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Homework Assignment; Additional Reading: Appendix D1 to D3 Problems 7-3 and 7-6 Due October 13

Homework Assignment; Additional Reading: Appendix D1 to D3 Problems 7-3 and 7-6 Due October 13. Water Balance Approach to Estimating Lake Evaporation Inputs - Outputs = Change in Storage Inputs: Outputs: Precipitation to Lake (W) Evaporation (E)

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Homework Assignment; Additional Reading: Appendix D1 to D3 Problems 7-3 and 7-6 Due October 13

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  1. Homework Assignment; Additional Reading: Appendix D1 to D3 Problems 7-3 and 7-6 Due October 13

  2. Water Balance Approach to Estimating Lake Evaporation Inputs - Outputs = Change in Storage Inputs: Outputs: Precipitation to Lake (W) Evaporation (E) Surface water input (SWin) Surface Water Output (SWout) Groundwater Input (GWin) Groundwater Output (GWout) W + SWin + GWin – E – SWout – GWout = V Solving for E gives: W + SWin + GWin– SWout – GWout – V = E Where V is change in lake volume, with all variables expressed in units of depth or depth per time.

  3. Pan Coefficient = Lake Evaporation/Pan Evaporation

  4. Energy Balance Approach to Estimating Lake Evaporation Inputs - Outputs = Change in Storage Inputs (MJ m-2 day-1) Outputs (MJ m-2-day-1) Solar (short wave) radiation (K) Reflected short wave radiation (K) Long wave radiation from atm. (L) Long wave radiation to atm. (L) Convection from atmosphere (H) Convection to atmosphere (H) Conduction from earth (G) Conduction to earth (G) Sensible heat from water inputs (Aw) Sensible heat to water outputs (Aw) Latent heat of evaporation (LE) With inputs and outputs expressed in units of energy per unit area per time (MJ m-2 day-1) and change in storage per unit area per time =Q/t, the energy balance can be written as: K +L –G – LE – H + Aw = Q/t Solving for LE: K +L –G– H + Aw –Q/t = LE

  5. Latent Heat Transfer with Evaporation LE = wv E where LE = rate of latent heat transfer from liquid to the atmosphere w = density of water v= heat of vaporization = 2.45 MJ/kg @ 20oC E = evaporation rate expressed as depth per unit time. LE = wvKEva (es - ea) E = evaporation rate KE = mass transfer coefficient va = wind speed es = water vapor pressure at water surface ea = water vapor pressure in the atmosphere

  6. Sensible Heat Transfer between water and atmosphere H = KHva (Ts - Ta) H = rate of convective sensible heat transfer between the water surface to the atmosphere (positive value is from lake to atmosphere) KH = sensible heat transfer coefficient va = wind speed Ts = temperature of the water surface Ta = temperature of the atmosphere KH = caa 6.25 { ln (zm - zd)/zo}2 Where ca = heat capacity of the atmosphere a = density of air zm = height at which wind velocity is measured zd = zero plane displacement zo = roughness height

  7. Bowen Ratio (B): • Sensible Heat to Latent Heat Transfer Ratio • B = H/LE • H = KHva (Ts - Ta) =  (Ts - Ta) • LE wvKEva (es - ea) (es - ea) • = psychrometric constant = caP 0.622 v H= B • LE = B wvE H= sensible heat exchange with atmosphere LE = latent heat exchange with atmosphere E = evaporated water

  8. Lake Energy Balance LE = K +L –G– H + Aw –Q/t LE +H = K +L –G+ Aw –Q/t LE + B•LE= LE(1+B) = K + L –G+ Aw –Q/t LE = K + L –G+ Aw –Q/t (1+B) E = LE = K + L –G+ Aw –Q/t (wv) wv(1+B) LE = rate of latent heat transfer to the atmosphere H= rate of sensible heat transfer to or from the atmosphere K = sortwave radiation input L = longwave radiation input G= energy from the earth Aw = energy input with precip, surface water, and groundwater Q/t = change in energy stored in the lake B = Bowen ratio =  (Ts - Ta)/(es - ea)

  9. Net Longwave Radiation input or output (L) L = Lin – Lout Lin = εwεat σ (Tat + 273)4 εw = emissivity of lake surface  0.95 εat = emissivity of atmosphere σ =Stefan-Boltzman Const. = 4.9 * 10-9 MJ m-2 day-1oK-4 Tat = effective radiating temperature of the atmosphere Lout = εw σ (Ts + 273)4 Ts = temperature of the water surface (oC)

  10. Shortwave Radiant Energy Reflection Kreflected= a Kin Kin=incoming radiant energy to the receiving material a = albedo (dimensionless) Kreflected= portion of incoming radiant energy that is neither transmitted or absorbed by the receiving material Shortwave radiation input to a lake K = Kin - Kout = Kin (1- a) K = net flux of solar radiation entering the lake Kin = incident solar radiation Kout = reflected solar radiation

  11. Water Advected Energy (Sensible Heat in water inputs and outputs) Aw =cww(Qi*Ti) cw= specific heat of water =1 cal/g-oC = 4.2 kJ/kg-oC w = density of water Qi = volume of water inputs and outputs Ti = temperature of each input and output Aw =cww(W •Ta +SWin•Tswin- SWout • Tswout +GWin • Tgwin- GWout • Tgwout)

  12. Change in lake sensible heat stored between time 1 to time 2 Q = cww (TL2V2 – TL1V1) AL Where: cw = specific heat of water =1 cal/g-oC = 4.2 kJ/kg-oC w = density of water TL1 = lake temperature at time 1 TL2 = lake temperature at time 2 V1 = lake volume at time 1 V2 = lake volume at time 2 AL = lake area

  13. Mass Transfer Approach to Estimating Evaporation E = KEva (es - ea) E = evaporation rate KE = mass transfer coefficient va = wind speed es = water vapor pressure at water surface ea = water vapor pressure in the atmosphere

  14. Saturation Vapor Pressure The saturation vapor pressure is the partial pressure of water vapor in the air at 100% relative humidity. It is a function of air temperature: e* = 0.611 • exp 17.3 T (T + 237.3) Where: e*= saturation vapor pressure (K Pa) T = temperature (oC) The relationship can also be written: e* = 0.611 • ey Where y = 17.3 T (T + 237.3)

  15. Relative Humidity is the ratio of actual vapor pressure to saturated vapor pressure at the prevailing temperature Wa = ea/e*a Where Wa = Relative Humidity (fraction) ea = water vapor pressure in the atmosphere e*a = saturation vapor pressure at the air temperature e*a= 0.611 • exp 17.3 Ta (Ta + 237.3) Where: Ta = air temperature (oC) measured at the same location as ea

  16. Vapor Pressure at the Water Surface The vapor pressure at a water surface equals the saturation vapor pressure at the temperature of the water surface (es = e*s) e*s = 0.611 • exp 17.3 Ts (Ts + 237.3) Where: es = water vapor pressure at the water surface (K Pa) e*s = saturation vapor pressure at water surface temp. (K Pa) Ts = temperature at the surface (oC) If vapor pressure in the atmosphere is less than e*s, then evaporation will occur.

  17. Mechanisms of • vapor • transport • in the atmosphere: • molecular • diffusion

  18. 2) Turbulent or convective diffusion

  19. Turbulent diffusion is influenced by the atmospheric temperature profile or lapse rate which can cause stable or unstable condition.

  20. Mass Transfer Approach to Estimating Evaporation E = KEva (es - ea) E = evaporation rate KE = mass transfer coefficient va = wind speed es = water vapor pressure at water surface ea = water vapor pressure in the atmosphere

  21. Mass Transfer Coefficient: KE KE = 0.622 ρa k2= 0.622 ρa P ρw [ln{(za-zd)/zo}]2P ρw6.25 [ln{(za-zd)/zo}]2 ρa = density of air k = 0.4 P = atmospheric pressure rw = density of water za= height of temperature and velocity measurement zd = zero plane displacement (approximate elevation of wind vel.=0) zo= surface roughness height For lakes when za = 2 m, zo= 0.23 mm, zd= 0 and typical values for ρaand P, and assuming velocity is expressed in km day-1 and evaporation rate in m day-1KE = 1.46 • 10-5 m km-1 kPa-1

  22. Apparent mass transfer coefficient (KE) as a function of lake area

  23. Mass Transfer Approach to Estimating Evaporation E = KEva (es - ea) = KEva (e*s – Wa e*a)  KEva e*s (1– Wa) E = evaporation rate KE = mass transfer coefficient va = wind speed es = vapor pressure at water surface ea = vapor pressure in the atmosphere e*s = saturation vapor pressure at the surface temp Wa = relative humidity e*a = saturation vapor pressure at the air temp.

  24. Penman “Combination” Equation • describing evaporation from an open pan • E = K + L - H H = KHva (Ts - Ta) = KH va {e*s – e*a} • vw • After additional substitutions and algebraic rearrangement… • E = •(K + L) + vw KEva {e*a– ea} • vw { + } • Where: • = slope of the saturation vapor pressure vs. temperature relationship at Ta • K = short wave radiation net input H = convective heat exchange. • L = longwave radiation net input KH= sensible heat transfer coef. • = psychrometric constant v = latent heat of vaporization w= density of water KE= mass transfer coefficient va = velocity of air e*a= saturation vapor pressure of atm. ea= water vapor pressure in atm.

  25. =Slope of the saturation vapor pressure vs temperature relationship • = d e* d T • e*= 0.611 • exp 17.3 T • (Ta + 237.3) • Where: • e* = saturation vapor pressure at T (kPa) • T = temperature (oC) • = d e* = 2508.3 • exp 17.3 T d T (T +237.3)2 (T +237.3)

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