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17 Introduction; Atmospheric Evaporativity
17.1 INTRODUCTION
Relevance "On a continental scale approximately 75% of the total annual precipitation is returned to the atmosphere by evaporation and transpiration and in many climatic regions the annual evaporative demand exceeds precipitation. For example, throughout a large part of the semi-arid prairie region of central Canada, annual free-surface evaporation is on average about double annual precipitation, 750-1000 mm compared to 350-500 mm, and in many years aveage montly summer evaporation may exceed rainfall by a factor of five or more. In this region and other climatically-similar zones water lost to evaporation and transpiration is a major factor, affecting agricultural production, water resource management, wildlife habitat and the planning and design of hydroelecric power and water supply facilities." (Gray, 1993)
Process "Evaporation involves the change in state of a liquid to a vapour. The process occurs when water molecules, which are in constant motion, possess sufficient energy to overcome the surface tension at the liquid surface and escape into the atmosphere. Concurrently, some of the water molecues in the atmosphere, which are also in motion, penetrate the water surface and are retained by the liquid. It is the net exchange of water molecules between liquid and atmosphere per unit area per unit time that establishes the evaporation rate." (Gray, 1993). The rate of evaporation is directly proportional to temperature, wind speed, solar radiation, and inversely proportional to relative humidity and is dependent upon the supply of water to the evaporating surface. The proper quantification of evaporation must thus consider these factors.
17.2 PENMAN'S COMBINATION FORMULA
Penman's method is a combination of the aerodynamic and energy budget methods. The original equation was just for evaporation from a free water surface rather than actual evapotranspiration. The equation has been developed further by others to allow for determination of actual evapotranspiration. Penman's equations remove some of the limitations of the previous equations, while maintaining a sound physical basis. All measurements may be made at standard weather stations and only one height is needed (2 m). The method retains considerable accuracy as long as the parameters used are measured and not estimated.
The rate evaporation of water from a surface is dependent upon the temperature of the surface, the temperature of the air it is evaporating into and the relative humidity, RH. The evaporation of water is a phase change which requires energy to move from the liquid state into the vapour state. The energy is taken from the surface the water is resting upon (whether it be soil, a plant leaf, or your skin), also from incoming solar radiation, and from the air temperature itself. Overall the effect of water evaporation is a loss of heat from the surface it rests upon (i.e., the cooling effect received when one leaves a swimming pool). The evaporation of 1 kilogram of water would utilize 2.45 MJ of energy at 20°C. This is approximately equilivalent to the amount of solar energy received in one day upon an horizontal surface 0.1 m2.
Penmans: Actual Evapotranspiration
The following equation is a form of Penman's equation and allows for estimation of water loss from a crop or soil surface in which evaporation is limited by physical properties of the plant or the soil. It enables estimation given the measurement of vapour pressure, wind, and temperature at only one level, usually at 2 m above the ground.
[17.1]
This equation describes the energy used in evapotranspiration (Qe) in terms of environmental parameters (energy budget, vapor density, temperature) and diffusion resistances of both heat (rH ) and vapour (rv) providing the ability for stomatal resistance to control transpiration. The components of the equation are:
Qe latent heat flux density (W m-2)
D slope of saturation vapor density curve at Tm = (TL + Ta)/2 and e*m = (e*a + e*L)/2; TL and Ta are leaf and air temperatures; e*a and e*L are the saturation vapor densities at air and leaf temperature. If the leaf temperature is lacking, D can be evaluated at Ta with minimal loss of accuracy.
D = 5311 e*m/(Tm)2 [17.1a]
Where e*m is in pascals and Tm is expressed in Kelvin.
At 20°C D is 144.5 Pa/K
Qg soil heat flux density per unit area (W m-2)
Qn net radiant flux density per unit area (W m-2)
ra density of air (1.204 kg m-3 at 20°C)
cp specific heat of air (1.01 kJ kg-1 °C-1)
rH the resistance to heat transfer for a flat surface with forced or free convection. Two methods can be used to calculate this; both of which are semi-physically based. The first method considers the interplay of inertial, viscous and buoyant forces in whether heat transfer is via forced or free convection while the other method uses wind and temperature profile measurements which already account for whether convection is free or forced.
For conditions of air temperature at 20°C; a surface temperature of 30°C; a wind speed of 5 m/s and a leaf width of 2 cm a value of about 15 s/m for rH can be approximated.
g* apparent psychrometer constant:
g rv/rH [17.1b]
g is the thermodynamic psychrometric constant and is an expression of:
[17.1c]
where pa is atmospheric pressure. At 20°C and 101 kPa g = 66 Pa °C-1
rv is the resistance to vapor flow from the surface of a leaf that considers the resistance of the epidermis, rvs, (stomate and cuticle) and the boundary layer resistance, rva . For further information refer to Appendix: rv Method.
For relatively free transpiration from a leaf in which the stomates are open an rvs of 100 s m-1 is an reasonable approximation. For closed stomate conditions an rvs of 3200 s m-1 can be used.
19 AIR AND WATER VAPOUR PRESSURE
19.1 PRINCIPLES
Gases are often quantified by their temperature and pressure which can be related to the mass and velocity of its molecules as described by the Kinetic Theory of Gases, which are based upon Newton's Laws of Motion. When a force is applied to a body, its momentum, the product of mass and velocity, changes at a rate proportional to the magnitude of the force. The pressure, p, which a gas exerts on the surface of a liquid or solid is a measure of the rate at which momentum is transferred to the surface from the kinetic motion of the molecules, which strike it and rebound.
Gas law: by assuming that the kinetic energy of all the molecules in an enclosed space is constant and assuming a perfect gas the following relationship will hold:
V = nRT/p [19.1]
where
V is volume of the gas (m3)
R is 8.314 J mol-1 K-1, called the molar gas constant.
T temperature in Kelvin (K); n is number of moles of gas; p is pressure (kPa) of the gas
Thus, one mole of gas at STP (standard pressure, 101.3 kPa, and temperature, 273.2 K) is 0.0224 m3.
Example 19.1 Find the volume of one mole of 'air' at STP and at 20°C (293K)Solution: In SI units, p = 101.3 kPa, T = 273.2 K and using [Eq 19.1]
V = nRT/p = 1 mol (8.314 J mol-1 K-1) [273.2 K/(1.013 x 105Pa)]
V = 0.02242 J/Pa = 0.0224 m3 at 273.2 K
V (293.2 K) = 0.0241 m3.
Partial pressures Total atmospheric pressure is the sum of the pressure of each individual gas, which is a function of its molar concentration and temperature. This includes water vapour. Total atmospheric pressure may be calculated by summing the partial pressure of each gas:
pa = n1RT/V + n2RT/V + n3RT/V [19.2]
where pa is total gas pressure; n1, n2, n3 are moles of different individucal gases. If gases listed in Table 19.1 are substituted the standard atmospheric pressure of 101.3 kPa can be calculated.
Table 19.1. Composition of dry air (Monteith and Unsworth, 1990)
Gas / Molecular weight (g) / Density at STP(kg m-3) / Per cent by volume / Concentration
(kg m-3)
Nitrogen / 28.01 / 1.250 / 78.09 / 0.975
Oxygen / 32.00 / 1.429 / 20.95 / 0.300
Argon / 38.98 / 1.783 / 0.93 / 0.016
CO2 / 44.01 / 1.977 / 0.03 / 0.001
Air / 29.00 / 1.292 / 100.00 / 1.292
19.2 ATMOSPHERIC MOISTURE
Saturated vapour pressure (e*a) is the highest concentration of water vapor that can exist in equilibrium with a flat, free water surface at a given temperature (Fig. 19.1).
If a container of pure water is uncovered in a closed space water will evaporate and the amount of water vapour in the gas phase will increase in concentration until an equilibrium between the number of water molecules in the gas phase being captured by the liquid is equal to the number leaving. An increase in temperature increases the random kinetic energy of the molecules resulting in more molecules escaping the liquid, thus increasing the saturation vapor pressure.
Saturated vapour pressure (e*a) may be related to ambient air temperature (Ta) by:
For T ³ 0°C (note that really is 237.3)
, or [19.3]
For T < 0°C (note that really is 273.2)
[19.4]
in which e*a is in Pa when T is in °C.
Non-saturated vapour pressure (ea, Fig. 19.1) also known as ambient vapour pressure is that which occurs at ambient conditions.
Relative Humidity (RH; Fig. 19.1) is the ratio of ambient vapor pressure (ea) to the saturation vapor pressure at the ambient air temperature (e*a). Relative humidity is sometimes multiplied by 100 to express it as a percent rather than as a fraction.
[19.5]
Absolute humidity (AH, rv) is the vapour density per cubic meter of air. The density of water vapour per unit volume of air may be expressed as a function of the vapour pressure by the perfect gas law:
rv = 2.164 ea /(Ta + 273.2) [19.6]
Vapor pressure (ea) is in pascals when vapor density, rv, is in g/m3 and Ta is in °C.
Specific Humidity (q) is the mass of water vapour per unit mass of moist air (kg/kg).
Dew Point Temperature (Td, Fig. 19.1) is the temperature at which air with vapour pressure ea when cooled without changing its water content, just saturates:
[19.7]
where Ta is ambient air temperature (K)
RH is relative humidity
A is 5311/Ta
Td is in °C
Wet Bulb Temperature (Tw, Fig. 19.1) is a measure of the maximum cooling effect of evaporation, i.e., the temperature drop achieved by adiabatic evaporation of water into unsaturated air. If the air was saturated than no water would be evaporated from the surface of the wet bulb and the temperature would be the same; thus if RH = 1.0 than T = Tw = Td; otherwise T > Tw > Td.
Psychrometric 'Constant' (g) relates the heat capacity of the atmosphere to the energy used in the evaporation of water. The evaporation of water requires heat. Air is cooled by evaporating water into it and at the same time the vapour density of the air increases as the water evaporates. The heat content of the air thus changes by the amount of the temperature drop. This must equal the latent heat of evaporation for the amount of water evaporated into it. This relationship between air heat content and the latent heat of evaporation may be expressed as:
Heat lost by air = Heat used in evaporation
racp (Ta - Tw) = 0.622 ra hv[e*w - ea]/pa [19.8]
where hv is the latent heat of vaporizaton for water, 0.622 is the ratio of the molecular weights of water vapour and air, ra is the density of air, pa is standard air pressure, cp is its specific heat (1.01 kJ kg-1 °C-1), and e*w is the saturation vapor pressure at Tw.
Equation 19.8 can be rearranged to give the psychrometric equation:
[19.9]
with g defined as a thermodynamic psychrometric "constant". The value for g depends upon temperature and atmospheric pressure and thus cannot be strictly viewed as a constant. At standard atmospheric pressure, 101.3 kPa, g=66 Pa K-1 at 0°C increasing approximately linearly to 67 Pa K-1 at 20°C. Equation 19.9 can be arranged to solve for ea which defines the family of straight diagonal lines in Fig. 19.1:
ea = e*w - g(Ta - Tw) [19.10]
Fig. 19.1. Temperature-vapour pressure-relative humidity diagram for atmospheric pressure of 101.3 kPa. The inset is for temperatures below )°C. Diagonal lines are for wet bulb temperature and are spaced at 2°C increments.
Example 19.2 Given an air temperature of 30°C and an RH of 0.5 find e*a, ea, Td, and Tw using Fig. 19.1Solutions:
e*a (Saturated vapour pressure) is found from the intersection of 30°C (vertical line) with the RH=1.0 curve and obtaining the pressure from following a horizontal line to the y-axis:
e*a (30°C) = 4,250 Pa
ea (ambiant vapour pressure) is found from the intersection of 30°C (vertical line) with the RH=0.5 curve and obtaining the pressure from following a horizontal line to the y-axis:
ea (30°C, 0.5) = 2100 Pa
Td (Dew point temperature) is found from the intersection of 30°C (vertical line) with the RH=0.5 curve, following a line horizontal from this intersection to where it intersects with the RH=1.0 curve and then reading the temperature from the x-axis:
Td (30°C, 0.5) = 18°C
Tw (Wet bulb temperature) is found from the intersection of 30°C (vertical line) with the RH=0.5 curve, following a diagonal line horizontal from this intersection to where it intersects with the RH=1.0 curve and then reading the temperature from the x-axis:
Tw (30°C, 0.5) = 22°C
The Psychrometer
The psychrometer consists of two thermometers placed in the air side by side. One is an ordinary thermometer and measures the air temperature and is known as the dry bulb. The other is coverd with a thin wet cloth or with a continuous film of water and as such is known as the wet bulb. The drier the air the greater the evaporation and thus the greater the heat loss from the water and the more the depression of temperature of the wet bulb. If the vapour pressure in the air is saturated then the wet and dry bulbs would read the same temperature.