Solar Updraft Tower Questions

[StoffStoff]

Hello Everyone,

I am currently working on a design project that deals with solar updraft concepts and I would urgently need some feedback and advice on two Problems i am encountering.

Problem 1) - I need to calculate the daily (preferably hourly) global solar insolation in W/m² for Ciudad Real in Spain. I am using the following equations and data to do so:

Extraterrestrial Solar Radiation Calculation

Ho = (1/p) * I_sc *E_o *[cos (λ)*cos(δ)*sin(ωs) +(p/180)*sin(λ)* sin(δ)* ω_s]

where
Eo = 1.00011 + 0.034221*cos(Γ) + 0.00128*sin(Γ) + 0.000719*cos(2Γ) + 0.000077*sin(2Γ)
δ = (180/p) *(0.006918 - 0.399912*cos(Γ) + 0.07025*sin(Γ) - 0.006758*cos(2Γ) + 0.000907*sin(2Γ) - 0.002697*cos(3Γ)+0.00148*sin(3Γ))
Γ = 2*p *(n_day - 1)/365
ωs = cos-1 [-sin(λ)* sin(δ)/cos(λ)* cos(δ)] or ωs = cos-1 [-tan(λ)* tan(δ)]

H_o = Daily Solar Radiation above the Atmosphere [MJ/m²d]
E_o = Eccentricity correction factor of Earth's orbit [-]
δ = Solar Declination [degrees]
Γ = Day Angle [radians]
ω_s = Hour Angle of the Sun [degrees]
pi = 3.14159
n_day = Day of the year ,e.g. January 1 = n_day = 1 , February 1 = n_day = 32, etc.
λ = Latitude of the site [degrees]
I_sc = Solar constant = 118.11 [MJ/m2d]

Global Solar Radiation Calculation using the Hargreaves and Samani Model

H = H_o [A_Model1(T_max - T_min)1/2] (H is also known as G)
where
H = Daily Global Solar Radiation [MJ/m²d]
H_o = Daily Solar Radiation above the Atmosphere [MJ/m²d]
T_max = Daily Maximum Temperature [^oC]
T_min = Daily Minimum Temperature [^oC]
A_Model1 = Empirical Coefficient equal to 0.16 for interior regions [^oC -0.5]


Conversion from MJ/m2[/sup]d to W/m²

1 MJ/m²d = 1E6 J/ m2[/sup]d = (1.000.000 J / m²d )*(d/60*60*24 s) = 11.57 J/ s m² = 11.57 W/m²
1 MJ/m²d = 11.57 W/m²

Questions:

1)
Are the equations for Global Solar Radiation, H (or G) and Extraterrestrial Solar Radiation, H_o correct ?

2)
Do the equations for H and Ho calculate the daily average values for Global Solar Radiation (G_average) and Extraterrestrial Solar Radiation (Ho,average)?

3)
Is the conversion from MJ/m²d to W/m² correct ?

4)
Are there more efficient or simpler yet accurate ways to calculate Global Solar Radiation for a given location ?

5)
Are my calculations of H_o and H correct ?

6)
Are there freely available data sets for the daily global horizontal solar radiation, (H or G) available for different Spanish towns and cities (Ciudad Real and Madrid in particular ? If so , where can I find them ?

I used Excel to process the following data and attached the file to the thread/post

Input Data:

Set 1
n_day = 1 (January 1st)
T_max = 12 C
T_min = 6 C
λ = 39 degrees
Set 2
n_day = 200 (July19th)
T_max = 30 C
T_min = 12 C
λ = 39 degrees

Results:

Set 1
E_o = 1,04 [-]
δ = -22.72 degrees / -0.40 radians
Γ = 0 degrees /radians
ω_s = 70.18 degrees / 1.22 radians
H^o = 26.04 MJ/m²d
H = 10.21 MJ/m²d = 118.10 W/m²
Set 2
E_o = 0,97 [-]
δ = 21.30 degrees / 0,37 radians
Γ = 3.43 degrees / 0.06 radians
ω_s = 108.40 degrees / 1.89 radians
H_o = 25.25 MJ/m2d
H = 17.14 MJ/m²d = 198.31 W/m²

Problem 2) The calculation of the power output P_turbine of the turbine a solar updraft tower

I am using the following equations to calculate P_turbine:

Equation 1 [(a*t*G - β*ΔT) *p*d²]/4 = Q*ρ*c_p* ΔT
Equation 2 Q = K*A*[2*g*h*(ΔT)/ T₁]0.5
Equation 3 P_turbine = c_p*ρ*Q*(ΔT)

where
G = global solar radiation [W/m²]
a = soil absorptance of solar irradiation (= 0.89) [-]

t = glass transmittance for solar irradiation (=0. 88) [-]
β = equivalent heat losses [W/K]
ρ = density of air [kg/m3]
ΔT = T₂ -T₁ [K]
T₁ = Temperature at the collector entrance [K]
T₂ = Temperature at the collector exit at the turbine [K]
d = diameter of the solar collector [m]
c_p = specific heat of air [J/Kg*K]
g = acceleration due to gravity [m/s²]
h = height of the solar tower stack [m]
K = discharge coefficient (= 0.65)
A = cross-sectional area of the solar updraft tower stack [m²]
Q = air mass moving speed [Kg/s]

Questions:

1)
Are equations 1, 2 and 3 correct?

2)
Can equation 1, 2 and 3 be used to calculate the daily power produced by the turbine ?

3)
Equations 1 and 2 are supposed to be solved simultaneously to calculate ΔT. How can this be done properly ?

I attempted to solve for ΔT and ended up with the following equations:

for equation 1, ΔT = (a*t*G*p*d²) / (4*Q*ρ*c^p + β*p*d²)
for equation 2, ΔT = (Q2*T₁) / [(K*A)²*2*g*h]

I set T1 equal to the average environmental temperature that exists outside the collector
Since Q is unknown, I attempt to solve for Q and got the following results:

(4 * ρ* cp* T₁) *Q³ + ( β*pi*d²*T₁) *Q² - (K*A)2 * (2*g*h) * (a*t*G*p*d2) = 0

The equation above is of the following form:

a*x³ + b*x² + c*x +d = 0

where
a = 4 * ρ* c_p* T₁
b = β*pi*d²* T1
c = 0
d = - (K*A)2 * (2*g*h) * (a*t*G*p*d²)

This cubic equation will yield 3 solutions/roots.


Sources:

1)

http://demonstrations.wolfram.com/SolarUpdraftTower/

2)

http://www.floatingsolarchimney.gr/Downloads/Documentation/paper3_2004.pdf

3)

http://www.fem.unicamp.br/~phoenics/EM974/PROJETOS/Temas Projetos/Solar Chimneys/solar_updraft.pdf

4)

http://www.arch.hku.hk/teaching/lectures/airvent/

 

[JStephen]

1) Don't know. You should be able to find similar equations elsewhere to confirm. I would think the numbers would be reasonably close on that kind of thing.
2) Refer to the source of the equations for the definitions. If not shown, see if they show technical papers as references, and hunt those up in a good university library.
3) Don't know, shouldn't be too hard to work through it, though. Make sure you're not multiplying peak power x 24 hours for energy per day or something like that.
4) Long long ago, I took a solar engineering class, and I recall stuff similar to this. The calculations are a pain, but you can program them into a spreadsheet (or in the days of yore, a programmable calculator).
5) Haven't checked.
6) You may have trouble finding exactly what you're looking for. For example, you may find solar heat gain, or average including cloudy days or who knows what. There again, check a good university library for other books on solar engineering.

1) Don't know if the equations are correct.
2) Maybe
3) Any way that works is good. Quite often, for real situations, only one of the three roots is required. In a lot of cases, iteration using the equations is quicker than coming up with a direct solution. With a spreadsheet, rootfinding by bisection or similar means is quite often quicker than direct methods, and less prone to error.

I don't know the entire physical situation here. I'm visualizing a big flat round "roof" that the sun shines on, it heats the air underneath, and that air going up the stack is what generates the power. In that case, you could potentially have some major flow losses in the flow under the big flat roof, or due to wind, or due to transient effects (warm-up in the mornings, for example). So your final answer could be considerably off from the actual power available, and it could be a lot of work to come up with better answers. The atmosphere cools as you go up, so if this is 100' high, it's negligible, if it's a 1,000' high, you need to factor that in as well.

This sounds sort of like homework (which isn't allowed at the site by the way), but I struggle finding time to check my own work, so pardon me if I don't check through somebody else's in detail.

 

[GregLocock]

Just as a reasonableness check daily average insolation should be at least 7kWh/m^2 in Spain , so your H values look low.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

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[IRstuff]

http://pveducation.org/pvcdrom/properties-of-sunlight/average-solar-radiation

this says that value is more like the peak, which I would tend to agree with. MIL-STD-310 maximum value is about 9 kWh/m^2

I haven't seen anything that suggests that solar chimneys are particularly efficient. There seems to be a very low conversion rate of solar energy to electrical energy.

TTFN
faq731-376

Need help writing a question or understanding a reply? forum1529

Of course I can. I can do anything. I can do absolutely anything. I'm an expert!

 

[racookpe1978]

1. Ciudad Real in Spain. Can you confirm: 38.9833° N, 3.9167° W for
Ciudad Real, Coordinates

2. Horizontal surface flat surface, or inclined flat surface -> At what angle is the assumed flat surface tilted up from horizontal?

3. Trained surface (rotating so it always faces the sun every day?)

4. Round or cylinder or what shape surface? What color? What material?

5. Theoretical direct sunlight only? Global (direct & diffuse radiation combined) radiation? Diffuse only? What percent clouds are you assuming/have you verified for every month of the year?

Yes. I have such a spreadsheet (yielding daily and hourly radiation each day-of-year), but it uses different source papers than yours - which may or may not be a problem.