92 lines
No EOL
3.9 KiB
ReStructuredText
92 lines
No EOL
3.9 KiB
ReStructuredText
=============
|
|
Energy budget
|
|
=============
|
|
|
|
.. figure:: figs/energy_1.png
|
|
:align: center
|
|
|
|
The *energy* module computes the temperature of individual leaves based on a detailed energy balance model (see
|
|
Supporting Information S3 in **Albasha et al., 2019**).
|
|
|
|
Energy gain of each leaf comes from:
|
|
|
|
1. the absorbed shortwave;
|
|
2. thermal longwave radiation from the sky;
|
|
3. thermal longwave radiation from the sky;
|
|
4. thermal longwave radiation from the soil;
|
|
5. thermal longwave radiation from the neighbouring leaves.
|
|
|
|
Energy loss of each leaf is due to
|
|
|
|
1. thermal longwave radiation emitted by the leaf
|
|
2. latent heat due to transpiration (evaporative cooling)
|
|
|
|
Energy gain or loss may result from heat exchange between the each leaf and the surrounding air by thermal
|
|
conduction-convection.
|
|
|
|
The resulting leaf-scale energy balance equation writes:
|
|
|
|
.. math::
|
|
0 & = \alpha_{i, \ R_g} \cdot \Phi_{i, \ R_g} \\
|
|
& + \epsilon_{i, \ leaf} \cdot \sigma \cdot
|
|
\left(
|
|
k_{i, \ sky} \cdot \epsilon_{sky} \cdot T_{sky}^4
|
|
+ k_{i, \ soil} \cdot \epsilon_{soil} \cdot T_{soil}^4
|
|
+ \left[1 - \left(k_{sky} + k_{soil} \right) \right] \cdot T_i^4
|
|
\right) \\
|
|
& - 2 \cdot \epsilon_{i, \ leaf} \cdot \sigma \cdot T_i^4 \\
|
|
& - \lambda \cdot E_i \\
|
|
& - 2 \cdot K_{air} \cdot \frac{T_i - T_{air}}{\Delta x_i}
|
|
|
|
|
|
where
|
|
:math:`i` refers to leaf identifier,
|
|
:math:`j` refers to neighbouring leaves identifier,
|
|
:math:`\Omega` denotes the upper hemisphere surrounding the leaf :math:`i`,
|
|
:math:`\alpha_{R_g} \ [-]` is lumped leaf absorptance in the shortwave band,
|
|
:math:`\Phi_{R_g} \ [W \ m_{leaf}^{-2}]` flux density of shortwave global irradiance,
|
|
:math:`\epsilon_{leaf} \ [-]` emissivity-absorptivity coefficients of the leaf,
|
|
:math:`\epsilon_{sky} \ [-]` emissivity-absorptivity coefficients of the leaf,
|
|
:math:`\epsilon_{soil} \ [-]` emissivity-absorptivity coefficients of the soil,
|
|
:math:`\lambda \ [W \ s \ {mol}^{-1}]` is latent heat for vaporization,
|
|
:math:`\sigma \ [W \ m^{-2} \ K^{-4}]` the Stefan-Boltzmann constant,
|
|
:math:`k_{sky} \ [-]` form factor of the sky,
|
|
:math:`k_{soil} \ [-]` form factor of the soil,
|
|
:math:`T \ [K]` leaf temperature,
|
|
:math:`T_{air}` air temperature,
|
|
:math:`T_{sky} \ [K]` sky temperature,
|
|
:math:`T_{soil} \ [K]` soil temperature,
|
|
:math:`K_{air} \ [W \ m^{-1} \ K^{-1}]` the thermal conductivity of air,
|
|
:math:`E \ [mol \ m_{leaf}^{-2} \ s^{-1}]` transpiration flux, and
|
|
:math:`\Delta x_i \ [m]` thickness of the boundary layer.
|
|
|
|
|
|
.. note::
|
|
Only the forced convective heat transfer is currently considered in HydroShoot since forced convection dominates
|
|
free convection once wind speed exceeds roughly 0.1 :math:`m \ s^{-1}` **(Nobel 2005)**. This wind speed threshold
|
|
is generally exceeded during diurnal hours. However, under low wind conditions heat transfer may be underestimated.
|
|
|
|
Sky and soil form factors: the *Pirouette Cacahuete* issue!
|
|
-----------------------------------------------------------
|
|
|
|
You may notice once you do your first run something like this:
|
|
|
|
.. figure:: figs/energy_2.png
|
|
:align: center
|
|
|
|
|
|
This refers to the method used to calculate the lumped sky and soil form factors (respectively :math:`k_{sky}` and
|
|
:math:`k_{soil}`).
|
|
|
|
In order to reduce calculation costs, :math:`k_{sky}` and :math:`k_{soil}`) are obtained by flip flopping the canopy:
|
|
|
|
.. figure:: figs/energy_3.png
|
|
:align: center
|
|
|
|
At first, the canopy is turned downwards. A unit irradiance is emitted from each sky sector and irradiance that is
|
|
intercepted by a leaf :math:`i` is assumed equivalent to the form factor between that leaf and the "soil".
|
|
In the second step, the canopy is turned upwards again and similarly, a unit irradiance is emitted from each sky sector.
|
|
In this case, irradiance that is intercepted by a leaf :math:`i` is assumed equivalent to the form factor between that
|
|
leaf and the "sky".
|
|
|
|
This method is clearly not 100% precise. It may need further improvements in the future. |