Energy Balance
Energy in = Energy out + Δ Storage
Bio 164/264January 11, 2007
C. Field
Radiation: Reminders from last time
• Energy of a photon depends on 1/wavelength– E = hc/– h is Planck’s constant (6.63*10-34 Js), c is the speed of
light (3*108m s-1), and is wavelength (m).• Thermal radiation depends on T4: Stefan-Boltzmann law
–
– = 5.67 * 10-8 W m-2 K-4
• Wavelength of maximum energy depends on 1/temperature (Wien Law)–
• Solar “constant” ~ 1360 W m-2, over sphere = 342 W m-2
B =T4
m = 2897T
Energy balance
• Conservation of energy• Energy in = Energy out + Δ Storage• Energy transport
– Radiation– Conduction– Convection = Sensible heat– Evaporation = Latent heat
• Δ Storage– Change in temperature– Change in the energy stored in chemical bonds– Change in potential energy
Radiation balance
• Thermal– In = IR down + IR up– Out = IR down + IR up– =461 + 346 – 397 – 397 = 63
• SW– In = direct*cos*a
diffuse down*adiffuse up *a= 282 + 120 + 50 W m-2
Out = reflected up + reflected down+ transmitted down+
transmitted up = already included in in
T = 25, = .95, a = 0.5
T = 35, = .95
T = 10, = 1.0*
ST = 426 W m-2
ST = 426 W m-2
ST = 365 W m-2
ST = 486 W m-2
SS = 600 W m-2, = 20
Sd = 100 W m-2
a = 0.6
Conduction
• Not very important in this class.
Convection
• Rate of transport = driving force * proportionality factor– Fick’s law – diffusion F’j = -Dj (dj/dz)
• D = molecular diffusivity
– Fourier’s law – heat transport H = -k (d/dz)• k = thermal conductivity (m2 s-1)
– Darcy’s law – water flow in a porous medium• Jw = -K() (d/dz)
• K() = 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 1 mole = 28.01 g– O2 1 mole = 32.00 g– CO2 1 mole = 44.01 g– H2O 1 mole =
• Molar density (mol m-3) ® = j/Mj is the same for all gases– Ideal gas law pjV = njRT– = 44.6 mol m-3 @ 0C and 101.3 kPa (STP)– ® = j/Mj
First – get mass flux in molar units
• Convert Fick’s law to molar units– diffusion F’j = -Dj (dj/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 (d/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– V = IR– I = V/R
• Conductances – mol m-2 s-1
• Resistances -- m2 s mol-1
series
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)
• Free convection– gH = (.54®DH(GrPr)1/4/d
– gHa = .05((Ts-Ta)/d)1/4 (mol m-2 s-1)
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 (): energy required to convert one mol of liquid water to a mol of water vapor 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 pressurewhere 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
vsat = 101325e 13.3185t - 1.976 t2 - 0.6445 t3 - 0.1229 t4
Evaporation and Latent heat
• E = gw(wl – wa)
• Latent heat = E• 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
E = .01*44*10^3 = 440 W m-2
Energy balance
• Net radiation + Convection + Latent heat + storage= 0
– Or
• Rn + H + E + 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 215:1498-1501.
• 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.
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