The passive solar cooling effect promoted by the solar chimney is a powerful element of building design especially in warm climates. However, the flow of air is not harnessed to generate useful work. Remarkably, Leonardo da Vinci (14521519) envisioned a spit that was turned by the rising air in a fireplace as shown in Fig. 7.4, that can transform the air temperature rise into work.
The idea of a solar tower is to put a flat, circular cover or collector a distance h above the ground, typically 12 m, that will heat the air underneath it, as shown in Fig. 7.5. The cover radius r can be quite large and of order hundreds of meters to produce utility scale electricity. This necessitates that the land where the tower is built should be very flat and devoid of obstructions.
A turbine is placed at the base of the tower (chimney) that is turned by the rising air within the tower of diameter D and height H. Some proposed solar towers are so massive that there are several turbines arranged circumferentially around the entrance to the tower. Furthermore, the tower height would make it one of the largest structures in the world!
In terms of an actual solar tower a pilotscale plant was built in Man – zanares, Spain and used between 1982 and 1989 to produce 50 kW of electrical power. The collector radius was 122 m and the tower diameter and height were approximately 10 m and 195 m, respectively. So, even a small, pilotscale solar tower is big.
Why are there no utilityscale solar towers used today? Part of the answer may center around the size and scale of these power plants. They will be huge, yet, several have been proposed around the world. Also,
because of their size, they can only be placed in desolate areas far away from where the electricity is needed. Finally, the power generated will be somewhat intermittent with little or none produced at night. A solution to this is to put material, like bags of water, under the cover to absorb radiation during the day, which is released at night to continue operation.
There are some advantages to these electrical power plants. An obvious one is that there will be few maintenance issues as the only moving parts will be the turbines. There is no burning of fossil fuels to power a steam cycle that turns a steam turbine. Another, perhaps less obvious advantage, yet, one that will surely become more important over the next century, is that water is not required (apart from the water used to store heat). Coalfired and nuclear fission power plants use a lot of water and thermal (as well as air) pollution is a challenge to their ecological – based operation. Water is expected to become a rarer commodity in the future and a solar tower could alleviate some of the environmental strain imposed on the Earth. Imagine a solar tower farm in northern Africa to produce electricity, transmitted to Europe; this was proposed although it never came to fruition.6
The analysis presented below is simplified and aimed at understanding the variables that affect production of electricity from the solar tower. Engineering details of the building structure and mass flow rates and pressures required to turn the turbine are not considered and are certainly needed for a true design.
The First Law of Thermodynamics is applied between station 1, far from the collector, and station 3, just at the exit of the tower, to determine the operating line for the solar tower resulting in
Cp [T3 – Ti] + 2v + gH
qwtAT Wm
Here m is the air mass flow rate, CP, the air heat capacity which is assumed to be constant over the temperature range considered, Ti at station 1 and T3 at station 3, v3, the air velocity at station 3, H, the tower height, PD(0), the insolation absorbed within the device, qrc, the rate of heat transfer due to radiation from the cover to the surroundings, qwc, the rate of heat transfer due to wind convection from the cover to the surroundings, qwt, the rate of heat transfer due to wind convection from the tower to the surroundings, AC, the cover area, AT, the tower external area and Wm, the amount of electrical power generated by the turbine. The radiative heat transfer losses from the tower are ignored, as will be justified after reading the discussion below. All signs for the rate of heat transfer and work have been explicitly written.
Before considering the magnitude of the heat transfer terms it is useful, in order to determine the operating line, to look at the magnitude of the terms on the lefthand side of the above equation. The heat capacity of air is approximately 1 kJ/kgK and the temperature rise will be of order 10 K, making this term, order 10 kJ/kg. The potential energy term gH will be of order 3 kJ/kg assuming the tower height to be 300
m, a good assumption.7 Now the kinetic energy term is much smaller as the velocity in these devices is of order 10 m/s, making it of order 0.03 kJ/kg, so it can be neglected. This allows us to write, with some generality, the operating line for a solar tower
m [Cp [T3 – Ti] + gH] = [PD (0) – <qrc – qwc] Ac – qwt At – [=] W
Some of the heat transfer terms can be neglected since they are very small. Consider radiant heat transfer from the cover to the sky
qrc = asec [TC – Ts4ky] (7.25)
with as being the StefanBoltzmann constant, eC, the emissivity of the cover, TC, the cover temperature and Tsky, the effective sky temperature for radiative heat transfer (an approximate relation is Tsky = 0.0552 Ta1rn5b where Tamb is the ambient air temperature). Assume the ambient air temperature is 35 °C and that the air flowing under the cover rises by 20 °C making the cover temperature 55°C. If the emissivity of the cover is 0.05, one can determine qrc = 10.3 W/m2 which, as it turns out, is quite small and negligible.
The next heat transfer term is due to wind blowing over the cover, which can be written
qwc = hw [Tc – Tamb] (7.26)
where hw is the heat transfer coefficient from a flat surface. There are many correlations for this and after an extensive survey of the literature the following relation best represents all data (see the article written by Palyvos, referenced at the end of this chapter)
hw (W/m2K) = 7.4 + 4.0vw (m/s) for 0 < vw < 4.5 m/s (7.27)
where vw is the free wind velocity 10 m above the surface. Assuming wind velocities of 0 and 3 m/s, as well as the cover temperature being 20°C above ambient, one finds qwc equal to 74.0 W/m2 and 194 W/m2, respectively. Clearly, this rate of heat transfer is much greater than radiative and is not in fact negligible even when the wind is not blowing and heat transfer is solely from natural convection.
The final heat transfer term to consider is heat transfer from the tower to the surroundings,
qwt = hwt [TT – Tamb] (7.28)
where hwt is the heat transfer coefficient for wind blowing on the outside of the cylindrical tower and TT is the tower external surface temperature. We use a standard Nusselt number (Nu)Reynolds number (Re) dimensionless correlation for air blowing past a cylinder to estimate hwt, see Appendix C where dimensionless correlations are explained,
Nu = hwtD = 0.35 + 0.56Re052
kair
pairvw D
Re =————
Tair
where kair, pair and pair are the thermal conductivity, density and viscosity of air. Again, for a 20 °C temperature rise of the air in the tower, assuming the tower diameter to be 10 m and using values for the physical parameters of air from Appendix B, one finds hwt changes from 9.23 x 104 W/m2K to 2.71 W/m2K as the wind velocity increases from 0 to 3 m/s. This is an over three order of magnitude change in the heat transfer coefficient, a vastly different change than occurs for heat transfer from a flat surface like the cover. The rate of heat transfer increases by the same amount from 0.0185 W/m2 to 54.2 W/m2. So, when there is no wind this heat transfer rate is truly negligible, while in a moderate wind it is significant, albeit much less than that from the cover. Furthermore, the tower area is much smaller than the cover so the overall magnitude is even less.
Some solar towers have been proposed, making the chimney almost the highest structure in the world, perhaps 800 m high. Will the wind conditions change for such a tall structure? As one might guess the velocity of wind changes with height above ground level and is usually represented by
where z is height above ground level and vw0 is the velocity at the reference height z0, usually taken as 10 m. The power law parameter n has values of 0.100.15 for smooth, uninhabited places on Earth, which is how the terrain would be at a solar tower installation. The Manzanares facility in Spain had a tower that was almost 200 m high. So, if the wind velocity was 3 m/s at a height of 10 m it would be 45 m/s at the top of the tower making the heat transfer coefficient increase by about 25%. An 800 m tower would experience winds of order 56 m/s and a 60% increase in hwt. Even with this increase the rate of heat transfer is not that significant when compared to the rate from the much larger cover.
Find the operating point for the solar tower pilot plant in Manzanares, Spain, assuming wind speeds of 0 and 3 m/s. The insolation is 8f0 W/m2 and the surrounding air temperature is 35 °C. Use eqn (7.8) rather than (7.15) for the design line to determine how much wind affects the heat transfer rate.
tower height (H) 
194.6 m 
tower diameter (D) 
10.16 m 
collector radius (r) 
122.0 m 
typical air temperature rise (AT) 
20 0C 
typical power output (IFm) 
50 kW 
Table 7.1 Physical dimensions and typical operating conditions for the solar tower at Manzanares, Spain. 
This facility operated under the conditions given in Table 7.1, with reference to Fig. 7.5. The design line for this power plant is easily calculated by using eqn (7.8) together with the data in the table and is presented in Fig. 7.6. Obviously, as the rate of the temperature rise increases the greater will be the mass flow rate.
The operating line is calculated from eqn (7.24) using the equations for the various heat losses given in eqns (7.25) to (7.29). Two wind velocities were assumed, 0 and 3 m/s with the results given in Fig. 7.6.
The effect of heat transfer is first discussed assuming a constant wind velocity of 3 m/s. The upper graph is used to show how the various heat transfer rates affect the calculation. First, if qrc and qwt are neglected the design line and operating point move a small amount. For example, the operating point moves from a temperature rise of 19.6 0C and 1380 kg/s to 19.9 0C and 1390 kg/s, a relatively small change. In fact, the curve labeled only qwc is two curves: one including qwc and qrc and another including only qwc. So, qrc has an extremely small contribution compared to the other terms, as expected. Finally, if all heat transfer terms are neglected the operating point changes, a much larger amount to a temperature rise of 24.1 0C and mass flow rate of 1510 kg/s. Remarkably, for all the cases studied, the temperature rise is calculated to be almost exactly that given in Table 7.1 regardless of the assumptions made.
Next consider how wind will affect the solar tower performance that is shown in the lower graph. Even though the wind is not blowing there will be contributions from qwc and qwt due to natural convection instead of forced convection. In this case the operating point changes to 22.3 0C and 1460 kg/s, demonstrating how important heat transfer is to operation. However, when there is little wind, neglecting all heat transfer considerations is not that poor an assumption since the mass flow rate is within 34% between the two scenarios.
It was shown in the above example that the operating line is given by eqn (7.24) while the design line can be determined from either (7.8) or (7.15). The above example shows that heat transfer from the solar tower to the surroundings is important under most conditions, however, if heat transfer considerations are neglected it is possible to write an equation very similar to eqn (7.18) with p and q given by


















where the approximation is valid for most solar towers based on their typical dimensions.
Mass flow rate is the most important variable in the solar chimney to provide cooling, however, in the solar tower it is the maximum work which is most important. The maximum power scales as
~ Pd(0)AcH (7.36)
Now it is apparent that the maximum power generated by a solar tower depends explicitly on the amount of power absorbed by the working fluid (air) Pd (0)Ac and the tower height H, as might be expected. More precisely, it is obvious that the maximum power will scale directly with Pd (0)Ac since this is the amount of energy available to it over a given time period. The direct scaling with height is strictly due to the buoyancy effect. The potential energy inside the tower is p3gH and that outside the tower is pi gH; their difference is the driving force for the chimney effect which scales directly with H. Since air has such a low density the height must be extreme to generate enough potential energy. Would water be a better working fluid? Do Exercises 7.7 to 7.9 to find out!
The tower diameter does not affect the maximum power generated by a solar tower, demonstrating that this variable is not as important as implied by eqn (7.33). However, the diameter affects, among other variables, the velocity of air in the tower, which is important to improving turbine efficiency so its influence is certainly not negligible.
The overall efficiency of a solar tower is actually quite small which ultimately contributes to the large size required. Taking the maximum power and dividing it by the power supplied to the air Pd (0)Ac one finds the solar tower efficiency n
A large solar tower would only have a 1% efficiency!
Find the operating point for the solar tower pilot plant in Manzanares, Spain using the approximate operating point given in eqn (7.32). Data for the power plant are given in Example 7.3. Also determine the maximum power that can be generated with this power plant using eqn (7.35) and compare it to the typical power output. Neglect the effect of wind and use eqn (7.8) as the design line.
The approximate operating point is found when heat losses are not considered and the temperature rise will be greater than that encountered in the real power plant. As such, the mass flow rate will also be overestimated since the buoyancy effect offered to the air is more than the true situation. Regardless, the equation to be used is
together with eqn (7.8) written as
The second equation is needed because p3 is a variable in the first equation, which is the air density at T3, and is unknown, and an iterative procedure must be performed. Air density as a function of temperature is given in Appendix B as p3(kg/m3) = 1.2973.757x 103 T3(°C). The solution procedure is to assume a value for T3 and calculate Р3 then determine m from eqn (7.32) and finally calculate the temperature rise from eqn (7.38). Continue this iteration until the assumed T3 and the calculated T3 = T1 + AT are equal. When this is done a mass flow rate of 1540 kg/s and a temperature rise of 25.3 °C are found. If eqns (7.24) and
(7.8) (or (7.38)) are solved simultaneously, neglecting all heat transfer considerations, then one finds 1510 kg/s and 24.1 °C, which is fairly close to the approximate operating point.
The maximum power that can possibly be generated by the solar tower is given by eqn (7.35). Inserting numbers and performing the calculation results in a maximum power of 242 kW. As shown in Table
7.1, the typical power output is 50 kW, so, the efficiency is approximately 20% based on the possible power that could be developed. Of course, the overall efficiency is only 0.6% based on the insolation falling on the cover using eqn (7.37).
A novel solar tower was discussed, building on the concepts that make a chimney and solar chimney work. Although it would be massive in size to generate industrialscale electricity, and there are no known power plants in operation, the physics and engineering involved in its concept are so simple there would seem to be potential for their use in the future. Furthermore, as water becomes a more valuable and rarer commodity, the fact that water is not used at all is a great advantage. The poorest land can also be used, freeing more valuable land and other natural resources for other uses.
The solar chimney has been used for many years to help to cool a house in a hot climate. Solar energy is stored in the chimney during the day and when the outside temperature falls at night a damper in the chimney is opened to allow air circulation through the house. This simple design is extremely effective, especially when there is no power available to cool the house.
Massive solar towers take this principle to another level and convert heat into electricity. Although conversion of rising hot air within a chimney to useful work was envisioned by Leonardo da Vinci many years ago, no significant, commercial solar tower has been erected. However, the incentive to build one may change, driven by energy security concerns, among other reasons.
These solar powered devices are only 1% efficient necessitating that they must be huge in scale. For example, in the USA, electrical consumption is approximately 4 x 1012 kWh/year which is an instantaneous power of 4.6 x 1011 W. Assuming that there is an average of 250 W/m2 of insolation available to the tower then the collector area must be 425 km x 425 km! Doubling the insolation reduces this to approximately 300 km x 300 km, so, this technology will certainly not replace coalfired power plants.
Yet, there are targeted places that could truly warrant use of this technology. Furthermore, water is not required and all the engineering principles are present, the technological risk is not as large as rolling out a completely new technology and building a new infrastructure. It could be useful for isolated, desolate places that do not require much power, and in the evening could be used for shelter. However, the economics of building this compared to installing a solar photovoltaic array is not clear.