On a Model for a Solar Updraft Tower

Kinetic and Mean-Field Models in the Socio-Economic
Sciences, ICMS, Edinburgh, July 2009
On a Model for a Solar Updraft Tower
Ingenuin Gasser
Department Mathematik, University of Hamburg, Germany
Overview: Solar Updraft Towers
• Introduction
• Modelling
• Analysis
• Numerical Simulations
• Validation of the Model
Solar Updraft Towers: Manzanares (Spain)
J. Schlaich, R. Bergermann, W. Schiel, G. Weinrebe, Journal of Solar Energy
Engineering 127 (1), 117-124, 2005
J. Schlaich, R. Bergermann, W. Schiel, G. Weinrebe, Journal of Solar Energy
Engineering 127 (1), 117-124, 2005 Wikipedia
Quelle: J. Schlaich, R. Bergermann, W. Schiel, G. Weinrebe, Journal of Solar
Energy Engineering 127 (1), 117-124, 2005
J. Schlaich, R. Bergermann, W. Schiel, G. Weinrebe, Journal of Solar Energy
Engineering 127 (1), 117-124, 2005
Solar Updraft Towers:
History:
• 1903 - Isidoro Cabanyes: Idea
• 1931 - Hanns Günther: Precise description
• 1982 - Jörg Schlaich: Prototye in Manzanares (Spain)
Facts:
•
•
•
•
•
Renewable solar energy
No water required
Low-tech plant
Little maintenance
Low efficiency
Modelling issues :
Velocities:
≈ 0 − 20 m/s
Temperatures:
≈ 10◦ − 50◦ Celsius ?
• Gas dynamics
• Energy source (greenhouse effect)
• Dynamics is driven by small changes in density (chimney effect)
• Small Mach number!
• Need of a precise description of the energy transport
• Compressible vs. incompressible model
• Multi-d vs. 1 d model
State of the art:
• 3d CFD Models (Müller 02)
• 3 different models (collector, turbine, chimney)
(Dos Santos Bernardes, Voss, Weinrebe 03,04)
Our aim:
*) to set up a simple (1d) model for the full plant,
*) which decribes the main features
*) which allows control and optimisation procedures.
Order of magnitude of the buoyancy forces in the chimney
• Height/radius of the chimney 200 m/5 m
• Volume of the chimney 15707 m3
• Pressure difference per meter altitude 0.12 mbar
• Buoyancy force: an elevation of the temperature of 10 Grad Celsius gives
about 0, 32 Newton buoyancy force for 1 m3 air.
• the full chimney volume gives 5026 Newton
• this corresponds to the weight of 500kg
• i.e. 64 Newton per square meter cross section
• i.e. an overpressure of 0, 64 mbar
Starting point:
1d compressible Euler equations
(Ãρ̃)t̃ + (Ãρ̃ũ)x̃ = 0,
(Ãρ̃ũ)t̃ + (Ã(ρ̃ũ2 + p̃))x̃ = Ãx̃p̃ −
˜ turbine (Ãũ)
−Ãgρ̃ sin(α) + ∆p
λco
λch p
à +
π Ã ρ̃ũ|ũ|
4hc
4
ρ̃ũ2
ρ̃ũ2
=
Ã(cv ρ̃T̃ +
+ gρ̃h̃) + Ãũ(cv ρ̃T̃ +
+ gρ̃h̃)) + Ãũp̃
2
2
t̃
x̃
p
Ã
Ã
αs Q̃s − αco 2 (T̃ − T̃co ) − αch2 π Ã(T̃ − T̃ch)
hc
hc
Variables: x̃, t̃, ρ̃ = ρ̃(x̃, t̃), ũ = ũ(x̃, t̃), p̃ = p̃(x̃, t̃), T̃ = T̃ (x̃, t̃)
Variable cross section: Ã = Ã(x̃)
• ideal gas law: p̃ = Rρ̃T̃
• collector and chimney pressure loss coefficients λco, λch
• collector height hc
• gravitational constant g
• specific heat cv
• ratio of absorbed solar energy αs
• collector and chimney heat transfer coefficients αco, αch
• collector and chimney temperature T̃co, T̃ch
• solar power per surface aerea Q̃s
• slope profile α = α(x), h̃x̃ = sinα
Dimensional analysis:
Quantity
u
ρ
p
T
x, h
t
A
hc
λco,λch
˜ turbine
∆p
αs
Q̃s
αco, αch
Tco , Tch
g
cv
Set ã = ar · a
Reference value
ur
ρr
pr
r
Tr = ρpr R
L
tr = L/ur
Ar
Machnumber: M 2 =
ρr u2r
pr
Typical reference value
10 m s−1
1.17 kg m−3
101328 Pascal
300 K
320 m
32 s
100 m2
2 m
10−2
0-50 Pascal
0.3
1000 W m−2 s−1
40 W m−2 K−1
300 K
9.81 kgm s−2
718 m2 s−2 K−1
Scaled compressible Euler equations: ǫ = γM 2
(Aρ)t + (Aρu)x = 0
1
1
(Aρu)t + (A(ρu2 + p))x = Ax p + ∆pturbine
ǫ
ǫ
√
−(ξcoA + ξch A)ρu|u|
−
1
Aρ sin(α)
F r2
ρu2
1
+
A(ρT + ǫ(γ − 1)(
ρh)) +
2
F r2
t
2
ρu
1
ρh)) + (γ − 1)Apu
= Aq − kcoA(T − Tco)
+
Au(ρT + ǫ(γ − 1)(
2
F r2
x
√
−kch A(T − Tch)
Variables: space x, time t,
density ρ = ρ(x, t), velocity u = u(x, t), temperature T = T (x, t), pressure
p = ρT
• Mach number M =
q
ǫ
γ
≪1
• Froude number F r
Stationary solution: adiabatic atmosphere formulas (not the barometric formulas)
1
1
γ−1
γ−1 ǫ
γ−1
ρh = ρh0 1 − hhr
h
= ρh0 1 −
γ F r2
γ
γ
γ−1
γ
−
1
ǫ
γ−1
h
ph = ph0 1 − hr
= ph0 1 −
h
γ F r2
γ
−
1
ǫ
Th = Th0 1 − hhr
= Th0 1 −
h
γ F r2
Small Mach number asymptotics:
p = p0 + ph + εp1 + O(ε2 )
Initial boundary value problem (I.G. ’09)
Ax
ρt + (ρu)x = − ρu
A
1
Ax
1
ut + uux + (p1 )x = − u2 − (ξco + ξch √ )u|u|
ρ
A
A
1 sin(α)
11
−
∆pturbine
(ρ
−
1)
+
ρ F r2
Aρ
γp0 ux = −
Ax
1 p0
p0
γp0u + q − kco ( − Tco ) − kch √ ( − Tch)
A
ρ
A ρ
Initial values:
u(x, 0) = u0(x) ρ(x, 0) = ρ0(x)
Boundary values:
p1(0, t) = 1, p1(1, t) = 1
ρ(0, t) = ρ0 (u(0, t) > 0), ρ(1, t) = ρ1 (u(1, t) < 0) (inflow condtions)
Alternative formulation (I.G. ’09)
v(t)
1
u(x, t) =
+
A(x)
γA(x)
Z
x
A(y)q(y)dy.
0
We obtain a PDE for the density ρ
1
ρt + uρx = − ρq.
γ
and an ODE for the only time dependent part v of u
Z 1
Z 1
Z 1
1
1
Ax 2
ρu dy +
(ξco + ξch √ )ρu|u|dy
vt = − R 1
ρuuxdy +
A
ρA−1dy
0
0
0 A
0
Z 1
Z 1
1
sin(α)
∆pturbine dy
(ρ
−
1)dy
+
(p
−
p
)
−
+
r
l
2
Fr
0 A
0
Analysis
Very similar problem: Tunnel fire model
(I.G., J. Struckmeier ’02, I.G., H. Steinrück ’06)
• stationary problem with multiple solutions
• transient problem: global existence and uniqueness
Solutions of the type:
v ∈ C 1 [0, T ] but in ρ we have to admit discontinuities.
These are natural due to the inflow conditions.
Idea of the proof:
Fixed-point- argument in the ODE
Use estimates on the density from the PDE in the ODE
• (linear) stabilty anaylsis
Numerical simulations: Manzanares
Parameter
Collector height
Collector radius
Chimney height
Chimney radius
Initial velocity
Initial density
ξco = ξch
Solar radiation energy
∆pturbine
Value
2m
120 m
200 m
5m
2 (20) ms−1
1, 17 kgm−1
0.1
1000 W m−2
0 (20 Pascal)
velocity u [m/s]
velocity u [m/s]
t [minutes]
15
20
10
0
20
200
0 0
x [m]
rho [kg/m³]
1.2
1.1
1
20
10
200
0 0
t [minutes]
x [m]
temperature T [C]
5
400
64
t [minutes]
10
t [minutes]
10
128 192 256
x [m]
rho [kg/m³]
320
128 192 256
x [m]
temperature T [C]
320
15
10
5
400
64
60
40
20
20
10
t [minutes]
0 0
200
x [m]
400
t [minutes]
15
10
5
64
128 192
x [m]
256
320
velocity u [m/s]
velocity u [m/s]
t [minutes]
15
20
10
0
20
200
0 0
x [m]
rho [kg/m³]
1.2
1.1
1
20
10
200
0 0
t [minutes]
x [m]
temperature T [C]
5
400
64
t [minutes]
10
t [minutes]
10
128 192 256
x [m]
rho [kg/m³]
320
128 192 256
x [m]
temperature T [C]
320
15
10
5
400
64
100
50
0
20
10
t [minutes]
0 0
200
x [m]
400
t [minutes]
15
10
5
64
128 192
x [m]
256
320
Numerical simulations: Buronga (planned)
Parameter
Collector height
Collector radius
Chimney height
Chimney radius
Initial velocity
Initial density
ξco = ξch
Solar radiation energy
∆pturbine
Value
2m
3500 m
1000 m
65 m
2 ms−1
1, 17 kgm−1
0.1
1000 W m−2
0 Pascal
velocity u [m/s]
velocity u [m/s]
t [minutes]
40
50
0
−50
40
5000
900 1800 2700 3600 4500
x [m]
rho [kg/m³]
0 0
x [m]
rho [kg/m³]
t [minutes]
20
t [minutes]
20
1.5
1
0.5
40
20
t [minutes] 0 0
x [m]
temperature T [C]
40
20
5000
900 1800 2700 3600 4500
x [m]
temperature T [C]
400
200
0
40
20
t [minutes]
5000
0 0
x [m]
t [minutes]
40
20
900 1800 2700 3600 4500
x [m]
Outlook
• Thermal model
• Turbine model
• Analysis
• Optimisation
• Down draft tower
Source: T. Altman, D. Zaslavsky, R. Guetta, G. Czich, Preprint 2006
History:
• 1975 - P.R. Carlson: US Patent No.3894393
• 1999 - D. Zaslavsky: Description and output estimates
Facts:
•
•
•
•
•
•
Renewable energy
Water required
No collector, big chimney required
No existing prototype
Discussion about the electrical output power
Discussion about the costs for the construction of big chimney’s
Thank you !