Preparation of Free-Surface Hyperbolic Water Vortices

Our lives are closely connected with spiral structures. They exist in almost everything and everywhere, including the structure of shells and ammonites and the formation of hurricanes, tornadoes, and whirlpools^1,2. On a cosmological scale, galaxies form and evolve according to the principle of the logarithmic spiral³. The best-known spirals are the golden and the Fibonacci spirals⁴, which have many applications ranging from describing plant growth and the crystallographic structure of certain solids to developing computer database search algorithms. The Fibonacci sequence is characterized as a numerical series that starts with 0 and 1 and has subsequent numbers corresponding to the sum of the previous two. This sequence can also be found when counting the reproduction rate of rabbits. Spirals are among some of the oldest geometric shapes drawn by Homo sapiens, such as the concentric circles found in Colombia and Australia (40,000-20,000 BC¹). Leonardo da Vinci⁵ tried to create a helicopter-shaped flying machine using a spiral blade (from the Greek word ἕλιξ πτερόν, or helix pteron, meaning spiral wing). Following the same principle, an aircraft designer, Igor Sikorsky, constructed the first helicopter in series production 450 years later⁶.

Many other examples point to the fact that helical flow structures might be very efficient and expense-saving because this type of flow is preferentially seen in nature. At the beginning of the 20th century, the Austrian forester and philosopher Viktor Schauberger realized this. He said that humans should study nature and learn from it rather than trying to correct it. Based on his ideas, he built rather unusual log flumes to float timber; the flumes did not take the straightest path between two points but followed the meandering of valleys and streams. This design made the water flow by twisting in a spiral along its axis, thus forming a vortex, which thereby reduced the amount of water used and produced a transport rate that significantly exceeded what was considered normal⁷.

Following his father's footsteps, Viktor's son Walter developed new technologies using the water vortex⁸ for various purposes: the treatment of drinking water, industrial process, the restoration of ponds and water courses, the oxygenation of ponds and little lakes, and river regulation and restoration. One of these ideas has recently gained considerable interest, namely water treatment using a hyperbolic funnel⁸, in which a vortex is created only by the flow of water without any stirring devices. It has been proven to be a very effective method for oxidizing iron in groundwater^9,10. A limitation of this technology is that it is less efficient for low-pH water¹¹.

Large amounts of drinking water in the Netherlands are obtained from underground sources¹², in which the concentration of iron can reach several tens of milligrams per liter¹³, whereas 0.2 mg/L is considered acceptable by the standards¹⁴. Most drinking water plants use aeration as one of the first steps to reduce the iron concentration in the water purification process. In most cases, the purpose of aeration is to increase the dissolved oxygen content, to remove gases and other related substances from the water, or both¹⁵. There are various methods by which aeration can introduce oxygen into liquid media. These methods include agitating the liquid surface using a mixer or turbine and releasing air through either macroscopic orifices or porous materials¹⁶.

The chemical process of iron oxidation was demonstrated by van de Griend¹⁷, in which an oxygen molecule takes an electron from ferrous iron and reacts with a free proton to form water, while the iron ion is oxidized (equation [1]):

,     (1)

The iron ion then precipitates as Fe(OH)₃ because of its reaction with water, which releases protons (equation [2]):

     (2)

The total reaction is given by equation (3):

.     (3)

In aeration, the techniques most often applied are cascades, tower, spray, and plate aeration systems^18,19. The disadvantage of these technologies is that they consume from 50% to 90% of all energy²⁰ and up to 40% of the budget for the operation and maintenance of the treatment facilities²¹.

Using a hyperbolic funnel for aeration can significantly decrease the costs and increase the efficiency of this process. Hyperbolic funnels are less sensitive to clogging due to their geometry and the fact that there are no moving parts, meaning the energy is spent only on pumping water. Such a system can be characterized by several parameters, such as the water flow rate of the funnel per hour (φ), the mean residence time (MRT), the hydraulic retention time (HRT), the oxygen volumetric mass transfer coefficient (K_La₂₀) (corrected to a standardized temperature of 20°C), the standard oxygen transfer rate (SORT), and the standard aeration efficiency (SAE). The flow rate of the funnel is needed to calculate the volume of water that can be processed in a certain time. The MRT is calculated from the ratio of the water flow rate to its volume in the funnel for a certain regime using equation (4):

     (4)

where V represents the liquid volume in the reactor.

The HRT can be determined experimentally using tracer technologies²² via its residence time distribution function. HRT provides fundamental insight into mixing processes, hold-ups, and segregation phenomena²³. It was shown by Donepudi²⁴ that the farther away the water jet is from the inlet, the faster it moves toward the outlet. At the initial moment, water is pumped tangentially to the upper cylindrical part of the funnel. Then, under the influence of gravity, together with the system's geometry, the tangential velocity decreases, and the axial velocity increases. The oxygen volumetric mass transfer coefficient, K_La₂₀ (unit reciprocal time), indicates the capability of a system to facilitate oxygen transfer to the liquid phase¹⁰. It can be calculated^25,26 according to equation (5):

     (5)

where C_out  is the dissolved oxygen (DO) concentration in the bulk liquid, C_で is the DO concentration in the feed, C_s is the DO concentration at saturation, and T is the water temperature.

The SORT value is the standard rate of oxygen transferred to the liquid phase by the system and is determined by equation (6)²⁷:

     (6)

where is the DO at saturation for a temperature of 20 °C. The SOTR value can be defined for a certain process, in which case the volume used in equation (6) is normalized by assuming 1 h of treatment time (process-specific SOTR), so that pilot scale aeration methods can be compared with real-scale systems. For the capability of a certain regime in the funnel, the system-specific SOTR must be calculated, which uses the volume of water inside the funnel for a (regime-specific) hydraulic retention time. This value is important when calculating the actual aeration capabilities of a regime in a given funnel.

The SAE is the ratio between the SOTR and the power expended for aeration. Since energy is spent only on pumping water to the top of the funnel and giving it the necessary flow to form a vortex, it is calculated as the sum of the potential energy of the volume of water pumped per hour at a height corresponding to the length of the funnel and the kinetic energy needed by the water to create a vortex²⁷ using equation (7):

     (7)

where P_p is the potential power (in kW) required to lift the water pumped to the height of the funnel, and P_k is kinetic power (in kW) required for the water pumped at the top of the funnel to gain enough flow to create a vortex. Normally, for equation (7), the system-specific SOTR should be used. If the process-specific SOTR is applied instead, it yields the energy consumption of a (theoretical) system with 1 h of hydraulic retention time.

These parameters are sufficient to assess the effectiveness and feasibility of using this technology but not to describe the process itself. It should be mentioned that vortices are among the least understood phenomena in fluid dynamics. Therefore, a lot of research efforts are invested in this direction. One of the main challenges in finding the general laws and rules of vortices in fluid dynamics is that there are always variations in the geometric boundary conditions, which influence the development of vortices and significantly influence their formation and dynamics. Thus, it is reasonable to assume that a free-surface vortex (FSV) cannot be considered analogously to a laboratory-type confined one. However, it was shown by Mulligan et al.²⁸ for the Taylor-Couette flow (TCF) that if the air-core of the FSV is considered as a virtual inner cylinder rotating at the same speed as the air core, both can be treated similarly. By doing so, equations that represent the free-surface vortex flow field can be substituted with the angular velocity conditions of the virtual cylinder, resulting in equations for the TCF system. It was also demonstrated that if the rotation speed of an imaginary cylinder is increased, at some point, Taylor-like vortices²⁸ appear as a secondary flow field and then disappear when approaching the walls.

After it was shown by Niemeijr²⁹ that it is possible to obtain three different types of water vortices in a Schauberger funnel (twisted, straight, and restricted) (Figure 1 and Figure 2), which are characterized by other hydraulic parameters, Donepudi²⁴ used the same approach as Mulligan et al.²⁸ to simulate vortex regimes using computational fluid dynamics (CFD) and thereby analyze the organization of their flow field to understand the underlying physical mechanisms. The system is very turbulent, and the secondary flow field is very unstable and is characterized by the appearance of a large number of Taylor-like vortices. Gas transport from the gas phase into the liquid phase is governed by diffusion, advection, and reaction. Therefore, to increase the efficiency of this process, it is necessary either to increase the gas concentration gradient or the volumetric motion of the liquid. The latter directly depends on the turbulence of the system in the form of Taylor-like vortices, which facilitate the transport of saturated fluid elements from the interface into the bulk liquid. In another work on this topic⁹, the main parameters for different vortex regimes, such as the water flow rate, K_La₂₀, and SOTR, were compared. This study showed great promise for this technology because the system enables very fast gas transfer compared to other methods that are used for water aeration.

The purpose of this article is to provide and demonstrate this method for creating different water vortex regimes in hyperbolic Schauberger funnels (small: 26 cm high and 15 cm top diameter; medium: 94 cm high and 30 cm top diameter; large: 153 cm high and 59 cm top diameter) with the goal of efficient water aeration.