## 11.3 Leaf Energy Budget

Radiation incident upon a leaf includes shortwave radiation in the form of direct sunlight, skylight and reflected light, and longwave radiation emitted by the atmosphere and clouds, or by the surfaces of ground, nearby plants, and other objects.

A plant may absorb 40 to 80 percent of incident sunlight and skylight and it will absorb 96 percent of incident longwave thermal radiation. A plant leaf redistributes this absorbed radiation by emitting a substantial amount of it as longwave radiation from the leaf surface. The remainder is partitioned into convective and evaporative heat exchange and a very small amount is utilized for photosynthesis. Only certain wavelengths of light are effective for photosynthesis. The amount of energy converted to biomass is very small (a few percents of the incident sunlight at most) and may be neglected in considerations of the energy budget of a leaf. The amount of radiation emitted by a leaf is proportional to the fourth power of the absolute temperature of the leaf surface. This phenomenological relation is known as the black body radiation law and is written $$R = \sigma[T+273]^4$$.

The flow of heat by convection is proportional to the temperature difference between the leaf and the air. Convective heat transfer is proportional to the wind speed and varies inversely with the characteristic dimension of the leaf which affects the boundary layer thickness. Because of the viscosity of air, there is an air layer which adheres to any surface, including that of a leaf known as a “boundary layer.” When the wind blows, the boundary layer of viscous air represents a transition zone between zero air flow at the leaf surface to free air flow at some distance from the leaf. A temperature difference may exist across a boundary layer of air and it is across this that heat moves from the leaf to the air by conduction and convection. The larger a leaf, the thicker the boundary layer of air adhering to the surface. A small leaf will have a boundary layer about 1 mm thick and a large leaf, like a banana leaf, will have a boundary layer of 1 or 2 cm thickness. It happens that for broad flat leaves the rate of heat transfer by convection is proportional to the square root of the ratio of wind speed to leaf dimension. Increase wind speed fourfold and the rate of convective heat exchange will increase twofold or increase leaf dimension fourfold and convection will decrease twofold. The characteristic dimension is approximately the average width of the leaf. Although the length of a leaf affects convection, we use a single characteristic dimension given by the leaf width here for simplicity. The more complex relationship involving leaf width and length is introduced later.

The transpiration rate is determined by the water vapor pressure or density difference between leaf and air. This relationship is described in more detail in the next two sections. Energy is required to convert liquid water to vapor and the amount of energy is known as the latent heat of water. It is temperature-dependent but at 30°C its value is $$2.43 \times 10^6 J kg^{-1}$$. Each of the environmental variables affect the leaf energy status simultaneously. They are radiation, air temperature, wind speed, and water vapor pressure, density, or relative humidity. A plant leaf responds to these by assuming a certain temperature and by losing water at a particular rate depending upon the properties of the leaf. Once the energy budget relationship is written, the student will see how it is that each of these environmental variables enter the common energy pool for the leaf.

The steady state energy budget for a leaf is where
• Thermal energy gained = Thermal energy lost

$$$Q_a = \varepsilon \sigma T_l + 273^4 + k_1 + \bigg(\frac{V}{D}\bigg)^{\frac{1}{2}} [ T_l - T_a ] + L (T_l) E \tag{11.1}$$$

where
• $$Q_a$$ = total amount of radiation absorbed in $$W m^{-2}$$,
• $$\varepsilon$$ = emissivity of leaf surface to longwave radiation,
• $$\sigma$$ = (Stefan-Boltzmann constant) $$5.67 \times 10^{-8} W m^{-2} K^{-4}$$,
• $$T_l$$ = leaf temperature in $$^{\circ}C$$,
• $$k_1$$ = $$9.14 J m^{-2} s^{-1/2}-11 ^{\circ}C^{-1}$$,
• $$v$$ = wind speed in $$m s^{-1}$$,
• $$D$$ = leaf width in $$m$$,
• $$T_a$$ = air temperature in $$^{\circ}C$$,
• $$L(T_l)$$ = latent heat of vaporization of water in $$Jkg^{-1}$$ as a function of leaf temperature and is equal to $$2.43 \times 10^6$$ at $$30^{\circ}C$$ and $$2.50 \times 10$$ at $$0^{\circ}C$$,
• $$E$$ is transpiration rate in $$kg m^{-2} \times s^{-1}$$

For any given value of $$E$$, a specific value of $$T_l$$ will balance Equation (11.1), provided all other quantities are known. However, $$E$$ depends on a unique set of environmental variables. It is seen in the next paragraphs that $$E$$ is a function of $$T_l$$. Therefore, it is possible to determine $$E$$ and $$T_l$$ simultaneously. If the air temperature is warmer than the leaf temperature, heat is gained by convection rather than lost and this term becomes negative on the right-hand side of the energy budget equation.

### 11.3.1 Resistance to Water Loss

There are three requirements for the loss of water from a leaf. There must be water available in the leaf, there must be energy available to convert liquid water to vapor, and finally, there must be a vapor pressure or density gradient along which water vapor may flow from inside to outside the leaf beyond the boundary layer of air which adheres to the leaf surface. Liquid water at the mesophyll cell walls within the leaf is vaporized and from the intercellular spaces, this water vapor passes out of the leaf by diffusion through the stomates or through the leaf cuticle. Usually, the cuticle is coated with a wax layer which is relatively impervious to water and as a result most of the water lost from a leaf is through the stomates. As with fluid passing through any tube or pipe, there is resistance to vapor flow by viscous drag with the walls. The resistance to water vapor diffusion through the substomatal and stomatal spaces with the leaf is $$r_l$$ in $$s m^{-1}$$ and represents an average value for the entire leaf. In addition, the water vapor must diffuse across a boundary layer of air adhering to the leaf surface and this offers a resistance $$r_a$$ given in $$s m^{-1}$$. The boundary layer resistance is affected by the wind speed across the leaf. In fact, the greater the wind speed the thinner the boundary layer thickness; and hence, diffusion resistance is greater on large leaves than on small leaves. Experimental results show that the boundary layer resistance varies directly as the square root of the leaf dimension (width) $$D$$ and inversely with the square root of the wind speed $$V$$, a relationship which is precisely the inverse of the influence of wind speed and leaf dimension on the exchange of heat by convection. Therefore, the boundary layer resistance $$r_a$$ is given by $$$r_a = k_2 \bigg(\frac{D}{V}\bigg)^{1/2} \tag{11.2}$$$

Although there are complexities concerning air flow about a leaf which cause $$k_2$$ to change its value for leaves of characteristic dimension less than $$0.005 m$$, we shall use a single value here for all leaf dimensions. $$k_2 = 200 s^{1/2-1}m^{-1}$$ when $$D$$ is in $$m$$ and $$V$$ is $$m s^{-1}$$. Since the ratio of $$D/V$$ is involved here, nothing is changed if $$D$$ is in $$cm$$ and $$V$$ in $$cm s^{-1}$$.

### 11.3.2 Transpiration Rate

The rate at which water vapor will diffuse out of a leaf will depend directly on the difference in water vapor density inside and outside the leaf, just as the rate at which water will flow downhill depends on the difference in height between the top and bottom of the hill.

For the moment, it is assumed that the air in the intercellular air spaces inside the leaf is at saturation at the temperature of the leaf and has a saturation density $$_s d_l (T_l)$$. The air outside the leaf and beyond the boundary layer has a water vapor density equal to $$[r.h.] \times _sd_a (T_a)$$, where $$r.h$$. is the relative humidity and $$_sd_a (T_a)$$ is the saturation water vapor density of the air as a function of the air temperature. The relative humidity is, by definition, the ratio of the actual water vapor density of the air to the saturation density. Usually, it is given in percent, but as used here, it is a decimal fraction. The rate of water loss from a leaf per unit area per unit time is equal to the gradient for water vapor density divided by the resistance to water vapor movement and is given by $$$E = \frac{_s d_l (T_l) - [r.h.]_s d_a (T_a)}{r_l + r_a} \tag{11.3}$$$

where $$r_a$$ is given by Equation (11.2). Here, if $$_s d_l (T_l)$$ and $$_s d_a (T_a)$$ are given in kg m-3 and $$r_l$$, and $$r_a$$ in s m-1, then $$E$$ is in kg m-2s-1.