Summary [MSc AP] Formula Sheet Environmental Physics

Environmental Physics Formula Sheet by Ruben Tol
Atmospheric Physics 7. Dimensional Analysis Gaussian Plumes
2 2
When the equation between variables is q (z−H)
− y 2 − 2σ2
1. Global Climate C(x, y, z) = e 2σy z ,
unknown, non-dimensionality can still 2πσy σz U
Radiative Flux: Stefan-Boltzmann yield equations by multiplying all de-
Z ∞ grees of freedom (a, b, c, d, e, . . . ) and H being the chimney stack height; sur-
I(λ) dλ = σT 4 [W/m2 ] having them equal a dimensionless con- face concentration: C(x, y = 0, z = 0).
0 stant, and then solving a system of equa-
(Mean) Solar Energy Flux tions of all unknowns (α, β, γ, δ, ϵ, . . . ) Pasquill-Giffort-Turner Classes
 2 to reach dimensionlessness and to obtain See lecture slides for tables.
RS S0 the relation between the variables.
S0 = σT 4 , S¯0 =
rES 4 σy =axα ;
aα bβ cγ dδ eϵ · · · = C [-]
Radiative Equilibrium Temperature σz =bxβ .
s 
1 − R S0 13. Weather, Wind, Clouds
TE = 4 , Energy Generation
1 − ϵ/2 4σ Wind Driving Force
atmospheric emissivity ϵ ≈ α = 0.30, Wind speeds (U horizontal, V vertical) 3. Wind Energy
Earth’s albedo (combination of atmo- are driven by temperature differences
spheric absorption and total reflection, due to its effect on pressure, and the Turbine Power
affects solar and not infrared radiation). Coriolis force generated by the rotation 3
of the Earth. P = 2ρUatm AT a(1 − a)2

2. Energy & CO2 Production dU 1 dp Efficiency: Betz Limit
=f V − ;
Ideal Gas Law dt ρ dx
16
dV 1 dp η = 4a(1 − a)2 , ηmax |a=1/3 =
nRg = − fU − ; 27
p= T, dt ρ dy
V
f =2Ω sin φ, Cut-in/Rated/Cut-Out Wind Speed
Rg = 8.3143 J K−1 mol−1 . Cut-in: Uatm exerts insufficient torque,
Ω being the Earth’s rotational speed, so no power is generated: P = 0.
Eq. Form for Atmospheric Air Parcel and φ one’s latitude.
Rated: P follows above turbine power
p = ρRd T, equation; P becomes constant at certain
Effect of Turbulence on Wind
Uatm when generator limits are reached.
Rd = 287.04 J K−1 kg−1 .
1 1 ∂p Cut-out: Uatm too high and could
u=− ;

k
2
ρf ∂y damage rotor, standstill: P = 0.
5. Boundary Layers 1+ 1+ hf

Day-time: air is mixed mostly due to k Wake Effects
v = u,
the thermal gradient. hf Just behind turbine:
Night-time: air is mixed mostly due r
k being a roughness factor; turbulence 1−a
to the density gradient: wind has to do rout = γR T , γ =
acts as a drag force. 1 − 2a
the mixing. Weak winds yield no tur-
bulence and high T differences, strong Wake Radius at x:
winds yield high turbulence and a more 14. Air Quality
uniform T distribution. Emission rx − rout = αx,
Eddy diffusivity is much larger than Strongly diluted through the day, mix- α = 0.082 (empirical).
molecular diffusivity, has smaller values ing dominated by convection; weak mix-
near the surface (where turbulent trans- ing at night, turbulent mixing only Ratio of Wind Speed at x:
port is more difficult), and is higher driven by wind shear.
in the day-time compared to the night- u(x) 2a
Atmospheric Stability =1−  2
time due to large temperature gradients Uatm αx
1 + γR
near the surface; eddy viscosity varies T
∂T
with height and time. Γ= (neutral);
∂z Power Ratio at x:
∂T
6. Turbulence Γ≈ (near-neutral); P (x)

u(x)
3
∂z =
Turbulent Flux ∂T P Uatm
Γ> (unstable);
∂z
F = ρcp w′ T ′ ∂T
Γ< (stable);
Momentum Flux/Reynolds Stress ∂z

τ = −ρu′ w′