When applied mathematicians step up, a promising 30-year-old solar energy concept finds new life

    For Vakhtang Putkaradze (mathematical and statistical sciences), applied mathematician and centennial professor in the Faculty of Science, the expansive universality of mathematics has allowed him to work in such diverse fields as nanotechnology, biomedical engineering, computer animation and most recently—renewable energy.

    By Taylor Robertson on June 23, 2014

    For Vakhtang Putkaradze (mathematical and statistical sciences), applied mathematician and centennial professor in the Faculty of Science, the expansive universality of mathematics has allowed him to work in such diverse fields as nanotechnology, biomedical engineering, computer animation and most recently—renewable energy.

    As part of his ongoing research, Putkaradze has joined collaborators from the department of mechanical engineering at the University of New Mexico to make an innovative leap in solving an age-old problem that has both inspired and plagued engineers for centuries.

    How do you build a really tall, free standing structure that can withstand the unpredictable and destructive force of wind?

    Solving this problem is of critical importance for a form of renewable energy production known as solar updraft. Solar updraft facilities operate quite differently than other forms of solar powered electricity production like photo-voltaics, which employ the photoelectric effect or concentrated solar power facilities which use mirrors and lenses to focus heating. Solar updraft technologies simply trap air heated by the sun in a large solar collector (or greenhouse with obvious benefits for food production) and provide an outlet for this air via a central solar chimney (or tower). The solar towers that Putkaradze’s group looked at capture energy by driving turbines with the convective flow of heated air as it rises into the cooler and rarefied higher atmosphere.  

    Producing electricity in this way has many benefits over other renewable methods including low variability, low maintenance, no need for a continuous water supply, and the ability to source a variety of local/inexpensive building materials.   However, making these towers efficient requires increasing the temperature differential between the collector and the top of the chimney. The best way to do this is to build taller towers.

    “The challenges of building a very tall structure soon begins to outweigh the thermodynamic benefits. Moreover the costs of erecting a tower soon begin to dominate the overall costs of the facility,” notes Putkaradze and his collaborators.

    Building a facility that could produce 200 megawatts of power, for example, would require a collector on the order of 10 km2 and a tower around 1 km tall. This would be comparable to, but below production capabilities of, the average coal plant (667 MW) or nuclear reactor (846 MW). In Alberta this would provide enough electricity for 240,000 residences; however, our climate makes these facilities unfeasible.

    But now, the future is looking up for renewable energy production… Way up!

    With this in mind, Putkaradze looked to challenge established design concepts by drawing on lessons from the past and using familiar elements in fundamentally new ways.

    The challenge when building a free-standing structure hundreds of meters tall is that the higher your tower, the greater the torque you have to support.  Many of the previous solar chimneys have drawn on antenna designs using guide-wires to stabilize rigid cylinders. This includes the original solar updraft facility in Manzanares, Spain which employed a 190m tall tower of iron sheeting. Unfortunately the facility was decommissioned in 1989 after the guide-wires failed in a storm causing the tower to collapse.  At a cost of $1 million with a peak power output of only 60kW the Spanish tower didn’t garner much attention from investors, but the project was important because it demonstrated the potential and the challenges of solar updraft technology.

    When thinking about the shape of this tower, Putkaradze found inspiration in the famous Eiffel Tower. Gustave Eiffel also had concerns about the challenges of wind loading when he designed his famous creation. He suspected that an exponential shape would minimize the torque on the structure, thereby reducing the need for strong diagonal supports—saving materials and giving his tower an airy look.

    Modern mathematical analyses have proven this hypothesis correct for a rigid structure, as the shape does indeed minimize the lateral forces the tower has to deal with. As an enduring testament to Gustave’s vision the tower that bears his name still stands over the city of lights today.

    However there was another important piece of the design that came to Putkaradze while having a discussion about physics with a friend over coffee.  Like many creative insights; the answer it seems was hiding in plain sight. Anyone who has used an inner tube has probably seen piles of them stacked on top of each other. Although you may not have appreciated it at the time, these stacks can become surprisingly tall while remaining stable. There is an interesting physical phenomenon here which Putkaradze realized he could exploit in the towers design to make it self-supporting.

    Modelling the forces between concentric inner tubes revealed what he suspected. Their interactions display a form of dynamics, well documented in the scientific literature, known as Hertzian-type chains. In simple terms the restoring forces between the tubes is singular and proportional to the square root of the angular displacement. This form of stability is much stronger than any linear restoring force—such as the familiar mass on a spring.

    The idea of allowing a tower to bend without breaking is not unique, considering that all modern skyscrapers allow for some lateral flexibility, but Putkaradze and his team are taking this idea into unexplored territory.

    Using a variety of optimization schemes, Putkaradze was able to find a shape that distributes the wind load evenly amongst the toroids when subjected to a constant, uniform wind. This was balanced with the requirement of maintaining flow through the chimney thereby not reducing power production significantly. The tower looks kind of like a hat from Dr. Seuss’s books.

    The next step is to translate their ideas into the real world and check that their model indeed performs as it should. When running computer simulations like theirs it is necessary to specify boundary conditions and make some linearizing simplifications, for example with their model of turbulence, in order for the calculations to converge on an approximate answer.

    “There is still uncertainty about what the air is doing at the top of the chimney. To be exact we would have to model the dynamics of the entire atmosphere. Which we obviously cannot do,” says Putkaradze

    At the University of New Mexico they have recently installed a 3m tower, built out of—as you might have guessed—inner tubes, which should prove their concept soon.

    The team continues to set their goals ever higher and are working on funding a bigger prototype that meshes their design with other groups at Arizona State University and a California company called Solar Tech LLC.  With this design, the group proposes a series of pumps and valves between the toroids which would allow operators at the facility to compensate for more realistic wind profiles. This includes scenarios where the wind velocity changes in time and with elevation.

    Putkaradze says creating a control scheme for the tower is the essential next step in the project.  “What we have is a very weak form of control since we can only directly adjust the time derivative of the pressure and not the pressure itself. Ideally in this situation we would like to know what the wind will do indefinitely into the future, but in reality the structure has to respond in real time to volatile conditions.” 

    Putkaradze says he is confident that he will be able figure out a reliable control method—noting that, along with his graduate student Michael Chi and a collaborator in Paris, Francois Gay-Balmaz, he has already discovered some very interesting mathematics that he hopes to publish soon.

    About energy alternatives

    The consensus among those thinking critically about energy policy is that in the 21st century renewable energy needs to play a more significant role. This has brought on a resurgence of interest in updraft facilities and in particular, the most troublesome part of their design, the solar towers. Other groups have proposed using Helium filled structures to offset the weight of their towers. Putkaradze’s group made a conscious decision to use air filled toroids and it allowed them to solve the problem of stability in a very economical way. Furthermore with the rising price of Helium and its inevitable loss to the atmosphere; air filled structures are looking more and more appealing. Using air also allows the tower to be deflated for maintenance or in the event of dangerous winds.

    About the writer:

    Taylor Robertson earned his a Specialization Physics BSc. from the U of A in 2013, where he studied space and mathematical physics.  While he hopes to design space technologies of the future, he is currently exploring research opportunities in electrical engineering for his graduate studies.  Taylor spends his free time outdoors hiking, biking, skiing and travelling the world.