Despite the advantages of nuclear technology to mankind, it attendance accident is always catastrophic. In this work, the energy balance equations associated with nuclear technology are structured in parametric form to form the basis of our Mathematical model solvable by the Conjugate Gradient Method (CGM).
As the nation of the world seek to meet the increasing demands for energy which lead to increases in environmental pollution from the burning of coal, oil and natural gas. The use of fossil fuels for transportation, generating electricity, heat and industrial production account for 85 percent [
Nuclear reactors produce energy through a controlled fission chain reaction. While most reactors generate electric power, some can also produce plutonium for weapons and reactor fuel. Power reactors use the heat from fission to produce steam, which turns turbines to generate electricity. In this respect they are similar to plants fueled by coal and natural gas. The components common to all nuclear reactors include a fuel assembly, control rods, a coolant, a pressure vessel, a containment structure and an external cooling facility.
Thermal reactors operate on the principle that uranium-235 undergoes fission more readily with slow neutrons than with fast ones. Light water (H2O), heavy water (D2O), and carbon in the form of graphite are the most common moderators. Since slow neutron reactors are highly efficient in producing fission in uranium-235, they use fuel assemblies containing either natural uranium (0.7percent U-235) or slightly enriched uranium (0.9 to 2percent U-235) fuel. Rods composed of neutronabsorbing materials such as cadmium or boron are inserted into the fuel assembly. The position of these control rods in the reactor core determines the rate of the fission chain reaction. The coolant is a liquid or gas that removes the heat from the core and produces steam to drive the turbines. In reactors using either light water or heavy water, the coolant also serves as the moderator. Reactors employing gaseous coolants (CO2 or He) use graphite as the moderator. The pressure, made of heavyduty steel, holds the reactor core containing the fuel assembly, control rods, moderator and coolant. The containment structure, compose of thick concrete and steel, inhibits the release of radiation in case of an accident and also secure components of the reactor from potential intruders. Finally, the most obvious components of many nuclear power plants are the cooling towers, the external components, which provide cool water for condensing the steam to water for recycling into the containment structure. Cooling towers are also employed with coal and natural gas plants [
Most of the nuclear power plants in the world are water-moderated thermal reactors. They are categorized as either light water or heavy water reactors. Light water reactors use purified natural water (H2O) as the coolant/ moderator, while heavy water reactors employ heavy water deuterium oxide (D2O). In light water reactors, the water is either pressured to keep it in superheated form (in a pressurized water reactor, PWR) or allowed to vaporize, forming a mixture of water and steam (in boiling water reactor, BWR). In PWR, superheated water flowing through tubes in the reactor core transfers the heat generated by fission to a heat exchanger, which produces steam in a secondary loop to generate electricity. None of the water flowing through the reactor core leaves the containment structure.
Other thermal reactors are Boiling Water Reactor (BWR), Heavy Water Reactor (HWR), Gas Cooled Reactor, Pebble Bed Reactor and Fast Neutron Reactor.
We have different types of non-nuclear electric power generation. Some are Wind-mills, Hydro-electric power (falling water) plants, Fossil-fuels plant and Solar cell plants.
This plant is device for tapping the energy of the wind by means of sails mounted on a rotating shafts. It is generally less reliable than the water power, but where water is deficient wind power is an attractive substitute. Wind-mill-plant can be use in areas that suffer from draught or from a shortage of surface water and also in a low-lying area where rivers offer little energy. They were mostly used, converting the energy of the wind into mechanical energy for grinding grain, pumping water and drainage. They produced less energy for electricity [
It is the electricity produced from generators driven by water turbines that convert the potential energy in falling or fast-flowing water to mechanical energy. They are usually located in dams that impound rivers; especially in areas with heavy rain fall and hilly or mountainous region. In generation, water is collected or stored at a higher elevation and led downward through large pipes or tunnels to a lower elevation. At the end of its passage down the pipes, the falling water causes turbines to rotate which in turn drive generators which convert the turbines mechanical energy into electricity [
The production of steam by fossil-fuels, produce the greatest amount of electrical energy compared to hydroelectric and wind-mills. Fossil-fuels are coal, petroleum, natural gas, oil shales, bitumens, tar sands and heavy oil. All contain carbon and were formed as a result of geologic processes acting on the remains of organic matter produced by photosynthesis. All fossil-fuels can be burned in air or with oxygen to provide heat. The heat may be utilized to produce steam to drive generators that can supply electricity [
This is the best plant that can only be conveniently use on satellites and space where the flow of energy out of the sun (the solar wind) can be harnessed without interference from the atmosphere or the rotation of the earth [
All the non-nuclear power generation discussed above produced low electricity compared to the nuclear reactors. The nuclear reactor released a large amount of energy during the process of fission. Most of this energy released takes form of heat, which is used to produce steam. The steam drives a turbine, the mechanical energy of which is converted to electricity by a generator. The whole process takes place in a safety environment that is not hazardous to the human. The nuclear reactor eliminates most of the disadvantages of the non-nuclear energy by generating enough energy for consumption; does not involved emission of carbon which is very dangerous to the environment; does not involved burning of coal and oils that are the causes of all manners of lung ailments and also does not dependent on fossil-fuels.
There were three major nuclear reactor accidents they are the three mole island in Pennsylvania, USA in 1979 [
With the above analysis on nuclear accidents, nuclear technology is necessary for power generation if the safety concerns can be adequately addressed and that is why this work is purely devoted to the safety aspect and not to the weapons production. The fission from the reactors produce energy in form of heat that releases steam to turn turbine to generate electricity, therefore it is necessary to find equations for rate of heat that can be use for the purpose of our work. See [12-14].
The following results were obtained through energy balance equation and were arranged in terms of rate of heats of the reactors (i.e. batch, continuous stirred tank, semi batch and plug flow). See [15,16].
If the above thirty-six parametric equations and nuclear tokens can be used for the structure model for the construction of nuclear reactors, then the safety will be maximized and disaster minimized. This will later be structured into mathematical model in form of quadratic functional. The model will be solved using Conjugate Gradient Method algorithm, with MATLAB as a support soft-ware. See [17,18].
In particular, we shall use the following quadratic functional for further work.
where i.e. X = (x1x2 x3x4x5x6)T, a = (111111)T, f0 = 1 and
where are the rate of heat of the reactors, and i, j = 1, 2, 3, 4, 5, 6. See [
The following nomenclatures are the nuclear tokens.
A = heat transfer area Cj = concentration of species j
= feed concentration of species j
= steady-state concentration of species j
= constant-pressure heat capacity
= partial molar heat capacity
= heat capacity per volume
= constant-pressure heat capacity per mass
= constant-volume heat capacity per mass
= heat capacity change on reaction,
E = total energy
= total energy per mass
= partial molar enthalpy
= enthalpy per unit mass
= enthalpy change on reaction,
= reaction rate constant for reaction i
= reaction rate constant evaluated at mean temperature Tm
= equilibrium constant for reaction i
= kinetic energy per unit mass
= total mass fow of stream k n = reaction order
= moles of species j,
= number of reactions in the reaction network
ns = number of species in the reaction network
P = pressure
Pj = partial pressure of component j
Q = volumetric flow rate Qf = feed volumetric flowrate
= heat transfer rate to reactor
= reaction rate for i th reaction
= production rate for j th species
= time
= temperature
= temperature of heat transfer medium
= mean temperature at which k is evaluated
= overall heat transfer coefficient
= internal energy per mass
= velocity of stream k
= reactor volume variable
= partial molar volume of species j
= reactor volume
= change in volume upon reaction i,
= rate work is done on the system
= coefficient of expansion of the mixture,
= reactor residence time,
= stoichiometric coefficient for species j in reaction i
= mass density
= mass density of stream k