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MATHEMATICAL MODELS FOR GENERATING ELECTRICITY FROM OCEANIC BODY Abstract In this paper we present some mathematical models for the generation of electrical energy from sea bodies and thunder strikes. Pictorial models are presented and their mathematical analyses are given. Introduction It is said that for Nigeria to be industrialized, she will need about 120,000 kilo waltz of electricity as against the present 3to 5 thousand kilo waltz of electricity generated in the country. Thus for a quick industrialization of the country, we would need to depend on nuclei fusion to be able to generate the expected 120,000 kilo waltz of electricity. However, generating electricity from nuclei fusion has the following disadvantages: •It will place Nigeria at a position of high nuclei risk which could be as a result of earthquake as it happened in Japan and China, thus leading to problem of nuclei radiation ; •Terrorist gaining control over such nuclei reactor could hold the nation to ransom; •It will aggravate the problem of Global warming if most nations would have to depend on nuclei fusion for electricity generation. Consequent upon these and coupled with the quest for industrialization, we need to think and design modalities for the generation of the 120,000 kilo waltz of electricity that is needed. Hence the need for the topic “mathematical models for generating electricity from oceanic body. “ Generating Electricity From The Sea Wave Our aim is to be able to control the sea water and wave in a way to optimize it for electricity generation. But there are some constraints to be considered before the issues of electricity generation can be possible. The constraints are; •The violent nature of the sea; •The sea coast low level; •The sandy nature of the sea; and •The salty nature of the sea. FIG.1 This tower was constructed by solving equation below on the 7th of February 1887; FIG.2 FIG.3 FIG.4 FIG.5 FIG.6 FIG.7 FIG.9 FIG.10 FIG.11 FIG.12 Because of the violent nature of the sea, we need to impose a system structure that is •Controllable •Admissible •Observable An admissible system is one to which corrective measures can be administered. A controllable system is a system that is mathematical expressed as that is a system that is bounded below and above. , which transfers the state If there is a finite time and a control to the origin at time ,the state is said to be controllable at time if all values of controllable. are controllable for all the system is completely . Observable system: is one that is analytic, that is, its first derivative exists within the interval. If by observing the output the state observable at time during the finite time interval can be determined, the state is said to be Kalman in 1960 showed that a necessary and sufficient condition for controllability and observability is that the system should be a nonsingular matrix  This is a system that is subject to us in a way thus successfully bypassing the turbulent and violent nature of the sea. Another concern is the salty nature of the ocean water. This can be prevented by coating the propeller with polythene material. Our intention can be conceptualized as follows; We want to optimize the ocean → Optimize electricity generation= Objective is to generate electricity= (Z) Subject to; •The violent nature of the sea; •The low land of sea coast; •The sandy nature of the sea; and •The salty nature of the sea. The state variable is the wave of the sea = x(t); The control variable is the structure that is introduced into the sea to control it flow= u(t) Let the coefficient of the x(t) and u(t) be 𝛼 and 𝛾 respectively ,such that we plot 𝛾 u(t) against 𝛼 x(t) . Then we can write; 𝛼 x(t) + i 𝛾 u(t) Multiplying 𝛼 x(t) + i 𝛾 u(t) by its conjugate will give; It follows that we want to (2) Thus we now have an optimization problem as; Subject to; •The violent nature of the sea; •The low land of sea coast; •The sandy nature of the sea; and •The salty nature of the sea Analysis of the Control Structure For us to obtain the control structure and to be able to examine its functionality (what goes on in the structure), we make use of the finite element method. This is because the stiffness matrix is defined in the fluid flow problem to relate nodal volumetric fluid-flow to nodal potentials, and in structure problem the stiffness matrix is defined to relate nodal forces to nodal displacement. For the discretized structure in Fig.10, we want to derive the element stiffness matrix and equation by using one dimensional finite element formulation. This will enable us to determine; •the potential at the junctions; •the velocities in each section of the structure and; •the volumetric flow rate. It follows that for a smooth pipe , the permeability coefficient, . We find that the element stiffness matrices are: Where the units of are meters for fluid flowing through a structure and is the area of each portion of the discretized structure. If a structure is discretized into parts then there will be nodal potentials or fluid heads. The determinant given as , , … The fluid velocity in element 1 is The Volumetric flow rates Generating Electricity From The Sound of The Sea Wave FIG.13 HOW THE STRUCTURE IN FIG.13 WORKS (i)The dish receives the sound of the sea wave; (ii) the sound sensitive material receives the sound and amplify it; (iii) a piezoelectric transducer converts the sound signal into an electrical signal; (iv) an ultrasonic transducer converts the electrical signal into an ultrasonic sound signal; (v) again the amplified ultrasonic sound signal is converted into an electric signal via a Piezoelectric transducer (vi)an amplifier that will be able to amplify electrical signal is needed at this point and the found one is a Biomolecular Transistor called Microtubules (MTs). MICROTUBULES (MTs): Taxol-stabilized Microtubules (MTs) behave as Biomolecular Transistors capable of amplifying electrical signal. We want all these structures to be isomorphic, that is, to hold together as one. Then we are to invoke an integration such that; represents the transformer If the electrical energy is very high, we will nee d to be a step down transformer such that But if the electrical energy is low, we will need to bust it to the order of our desire such that Thus (10) becomes; For a step up case. While (10) becomes It is obvious that since systems of operations, can be written as matrices which must be square and all of equal order. This will allow for easy multiplication. FIG.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . If the human’s ears are viewed as 2 cylinders, then the volume of air that goes into the brain is given as By also considering air flow to the brain through the noisy as a cylinder the volume of air flow is Thus the volume of air flow to the brain is It follows that if one wears an ear piece on the two ears, the air flow is reduced to This is dangerous to the brain ! And Generating Electricity From Thunder Strike Electrons are emitted when there is a thunder strike. To overcome the problem of electricity generation, it is high time we considered how to gather the electrons that are emitted. There is an existing technology by which these electrons can be gathered. This is by using a thunder catcher. A Mathematical View of Thunder Strike and a Thunder Catcher In Mathematics, the thunder catcher can be viewed as a point of convergence because the electrons significantly meet on the thunder catcher. By using an approximate Mathematical Method, one can represent the pictures of the thunder strikes by graph theory as follows; (i) Fig.17 and fig. 19 can be represented approximately as binary and ternary trees respectively; (ii) Fig.18 can be studied using the Scattering velocity or complex system Analysis; (iii)the Schrodinger Equation for n electrons can be used and it is given as; According to the data collected in this month of April 2013 in Ado-Ekiti, there was only one thunder strike in my area as shown below. 14/4/2013 Thunder strike 18/4/2013 lightening but no thunder strike 19/4/2013 no thunder strike 20/4/2013 no thunder strike 25/4/2013 no thunder strike 28/4/2013 no thunder strike The steps by which electricity from thunder strikes can be generated are as follows; •first find a means of measuring the electrons emitted when there is a thunder strike; •we are to make use of some thunder catcher that are connected to some capacitors since capacitors can receive and store electrons, • if the quantity of electrons trapped by the thunder catcher is as the capacitors serve as electrons banks, we envisage that the electrons harvested therein could be stored until the next raining season since thunder strikes do not always occur every time it rained.