Environmental Physics Formula Sheet by Ruben Tol

Environmental Physics Formula Sheet by Ruben Tol Atmospheric Physics

• Global Climate

Radiative Flux: Stefan-Boltzmann

Z ∞

I(λ) dλ=σT 4 [W/m 2 ] (Mean) Solar Energy Flux S0= ` RS rES ´ 2 σT 4 , ¯ S0= S0 4 Radiative Equilibrium Temperature TE= 4 s ` 1−R

1−ϵ/2

´ S0 4σ , atmospheric emissivityϵ≈α= 0.30, Earth’s albedo (combination of atmo- spheric absorption and total reflection, affects solar and not infrared radiation).

• Energy & CO2Production

Ideal Gas Law p= nRg V T, Rg= 8.3143 J K −1 mol −1 .Eq. Form for Atmospheric Air Parcel p=ρRdT, Rd= 287.04 J K −1 kg −1 .

• Boundary Layers

Day-time:air is mixed mostly due to

the thermal gradient.

Night-time:air is mixed mostly due

to the density gradient: wind has to do

the mixing. Weak winds yield no tur- bulence and highTdifferences, strong winds yield high turbulence and a more uniformTdistribution.Eddy diffusivity is much larger than molecular diffusivity, has smaller values near the surface (where turbulent trans- port is more difficult), and is higher in the day-time compared to the night- time due to large temperature gradients near the surface; eddy viscosity varies with height and time.

• Turbulence

Turbulent Flux F=ρcpw ′ T ′ Momentum Flux/Reynolds Stress τ=−ρu ′ w ′

• Dimensional Analysis

When the equation between variables is unknown, non-dimensionality can still yield equations by multiplying all de- grees of freedom (a, b, c, d, e, . . .) and having them equal a dimensionless con- stant, and then solving a system of equa- tions of all unknowns (α, β, γ, δ, ϵ, . . .) to reach dimensionlessness and to obtain the relation between the variables.a α b β c γ d δ e ϵ

· · ·=C[-]

• Weather, Wind, Clouds

Wind Driving Force Wind speeds (Uhorizontal,Vvertical) are driven by temperature differences due to its effect on pressure, and the Coriolis force generated by the rotation of the Earth.dU dt =fV− 1 ρ dp dx ; dV dt =−fU− 1 ρ dp dy ; f=2Ω sinφ, Ω being the Earth’s rotational speed, andφone’s latitude.Effect of Turbulence on Wind u=− 1

• +

ı

• +

k hf ȷ 2 1 ρf ∂p ∂y ; v= k hf u, kbeing a roughness factor; turbulence acts as a drag force.

• Air Quality

Emission Strongly diluted through the day, mix- ing dominated by convection; weak mix- ing at night, turbulent mixing only driven by wind shear.Atmospheric Stability Γ = ∂T ∂z (neutral); Γ≈ ∂T ∂z (near-neutral); Γ> ∂T ∂z (unstable); Γ< ∂T ∂z (stable); Gaussian Plumes C(x, y, z) = q 2πσyσzU e − y 2 2σ 2 y − (z−H) 2 2σ 2 z, Hbeing the chimney stack height; sur-

face concentration:C(x, y= 0, z= 0).

Pasquill-Giffort-Turner Classes See lecture slides for tables.σy=ax α ; σz=bx β .Energy Generation

• Wind Energy

Turbine Power

P= 2ρU

3 atmATa(1−a) 2

Efficiency: Betz Limit

η= 4a(1−a) 2 , ηmax| a=1/3= 16 27 Cut-in/Rated/Cut-Out Wind Speed

Cut-in:Uatmexerts insufficient torque,

so no power is generated:P= 0.

Rated:Pfollows above turbine power

equation;Pbecomes constant at certain Uatmwhen generator limits are reached.

Cut-out:Uatmtoo high and could

damage rotor, standstill:P= 0.

Wake Effects

Just behind turbine:

rout=γRT, γ= r 1−a 1−2a

Wake Radius atx:

rx−rout=αx, α= 0.082 (empirical).

Ratio of Wind Speed atx:

u(x) Uatm

= 1−

2a ı

• +

αx γRT ȷ 2

Power Ratio atx:

P(x) P = ` u(x) Uatm ´

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