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Homework No. 4 - Solution Homework No. 4 - Solution 1. For Lake Nasser, Egypt, in July, the average net radiation is 170 W m-2; the mean daily air temperature is 28.5 ºC, the relative humidity is 55 percent and the wind speed is 2.7 m s-1 at z2=2 m. D At the above temperature is equal to 3.6. Assume zo=0.0003 m and p=101300 Pa. g a. Obtain the open water evaporation rate in millimeters per day using the Thornthwaite- Holzmann aerodynamic or mass transfer method. b. Obtain the open water evaporation rate in millimeters per day using the Penman equation. Use both the aerodynamic method and the empirical Penman equation for the evaluation of the drying power of the air, and compare your results. For each of these conditions, what is the Bowen Ratio? c. Obtain the equilibrium evaporation rate, Ee. d. Obtain the Bowen Ratio implied by the equilibrium evaporation rate and compare with the Bowen Ratio obtained in part b. e. Obtain estimates of the sensible heat flux for these conditions. f. Obtain estimates of the evaporation rate under conditions of minimal advection, Epe. Useae = 1.28. Obtain the Bowen Ratio implied by this rate of evaporation under conditions of minimal advection. g. Obtain the evapotranspiration rate if the above conditions apply to a (well-watered) vegetated surface for which the stomatal resistance equals 200 s/m. For the aerodynamic resistance use the following expression: 2 z2 ln zo r := av k 2 u O restart;with(student); D, Diff, Doubleint, Int, Limit, Lineint, Product, Sum, Tripleint, changevar, completesquare, distance, equate, integrand, intercept, intparts, leftbox, leftsum, makeproc, middlebox, middlesum, midpoint, powsubs, rightbox, rightsum, showtangent, simpson, slope, summand, trapezoid Define evaporation functions needed for this solution: Saturation vapor pressure function (Clausius-Clapeyron eqn.) in Pascals. Note: To = 273.15 K.. 1 1 Lv K T To K Rv es = 611 e . O es:=(TT)->611*exp(-Lv/Rv*(1/TT-1/273.15)); 1 1 Lv K TT 273.15 K es := TT/611 e Rv 1 Define slope of saturation vapor pressure function Lv es T Delta = Rv T 2 O Delta:=T->Lv*es(T)/Rv/T^2; Lv es T D := T/ Rv T 2 Define psychrometric constant Cp p gamma = epsilon Lv O Gamma:=Cp*p/epsilon/Lv; Cp p G := e Lv Define moist air density function 1 K epsilon e p 1 K p rho := Rd T O rho:=(p,Ta,e)->p/Rd/Ta*(1-0.378*e/p); 0.378 e p 1 K p r := p, Ta, e / Rd Ta Define Thornthwaite-Holzmann equation (e.g., Aerodynamic method) 2 rho k u2 epsilon es2 K e2 E = 2 z2 p ln zo O Ea:=(p,Ta,e)->rho(p,Ta,e)*k^2*u*epsilon*(es(Ta)-e)/p/(ln(z2/zo)) ^2; 2 r p, Ta, e k u e es Ta K e Ea := p, Ta, e / z2 2 p ln zo Define equilibrium evaporation equation Delta Rn E = e DeltaC gamma Lv O Ee:=(T)->(Delta(T)/(Delta(T)+Gamma))*Rn/Lv; 2 D T Rn Ee := T/ D T C G Lv Define Partial Equilibrium Evaporation equation Epe = alphae Ee O Epe:=1.28*Ee; Epe := 1.28 Ee Define Penman Equation Delta Rn gamma EA E = C DeltaC gamma Lv DeltaC gamma O E:=T->(Delta(T)/(Delta(T)+Gamma))*Rn/Lv+(Gamma/(Delta(T)+Gamma))* EA; D T Rn G EA E := T/ C D T C G Lv D T C G Define the drying power of the air using the equation proposed by Penman (1948) EA = f u es2 K e2 where the winf function f(u) is: f(u) = 0.26(1+0.54u). This equation for the drying power of the air yields an evaporation rate in mm/day for u in m/s and vapor pressures in mb. Thus, divide by 100 to convert from Pa to mb, divide by 1000 to convert to m, multiply by density of liquid water, and divide by 86400 to convert from days to seconds. O f:=u->0.26*(1+0.54*u);Eaa:=u->f(u)*(es(Ta)-e)/100/86400/1000* 1000; f := u/0.26 1 C 0.54 u 1 Eaa := u/ f u es Ta K e 8640000 Define modified Penman equation for vegetated surfaces (Penman-Monteith equation) Delta Rn rho Cp es2 K e2 C Lv Lv rav E = rst DeltaC gamma 1 C rav O Ev:=T->(Delta(T)*Rn/Lv+rho(p,T,e)*Cp*(es(Ta)-e)/Lv/rav)/(Delta(T) +Gamma*(1+rst/rav)); D T Rn r p, T, e Cp es Ta K e C Lv Lv rav Ev := T/ rst D T C G 1 C rav Define input data for this problem: O Ta:=28.5+273.15; 3 Ta := 301.65 O u:=2.7; u := 2.7 O p:=101300;z2:=2;zo:=0.0003;epsilon:=0.622;Rd:=287;k:=0.4;Lv:= 2500000;Rv:=461.5;Rn:=170;Cp:=1004;rst:=200;Rs:=170;rh:=0.55; p := 101300 z2 := 2 zo := 0.0003 e := 0.622 Rd := 287 k := 0.4 Lv := 2500000 Rv := 461.5 Rn := 170 Cp := 1004 rst := 200 Rs := 170 rh := 0.55 a) Obtain the open water evaporation rate using the Thornthwaite-Holzmann aerodynamic or mass transfer method (kg/m2/s).Note: the evaporation rates below are given in kg/m2/s and in mm/day. O e:=rh*es(Ta);es(Ta);rho(p,Ta,e); e := 2188.555337 3979.191521 1.160547605 O EA:=evalf(Ea(p,Ta,e));%*86400; EA := 0.00007110310828 6.143308555 b) Obtain the open water evaporation rate in using the Penman equation. Use both the aerodynamic method and the empirical Penman equation for the evaluation of the drying power of the air, and compare your results. For these conditions, what is the Bowen Ratio and the impied sensible heat? Compute Delta Pa/K O evalf(Delta(Ta)); 236.8953229 Compute gamma Pa/K 4 O evalf(Gamma); 65.40527332 Delta Compute gamma O Delta(Ta)/Gamma; 3.621960599 Delta Compute DeltaC gamma O evalf(Delta(Ta)/(Delta(Ta)+Gamma)); 0.7836416000 gamma Compute DeltaC gamma O evalf(Gamma/(Delta(Ta)+Gamma)); 0.2163584000 Compute Evaporation using Penman equation and an estimate of EA based on the Thornthwaite- Holzmann equation (kg/m2/s): O Epen:=E(Ta);%*86400; Epen := 0.00006867138354 5.933207538 O Bo:=Rn/Lv/Epen-1; Bo := K0.0097767586 Obtain estimates of the sensible heat flux for these conditions, Watts/m2. O H:=Bo*Lv*Epen; H := K1.678458849 Define the drying power of the air using the empirical equation proposed by Penman (1948) (kg/m2/s) O EA:=Eaa(u);%*86400; EA := 0.0001324490477 11.44359772 Recompute Penman evaporation using the new value of drying power of air (kg/m2/s). O Epen:=evalf(E(Ta));%*86400; Epen := 0.00008194409284 7.079969621 O Bo:=Rn/Lv/Epen-1; Bo := K0.1701659309 5 O Obtain estimates of the sensible heat flux for these conditions, Watts/m2. O H:=Bo*Lv*Epen; H := K34.86023210 c) Obtain the equilibrium evaporation rate, Ee (kg/m2/s). O Eeq:=evalf(Ee(Ta));%*86400; Eeq := 0.00005328762880 4.604051128 d) Obtain the Bowen Ratio implied by the equilibrium evaporation rate and compare with the Bowen Ratio obtained in part b. O Bo:=Rn/Lv/Eeq-1;Bo:=(Delta(Ta)+Gamma)/Delta(Ta)-1; Bo := 0.276093561 Bo := 0.276093561 Obtain estimates of the sensible heat flux for these conditions Watts/m2. O H:=Bo*Lv*Eeq; H := 36.78092798 f) Obtain estimates of the evaporation rate under conditions of minimal advection, Epe (kg/m2/s). Use ae = 1.28. Obtain the Bowen Ratio implied by this rate of evaporation under conditions of minimal advection. O Epeq:=evalf(Epe(Ta));%*86400; Epeq := 0.00006820816486 5.893185444 O Bo:=Rn/Lv/Epeq-1; Bo := K0.0030519053 Obtain estimates of the sensible heat flux for these conditions, Watts/m2. O H:=Bo*Lv*Epeq; H := K0.5204121496 g) Obtain the evapotranspiration rate if the the above conditions apply to a (well-watered) vegetated surface (kg/m2/s). Compute the aerodynamic resistance (s/m): O rav:=(ln(z2/zo)/k)^2/u; rav := 179.4579361 O Evv:=evalf(Ev(Ta));%*86400; Evv := 0.00005532997800 4.780510099 6.
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