
Calculations
Example for calculations. 1. Air Flow Calculations, according to Natural Draft Pressure Calculator ^{1}
q = Π d_{h}^{2} /4 [ (2 g (p_{o}  p_{r}) h ) / ( λ (l p_{r} / d_{h}) + ∑ξ p_{r} ) ]½
We can expect the natural draft of air flow at the speed of 139.2 m/s. This translates into 109,357,369 m^{3}/s of air going through the superchimney (50,916 m^{3}/s for 20m diamterer). As the air density equal 1.164 kg/m^{3} at 30°C^{2}, we can calculate that 127,292,000 kg/s of air moves through the superchimney(50,916 kg/s for 20m diameter). 2. Producing electric power. Calculate amount of electric energy produced.
Wind Turbine Power ^{7}: 3. Calculation of amount of water condensate. Suppose the relative air humidity at the bottom of the superchimney is 30%, which is typical for deserts. Thus, the air contains roughly 9g of water per kg of air. At elevation of 5000m the temperature is roughly 20C and the pressure is about 55,000 Pa (and this is roughly half of the pressure existing at sea level). At such conditions, air can hold at most 1.7 g of water per kg of air (point of saturation). Therefore, once the air from the superchimney is expelled out, water from the air will condensate in the amount of 7.3g per kg of air. In the given system, where 127,292, 000 kg/s of air is going through the superchimney, this means that 929,231 kg of water condensate per second(371 kg for 20m diamterer). 4. Calculation of CO_{2} uptake by irrigated desert. There are different estimates of CO_{2} uptake capacity given by the EPA. For example, it is estimated that each acre of reforested land will absorb up to 2.1 tons of carbon per year for period of 120 years. In fact, this number does not count additional CO_{2} which will be fixed in soil. Thus, one superchimney will allow to trap: 300* 640 Sq.Acres/ sq.mile* 2.1=403, 200 tons of carbon, 1 ton Carbon equivalent = 3.667 ton CO_{2}, thus one superchimney will allow to trap 1,478,354 tons of CO_{2} per year(592tons for 20m diamterer) 5. Calculation of number of superchimneys needed to cool the atmosphere. There are many factors to analyze. However, it is clear that in principle, the superchimney will facilitate heat exchange and, given the enormous amount of air coming through the superchimney, it will have effect on the heat balance of the Earth atmosphere. According to Kiehl^{3}, annually Earth receives 492 W/m^{2} of radiation combined (direct solar and due to the green house effect) . Global Warming is attributed to the fact that the Earth is presently absorbing 0.85 ± 0.15 W/m^{2} more than it emits into space^{4} (Hansen et al. 2005). So the planet absorbs approximately 0.2% of radiation energy more than it should to maintain constant temperature. According to Wikipedia, Earth surface heat captured by the atmosphere. More than 75% can be attributed to the action of greenhouse gases that absorb thermal radiation emitted by the Earth's surface. The atmosphere in turn transfers the energy it receives both into space (38%) and back to the Earth's surface (62%), where the amount transferred in each direction depends on the thermal and density structure of the atmosphere.^{5} The superchimney will emit air at the 5000m, which is roughly the point in the atmosphere where half the amount of air is below and half is above. Thus, it can be assumed that reabsorption will be cut in half. So, instead of normal distribution we will have ~70 % of energy lost to space and only 30% reabsorbed into atmosphere. In other words, the air, which will go through the superchimney, will loose 30% more heat than a normal air. According to the above calculation 127,292, 000 kg of air will go through the superchimney every second. In a year it comes to 4x10^{15}kg . According to theNational Center for Atmospheric Research, "The total mean mass of the atmosphere is 5.1480×10^{18} kg...". Therefore, annually 7.7 ×10^{4} of the whole atmosphere will go through the superchimney. As shown above, that air will lose 30% more heat than normal air. Thus, the whole atmosphere will lose 2.31×10^{4} or 2.31×10^{2}% more heat than it would otherwise. Since the planet absorbs approximately 0.2% of radiation energy more than it should to maintain constant temperature, we can aproximate that 10 superchimneys will offset Global Warming(25,000 for 20m diamterer). 6. Calculation of building a flexible chimney There will be three limitations to consider:
1. Whether flexible chimney will hold its shape. 1. Whether a flexible chimney will hold its shape.
In order to guarantee that chimney holds its shape, its theoretical mass needs to be less than pressure multiplied by inner area surface. In order to guarantee that chimney holds its shape, its theoretical mass needs to be less than pressure multiplied by inner area surface.
http://en.wikipedia.org/wiki/Dynamic_pressure#cite_note5 Thus we can estimate the additional pressure due to the flow of air in the chimney. It will be 8400Pa. It means that as long as each square meter of the surface weighs less then 840 kg, the chimney will hold its shape and can be standing upright. 2. Whether it will be standing upright.
There are three forces which will be pushing the chimney up: reaction force at the mushroom cap, buoyancy and drag force along the surface.The sum of these forces will be indicative of the maximum allowed weight of the chimney. a. Reaction force.
In order to overcome reactive force of the exiting air the chimney will need to be equipped with "Mushroom cap" where exits for air will be on the down facing surface of the cap. Such assembly will ensure that exiting air will be pushing chimney upwards.
The force will be equal to
A = ΠR^{2} area of the opening b. Buoyancy force
The air inside the chimney is warmer than the outside air. Yet, pressure is almost the same. Thus air inside is less dense. Therefore, chimney will act like a hot air balloon and support itself by buoyancy force. In order to average up our estimates for the whole system, we will calculate the buoyancy force for the chimney at the mid level altitude of 2500 m.
F = V*g*(p_{cool}  p_{hot}) c. Air Friction
It is difficult to estimate air resistance as it will greatly depend on the inner surface property of the chimney. Since our objective to have a higher air flow, we will always try to keep friction force at minimum by providing smooth polished surface inside.
Total Force:
Total Upward Force 7. Erecting the chimney. Whether resulting buoyancy will be enough to support such structure. In order to average up our estimates for the whole system we will calculate that buoyancy force for the chimney at the mid level altitude 2500 m and inside air at 70°C.
F = V g (p_{cool}  p_{hot})
 