Formula memento ME-460

Solar energy Solar energy harnessed in 1 day, on horizontal extraterrestrial surface: 1.Solar constant correction: (i.e. day D of the year) 2. Declination  correction: (day J of the year) 4. Duration H D/2 of half a day of sunlight: (expressed as an angle, where 1 h = 15° or  /12) 1367 W/m 2 3.Latitude  correction (location) :

K = conduction/convection loss coefficient (W/m 2.K) A 0 = absorption coefficient (high absorption, low emission) Solar power transmitted to fluid per m 2 of collector TypeA 0 (%)K (W/m 2.KPrice selective coating, no glazing /m 2 selective coating, glazing vaccum tubes summer mid-season winter summer mid-season winter E in T amb T coll TTx =  T  / E in 80030°70°C °50°C °C30°C W/m 2  Solar thermal flat collector

Photovoltaics short circuit-current leakage current

Process / Maximal solar input on ground level: 1400 kWh/m 2.yr or 160 W/m 2 =100% Solar radiation energy  photosynthetic active part, PAR ( nm)43% Maximum capture by leafs (canopy) = 80%34.4% Maximum photonic energy capture efficiency into glucose = 28.6%9.8% ⅓ on average of the glucose energy is used for the plant metabolism (respiration)6.6% Max. practical efficiency of ‘ C-4 ’ ‘energy’ plants (corn, sorghum, sugar cane), daily basis (24h) 5% Max. practical efficiency of ‘ C-3 ’ common plants (=95% of biomass, e.g. wheats, rice, trees,…), daily basis (24h) 3%  from the available 5.1 E+23 J/yr radiation (1400 kWh/m 2.yr), thus 3% is theoretically captured by common biomass (42 kWh/m 2.yr = 150 MJ/m 2.yr) 1.5 E+22 J/yr (4.8 W/m 2 ) Climate factors, shading, and biomass density per m 2 drop this capture efficiency by another factor 5 (  1 W/m 2 = 8.4 kWh/m 2.yr = 30 MJ/m 2.yr ≈ 2 kg wood/m 2.yr) 0.6% 3 E+21 J/yr Biomass photosynthetic efficiency 1 W/m 2 is a rather poor storage density ! Even for the ‘2 kW-society’, every citizen would then need his personal 2000 m 2 ‘storage’

Biomass fuel energy content where C = wt% carbon content where C = cellulose+hemicellulose wt% content; 1-C = lignine content W = wt% water content

Biogas generation from organic waste Remark: CO 2, NH 3, H 2 S dissolve better in H 2 O than CH 4, hence the recovered gas is actually methane-enriched Buswell-Boyle (with N, S): Buswell formula: e.g. for manure, approximated as C 4 H 8 O 2 (butyric acid):

36 HYDROPOWER A Hydro-turbine power The energy given to the turbine blades is given by (Bernoulli): + v1v1 2 2 g·Z 1 + P1P1  - g·Z 2 + P2P2  - v2v2 2 = w 12 Components: potential (pressure) kinetic (speed) 1  inlet gate of the wheel 2  outlet gate of the wheel 1000 kg/m 3 volume flow m 3 /s 9.81 m 2 /s 2 head (height drop in m)

35 HYDROPOWER A Specific turbine speed The essential characteristics of a water turbine are summarized in a single parameter, the specific speed   = .  V 1/2 2w2w 3/4 volumetric flow [m 3 /s] angular speed [rad/s] mass energy at disposal for the turbine [J/kg] – given by the height of the water drop and the flow volume  = n. 2  rotation speed [rotations per second] n = 50/p (p: # alternator pairs of poles) If  is low, V vs. w (=V x H) is low, hence the water drop H is high

46 WIND ENERGY A P = probability for ( v > v *) = exp[-( v */c) k ] Wind Weibull parameters k and c Slope = k ln(v*) Intercept = - k ln(c) log Approximate formulas: for 1.6 ≤ k ≤ 3.0: for 1.8 ≤ k ≤ 2.3:

Wind power (Betz theory) chord radius

Wind turbine  =  +   = arctan (2R / 3 r ) N:number of blades position (r) between hub and blade tip turbine radius Tip speed ratio angular speed rad/s wind speed torque angle of attackpitch angle chord length

Tidal turbine 35% max spring tide power = 4 x max neap tide power average max power = (spring max + neap max ) / 2 average power (W/m 2 ) = 0.4  avg max power

Wave power W wave = M z ·g·2y c = 2  ·y 0 ·  ·g·  ·y0·y0 8 =  ·g·y 0 · 2 4 P [W/m] = 3064·y 0 1/2 2 · (wave amplitude y 0 in m, wave length in m) Energy J/m per m wide wave front 1000 kg/m m 2 /s 2 Period (s) of the wave (inverse of frequency with which waves roll in) wavelength (m) wave speed (m/s) (= also the wind speed)

Geothermal power Maximum available power (exergy): Q = massflow (kg/s) * Cp *  T 4184 J/kg 1 st law efficiency2 nd law efficiency