Thermoelectric Energy Conversion of a Drinking Bird by Disk-Magnet Electromagnetic Induction

We discuss a thermoelectric energy generation (TEG) technique by employing a thermomechanical model of a drinking bird (DB). The motion of a drinking bird is produced by the entropy-flow explained by the second law of thermodynamics, which is one of the fundamental laws of heat engines. We propose a disk-magnet electromagnetic induction (DM-EMI) employed to the motion of a drinking bird. The generalization of DM-EMI to heat engines for mechanoelectric energy conversions and properties of extracted electric powers are specifically discussed. The electric power of DM-EMI has a li-mited power generation characteristic to a mechanical rotation produced by heat engines, but it will be very useful for practical applications to wind turbines, coal-fired and nuclear power plant for mechanoelectric energy conversions. The DM-EMI will contribute to environmental problems to maintain clean and susceptible energy as one of energy harvesting technologies.


Introduction
A mathematical expression of motion for a thermomechanical drinking bird (DB) was discussed and numerical solutions were explicitly derived and shown [1]. A drinking bird is a model of irreversible thermodynamics and manifests basic principles of heat engines, which is useful for mechanoelectric energy conversions. The model reproduced a DB's periodic motion reasonably well with the assumption of a constant speed, 0 v , of volatile liquid in a DB's glass tube. The result is somewhat surprising, because DB's motion is a simple back-and-force oscillating motion and the speed, 0 v , of volatile water in the glass tube should not be constant. The result indicates that the mean-value assumption to the constant speed of upward moving liquid is reasonable for solving a DB's equation of motion as the first step.
Drinking birds and Stirling engines are excellent examples of heat engines or thermomechanical phenomena and should be scrutinized in terms of the theory of nonequilibrium thermodynamics or irreversible process of thermodynamics [2] [3]. The sensitive heat-flow or entropy-flow of a DB will help understand irreversible thermodynamic phenomena and testability of theoretical concepts, which could be, for example, extended and interpreted as a role of mitochondria in a cell or photo electrochemical processes in semiconductors [4] [5].
The upward moving liquid is essential for a DB's periodic motion, and the immediate driving force of upward moving liquid is a pressure in the glass tube, originated from a heat-flow, a temperature difference between upper head-bulb and lower bulb, or an entropy-flow. The analysis of sensitive heat engines is essential for effective and optimal uses of energies in mechanical applications and fundamental understanding of thermodynamic phenomena. One can learn fundamental principles of thermal reactors, or even applications to nuclear reactors from dynamics of heat engines of drinking birds and Stirling engines as one of energy conversion techniques of energy harvesting technologies [6] [7].
As explained in the previous paper for testability and applicability [1], we will show a specific thermally-driven mechanoelectric energy conversion technique derived from the sensitive irreversible thermodynamic structure of a drinking bird. The fundamental difference between nonlinear differential equations of Newtoninan and thermomechanical drinking birds is briefly discussed in Section 2. The disk-magnet electromagnetic induction by a drinking bird and an extended disk-magnet electromagnetic energy conversion technique for heat engines, in general, are discussed in Section 3. Discussions and perspectives are in Section 4.

The Nonlinear Differential Equation of a Thermomechanical Drinking Bird
The equation of motion for a mechanical drinking bird could be first constructed by employing Newtonian mechanics, Lagrangian or Hamiltonian dynamics with the rotational kinetic energy and potential energy: where the DB's length l a b = + , the moment of inertia given by the sum,

I I I I = + +
which is moment of inertia of head 3 I , glass tube 1 I , and lower bulb 2 I , respectively. By employing the idea of the center of mass assumption, the moment of inertia of head and lower bulb is given by 1 I is calculated from the axis of rotation (see Figure 1(b)). One should note that the mass, 2 m , is the sum of masses of liquid and lower glass bulb. The mechanical equation of motion for DB is derived from the Euler-Lagrange equation as: The velocity-dependent friction term, cθ  (c is a free parameter), is externally included in the equation of motion for accounting an oscillation damping, and by defining the effective mass, m * , for convenience as: the mechanical equation of motion for a DB is written as: The motion of a drinking bird is a combination of the solutions converging to 0 θ = and θ = π . However, the mechanical solutions derived from (2.4) are linearly independent and cannot produce the motion of a drinking bird. The continuous drinking motion from 0 θ = to θ = π does not exist in (2.4), because the solutions converging to 0 θ = and θ = π are respectively bifurcation solutions [8].
We proposed a thermomechanical model of a DB's periodic motion assuming a constant speed, 0 v , of volatile liquid in a DB's glass tube. The equation of motion for the thermomechanical DB results in, and the thermomechanical solution of (2.5) produces a DB's periodic motion reasonably well. The nonlinear differential Equation , and inertial masses, ( ) I t , producing a periodic DB's solution, and the speed of liquid 0 v is essential for the entire motion driven by entropy flow of DB's mechanical system. The solutions of nonlinear differential equation with time-dependent coefficients are fundamentally different from those of nonlinear differential equation with time-independent coefficients (2.4). This is one of the important properties shown by DB's equation of motion. The numerical solutions to a drinking bird motion (2.5) are shown in Figure 2. We will use the result for a mechanoelectric energy conversion of the drinking bird motion in the next section.

The Disk-Magnet Electromagnetic Induction by a Drinking Bird
The energy generation from a drinking bird is a sensitive mechanoelectric energy conversion process, and it could suggest useful ideas for applications to modern energy harvesting technologies. The motion of a drinking bird was recorded with the use of modern equipments by experimentalists, but was not solved theoretically. We solved a thermomechanical model of nonlinear motion of a drinking bird [1] and are ready to propose a direct mechanoelectric conversion of DB's motion for applications. In this section, we discuss and show an optimal energy conversion technique by employing thermomechanical solutions of a drinking bird. A generation of electric energy from a drinking bird was only experimentally tested by attaching a magnet and a coil on the base of a drinking bird [9], however the energy conversion method had a couple of unavoidable problems: 1) A coil-magnet dynamo on the base of a drinking bird will act as magnetic force to brake and damp the swinging motion. Though a better improvement of dynamo on the base of a DB is possible, the extraction of energy would ultimately yield diminishing results. 2) A coil-magnet dynamo on the base of a drinking bird will be soaked and affected by water dripping through the head to the lower bulb in drinking motion.
The difficulties (a magnetic braking of motion and water dripping) seem to be sufficient enough not to consider any practical applications in electric power generations.
However, we will show an optimal thermoelectric energy conversion technique by resolving above difficulties and extend the power conversion technique to heat engines from milli-watt to killo-watt electric power generations. The irreversible thermodynamic process expressed by a drinking bird and Stirling engines is useful for foundations and the progress of energy harvesting technologies.
The disk-magnet electromagnetic induction of symmetric settings of coils and magnets is shown in Figure 3, in order to solve the essential problem of coil-magnet dynamo on the base of a drinking bird. The magnets and coils are arranged horizontally on the right and left of the lower bulb of the drinking bird.
The forces between a magnet and a coil exerted by right and left coils on DB's body work to cancel each other. Therefore, the braking or damping effects coming from electromagnetic induction forces can be eliminated. In other words, the produced magnetic forces which caused magnetic braking and damping of swinging motion cancel each other with the symmetric setting of magnets and coils.
For the disk-magnet electromagnetic induction when the magnet crosses near the cross-sectional area of a corresponding coil, a simplified assumption on the magnetic flux is employed. As shown in Figure 3(b), the magnetic flux is perpendicular to the magnetic disk, and the maximum magnetic flux penetrating a coil is given by magnetic flux density B (Wb/cm 2 ) multiplied by the cross-sectional The assumption is physically reasonable when the magnet is strong (e.g. neodymium magnet) and slides through near the cross-sectional area of the coil (see Figure 3(b)). The

Numerical Simulations of the Disk-Magnet Mechanoelectric Energy Conversion
The

The Disk-Magnet Electromagnetic Induction (DM-EMI)
The results shown in the numerical simulations and DM-EMI technique should be considered qualitative and checked quantitatively by experiments and applied to direct mechanoelectric energy conversions, such as wind turbine systems, Stirling engines, even to coal-fired and nuclear radiation plants as one of energy harvesting technologies. Possible basic extensions of the disk-magnet electromagnetic induction (DM-EMI) could be applied to, for example, technique for heat-energy conversions of coal-fired and decommissioned nuclear plants. The sensitive energy-conversion technique derived from a water drinking bird would provide us with an important ideas and impacts to heat engines in our social life [10].
A generalization of DM-EMI is shown in Figure 8. The four disk magnets are positioned on a rotator, and the rotator moves among symmetrically positioned four coils (4DM-EMI). Each coil of right and left disks generates pulses of electric currents and powers. Similarly, one can introduce the 8 disk-magnet electromagnetic induction (8DM-EMI) and so on. When an angular speed increases or pulse responses in time become shorter and shorter, the direct numerical evaluations of DM-EMI may become difficult. One should note that an optimal angular speed would exist, resulting in maximum pulses of electric currents and powers, which should be experimentally checked with physical parameters of a specific mechanical system.

Discussions and Perspectives
Because the dynamics of a drinking bird and a Stirling engine is a very sensitive physical problem of the irreversible thermodynamics, it helps us understand natural phenomena in terms of energy and heat flows, resulting in conspicuous applicability and testability. The heat engine suggests that energy can be more efficiently produced and used so that the waste of energy can be dramatically decreased. The structure of a drinking bird and a Stirling engine as heat engines could advance technologies for our energy productions and consumptions into more energy efficient systems.
Energy efficiencies of heat engines should be compared to those of plants and animals, such as photosynthesis of mitochondria, adenosine triphosphate (ATP), etc, as a source of heat and electric energy conversion system employed by biochemical processes. In views of science, drinking birds, Stirling engines, mitochondria and coal-fired plants, even nuclear power stations are regarded as having similar mechanism as heat engines using energy or conversions of heat flow to electric energy. Precisely speaking, systems mentioned above should be generally explained as transport engines, and even milli-watt to 100-watt electric power can be extracted from warm water discharged from nuclear power plans to rivers and oceans at the end operations. Hence, the sensitive mechanism of plant and animals as transport engines would be very useful for the progress of modeling technologically efficient heat engines.
The sensitive heat engines have guided us to the idea of a direct mechanoelectric conversion technique, which should be applied mechanical rotation and heat transport systems in our daily life, such as horizontal and vertical wind turbines on housetops, vibration and rotation systems of automobiles [6] [7]. The generalization of disk-magnet electromagnetic induction, 4DM-EMI, 8DM-EMI, ..., rotating disk, would contribute to practical applications for energy harvesting technologies. The rectification and energy storage of pulse-power of EMI should be consulted with experts in those fields [11] for efficient use of DM-EMIs, and in addition, advanced thermal machines and technologies in the field of cryogenics [12], Stirling coolers [13], pulse-tube-refrigerators [14], thermoacoustic engines [15] and so forth, could be applied to make refined heat engines for sustainable and renewable energies.
We solved the equation of motion mathematically in the first paper on drinking bird's motion and explained fundamental logical structure of science, regarding reproducibility, self-consistency and testability [1]. The numerical experiment or reproducibility is exhibited in detail by nonlinear equation of motion with time-dependent coefficients [1]. The current paper is regarded as testability or applicability of sensitive irreversible thermodynamic mechanism of a drinking bird as transport engines, and we are directed to investigate self-consistency of our thermomechanical model of drinking birds in terms of entropy-flow in irreversible thermodynamics. It is to investigate physical reasons why a nonlinear model with time-dependent coefficients worked reasonably well, which will be pursued in following research.
The symmetric settings of right and left coils of DM-EMI are essential to avoid braking mechanism of EMI, and the optimal electric power production and energy efficiency of disk-magnet electromagnetic induction should be checked.
The technology can be applied to environmental problems to support clean and sustainable energy (CSE) [10]. We will apply the direct mechanoelectric conversion method for resolving nuclear radiation wastes in the near future.

Appendix: The Energy Generation from Swinging Motion of a Drinking Bird by Employing the Disk-Magnet Electromagnetic Induction (DM-EMI)
Let us calculate the time-dependent crossing surface, ( ) S θ , which is shown as the shaded area in Figure 9. The length R is given by where ( ) ( )