CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. application Ser. No. 13/405,012, now U.S. Pat. No. 8,398,296, filed Feb. 24, 2012; U.S. Provisional Patent Application No. 61/575,644, filed Aug. 24, 2011; and U.S. Provisional Patent Application No. 61/447,004, filed Feb. 25, 2011, the disclosures of each of which are incorporated herein by reference. This application incorporates Group I claims from parent U.S. application Ser. No. 13/405,012, which group of claims was defined in a requirement for restriction that issued in the parent application and which claims were not elected for examination in the parent application. A utility patent application titled “Magnetically Coupled System For Mixing” is being filed by the same applicant for this application on the same date of the filing of this application and incorporates Group III claims from parent U.S. application Ser. No. 13/405,012, which group of claims was defined in the requirement for restriction issued in the parent application and which claims were not elected for examination in the parent application. This application and the utility patent application titled “Magnetically Coupled System For Mixing” being filed by the same applicant for this application on the same date of the filing of this application are divisional applications. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     This invention was made in part with United States government support under the Department of Energy grant number DE-EE0002867. The government has certain rights in this invention. 
    
    
     BACKGROUND 
     The present invention relates generally to mixing systems for use in enclosed vessels, such as rigid or flexible enclosures, or open vessels, such as pond systems, which may serve as reactors, bioreactors or photobioreactors. Systems in accordance with the present invention may be used to cultivate algae and other microorganisms in water for purposes such as producing biofuels, bulk chemicals, pharmaceutical compounds or other products or treating wastewater. 
     High density, high pigment aqueous algae cultures require mixing to evenly distribute nutrients to microorganisms in the culture and to ensure that the microorganisms in the culture are cyclically exposed to light needed for photosynthesis. One of the key challenges for commercial-scale mixing systems is to minimize the use of energy and capital expense while providing optimal production conditions. 
     Large, open pond systems typically use large paddle wheel mixers to move water around a raceway, but paddle wheel mixers are inefficient and require significant energy inputs, which may be cost-prohibitive for use in cultivation of microorganisms for the production of biofuel or other commodities. In addition, paddle wheel mixers are designed to move water in a path of horizontal flow and do not effectively move algae in a vertical plane, which is needed to ensure even exposure of the algae to light at the surface of the aqueous culture. 
     In “Biotechnology of Algal Biomass Production: A Review of Systems for Outdoor Mass Culture,” Journal of Applied Phycology 5: 593-604 (1993), Chaumont reviews mixing techniques proposed for use with algae cultures, including forcing culture through a slit in a board dragged through an open pond; “a mixing system consisting of a continuous flume containing arrays of foils similar in design to segments of airplane wings”; airlift; injectors; propellers; pump and gravity flow devices using natural energy sources; open pond loop “raceways” incorporating paddlewheel stirring devices; and sloped ponds and other cultivation units having parallel troughs or baffles, for example. 
     In “Photobioreactors for Mass Cultivation of Algae,” Bioresource Technology 99: 4021-4028 (2008). Ugwu et al. note that inefficient stirring mechanisms in open cultivation systems yield poor mass transfer rates that result in low biomass productivity. 
     Vertical photobioreactor systems use pumps, blowers or compressed air to introduce rising air bubbles and produce turbulent fluid motion in the aqueous algal culture for the purpose of mixing. Horizontal photobioreactor systems typically use pumps to circulate the culture and create turbulence in the aqueous algae culture to provide mixing. 
     Ugwu et al. (“Photobioreactors for Mass Cultivation of Algae”) describe the use of air-pump, bubble column and airlift systems to mix cultures in tubular and vertical-column photobioreactors. 
     The effects and performance of mixing in vessels such as bioreactors have also been investigated for numerous configurations of other mixing elements, such as the combination of radial impellers with axial up-pumping hydrofoils (Vrabel et al., “Mixing in Large-Scale Vessels Stirred With Multiple Radial or Radial and Axial Up-Pumping Impellers: Modelling and Measurements,” Chemical Engineering Science, Vol. 55, No. 23: 5881-5896 (2000)); a rotating impeller in combination with glass tubes acting as baffle plates (Ogbanna et al., “A Novel Internally Illuminated Stirred Tank Photobioreactor for Large-Scale Cultivation of Photosynthetic Cells,” Journal of Fermentation and Bioengineering, Vol. 82, No. 1: 61-67 (1996)); up-pumping impellers (Nienow et al., “The Versatility of Up-Pumping Hydrofoil Agitators,” Chemical Engineering Research and Design, Vol. 82, No. 9: 1073-1081 (2004)); axial and mixed dual-impeller systems (Bouaifi et al., “Power Consumption, Mixing Time and Homogenisation Energy in Dual-Impeller Agitated Gas-Liquid Reactors,” Chemical Engineering and Processing, Vol. 40, No, 2: 87-95 (2001)); the combination of airlift with hydrofoil impellers (Chisti et al., “Oxygen Transfer and Mixing in Mechanically Agitated Airlift Bioreactors,” Biochemical Engineering Journal, Vol. 10, No. 2: 143-153 (2002)); and turbines, down-pumping hydrofoils and up-pumping hydrofoils (Boon et al., “Comparing a Range of Impellers for ‘Stirring as Foam Disruption’,” Biochemical Engineering Journal, Vol. 10, No. 3: 183-195 (2002)). 
     In “A Simple Algal Production System Designed to Utilize the Flashing Light Effect,” Biotechnology and Bioengineering, Vol. XXV: 2319-2335 (1983) and in “High Algal Production Rates Achieved in a Shallow Outdoor Flume,” Biotechnology and Bioengineering, Vol. XXVIII: 191-197 (1986), Laws et al. describe gains in solar energy conversion efficiency and algae production yielded by emplacing arrays of foils similar in design to airplane wings to create vortices and systematic mixing in an algal culture flume. 
     Many of these methods provide mixing regimes for high-density algae cultures but consume too much energy to be cost effective in the production of biofuel, bulk chemicals or other commodities on an industrial scale. The required energy inputs for such methods and configurations exceed the energy yield that can be produced by the algae culture in the form of, for example, biofuel. Accordingly, a need exists for a mixing system that provides sufficient mixing and gas transfer for optimal production of biofuel and other materials while maintaining acceptable energy consumption in the context of operating costs for the reactor system and minimizing capital expense. 
     In addition, a need exists to provide effective mixing and gas transfer in vessels such as bioreactors and photobioreactors while maintaining structural integrity of the vessel, minimizing risk of contamination of the contents of the vessel, minimizing exposure of pumps and other mixing drive system components to corrosive agents in the vessel and facilitating ease of maintaining the drive system components. Various mixing apparatuses rely on the use of rotary impellers and similar elements that are not physically connected to a drive motor but instead are driven by magnetic coupling. 
     U.S. Pat. No. 7,824,904 (Dimanshteyn for “Photobioreactors for Production of Algae and Methods Therefor”) discloses mixing a liquid microbial culture using a rotary or oscillatory system comprising one or more motors, one or more shafts connected to the one or more motors and a plurality of mixing blades attached to the one or more shafts. 
     U.S. Pat. Appl. Pub, No. 2009/0035856 (Galliher et al. for “Continuous Perfusion Bioreactor System”) discloses vessels such as a disposable, collapsible bag having an integrated magnetically-driven rotating impeller that provides mixing for cell culture, cell containment, bioreactor and/or pharmaceutical manufacturing systems. 
     U.S. Pat. Appl. Pub. No. 2009/0130757 (Terentiev for “Bioreactor With Mixer and Sparger”) discloses a bioreactor that comprises an impeller positioned within an interior compartment of the vessel that is rotated by way of a magnetic coupling. 
     U.S. Pat. Appl. Pub, No. 2011/0003366 (Zeikus for “Methods of Using Pneumatic Bioreactors”) discloses a pneumatic bioreactor containing a fluid to be mixed that includes a floating impeller that rises in the fluid as gas bubbles carry it upward to the surface and falls when the gas is then vented, wherein the mixing speed is controlled with electromagnets in the vessel acting upon magnetic material in the impeller or its guides. 
     PCT Published Patent Application WO 2005/121310 (Johnson et al. for “Creation of Shear in a Reactor”) discloses the use of a applying a magnetic field to a magnetically-activated element to generate shear in a liquid sample. 
     U.S. Pat. Appl. Pub. No. 2009/0219780 (Castillo et al. for “Mixing System Including a Flexible Bag, Specific Flexible Bag and Locating System for the Mixing System”) discloses a mixing system comprising a flexible bag with a rotary magnetic impeller and an alignment facilitation device adapted to facilitate alignment between the magnetic impeller and a magnetic driver located external to the system. 
     In “Design, Construction and Testing of Pilot Scale Photobioreactor Subsystems,” Master of Science (MS) Thesis, Ohio University, Mechanical Engineering (Engineering and Technology), 2008, Mears describes the work of Tsygankov (2001) involving a coaxial cylinder reactor in which two coaxial tubes are placed one inside the other with algae fluid located in the annular space between the surfaces of both tubes. Mears further describes the reactor of Tsygankov incorporating a ferromagnetic ring in the section containing the algae and applying a magnetic field to move the ring back and forth, mixing the algae liquid. 
     In “Microbioreactors for Bioprocess Development,” Journal of the Association for Laboratory Automation, Vol. 12, No. 3: 143-151 (2007), Zhang et al. describe the use of a magnetic stir bar to mix a microbial solution in a cylindrical reactor chamber. 
     A need exists to incorporate a magnetic coupling drive system with a mixing configuration that is effective in a photobioreactor while maintaining the structural integrity of the photobioreactor and ability to service the components of the drive system without compromising the algae culture therein. 
     The above discussion includes both information known to the art prior to the filing date and information forming part of the present inventive disclosure. Inclusion of any statement in this section, whether as a characterization of a published reference or in a discussion of technical problems and their solutions, is not to be taken as an admission that such statement is prior art. 
     SUMMARY OF INVENTION 
     An object of this invention is a magnetically coupled mixing system adapted to provide vertical mixing in an open or enclosed vessel while advantageously maintaining low energy usage requirements. 
     A further object of this invention is a magnetically coupled mixing system adapted to provide gas transfer in an open or enclosed vessel while advantageously maintaining low energy usage requirements. 
     A further object of this invention is a magnetically coupled mixing system wherein components used to drive the mixing system are located outside a sealed mixing vessel, permitting access outside the sealed mixing vessel for greater ease of maintaining the drive components, while maintaining the integrity of the sealed mixing vessel. 
     Accordingly, this invention provides for a magnetically coupled mixing system comprising a mixing vessel; a liquid mixture disposed within the reactor vessel; a drive conduit; a drive element disposed within the drive conduit and adapted to move in a longitudinal direction within the drive conduit; a follower member having a first distal end and a second distal end, wherein the follower member is disposed around the perimeter of the drive conduit and is adapted to move longitudinally along the drive conduit; a magnetic follower element disposed within the follower member, wherein the magnetic follower element is adapted to couple magnetically with the drive element and is proximally disposed outside the drive conduit; a foil having a surface shaped or configured to provide hydrodynamic lift, wherein the foil is disposed at least partially in the liquid mixture; and a support member connecting the foil and the follower member. 
     This invention also provides for a magnetically coupled mixing system comprising a mixing vessel; a liquid disposed within the mixing vessel; a gas disposed within the mixing vessel; a drive conduit; a drive element disposed within the drive conduit and adapted to move in a longitudinal direction within the drive conduit; a follower member having a first distal end and a second distal end, wherein the follower member is disposed around the perimeter of at least a portion of the drive conduit and is adapted to move longitudinally along the drive conduit; a magnetic follower element disposed within the follower member, wherein the magnetic follower element is adapted to couple magnetically with the drive element and is proximally disposed outside the drive conduit; and a crossbar attached to the follower member, wherein the crossbar is at least partially disposed in the liquid, the crossbar mixes the liquid, the gas, or an interface between the liquid and the gas, the crossbar has a surface shaped or configured to generate a breaking wave front in the liquid, and the breaking wave front is generated by movement of the crossbar along a linear path in the mixing vessel. 
     This invention also provides for a magnetically coupled mixing system comprising drive fluid contained within the drive conduit and a pump in communication with the drive conduit, wherein the pump is adapted to move the drive fluid and the drive element within the drive conduit. 
     This invention also provides for a magnetically coupled mixing system wherein the drive fluid is air, water, mineral oil or polyethylene glycol. 
     This invention also provides for a magnetically coupled mixing system wherein the drive fluid contains corrosion-inhibiting agents. 
     This invention also provides for a magnetically coupled mixing system wherein the pump is a reversible flow pump. 
     This invention also provides for a magnetically coupled mixing system wherein the pump is a positive displacement pump or a velocity pump. 
     This invention also provides for a magnetically coupled mixing system wherein the pump is a diaphragm pump or a centrifugal pump. 
     This invention also provides for a magnetically coupled mixing system comprising a flow control valve. 
     This invention also provides for a magnetically coupled mixing system wherein the pump is located outside of the mixing vessel. 
     This invention also provides for a magnetically coupled mixing system wherein the mixing vessel is an enclosed vessel, an open vessel, a reactor, a bioreactor, a photobioreactor or an open pond. In the context of a photobioreactor comprising cells in a liquid suspension, the magnetic mixing system of the present invention employing a linearly moving foil provides gentle vertical mixing which allows for particle distribution and movement within a light field but without damaging cells. For situations wherein the cells are producing a target molecule, the cost of the energy consumed by the mixing system of the present invention is less than the value of the target molecule, with energy consumed and molecules produced averaged over the same time period. In a preferred embodiment, the energy cost is 10% or less of the value of the target molecules produced. 
     This invention also provides for a magnetically coupled mixing system wherein the liquid mixture comprises algae and water. This invention also provides for a magnetically coupled mixing system that may be used in other commercial processes that require low energy input and regular, gentle mixing in elongate reactors, including but not limited to pharmaceutical cell culture, food processing and waste water treatment. This invention also provides for a magnetically driven skimmer that can economically remove surface solids that accumulate in algal ponds and waste water. This invention also provides for a magnetically driven foil that may be used in a vapor phase to increase the efficiency of a solar still. 
     This invention also provides for a magnetically coupled mixing system wherein the drive conduit is disposed inside the reactor vessel. 
     This invention also provides for a magnetically coupled mixing system wherein the drive conduit is disposed at least partially within the liquid mixture. 
     This invention also provides for a magnetically coupled mixing system wherein the drive conduit is disposed outside the reactor vessel. 
     This invention also provides for a magnetically coupled mixing system comprising a blocking element disposed within the drive conduit, wherein the blocking element is adapted to restrict the movement of the drive element within a desired range within the drive conduit. 
     This invention also provides for a magnetically coupled mixing system comprising a longitudinal vane disposed on an inner surface of the drive conduit such that channels adapted to permit flow of the drive fluid are formed on the inner surface of the drive conduit, wherein the channels are bounded by the surface of the drive element, the surface of the longitudinal vane and the inner surface of the elongate tubular hollow member. 
     This invention also provides for a magnetically coupled mixing system comprising a longitudinal groove formed in an inner surface of the drive conduit such that a channel adapted to permit flow of the drive fluid is formed in the inner surface of the drive conduit, wherein the channel are bounded by the surface of the drive element and the surfaces of the groove formed in the inner surface of the drive conduit. 
     This invention also provides for a magnetically coupled mixing system wherein the drive element is ferromagnetic or magnetic. 
     This invention also provides for a magnetically coupled mixing system comprising materials that suppress corrosion or wear, wherein the materials coat the drive element. 
     This invention also provides for a magnetically coupled mixing system wherein the drive element is spherical or cylindrical. 
     This invention also provides for a magnetically coupled mixing system wherein the follower member comprises a hollow elongate tubular enclosure. 
     This invention also provides for a magnetically coupled mixing system wherein the magnetic follower element is adapted to move longitudinally within the follower member. 
     This invention also provides for a magnetically coupled mixing system comprising a flotation member and a support member connecting the flotation member with the follower member or the foil. 
     This invention also provides for a magnetically coupled mixing system wherein the flotation member is configured to provide surface mixing of the liquid mixture. 
     This invention also provides for a magnetically coupled mixing system wherein the flotation member comprises a pontoon. 
     This invention also provides for a magnetically coupled mixing system comprising a tracking member, wherein the tracking member is proximal to a wall of the mixing vessel and is adapted to prevent the foil and the flotation member from contacting the wall, wherein the support member connects the foil, the follower member and the tracking member. 
     This invention also provides for a magnetically coupled mixing system comprising a carrier member in which the magnetic follower element is contained, wherein the carrier member is proximally disposed outside the drive conduit. 
     This invention also provides for a magnetically coupled mixing system comprising a bumper element disposed within the follower member. 
     This invention also provides for a magnetically coupled mixing system comprising a flexible elongate tension member that connects the foil or support member to the follower member. 
     This invention also provides for a magnetically coupled mixing system wherein the foil is uncambered, has a quadrangular planform shape and is configured at an angle of attack sufficient to generate hydrodynamic lift and trailing vortices. 
     This invention also provides for a magnetically coupled mixing system wherein the foil comprises a cambered surface. 
     This invention also provides for a magnetically coupled mixing system wherein the foil is substantially vertically oriented and has a surface configured and angled to provide hydrodynamic lift. 
     This invention also provides for a magnetically coupled mixing system comprising an axle attached to a support member, wherein the foil is rotatably mounted on the axle. 
     This invention also provides for a magnetically coupled mixing system wherein the foil further comprises a weight or a cavity embedded in the foil proximal to a trailing edge of the foil. 
     This invention also provides for a magnetically coupled mixing system wherein the foil further comprises a steering element disposed on the surface of the foil proximal to a trailing edge of the foil. 
     This invention also provides for a magnetically coupled mixing system wherein the planform shape of the foil is bilaterally symmetric and is triangular or quadrangular. 
     This invention also provides for a magnetically coupled mixing system comprising a support member that is substantially horizontally oriented, wherein a top edge of the support member is cambered and is adapted to induce a hydraulic jump in the liquid mixture. 
     This invention also provides for a magnetically coupled mixing system comprising mixing structures attached to a horizontal support member, wherein the mixing structures are configured to stir the surface of the liquid mixture. 
     This invention also provides for a magnetically coupled mixing system comprising a flexible dredging member attached to the foil, the pontoon or a support member, wherein the flexible dredging member is at least partially suspended in the liquid mixture and is configured to induce vertical mixing of the liquid mixture. 
     This invention also provides for a magnetically coupled mixing system wherein the foil is pivotally attached to a support member, such that the angle of attack of the foil is variable and selectively adjustable. 
     This invention also provides for a magnetically coupled mixing system comprising a poppet valve disposed inside the drive element. 
     This invention also provides for a magnetically coupled mixing system comprising a bypass conduit connected to the drive conduit. 
     This invention also provides for a magnetically coupled mixing system comprising a foil disposed only in gas. 
     This invention also provides for a magnetically coupled mixing system comprising a gas sparging hose attached to the follower member, the follower element, the foil or the support member. 
     This invention also provides for a magnetically coupled mixing system comprising a Venturi tube formed in the support member, with one opening of the Venturi tube disposed above the surface of the liquid and the opposite opening of the Venturi tube disposed below the surface of the liquid. 
     This invention also provides for a magnetically coupled mixing system comprising a vertically-oriented foil attached to a horizontally-oriented foil. 
     This invention also provides for a magnetically coupled mixing system wherein the mixing system mixes only a portion of the depth of the liquid disposed within the mixing vessel. 
     This invention also provides for a magnetically coupled mixing system wherein the mixing system mixes only a portion of the length of the liquid disposed within the mixing vessel. 
     This invention also provides for a magnetically coupled mixing system comprising a mixing vessel; a liquid disposed within the mixing vessel; a foil having a surface shaped or configured to provide hydrodynamic lift, wherein the foil is disposed at least partially in the liquid, and wherein vertical mixing of the liquid is achieved by linear motion of the foil in the mixing vessel; a cable attached to the foil; and a reversible motor adapted to pull the cable. 
     This invention also provides for a magnetically coupled mixing system comprising a drive conduit; a follower member having a first distal end and a second distal end, wherein the follower member is disposed around the perimeter of at least a portion of the drive conduit and is adapted to move longitudinally along the drive conduit, and wherein the cable is attached to the foil or the follower member; and a support member connecting the foil and the follower member. 
     This invention also provides for a system for achieving vertical mixing within a liquid in a reactor comprising a foil moving by magnetic means in a linear direction. 
     This invention also provides for a method to achieve vertical mixing in a liquid in a reactor comprising the steps of moving a drive element in a linear direction within the reactor; magnetically coupling a follower element to the drive element; and coupling a foil to the follower element, such that the foil produces vertical mixing in the liquid in the reactor. 
     This invention also provides for a system for the production of a target molecule or accumulation of biomass comprising a suspension of cells in a liquid, wherein the cells are capable of producing a target molecule or accumulating biomass; and a foil moving by magnetic means in a linear direction in the liquid and producing vertical mixing of the suspension; wherein production of the target molecule or accumulation of biomass is greater in the presence of the moving foil than in the absence of the moving foil; and wherein the system is a bioreactor. 
     This invention also provides for a system for the production of a target molecule or accumulation of biomass comprising a suspension of cells in a liquid, wherein the cells are capable of producing a target molecule or accumulating biomass; and a foil moving by magnetic means in a linear direction in the liquid and producing vertical mixing of the suspension; wherein accumulation of biomass is greater in the presence of the moving foil than in the absence of the moving foil; and wherein the system is a bioreactor. 
     This invention also provides for a system for achieving vertical mixing within a fluid in a reactor comprising a foil moving by magnetic means in a linear direction. 
     This invention also provides for a method to achieve vertical mixing in a fluid in a reactor comprising the steps of moving a drive element in a linear direction within the reactor; magnetically coupling a follower element to the drive element; and coupling a foil to the follower element, such that the foil produces vertical mixing in the fluid in the reactor. 
     This invention also provides for a magnetically coupled mixing system comprising a mixing vessel; a fluid disposed within the mixing vessel; a drive conduit; a drive element disposed within the drive conduit and adapted to move in a longitudinal direction within the drive conduit; a follower member having a first distal end and a second distal end, wherein the follower member is disposed around the perimeter of at least a portion of the drive conduit and is adapted to move longitudinally along the drive conduit; a magnetic follower element disposed within the follower member, wherein the magnetic follower element is adapted to couple magnetically with the drive element and is proximally disposed outside the drive conduit; a foil having a surface shaped or configured to provide lift, wherein the foil is disposed at least partially in the fluid, and wherein vertical mixing of the fluid is achieved by linear motion of the foil in the mixing vessel; and a support member connecting the foil and the follower member. 
     This invention also provides for a magnetically coupled mixing system comprising a mixing vessel; a fluid disposed within the mixing vessel; a foil having a surface shaped or configured to provide lift, wherein the foil is disposed at least partially in the fluid, and wherein vertical mixing of the fluid is achieved by linear motion of the foil in the mixing vessel; a cable attached to the foil; and a reversible motor adapted to pull the cable. 
     This invention also provides for a magnetically coupled mixing system comprising a mixing vessel; a liquid disposed within the mixing vessel; a drive conduit; a drive element disposed within the drive conduit and adapted to move in a longitudinal direction within the drive conduit; and a magnetic follower element disposed around the perimeter of at least a portion of the drive conduit and adapted to move longitudinally along the drive conduit, wherein the magnetic follower element is adapted to couple magnetically with the drive element and is proximally disposed outside the drive conduit. 
     This invention also provides for a system for the production of a target molecule or accumulation of biomass comprising a suspension of cells in a liquid, wherein the cells are capable of producing a target molecule or accumulating biomass; and a foil moving by magnetic means in a linear direction in the liquid and producing vertical mixing of the suspension; wherein the total cost of mixing per unit weight of the target molecule or biomass produced is lower using the foil to induce vertical mixing than using a paddlewheel mixer. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       These and other features, aspects and advantages of this invention will become better understood with regard to the following description, appended claims and accompanying drawings where: 
         FIG. 1  shows a sectional view of a portion of a mixing system in accordance with certain embodiments of the present invention; 
         FIG. 2  shows an axial view of a portion of a mixing system in accordance with certain embodiments of the present invention; 
         FIG. 3  shows a perspective view of a portion of a mixing system in accordance with certain embodiments of the present invention; 
         FIG. 4  shows a perspective view of a portion of a mixing system in accordance with certain embodiments of the present invention; 
         FIGS. 5A  and B show a perspective and sectional view, respectively, of a follower element in accordance with certain embodiments of the present invention; 
         FIG. 6  shows a sectional view of a portion of a mixing system in accordance with certain embodiments of the present invention; 
         FIGS. 7A-H  show sectional views of exemplary suitable configurations of drive elements and follower elements; 
         FIG. 8  shows a sectional view of a drive element in accordance with certain embodiments of the present invention; 
         FIG. 9  shows a perspective view of a drive element in accordance with certain embodiments of the present invention; 
         FIGS. 10A-C  show sectional views of a drive element in accordance with certain embodiments of the present invention; 
         FIGS. 11A-C  show sectional and perspective views of a drive element in accordance with certain embodiments of the present invention; 
         FIGS. 12A-D  show sectional and perspective views of a drive element in accordance with certain embodiments of the present invention; 
         FIG. 13  shows a portion of a mixing system in accordance with certain embodiments of the present invention; 
         FIG. 14  shows a sectional perspective view of a mixing system and idealized fluid flow in accordance with certain embodiments of the present invention; 
         FIG. 15  shows a sectional perspective view of a mixing system in accordance with certain embodiments of the present invention; 
         FIG. 16  shows a sectional perspective view of a mixing system in accordance with certain embodiments of the present invention; 
         FIGS. 17A-C  show side and front views of portions of mixing systems in accordance with certain embodiments of the present invention; 
         FIG. 18  shows a sectional perspective view of an embodiment of a drive conduit; 
         FIGS. 19A  and B show sectional views of a drive element in accordance with certain embodiments of the present invention; 
         FIGS. 20A  and B show sectional and end views of a drive element in accordance with certain embodiments of the present invention; 
         FIGS. 21A  and B show sectional and end views of a follower member and bypass conduit in accordance with certain embodiments of the present invention; 
         FIG. 22  shows an embodiment of a hydraulic mixing system in accordance with certain embodiments of the present invention; 
         FIG. 23  shows an embodiment of a pneumatic mixing system in accordance with certain embodiments of the present invention; 
         FIG. 24  shows a planform schematic view of a mixing system in accordance with certain embodiments of the present invention; 
         FIG. 25  shows a planform schematic view of a mixing system in accordance with certain embodiments of the present invention; 
         FIG. 26  shows a cable driven mixing system in accordance with certain embodiments of the present invention; 
         FIG. 27  shows a cable driven mixing system in accordance with certain embodiments of the present invention; 
         FIG. 28  shows an embodiment of a foil in accordance with certain embodiments of the present invention; 
         FIG. 29  shows trailing vortices and vertical mixing generated by horizontal movement of a foil in a mixing system in accordance with certain embodiments of the present invention; 
         FIG. 30  shows a computational simulation of trailing vortices and vertical mixing generated by horizontal movement of a foil in a mixing system in accordance with certain embodiments of the present invention; 
         FIG. 31  shows a foil and a flotation member in accordance with certain embodiments of the present invention; 
         FIGS. 32A  and B show foils in accordance with certain embodiments of the present invention; 
         FIGS. 33A  and B show foils, flotation members, a drive system and support members in accordance with certain embodiments of the present invention; 
         FIGS. 34A-D  show foils in accordance with certain embodiments of the present invention; 
         FIGS. 35A-D  show flotation members, support members, foils, flexible dredging members and brushes in accordance with certain embodiments of the present invention; 
         FIG. 36  shows a foil, flotation member and support members that are rotatably connected in accordance with certain embodiments of the present invention; 
         FIGS. 37A  and B show a portion of a mixing system having a cambered horizontal support member in accordance with certain embodiments of the present invention; 
         FIG. 38  shows foils, flotation members, support members and a surface agitating comb in accordance with certain embodiments of the present invention; 
         FIG. 39  shows foils, flotation members and support members in accordance with certain embodiments of the present invention; 
         FIG. 40  shows a portion of a mixing system and an airfoil in accordance with certain embodiments of the present invention; 
         FIG. 41  shows a comparison of calculated energy requirements for mixing systems of the present invention with a mixing system known in the art; 
         FIG. 42  shows a comparison of energy consumption in pneumatic and hydraulic mixing systems in accordance with certain embodiments of the present invention; 
         FIG. 43  shows an exemplary graphical representation of the dependence of biomass accumulation on mixing type in closed photobioreactors; 
         FIG. 44  shows an exemplary graphical representation of capital expenditure per hectare for different mixing systems; 
         FIG. 45  shows a perspective view of tethered drive elements in accordance with certain embodiments of the present invention; 
         FIG. 46  shows a side view of an embodiment of the present invention adapted to produce a breaking wave in shallow liquid; 
         FIG. 47  shows a side view of an embodiment of the present invention producing a breaking wave in shallow liquid; and 
         FIG. 48  shows a perspective view of an embodiment of the present invention producing a breaking wave in shallow liquid. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Mixer Drive System Design 
       FIGS. 1-4  show sectional, perspective and front views of a tubular follower member  120 . In the exemplary embodiment, the tubular follower member  120  is disposed around a portion of a drive conduit  100 , such that the follower member  120  slides along the surface of the drive conduit  100  in a longitudinal direction. The follower member  120  partially encloses a magnetic follower element  110 , which is adapted to slide within the follower member  120  along the surface of the drive conduit  100  in a longitudinal direction. The distal end portions  192  of the follower member  120  are partially enclosed, thereby restricting the movement of the follower element  110  and providing surfaces against which the follower element  110  can exert force. 
     The follower member  120  and drive conduit  100  can be constructed from, for example, blow-molded or injection-molded thermoplastic, or any other material that is suitably rigid and light-weight. 
       FIGS. 5A  and B show perspective and sectional views of an exemplary follower element  110 , comprising magnets  140 , a bushing  150  and an enclosure  160 . The annular axial cross-section of the bushing  150  enables the follower element  110  to slide axially along the drive conduit  100 . 
     The bushing  150  can be constructed from, for example, stainless steel or any other material that is suitably resistant to wear and has a low coefficient of friction. The enclosure  160  can be constructed from, for example, polyethylene or any other material that is suitably durable and has a low coefficient of friction. 
       FIG. 6  illustrates a sectional view of a portion of an exemplary foil assembly  180  of the present invention. In certain embodiments, the drive system utilizes magnetic coupling between a drive magnet  142  or drive ferromagnet  144  contained within the drive conduit  100  and a follower magnet  140  disposed outside the drive conduit  100 , within the follower member  120 , wherein the magnetic coupling is used in conjunction with a motive force, such as a pneumatic force or a hydraulic force, to propel the foil assembly  180  through a mixing vessel. 
     In the exemplary embodiment, a tubular follower member  120  is disposed around a portion of a drive conduit  100 , such that the follower member  120  slides along the surface of the drive conduit  100  in a longitudinal direction. The follower member  120  is connected to a hydrodynamic cambered foil  170  by a support member  130 . The distal end portions  192  of the follower member  120  are partially enclosed. 
     In the exemplary embodiment, the follower magnet  140  is a ring magnet that encompasses the circumference of the exterior surface of the drive conduit  100 . The follower magnet  140  is adapted to slide within the follower member  120  along the surface of the drive conduit  100  in a longitudinal direction. The partial enclosures of the end portions  192  of the follower member  120  restrict the movement of the follower magnet  140  relative to the follower member  120  and provide surfaces against which the follower magnet  140  can exert force. 
     The drive magnet  142  or drive ferromagnet  144  is disposed within the drive conduit  100  and is adapted to move longitudinally within the drive conduit  100  when motive force is applied to the drive magnet  142  or drive ferromagnet  144 . 
     In operation, a motive force is applied to the drive magnet  142  or drive ferromagnet  144 , which traverses the drive conduit  100  in a longitudinal direction. The follower magnet  140  is magnetically coupled with the drive magnet  142  or drive ferromagnet  144  and moves in unison with the drive magnet  142  or drive ferromagnet  144 . When the follower magnet  140  comes into contact with the partial enclosure of either end portion  192  of the follower member  120 , the momentum of the follower magnet  140  is transferred to the follower member  120 , which is propelled by the follower magnet  140  and slides along the length of the drive conduit  100 . The foil  170  is connected to, and moves with, the follower member  120 , such that the foil  170  traverses through the mixing vessel and the liquid mixture contained therein in a linear path. 
     The direction in which motive force is applied to the drive magnet  142  or drive ferromagnet  144  can be reversed, inducing the drive magnet  142  or drive ferromagnet  144  to move in the opposite longitudinal direction. The follower magnet  140  is magnetically coupled with the drive magnet  142  or drive ferromagnet  144  and correspondingly changes direction of motion with the drive magnet  142  or drive ferromagnet  144 . Immediately after the initial change of direction of motion, the follower magnet  140  disengages from contact with the partial enclosure of the end portion  192  of the follower member  120  and traverses the open interior portion of the follower member  120 , during which the follower member  120  will remain stationary, or will not otherwise undergo motion attributable to the follower magnet  140 . If movement of the follower magnet  140  is sustained, the follower magnet  140  subsequently comes into contact with the opposite partially enclosed end portion  192  of the follower member  120  and transfers momentum to the follower member  120 . The follower member  120  and attached foil  170  consequently undergo a change of direction of motion. 
     In the exemplary embodiment, the distance between the position of the partially enclosed end portion  192  of the follower member  120  and the vertical centerline of the support member  130  creates a locus of force, or tow point, between the follower magnet  140  and follower member  120  that is forward of the foil  170  relative to the direction of motion of the foil assembly  180 . When the direction of motion reverses, the tow point becomes the point of contact between the follower magnet  140  and the opposite end of the follower member  120 , which is likewise forward of the foil  170  relative to the direction of motion of the foil assembly  180 . This configuration enhances guidance and stability of the follower member  120  and the attached foil  170  by preventing yaw of the foil assembly  180  while the follower member  120  is motion in either direction. 
       FIGS. 7A-H  illustrate exemplary suitable drive magnets  142  or drive ferromagnets  144  and follower magnets  140 . Each drive magnet  142  or drive ferromagnet  144  is disposed within a drive conduit  100  and is adapted to move within the drive conduit  100  in a longitudinal direction in response to a motive force applied to the drive magnet  142  or drive ferromagnet  144 . The drive magnets  142  or drive ferromagnets  144  comprise, for example, one or more ferromagnetic ball bearings, one or more axially-magnetized cylindrical magnets, or one or more spherical magnets. One of skill in the art will understand that the drive magnet  142  or drive ferromagnet  144  can be made of any material and have any shape suitable to promote magnetic coupling with the follower magnet  140  and range of motion within the drive conduit  100 . In some embodiments, the drive magnet  142  or drive ferromagnet  144  comprises steel, a neodymium iron boron magnet or another rare earth magnet. In some embodiments, the drive magnet  142  or drive ferromagnet  144  is coated with felt or other materials that are suitable for suppressing corrosion or wear of the drive magnet  142  or drive ferromagnet  144  and other surfaces that come into contact with the drive magnet  142  or drive ferromagnet  144 . 
     In certain embodiments, the drive conduit  100  is a tube having a circular cross section and is made of low density polyethylene, high density polyethylene, cross-linked polyethylene, polyvinyl chloride, copper, steel or any other suitable material. In some embodiments, the construction of the drive conduit  100  provides positive buoyancy to help maintain the position of the drive conduit  100  relative to the surface  320  of the liquid mixture. 
     The follower magnet  140  is disposed on or around the external surface of the drive conduit  100  in a manner that allows the follower magnet to move along the longitudinal axis of the drive conduit  100 . An example of a suitable follower magnet  140  is one axially-magnetized ring magnet, wherein the ring magnet circumferentially encompasses a portion of the drive conduit  100 . Another suitable configuration is a plurality of follower magnets  140  embedded in a follower element  110  in the form of a sliding sleeve. In some embodiments, the follower member  120  does not fully encircle the drive conduit  100 . In some embodiments, the follower magnets  140  are disposed on opposite sides of the drive conduit  100  and are equidistant apart relative to the circumference of the drive conduit  100 . One of skill in the art will understand that the follower magnet  140  may be made of any material and have any shape suitable to promote magnetic coupling with the drive magnet  142  or drive ferromagnet  144  and longitudinal range of motion along the exterior of the drive conduit  100 . In some embodiments, the follower magnet  140  comprises a neodymium iron boron magnet or another rare earth magnet. 
     In the present invention, the gap between the surface of the drive magnet  142  or drive ferromagnet  144  and the interior surface of the drive conduit  100  is minimized in order to reduce hydraulic or pneumatic fluid flow around the drive magnet  142  or drive ferromagnet  144  and maximize motive force applied to the drive magnet  142  or drive ferromagnet  144  and the foil  170  for a selected flow rate of drive fluid. In certain embodiments comprising a pneumatic fluid used to apply motive force to the drive magnet  142  or drive ferromagnet  144 , a low friction seal between the surface of the drive magnet  142  or drive ferromagnet  144  and the interior surface of the drive conduit  100  is utilized. The low friction seal can be created by, for example, dispersing oil along the length of the interior surface of the drive conduit  100  or by applying a ferromagnetic fluid to the surface of the drive magnet  142  or drive ferromagnet  144 . In certain embodiments, felt or another suitable material or coating is adhered or applied to the surface of the drive magnet  142  or drive ferromagnet  144  to reduce friction between the drive magnet  142  or drive ferromagnet  144  and the inner surface of the drive conduit  100 . 
       FIG. 8  shows a sectional view of a drive element  190  disposed within a drive conduit  100 . The drive element comprises a drive magnet  142  or drive ferromagnet  144  embedded in a plug of closed cell foam  200 , wherein the shape of the plug of closed cell foam  200  conforms to the inner surface of the drive conduit  100 . The drive element  190  further comprises inserts of open cell foam  210  embedded within the plug of closed cell foam  200 . The inserts of open cell foam  210  are positioned and adapted to expand and exert outward pressure on the plug of closed cell foam  200  in order to improve sealing and decrease empty space between the outer surface of the drive element  190  and the inner surface of the drive conduit  100 . 
     FIGS.  9  and  10 A-C show another embodiment of a drive element  190 , comprising a drive magnet  142  or drive ferromagnet  144 , an o-ring  220 , an insert  230 , a ring  240  and an end cap  250 . The o-ring  220  may be made of, for example, nitrile rubber. The insert  230  may be made of, for example, nylon 6-6. The ring  240  may be made of, for example, polytetrafluoroethylene. The end cap  250  may be made of, for example, nylon 6-6. The outside diameter of the o-ring  220  is slightly larger than the inside diameter of the drive conduit  100 . The o-ring gland  232  in the insert  230  in which the o-ring  220  sits is wider than the diameter of the o-ring  220 , and the inside diameter of the o-ring  220  is slightly larger than the diameter of the o-ring gland  232  in which the o-ring  220  sits, such that the o-ring  220  sits loosely in the o-ring gland  232 . When the drive element  190  is disposed in the drive conduit  100 , the o-ring  220  is squeezed against the inside wall of the drive conduit  100  but is free to move laterally in the o-ring gland  232  ( FIG. 10B ). Pneumatic motive force applied to the drive element  190  forces the o-ring  220  to move in the direction of the motive force until it contacts the wall of the o-ring gland  232 , forming a floating seal between the o-ring  220 , the wall of the o-ring gland  232  and the inner wall of the drive conduit  100  ( FIG. 10C ). 
       FIGS. 11A-C  and  12 A-D show a drive element  190  that incorporates a bypass adapted for use with a mixing system that incorporates a hydraulic form of motive force. The drive element  190  has an annular shape, and a plunger  260  is disposed at least partially within the drive element  190  such that the plunger  260  can slide laterally within the drive element  190 . When the plunger  260  is positioned such that neither head  262  is in contact with the drive element  190 , fluid can flow around the plunger  260  and through the annular opening in the drive element  190  ( FIG. 11C ). 
     A cap  270  may be sized and positioned within the drive conduit  100  such that the cap  270  restricts the lateral motion of the drive element  190 . The cap  270  may be configured so as to hold the plunger  260  in the open position when the cap contacts the drive element  190 , such that fluid flows through the annular opening in the drive element  190  and through corresponding channels  272  formed in the cap  270 . If any blockage occurs in the drive system, the configuration shown in  FIGS. 11A-C  and  12 A-D prevents excess fluid pressure from accumulating. 
     As illustrated in  FIG. 13 , the towing point for rigidly mounted foils  170  is shifted forward of the flotation members  300 , which act as rudders to provide steering, when the direction of movement of the foil assembly  180  is reversed. The exemplary embodiment illustrated in  FIG. 13  allows the tow point of the foil assembly  180  to slide to a stop beyond the flotation members  300  by allowing the follower element  110  to slide along the drive conduit  100  between end caps  122  that are mounted in fixed positions on the drive conduit  100 . Pads  280  may be attached to the follower element  110  or end caps  122  to reduce the force of impact caused by acceleration of the foil assembly  180  during a reversal in the direction of motion. The impact force may also be reduced by incorporating an alternative attachment between the foil  170 , flotation member  300  or support members  130  and the end caps  122 . The attachment may incorporate flexible tension members  290  made of, for example, 0.125 inch-diameter silicon rubber, under slight tension. The tow point in this exemplary embodiment also alternates as a steering element or stabilizing rudder when it is shifted aft of the foil  170 . 
       FIGS. 14 ,  15 ,  16  and  17 A-C illustrate alternative embodiments of mixing vessels containing liquid algae cultures and mixing systems in accordance with the present invention. In  FIGS. 14 and 15 , the drive conduit  100  is disposed in or above the surface  320  of the algae culture so that it floats in the photobioreactor  310 . The drive conduit  100  may be made from, for example, high density polyethylene or any other suitable material that is inexpensive and is durable in saltwater, volatile compounds, sterilizing agents and moderate heat. In the exemplary embodiments, the follower element  120  contains follower magnets  140 . Magnets of 0.5 inch diameter and 0.5 inch length can achieve a coupling force of four pounds with minimal lateral force acting on the drive magnet  142  or drive ferromagnet  144 , which would be exhibited as friction between the drive magnet  142  or drive ferromagnet  144  and the inner wall surface of the drive conduit  100 . The exemplary embodiments also incorporate horizontal support members  130  positioned above the surface  320  of the algae culture to reduce hydrodynamic drag, and vertical support members  130  connecting the flotation members  300  to the foils  170  that are adapted to minimize interference with fluid flow around the foils  170 . 
     In some embodiments, the drive conduit  100  is located outside the mixing vessel, and the drive element  190  contained therein is magnetically coupled to a foil assembly  180  disposed inside the mixing vessel. As shown in  FIG. 16 , the drive conduit  100  is disposed underneath the bottom surface of the photobioreactor  310 . The film of the photobioreactor  310  lays over the drive conduit  100  and a track  352 . The weight of the algae culture provides sufficient hydraulic pressure to fix the position of the drive conduit  100  and the track  352 . A wheeled carrier  350  inside the photobioreactor  310  longitudinally traverses the track  352  and carries a follower magnet  140  that is magnetically coupled with a drive magnet  142  or drive ferromagnet  144  disposed within the drive conduit  100 . An array of foils  170  are attached to the wheeled carrier  350 . In some embodiments, the drive conduit  100  and the track  352  are welded or otherwise fixed to the film of the bioreactor  310 . 
     Sufficient vertical and horizontal clearances between the wheeled carrier  350  and the track  352  are maintained to accommodate for the conformation of the film around the track  352 . In the exemplary embodiment, a set of 3, 0.5 inch long follower magnets  140  provides 2 pounds of coupling force in the axial direction, which is sufficient to maintain coupling during sudden decelerations of the drive element  190  when stopped or started at the ends of the photobioreactor  310 . This arrangement of follower magnets  140  also provides a torque and downward attraction force of approximately 4 pounds that prevents the wheeled carrier  350  from separating from the track  352 . 
     The embodiments illustrated in  FIGS. 14 and 15  eliminate the need to lay the film of the photobioreactor  310  carefully over the drive conduit  100  in the embodiment of  FIG. 16  in order to avoid wrinkles that might structurally compromise the film of the photobioreactor  310  and the need to have a flat track  352  underneath the photobioreactor  310 . In the embodiments of  FIGS. 14 and 15 , there is no wheeled carrier  350  rolling on the floor of the photobioreactor  310 , so the thin film of the photobioreactor  310  is not exposed to continual mechanical stress on its surface, which eventually may lead to failure of the film. 
     Photobioreactors  310  also may deflect horizontally over lengths of, for example, 50 feet. With the exemplary embodiment illustrated in  FIG. 16 , deflection of more than a few inches may cause a foil  170  to impact the film on the side of a photobioreactor  310  and potentially tear the film, if no guard mechanism is in place, and if the drive conduit  100  and the track  350  are not fixed to an inside surface of the bioreactor  310 . In the embodiment illustrated in  FIG. 15 , if the photobioreactor  310  curves excessively, the wall of the photobioreactor  310  would push the tracking member  340  above the surface  320  of the algae culture and displace the foil assembly  180  laterally, since the drive conduit  100  would offer minimal bending resistance. The tracking members  340  thereby guard against accidental contact between the foils  170  and the film on the sides of the photobioreactor  310 . 
       FIGS. 17A-C  illustrate exemplary embodiments having a drive conduit  100  positioned underneath, or on any side of, a photobioreactor  310 . One or more drive magnets  142  or drive ferromagnets  144  are contained in the drive conduit  100 . One or more follower magnets  140  are attached to a flotation member  300  by support members  130  and are magnetically coupled with the drive magnets  142  or drive ferromagnets  144 . 
     Motive Force System 
     The present invention utilizes a pump  420  to provide motive force to the drive elements  190  in an array of mixing vessels, such as photobioreactors  310 . In some embodiments, one pump  420  can drive mixing in multiple vessels. 
     The pump  420  moves drive fluid through the drive conduit  100 . The drive fluid may be, for example, air, water, mineral oil or polyethylene glycol, and the drive fluid may be suffused with agents that inhibit corrosion. In some embodiments, a drive fluid of low density is selected for use in the drive conduit  100  to promote positive buoyancy of the drive conduit  100 , in particular if the drive conduit  100  is constructed of materials having high density, such as steel or iron. 
       FIGS. 22 and 24  show exemplary embodiments in which a drive conduit  100  served by a single pump  420  and a switching valve  430  is routed through multiple photobioreactors  310  in a serpentine configuration. One or more drive elements  190 , follower elements  110  and foil assemblies  180  can be disposed inside each photobioreactor  310 . In the exemplary embodiment, the pump  420  and switching valve  430  are located outside the photobioreactors  310  and the drive system is closed. 
       FIG. 22  illustrates an exemplary hydraulic mixing system in which one pump  420  drives mixing in four rows of at least 70 photobioreactors  310  per row. Each row of photobioreactors  310  is partitioned into groups of 12 photobioreactors  310  through which a single drive conduit  100  of 0.5 inch diameter passes. Row is distributed to the rows of photobioreactors  310  via the drive conduits  100  using a two inch diameter pipe  410 . A pump  420  pushes water through the pipes  410  and drive conduits  100  to drive one or more drive elements  190  in the drive conduits  100  in each photobioreactor  310 . The drive elements  190  are magnetically coupled to foil assemblies  180 , which comprise follower elements  110 , follower members  120 , foils  170  and connecting support members  130 . 
     The exemplary hydraulic mixing system of  FIG. 22  comprises a pump  420  rated at one horsepower, 48 gallons per minute and 200 foot head (approximately 100 pounds per square inch). The exemplary hydraulic mixing system further comprises a four-way air piloted valve  430  adapted to switch the direction of hydraulic flow, a flow control valve  440 , a flow meter  450  and an air bleed tank  460 . The hydraulic pressure in the pipe  410  is approximately 30 pounds per square inch. 
     The exemplary hydraulic mixing system of  FIG. 22  moves the drive elements  190  and foil assemblies  180  at a constant speed for a preselected length of time. In some embodiments, a length of time greater than 30 seconds is expected to be sufficient for the drive elements  190  and foil assemblies  180  to traverse the full length of a photobioreactor  310 . Mechanical stops (not shown) at the ends of each photobioreactor  310  restrict the movement of the drive elements  190  and foil assemblies  180  while hydraulic flow continues for a short period of time to enable the drive elements  190  and foil assemblies  180  to reach the end of each photobioreactor  310 . In some embodiments, the short period of time is five to ten seconds. 
     In the exemplary hydraulic mixing system, the four-way air piloted valve  430  subsequently is activated to reverse the hydraulic flow throughout the drive conduit  100 . The activation of this valve  430  preferably occurs slowly enough to avoid dislodging the magnetic coupling between the drive elements  190  and follower elements  110 . In some embodiments, the length of time for activation of the valve  430  is greater than 100 milliseconds. 
     In alternative embodiments, a reversible positive displacement pump  420  with a variable frequency drive is used and the four-way air piloted valve  430  is omitted. The hydraulic flow through the system is controlled to a specified rate. 
     Energy usage can be reduced by reducing the cross-sectional area of the drive conduit  100  or changing the drive fluid to air. Reducing the cross-sectional area of the drive conduit  100  requires closer tolerances between the drive element  190  and follower element  110  so that smaller magnets can be used. Changing the drive fluid to air requires using a sealing fluid around the drive magnet  142  or drive ferromagnet  144  to avoid wasting energy. 
       FIG. 23  illustrates an exemplary pneumatic mixing system in which one pump  420  drives mixing in 240 photobioreactors  310  connected in parallel. Drive conduits  100  of 0.5 inch diameter pass through each row of photobioreactors  310 . Flow is distributed to the rows of photobioreactors  310  via the drive conduits  100  using a two inch diameter pipe  410 . A pump  420  pushes air through the pipe  410  and drive conduits  100  to drive one or more drive elements  190  in the drive conduits  100  in each photobioreactor  310 . The drive elements  190  are magnetically coupled to foil assemblies  180 , which comprise follower elements  110 , follower members  120 , foils  170  and connecting support members  130 . 
     In some embodiments, a system of the present invention incorporates multiple drive elements  190  and foil assemblies  180  in one mixing vessel. The foil assemblies  180  are disposed on the drive conduit  100  at selected distance intervals and are driven by the same motive system. The foil assemblies  180  may be configured and spaced to provide mixing over the full length of the mixing vessel, wherein each foil assembly  180  provides mixing for a selected portion of the mixing vessel. One of ordinary skill will appreciate that the lowest energy consumption required to achieve a desired degree of vertical mixing in a photobioreactor  310  can be determined by varying the number and configuration of foil assemblies  180  that are used in the photobioreactor  310 . 
       FIG. 24  illustrates an embodiment in which multiple photobioreactors  310  are serviced by one pump  420  and one drive conduit  100 . The photobioreactors  310  are connected in series. In this embodiment, multiple foil assemblies  180  are disposed inside each photobioreactor  310  and are configured to traverse the length of the bioreactor in opposite directions. 
       FIG. 25  illustrates an embodiment in which multiple photobioreactors  310  are serviced by one pump  420  and multiple drive conduits  100  that are fed by a header pipe  410 . Pairs of photobioreactors  310  are connected in parallel, and the photobioreactors  310  in each pair are connected in series. In this embodiment, multiple foil assemblies  180  are disposed inside each photobioreactor  310  and are configured to traverse the length of the bioreactor in the same direction. 
     In embodiments of the present invention that utilize pneumatic force to move foil assemblies  180 , multiple foil assemblies  180  may be disposed inside each photobioreactor  310 , while multiple photobioreactors will be connected in parallel, instead of in series. As shown in  FIG. 45 , multiple drive elements  190  are tethered by a connector  252 , which may be any flexible or rigid elongate member made of, for example, plastic, nylon or elastomer, that physically connects the drive elements  190 . Unidirectional floating seal grooves  222  are formed in the drive elements  190  positioned at the distal end of each chain of drive elements  190 . When pneumatic motive force is applied, the unidirectional floating seal groove  222  formed in the drive element  190  in the upstream position allows gas to slip underneath the o-ring  220  and prevent the formation of a seal. A seal is instead formed by the o-ring  220  present in the downstream drive element  190 . The downstream drive element  190  moves in the direction of the pneumatic motive force and pulls the upstream drive element  190  in the same direction. If the chain of drive elements  190  includes three or more drive elements  190 , then only the drive element  190  at the upstream end of the chain and the drive element  190  at the downstream end of the chain have o-rings  220 , so that no seals are formed by the drive elements  190  positioned in the interior of the chain. 
     In embodiments of the present invention that utilize hydraulic force to move foil assemblies  180 , a disadvantage of servicing multiple vessels in series using one pump  420  is that an increased pressure drop caused by a decreased rate of mixing in one vessel, such as due to a stopped drive element  190 , affects mixing speed in all other vessels in the array. In some embodiments, a system of the present invention comprises a positive displacement pump  420 , such as a vane pump, for the purpose of rendering flow rate through the pump  420  independent of pressure variations and providing consistent mixing speed in the array of vessels. 
     Operating pressure in a system of the present invention utilizing hydraulic force is approximately 32 pounds per square inch, while maximum pressure induced by stoppages of the drive elements  190  in twelve mixing vessels may be as high as 360 pounds per square inch. A large pump motor  420  may generate sufficient torque to maintain constant mixing speed under temporary large pressure drops in the mixing system but will not operate efficiently under smaller pressure drops that are typical during normal operation of the mixing system, when no drive elements  190  are stopped. With reference to  FIG. 18 , in some embodiments, the conduit used in a system of the present invention allows flow past a stopped drive element  190  and reduces the pressure drop, which reduces the size of the pump motor  420  needed for the system, by incorporating a small channel  360  between the surface of the drive element  190  and the inner surface of the drive conduit  100 . The channel  360  is sufficiently small to maintain motive force in the form of pressure from a drive fluid applied to the drive element  190  under normal operation, and the change in outside diameter of the drive conduit  100  is sufficiently small to avoid interference with a follower element  110 . If channels  360  are formed in the inner surface of the drive conduit  100  and the outside diameter of the drive conduit  100  is unchanged, a higher schedule drive conduit  100  must be selected to compensate for the loss of pressure capability in the drive conduit  100 . 
     As illustrated in  FIGS. 19A  and B and  FIGS. 20A  and B, in some embodiments, the drive element  190  incorporates a poppet valve  370 . If a misaligned or otherwise impeded drive element  190  creates a blockage in the drive conduit  100 , the poppet valve  370  is adapted to allow drive fluid to flow through the drive element  190  and provides an alternative means of releasing pressure. In some embodiments, the internal surface of the drive conduit  100  incorporates drive stops  380  that restrict the motion of the drive element  190  in a longitudinal direction within the drive conduit  100 . In some embodiments, a valve opening pin  390  is adapted to actuate the poppet valve  370 . 
     As illustrated in  FIGS. 21A  and B, some embodiments comprise a bypass conduit  400  that is formed in the drive conduit  100 . The bypass conduit  400  is adapted to relieve fluid pressure by diverting the flow of drive fluid in the event that a blockage occurs in the drive conduit  100  between the inlet and outlet of the bypass conduit  400 . In some embodiments, the follower member  120  is shaped to accommodate the placement of the bypass conduit  112 . 
     As illustrated in  FIGS. 26 ,  27  and  28 , in some embodiments the motive system is a cable connected system. A cable  470  attaches to a foil  170  and/or to flotation members  300  in a foil assembly  180 . The cable  470  is pulled by a motor  472  that is configured to pull the cable  470  in more than one direction. The foil  170  is moved by actuating the motor  472  to pull the cable  470 , and the cable  470  is wound on a spool  474  at either end of the mixing vessel depending on the direction of travel. 
     Foil Design 
     A system in accordance with certain embodiments of the present invention comprises a surface, or foil  170 , that is shaped to provide hydrodynamic lift, wherein the foil  170  can be moved through an aqueous culture of algae to efficiently generate a vertical movement of algae.  FIGS. 14 ,  15 ,  16  and  28  illustrate exemplary foils  170  used in a mixing system of the present invention. It is desirable to induce flow with the lowest velocity needed to provide satisfactory mixing while minimizing energy consumption. Net vertical flow of the aqueous culture in the mixing vessel is zero. In the lowest energy case, upward and downward velocities are equal and act over equal areas. 
       FIG. 14  illustrates idealized vertical fluid motion in trailing vortices  330  generated by foils  170  in a photobioreactor  310 . The trailing vortices  330  remain stationary in the middle of the photobioreactor  310  due to net cancellation of induced velocity from image vortices (not shown) that are created at the top and bottom of the aqueous culture. Generating trailing vortices  330  in the center of the photobioreactor  310  may maximize the mixing length, or vertical particle displacement, over which the vortices  330  can transport flow. Trailing vortices  330  in the center of the photobioreactor  310  form a stable arrangement and tend not to migrate in location. In contrast, pairs of counterrotating vortices  330  move vertically in unbounded flows or move laterally near a horizontal surface. 
     In the exemplary embodiment, foils  170  distributed along the span of a photobioreactor  310  provide a regular arrangement of mixing vortices  330  that minimize the presence of dead zones with no mixing. Such mixing is sufficient to sustain algae cultures over extended periods of time. Aqueous algae cultures can die from anoxia due to reduced gas exchange and sedimentation of the algae if mixing is stopped for a period of several hours. 
     A foil assembly  180  generally may be moved at a higher speed with downwardly concave foils  170 , instead of upwardly concave foils  170 . However, downwardly concave foils  170  produce trailing vortices  330  that induce sedimentation of algae in regions away from the foils  170 , while upwardly concave foils  170  produce trailing vortices  330  that rotate in the opposite direction and induce sedimentation of algae in the region underneath the foil  170 . Additional mixing and resuspension of settled algae may be facilitated to a greater extent by a pattern of sedimentation underneath the foil  170  rather than sedimentation away from the edges of the foil  170 . 
     The migratory speed of trailing vortices  330  can be predicted by determining the velocity a vortex  330  will induce on its neighbor, according to potential flow theory and in situations with surfaces, including the effect of image vortices. For a vortex  330  near a single horizontal surface, such as the bottom of the photobioreactor  310 , the image vortex (not shown), which is a vortex  330  with opposite rotation placed equidistant but on the opposite side of the horizontal surface, will induce the actual vortex  330  to move laterally with a certain speed. When the actual vortex  330  is bounded on the opposite side by a second horizontal surface, which is the top surface  320  of the liquid mixture in exemplary embodiments of the present invention, the image vortex from this surface  320  will induce a directed motion countering the induced motion from the lower surface. If the vortex  330  is centered between these two surfaces, then the induced effect cancels completely and the vortex  330  remains stationary. The stability of the vortex  330  position allows for subsequent passes of a foil  170  to reinforce the strength of the vortex  330 . 
     The trailing vortices  330  decay slowly and are continually reinforced as the foils  170  longitudinally traverse the photobioreactor  310  in either direction, allowing the foils  170  to effectively mix a large area of aqueous culture in the vessel compared to the planform area of each foil  170 . This efficiency reduces the amount of equipment needed for mixing and capital costs. 
     The preferred placement of the foil  170  is middepth in the aqueous culture, with the span of the foil  170 , and the lateral spacing between foils  170 , equal to the depth of the culture. This placement produces a stable configuration of trailing vortices  330 . 
     The trailing vortices  330  are strengthened by increasing the lift generated by the foil  170 . Lift is controlled by the planform area, angle of attack, camber and speed of the foil  170 . The trailing vortices  330  are also reinforced by multiple passages of each foil  170 , which can be increased by increasing foil  170  speed for a photobioreactor  310  of a fixed length or by employing multiple foil assemblies  180  along the length of one photobioreactor  310 . Foils  170  can be spaced equidistant along the transverse axis of the photobioreactor  310  to reduce the time over which a trailing vortex  310  will decay before reinforcement and thereby achieve desired recirculation with uniformity and low power requirements. 
       FIG. 29  illustrates measurements of a trailing vortex  330  pair generated by a single foil  170  in an algae culture having a depth of eight inches. The particle traces shown cover the depth of a photobioreactor  310 , eight inches, and show the flow pattern of the particles after the passage of one foil  170 . Movement of the foil  170  (not shown) is in a plane perpendicular to the plane of the image. The trailing vortices  330  were generated by the distal ends of a foil  170  designed to produce similar recirculation times in the aqueous culture when the foil  170  longitudinally traverses the photobioreactor  310  at mid-depth in the algae culture. The foil  170  shape is symmetric and cambered to generate a vortex  330  system with rotations that are independent of the direction of traverse. The trailing vortices  330  were measured through laser diagnostics. The particles are neutrally buoyant and were illuminated with a laser light sheet. Four consecutive exposures are superimposed in this image to show the movement of the particles over intervals of 66.7 milliseconds. Total plotted time for each particle is therefore 4×66.7 milliseconds=0.26 seconds. 
       FIG. 30  illustrates a computational simulation of trailing vortices  330  generated by a foil  170  passing through an algae culture with a depth of eight inches. This image provides the same view of particle traces shown in  FIG. 29 . Each trace in  FIG. 30  represents a total travel time of 8.21 seconds per particle. Particles in this simulation are nonuniformly distributed at an initial time measurement, with most of the particles being located at mid-depth in the algae culture in a photobioreactor  310  at time=0 seconds. For a given position along the length of the algae culture in the photobioreactor  310 , the vortices  300  decay over time and must be reinforced or regenerated by the periodic passage of the foil  170 . The maximum time to reinforce trailing vortices  330  is roughly 30 seconds between passages for a foil  170  travelling at 0.5 meters per second. 
     In some embodiments, the foil assembly  180  comprises a symmetric, cambered foil  170 . One of skill in the art will recognize that other configurations are also suitable to produce effective mixing and generate trailing vortices  330 . Trailing vortices  330  can be generated by hydrodynamic drag on flat foils  170  that have no camber and are held at a constant angle of attack as the foils  170  traverse a photobioreactor  310 ,  FIG. 31  illustrates a foil  170  with a quadrangular planform shape that can produce hydrodynamic lift. 
       FIGS. 32A  and B illustrate exemplary foils  170  that are mounted on axles  510  connected to the foil assembly  180  and are adapted to swivel or pivot to an inclined orientation. The angle of attack, or pitch, of the foil  170  may be maintained by differential weighting along the chord of the foil  170 . Triangular and quadrangular planform shapes are suitable for foils  170  of these embodiments. Differential weighting between the sum of the hydrodynamic lift and the weight of the foil  170  may be adjusted by incorporating a weight  480 , or an interstitial space  490 , embedded in the foil  170 . The angle of attack and the amount of circulation generated may thereby be varied and adjusted. 
     With respect to foils  170  that incorporate embedded weights  480 , care must be taken to ensure that the trailing edges of foils  170  with long chords do not scrape the bottom of the mixing vessel during a reversal in travel direction of the foil  170 , as the foil  170  swings underneath the axle  510 . With a buoyant trailing edge incorporating an interstitial space  490 , this is avoided since the trailing edge of the foil  170  would swing over the axle  510 . 
     With reference to  FIGS. 32A  and B, foils  170  that swivel preferably incorporate a rudder  500  to prevent yaw while avoiding the need to pull the foil  170  from a shifting forward location. 
       FIGS. 33A  and B illustrate an embodiment comprising foils  170  positioned in a vertical orientation. Trailing vortices  330  can be generated by vertical foils  170  that are oriented at slight angles to the direction of the foil  170  movement. In some embodiments, the vertical foils  170  have a triangular planform shape. 
     In some embodiments, foils  170  are made from molded plastic, fiberglass, sintered nylon, glass-reinforced plastic, or any other material that is suitable to provide rigidity, durability, and positive or neutral buoyancy. 
       FIGS. 34A-D  illustrate an embodiment comprising vertically-oriented foils  170  that are designed increase the extent of mixing directly beneath a horizontally-oriented, upwardly concave foil  170  in region where settling may occur. The two vertically-oriented foils  170  are set in opposition to yield net zero horizontal lift. The two vertically-oriented foils  170  create a set of trailing vortices  330  that impinge on the liquid culture directly beneath the vertically-oriented foils  170  and create a high shear zone that promotes mixing and reduces sedimentation beneath the foils  170 . 
       FIGS. 35A  and B illustrate the use of a chain  520  or other suitable flexible dredging member to reduce sedimentation in the aqueous culture. In this exemplary embodiment, a chain  520  is attached to, and suspended from, the foil  170  and drags through the space beneath the foil  170  where sediment accumulates. 
     If sedimentation is a more severe problem than the need for vertical mixing, foils  170  can be omitted from the foil assembly  180  and a uniform brush  530  can be attached to the foil assembly  180  instead, as illustrated in  FIGS. 35C  and D. The density of the brush  530  bristles must be sparse enough to avoid excessive hydrodynamic drag forces. Sufficient downward force must be applied to the brush  530  by, for example, utilizing the weight of the support to scrape settled algae from the bottom of the mixing vessel. 
     As illustrated in  FIG. 36 , a foil  170  can be rotatably attached to a vertical support member  130  at a pivot point  540 . When the foil  170  is traversing the photobioreactor  310 , the opposing force exerted by the algae culture causes the foil  170  to rotate away from the direction of travel, thereby tilting the foil  170  and holding the foil  170  at an angle of attack. This effect creates hydrodynamic drag and turbulence at the trailing edge of the foil  170 , which enhances mixing. When the foil  170  reverses direction, the foil  170  swings in the opposite direction in a pendulum fashion and correspondingly creates turbulence and mixing in the same manner. 
     Supporting Structure 
     There are several ways to maintain the foil  170  at a constant pitch and with minimal roll. In an exemplary embodiment illustrated in  FIG. 28 , the foil  170  is attached to tensioned guidelines  470 . Thin rods are attached to the leading and trailing edges of the foil  170 . 
     In embodiments shown in  FIG. 31  and  FIGS. 35-D , the foil  170  is attached to a flotation member  300 , such as, for example, a pontoon, which reduces frictional losses. The wave pattern generated by the leading end of the flotation member  300  additionally increases mixing at the surface  320  of the algae culture to enhance gas exchange, light penetration and photosynthesis. Counter-rotating longitudinal vortices  330  near the surface  320  are also generated in the wake of the flotation member  300  and can be enhanced by specific hull designs. The width and lateral weight distribution of the flotation member  300  control roll, which is essential to maintain the foil  170  at a constant depth, while the length of flotation member  300  controls pitching and thus the angle of attack of the foil  170 . 
     As illustrated in  FIGS. 37A  and B, agitation of the surface  320  can be enhanced by inducing a small hydraulic lump  550  by towing a foil  170  or a cambered horizontal support member  130  just below the surface  320  of the algae culture. 
     As illustrated in  FIG. 38 , in some embodiments, a surface agitating comb  560  or other suitable ancillary structure is attached to a horizontal support member  130  to agitate the surface  320  of the algae culture and increase gas transfer rates between vapor and liquid phases contained in a photobioreactor  310 . 
     As illustrated in  FIG. 39 , in some embodiments, one or more vertical support members  130  connecting a foil  170  to a flotation member  300  serve as rudders to prevent yaw. To be effective as steering devices, the foil  170  needs to pulled from a point forward of the center of pressure for the vertical support members  130  acting as rudders. A larger distance between the tow point and the center of pressure for the vertical support members  130  requires a stronger restoring force to align the foil  170  and counter any imbalance caused by imperfect foil  170  manufacturing, fouling from algae growth on the foil  170  or friction from contacting a surface of the photobioreactor  310 . The vertical support members  130  additionally agitate the surface  320  of the algae culture. 
     In some embodiments, flotation members  300  and support members  130  are made from molded plastic, fiberglass, sintered nylon, glass-reinforced plastic, or any other material that is suitable to provide rigidity, durability, and positive or neutral buoyancy. 
     As illustrated in  FIG. 40 , in some embodiments, an airfoil  580  is positioned above the surface  320  of liquid in a photobioreactor  310 . This configuration advantageously induces circulation of the vapor phase in the photobioreactor  310 , which helps to increase the efficiency of a solar still by enhancing vapor transport from the surface  320  of the water to the walls of the photobioreactor  310 . 
     In some embodiments, the longitudinal axis of the photobioreactor  310  or other mixing vessel is vertical. The drive conduit  100  likewise is vertically oriented, and the foil assembly  180  moves in a vertical direction along the drive conduit  100  and is fully submersed in the contents of the mixing vessel over at least a portion of the mixing vessel. 
     In some embodiments, the foil assembly  180  is used to skim the surface  320  of an open pond. The drive conduit  100  is positioned at or near the surface  320  of the pond, and the position and angle of attack of the foil  170  is adapted to maintain the foil  170  at or immediately beneath the pond surface  320 . 
     In an exemplary embodiment, a gas sparging hose  700  is attached to a foil  170 . Gas  702  may be pumped through the hose  700  and bubbled into the algae culture, with the outlet of the hose  700  located, for example, underneath the foil  170 , near an edge of the foil  170  or immediately underneath the surface  320  of the algae culture. The hose  700  is adapted to move along the length of the photobioreactor  310  with the foil  170 . Movement of the foil  170  creates shear in the algae culture near the lateral edges of the foil  170 , which shears and reduces the size of the gas bubbles  702 . Additionally, the gas bubbles  702  are entrained in the trailing vortices  330 , which increases the residence time of the gas bubbles  702  in the algae culture and improves efficiency of mass transfer between the liquid phase of the algae culture and the gas phase present in the photobioreactor  310 . 
     The hose  700  supplying the gas may be made of any suitable materials that are impermeable to the gas, that provide buoyance for the hose  700  to float on the surface  320  of the algae culture and that are sufficiently pliable to enable the hose  700  to fold or coil on the surface  320  of the algae culture. In some embodiments, a cage additionally is attached to the foil assembly  180  to capture slack in the hose  700  and prevent tangling. 
     In an exemplary embodiment, one or more Venturi tubes are incorporated in support members  130 , with one opening of each Venturi tube disposed above the surface  320  of the algae culture and the opposite opening of each Venturi tube disposed below the surface  320  of the algae culture. The diameter of each Venturi tube may be, for example, approximately 1 to 2 millimeters. 
     As the foil  170  moves through the algae culture, a pressure gradient develops across the length of the Venturi tube. The pressure gradient pulls gas into the algae culture from above the surface  320  of the algae culture, creating small bubbles that are expelled into the algae culture. In some embodiments, Venturi tubes in the support members  130  are formed symmetrically to provide equivalent gas bubbling in each direction of longitudinal motion of the foil  170 . 
     In an exemplary embodiment using pneumatic motive force to drive the drive element  190 , the foils  170 , support members  130  and other parts of the foil assembly  180  are omitted and only a follower element  110 , which is magnetically coupled to the drive element  190 , moves along the length of the drive conduit  100 . When driven at high speeds, a cylindrical follower element  110  creates an air cavity in the shape of a bell behind the follower element  110  as it moves through the algae culture. The plunging jet of the liquid bell causes the formation of small bubbles in the algae culture, which increases mixing and mass transfer from the algae culture. 
     In some embodiments in which the foil assembly  180  is omitted, a gas sparging hose  700  is attached to the follower element  110 . The hose  700  introduces gas bubbles  702  into the algae culture. The end of the hose  700  that is not attached to the follower element  110  may be connected to the photobioreactor  310  along the side in the center of the photobioreactor  310 . 
     Mixing Operations 
     A mixing system in accordance with some embodiments of the present invention is capable of generating vertical mixing that is essential to the cultivation of algae in a photobioreactor  310  while minimizing capital investment and energy usage. In shallow depths with vertical recirculation, the necessary minimum vertical velocity needed to maintain a culture comprising certain strains of algae and to prevent visible sedimentation is approximately 20 to 30 seconds for turnover of algae in a circular cross section of six to eight inches contained in a culture having a depth of eight inches. For circulation rates that exceed this threshold, production in these systems increases only slightly, while energy consumption increases significantly. 
       FIG. 41  illustrates calculated energy requirements for mixing in a raceway pond system known in the art, as compared with energy requirements for certain embodiments of the present invention, to account for sliding friction, hydraulic and pneumatic losses and motor efficiency. The system of the present invention advantageously consumes energy at a lower rate while providing sufficient mixing of an aqueous algae culture. The smaller and larger columns for the raceway system represent, respectively, energy consumption required for 40 second and 20 second turnover times. This range of circulation times corresponds to the initial circulation time (20 seconds) after a foil assembly  180  passes through a static point in the mixing vessel and the decayed circulation time (40 seconds) immediately before the foil assembly  180  subsequently passes through the same point again, when the foil assembly  180  is towed at 0.5 meters per second. 
     A major distinction between rotary impeller motion known in the art and the linear foil  170  motion comprised by the present invention is the functionality and performance of the type of motion. Rotary impeller, or paddle wheel, motion is generally utilized to generate a directed flow, typically along the length or circumference of the vessel. Fluctuations in motion that are transverse to the selected direction of flow are chaotic and occur due to turbulence that is generated in the boundary layer of the vessel. Components of turbulence that are not directed vertically and do not contribute to vertical transport, which is important in the embodiment of a photobioreactor  310  where light is received from overhead, are nonetheless generated. The chaotic vertical motion and extra fluctuation components thus make this rotary motion energetically inefficient for bulk vertical transport. 
     The efficiency of vertical transport may be increased by placing foils  170  in a stream (as taught by Laws et al.) to induce vertical recirculation. However, the majority of flow kinetic energy is still dissipated through overcoming boundary friction in sustaining the relative motion required. 
     In accordance with embodiments of the present invention, by moving the foil  170  in a linear path through an algae culture contained in a photobioreactor  310  or other mixing vessel, the problem of dissipation of flow kinetic energy is overcome. The decay of recirculation, which is created by passing a foil  170  through the algae culture in a linear path to generate trailing vortices  330 , is also ameliorated by repeatedly passing the foil  170  through the same path to continually reinforce the vortices  330 . 
     Vertical mixing systems known in the art may be compact, at the expense of elevated energy consumption. In accordance with embodiments of the present invention, components of the vertical mixing system may be manufactured from lightweight materials, such as plastics, which can help minimize capital expenditures and energy consumption for the system. 
       FIG. 42  details energy in pneumatic and hydraulic mixing systems of the present invention. In the exemplary embodiment, the pneumatic mixing system uses air and the hydraulic mixing system uses water. Consumption data attributable to drive element friction/fluid leakage and to drive conduit/pipe friction are specific to the motive force used in each mixing system. Drive element friction/fluid leakage indicates energy loss due to friction between the outer surface of the drive element  190  and the inner surface of the drive conduit  100 , in conjunction with head loss due to fluid leaking past the drive element  190  inside the drive conduit  100 . Drive conduit/pipe friction indicates energy loss due to friction as the drive fluid flows through the pipes  410  and drive conduit  100  of the mixing system.  FIG. 42  illustrates that combined drive element friction/fluid leakage and drive conduit/pipe friction are substantially lower for the pneumatic mixing system than for the hydraulic mixing system. 
     In  FIG. 42 , foil assembly indicates energy consumption attributable to hydrodynamic drag on the foil assembly  180  moving through the algae culture, which is independent of the drive system used. Comparison of total energy consumption and energy losses attributable to each category shows that a substantially higher proportion of energy is translated directly to moving the foil assembly  180  using the pneumatic mixing system than using the hydraulic mixing system. 
     In some embodiments, a mixing system of the present invention is used to agitate algae culture in a photobioreactor  310  intermittently. According to the present invention, intermittent operation of the mixing system provides sufficient vertical mixing while economizing energy consumption. 
     In exemplary embodiments, mixing systems of the present invention are used to agitate only portions of algae culture contained in a photobioreactor  310 . In some embodiments, horizontally oriented foils  170  generate trailing vortices  330  that agitate the algae culture from the surface  320  to a depth that is less than the total depth of the algae culture. The span of each foil  170  may be less than the total depth of the algae culture, so that the foil  170  generates trailing vortices  330  having diameters less than the depth of the algae culture  113 . Lateral spacing between foils  170  may be greater than the span of each foil  170 . 
     In some embodiments, vertically oriented foils  170 , each having a span less than the depth of the algae culture, agitate the algae culture from the surface  320  to a depth that is less than the total depth of the algae culture. 
     In some embodiments, the foil assembly  180  traverses only a portion of the length of the photobioreactor  310 , agitating the algae culture contained in that portion while leaving the algae culture in the remaining portion of the photobioreactor  310  unmixed. 
     Example 1 
     Both hydraulic and pneumatic drive fluids have been used to propel a magnetically coupled foil assembly  180  through a photobioreactor  310  at 0.5 meters per second. In a preferred embodiment, the foil assembly  180  traverses the 50 foot length of a commercial scale photobioreactor  310  at 0.5 meters per second for a defined time interval (usually 30 seconds) before the foil assembly  180  reverses direction of motion. The minimum steady state power requirement for motion of a foil assembly  180  according to these specifications has been determined by measuring fluid pressures and flow rates while the foil assembly  180  moves at a constant speed. This determination neglects any additional energy consumption or efficiency that occurs during the few seconds when the foil assembly  180  is moving at less than 0.5 meters per second while reversing direction of motion. 
     Using hydraulic drive fluid and a drive ferromagnet  144 , measurements indicate a pressure drop of 8 to 9 pounds per square inch per photobioreactor  310  at a fluid flow rate of 2 gallons per minute. Without accounting for pump  420  and drive efficiency, the minimum power requirement using water as a drive fluid is determined by Equation 1:
 
Power min =ΔPressure×Flowrate=7.8 Watts per photobioreactor 310
 
     which is equivalent to 0.34 Watts/m 2  for a 23 m 2  photobioreactor  310 . 
     Power requirements have not varied in photobioreactors  310  containing either freshwater, seawater or algae culture. Accordingly, power consumption by mixing systems of the present invention used in liquid not containing algae culture is substantially equivalent to power consumption by mixing systems of the present invention used in liquid containing algae culture. According to the present invention, drag on the foil assembly  180  is mostly due to inertial forces, rather than viscous forces, and the densities of all fluids tested in the photobioreactor  310  are roughly equivalent. 
     A preponderance of power loss in mixing systems of the present invention that use hydraulic drive fluid were attributable to fluid leakage past the drive ferromagnet  144  and the resulting increased pressure drop to force the hydraulic fluid, typically water, through the drive conduits  100  and pipes  410 . Using a pneumatic drive fluid significantly reduced pressure drop over the drive ferromagnet  144  due to the lower viscosity of gas, typically air, compared to water and other fluids and due to the use of seals to reduce fluid leakage past the drive ferromagnet  144 . Thus, mixing systems of the present invention that incorporate a floating pneumatic seal around the drive ferromagnet  144  and use lubricated aft typically have operated at 3 to 4 pounds per square inch at 5 to 6 standard liters per minute. 
     Four pneumatically driven mixing systems have been operated outdoors with 6.16±6% standard liters per minute of air consumed by each (referenced at 25 degrees Celsius) under 2.75±0.05 pounds per square inch. Using the standard equation for calculating the power required to compress gas adiabatically, the minimum energy requirement is thus 1.82 Watts/photobioreactor  310 , or 0.08 Watts/m 2 . 
     Net energy usage for a commercial mixing system of the present invention to power  240  photobioreactors  310 , based on distribution losses of 5%, a compressor of 50% efficiency and an electrical drive of 90% efficiency yields a minimum energy requirement of 4.26 Watts/photobioreactor  310 , or 0.185 Watts/m 2 . An array of 240 photobioreactors  310 , as illustrated in  FIG. 23 , would require the use of a rotary lobe compressor rated at 1.5 horsepower. A commercial plant with a centralized compressor system may utilize a more efficient compressor to reduce the power requirements. It is also possible to use bleed air from the first compression stages of a power generating gas turbine as the pneumatic drive fluid, which may reduce net energy cost if the efficiency of conversion of fuel energy to electricity is considered. 
     The energy required to run a pneumatically driven foil assembly  180  is roughly equivalent to that required for a large raceway system running at 0.25 meters per second, with an 8 inch algae culture depth. The net power requirements for both systems are approximately 0.2 Watts/m 2 . Thus, the operating expense for both systems is $1230 per hectare per year assuming an energy cost of $0.07 per kilowatt hour. The operating expense is higher for small raceway systems, however. One major manufacturer of raceway paddlewheels (Waterwheel Factory, Inc.) estimates that motors rated for at least 20-40 Watts would be required to provide mixing in photobioreactors  310  measuring 5 feet by 50 feet under the most optimistic conditions, which is a factor of 5-10 times higher power requirement than the exemplary foil mixed systems. 
     Example 2 
     Growth of cyanobacteria in response to mixing was compared in two reactor types that varied in mixing system design. Two closed foil-mixed photobioreactors and two closed flume-style raceway pond photobioreactors were tested. The oval-shaped raceway pond photobioreactors and the foil-mixed photobioreactors were constructed and enclosed using the same thin, flexible polymeric film. 
     Inoculum cultures of a unicellular cyanobacterium were scaled in 50-liter flat-panel culture vessels. The inoculum cultures were then transferred into the two raceway pond photobioreactors and the two foil-mixed photobioreactors containing seawater and BG-11 nutrient mix. Each raceway pond photobioreactor contained approximately 460 liters of seawater and each foil-mixed photobioreactor contained approximately 900 liters of seawater. Sunlight entered each reactor across the top surface only, and the culture depth in each reactor was 8 inches (20 centimeters), yielding equal surface area to volume ratios for the four reactors. The seawater in each photobioreactor was pre-filtered to 0.2-μm and had salinity of 35. 
     Air was delivered to each photobioreactor at a rate of 5 liters per minute, and carbon dioxide was added from 08:00 to 18:00 local standard time, controlled to a volumetric ratio of 10% carbon dioxide to air. Each photobioreactor was maintained under ambient irradiance and temperature conditions. 
     The culture contained in each foil-mixed photobioreactor was mixed using a foil assembly comprising four foils positioned at a depth of four inches in the algae culture contained in the photobioreactor. Each foil had a span of seven inches and a chord of four inches. The foils were spaced with their centerlines 14 inches apart. 
     The foil speed traversing the length of the foil-mixed photobioreactor was maintained at 0.5 meters per second while the foil was in motion. When the foil reached each end of the photobioreactor, the motion of the foil was paused for 20 to 22 seconds in order to simulate the period of the foil, and thus vortex reinforcement frequency, in a 50 foot-long commercial-scale, foil-mixed photobioreactor. 
     Each raceway photobioreactor was operated similarly to a paddlewheel raceway in which horizontal motion of culture was maintained through pumping and recirculation of the flow. Rather than using a paddlewheel, however, pumping in the raceway photobioreactors was accomplished using four Tunze® Turbelle® stream  6085  pumps in each bioreactor. These are propeller pumps with a 90 mm (3.5 in.) ball design generally used for water circulation in aquariums or tanks. Each Tunze® Turbelle® stream  6085  produces flow rate of about 8 liters per hour at power consumption of about 14 Watts, but the particular power consumption of these pumps is of secondary importance. Rather, the horizontal flow that they produced in the raceway photobioreactor was the target. The pumps were arranged to provide a flow rate of 0.25 m/s in an 8 inch (20 cm) deep culture in the photobioreactor raceways. This flow rate was calculated following Weissman et al., “Photobioreactor Design: Mixing, Carbon Utilization, and Oxygen Accumulation,” Biotechnology and Bioengineering, Vol. 31, Pp. 336-344 (1988), equating the electrical energy consumption of a commercial scale paddlewheel system to generate this flow (0.21 Watts/square meter) to the power to drive the pneumatic foil system. The power requirement was determined using Manning&#39;s equation for hydraulic loss. Drive efficiency was estimated as 0.31 for a paddlewheel operating at the specified speed and depth on a 100 square meter raceway. The drive efficiency could be as high as 0.5 for a very large system, but the increased efficiency would not significantly increase the flow rate, i.e., to 0.28 meters per second, under the stated power consumption. Raceway reactors are also typically operated at mixing speeds of 0.15-0.25 meters per second to minimize settling of algae in the culture. 
     As shown in Table 1, volumetric dry weight of the unicellular cyanobacterium in each algae culture was measured for each photobioreactor three times per week for three weeks, as each culture matured from growth phase to early stationary phase. Dry weight areal biomass of the unicellular cyanobacterium in each algal culture, shown in Table 1, was calculated based on measured culture volumes and the surface areas of the culture in each photobioreactor, shown in Table 2. 
     
       
         
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                   
                 TABLE 1 
               
             
             
               
                   
                   
               
               
                   
                 Raceway 
                 Raceway 
                 Foil Mixed 
                 Foil Mixed 
               
               
                   
                 PBR 1 
                 PBR 2 
                 PBR 1 
                 PBR 2 
               
             
          
           
               
                 Time (day) 
                 mg/L 
                 g/m 2   
                 mg/L 
                 g/m 2   
                 mg/L 
                 g/m 2   
                 mg/L 
                 g/m 2   
               
               
                   
               
             
          
           
               
                 0.7 
                 59.33 
                 11.67 
                 14.67 
                 2.98 
                 57.26 
                 12.02 
                 62.07 
                 13.35 
               
               
                 2.4 
                 85.83 
                 16.88 
                 72.50 
                 14.75 
                 67.70 
                 14.22 
                 97.22 
                 20.90 
               
               
                 4.4 
                 218.52 
                 42.98 
                 209.63 
                 42.66 
                 209.63 
                 44.02 
                 194.81 
                 41.89 
               
               
                 7.3 
                 380.00 
                 74.73 
                 351.11 
                 71.45 
                 455.56 
                 95.67 
                 443.33 
                 95.32 
               
               
                 10.3 
                 397.64 
                 78.20 
                 435.56 
                 88.64 
                 513.33 
                 107.80 
                 475.56 
                 102.24 
               
               
                 11.3 
                 379.17 
                 74.57 
                 369.44 
                 75.18 
                 433.33 
                 91.00 
                 419.44 
                 90.18 
               
               
                 14.4 
                 390.00 
                 76.70 
                 460.00 
                 93.61 
                 466.67 
                 98.00 
                 490.00 
                 105.35 
               
               
                 16.3 
                 425.00 
                 83.58 
                 460.00 
                 93.61 
                 580.00 
                 121.80 
                 536.67 
                 115.38 
               
               
                 18.3 
                 420.00 
                 82.60 
                 540.00 
                 109.89 
                 660.00 
                 138.60 
                 636.67 
                 136.88 
               
               
                 21.3 
                 N/A 
                 N/A 
                 N/A 
                 N/A 
                 587.73 
                 123.42 
                 541.67 
                 116.46 
               
               
                   
               
             
          
         
       
     
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 Reactor 
                 SA (m 2 ) 
                 Volume (L) 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Raceway PBR 1 
                 2.31 
                 454 
               
               
                   
                 Raceway PBR 2 
                 2.31 
                 470 
               
               
                   
                 Foil Mixed PBR 1 
                 4.23 
                 889 
               
               
                   
                 Foil Mixed PBR 2 
                 4.31 
                 926 
               
               
                   
                   
               
             
          
         
       
     
     The logistic growth model stated in Equation 2 (Kot, “Elements of Mathematical Ecology” Cambridge University Press (2001)) was parameterized from the areal data in each photobioreactor as listed in Table 1. 
     
       
         
           
             
               
                 
                   
                     B 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         KB 
                         0 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           μ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                       
                     
                     
                       K 
                       + 
                       
                         
                           B 
                           0 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ⅇ 
                               
                                 μ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                             
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     wherein 
     t=time (day); 
     μ=specific growth rate (1/day); 
     B 0 =biomass (g-DW/m 2 ); and 
     K=biomass abundance at stationary phase (g-DW/m 2 ). 
     Values of the parameters in Equation 2 obtained from nonlinear least square fit of the logistic growth model to the data in Table 1 are shown in Table 3: 
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
               
                   
                   
                   
                 Max. Growth Rate 
                   
               
               
                 PBR 
                 μ 
                 K 
                 (g/m 2 /d) 
                 R 2   
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Raceway PBR 1 
                 0.6741 
                 79.97 
                 13.4783 
                 0.987 
               
               
                 Raceway PBR 2 
                 0.4740 
                 96.14 
                 11.3931 
                 0.952 
               
               
                 Foil Mixed PBR 1 
                 0.4911 
                 117.81 
                 14.4658 
                 0.922 
               
               
                 Foil Mixed PBR 2 
                 0.4307 
                 116.90 
                 12.5867 
                 0.935 
               
               
                 p value 
                 0.3921 
                 0.069 
                 0.519 
               
               
                 t-stat* 
                 1.083 
                 −3.618 
                 −0.777 
               
               
                 Significant? 
                 No 
                 Yes (90%) 
                 No 
               
               
                   
               
               
                 *2-tailed t-test 
               
             
          
         
       
     
     Plots of the average dry weight areal biomass data for each replicate photobioreactor type in Table 1 and the parameterized logistic growth models in Equation 2 for averaged data from both reactor types are overlain in  FIG. 43 . If an exponential model of growth is assumed instead of a logistic model and specific growth, μ, is calculated as ln(B 2 /B 1 )/(t 2 −t 1 ), where t 1 =2.4 days and t 2 =7.3 days, then specific growth rates average 0.343±0.055 and 0.307±0.013, respectively to foil-mixed and raceway PBRs, and are not significantly different (t-test stat=0.8969, p=0.4644). 
     As shown in Table 4, both measured and modeled cumulative biomass growth at day 16 are significantly greater in the foil-mixed photobioreactors than in the raceway photobioreactors: 
     
       
         
               
             
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 16-day cumulative biomass growth (g/m 2 ) 
               
             
          
           
               
                   
                 Measured Cum. 
                 Modeled Cum. 
               
               
                   
                 Growth (g/m 2 ) 
                 Growth (g/m 2 ) 
               
             
          
           
               
                 Reactor 
                 Mean 
                 Std. Dev. 
                 Mean 
                 Std. Dev. 
               
               
                   
               
             
          
           
               
                 Foil Mixed PBRs 
                 118.6 
                 4.5 
                 116.5 
                 1.01 
               
               
                 Raceway PBRs 
                 88.6 
                 7.1 
                 87.5 
                 10.7 
               
             
          
           
               
                 t-stat 
                 5.0375 
                 3.6612 
               
               
                 p value 
                 0.0372 
                 0.0672 
               
               
                 Significant? 
                 Yes (95%) 
                 Yes (90%) 
               
               
                   
               
             
          
         
       
     
     Example 3 
     Capital expense is compared for conventional mixing technology using paddlewheels to mix shallow algae cultures and for a mixing system in accordance with the present invention using a magnetically coupled, moving foil assembly. Details of the capital expense for large scale paddlewheel systems have been published in Weissman et al. Two paddlewheel systems described therein are adapted for 0.4 hectare and 8 hectare ponds. The paddlewheel mixer capital expense totals adjusted to present day values are approximately $11,000 and $36,000 for the 0.4 hectare and 8 hectare ponds, respectively. As noted in Table 5, these capital expense totals are equivalent to approximately $53,000 and $9,000 per hectare, respectively, 
     
       
         
               
             
               
               
               
               
             
               
               
               
               
             
               
             
               
               
               
               
             
           
               
                 TABLE 5 
               
             
             
               
                   
               
               
                 5 × 50 ft reactor area (23 m 2 ) 
               
             
          
           
               
                   
                   
                 Open 
                 Sealed Film 
               
               
                   
                 Component 
                 channel 
                 Photobioreactor 
               
               
                   
                   
               
             
          
           
               
                   
                 center barrier 
                   
                 $116 
               
               
                   
                 paddlewheel 
                 $37 
                 $37 
               
               
                   
                 support 
               
               
                   
                 guard poles 
                   
                 $2 
               
               
                   
                 flange 
                 $40 
                 $40 
               
               
                   
                 bearings 
               
               
                   
                 lip seals 
                   
                 $40 
               
               
                   
                 paddlewheel 
                 $22 
                 $22 
               
               
                   
                 shaft 3/8 SS 
                 $45 
                 $45 
               
               
                   
                 end fairings 
                   
                 $15 
               
               
                   
                 gear motor 
                 $40 
                 $40 
               
               
                   
                 Capex/PBR 
                 $184 
                 $356 
               
               
                   
                 Capex/hectare 
                 $80,122 
                 $154,966 
               
             
          
           
               
                 Industrial scale raceway ponds 
               
               
                 Weissmann &amp; Goebel 1987* 
               
             
          
           
               
                   
                 Capex/hectare 
                 $52,770 
                 (0.4 hectare) 
               
               
                   
                 Capex/hectare 
                 $8,850 
                   (8 hectare) 
               
               
                   
                   
               
               
                   
                 *Cost adjusted for 2011 
               
             
          
         
       
     
     An independent estimate for the lowest cost of a paddlewheel mixing system suitable for use in an open 23 square meter (0.0023 hectare) raceway pond, which is equivalent to the typical size of an enclosed bioreactor measuring 5 feet by 50 feet, was determined by estimating the lowest material costs for a design provided by Waterwheel Factory, Inc., a major waterwheel manufacturer. The estimated material costs are shown in Table 2. 
     At $184 per raceway photobioreactor, the capital expense per hectare for a small-scale paddlewheel mixing system is approximately $80,000 per hectare. This estimate is plotted in  FIG. 44  along with historical published data. As show by the dotted line plotted in  FIG. 44 , there is a trend toward increased capital expense per area with smaller raceway systems. 
     The capital expense for a foil mixing system in accordance with the present invention used with a 23 square meter enclosed photobioreactor is detailed in Table 6. 
     
       
         
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 6 
               
               
                   
               
               
                 Compo- 
                   
                 feet of 
                   
                   
                   
               
               
                 nent 
                 Specification 
                 piping 
                 lb/ft 1   
                 $/ft 2   
                 total cost 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 header 
                 1.5″ SDR11 
                 610 
                 0.41 
                 0.492 
                 $300 
               
               
                 piping 
               
               
                 (central 
               
               
                 row) 
               
               
                 header 
                 1.25″ 
                 1420 
                 0.16 
                 0.192 
                 $273 
               
               
                 piping 
                 SDR15.3 
               
               
                 (sides) 
               
               
                 mixer 
                 ½″ SDR10.1 
                 12,480 
                 0.048 
                 0.0576 
                 $719 
               
               
                 tubing 
               
               
                   
               
               
                   
                   
                 quantity 
                 $/item 
               
               
                   
               
               
                 com- 
                 1.5 hp rotary 
                 1 
                 1000 
                   
                 $1,000 
               
               
                 pressor 
                 lobe blower 
               
               
                 elec- 
                 ABB VFD or 
                 1 
                 350 
                   
                 $350 
               
               
                 tronics 
                 4-way valve 
               
               
                 and 
               
               
                 control 
               
               
                 external 
                 NdFeB Grade 
                 240 
                 6.4 
                   
                 $1,536 
               
               
                 magnets 
                 N42 
               
               
                   
                 1.25″OD × 
               
               
                   
                 0.75″ ID × 
               
               
                   
                 ⅛″ 
               
               
                 internal 
                 NdFeB Grade 
                 480 
                 2.37 
                   
                 $1,138 
               
               
                 magnets 
                 N40 0.5″OD × 
               
               
                   
                 0.25″ ID 
               
               
                   
                 ¼″ 
               
               
                 mixer 3   
                 2 lbs HDPE 
                 240 
                 4.38 
                   
                 $1,051 
               
               
                   
                   
                   
                   
                 cost/module 4   
                 $6,366 
               
               
                   
                   
                   
                   
                 cost/PBR 5   
                 $27 
               
               
                   
                   
                   
                   
                 cost/hectare 
                 $11,533 
               
               
                   
               
               
                   1 Performance Pipe IPS size data for PE 4710. 
               
               
                   2 $1.20/lb in accord with pricing from Ferguson Enterpries, Inc. 
               
               
                   3 Cost calculated by 3X cost of plastic for standard blow molding, $0.73/lb in accord with ICIS pricing. 
               
               
                   4 1 module consists of 240 photobioreactors arranged as in FIG. 23. 
               
               
                   5 Each photobioreactor is 5 feet wide by 50 feet long and has a wet area of 23 square meters. 
               
             
          
         
       
     
     The capital expense illustrated in Table 6 includes the drive and distribution components of the pneumatic drive foil mixing system for a set of 240 photobioreactors (in 4 rows of 60). Here, the capital expense per bioreactor is $27, or approximately $11,500 per hectare. Thus, the capital expense per area for a facility constructed using photobioreactor modules of this size is independent of the size of the facility. 
     Capital expense is compared for foil mixing systems and raceway mixing systems in Table 7 and  FIG. 44 . 
     
       
         
               
               
               
             
               
               
               
               
             
           
               
                   
                 TABLE 7 
               
               
                   
                   
               
               
                   
                 individual 
                   
               
               
                   
                 reactor size 
               
               
                   
                 (hectare) 
                 Capex/hectare 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 open paddlewheel 
                 8 
                 8,848 
               
               
                   
                 Weissman&amp;Goebel 
               
               
                   
                 1987 
               
               
                   
                 open paddlewheel 
                 0.4 
                 52,772 
               
               
                   
                 Weissman&amp;Goebel 
               
               
                   
                 (1987) 
               
               
                   
                 open channel 
                 0.0023 
                 80,122 
               
               
                   
                 paddlewheel estimate 
               
               
                   
                 sealed film PBR 
                 0.0023 
                 154,966 
               
               
                   
                 paddlewheel estimate 
               
               
                   
                 foil mixer 
                 0.0023 
                 11,533 
               
               
                   
                   
               
             
          
         
       
     
     The data in Table 7 and  FIG. 44  show that the currently sized foil mixing system is approximately one order of magnitude less expensive than raceway paddlewheel mixing systems on an areal basis, except for raceways which approach a size of 8 hectares. Thus, foil mixing reduces the capital cost of mixing on all scales which can be enclosed at reasonable expense. Even if the cost to enclose a 0.4 hectare (approximately 1 acre) pond was negligible, foil mixing systems are still 4.5 times less expensive than raceway paddlewheel mixing systems. The cost for a sealed raceway system would exceed the estimates shown in Table 6 due to the cost of seals, guards, central barriers and end pieces that would need to be manufactured for the photobioreactors. 
     The foregoing Examples 1 through 3 demonstrate that the biomass output of a foil mixed system was comparable or better than a raceway pond, while the operating expense of the foil system was equivalent or lower. The most distinctive advantage of the foil mixed system is its low capital expense, which makes up a large fraction of the total cost even when spread over a 15 year operating period as shown in Table 8. Capital expense is a much larger fraction of the total expense for the paddlewheel raceway systems for areas that are practical to enclose (e.g. 23 square meters). Thus the total cost of mixing to generate biomass is much higher in the paddlewheel system, by a factor as large as 8. Here the cost of mixing is only about 4 cents per kilogram of dry weight, while cost of mixing for an enclosed paddlewheel system could be 34 cents per kilogram. 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                   
                 TABLE 8 
               
               
                   
                   
               
               
                   
                   
                 Pneumatically 
                 Enclosed 
               
               
                   
                   
                 driven foil 
                 paddlewheel 
               
               
                   
                 Annual expenditure/ 
                 mixed reactors 
                 reactors 
               
               
                   
                 hectare 
                 (23 sq. m) 
                 (23 sq. m) 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Capex/15 years 1   
                 769 
                 10,331 
               
               
                   
                 Opex 
                 1,230 
                   (2) 5,775 
               
               
                   
                 Total 
                 1,999 
                 16,106 
               
               
                   
                 Biomass yield per year 3   
                 47,450 
                 47,450 
               
               
                   
                 Total cost 4  ($/kg of 
                 0.042 
                 0.339 
               
               
                   
                 biomass) 
               
               
                   
                   
               
               
                   
                   1 Interest on loan not considered in either case. 
               
               
                   
                   (2) 20 Watts per photobioreactor as the minimum suggested by Waterwheel Factory, Inc. for 0.25 meters per second flow. 
               
               
                   
                   3 Assuming maximum growth rate (13 grams per square meter per day) demonstrated in Example 2 is sustained over a year. 
               
               
                   
                   4 Considering only the cost of mixing. 
               
             
          
         
       
     
     In certain embodiments, as illustrated in  FIGS. 46 ,  47  and  48 , the depth of the algae culture is shallow and a crossbar  590  is used to create a shallow breaking wave front  600  on the surface  320  of the algae culture. In certain embodiments, the depth of the algae culture is preferably approximately 1 to 2 inches. In certain embodiments, the depth of the algae culture is preferably approximately 1 inch. The crossbar  590  is attached to a follower element  110  that is disposed on a drive conduit  100 . The crossbar  590  is propelled using magnetic coupling between the follower element  110  and a drive element  190 , to which pneumatic or hydraulic motive force is applied. 
     The crossbar  590  is propelled at sufficient speed to displace fluid along the length of the photobioreactor  310 , such that a wave front  600  is generated that moves at a higher speed than the shallow water wave in the direction of travel of the crossbar  590 . For an algae culture of 1 inch depth, a wave front  600  may be generated by a crossbar traveling at 0.5 meters per second. 
     The wave front  600  generated at the surface  320  of the algae culture provides enhanced mass transfer between the liquid algae culture and the air above the surface  320  of the algae culture. Dissolved oxygen content in the algae culture would be lower when a breaking wave front  600  is generated than when a breaking wave front  600  is not generated in the algae culture. The wave front  600  also generates provides vertical mixing of the algae within the culture, which enhances photosynthetic productivity and diffusion of nutrients. 
     The range of traverse, or stroke length, of the crossbar  590  may be substantially shorter than the length of the photobioreactor  310 . Stroke length can be controlled by placing stoppers inside the drive conduit  100  to restrict the motion of the drive element  190  or around the outside of the drive conduit  100  to restrict the motion of the follower element  110 . In one embodiment, a stroke length of 1 meter is sufficient to generate a wave front  600  that can propagate for 80% of the length of the photobioreactor  310 . Shorter stroke length of the wave front  600  produces shorter propagation distance of the wave front  600 . 
     A wave front  600  can be created in a photobioreactor  310  that is 50 feet long using 8.5-9.5 pounds per square inch hydraulic motive force in a drive conduit  100  of 0.52 inches inside diameter, which propels the crossbar  590  at 0.33-0.5 meters per second. If the crossbar  590  is located in the center of the photobioreactor  310 , a cycle time of approximately 10 seconds for the crossbar  590  to complete one oscillation allows for one breaking wave front  600  to be present on either side of the crossbar  590  at any time, while maintaining low energy usage, approximately 4 Watts per photobioreactor  310 , equivalent to a mixing system using a foil assembly  180 . 
     The crossbar  590  used to generate a wave front  600  may be buoyant and may have chamfered edges on the lower surfaces of the crossbar  590  to generate lift, which prevents the crossbar  590  from touching the bottom of the photobioreactor  310  and reduces friction and wear on the components of the crossbar  590  and the photobioreactor  310 . The crossbar  590  will be optimally designed such that reflections of the wave front  600  are minimized. The wave front  600  preferably dissipates when it reaches the end of the photobioreactor  310  and does not disrupt the motion of the crossbar  590 . Resonant operation of the crossbar  590  is possible but may be difficult to control at low capital expense. 
     Multiple crossbars  590  in one photobioreactor  310  can be driven in the same manner as a system using multiple foil assemblies  180 . In combination with reducing the stroke length of each crossbar  590 , a configuration employing multiple crossbars  590  can be used to generate wave fronts  600  at higher frequencies, so that more than two breaking wave fronts  600  would be present in the photobioreactor  310  at any time. 
     Although the present invention has been described in considerable detail with reference to certain embodiments thereof, other embodiments are possible. Therefore, the spirit and scope of the appended claims should not be limited to the description of the embodiments contained therein.
The invention provides a mixing system comprising a magnetically coupled drive system and a foil for cultivating algae, or cyanobacteria, in an open or enclosed vessel. The invention provides effective mixing, low energy usage, low capital expenditure, and ease of drive system component maintenance while maintaining the integrity of a sealed mixing vessel.