Energy Balance
Energy in = Energy out +
Δ Storage
Bio 164/264
January 11, 2007
C. Field
Radiation: Reminders from last time
• Energy of a photon depends on 1/wavelength
– E = hc/l
– h is Planck’s constant (6.63*10-34 Js), c is the speed of light
(3*108m s-1), and l is wavelength (m).
• Thermal radiation depends on T4: Stefan-Boltzmann law
– B =sT4
–
s = 5.67 * 10-8 W m-2 K-4
• Wavelength of maximum energy depends on 1/temperature
(Wien Law)
–
lm = 2897
T
• Solar “constant” ~ 1360 W m-2, over sphere = 342 W m-2
Energy balance
• Conservation of energy
• Energy in = Energy out + Δ Storage
• Energy transport
–
–
–
–
•
Δ
Radiation
Conduction
Convection = Sensible heat
Evaporation = Latent heat
Storage
– Change in temperature
– Change in the energy stored in chemical bonds
– Change in potential energy
Radiation balance
SS = 600 W m-2, q = 20
• Thermal
– In = IR down + IR up
– Out = IR down + IR up
– =461 + 346 – 397 – 397 = 63
Sd = 100 W m-2
T = 10,  = 1.0*
• SW
– In = direct*cosq*a +
diffuse down*a +
diffuse up *a
ST = 426 W m-2
= 282 + 120 + 50 W m-2
Out = reflected up +
reflected down+
transmitted down+
transmitted up
= already included in in
a = 0.6
ST = 365 W m-2
T = 25,  = .95, a = 0.5
ST = 426 W m-2
ST = 486 W m-2
T = 35,  = .95
Conduction
• Not very important in this class.
Convection
• Rate of transport = driving force *
proportionality factor
– Fick’s law – diffusion F’j = -Dj (drj/dz)
• D = molecular diffusivity
– Fourier’s law – heat transport H = -k (dT/dz)
• k = thermal conductivity (m2 s-1)
– Darcy’s law – water flow in a porous medium
• Jw = -K(y) (dy/dz)
• K(y) = hydraulic conductivity
Keeping units straight - Moles
• Most of the mass fluxes in this class will be in
moles, where 1 mole = m.w. in g
–
–
–
–
N2
O2
CO2
H2O
1 mole = 28.01 g
1 mole = 32.00 g
1 mole = 44.01 g
1 mole =
• Molar density (mol m-3) ® = rj/Mj is the same
for all gases
– Ideal gas law pjV = njRT
– = 44.6 mol m-3 @ 0C and 101.3 kPa (STP)
– ® = rj/Mj
First – get mass flux in molar units
• Convert Fick’s law to molar units
– diffusion F’j = -Dj (drj/dz)
– Fj = F’j/Mj= - ® Dj (dCj/dz)
• D = molecular diffusivity
• Cj = mole fraction of substance j
Convection – moving heat in air
• Start with Fourier’s law
– Heat transport H = -k (dT/dz)
• k = thermal conductivity
• cp = molar specific heat of air 29.3 J mol-1 C-1
• k/cp = DH = thermal diffusivity
– Heat transport H = - ®cpDH(dT/dz)
• In discrete form
– Mass Fj = gj (Cjs – Cja) = (Cjs – Cja)/rj
– Heat H = gHcp(Ts-Ta) = cp(Ts-Ta)/rH
Conductances and resistances?
• Ohm’s law
series
– V = IR
– I = V/R
• Conductances – mol m-2 s-1
• Resistances -- m2 s mol-1
parallel
Physics of the conductance gH
• Dimensionless groups
– Re = ratio of inertial to viscous forces
– Pr = ratio of kinematic viscosity to thermal
diffusivity
– Gr = ratio of bouyant*inertial to viscous2
• Forced convection
– gH = (.664®DHRe1/2Pr1/3)/d
– gHa = 0.135 √(u/d)
(mol m-2 s-1)
– gH = (.54®DH(GrPr)1/4/d
– gHa = .05((Ts-Ta)/d)1/4
(mol m-2 s-1)
• Free convection
Heat transport by convection
• If:
– Ta = 20,Tl = 25, u = 2, d = .2
• Then
– gHa = .135(3.16) = .427
– H = gHa*2*cp*(Tl-Ta) = .427*2*29.3*5 = 125 W m-2
Latent heat: Energy carried by water
• Latent heat of vaporization (l): energy required to convert one
mol of liquid water to a mol of water vapor
l is a slight function of temp, but is about 44*103 J mol-1 at normal
ambient
– (this is 585 cal/g!)
• Latent heat of fusion: energy required to convert one mol of
solid water to a mol of liquid water 6.0*103 J mol-1
• Latent heat plays a dramatic role in temperature control.
– Water temperature won’t rise above boiling
– Frozen soil or snow won’t rise above zero
– Evaporating water requires a large amount of energy.
• 1 mm/day = 1kg/m2day, requires 2.45*106 J/m2
• since a day is 86,400 s and a Watt is a J/s, this amounts to
2.45*106/8.64*104 = 28.3 W/m2
•
• when the atmosphere is dry, evaporation can be 6 mm/day, or
even more
Evaporation
• Here, we can return directly to Fick’s law
– Fj = F’j/Mj= - ® Dj (dCj/dz)
– Fj = gj (Cjs – Cja) = (Cjs – Cja)/rj
• Where the driving gradient (Cjs – Cja) is the
difference between the water vapor inside
and outside the leaf (mol mol-1)
• And gw is a theme for another lecture
Water vapor concentration
• The amount of water vapor the air can hold is
a function of temperature = saturation vapor
pressure
• Relative humidity = ratio of actual vapor
pressure to saturation vapor pressure
Saturation vapor pressure
2
3
4
13.3185t
1.976
t
0.6445
t
0.1229
t
vsat = 101325e
where t = 1 - (373.16/T)
•
•
•
•
•
T = absolute temperature = T (ºC) + 273.16
Vsat is in Pascals – 101325 Pascals = 1 atm
Vapor pressure of the air V = Vsat*RH
Vapor pressure deficit = Vsat – V
Mol fraction (wi) = V/P where P = atmospheric pressure
Evaporation and Latent heat
• E = gw(wl – wa)
• Latent heat = lE
• Example
– If gw = .5 mol m-2 s-1, wl = 0.03 mol mol-1, wa = 0.01
mol mol-1
– Then E = .5*.02 = .01 mol m-2 s-1
 lE = .01*44*10^3 = 440 W m-2
Energy balance
• Net radiation + Convection
+ Latent heat
+ D storage
=0
– Or
• Rn + H + lE + D storage = 0
Functional role of energy balance
• Ehleringer, J., O. Björkman, and H. A. Mooney. 1976. Leaf
pubescence: effects on absorptance and photosynthesis in a
desert shrub. Science 192:376-377.
Energy balance classics – leaf scale
• Parkhurst, D. F., and O. L. Loucks, 1972: Optimal leaf
size in relation to environment. Journal of Ecology,
60, 505-537.
• Mooney, H. A., J. A. Ehleringer, and O. Björkman,
1977: The energy balance of leaves of the evergreen
desert shrub Atriplex hymenelytra. Oecologia, 29,
301-310.
• Gates, D. M., W. M. Heisey, H. W. Milner, and M. A.
Nobs, 1964: Temperatures of Mimulus leaves in
natural environments and in a controlled chamber.
Carnegie Inst. Washington Ybk., 63, 418-426.
Energy balance classics – large scale
• Charney, J., P. H. Stone, and W. J. Quirk. 1975. Drought in the
Sahara: A biogeophysical feedback. Science 187:434-435.
• Shukla, J., and Y. Mintz. 1982. Influence of land-surface
evapotranspiration on the earth's climate. Science 215:14981501.
• Bonan, G. B., D. B. Pollard, and S. L. Thompson. 1992. Effects of
boreal forest vegetation on global climate. Nature 359:716-718.
• Sellers, P. J., L. Bounoua, G. J. Collatz, D. A. Randall, D. A.
Dazlich, S. Los, J. A. Berry, I. Fung, C. J. Tucker, C. B. Field, and
T. G. Jenson. 1996. A comparison of the radiative and
physiological effects of doubled CO2 on the global climate.
Science 271:1402-1405.