1
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.
2
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.
Qe =
(Qn - Qg ) + [acp (e*a - ea) / rH]
 + *
[17.1]
This equation describes the energy used in evapotranspiration (Q e) 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)

slope of saturation vapor density curve at T m = (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,  can be evaluated at Ta with minimal loss of accuracy.
 = 5311 e*m/(Tm)2
[17.1a]
Qg
Qn
a
cp
rH
*

Where e*m is in pascals and Tm is expressed in Kelvin.
At 20°C  is 144.5 Pa/K
soil heat flux density per unit area (W m-2)
net radiant flux density per unit area (W m-2)
density of air (1.204 kg m-3 at 20°C) 
specific heat of air (1.01 kJ kg-1 °C-1)
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.
apparent psychrometer constant:
rv/rH
[17.1b]
is the thermodynamic psychrometric constant and is an expression of:
e*w - ea = cp pa = 
Ta - Tw 0.622 hv
[17.1c]
where pa is atmospheric pressure. At 20°C and 101 kPa  = 66 Pa °C-1
rv
is the resistance to vapor flow from the surface of a leaf that considers the resistance of the epidermis, r vs,
(stomate and cuticle) and the boundary layer resistance, rva . For further information refer to Appendix: r v
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 r vs of 3200 s m-1 can be used.
3
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 m 3.
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)
Nitrogen
Oxygen
Argon
CO2
Air
28.01
32.00
38.98
44.01
29.00
Density at STP
(kg m-3)
1.250
1.429
1.783
1.977
1.292
Per cent by
volume
78.09
20.95
0.93
0.03
100.00
Concentration
(kg m-3)
0.975
0.300
0.016
0.001
1.292
4
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  0°C (note that really is 237.3)
e*a = 611 exp
17.27 Ta
Ta + 237.3
,
or
[19.3]
For T < 0°C (note that really is 273.2)
e*a = 611 exp
19.59 Ta
Ta + 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.
ea
[19.5]
RH =
e*a
Absolute humidity (AH, v) 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:
v = 2.164 ea /(Ta + 273.2)
Vapor pressure (ea) is in pascals when vapor density, v, 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).
[19.6]
5
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:
Td =
Ta
- 273.2
1 - (lnRH)/A
[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' () 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
acp (Ta - Tw) = 0.622 a 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, a 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:
e*w - ea = cp pa = 
Ta - Tw 0.622 hv
[19.9]
with  defined as a thermodynamic psychrometric "constant". The value for  depends upon
temperature and atmospheric pressure and thus cannot be strictly viewed as a constant. At
standard atmospheric pressure, 101.3 kPa,  = 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 - (Ta - Tw)
[19.10]
400
0. 6
-10
300
7000
200
0. 4
-20
0. 2
100
0
0
-10
0.4
-20
30
5000
te
m
pe
ra
tu
r
e(
°C
)
0.3
4000
W
et
bu
lb
3000
0.1
10
2000
0.2
20
1000
0
0
10
20
30
Air temperature (°C)
40
50
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.
Relative humidity
6000
Vapour pressure (Pa)
0.7
0.
0.5
8000
8
500
0.6
600
40
9000
1. 0
10000
0.8
1.0
0.9
6
7
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. 19.1
Solutions:
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 xaxis:
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.
From reading both the wet and dry bulbs simultaneously it is possible to calculate the
amount of water vapour in the air:
ea = e*a - 0.066 pa (Ta - TW) (1 + 0.00115TW)
where Ta and TW are in °C.
Given that the two bulbs are placed in the same air stream with a wind of at least 3 m s-1,
than the wet bulb will achieve maximum depression. The sling psychrometer is twirled by hand
about a pivot to achieve this. The temperature difference, Ta - TW, is called the wet bulb
depression. (Gray 1991).
8
20 Wind and Aerodynamic Expressions
20.1 INTRODUCTION
Wind has numerous important effects within nature: it exerts a force (momentum, ); it transports
energy (heat, H); water vapour (E) and other matter (eg. soil, water, pollutants); and it effectively
mixes various atmospheric layers. Without wind, also known as convective currents (in both the
horizontal and vertical sense), heat from the exchange surface, water vapor from an evaporating
surface, or CO2 from a transpiring plant could only be transported away by the slow process of
diffusion.
To determine the effect of wind upon heat transfer away from exchange surfaces (i.e., soil and
plant surfaces that intercept solar radiation) it is necessary to know the wind speed in the vicinity
of the organism. This requires knowledge about the behavior of wind near solid surfaces and
knowledge of turbulent mixing. Turbulent transfer theory allows the derivation of equations for
wind, temperature, vapor density, and CO2 profiles and fluxes, thus enabling further calculations
concerning water evaporation and plant transpiration.
20.2 LAMINAR AND TURBULENT FLOW (GRAY, 1991)
"When air passes over a natural surface an atmospheric boundary layer develops starting with the
formation of a layer of laminar flow at the leading edge followed by the development of
transition and turbulent flow zones (Fig. 20.1). In fully-developed turbulent flow the lower
atmosphere may be divided into three distinct layers: the laminar layer (nearest to the surface),
the turbulent layer and the outer layer of frictional influence. The laminar layer, in which flow
is laminar, is only a few millimetres in thickness. In this layer temperature, humidity and wind
speed vary almost linearly with height and the transfer of momentum, heat, and water vapor are
essentially molecular processes. Conversely, the overlying turbulent layer can be several metres
thick depending on the level of turbulence. In this layer, temperature, humidity and wind
velocity tend to vary linearly with the logarithm of height, and the transfer of momentum, heat
and vapour are turbulent processes, which are analogous to the process of eddy diffusion.
Wind has a large random component to it, in both time and space. The description of
fluctuations, or eddies, in the atmosphere can approached as if they were molecules in gas; they
bounce about with random motion but in the general direction of the mean gradient, here taken as
the wind. The fluctuations are responsible for the transport of heat, gas, and moisture within the
atmosphere and it is possible to determine the mean movement of numerous fluctuations as with
the diffusion process in a gas.
9
Uniform air
stream
Laminar boundary
layer
Turbulent boundary
layer
z
height
u (z)
wind velocity
Onset of turbulence
Fig. 20.1. Development of laminar and turbulent boundary layers over a smooth flat plate
(after Monteith and Unsworth, 1990)
20.3 AERODYNAMIC EXPRESSIONS (CAMPBELL 1977 AND GRAY 1991)
Vertical flux of momentum, heat, and water vapour from a surface with a horizontal wind
component and that is being heated resulting in convective currents may be represented by the
following equations:
Flux of momentum (Q, kg s-2 m-1),
Q = - K M a du
dz
[20.1]
where KM is the eddy viscosity (m2/s)
a is density of air (1.2 kg/m3)
u is wind velocity as a function of height (m/s)
Heat flux, (QH, J s-1 m-2),
QH = - K H a cp dT
dz
[20.2]
(m2/s)
where KH is the eddy thermal diffusivity
cp is mass heat content of air (1.01 kJ kg-1 °C-1)
T is air temperature as a function of height (°C)
Vapour flux (Qe, kg s-1 m-2)
QE = - K e
dv
dz
[20.3]
(m2/s)
where Ke is the eddy vapour diffusivity
v is density of water vapour in air as a function of height (kg/m3)
and the overbars indicate averages taken over a 15-30 minute time interval
10
These above equations are for steady-state flux for turbulent conditions, where the transport
coefficients, K, are analogous to molecular diffusion. If flux is assumed to be constant with
height then K will proportionally change with height above the surface, with wind speed, surface
roughness, and heating at the surface. At steady-state, we assume the flux densities, Q, QH, and
Qe, to be independent of height.
Transport coefficients (K, are analogous to diffusion coefficients, m2/s) can be used to calculate
vertical flux of momentum, heat, and vapour within atmospheric conditions. Although the
mechanism for turbulent transport is different than that of molecular transport the approach
mathematically is similar. Increases in the K coefficients with z must therefore be balanced by
corresponding decreases in gradients in order to maintain the equations as steady-state. The K
coefficients may be estimated from knowledge of certain aerodynamic properties and may be
assumed to increase linearly with u* and z:
KM = ku*(z + zM - d)
KH = ku*(z + zH - d)
Ke = ku*(z + ze - d)
[20.4]
[20.5]
[20.6]
where
u* is the friction velocity and has units of m/s and is equivalent to (a)1/2;
k is von Karman's constant, generally taken as 0.4;
d is the exchange surface; and
zM, zH, and ze are roughness parameters for momentum, heat, and water vapor.
If Equation 20.4 is substituted into Equation 20.1 and the resulting equation integrated from the
height of the exchange surface, d, to some height, (z + zM - d), the resulting equation describes u
as a function of height:
u = u ln z + zzM - d
M
k
*
[20.7]
The other profile equations for temperature (Eq 20.11) and water vapor (Eq. 20.12) can similarly
be obtained.
Exchange surface or zero plane displacement (d, m) can be thought of as the distance from the
arbitrarily chosen height, zero, to the average height of heat, vapor, or momentum exchange. This
occurs where the distribution of shearing stress over the ground is aerodynamically equivalent to
the imposition of the entire stress (it may also be described as a 'centre' of pressure). For a
smooth surface, with z measured from the surface, d = 0. For dense vegetation (agricultural
crops), d can be estimated from the average crop height, h:
d = 0.64h
If roughness elements are more sparsely spaced, Equation 20.8 does not hold.
[20.8]
11
Momentum roughness parameter (zM, m) is a length characteristic of the form drag at the
momentum exchange surface. It is dependent upon shape, height, and spacing of the roughness
elements and for uniform surfaces may be estimated by empirical correlations:
zM = 0.13 h
[20.9]
Heat and Vapour roughness parameters (zH, ze, zm) The roughness parameters for the other
profile equations can be generally expressed as functions of the momentum roughness parameter.
For simplification these can be set as:
zH = ze = 0.2 zM
[20.10]
The above relationship sufficiently describes most vegetated surfaces, but should not be used for
very smooth surfaces (ice, water, mud flats, etc.). Table 20.1 displays some roughness values
collected from the literature.
Table 20.1. Roughness height (summarized from Gray, 1991)
Surface
Smooth mud flats, ice
Large water surfaces (average)
Snow on prairie
Mown grass, 1.5 cm high
Mown grass, 4.5 cm
Grass with few scattered bushes
Long grass (60-70 cm)
Alfalfa (30-40 cm)
Wheat stubble (18 cm)
1-2 m high vegetation
Pine Forest
5m
27 m
Deciduous forest (17 m)
Roughness zM (cm)
0.001
0.005
0.01
0.2
2.0
4
6.1
1.3
2.4
20
65
300
270
20.4 AIR TEMPERATURE PROFILES (CAMPBELL, 1977)
Solar radiation is intercepted by the soil and the plant canopy where it is transformed into heat.
The surface which intercepts solar radiation, whether it be the bare ground or a plant canopy is
referred as the exchange surface. The heat is transferred away from the surface by conductance
into the soil and be convection into the air layers above. With increasing height and depth the
temperatures decrease and approach an overall average (Fig. 20.1).
The theory of turbulent transport specifies the shape of the temperature profile over a uniform
surface with steady-state conditions. The temperature profile may be described by the following
equation:
12
T = To -
H
ln z + zzH - d
*
H
a cp ku
[20.11]
where T is the mean air temperature at height z, To is the temperature at the exchange surface
(where z = d), d is the zero-plane displacement height, zH is a roughness parameter for heat
transfer, H is the sensible heat flux from the surface to the air, a and cp are air density and the
specific heat of air (acp = 1200 J m-3 K-1), k is the von Karman constant (0.4), and u* is the
friction velocity, a windspeed and surface roughness parameter. This equation is useful for
predicting temperatures at the exchange surface or higher up in the atmosphere given several air
temperatures at known heights. The log plot of Fig. 13.1 produces a straight line which can be
extrapolated to ln zH to determine the mean surface temperature.
2
6
1
5
0
ln (z - d + zH)
7
Height (m)
4
-1
3
-2
2
-3
1
-4
0
zH
-5
To
29 30 31 32 33 34 35
Temperature (°C)
Fig. 20.1. Typical daytime temperature profile plotted as a function of height (left) and logarithm of height (right).
The log plot shows the extrapolation of the measured profile to z-d = zH to determine the surface temperature.
26
27
28
29 30 31 32
Temperature (°C)
33
34
35 26
27
28
Some of the important things that this equation points out about the temperature profile are:
1. Near the surface the temperature profile is logarithmic;
2. Temperature increases with height when H is negative (heat flux toward the surface) and
decreases with height when H is positive. During the day, sensible heat flux is generally
away from the surface so T decreases with height.
3. The temperature gradient at a particular height increases in magnitude as the magnitude of H
increases, and decreases as wind or turbulence increase.
13
20.5 WIND PROFILES
The profile equation for wind (Eq. 20.7) is useful for extrapolating or interpolating to find wind
speeds at heights where they were not measured (Fig. 20.2). The actual wind profiles are curved
near the surface when plotted in conventional linear fashion, but when ln(z + zM - d) is plotted as
a function of wind velocity, a straight line results, from which measurements can be extrapolated
to other heights. At least two measurements are required, but of course accuracy increases if
more measurements are available.
2
4
1
(d
rop
c
ll
Ta
0
z M = 0.09 m
s(
d
=
0
m)
-1
-2
-3
rop
Ta ll c
1
Sh
or
tg
2
m)
ra
s
ln(z - d +zM)
Height (m)
3
.5
=0
ra ss
-4
S hort g
z M = 0.01 m
-5
0
0
2
4
6
Windspeed (m/s)
8
10
12
0
2
4
6
Windspeed (m/s)
8
10
12
Fig. 20.2. Wind profiles as a function of height (left) and as a function of logarithm of height (right) for short
grass and tall crop. The dashed lines on the log plot show the extrapolation of the measured wind profile to zero
wind speed to determine the momentum roughness parameter (adapted from Campbell, 1977).
If the heights at which the wind is measured are greater than the values for zM, then ln(z - d) can
be plotted rather than ln(z - d + zM). The plots of the measured points can be extrapolated to u =
0 to give an experimentally determined estimate of zM. For a rough estimate Eqs 20.8 and 20.9
can be used to estimate zM and d, and to construct a profile from a single wind measurement.
As an example, assume that the average wind at 2 m height is 3 m/s over a grass surface which has an average
height of 20 cm above the soil surface. We would like to know the average wind speed at a height of 1 m
where the air temperature recorder is located. Using Equations 20.12 and 20.15 d is calculated as 13 cm and
zM = 2.6 cm. Inserting these values along with z = 2.0 m into the ln term of Equation 20.7 we calculate a value
for the ln term of 4.29. The mean velocity of wind is known at 2 m (3 m/s) and this value is the u term of
Equation 13.7. As the k value is known it is possible to calculate u* at 0.28 m/s. Now all the terms on the
right side of the equation are known it is possible to calculate u at a height of 1.0 m which is 2.5 m/s.
14
20.6 MEASUREMENT OF WIND
Wind is air in motion; this motion is a vector quantity, a directed magnitude and as thus
needs to be properly expressed by two numbers, the direction and speed (Middleton and
Spilhaus, 1953). The direction of the wind is universally considered to be the direction from
which it is blowing. Wind velocity is a vector, as opposed to a scalar quantity and this is taken as
both speed and direction. In Canada and the United States the direction of the wind is normally
stated in terms of eight (i.e., N, NE, E, etc.), sixteen (i.e., N, NNE, NE, ENE, E, etc.), or thirtytwo compass points. The speed of the wind in Canada is presently indicated in meters per second
or kilometers per hour. Often day wind is expressed as wind run, the distance travelled by air in
motion, expressed in terms of kilometres. To obtain average speed, distance travelled by be
divided by time.
Wind Direction; Vanes
The oldest meteorological instrument is the wind vane. Basically, a wind vane is a body
mounted unsymmetrically about a vertical axis, on which it is free to turn; the end offering the
least resistance to the motion of wind points into the wind. There are many different types of
wind vanes, their design being related to sensitivity. Many have a dampening mechanism in built
so that they will take an average direction and not switching direction at the slightest turbulence.
Wind Speed; Cup anemometers
The original cup wheel consisted of four plain hemispherical cups with their diametral
planes vertical and arranged radially at equal angles about a vertical axis. Later three cups came
to be preferred with a conical design which increases the strength and performance of the
anemometer. The size of the anemometers has also decreased increasing the accuracy and
response time. The revolutions of the cup wheel are attached to a counting mechanism or an
electronic device for indicating instantaneous speed
Wind speed; Propeller or windmill anemometers
Propellor anemometers are rarely used. To properly operate they must turn into the wind,
but this also enables them to be both a speed and direction indicators.
Windmill anemometers are not used very much in meteorology any longer; however their
design makes them very useful for measurement of low speed air currents in building air ducts.
Their design enables a nearly linear relation between the speed of the wind and the angular
velocity of the windmill.
Wind speed; Thermal anemometers
Thermal anemometers make use of the cooling power of moving air upon a heated wire.
Hot-wire anemometers are mainly used for duct air flow measurements.
15
22 Radiation Energy Budget
22 SHORT AND LONG WAVE RADIATION
22.1 RADIANT ENERGY BUDGETS
Radiant energy flux at the earth's surface is a combination of incoming and outgoing short- and
longwave radiation. The net radiation for a surface is the algebraic sum of these streams of
energy and can be approximated by equations described in the appendices of this section or
measured with field instrumentation. For a flat horizontal surface at ground level, such as a soil
surface, the net radiant flux density (Qn, W m-2) is:
Qn = Qdrs + Qdfs - r (Qdrs + Qdfs) + Ql - Ql
[22.1]
Qn = Qsn (1-r) + Ql - Ql
[22.2]
or
where
Ql is the outgoing long-wave terrestrial radiation
Ql is the incoming long-wave atmospheric radiation;
Qdrs is the shortwave direct radiation
Qdfs is the shortwave diffuse radiation
r is the albedo of the surface.
For an horizontal object suspended above the ground surface, such as a leaf, the underside of the
leaf must be considered as it will receive reflected short wave radiation from the ground surface
and longwave radiation from the ground. As the leaf has two sides it will emit long wave
radiation from both sides.
The resulting radiant energy balance for a small, flat object
suspended horizontally above a surface is:
Qn = Qs + rgQs + Qld + Qlu - Qle- Qle - rlQs - rlrgQs
[22.3]
Qn = Qs(1+ rg)(1 - rl) + Qld + Qlu - Qle- Qle
[22.4]
or
where Qleand Qle is the long-wave emittance for each side of the object
Qld and Qlu are the long-wave irradiance flux received at the up- and down-facing
surfaces;
Qs is the global radiation = Qdrs + Qdfs
rg is the albedo of the ground surface below the object; and
rl is the albedo of the leaf
rl Qs
Qdfs
Ql
Sun
Qlu Qle
Qdfs
Sun
Qdrs
rQs
Qdrs
rg Qs
Ql
Qld
Soil Surface
Qle rl rg Qs
Soil Surface
Fig. 22.1. Radiant energy flux exchange for a soil surface and a leaf.
16
22 Radiation Energy Budget
Table 22.1. Radiation components at Saskatoon (52°N) and Hawaii (20°N).
Radiation
component
Shortwave
Direct (Qdrs)
Diffuse (Qdfs)
Reflected (Qr)
Longwave
Terrestrial (Ql)
Atmospheric (Ql)
Net
June, Ta 27°C, Ts 40°C
Black soil
Crop
r = 0.1
r = 0.2
(W m-2)
(W m-2)
Jan; Ta -20°C, Ts -15°C
Black soil
Fresh snow
r = 0.1
r = 0.85
(W m-2)
(W m-2)
Hawaii; Jan; Ta 35°C
Beach sand, Ts 40°C
r = 0.35
(W m-2)
972
103
-107
972
103
-215
184
83
-27
184
83
-227
810
101
-319
-529
393
-529
393
-244
144
-244
144
-529
458
831
724
140
-60
521
Negative numbers represent flux away from the earth's surface while positive is towards from the earth's surface.
Latitude of Hawaii is 20°N
Ta
Air temperature
Ts
Surface temperature
r
Albedo
REFERENCES USED IN WRITING OF THIS SECTION
Campbell, G.S. 1977. An Introduction to Environmental Biophysics. Springer-Verlag, New
York.
Gray, D.M. 1991. Handbook of Hydrology - 1991; Chapter 6, Evaporation. Division of
Hydrology, University of Saskatchewan.
Monteith, J.L. and M.H. Unsworth. 1990. Principles of Environmental Physics. Second Edition.
Edward Arnold, a division of Hodder & Stoughton. 291 pp.
Maidment, D. (ed.) 1993. Handbook of Hydrology. McGraw-Hill, Inc.
22 Radiant Energy Appendices
17
APPENDICES FOR RADIATION ENERGY BALANCE
22A. Basic Terminology
Short wave radiation is defined as that recieved from the sun. This radiation approximates that received from a
6000 K blackbody, is defined as the wave lengths of 0 to 4000 nm. The mean radiant flux density outside the earth's
atmospher and normal to the solar beam is about 1360 W m-2. The ultra violet range, <400 nm, accounts for 9.0 %
of the energy; the visible range, 400 - 700 nm, accounts for 39.8 %; the near infra-red, 700-1500 nm, for 38.8%, and
the far infra-red, >1500 nm, accounts for 12.4% of the solar energy.
Long wave radiation is that radiation emitted by the earth, as represented by a 288 K blackbody. This radiation has
wave lengths between 4 µm and 80 m. The average emittance at the earth's surface is 390 W m-2.
Thermal radiation or radiant energy; that type of radiation emitted by a body due to its temperature. If the
temperature of an object is above absolute zero (0°K or -273 °C) it radiates energy. The wavelength of the energy
emitted is controlled by temperature; thus the sun, a relatively hot object, radiates thermal energy in wavelengths
visible to our eyes; however the earth being much cooler radiates thermal energy at much longer wavelengths, within
the infra red zone.
Radiant energy is transferred by photons, discrete bundles of energy that travel at the speed of light:
c = 3 x 108 m/s
[22A.1]
The amount of radiant energy e for any specific wavelength may be described by an equation due to Planck:
e = hc/

[22A.2]
where h is Planck's constant (6.63 x 10-34 J s) and  is the wavelength. Thus green photons having a wavelength of
0.55 µ would have an energy e = 3.6 x 10-19 J. This is the amount of energy available for photochemical reactions
that use green light. For practical purposes the energy in a mole of photons (6.02 x 10 23) is used. A mole of photons
is called an Einstein (E). Thus the energy of a mole of photons at the green wavelength is 2.2 x 10 5 J E-1.
Black body is a body which absorbs all radiation falling on it. No material is a perfect black body, however some
materials approach this over parts of the electromagnetic spectrum. Snow is a very poor absorber of visible
radiation, but almost a perfect blackbody in the far infrared.
Stephan-Boltzmann law: describes the radiant energy emitted by a unit area of surface of a blackbody radiator.







QB =  T 4
[22A.3]
where

QB is the emitted flux density (W m-2),

 is the Stephan-Boltzmann constant (5.67 x 10-8 W m-2 K-4), and
T is the Kelvin temperature.
The earth can be considered as approximating a blackbody radiator emitting at 288K. The average emittance of the
Earth is therefore 390 W m-2. Using a temperature of 6000K for a blackbody approximating the sun, the energy
emitted at the sun's surface is 73 MW m-2.
Gray bodies; the energy emitted by nonblackbodies or gray bodies is given by:



 =  T4
[22A.4]
where is the emissivity of the surface.
For a blackbody  = 1. Most natural surfaces have long-wave emissivities between 0.90 and 0.98. The emissivity is
a function of wavelength, though it can often be treated as constant and equal to some average value for fairly large
wavebands.
18
22 Radiant Energy Appendices
Absorptivity: The fraction of incident radiation at a given wavelength that is absorbed by a material.
Emissivity: The fraction of blackbody emission at a given wavelength emitted by a surface.
Reflectivity: The fraction of incident radiation at a given wavelength reflected by a surface.
Transmissivity: The fraction of incident radiation at a given wavelength transmitted by a material.
Radiant flux: The amount of radiant energy emitted, transmitted, or received per unit time.
Radiant flux density: Radiant flux per unit area (W m-2).
Radiant emittance: The radiant flux density emitted by a surface.
Watt (W) = 1 J s-1 = 1 kg m2 s-3
Sun; 6000 °K at surface;
energy emitted at surface = 72 MW m-2
Top of earth's atmosphere;
solar energy received = 1360 W m-2
Scattering and diffuse reflection = 5%
Absorption by molecules and dust = 15%
Cloud reflection: 30-60%
For cloudy sky:
Clear
Sky
Reflection at exchange surface:
by full crop canopy = 20 %
by bare black soil, moist = 10%
Absorption in
clouds: 5% - 20%
0-45% reaches exchange surface
Correction for sun's angle (dependant upon time of day, season, and latitude)
For Saskatoon (52°N), June, at midday this is about = 80%
For Saskatoon (52°N), Jan., at midday this is about = 20%
Fig. 22A.1 Losses of incoming solar radiation by scattering, reflection, and absorption.
Lambert's cosine law: the radiant flux density received at a surface depends upon the orientation of the surface to
the radiant beam. Although the radiant flux of the beam itself is constant the amount received by the surface will
decrease as the surface is orientated away from the perpendicular to the beam. The beam covers a larger and larger
surface. Thus the amount of energy a north facing hill slope receives per unit area is smaller than that received by a
slope facing the sun. The variation of flux density with the angle is described quantitatively by Lambert's cosine law:
Q = Qo cos
[22A.5]
where



Q is the flux density normal to the beam,
Qo is the flux density at the surface, and
 is the zenith angle (between the light beam and a normal to the surface).
The sun's elevation angle () is more convenient to use than the zenith angle . As they are complimentary angles,
cos  = sin .
Bouguer's law: describes the attenuation of the flux density of a beam of radiation as it propagates through a
homogeneous medium, such as light penetration in the atmosphere, in crop canopies, in water, and in snow.



Q = Qo e-kx
where
Qo is the unattenuated flux density,
x is the distance the beam travels in the medium, and
[22A.6]
22 Radiant Energy Appendices
19
k is the extinction coefficient (m-1) for the medium.
The law applies only for wavebands narrow enough that k remains relatively constant over the waveband.
Plancks law: describes the radiant energy spectrum from a blackbody (Eq. 22A.7). Energy from a radiant source is
emitted in a spectrum of differing wavelengths. Photochemical reactions in biological systems, such as
photosynthesis, sight, and sunburn respond only to radiant energy in limited wavebands.
i B = 2 hc2 / [5 (exp (hc / (k T )) - 1]
[22A.7]
where
i B (W m-3) is the energy flux density per unit wavelength, or spectral emittance
k is Boltzmann constant (1.38 x 10-23 J/K)


 is the wavelength
h is Planck's constant (6.63 x 10-34 J s)
c is the speed of light (3 x 108 m s-1)
Wien's law: describes the relationship between the wavelength of peak emittance (on a wavelength basis) and
temperature:
m = 2897/T
[22A.8]
where  is in µ and T in Kelvins. Any given spectrum for a specific blackbody temperature as described by Planck's
law (Eq. A4.6) has a maximum spectral emittance (m) at some particular wavelength. For a 6000 K source, the
sun, this occurs at 0.48 µ and for a 288K source, the earth, this occurs at 10.06 µ.
20
22 Radiant Energy Appendices
B. Calculation of short wave radiation at the earth's surface
Total shortwave (Qs ); solar radiation upon entering the earth's atmosphere is absorbed, reflected, and transmitted via
particles and gases within the atmosphere. That absorbed may be emitted again in the long wavelength range. Thus
the total amount of short-wave radiation reaching the earth's surface will be less than that outside the earth's
atmosphere:
Qs = Qdrs + Qdfs
[22B.1]
where
Qs is total shortwave radiation received on a horizontal plane at the earth's surface,
Qdrs is direct short wave recieved on a horizontal plane, and
Qdfs is diffuse short wave recieved on a horizontal plane.
Direct shortwave (Qdrs ); the amount of direct shortwave reaching the earth's surface is dependent upon the
transmissivity of the atmosphere, the distance travelled in the atmosphere and the incident flux density. The
atmosphere absorbs and redirects. The following expression simply combines these factors:
Qdrs = am QA sin 
where
QA
a
m
sin 
[22B.2]
is the flux density normal to the solar beam just outside the earth's atmosphere (1360 W m-2)
is an atmospheric transmission coefficient,
is the optical airmass number, the ratio of slant-path elevation length through the atmosphere
to zenith path length, and
uses the sun's elevation angle from the horizon () to convert from radiance perpendicular to
the solar beam.
The sun's elevation angle  can be determined from:
sin  = sin sin  + cos cos  cos [15(t - to )]




[22B.3]
where all angles are in degrees.
 is the latitude,
 is the solar declination corresponding to the time of observation (Table 22B.1),
t is time of day in hours, and to is the time of solar noon.
Table 22B.1 Solar declination angles (in degrees) on the first day of each month
Jan
Feb
Mar
-23.1
-17.3
-8.0
Apr
May
June
+4.1
+14.8
+21.9
July
Aug
Sept
+23.2
+18.3
+8.6
Oct
Nov
Dec
-2.8
-14.1
-21.6
The optical air mass number, m, for elevation angles greater than 10 degrees may be representated by:
m = (pa/po )/ sin 
[22B.4]
The ratio pa/po is atmospheric pressure at the observation site divided by sea level atmospheric pressure.
The transmission coefficient (a ) varies around 0.9 for a very clear atmosphere, to around 0.6 for a hazy or smoggy
atmosphere. A typical value for clear days would be around 0.84.
21
22 Radiant Energy Appendices
Diffuse sky irradiance (Qdfs ) may be estimated by using half the difference between the irradiance on an horizontal
surface below and above the atmosphere. Thus,
Qdfs = 0.5 QA (1 - am ) sin 
When clouds obscure the sun,Qs = Qdfs, since there is no direct radiation component.
[22B.5]
The net solar radiation exchanged at the surface accounts for the solar radiation reflected:
Qsn = Qdrs + Qdfs - r(Qdrs + Qdfs) = Qs (1-r)
[22B.6]
where
Qsn is net solar radiation exchanged at a horizontal surface, and
r is the surface reflectivitly coefficient or albedo.
Ordinarily only the total incoming flux Qs (global radiation) is measured.
An albedo of 0 is equivalent to a black body that absorbs all short wave and reflects nothing in the short wave. An
albedo of 1 is for an object that reflects all incoming short wave.
Table 22B.2 Typical short-wave reflectivity (albedo) of soils and vegetation.
Surface
albedo
Open water
Swamp forest
Coniferous forest
Deciduous woodland
Lawn grass, dry
Lawn grass, wet
Spring wheat
Winter wheat
Winter rye
Cotton
Lettuce
0.05
0.16
0.16
0.18
0.21
0.35
0.10 - 0.25
0.16 - 0.23
0.18 - 0.23
0.21
0.18
Surface
albedo
black soil, moist
black soil, dry
grey soil, moist
grey soil, dry
sand, moist fine
sand, dry fine
snow, wet, greyish
snow, wet, coarse
snow, dry, clean
0.08
0.14
0.11
0.27
0.24
0.37
0.46
0.61
0.88
Potatoes
0.19
Note these values are for high elevation angles of the sun and should be used
with caution if the sun is below 30°.
22
22 Radiant Energy Appendices
C. CALCULATION OF LONG-WAVE RADIATION RECEIVED AT THE EARTH'S SURFACE
Longwave radiation includes both incoming atmospheric or counter radiation (Ql) emitted continuously by
atmospheric gases (primarily water vapor and CO2), aerosols, and clouds, and outgoing terrestrial or thermal
radiation (Ql) emitted by surface elements. In accordance with Stefan's law, the emitted flux densities are:
Ql= a Ta 4
[22C.1]
Ql = s Ts 4
[22C.2]
and
where a and s are atmospheric and surface emissivity coefficients,  the Stefan-Boltzmann constant, and Ta and
Ts the temperature in of the air and the surface in Kelvins. Clear sky emissivity may be calculated as a function of
vapor density at 1-2 m height:
a = 0.58 va1/7
[22C.3]
or of temperature (above freezing in °C) at 1-2 m height
a = 0.72 + 0.005 T a
[22C.4]
Clouds have an emissivity of 1.0, so when clouds are present, atmospheric emissivity is higher than for a clear sky.
The atmospheric emittance on cloudy days can be estimated by adding the energy emitted by the clear sky portions
of the sky to the energy emitted by the clouds. The atmospheric emissivity for cloudy days is therefore:
ac = a + C (1 - a - 4 dT/Ta)
[22C.5]
with a given by equation [C.4], C the fraction of the sky covered by clouds, and dT the difference between T a and
cloud base temperature. Typically dT is around 2°K. Equation [C.5] predicts ac = a when C = 0, and ac = 1-4
dT/Ta when C = 1. Estimates of ac are relatively insensitive to errors in estimating C and dT since the range of
ac is relatively small. Estimates accurate to at least ±10% should be possible with very crude estimates of C and
dT.
23
22 Radiant Energy Appendices
Laminar
wind
direction
Transition
Fully Turbulent
outer
layer
turbulent
layer
laminar
sublayer
leading edge
Fig. 14.1. Schematic of development of flow regimes over an infinite plane (after Gray, 1991)
21 Resistance to heat and mass transport
21.1 DIFFUSION AND RESISTANCE
Fick's first law of diffusion: The flux (E, kg m-2 s-1) of a component i by diffusion in the
direction z is proportional to the gradient of concentration in that direction (dCi/dz, kg m-3 m-1),
where the proportionality factor is the diffusion coefficient (Di, m2 s-1):
Eiz = - Di dCi = - m2 s-1
dz
kg m-3
kg
m = - m2 s
[21.1]
The flux of material can be expressed in terms of the resistance to transport within the medium.
Resistance is the ratio of the flow path length to the diffusion coefficient (dz/Di). If flux is
expressed in terms of volume or mass or heat flowing through a unit area per unit time, then the
resistance takes on units of s/m or
rate of transfer of entity = potential difference/resistance
i
Eiz = dCi = dC
ri
dz
Di
[21.2]
where Eiz is the flux of material 'i' in the z direction
dCi is the change in concentration of material 'i' as measured over the distance dz
Di is the diffusion coeficient (m2/s) of material 'i'; and
ri is the resistance of material 'i' (s/m).
Diffusion resistances for the transport of momentum, heat, water vapour and CO2 in the
atmosphere are commonly employed;
22 Radiant Energy Appendices
24
rM resistance for momentum transfer at the surface of a body
rH resistance for convective heat transfer
rv resistance for water vapour transfer
rc resistance for CO2 transfer
Example: The rate of water loss from your skin just after you get out of a swimming pool?
Tskin = 33°C, Ta = 30°C, RH = 0.20, Dv (30)=25.7 m2/s x 10-6, dz= 2.5 mm
Ev = - dpv (dz/Di)-1
assume that at the skin surface RH = 1, therefore
e*a(33°C) = 5032 Pa
ea(30°C) = e*a(30°C) x RH = 4245 Pa x 0.20 = 849 Pa
rv = dz/Dv = 0.0025 m (25.7 x 10-6 m2/s)-1 = 97 s/m
Ev = (5032 - 849 Pa) (97 s/m)-1 = 43 Pa of vapour s-1 m-2
which when converted to amount of water evaporated this is 0.3 g (m2 s)-1
21.2 DETERMINATION OF RESISTANCE TO HEAT TRANSFER
Convection is basically the transfer of heat and mass by moving air. Although this may involve
turbulent transfer, as opposed to laminar transfer, the mathematical function describing flux
densities is analogous to diffusion. The expression describing the transfer process of heat and
mass in air, must however consider whether convection is forced or free; the degree of
turbulence and the shape of the object within the air stream.
Forced convection refers to the condition in which the medium (air or liquid) is moved past a
surface by some external force (i.e. wind). The rate of transfer at right angles to the airstream is
dependent upon the wind speed and surface roughness. The vertical velocity increases with
distance away from the exchange surface to a constant value.
Free convection refers to the ascent of air above an hot object (when placed in cooler air) or the
descent of air above a cold object. Free convection occurs due to density gradients in the air or
liquid as it is heated or cooled by the exchange surface. These density gradients cause the air
mass to mix. With free convection the vertical velocity of the air mass increases with distance
away from the exchange surface, but at a certain distance it reaches a maximum and then begins
to decrease until it is eventually zero.
Due to the different velocity profiles between forced and free the resistance to heat transfer and
thus vapour transfer will vary according to whether convection is forced or free.
To determine whether forced or free convection is occurring the inertial, viscous, and buoyant
forces must be considered. This can be done by using dimensionless groups of transport
processes. Two such groups, the Reynolds number and the Grashof number, are routinely used
for engineering problems.
25
22 Radiant Energy Appendices
Reynolds number (Re ) provides an indication of whether flow is laminar or turbulent. At low
Re, viscous forces predominate, and the flow is laminar. At high Re, inertial forces predominate
and flow becomes turbulent:
Re = u d v -1
u
v
d
[21.3]
is velocity of the fluid or gas (m s-1);
is kinematic viscosity of air = 151 x 10-7 m2 s-1
is characteristic dimension (dia of cylinders and spheres, length of plates). For leafs this is
usually taken as the average width
Grashof number (Gr ) provides an indication of the effect of a buoyant force (an air mass
rising due to heating) upon the effects of whether flow is laminar or turbulent:
Gr = a g d 3(Ts - Ta) v-2
a
g
d
Ts
Ta
is coefficient of thermal expansion of air = 1/273
is gravitational acceleration = 9.8 m s-2
is characteristic dimension
is surface temperature (°C)
is air temperature (°C)
[21.4]
26
22 Radiant Energy Appendices
To determine whether forced or free convection is dominant in atmospheric processes the
Grashof - Reynolds ratio is used:
Gr/Re2
[21.5]
If this ratio is much below one, forced convection dominates. When the ratio is near one both free and
forced convection must be considered.
To determine the resistance coefficient for heat transfer (rH) from the boundary layer may be
determined by two different methods. The first method utilizes non-dimensional numbers to
determine whether convection is forced or free and the second method utilizes aerodynamic
measurements and equations.
1. Non-dimensional method for determination of rH. First determine whether convection is
dominantly forced or free than use one of the following equations.
Forced convection the resistance to heat transfer for a flat surface is:
1.5 d
rH =
1/2
DH Re (v/DH) 1/3
[21.6]
DH is the diffusion of sensible heat in air = 2.15 x 10-5 m2 s-1 (at 20°C)
and for general conditions (air at 20°C and 100 kPa) this equation can be reduced to:
rH = 307 (d/u )1/2
[21.6b]
where d is in m and u is in m/s
Free convection the resistance to heat transfer for a flat surface is:
d
rH =
0.54DH Gr (v/DH) 1/4
[21.7]
and for general conditions (air at 20°C and 100 kPa) this equation is:
rH = 840
d
Ts - Ta
1/4
[21.7b]
where temperature is in °C.
2. Determination of rH by aerodynamic measurements
The boundary layer resistance of a surface can be described by:
ln z - dz + zH ln z - dz + zM
H
M
rH =
k2 u
[21.8]
27
22 Radiant Energy Appendices
21.3 DETERMINATION OF RESISTANCE TO VAPOUR TRANSFER
The resistance of vapour flow from a leaf surface (rv) must consider two resistances that of
vapour transport throught the leaf epidermis, rvs, (stomate and cuticle) and the boundary layer
resistance, rva . For a leaf with equal resistances on both sides:
rv= (rvs + rva )/2
[21.9]
rva may be approximated by a resistance equation for diffusion of water vapour in laminar flow from
a flat plate:
rva =
1.5 d
Dva Re 1/2 v/Dva
1/3
[21.10]
Dva is the diffusion of water vapour in air = 2.42 x 10 -5 m2 s-1 at 20°C
at 20°C and 100 kPa this equation may be reduced to:
rva = 283 (d/u )1/2
[21.10b]
rvs varies according to a combination of factors; the intensity of radiation, the air temperature, and
the moisture potential of the leaf. Optimum conditions for maximum exchange between the leaf
and the atmosphere is when the stomates are open the widest and thus the resistance is the least.
Given there is no other limiting conditions least resistance (rvs of about 200 s m-1) occurs at air
temperatures of 27°C, received short-wave irradiance of between 200 and 400 W m-2. When
temperatures or plant water status become limiting the stomates close and the rvs will increase to
beyond 3200 s m-1. The stomate resistance will generally vary between these two values
depending upon environmental conditions.
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23 Penman's Revisited
Penmans: Potential Evapo-transpiration
Equation [23.1] describes transpiration in terms of environmental parameters (energy budget,
vapor pressure, and temperature) and diffusion resistances of both heat (rH ) and vapour (rv)
providing the ability for stomatal resistance to control transpiration. Given a well watered
dense crop surface or wet soil surface the heat exchange and vapour exchange surfaces are
the same and the vapor resistance becomes neglible when compared to boundary resistance
so, rH = rv. Converting vapour pressures to vapour densities using Eq [19.6] (g/m3), using
the aerodynamic rH method to express rH and setting rH = rv; Equation [21.8] can be given
as:
(Qn - Qg)] + [ f(u) ( *va -  va)]
Qep =
+
ƒ(u )
=
[23.1]
the wind function which is
ƒ(u ) = hv/rH
[23.1a]
and thus has units of J m g-1 s-1.
If the equation is used for daily evapotranspiration, Qg may be set to zero and ƒ(u) can be
taken as:
ƒ(u) = 0.458(1 + u )
[23.1b]
where u is the daily average windspeed in m/s as measured at 2 m above the ground
surface (z) and resulting units are MJ m g-1 d-1 if Qn and Qg are in MJ m-2 d-1.
Further simplifications of Penmans Equations
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The data requirements for Penman's combination formulaes for potential evaporation can still
be relatively extreme. Further simplification is possible if we assume that Qep depends
primarily on energy supply to the evaporating surface and thus vapour gradients and wind
effects are neglible:
Qep = ∆ Qn / (∆ + )
[23.2]
Further simplification has been done by setting Qn proportional to total incoming short wave
radiation Qs and the air temperature Ta (°C) :
Qep = 0.025 °C-1 Qs (Ta + 3°C)
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[23.3]
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24 Other Evapotranspiration Methods
24.1 PHYSICAL BASED METHODS
Aerodynamic-Profile Method
In the profile methods the turbulent transport equations (20.1, 20.2, 20.3) are used to estimate the
flux density of water vapor (E ) and heat (H ) from an exchange surface. These methods require
that the vapour density, temperature, and wind speed be measured at two heights and that the
eddy transfer coefficients, KH and Ke be estimated. The eddy transfer coefficients vary greatly
with any daily fluctuations in windspeed and thus often result in poor estimations. The eddy
transfer coefficients can be simplified and replaced by an approximation (Eq. 24.1a) and the
energy transfer due to evaporation of water can then be calculated:
Qe = - hv
a 0.622
Ke ez1 -- ez2
pa
1
2
[24.1]
where
Qe is the energy involved in evaporating water (J s-1 m-2), and
u 2 - u 1 z2 - z1
[24.1a]
ln zz 2 2
1
The amount of heat transfer to the atmosphere (QH) can also be estimated with equation [20.2]
and assuming that KH = Ke.
Ke = k2
The Energy Budget (Monteith and Unsworth, 1990; Gray, 1991)
The amount of energy transfer due to evaporation of water can also be estimated by using an
energy budgeting technique (kJ s-1 m-2) for a crop, soil, or water surface:
Qe = Qn - Qh - Qg
[24.2]
Qe =
turbulent flux of latent energy. This term may be calculated by difference or
may be calculated using resistance or aerodynamic equations.
Qn =
net radiation (net shortwave and net longwave).
Qh =
is the sensible heat loss which is the energy flux that goes into heating the air
and depends upon turbulent transfer and can be estimated by resistance or aerodynamic
equations.
Qg =
is the rate at which energy utilized in heating the soil or water . A reasonable
daytime approximation for crops is :
Qg = 0.1 Qn
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For soils Qg can be relatively large at any one time during the day, but net
storage over a 24 hr period is neglible as nighttime loss of soil heat negates that gained during
the day. For daily averages, Qg can be taken as zero.
More accurate forms of this equation considers advection energy and energy storage in the
control volume. As both the Qh and Qe terms are affected by turbulent transfer and their
resistance terms may be related then a ratio of these ( the Bowen ratio ) can be used to simplify
calculations
=
Qh a cp (T1 - T2 ) rv
=
rH
Qe
(v1 - v2 )
[24.3]
Qh and Qe may be estimated by resistance or aerodynamic methods. Measurements of
temperature and vapour density are needed at two heights along with estimates of vapour and
heat transfer resistances (rv and rH) between these two heights. Using the Bowen ratio to
substitute for Qh in Equation [24.2] , the rate of energy utilization by evapotranspiration may be
estimated:
Qe = (Qn - Qg ) / (1 + )
[24.4]
For the atmosphere just above the surface the assumption that rv = rH provides reasonable
accuracy and removes the need for obtaining values for these terms as they cancel out:
Data for this method must be collected over short periods of time, thus making data collection
difficult. Bowen's ratio is especially difficult to estimate and can contribute the largest amount of
error to the estimate of LE. The value of  has little influence on LE provided -0.1 <  <0.1,
which is normally the case for humid conditions when water is not limiting and most of Qn goes
into Qe rather than Qh.
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24.2 EMPIRICAL METHODS
Thornthwaite (Gray 1970)
Thornthwaite (1948) and Thornthwaite and Mather (1954) developed a complicated expression
for Etp (potential evapotranspiration) in terms of mean air temperature:
Etp = 16.2 df 10Ta
I
where
Etp
a
[24.5]
is the potential evapotranspiration for a 30-day month (mm)
df
is the daylight factor (the possible number of hours of bright sunshine for the month divided by
360). This is a function of latitude and solar declination (time of the year) Table 16B.1.
Ta
is the mean monthly air temperature (°C)
I
is a heat index, which is the sum of all the months (n) with mean monthly temperature above
0°C:
n
I=
•
m=1
Tm
5
1.51
[24.5a]
and the exponent a is a cubic function of I, expressed as:
a = 6.75 x 10-9I 3 - 7.71 x 10-6I 2 + 1.179 x 10-2I + 0.492
[24.5b]
The procedure for calculation of monthly Etp is to:
1. Calculate I for all the months in which the mean monthly temperature is above 0°C
(Equation 24.5a); for the Canadian Prairies this can be considered to be April through
October;
2. Use Equation 24.5b to obtain a ;
3. Use Equation 24.5 with the monthly air temperature and the appropriate daylength factor to
obtain Etp .
For Saskatoon (Latitude 52 °N) the daylight factors, df, are;
April 1.17, May 1.33, June 1.36, July 1.37, Aug 1.25, Sept 1.06, and Oct 0.88.
The dominant parameters are temperature and length of day. Used together they account for the
balance of radiation exchanges, air movement, humidity and other meteorological parameters
that affect evaporation. The formulae developed by Thornthwaite are based upon catchment-area
data and controlled experiments. The above equation is deceptively simple as the monthly
indices have to be adjusted for the length of day which is dependent upon the latitude. The
equation is meant only for monthly estimates and although the values will be of the right order of
magnitude they are only approximate.
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Blaney and Criddle (1950)
This formula uses temperature, relative humidity and length of daytime hours. Based upon
measurements in New Mexico and Texas they found that the monthly evapotranspiration (mm)
from a growing crop was:
Et = 25.4 (1.8Ta + 32) k d (1.14-hr )
where
k
[24.6]
is a crop constant (Gray, 1991)
small grains, 3 months, 0.75
alfalfa, frost-free months, 0.85
corn, 4 months, 0.75
beans, 3 months, 0.65
potatoes, 3 months, 0.70
Tam
is the mean monthly air temperature (°C)
d
is the monthly ratio of daytime hours within the month of concern divided by
the total in the year;
hr
is the mean monthly relative humidity (ratio).
For Saskatoon (52°N) the monthly ratios of annual daytime hours (d) are;
April 0.0929, May 0.1085, June 0.1113, July 0.1120, Aug 0.1012, Sept 0.0849, and Oct 0.0739.
The formula gives in the simplest possible form recognition of two factors already mentioned,
namely that if the heat budget is shared in fixed proportions between evaporation and heating of
the air then air temperature will be a useful parameter for correlation. For normal crops the
transpiration is limited by day length (Penman 1963). This method tends to provide a more
reliable estimate of seasonal evapotranspiration than the Thornthwaite method for arid regions
(Gray 1970).
Example: What is the daily potential evaporation given the following daily
measurements:
Ta = 25°C, Ts = 20°C, RH = 0.30, u = 3 m/s, Qn = 20 MJ m-2 d-1, Qg
=0
From Eqs 13.3 and 13.5; e*a = 3169 Pa and ea = 950 Pa
From Eq 13.6; *a = 23.0 g/m3 and a = 6.9 g/m3
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Baier and Robertson (1965)
This technique estimates daily latent evaporation from simple meteorological observations and
astronomical data all simply obtained. The technique was developed from climatological records
taken at agricultural research stations at Ottawa, Normandin, Swift Current, Lacombe,
Beaverlodge, and Fort Simpson during a 5 year period. The method is a multiple regression
analysis using 3 to 6 variables. Some of the mutliple regression equations with their regressions
coefficients (r) are presented below:
Etb = (0.933Tr + 0.928Tm + 0.0486QAo - 87.03)/44.2
r = 0.68
[24.7a]
Etb = (1.09Tr - 0.0228Tm + 0.0506QAo + 2.99pd - 42.28)/44.2
r = 0.76
[24.7b]
r = 0.81
[24.7c]
r = 0.84
[24.7d]
Etb = (1.04Tr + 0.35Tm + 0.0403QAo + 2.31pd + 0.101u
- 69.30)/44.2
Etb = (0.531Tr + 0.337Tm + 0.0107QAo + 0.0512Qs + 1.77pd + 0.0977u - 53.39)/44.2
where
Etb
is the daily evapotranspiration from a Bellani plate (cm/d);
Tr
is the difference in °F between daily maximum and daily minimum temperature
Tm
is the maximum daily air temperature (1.2 m above ground);
is the daily solar radiation received just outside the earth's atmosphere on a
plane horizontal to the earth's surface (cal cm-2 d-1).
Qs
is the daily solar radiation (direct plus diffuse) received at the earth's surface (cal
-2
-1
cm d ).
pd
is the vapor pressure deficit (p'v - pva) as determined from the saturated vapor
pressure deficit measured from the mean daily temperature minus actual vapor pressure (mb);
u
is the daily windspeed (miles per hour).
QAo
The potential evaporation is from a black Bellani plate atmometer. Although it is possible to
obtain an estimate using only three variables (Eq. 16.3.1) the accuracy improves with a greater
number of variables. The reliability of the estimates are further improved if daily values of
estimated evaporation are accumulated for periods longer than 2 weeks.
24.3 Instrumental Methods
The direct measurement of evaporation using the depletion rate of a free water body
exposed to atmospheric conditions is routinely made, however the interpretation of the data must
always be in consideration of the method used. The size of the surface exposed, the type of
terrain surrounding the instrument, and shading of the instrument are but a few of the variables
that will affect the evaporation rate. For these reasons any instrumentation used must be made
with identical
instrumentation and in identical surroundings to those used at other
meteorological stations so as to insure that comparisons can be made. These instruments are
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termed relative as the measurements made are done so in comparison to the atmospheric
evaporativity. Units of expression are similar to that used in precipitation; mm.
There are four main classes of relative evaporation gauges: (1) large evaporation tanks sunk into
the ground or floating on protected waters; (2) small evaporation pans; (3) porous porcelain
bodies; (4) wet paper surfaces. Each has its advantages and disadvantages.
a) Large evaporation tanks (Gray 1970)
Large evaporation tanks are used routinely at major meteorological stations throughout
Canada and the U.S. The most commonly used one is the U.S. Weather Bureau Class A pan.
This pan is 4 feet (1.2 m) in diameter and 10 in (0.25 m) deep. It is constructed of galvanized
steel and water is maintained 5-7 cm from the top. It is set upon timbers so that the bottom is
about 15 cm above the surface thus eliminating difficulties caused by drifting soil and snow.
Earth is embanked up over the timbers leaving approximately a 2-3 cm air gap to permit air
circulation under the pan. Wind, temperature, and rain measurements should be taken from the
same local. Water level is maintained at a constant level by daily additions of water to the pan.
The evaporation rate is thus determined by the amount of water added. The results from these
tanks are only of value when compared with other similar tanks and are not representative of the
true evaporation rate from soil or lake surfaces. For accurate comparisons between met stations
and for proper use of correction coefficients if the data is to be applied to water bodies the pans
are generally kept within cut grassed areas of certain dimensions.
b) Small evaporation pans
Most recording atmometers use small evaporation pans (usually 15 to 30 cm in diameter)
which measure the variation of the weight or level of water with time. A simple cylindrical pan
with a pointed wire soldered to the bottom (for determination of constant water level) is most
commonly used. The amount of water necessary to bring the water level back to the point of the
wire is daily added. If a large number of these are to be used for any one project care must be
taken concerning color, size, height installed above ground.
c) Porous porcelain bodies
Porous porcelain spheres, cylinders, or plates have been used by various workers since
1813. The porous material and the shape is meant to represent soil or plant conditions. The one
most commonly used is the Bellani plate which is a thin black plate 7.5 cm in diameter. The
Bellani plate is attached to a reservoir of water that is filled with distilled water. It is considered
to be more sensitive to wind than the Class A evaporation pan. Bellani plate readings are
commonly presented in cm3 of water and to be converted to cm they must be divided by the plate
area (44.2 cm2).
d) Wet paper or cloth-wick surfaces
This type is represented by the Piché atmometer, which consists of a graduated tube closed
at the upper end while the bottom end, ground flat, is placed upon a circular piece of filter paper.
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The vessel is filled with distilled water and the filter paper exposed to the atmosphere. The
results are very sensitive to wind speed. A modern variation of this utilizes a wick to transfer the
water to the filter paper. The amount of evaporation is sensed by a float and recorded on the
chart by a pen-writing system.
Comparisons of Atmometer methods (Gray 1970)
In comparison of the relative importance of three major factors in evaporation; net
radiation, humidity, and wind, researchers have found that the ratio of components for the Class
A pan was 80:6:14, while that for the Bellani Plate was 41:7:52. To convert atmometer readings
to crop or lake evapotranspiration heights (cm) the following conversion coefficients are
generally applicable for Western Canada:
crops
0.0226 x Bellani reading (cm3)
lakes
Evaporation by soil water balance measurements
0.67 x Class A pan (cm)
0.70 x Class A pan (cm)
Determination of the amount of soil water lost due to evapotranspiration involves
measurement of other soil water processes such as soil water additions from precipitation and
capillary rise and soil water losses such as drainage. The ability of the soil to store water must
also be accounted for. The general equation for estimation of evapotranspiration is thus:
Et = P - D + d
[24.7]
where
Et
is the amount of water lost from the soil surface due to evaporation and from the plant
leaves due to transpiration;
P
is the amount of precipitation (mm);
D
is the amount of drainage (mm);
d
is the amount of soil water lost or gained within the depth interval (mm).
This equation assumes that there is no water gain from capillary rise and that no runoff
occurs. If the soil is below field capacity and no precipitation occurs than all decreases in the soil
moisture content can be attributed to evapotranspiration:
Et = d
[24.8]
Unless a soil lysimeter is used problems with the estimation of the drainage portion and
soil water rise from deeper wetter horizons can result in considerable error. In lysimeter studies
the waters leaching through the soil profile are collected from the bottom and changes in soil
moisture are determined by weighing or by direct soil moisture measurements.
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