CE 374 K – Hydrology

Evaporation

Daene C. McKinney

Evaporation

• Terminology– Evaporation: liquid water passes directly to

the vapor phase– Transpiration: liquid water passes from

liquid to vapor through plant metabolism– Sublimation: water passes directly from the

solid phase to the vapor phase

Factors Influencing Evaporation• Energy supply for vaporization (latent heat)

– Solar radiation

• Transport of vapor away from evaporative surface– Wind velocity over surface– Specific humidity gradient above surface

• Vegetated surfaces– Supply of moisture to the surface– Evapotranspiration (ET)

• Potential Evapotranspiration (PET) – moisture supply is not limited

Evaporation from a Water Surface

• National Weather Service Class A type

• Installed on a wooden platform in a grassy location

• Filled with water to within 2.5 inches of the top

• Evaporation rate is measured by manual readings or with an analog output evaporation gauge

h

Area, A

CSwρ

aρ

AEm wv ρ=&

dtdhE −=

nRsHSensible

heat to airNet radiation Vapor flow rate

Heat conductedto ground

G

Methods of Estimating Evaporation

• Energy method• Aerodynamic method• Combined method

Energy MethodContinuity of Liquid Phase

∫∫ ⋅+∫∫∫ ∀=−CS

wCV

wv ddtdm dAVρρ&

0=

dtdhAwρ=

No flow of liquid No flow of liquid water through CSwater through CS

AEm wv ρ=&

dtdhE −=

hwρ

aρ

vm&

dtdhE −=

nRsH

G

Energy Method (2)

Vapor Phase - Continuity

∫∫ ⋅=CS

avw qAE dAVρρ

0=Steady flow of airSteady flow of air

over waterover water

AEwρ=

∫∫ ⋅+∫∫∫ ∀=CS

avCV

avv qdqdtdm dAVρρ&

∫∫ ⋅=CS

avw

qA

E dAVρρ

1

∫∫ ⋅=CS

avv qm dAVρ&

hwρ

aρ

vm&

dtdhE −=

nRsH

G

∫∫ ⋅+++

∫∫∫ ∀++=−

CSu

CVu

dgzVe

dgzVedtd

dtdW

dtdH

AVrr

ρ

ρ

)2/(

)2/(

2

2

Energy Method (3)Energy Eq.

0=

hwρ

aρ

vm&

dtdhE −=

nRsH

G

.,0;0 consthV ≈=≈

∫∫∫ ∀=CV

wu dedtd

dtdH ρ

GHRdtdH

sn −−=

GHR sn −−=

Energy Method (4)Energy Eq. for Water in CV

Assume:1. Constant temp of water in CV2. Change of heat is change in internal

energy of evaporated water hwρ

aρ

vm&

dtdhE −=

nRsH

Gvvml

dtdH

&=

GHRdtdH

sn −−=

GHRml snvv −−=& AEm wρ=&

( )GHRAl

E snwv

−−=ρ1

Recall:Recall:

wv

nr l

REρ

= Neglecting sensible Neglecting sensible and ground heat fluxesand ground heat fluxes

Aerodynamic Method• Include transport of vapor

away from water surface as function of: – Humidity gradient above

surface– Wind speed across surface

• Upward vapor flux

• Upward momentum flux

nR

E

Net radiation

Evaporation

Air Flow

1212

zzqq

Kdz

dqKm vvwa

vwa −

−−=−= ρρ&

1212

zzuuK

dzduK mama −

−== ρρτ

( )( )12

21uuKqqKm

m

vvw−

−=

τ&

Aerodynamic Method (2)

• Log-velocity profile

• Momentum flux

( )( )12

21

uuKqqK

mm

vvw

−

−=τ&

⎟⎟⎠

⎞⎜⎜⎝

⎛=

oa

ZZ

ku ln1

ρτ

( )( )

2

12

12ln ⎥

⎦

⎤⎢⎣

⎡ −=

ZZuuk

aρτ

( )( )( )[ ]212

122

ln21

ZZK

uuqqkKm

m

vvaw −−=

ρ&

Thornthwaite-Holzman Equation

nR

E

Net radiation

Evaporation

Air Flow

Too many variables! Often only know qv and u at 1 elevation

Aerodynamic Method (3)

( )eeBE sa −=

( )[ ]22

22

ln622.0

ow

a

ZZPukB

ρρ

=

pressure vapor =e

nR

E

Net radiation

Evaporation

Air Flow

• Simplify

pressure vapor sat. =se

Combined Method• Evaporation could be calculated by

– Aerodynamic method: when energy supply is not limiting– Energy method: when vapor transport is not limiting

• Normally, both are limiting, so use a combination method

Combined Method (Cont.)• Combining

– Energy balance – Aerodynamic Methods

• Combined Method

– Well suited to small areas with detailed data

• Net Radiation• Air Temperature• Humidity• Wind Speed• Air Pressure

wv

nr l

REρ

=

( )eeBE sa −=

ar EEEγ

γγ +Δ

++ΔΔ

=wv

hp

KlpKC

622.0=γ

2)3.237(4098

Tes+

=Δ

rEEγ+Δ

Δ= 3.1 Priestly & Taylor

Example

z = 2 mP = 101.3 kPau = 3 m/sRn = 200 W/m2

T = 25 degCRh = 40%

• Use Combo Method to find Evaporation

kJ/kg 244110)25*36.22500(

237010501.23

6

=−=

−=

x

Txlv

mm/day10.7997*102441

2003

=

==xl

REwv

nr ρ

Example (Cont.)

z = 2 mP = 101.3 kPau = 3 m/sRn = 200 W/m2

T = 25 degCRh = 40%

• Use Combo Method to find Evaporation

( )mm/day45.7

)day1/s86400(*)m1/mm1000(*126731671054.4 11

=−= −xEa

( )[ ] ( )[ ]sm/Pa1054.4

1032ln997*3.101

3*19.1*4.0*622.0ln

622.0 1124

2

22

22

⋅=== −−

xxZZP

ukBow

aρ

ρ

Pa3167=se

Pa12673167*4.0* === sh eRe

Example (Cont.)

z = 2 mP = 101.3 kPau = 3 m/sRn = 200 W/m2

T = 25 degCRh = 40%

Pa/degC1.67102441*622.0103.101*1005

622.0 3

3===

xx

KlpKC

wv

hpγ

• Use Combo Method to find Evaporation

Pa/degC7.188)253.237(

3167*4098)3.237(

409822 =

+=

+=Δ

Tes

738.0=+ΔΔγ

mm/day2.745.7*262.010.7*738.0 =+=+Δ

++ΔΔ

= ar EEEγ

γγ

262.0=+Δ γγ

Example

• Use Priestly-Taylor Method to find Evaporation rate for a water body

rEEγ+Δ

Δ= 3.1 Priestly & Taylor

mm/day10.7=rE 738.0=+ΔΔγ

mm/day80.610.7*738.0*3.1 ==E

z = 2 mP = 101.3 kPau = 3 m/sRn = 200 W/m2T = 25 degCRh = 40%

Evapotranspiration

• Evapotranspiration– Combination of evaporation from soil surface and

transpiration from vegetation– Governing factors

• Energy supply and vapor transport• Supply of moisture at evaporative surfaces

– Reference crop• 8-15 cm of healthy growing green grass with abundant water

– Combo Method works well if B is calibrated to local conditions

Potential Evapotranspiration

• Multiply reference crop ET by a Crop Coefficient and a Soil Coefficient

rcs ETkkET =

3.10.2t;Coefficien Crop

≤≤=

c

ck

k

10t;Coefficien Soil

≤≤=

s

sk

k

ET Actual =ET

ET Crop Reference =rET

http://www.ext.colostate.edu/pubs/crops/04707.html

CORN

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100 120 140 160

Time Since Planting (Days)

Cro

p C

oeff

icie

nt, k

c

Combined Method• Evaporation could be calculated by

– Aerodynamic method: when energy supply is not limiting– Energy method: when vapor transport is not limiting

• Normally, both are limiting, so use a combination method

• Sensible heat flux is difficult to estimate– Assume it is proportional to the vapor heat flux– Where β = Bowen ratio– Energy balance equation (G=0)

( )vvs mlH &β=

( )

( )βρ

+=

−−=

1

1

mlR

GHRAl

E

vn

snwv&

Combined Method (2)• Transport equations for heat and vapor

dzdqKH v

was ρ−=dzdTKCm hpav ρ−=&

( ))(

12

12

vvw

hp

v

sqqKTTKC

mH

−

−=

&

( ))(622.0 12

12

eeKlTTpKC

wv

hp

−

−=β

( )vvs mlH &β=peqv 622.0=

wv

hp

KlpKC

622.0=γ

( ))( 12

12eeTT

−−

= γβ 1≈w

hKK

Recall Vapor Pressure

Temperature

Vap

or P

ress

ure

ese

Measured temp, TDew Point temp, Td

Saturate vapor pressurefor given temperature

Vapor Pressure

Saturation Vapor Pressure

sh e

eR =

⎟⎠⎞

⎜⎝⎛

+=

TTes 3.237

27.17exp6112)3.237(

4098T

edTde ss

+==Δ