1. Energy Balance Energy in = Energy out + Δ Storage Bio 164/264 January 11, 2007 C. Field

2. 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*10 8 m s -1 ), and is wavelength (m). Thermal radiation depends on T 4: 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

3. 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

4. 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*a   diffuse 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* S T = 426 W m -2 S T = 365 W m -2 S T = 486 W m -2 S S = 600 W m -2,  = 20 S d = 100 W m -2 a = 0.6

5. Conduction Not very important in this class.

6. Convection Rate of transport = driving force * proportionality factor –Fick’s law – diffusion F’ j = -D j (d  j /dz) D = molecular diffusivity –Fourier’s law – heat transport H = -k (d  /dz) k = thermal conductivity (m 2 s -1 ) –Darcy’s law – water flow in a porous medium J w = -K(  ) (d  /dz) K(  ) = hydraulic conductivity

7. Keeping units straight - Moles Most of the mass fluxes in this class will be in moles, where 1 mole = m.w. in g –N 2 1 mole = 28.01 g –O 2 1 mole = 32.00 g –CO 2 1 mole = 44.01 g –H 2 O 1 mole = Molar density (mol m -3 ) ® =  j /M j is the same for all gases –Ideal gas law p j V = n j RT –= 44.6 mol m -3 @ 0C and 101.3 kPa (STP) –® =  j /M j

8. First – get mass flux in molar units Convert Fick’s law to molar units –diffusion F’ j = -D j (d  j /dz) –F j = F’ j /M j = - ®  D j (dC j /dz) D = molecular diffusivity C j = mole fraction of substance j

9. Convection – moving heat in air Start with Fourier’s law –Heat transport H = -k (d  /dz) k = thermal conductivity c p = molar specific heat of air 29.3 J mol -1 C -1 k/c p = D H = thermal diffusivity –Heat transport H = - ®c p D H (dT/dz) In discrete form –Mass F j = g j (C js – C ja ) = (C js – C ja )/r j –Heat H = g H c p (T s -T a ) = c p (T s -T a )/r H

10. Conductances and resistances? Ohm’s law –V = IR –I = V/R Conductances – mol m -2 s -1 Resistances --m 2 s mol -1 series parallel

11. Physics of the conductance g H Dimensionless groups –Re = ratio of inertial to viscous forces –Pr = ratio of kinematic viscosity to thermal diffusivity –Gr = ratio of bouyant*inertial to viscous 2 Forced convection –g H = (.664®D H Re 1/2 Pr 1/3 )/d –g Ha = 0.135 √(u/d) (mol m -2 s -1 ) Free convection –g H = (.54®D H (GrPr) 1/4 /d –g Ha =.05((Ts-Ta)/d) 1/4 (mol m -2 s -1 )

12. Heat transport by convection If: –T a = 20,T l = 25, u = 2, d =.2 Then –g Ha =.135(3.16) =.427 –H = g Ha *2*c p *(T l -T a ) =.427*2*29.3*5 = 125 W m -2

13. 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*10 3 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*10 3 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/m 2 day, requires 2.45*10 6 J/m 2 since a day is 86,400 s and a Watt is a J/s, this amounts to 2.45*10 6 /8.64*10 4 = 28.3 W/m2 when the atmosphere is dry, evaporation can be 6 mm/day, or even more

14. Evaporation Here, we can return directly to Fick’s law –F j = F’ j /M j = - ®  D j (dC j /dz) –F j = g j (C js – C ja ) = (C js – C ja )/r j Where the driving gradient (C js – C ja ) is the difference between the water vapor inside and outside the leaf (mol mol -1 ) And g w is a theme for another lecture

15. 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

16. Saturation vapor pressure wheret = 1 - (373.16/T) T = absolute temperature = T (ºC) + 273.16 V sat is in Pascals – 101325 Pascals = 1 atm Vapor pressure of the air V = V sat *RH Vapor pressure deficit = V sat – V Mol fraction (w i ) = V/P where P = atmospheric pressure

17. Evaporation and Latent heat E = g w (w l – w a ) Latent heat = E Example –If g w =.5 mol m -2 s -1, w l = 0.03 mol mol -1, w a = 0.01 mol mol -1 –Then E =.5*.02 =.01 mol m -2 s -1 – E =.01*44*10^3 = 440 W m -2

18. Energy balance Net radiation + Convection + Latent heat +  storage = 0 –Or Rn + H + E +  storage = 0

19. 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.

20. 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.

21. 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 CO 2 on the global climate. Science 271:1402-1405.