7 Mathematical Modelling of Solar Updraft Tower
101
Here, mass flow rate of air unknown. It can be calculated using
˙
m = ρ i ∗ A chim ∗ v i .
(7.5)
In order to simplify the calculations, approximate ρ i = ρ o . Mass flow rate of air after
approximation is given by
˙
m ≈ ρ o ∗ A chim ∗ v i
(7.6)
By taking the above approximation, the equation to obtain temperature at turbine
inlet becomes a linear equation rather than a cubic equation. This simplifies the
calculations significantly. Velocity at the bottom of the chimney is given by (Schlaich
et al. 2005).
v i =
2 ∗ g ∗ H chim ∗
T i − T o
T o
(7.7)
Put the values of velocity in Eq. (7.6) to get the mass flow rate. Substitute ˙
m in
Eq. (7.4). Here, T i is the only unknown. Making T i the subject gives us a relation to
obtain the average temperature of air at chimney entrance.
T i = T o
1 +
(Q
∗ A coll ∗ R)
2
(P o ∗ A chim ∗ C p ) 2 ∗ 2 ∗ g ∗ H chim
1/3
(7.8)
Once T i is obtained, velocity of air at chimney entrance (v i ) can be calculated
using Eq. (7.7). Using air velocity, the power generated by the wind turbine is given
by
P out = C ∗
1
2
∗ ρ i ∗ A chim ∗ v
3
i
(7.9)
where C is assumed to be 0.45 for this wind turbine. Efficiency of the chimney is
defined as the ratio of kinetic energy of air at chimney entrance to the heat absorbed
by the air.
η chim =
1
2
∗ ρ i ∗ A chim ∗ v
3
i
˙
m ∗ C p (T i − T o )
(7.10)
Putting the values of ˙
m and v i as defined earlier in Eqs. (7.5) and (7.7), this can be
simplified to
η chim =
g ∗ H chim
C p ∗ T o
(7.11)
101
Here, mass flow rate of air unknown. It can be calculated using
˙
m = ρ i ∗ A chim ∗ v i .
(7.5)
In order to simplify the calculations, approximate ρ i = ρ o . Mass flow rate of air after
approximation is given by
˙
m ≈ ρ o ∗ A chim ∗ v i
(7.6)
By taking the above approximation, the equation to obtain temperature at turbine
inlet becomes a linear equation rather than a cubic equation. This simplifies the
calculations significantly. Velocity at the bottom of the chimney is given by (Schlaich
et al. 2005).
v i =
2 ∗ g ∗ H chim ∗
T i − T o
T o
(7.7)
Put the values of velocity in Eq. (7.6) to get the mass flow rate. Substitute ˙
m in
Eq. (7.4). Here, T i is the only unknown. Making T i the subject gives us a relation to
obtain the average temperature of air at chimney entrance.
T i = T o
1 +
(Q
∗ A coll ∗ R)
2
(P o ∗ A chim ∗ C p ) 2 ∗ 2 ∗ g ∗ H chim
1/3
(7.8)
Once T i is obtained, velocity of air at chimney entrance (v i ) can be calculated
using Eq. (7.7). Using air velocity, the power generated by the wind turbine is given
by
P out = C ∗
1
2
∗ ρ i ∗ A chim ∗ v
3
i
(7.9)
where C is assumed to be 0.45 for this wind turbine. Efficiency of the chimney is
defined as the ratio of kinetic energy of air at chimney entrance to the heat absorbed
by the air.
η chim =
1
2
∗ ρ i ∗ A chim ∗ v
3
i
˙
m ∗ C p (T i − T o )
(7.10)
Putting the values of ˙
m and v i as defined earlier in Eqs. (7.5) and (7.7), this can be
simplified to
η chim =
g ∗ H chim
C p ∗ T o
(7.11)