CHAPTER 3 - MECHANISMS OF ENERGY TRANSFER

Mass and energy in the air

To know the concepts of frost protection, it is important to have a good description of the constituents of air and their relationship to energy content. Numerically, nitrogen (N₂) and oxygen (O₂) molecules are the main constituents of the atmosphere, with water vapour (H₂O) being a minor (and variable) component. Within a cubic metre of air there are more gas molecules than stars in the universe (about 2.69 × 10²⁵), but the volume occupied by the molecules is less than about 0.1 percent of the total volume of the air (Horstmeyer, 2001). Therefore, while the number of air molecules within a cubic metre of atmosphere is immense, the Earth's atmosphere is mostly empty space. However, the molecules are moving at high velocity, so there is considerable kinetic energy (i.e. sensible heat) in the air. In this chapter, the methods of energy transfer that control sensible heat content and hence air temperature are discussed.

Energy transfer rates determine how cold it will get and the effectiveness of frost protection methods. The four main forms of energy transfer that are important in frost protection are radiation; conduction (or soil heat flux); convection (i.e. fluid transfer of sensible and latent heat properties) and phase changes associated with water (Figure 3.1).

Radiation is energy that comes from oscillating magnetic and electric fields and, unlike the other transfer mechanisms, can transfer through empty space. Good examples are the energy one feels from sunlight or from standing near a fire. Radiation that is intercepted by a surface is commonly expressed in terms of energy per unit time per unit surface area (e.g. W m^-2). In frost protection, the net radiation (R_n) is an important factor. The components that determine R_n, including short-wave (solar) radiation downward (R_Sd) and upward (R_Su), and long-wave radiation downward (R_Ld) and upward (R_Lu), are discussed later in this chapter.

Conduction is heat transfer through a solid medium, such as heat moving through a metal rod (Figure 3.1) or through the soil. Technically, soil heat can be measured with a thermometer, so it is sensible heat, but it moves mainly by conduction (i.e. from molecule to molecule) through the soil. When energy passes through the soil by conduction it is called soil heat flux density and it is commonly expressed as units of energy per unit time per unit surface area that it passes (e.g. W m^-2). In frost protection, the main interest is in soil heat flux density (G) at the surface of the soil.

FIGURE 3.1
The four forms of heat transfer

The four forms of heat transfer are:

conduction, where heat is transferred through solid material from molecule to molecule (e.g. heat passing through a metal bar);

sensible heat flux, where warmer air is transferred from one location to another (e.g. warm air rising because it is less dense);

radiation, where heat is transferred as electromagnetic energy without the need for a medium (e.g. sunlight); and

latent heat flux, where sensible heat is converted to latent heat when water vaporizes and converts back to sensible heat when the water molecules condense or deposit (as ice) onto a surface.

Sensible heat is energy that we can "sense", and temperature is a measure of the sensible heat content of the air. When the sensible heat content of air is high, the molecules have higher velocities and more collisions with each other and their surroundings, so there is more kinetic energy transfer. For example, a thermometer placed in warmer air will have more air molecule collisions, additional kinetic energy is transferred to the thermometer and the temperature reads higher. As sensible heat in the air decreases, the temperature drops. In frost protection, the goal is often to try to reduce or replace the loss of sensible heat content from the air and plants. Sensible heat flux density (H) is the transfer of sensible heat through the air from one place to another. The flux density is expressed as energy per unit time per unit surface area (e.g. W m^-2) that the energy passes through.

Latent heat is released to the atmosphere when water is vaporized and the latent heat content of the air depends on the water vapour content. Latent heat changes to sensible heat when water changes phase from water vapour to liquid water or ice. As water vapour moves, the flux density is expressed in units of mass per unit area per unit time (e.g. kg m^-2 s^-1). Multiplying by the latent heat of vaporization (L) in J kg^-1 converts the water vapour flux density from mass units to energy units. Therefore, the flux is expressed in energy per unit time per unit surface area or power per unit surface area (e.g. W m^-2). The water vapour content of the air is a measure of the latent heat content, so humidity expressions and the relationship to energy are discussed in this chapter.

Energy balance

Sign convention

Positive and negative signs are used in transfer and balance calculations to indicate the direction of energy flux to or from the surface. Any radiation downward to the surface adds to the surface energy and therefore is considered positive and given a "+" sign. Any radiation away from the surface removes energy and it is considered negative with a "-" sign. For example, downward short-wave radiation from the sun and sky (R_Sd) is positive, whereas short-wave radiation that is reflected upward from the surface (R_Su) is negative. Downward long-wave radiation (R_Ld) is also given a positive sign since it adds energy to the surface and upward long-wave radiation (R_Lu) is given a negative sign. Net radiation (R_n) is the "net" amount of radiant energy that is retained by the surface (i.e. the sum of all gains and losses of radiation to and from the surface).

These relationships are illustrated for (a) daytime and (b) night-time in Figure 3.2. Note, in the equation, that net radiation is equal to the sum of its components and the sign indicates whether the radiation is downward (positive) or upward (negative). If the sum of the component parts is positive, as happens during the daytime (Figure 3.2a), then R_n is positive and more energy from radiation is gained than lost from the surface. If the sum of the component parts is negative, as happens during the night (Figure 3.2b), then R_n is negative and more radiation energy is lost than gained.

FIGURE 3.2
Sign convention for radiation during (a) daytime and (b) night-time

R_n supplies energy that heats the air, plants and soil or evaporates water. In this book, the equation in Figure 3.3 is used for the surface energy balance. Note that energy storage in the plants, photosynthesis and respiration are generally ignored in vertical energy fluxes in frost protection. Assuming that all of the energy fluxes are vertical, energy from R_n is partitioned into the components G, H and LE, so R_n is set equal to the sum of G, H and LE (Eq. 3.1).

Again, the sign of the energy flux component indicates the direction of energy flow. Radiation adds energy to the surface, so it is positive to the surface. When G is positive, energy is going into the soil, and when H and LE are positive, the energy flux is upward to the atmosphere. Therefore, G, H and LE fluxes are positive away from the surface and negative towards the surface.

Although, most energy transfer on a frost night is vertical, a crop is three-dimensional, and energy can pass horizontally as well as vertically through a crop. Energy transfer through a crop is often depicted using an energy box diagram (Figure 3.4), which represents the volume of air to be heated in frost protection. The energy content of the box in the diagram depends on the sources and losses of energy (Figure 3.4), where most of the energy fluxes can be in either direction. The energy balance for the box is given by:

where R_n is a positive number when more energy from radiation is received than is emitted and reflected, and it is negative if more radiant energy is lost than gained. The variables G, H and LE are all positive when the energy is exiting from the box and are negative if the energy is entering the box. F₁ is horizontal sensible and latent heat flux into the box (a negative number) and F₂ is horizontal sensible and latent heat flux out of the box (a positive number). The sum of F₁ and F₂ is the net difference in horizontal flux of sensible and latent heat. The variable P_R is for photosynthesis (a positive number) and respiration (a negative number). However, P_R is small and commonly ignored for energy balance calculations. The variable DS is the change in stored energy (sensible heat) within the box, which is positive if the energy content increases (e.g. when the temperature increases) and it is negative when the energy content decreases (e.g. temperature falls).

FIGURE 3.4
A box energy diagram showing possible sources and losses of energy from a crop represented by the box

The symbols are:

net radiation (R_n),
sensible heat flux (H),
latent heat flux (LE),
soil heat flux or conduction (G),
sensible and latent energy advection in (F₁) and out (F₂),
and energy storage in the crop (DS)

During a typical radiation frost night, R_n is negative, F₁ and F₂ sum to near zero, and P_R is insignificant. If water is not used for protection and there is no dew or frost formation and minimal evaporation, then LE is insignificant. Both G and H are negative, implying that heat is transferring into the box, but the magnitude of G + H is less than R_n, so DS is negative and the air and crop will cool.

In many active and passive frost protection methods, the goal is to manipulate one or more of the energy balance components to reduce the magnitude of DS. This can be done by improving heat transfer and storage in the soil, which enhances soil heat storage during the day and upward G at night; by using heaters, wind machines or helicopters that increase the magnitude of negative H; by reducing the magnitude of the negative R_n; or by cooling or freezing water, which converts latent to sensible heat and raises the surface temperature. When the surface temperature is raised, the rate of temperature fall decreases. In this chapter, energy balance, radiation, sensible heat flux, soil heat flux or conduction, latent heat flux, humidity and water phase changes are discussed.

The energy from net radiation can also vaporize water and contribute to latent heat flux density (LE) or evaporation from the surface. Recall that when water is vaporized, sensible heat is converted to latent heat. When water condenses, the process is reversed and latent heat is converted to sensible heat. The E in LE represents the flux density of water molecules (kg s^-1 m^-2), so E is the mass per unit time passing through a square metre of surface area. The latent heat of vaporization (L) is the amount of energy needed to vaporize a unit mass of water (L » 2.45 × 10⁶ J kg^-1). Consequently, the latent heat flux density (LE), like R_n, H and G, has the same units (J s^-1 m^-2 = W m^-2). When water vapour is added to the air (i.e. the flux is upward), it is given a positive sign. When water vapour is removed from the air in a downward flux (i.e. during dew or frost deposition), the sign is negative.

In arid climates, during the morning, when the surface temperature is higher than air temperature, it is common for R_m G, H and LE to be positive, with LE considerably less than R_n (Figure 3.5). During the afternoon in arid climates, when the air temperature is higher than surface temperature, it is common for R_n to be positive, G to be small and negative, H to be negative and LE to be similar in magnitude to R_n (Figure 3.6). Note that H is often positive all day in humid climates where there is less horizontal advection of warm air over a cooler crop. During radiation frost conditions without dew or frost formation, typically R_n<0, G<0, H<0 and LE=0 (Figure 3.7). If and when condensation occurs, LE is negative and it supplies additional energy to help replace net radiation losses (Figure 3.8).

During a radiation frost night, there is a net loss of radiation (i.e. R_n < 0). Energy fluxes from the soil and air partially compensate for the energy losses, but as the sensible heat content of the air decreases, the temperature drops. Most active frost protection methods attempt to replace the energy losses with varying degrees of efficiency and cost.

FIGURE 3.5
Mid-morning summer energy balance with R_n, G, H, and LE

FIGURE 3.7
Pre-dawn radiation frost energy balance without condensation and with R_n, G and H

FIGURE 3.6
Mid-afternoon summer energy balance with R_n and LE (+) and G and H

FIGURE 3.8
Pre-dawn radiation frost energy balance with condensation and R_n, G, H and LE

Humidity and Latent heat

In addition to sensible heat, air also contains latent heat that is directly related to the water vapour content. Each water molecule consists of one oxygen atom and two hydrogen atoms. However, hydrogen atoms attached to the oxygen atom are also attracted to the oxygen atoms of other water molecules. As more and more water molecules form the hydrogen bonds, they form a crystalline structure and eventually become visible as liquid water. Not all of the molecules are properly lined up to form hydrogen bonds so clumps of joined water molecules can flow past one another as a liquid. When the water freezes, most of the molecules will make hydrogen bonds and will form a crystalline structure (ice).

To evaporate (i.e. vaporize) water, energy is needed to break the hydrogen bonds between water molecules. This energy can come from radiation or sensible heat from the air, water, soil, etc. If the energy comes from sensible heat, kinetic energy is removed from the air and changed to latent heat. This causes the temperature to decrease. When water condenses, hydrogen bonds form and latent heat is released back to sensible heat causing the temperature to rise. The total heat content (i.e. enthalpy) of the air is the sum of the sensible and latent heat content.

The water vapour content of the air is commonly expressed in terms of the water vapour pressure or partial (barometric) pressure due to water vapour. A parameter that is often used in meteorology is the saturation vapour pressure, which is the vapour pressure that occurs when the evaporation and condensation rates over a flat surface of pure water at the same temperature as the air reaches a steady state. Other common measures of humidity include the dew-point and ice point temperatures, wet-bulb and frost-bulb temperatures, and relative humidity. The dew-point temperature (T_d) is the temperature observed when the air is cooled until it becomes saturated relative to a flat surface of pure water, and the ice point temperature (T_i) is reached when the air is cooled until it is saturated relative to a flat surface of pure ice. The wet-bulb temperature (T_w) is the temperature attained if water is evaporated into air until the vapour pressure reaches saturation and heat for the evaporation comes only from sensible heat (i.e. an adiabatic process). Saturation vapour pressure depends only on the air temperature, and there are several equations available for estimating saturation vapour pressure.

By substituting the air (T_a), wet-bulb (T_w) or dew-point (T_d) temperature for T in Equation 3.3, one obtains the saturation vapour pressure at the air (e_a), wet-bulb (e_w) or dew-point (e_d) temperature, respectively.

If the water surface is frozen, Tetens (1930) presented an equation for saturation vapour pressure (e_s) over a flat surface of ice at subzero temperature (T) in °C as:

where e_s is the saturation vapour pressure (kPa) at subzero air temperature T (°C). By substituting the frost-bulb (T_f) or ice point (T_i) temperature for Tin Equation 3.4, one obtains the saturation vapour pressure at the frost-bulb (e_f) or at the ice point (e_i) temperature, respectively.

The latent heat content of air increases with the absolute humidity (or density of water vapour) in kg m^-3. However, rather than using absolute humidity, humidity is often expressed in terms of the vapour pressure. Vapour pressure is commonly determined using a psychrometer (Figure 3.9) to measure wet-bulb (T_w) and dry-bulb (T_a) temperatures. The dry-bulb temperature is the air temperature measured with a thermometer that is ventilated at the same wind velocity as the wet-bulb thermometer for measuring the wet-bulb temperature.

An equation to estimate the vapour pressure from T_w and T_a is:

where

is the psychrometric constant (kPa °C^-1) adjusted for the wet-bulb temperature (T_w), the saturation vapour pressure at the wet-bulb temperature (e_w) is calculated by substituting T_w for T in Equation 3.3, and P_b (kPa) is the barometric pressure, where all temperatures are in °C (Fritschen and Gay, 1979). Alternatively, one can find the value for e_w corresponding to the wet-bulb temperature in Tables A3.1 and A3.2 (see Appendix 3 of Volume I).

FIGURE 3.9
Fan aspirated (upper instrument) and sling (lower instrument) psychrometers, which measure dry-bulb and either wet-bulb or frost-bulb temperatures to determine various measures of humidity

Barometric pressure (P_b) varies with passage of weather systems, but it is mainly a function of elevation (E_L). For any location, P_b can be estimated using the equation from Burman, Jensen and Allen (1987) as:

with E_L being the elevation (m) relative to sea level.

When the temperature is subzero, the water on the wet-bulb thermometer may or may not freeze. Common practice is to freeze the water on the wet-bulb thermometer, by touching it with a piece of ice or cold metal. When the water freezes, there will be an increase in the temperature reading as the water changes state from liquid to solid, but it drops as water sublimates from the ventilated ice-covered thermometer bulb. Within a few minutes, the temperature will stabilize at the frost-bulb (T_f) temperature. From the air and frost-bulb temperatures, the vapour pressure of the air is then determined using:

where

is the psychrometric constant adjusted for the frost bulb temperature (T_f), and the saturation vapour pressure at the frost-bulb temperature (e_f) is calculated by substituting T_f into Equation 3.4. Alternatively, one can find the value for e_f corresponding to the frost-bulb temperature in Table A3.3 in Appendix 3 of Volume I.

Relationships between temperature, vapour pressure and several measures of humidity for a range of subzero temperatures are shown in Figure 3.10. The upper curve represents the saturation vapour pressure over water (Equation 3.3) and the lower curve represents the saturation vapour pressure over ice (Equation 3.4). Therefore, at any given subzero temperature, the saturation vapour pressure over ice is lower than over water. Given an air temperature of T_a = -4°C and a vapour pressure of e = 0.361 kPa, the corresponding temperatures are: T_d = -7.0, T_i = -6.2, T_w = -4.9 and T_f = -4.7 °C for the dew-point, ice point, wet-bulb and frost-bulb temperatures, respectively. The corresponding saturation vapour pressures are: e_d = 0.361, e_i = 0.361, e_w = 0.424 and e_f = 0.411 kPa. The saturation vapour pressure at air temperature is e_s = 0.454 kPa.

Sometimes it is desirable to estimate the wet-bulb temperature from temperature and other humidity expressions. However, because the vapour pressure is a function of T_w, e_w, T_a - T_w and P_b, it is difficult without complicated programming. The same problem arises for estimating the frost-bulb temperature (Equation 3.8) from other humidity expressions. However, an Excel application (CalHum.xls) for estimating T_w and T_f from other parameters is included as a computer application with this book.

For any given combination of subzero temperature and humidity level, the actual and saturation vapour pressures at the dew-point and ice point are equal (i.e. e_d = e_i = e). In addition, the dew-point is always less than or equal to the wet-bulb, which is less than or equal to the air temperature (i.e. T_d £ T_w £ T_a). A similar relationship exists for the ice point, frost-bulb and air temperature (i.e. T_i £ T_f £ T_a). At any subzero temperature, e_i £ e_d.

FIGURE 3.10
Saturation vapour pressure over water (upper curve) and over ice (lower curve) versus temperature

Figure 3.11 shows the corresponding air, wet-bulb, frost-bulb, ice point and dew-point temperatures at sea level for a range of dew-point temperature with an air temperature T_a = 0 °C. If the dew-point is T_d = -6 °C at T_a = 0 °C, both the wet-bulb and frost-bulb temperature are near -2 °C. In fact, there is little difference between the wet bulb and frost bulb temperatures for a given dew-point temperature in the range of temperature important for frost protection. However, the ice point and dew-point temperatures deviate as the water vapour content of the air (i.e. the dew-point) decreases. Because there is little difference between the wet-bulb and frost-bulb temperature, there is little need to differentiate between the two parameters. Therefore, only the wet bulb temperature will be used in further discussions.

The total heat content of the air is important for frost protection because damage is less likely when the air has higher total heat content. During a frost night, the temperature falls as sensible heat content of the air decreases. Sensible heat content (and temperature) decreases within the volume of air from the soil surface to the top of the inversion because the sum of (1) sensible heat transfer downward from the air aloft, (2) soil heat flux upward to the soil surface and (3) transfer of heat stored within the vegetation to the plant surfaces is insufficient to replace the sensible heat content losses resulting from net radiation energy losses.

If the air and surface cool sufficiently, the surface temperature can fall to T_d and water vapour begins to condense as liquid (i.e. dew) or to T_i and water vapour begins to deposit as ice (i.e. frost). This phase change converts latent to sensible heat at the surface and partially replaces energy losses to net radiation. Consequently, when dew or frost form on the surface, the additional sensible heat supplied by conversion from latent heat reduces the rate of temperature drop.

FIGURE 3.11
Corresponding wet-bulb (T_w), frost-bulb (T_f), ice point (T_i) and dew-point (T_d) temperatures as a function of dew-point temperature at an elevation of 250 m above mean sea level (i.e. air pressure (P_b) = 98 kPa) with an air temperature T_a = 0°C.

A good measure of the total heat content of the air is the "equivalent" temperature (T_e), which is the temperature the air would have if all of the latent heat were converted to sensible heat. The formula to calculate T_e (°C) from air temperature T_a (°C), vapour pressure e (kPa) and the psychrometric constant g(kPa °C^-1) is:

Calculated T_e values for a range of T_a and T_i are given in Table 3.1 and for a range of T_a and T_d in Table 3.2. Values for T_d and T_i depend only on the water vapour content of the air and hence the latent heat content of the air. When T_d or T_i is high, then T_e is often considerably higher than the air temperature, which implies higher total heat content (i.e. higher enthalpy). Therefore, when T_e is close to T_a, the air is dry, there is less heat in the air and there is more chance of frost damage.

TABLE 3.1
Equivalent temperatures (T_e) for a range of air (T_a) and ice point (T_i) temperatures at sea level with the saturation vapour pressure (e_a) and the psychrometric constant (g), which are functions of T_a

TABLE 3.2
Equivalent temperatures (T_e) for a range of air (T_a) and dew-point (T_d) temperatures at sea level with the saturation vapour pressure (e_a) and the psychrometric constant (g), which are functions of T_a

Sensible heat

The energy content of the air depends on the barometric pressure, temperature and the amount of water vapour present per unit volume. The energy (or heat) that we measure with a thermometer is a measure of the kinetic energy of the air (i.e. energy due to the fact that molecules are moving). When a thermometer is placed in the air, it is constantly bombarded with air molecules at near sonic speeds. These collisions transfer heat from the molecules to the thermometer and cause it to warm up. This makes the liquid in the thermometer expand and we read the change in the level of the liquid as the temperature. When the air temperature increases, the air molecules move faster and therefore have more kinetic energy. As a result more molecules strike the thermometer and at higher speeds, causing greater transfer of kinetic energy and a higher temperature reading. Thus, temperature is related to the velocity of air molecules and the number of molecules striking the thermometer surface. Like a thermometer, air molecules strike our skin at near sonic speeds and kinetic energy is transferred from the molecules to our skin by the impact. We "sense" this transfer of energy, so it is called "sensible" heat.

If the air were perfectly still (i.e. no wind or turbulence), then the temperature that we sense would depend only on molecular heat transfer, where energy is transferred due to high-speed collisions between air molecules travelling over short distances. However, because there is wind and turbulence, air parcels with different sensible heat content move from one place to another (i.e. sensible heat flux). For example, if you stand inside of a dry sauna with relatively still air you will feel hot mainly because of molecular heat transfer through a boundary layer of still air next to your body. However, if a fan is started inside the sauna, some of the hotter air (i.e. with faster moving molecules) will be forcefully convected through the boundary layer to your skin. Because mechanical mixing, due to the fan, forced transfer to your skin, it is called "forced" convection. Hotter air is less dense than colder air (i.e. the mass per unit volume is less), so if the heat source is in the floor of the sauna, air at the surface will be less dense and it will rise into the cooler air above. When the less dense warmer air rises, the heat transfer is called "free" convection. In nature, the wind mainly blows air parcels horizontally and if warmer air blows into an area, the process is called "warm air advection". Similarly, if cold air blows into an area, the process is called "cold air advection". In frost protection, both forced and free convection as well as advection are important.

Sensible heat flux is important for frost protection on both a field scale and on an individual leaf, bud or fruit scale. Downward sensible heat flux from the air to the surface partially compensates for energy losses due to net radiation at the surface. However, as sensible heat is removed at the surface, air from above the crop transfers downward to compensate. This causes a loss of sensible heat above the crop as well as in the crop. As a result the temperature falls at all heights within the inversion layer, but most rapidly near the surface. Some protection methods (e.g. wind machines and helicopters) mainly use enhanced sensible heat transport to provide more energy to the surface and slow the temperature drop. Also, methods such as heaters partially use sensible heat flux to transport energy to a crop and provide protection.

In addition to field-scale energy transfer, the sensible heat flux through boundary layers of leaves, buds and fruit to the surface is important for determining the temperature of sensitive plant parts. A boundary layer over plant surfaces is a thin layer of still air where much of the heat transfer is by molecular diffusion. This layer tends to insulate the plant parts from sensible and latent heat transfer with the air. For example, wind machines are known to provide some frost protection even when there is no temperature inversion above a crop. This occurs because increasing ventilation will reduce the depth of boundary layer over the leaf, bud, or fruit surfaces and enhances sensible heat transfer from ambient air to the surface.

According to Archimedes principle, a body totally or partially immersed in a fluid is subject to an upward force equal in magnitude to the mass of the fluid it displaces. Totally immersed materials with an average density smaller than that of the fluid will rise and denser materials will fall towards the bottom. A good illustration of how density works is a hot air balloon. When hot air is forced into a balloon, more molecules hit the inside than the outside of the balloon, so there is more pressure on the inside and the walls expand. Eventually, the balloon becomes fully expanded. As additional hot air is introduced into the hole in the bottom, air molecules inside the balloon move at higher velocities and some air is forced out of the hole at the bottom. More molecules leave than enter through the hole in the bottom, so the mass of air inside decreases, while the volume remains relatively fixed. Consequently, the density decreases. When the density (i.e. the mass of the balloon, gondola, heater, etc., divided by the volume occupied by the balloon and its parts) is less than the density of the ambient air, the balloon will rise. If the heater is stopped, then air inside the balloon will begin to cool and air from outside will enter the hole in the bottom, which causes the density of the balloon to increase. As it becomes denser, the balloon will descend. Clearly, density is an important factor determining whether air moves up or down and therefore it is important for frost protection.

Considering the balloon example, it is clear that warmer, less dense air rises and colder, denser air will descend. During a radiation frost night, cold air accumulates near the surface and, if the ground is sloping, it will begin to flow downhill much like water flows downhill. However, like water, the flow of cold air is controllable using obstacles (fences, walls, windbreaks, etc.) to funnel the air where it will do less damage. This has been effectively used as a frost protection method. At the same time, obstacles can also block the normal drainage of cold air from a crop and increase the potential for damage.

Conduction - Soil heat flux

Like molecules in the air, molecules in solids also move faster when energy is transferred to the solid and the temperature of the solid increases. This form of energy transfer is called conduction. A good example is the transfer of heat through a metal rod if one end is placed in a fire, where the heat is transferred from molecule to molecule to the other end of the rod. Conduction is an important transfer mechanism for energy storage in the soil and therefore it is important for frost protection.

The rate that energy transfers by conduction depends on the capacity for the material to conduct energy (i.e. thermal conductivity) and the gradient of temperature with distance into the material. The thermal conductivity of a soil depends on the type and relative volume occupied by soil constituents. Air is a poor conductor of heat, so dry soils with more air spaces have lower thermal conductivities. The thermal conductivity of dry soils varies, but it is approximately 0.1, 0.25 and 0.3 W m^-1 °C^-1 for organic, clay and sandy soils. If the soils are nearly saturated with water, the conductivity is approximately 0.5, 1.6 and 2.4 W m^-1 °C^-1 for the three general soil types.

There is positive conduction into the soil when the surface is warmer than the soil below and the conduction is negative when heat conducts upward to a colder surface. As the sun comes up, the surface is warmer than the soil below, so heat conducts downward and is stored in the soil. As net radiation decreases in the afternoon, the surface will cool relative to the soil below and heat is conducted upwards towards the surface (i.e. negative flux). This negative heat flux continues during the night as soil heat conducts upwards to replace lost energy at the cooler surface. On an hourly basis, the soil heat flux density can change considerably but, on a daily basis, the amount of energy going into the soil is generally about the same as the quantity leaving the soil. In the longer term, there is a slight deficit each day during the autumn, so the soil gradually loses energy and cools. In the spring, there is a slight increase in energy receipt and storage each day, so the mean daily soil temperature will gradually increase. One should always remember that soil selection and management has both short-term (i.e. daily) and long-term (i.e. annual) effects on soil temperature.

Soil flux heat density (G) is estimated as:

where K_s is the thermal conductivity (W m^-1 °C^-1) and the second term on the right hand side is the change in temperature with depth (°C m^-1) called the thermal gradient. It is not possible to directly measure soil heat flux density (G) at the surface. If a heat flux plate is placed on the surface, then sunlight striking the plate will cause considerably higher flux density data than the real conduction through the soil. Burying the flux plate within 0.01 to 0.02 m of the surface can lead to errors if the soil cracks and lets sunlight strike the plate, rainfall or irrigation water drain onto the plate, or condensation forms on the plate surfaces. Generally, it is best to bury heat flux plates between 0.04 and 0.08 m deep and correct for soil heat storage above the plates to avoid these problems.

Soil heat flux density at the surface (G = G₁) is estimated using:

where G₂ is the heat flux plate measurement (W m^-2) at depth Dz (m) in the soil, C_V is the volumetric heat capacity of the soil (J m^-3 °C^-1), T_sf and T_si are the mean temperatures (K or °C) of the soil layer between the flux plate level and the soil surface at the final (t_f) and initial (t_i) time (s) of sampling (e.g. t_f - t_i = 1800 s for a 30 minute period). Typically, a set of two to four thermocouples wired in parallel are used to measure a weighted mean temperature of the soil layer above the heat flux plates at the beginning and end of the sampling period to calculate the right-hand term of Equation 3.12. Based on de Vries (1963), a formula to estimate C_V (J m^-3 °C^-1) is:

where V_m, V_o and q are the volume fractions of minerals, organic matter and water, respectively (Jensen, Burman and Allen, 1990).

Thermal diffusivity (k_T) of the soil is the ratio of the thermal conductivity to the volumetric heat capacity:

This parameter is useful as a measure of how fast the temperature of a soil layer changes, so it is important when considering soil selection and management for frost protection. As a dry soil is wetted, K_s increases more rapidly than C_V, so k_T increases as the water content rises in a dry soil. However, as the soil pores begin to fill with water, the C_V increases more rapidly than K_s so k_T levels off near field capacity, and then decreases as the soil becomes saturated. The optimal heat transfer occurs at the peak k_T value, so one goal for frost protection is to maintain the water content of the surface soil layer at near field capacity to maximize k_T. For both sandy and clay soils, dry soils should be avoided and there is no advantage to have saturated clay soil (Figure 3.12). For soils ranging between clay and sand, water contents near field capacity generally have the highest6 value. Highly organic (peat) soils generally have a low thermal diffusivity regardless of the soil water content (Figure 3.12). Therefore, for frost protection, peat soils should be avoided when selecting a site for a new crop.

In addition to energy conduction into and out of the soil, there is also conduction into and out of plant materials (e.g. tree trunks, large fruit). Relative to soil heat flux density, energy storage in the plant tissues are small, but it may be important in some instances. For example, heat storage in citrus fruit causes the fruit skin temperature to fall slower and not as far as the air temperature. This requires consideration when determining when to protect citrus orchards.

FIGURE 3.12
Sample thermal diffusivities for sand, clay and peat (organic) soils as a function of volumetric water content (modified from Monteith and Unsworth, 1990)

Radiation

Electromagnetic radiation is energy transfer resulting from oscillation of electric and magnetic fields. A good example is sunlight or solar radiation, which transfers huge amounts of energy to the Earth's surface. Most of the distance between the Sun and Earth is a vacuum (i.e. empty space), so one property of radiation is that the heat transfer occurs even through a vacuum. Although much cooler, objects on Earth also radiate energy to their surroundings, but the energy content of the radiation is considerably less. The energy radiated from an object is a function of the fourth power of the absolute temperature:

where e is the emissivity (i.e. the fraction of maximum possible energy emitted at a particular temperature); s = 5.67 × 10^-8 W m^-2 K^-4, the Stefan-Boltzmann constant; and T_K is the absolute temperature (T_K= T_a + 273.15). Assuming that e = 1.0, the radiation flux density from the surface of the sun at 6000 K is about 73,483,200 W m^-2, whereas radiation from the surface of the Earth at about 288 K is approximately 390 W m^-2. However, because irradiance (i.e. radiation flux density in W m^-2) that is received by a surface decreases with the square of the distance from the Sun and the mean distance between the Earth and Sun is about 150 660 000 km, the solar energy has reduced to about the solar constant (G_sc = 1367 W m^-2) by the time it reaches the upper atmosphere of the Earth. As the radiation passes through the atmosphere, some is reflected and some is absorbed, so, on a clear day, only about 75 percent of solar radiation reaches the surface. Because the earth receives solar energy on a surface area (pr²) of a disk perpendicular to the sun's rays with a radius (r) the same as the earth but it emits from a surface area of a sphere (4pr²), the input and output of radiant energy are in balance and the Earth's temperature is relatively stable.

Radiant energy can be described in terms of wavelength of the radiation. Bodies with higher temperature emit shorter wavelengths of the electromagnetic energy. Energy emitted by a perfect emitter at 6000 K falls within the range of 0.15 to 4.0 mm, where 1.0 mm = 1.0 × 10^-6 m. Much of the high-energy (short wavelength) radiation is absorbed or reflected as it passes through the atmosphere, so solar radiation received at the Earth's surface mostly falls in wavelength range between 0.3 to 4.0 mm. The wavelength of maximum emission (l_max) is calculated using Wein's displacement law as:

where T_K is the absolute temperature of the emitting object. For the Sun at 6000 K, the l_max is about 0.48 mm. Most thermal (i.e. terrestrial) radiation from objects at Earth temperatures falls in the range between 3.0 and 100 mm, with a peak at about 10 mm for a mean temperature T_K » 288 K. There is overlap between 3.0 and 4.0 mm for the solar and terrestrial radiation, but the energy emitted in that range is small for both spectral distributions. Therefore, energy from the Sun is called short-wave (i.e. short-wave band) and that from the Earth is called long-wave (i.e. long-wave band) radiation. The two bands have insignificant overlap.

The net short-wave radiation (R_Sn) is calculated as:

where R_Sd and R_Su are the downward (positive) and upward (negative) short-wave radiation flux densities, respectively. Since the Earth is too cold to emit significant energy as short-wave radiation, R_Su comprises only reflected short-wave radiation. The fraction of short-wave radiation that is reflected from a surface is called the albedo (a), so the upward short-wave radiation is expressed as:

Therefore, the net short-wave radiation (i.e. the amount absorbed at the surface) can be expressed as:

Vegetated surfaces typically absorb most of the long-wave downward radiation that strikes them. However, a minute fraction is reflected back to the sky. The surface also emits long-wave radiation according to the fourth power of its absolute temperature. The net long-wave radiation is the balance between gains and losses of radiation to and from the surface as given by:

where the downward long-wave radiation (R_Ld) is a gain (i.e. a positive number) and the upward long-wave radiation (R_Lu) is a loss (i.e. a negative number). The apparent sky temperature is much colder than the surface, so downward long-wave radiation is less than upward long-wave radiation and net long-wave radiation is negative.

Downward radiation R_Ld is the energy emitted at the apparent sky temperature, which varies mainly as a function of cloudiness. Since the surface temperature and apparent sky temperature are usually unknown, many equations have been developed to estimate R_Ln as a function of the standard screen temperature T_a (°C).

The following equation for R_Ln gives good daytime estimates:

where f is a function to account for daytime cloudiness (Wright and Jensen, 1972):

where R_Sd is measured total solar radiation and R_So is the clear-sky solar radiation. The minimum is f = 0.055 for complete cloud cover (i.e. R_Sd/R_So = 0.3) and the maximum is f £ 1.0 for completely clear skies (Allen et al., 1998). In Equation 3.21, T_K = T_a + 273.15 is the absolute temperature (K) corresponding to T_a (i.e. the temperature measured in a standard shelter). The apparent net emissivity (e_o) between the surface and the sky is estimated using a formula based on Brunt (1932) and using coefficients from Doorenbos and Pruitt (1977):

where e_d is the actual vapour pressure (kPa) measured in a standard weather shelter. There is no known method to accurately estimate f during night-time; however, skies are commonly clear during radiation frost nights, so R_Ln can be estimated using Equations 3.21 and 3.23 with f = 1.0.

Depending on the temperature and humidity, R_Ln on a radiation frost night typically varies between -73 and -95 W m^-2 (Table 3.3). When skies are completely overcast, R_Ln depends on the cloud base temperature; but R_n = -10 W m^-2 is expected for low, stratus-type clouds. Therefore, depending on cloud cover, -95 W m^-2 < R_Ln < -10 W m^-2, with a typical value around -80 W m^-2 for a clear frost night.

Figure 3.13 shows an example of changes in net radiation, soil heat flux density and air temperature that are typical of spring-time in a California mountain valley. During the day the peak R_n » 500 W m^-2 and during the night net radiation fell to about -80 W m^-2. It increased after 0200 h as cloud cover slowly increased. Note that the night-time temperature starts to drop rapidly at sunset, which was shortly after R_n became negative. Starting at about two hours after sunset, the rate of temperature decrease remained fairly constant until the cloud cover increased and caused an increase in the temperature.

TABLE 3.3
Net long-wave radiation (W m^-2) for a range of air (T_a) and subzero dew-point (T_d) temperatures (°C) and saturation vapour pressure at the dew-point temperature (e_d) in kPa. The R_Ln values were calculated using Equations 3.21 and 3.23, and assuming f = 1.0

FIGURE 3.13
Net radiation (R_n), soil heat flux density (G), air temperature (T_a) at 1.5 m, and dew-point temperature (T_d) at 1.5 m in a walnut orchard with a partial grass and weed cover crop in Indian Valley, California, USA (latitude 39°N) on 14-15 March 2001

Latent heat flux

When water vapour condenses or freezes, latent is changed to sensible heat and the temperature of air and other matter in contact with the liquid or solid water will temporarily rise. Latent heat is chemical energy stored in the bonds that join water molecules together and sensible heat is heat you measure with a thermometer. When latent heat is changed to sensible heat, the air temperature rises. When ice melts or water evaporates, sensible heat is changed to latent heat and the air temperature falls. Table 3.4 shows the amount of heat consumed or released per unit mass for each of the processes. When the energy exchange is positive, then sensible heat content increases and the temperature goes up. The temperature goes down when the energy exchange is negative.

Subzero temperatures can lead to the formation of ice crystals on plant surfaces. For water vapour to condense as dew or ice to deposit onto surfaces as frost, the air in contact first becomes saturated (i.e. reaches 100 percent relative humidity). With a further drop in temperature, water vapour will either condense or deposit onto the surface. These are both exothermic reactions, so latent heat is converted to sensible heat during the condensation or deposition process and the released heat will slow the temperature drop.

TABLE 3.4
Energy exchange of water due to cooling, heating and phase changes

NOTE: Positive signs indicate release of sensible heat and negative signs indicate removal of sensible heat.

Water vapour flux density (E) is the flux of water molecules per unit time per unit area (i.e. kg s^-1 m^-2). When multiplied by the latent heat of vaporization (L » 2.501 × 10⁶ J kg^-1 at 0 °C), the water vapour flux density is expressed in energy units (i.e. W m^-2). Evaporation is important for all frost protection methods involving the use of water. The ratio of the latent heat of vaporization to the latent heat of fusion is 7.5, so considerably more water must be frozen than is vaporized to have a net gain of energy when using sprinklers for frost protection.

It is common for fruit growers to experience problems with spots of damage on the skin of fruit. While this may not damage the fruit to the point where it is completely lost, the spot damage reduces the value of fruit for table consumption. This problem is probably due to water droplets being on the fruit before going into a night with subzero air temperature. For example, if a light rain, fog or irrigation occurs during the day so that the fruit is covered by spots of water, this water will evaporate during the night and the fruit flesh near water droplets can cool as low as the wet-bulb or frost-bulb temperature, which is lower than the air temperature. As a result, damage can occur where there were water droplets on the fruit. If the dew-point temperature is low, damage can occur to sensitive crops, even if the air temperature remains above 0 °C.

Additional resources on energy balance

Readers who want more rigorous and detailed information on energy balance as it relates to frost protection are referred to Rossi et al. (2002), Barfield and Gerber (1979) and Kalma et al. (1992).