Micrometeorological Method

4.1.3.1. Mass Transport Approach

As we know that the evaporation depends on the incoming heat energy and vapor pressure gradient, which, in turn, depend on several weather variables such as temperature, wind speed, atmospheric pressure, solar radiation, and quality of water. The whole process of evaporation is complicated. A complete physical model can not accommodate all the factors which are not controllable and measurable. The mass-transfer approach is one of the oldest methods (Dalton, 1802; Meyer, 1915;

Penman, 1948) and is still an attractive method for estimating free water surface evaporation because of its simplicity and reasonable accuracy. The mass-transfer methods are based on the Dalton equation, which, for free water surface, can be written as:

E_o = C (e_s - e_a)…... (4.13) where, E_o is free water surface evaporation, e_s is the saturation vapor pressure at the temperature of the water surface, e_a is the vapor pressure

in the air and C is aerodynamic conductance. The term e_a is also equal to the saturation vapor pressure at the dew point temperature. The parameter C depends on the horizontal wind speed, surface roughness and thermally induced turbulence. It is normally assumed to be largely dependent on wind speed,

u

. Therefore, Eq. 4.13 can be expressed as:

E_o = f(u) (e_s - e_a) …... (4.14) where, f(u) is the wind function. This function depends on the observational heights of the wind speed and vapor pressure measurements. Penman (1948) proposed the following variation of the Dalton equation.

E_o = 0.40 (e_s - e_a) (1+0.17 u_d) …... (4.15) where,

E_o= potential evaporation, mm

e_s = saturation vapor pressure at the temperature of the water surface, mm of Hg

e_a = actual vapor pressure in the air, mm of Hg u_d = wind speed at 2m height, miles/day Example 4.3.

Determine the potential evaporation with Penman equation. The following data set is given:

e_a =11.50 mm of Hg e_s =13.20 mm of Hg u_d= 45 miles/day

Solution. The potential evaporation can be calculated by the following equation.

E_o = 0.40 (e_s - e_a) (1+0.17 u_d) = 0.40 (13.20-11.50) (1+0.17×45)

72 AN INTRODUCTION TO DRIP IRRIGATION SYSTEM

= 0.68 ×8.65 = 5.88 mm/day Example 4.4.

In a cropped field, five set of the micro-meteorological data were recorded and given in the table. Compute the potential evaporation by using the Penman Mass Transport Method.

Data Set Saturation vapor Actual vapor pressure, Wind speed at 2 pressure, mm of Hg mm of Hg m height, miles/day

1. 12.0 10.8 40.5

2. 10.2 8.2 12.8

3. 8.6 7.6 30.8

4. 7.2 7.0 45.6

5. 10.4 9.2 42.0

Solution. Penman (1948) equation based on Mass Transport Approach is given hereunder. All the required inputs are given.

E_o = 0.40 (e_s - e_a) (1+ 0.17 u_d) For data set 1.

Saturation vapor pressure at the temperature of the water surface,e_s= 12.0 mm of Hg

Actual vapor pressure in the air, e_a = 10.8 mm of Hg Wind speed at 2m height, u_d = 40.5 miles/day Now, applying the formula, we get

E_o = 0.40 (e_s - e_a) (1+0.17u_d) = 0.40(12.0-10.8)(1+0.17*40.5) = 3.78 mm/day

Similarly for other data set, the information is given in the table.

Data Set saturation Actual vapor Wind speed at Potential vapor pressure, e_s pressure, e_a 2m height, u_d evaporation, E_o

(mm of Hg) (mm of Hg) (miles/day) (mm/day)

1. 12.0 10.8 40.5 3.78

2. 10.2 8.2 12.8 2.54

3. 8.6 7.6 30.8 2.49

4. 7.2 6.5 45.6 2.45

5. 10.4 9.2 42.0 3.91

4.1.3.2. Aerodynamic approach

Evaporation from an open surface is a dynamic process, whose rate depends on ability of wind and the humidity gradient in the air above the evaporating surface to remove water vapor away from the evaporating surface via aerodynamic processes. Thornthwaite and Holzman (1939) proposed the following equation which included specific humidity gradient and logarithmic wind profile.

where,

E_O = evaporation, mm/hr P_a = density of air, kg/m³

κ = von Karman’s constant, 0.41

q₂ and q₁ = specific humidity in g/kg at heights z₂ and z₂ u₂ and u₁ = wind speed in m/s at heights z₂ and z₂

Example 4.5. Estimate the evaporation from field data collected in West Bengal by using aerodynamic approach. The data are given below.

ρ_a= density of air = 1.3 kg/m³ κ = von Karman’s constant, 0.41

q₂ and q₁ = specific humidity in g/kg at heights z₂ and z₁ =6.90 and 8.75 g/kg

and outgoing energy fluxes can be expressed as follows.

R_n = H + λE + G…... (4.20) where,

R_n = energy flux density of net incoming radiation (W/m²) H = flux density of sensible heat into the air (W/m²) λE = flux density of latent heat into the air (W/m²) G = heat flux density into the water body (W/m²)

To convert the above lE in W/m² into an equivalent evapotranspiration in units of mm/d, λE should be multiplied by a factor 0.0353. This factor equals the number of seconds in a day (86 400), divided by the value of l (2.45 x 10⁶ J/kg at 20°C). Density of water is assumed as 1000 kg/m³. Penman (1948) combined energy balance approaches and turbulent transport of water vapor away from an evaporating surface can be given as

.

E_o = evaporation from open water surface, mm/day R_n = net radiation in water equivalent, mm/day G = soil heat flux in water equivalent, mm/day

( )

(

ⁿ_N

)

^T

(

^ed

) (

ⁿ_N

)

R

R_n= _A1−α 0.32+0.46× −σ _a⁴0.55−0.92 0.10+0.90 ..(4.22) where,

α =Albedo

R_A = mean monthly radiation outside the atmosphere in water equivalent, mm/day

78 AN INTRODUCTION TO DRIP IRRIGATION SYSTEM

T_a = mean air temperature, ⁰K

s = slope of the saturation vapor pressure vs. temperature curve, mm of Hg/⁰C.

γ = psychometric constant (0.49 mm of Hg/⁰C)

n/_N = ratio of actual to maximum possible sunshine hours

σ = Stefan-Boltzman constant = 5.56 x 10^-8 w/m²/K⁴ = 2.06 x 10^-9 mm/day

ed= saturation vapor pressure of the atmosphere in mm of Hg at dew point temperature or actual vapor pressure.

Again

(

Td Tw

)

ea esw MeanRH

ed = × = −0.49 −

100 …... (4.23) where,

ea= saturation vapor pressure at mean air temperature, mm of Hg esw= saturation vapor pressure at wet bulb temperature (T_w) and dry bulb temperature(T_d), mm of Hg.

An aerodynamic component is explained by the following expression.

E_a = 0.35 (ea - ed) × (1+ 0.0098u₂) …... (4.24) Here u₂ is wind speed in miles/day at 2m height and u₂ can be determined as given below.

u₂ = log 6.6 × u_h /log h …... (4.25) where,

u_h = wind speed at miles/day at height ‘h’ in feet

Example 4.7. Estimate the net radiation in water equivalent with the help of following given collected data in the field by using Penman combination method. This example is taken from Ram Nivas et al. (2002).

Mean air temperature, T_a = 16.1⁰C Mean relative humidity = 68.5%

80 AN INTRODUCTION TO DRIP IRRIGATION SYSTEM

Finally, determine E_a with the given equation.

( ) ( 1 0 . 0098

2

)

By using Eq. 4.21, evaporation from open water surface in mm/day can be determined.

4.1.5.2. Modified Penman Approach

Monteith (1963, 1964) introduced resistance terms into the Penman method and proposed the following equation to estimate latent heat of evaporation.

( )

C_p = specific heat of air at constant pressure, 1008 J/kg/⁰C r_a = aerodynamic resistance, s/m

r_c = canopy resistance, s/m = r_s + 15

r_s = stomatal resistance, s/m

( ) ( )

r

ad= adaxial resistance r_ab = abaxial resistance LAI = leaf area index

ea = actual vapor pressure, mm of Hg es = saturation vapor pressure, mm of Hg

Aerodynamics resistance (r_a) is determined as follows.

( )

{ }

[ ]

2

ln 2

uk zo d

r_a = z− ... (4.28)

where, z = height

d = zero plane displacement = 0.63z zo = roughness parameter = 0.13z u = wind speed at height, z κ = von Karman’s constant (0.41)

Example 4.8. Determine the latent heat of evaporation by using Modified Penman Approach with the following data collected in the field.

Net radiation, R_n = 282.0 W/m² Soil heat flux, G = 10.1 W/m²

S/_γ = 1.860 at 17.5⁰C

Density of air, ρ_a= 1.3 kg/m³

Specific heat of air at constant pressure, C_p = 1008 J/kg/⁰C Psychometric constant, γ = 0.49 mm of Hg/⁰C

Dry bulb temperature = 17.0⁰C Wet bulb temperature = 9.8⁰C

Saturation vapor pressure, es = 5.0mm of Hg

= 505.734 + 567.013) / (2.86+2.95)

= 186.06 W/m² = 0.27 mm/hr

4.1.5.3. Van Bavel method

Van Bavel (1966) modified the Penman method and introduced a new term ‘

B

_V ’, and proposed the following formula.

( )

da = vapor pressure deficit, mm of Hg and is equal to difference of es and ea

P = air pressure, 0.75 mm of Hg u = wind velocity, m/s

z = crop height, m

d = zero plane displacement L = latent heat, J/kg, 2.346 zo = roughness parameter

ε = 0.622 (it is a ratio of molecular weight of water to molecular weight of dry air

4.1.5.4. Slatyer and Mcllroy Method

The Slatyer-McIlroy equations are based on evapotranspiration from a wet surface with minimum advection. This happens when the air above a surface is saturated due to vapor exchange with the wet surface.

They used wet bulb depression instead of vapor pressure deficit and

84 AN INTRODUCTION TO DRIP IRRIGATION SYSTEM

proposed the following modified equation.

( )

D_z= wet bulb depression at height ‘z’. It is a difference of dry and wet bulb temperature.

Example 4.9. Determine the potential evapotranspiration by using the Slatyer and Mcllroy method with the flowing data.

Net radiation (R_n) = 310.0 W/m² Soil heat flux (G) = 12.6 W/m²

Slope of the saturation vapor pressure vs. temperature curve, s = 0.91mm of Hg/⁰C

Psychometric constant, γ = 0.49 mm of Hg/⁰C Density of air,

ρ

_a^{= 1.3 kg/m3}

Specific heat of air at constant pressure, C_p= 1008 J/kg/⁰C Aerodynamic resistance, r_a = 14.30 s/m

Wet bulb depression (D_z) = 1.4⁰C

Solution. Putting all the given values in Eq. 4.31

( )

0.91 0.91 0.49 14.30

−

4.1.5.5. Priestley and Taylor Method

The Priestley-Taylor model (Priestley and Taylor, 1972) is a modification of Penman’s theoretical equation. An empirical approximation of the Penman combination equation is made by the Priestley-Taylor to eliminate the need for input data other than radiation.

The adequacy of the assumptions made in the Priestley-Taylor equation has been validated by a review of 30 water balance studies in which it was commonly found that, in vegetated areas with no water deficit or very small deficits, approximately 95% of the annual evaporative demand was supplied by radiation (Stagnitti et al., 1989). Priestley and Taylor (1972) found that the actual evapotranspiration from well watered vegetation was generally higher than the equilibrium potential rate and could be estimated by multiplying the ET_o by a factor (α) equal to 1.26. They demonstrated that potential evapotranspiration is directly related to the equilibrium evaporation in the absence of advection and proposed the following equation.

( )

where,

α

is the empirical constant and is taken as 1.26 for temperature ranging between 15-30⁰C. Although the value vary throughout the day, there is general agreement that a daily average value of 1.26 is applicable in humid climates (Shuttleworth and Calder, 1979). Morton (1983) observed that the value of 1.26, estimated by Priestley and Taylor, was developed using data from both moist vegetated and water surfaces. Morton recommended as 1.32 for estimation from vegetated areas as a result of the increase in surface roughness. Generally, the coefficient for an expansive saturated surface is usually greater than 1.0. This means that true equilibrium potential evapotranspiration rarely occurs; there is always some component of advection energy that increases the actual evapotranspiration. Higher values of ranging up to 1.74, have been recommended for estimating potential evapotranspiration in more arid regions (ASCE, 1990).

Example 4.10. Estimate the potential evapotranspiration by using the

86 AN INTRODUCTION TO DRIP IRRIGATION SYSTEM

Priestley and Taylor method. The following data were collected and given below.

Net radiation (R_n) = 310.0 W/m² Soil heat flux (G) = 12.6 W/m²

Slope of the saturation vapor pressure vs. temperature curve, = 0.91mm of Hg/⁰C

Psychometric constant, γ = 0.49 mm of Hg/^oC An empirical constant, α = 1.26

Solution. All the data required by the Priestley and Taylor method of computing potential evapotranspiration are given, so it easier to put all the values in Eq.4.32

= 1.26×0.91×(310-12.6)

0.91+0.49 = 243.57 W/m² = 0.36 mm/hr

In document AN INTRODUCTION TO DRIP IRRIGATION SYSTEMS (pagina 85-101)