps of τ, providing observations of local differences in thermal responsiveness of leaf surfaces. We propose that active thermography is an informative methodology to measure leaf heat transfer and derive spatial maps of thermal responsiveness of leaves contributing to improve models of leaf heat transfer processes.

The slope of the linear equation is given by the exponential function fm(u) and the y-intercept is given by the exponential function fy (u). Both the relationships and the respective exponential equations are given in figure S1.
To model hleaf according Eq.5 to Eq.8, the numerical constants a and b (see Eq. 8), which are related to the leaf geometry, are required to estimate the heat transfer coefficient for convective heat (hH). There are several values available in literature for example given by Monteith and Unsworth (2008) or Dixon and Grace (1983). However, in order to achieve a possibly accurate estimation of hleaf valid for our experimental conditions, we experimentally determined these parameters on a separate set of dark-adapted plants.
First, we had to model hH and the respective conductance to convective heat (gH), which cannot be easily measured directly. For this purpose, we used dimensionless numbers, which relate the mean leaf diameter (d) to the boundary layer thickness. This ratio is known as Nusselt number 2 (Nu) (e.g. Monteith and Unsworth, 2008;Nobel, 2009). Because, gH is a function of leaf diameter and boundary layer thickness, Nu can be used for calculations of gH with respect to leaf properties, such as size and shape. Using Nu, gH may be calculated as follows (Dixon and Grace, 1983): Here, k is the thermal conductivity of air (0.024 W m -1 K -1 ), ρa the density of air (1.204 kg m -3 ), and cp the specific heat capacity of air (1005 J kg -1 K -1 ).
In free convection, the heat transfer depends on upwelling air movement from the leaf surface into ambient air, which is maintained by the difference between leaf temperature and ambient air temperature (TL-Ta). Under these conditions, Nu is the function of another dimensionless number, the Grashof number (Gr). Using Gr, the boundary layer is mainly characterized by TL-Ta, the leaf dimension, and buoyancy forces of air. The detailed calculation of Gr is as follows: Here, g is the gravitational acceleration (9.81 m s -2 ), β the thermal expansion coefficient of air (0.0034 K -1 ), and ν the kinematic viscosity of air (15.6 * 10 -6 m² s -1 ).
With Gr, Nu becomes: Here, a and b are the numerical constants describing the geometry of a leaf.
For forced convection, conditions where wind occurs, Nu becomes a function of the Reynold's number (Re). Re relates the boundary layer composition to the wind speed (u) and d of a leaf.

Eq. S5
Similar to the formulation for free convection in forced convection Nu becomes: We have used moderate wind treatments in our experiments. Because at low wind speeds free convection still occurs, it is convenient to use mixed convection to describe convective heat 3 transfer. This implies both free and forced convection contributes to leaf heat transfer. In our study, the following formulation of the mixed convection was used (Bailey, 1993): To calculate gH according to equation S2, the parameters a and b are required, which we derived from our reference measurements, where dark-adapted leaves were exposed to step-wise increasing wind-speeds from 0 m s -1 to 1.2 m s -1 .
In a first step, we calculated gH of the dark-adapted leaves by rearranging Eq. 4 and making two assumptions: 1. The heat transfer coefficient for evapotranspiration (hλE) can be neglected for dark-adapted leaves. This assumption is supported by the measured values, i.e. gs was very low for barley and bean leaves in the dark ( Fig. 1).
2. gH behaves in a similar way irrespective of dark-or light-adaptation.
Using these assumptions, gH could be calculated as follows: To calculate gH according to Eq. S8 we first derived C Aleaf -1 using the model from the first series of reference measurements, where the relationship between τ and LWC was established.
In a second step, we used the calculated gH to calculate the respective Nu (Eq. S2). This calculated Nu was compared to theoretical Nu values, obtained by equation S7 with the variables a and b being 1. By plotting the logarithm of the measured Nu (log Numeasured) against the logarithm of equation S7 (log(Gr + 1.4 Re²)), we obtained both parameters, a and b (Fig. S2).
Note here that Eq. S7 equals Eq. S4 for leaves exposed to free convection conditions. The respective values for barley and bean and for free and mixed convection are given in table S1. By substituting a and b into equation S7, gH could be calculated at any prevailing conditions. Figure S3 summarizes the workflow of the whole modelling approach and the final experiment.
We tested the dimensionless numbers model by comparing calculated C A -1 leaf based on the linear relationships ( Fig. S1) with C A -1 leaf based on dimensionless numbers (Fig. S4). Generally, both models matched well to each other. For barley, we found a strong linear correlation, which was highly significant (p < 0.001 and r = 0.97). In contrast, the model using dimensionless numbers revealed some weaknesses for bean, particularly for C A -1 leaf calculations for nonwind conditions (Fig. 5.4b).
Here the dimensionless numbers model clearly underestimates C A -1 leaf compared to the model based on the linear regressions. Because the models did not match very well for 0.0 m s -1 , these values were excluded from further statistical analyses. However, for the wind speeds between 0.2 and 1.2 m s -1 we found a strong and highly significant linear correlation between the two models also for bean (p < 0.001 and r = 0.986).      The temperature of the water used for wetting the filter papers was kept at 22°C using a water bath. Different volumes of water (0, 1200, 1500, 2000 μl] were added to round filter papers (area = 95 cm²) by pipetting the water on the center of the paper. After homogenous distribution of the water, samples were transferred into the experimental setup as described in the manuscript. Measurements were conducted at wind speed of 0.0 m s -1 . Room temperature was around 22°C. Samples of different water content were measured randomized in order to minimize environmental effects.