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.