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## Energy Balance Energy in = Energy out + Δ Storage

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**Energy BalanceEnergy 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 • s = 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**Energy balance**• Conservation of energy • Energy in = Energy out + ΔStorage • Energy transport • Radiation • Conduction • Convection = Sensible heat • Evaporation = Latent heat • ΔStorage • 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 • SW • In = direct*cosq*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 Sd = 100 W m-2 T = 10, e = 1.0* ST = 426 W m-2 ST = 365 W m-2 T = 25, e = .95, a = 0.5 ST = 426 W m-2 ST = 486 W m-2 a = 0.6 T = 35, e = .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 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) ® = 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 • 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 (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**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 215:1498-1501. • 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.