## Simultaneous Effects of Temperature and Temporal Changes of Tritium Reduction on the Energy Gain of DT Fuel Pellet Using DCI

**Received Date:** March 16, 2022 **Accepted Date:** April 16, 2022 **Published Date:** April 18, 2022

**doi:** 10.17303/jmsa.2022.6.202

**Citation:** R Mirzaeean (2022) Simultaneous Effects of Temperature and Temporal Changes of Tritium Reduction on the Energy Gain of DT Fuel Pellet Using DCI. J Mater sci Appl 6: 1-16

### Abstract

NIn this paper, we study on the behavior of Deuterium-Tritium(D-T) plasma nuclear fusion reaction in terms of variations of time and temperature, in the presence of deuterium-tritium sources using DCI. One of the important problems in the human life is obtaining clean energy. Therefore, by solving the time and temperature dependent balance equations on the system of D-T fusion using DCI we determine the optimum physical conditions with low tritium consumption rate to obtain the total energy gain with the value of greater than 200.

**Keywords: **Plasma; Double Cone; Time; Gain; Temperature; Ignition

### Introduction

The ICF program has focused its attention on “central hot spot” ignition, whereby a hollow spherical shell of DT ice containing DT gas is compressed, creating a central hot spot surrounded by a dense shell of cold DT [1]. The alpha particles from fusion in the hot ignition spark create a propagating fusion burn in the cold fuel. The target gain that can be achieved is limited by the high investment of energy in compression of the fuel and the threshold energy for ignition is high because the spark density is much lower than the fuel density in isobaric ignition. These factors set the minimum size of the driver and push the energy input for high gain to a few mega joules. Gain values are higher for direct drive than for indirect drive due to coupling efficiency differences, but it appears difficult to obtain gains greater than 100 , and values of a few tens are more conservative. While there are several drivers that promise high enough efficiency for the driver efficiency-target gain product to lead to an acceptable commercial inertial fusion energy (IFE) power plant.

The fast ignitor (FI) may be that concept [2]. With FI the compression and ignition steps are separated. A target of DT is compressed to high density at low temperature by lasers or particle beams. A second, very high intensity beam delivers the energy used to ignite the compressed core and heats an ignition spark at the density of the cold fuel in isochoric ignition. The FI concept promises much higher gain for the same driver energy than isobaric hot spot ignition. In addition, the “ignition threshold” that occurs at about 1MJ for hot spot ignition may be reduced to ~100KJ[3]. These changes in the drive requirements for inertial fusion give rise to a cascading sequence of changes to the concept for inertial fusion energy that may lead to a much more attractive inertial fusion power plant with a much easier development pathway.

Deuterium-Tritium fusion appears to be the best and most effective way to produce energy. By fusing the two isotopes of hydrogen in to the heavier element helium large quantities of energy are released. One of the disadvantage of D-T fusion is that tritium must be created. Tritium has a half-life of a little more than ten years so there are no naturally occurring atoms left on earth. Tritium is normally bred from lithium 6 atoms by shooting them with a neutron. [4-8] In order to fusion to be economically an attractive energy source it is necessary that the energy from thermonuclear reaction exceeds both the energy invested in achieving it and the losses due to radiation, reactor wall, conversion inefficiencies, etc. A simple estimate of the driver efficiency and target gain can be obtained using energy bookkeeping approach (Fig.1).P_{th} , the power after converting the thermal energy to electrical energy.

Let P_{tn} be the electrical power that we intend to invest in achieving fusion and η_{d} let be the efficiency of conversion of this power to laser light power P_{d} . If we define the target gain as the ratio of the power obtained from the fusion reaction to the driver power,then: P_{g}=GP_{d}=Gη_{d}P_{in} taking into account the conversion efficiency from thermal to electrical power, the available electrical power is given by: P_{th}= η_{th}P_{g}=Gη_{th}η_{d}P_{in}. A fraction fGη_{th}η_{d}P_{in} of this power is used to run the driver and the remaining fraction (1-f)Gη_{th}η_{d}P_{in} is send to the customers. The closer the factor(1-f)to unity, the more electricity is available to customers. This can be achieved by using small values of , however, has a lower bound determined by the equation:P_{in}=f_{min}Gη_{th}η_{d}P_{in} →f_{min}=1/Gη_{th}η_{d}. Using a typical value η_{th} of 40% and recycling less than 1/4 of the power to run the driver, we get,Gη_{d}≥10.Solid state lasers for example have an efficiency of 1-10% [9] and to fulfill the available condition, gains of 100-1000 are needed.

^{-1}.Head-on collision of the compressed and accelerated fuel pellets from the cone tips convert the forward kinetic energy to the thermal energy of the colliding fuel pellets with an increased density. [11-18] The pre-heated high-density fuel pellets can keep its status for a time period of about 200 ps. Within this period, electrons with MeV energy produced by ps heating laser pulses, guided by a ns laser-generated strong magnetic field further heat the fuel efficiently. Then, the fuel pellets can gain an ignition temperature of greater than 5keV with magnetically assisted heating of MeV electrons produced by the heating laser pulses. [19-23]

In this work, the main goal is utilizing of DCI design for calculating the optimum total energy gain in D-T mixture with low tritium consumption rate. Because, as we know tritium fuel is rare, radioactive and expensive. Therefore, firstly we introduced the balance equations on the D-T plasma fusion secondly at the available physical conditions by computer programing we solve these nonlinear point kinetic equations and finally we compute the optimum value of total energy gain with selecting low tritium consumption rate.

### Balance equations and physical parameters in the D-T system

In a plasma including deuterium and tritium, the nuclear fusion product of these nuclides are neutron and alpha particles.During the process of fusion, produced neutrons nearly escape from plasma fusion without any interaction but alpha particles remain inside the fusion plasma and increase the plasma energy. Note that we consider the steady injection of deuterium and tritium into the core with rate of s_{D},and s_{T},also,we consider τ_{T},τ_{D},τ_{α} as a half-life of tritium, deuterium and alpha particles, respectively, such that τ_{D}=τ_{T}=τ_{p} thus the balance equations of particle density of deuterium n_{D}(x),tritum n_{T}(x)and alpha n_{α}(x) , respectively, are given by:

With defining relative quantity

_{e}(x) is plasma density and defined as, n

_{e}(x)=n

_{D}(x)+n

_{T}(x)+2n

_{α}so we will have f

_{D}+f

_{T}+2f

_{α}=1. Also, with assuming that f

_{D},f

_{T}are independent of time the equations (1)and (2) are converted to

We assume that S_{DT}=S_{D}+S_{T} and by adding the two equations of (4) and (5) we obtain:

Where<σ_{DT}> is the average reactivity of deuterium-tritium fusion reaction. Which is:

Figure 3 shows the variations of D-T average reactivity versus temperature

We solve the above equations in terms of time by assuming that S_{T}=S_{D}=2.2*10^{20}m^{-3}S^{-1} and τ_{p}=2 s and T=100KeV. In Figure. , the time variations of plasma density for some specific f_{T} is shown. From this figure ,we see clearly that by increasing the time and the plasma density is increased and from a special time by increasing the time for each value of f_{T}the plasma density is constant. Also, in figure.4b the three dimensional variations of plasma density are given.

Also, figures.5,6 and 7 show that the three dimensional variations of deuterium, tritium and alpha particle density in terms of time and tritium fraction.

From these figures and our calculated numerical values of n_{D},n_{T}, and n_{α} in terms of time and f_{T} we find that by increasing time n_{D},n_{T}, and n_{α} are increased also in a constant time by increasing f_{T},n_{T}, and n_{α} are increased but n_{D} is decreased. Because by increasing f_{T} from equation of f_{D}=1-f_{T}-2f_{α},f_{D}is decreased and thus n_{D} is decreased. But, if in system there are no any source injection of deuterium and tritium, then the density of deuterium and tritium are reduced while the density of alpha particles is increased.

The other parameter in this reaction is known as burn consumption fraction that is given by:

Where n^{0}_{e} is plasma density at time On the other hand mass density of plasma is defined as

Finally, we will have

P =n_{e}(t)× 1.66 ×10^{-24}[2f_{D}+3f_{T}+4f_{α}]

**Calculation of power and energy density for -T fusion reaction in the presence of the deuterium and tritium sources**

Another important issue that should be consider about fusion plasma of deuterium-tritium, is behavior of total power in fusion system.

Total power includes P_{α},P_{oh},P_{ext},P_{berms},P_{loss},P_{sync} which are defined in the following ,respectively:

a)P_{α}: is the portion of transmitted power to high energy alpha particles produced from D-T reaction that is deposited into the plasma.

b) P_{oh}: is a thermal power

c)P_{ext} : the total power that is given to the system by an external factor.

d)P_{berms} :is the portion of loss power that is due to Bremsstrahlung radiation

e)P_{loss} :is the portion of transmitted power to alpha particle, during the D-T reaction because of escaping alpha particle from fusion plasma that is not deposited in the chamber.

f)P_{sync} : is the portion of loss power that is due to synchrotron radiation.

So due to being loss P_{sync},P_{berms},P_{loss} and being productive P_{α},P_{oh},P_{ext} we have the following equilibrium energy density:

In these relations temperature is in keV and we have:[4,5]

_{α},P

_{berms}, and P

_{loss}in terms of time for several values of f

_{T}are given.

We see clearly from these figures ,by increasing f_{T},n_{e} is increased and thus by enhancement of n_{e} according to n_{α} is raised therefore P_{α} is increased. Also by increasing f_{T},f_{α} and f_{D} are decreased therefore by observing relation f_{D}=1-f_{T}-2f_{α}, therefore by increasing f_{α} and f_{T},f_{D} is reduced. On the other hand by increasing f_{α}is raised and is dominant on the f_{D}+f_{T},finally P_{loss} is increased by enhancement F_{T}. Also,P_{berms} by increasing time is increased because by raising time , plasma density is increased ,therefore the collisions of charged particles are increased thus the energy is dissipated in the form of radiation. In Figure 11, the variations of energy density in terms of time for several f_{T} is shown.

Also in Figure 12 , the three dimensional variations of energy density in terms of time and f_{T} is given.

**Study on the temperature effects on the tritium-deuterium fusion**

Another point that should be referred to in tritium-deuterium fusion is the effect of temperature changes in terms of time, to achieve this goal ,by doing time derivative from equation

By inserting relations (1),(2),(3) in this relation we have:

However, we know that energy density for a system with density n and temperature , is:

w = 3/2nT

where is T is in Kev. So the plasma energy density is given by:

In which T_{e} is electronic temperature and T_{i} is ionic temperature. By derivation of this equation respect to time we obtain:

and with assuming that quantity γ_{e}= T_{i}(x)/T_{e}(x) is constant and using relation

If we replace equation (11) in the left hand side of equation (22) we obtain:

By obtaining T_{i} from above equations and using the equation T_{e} = T_{i}/γ_{e} we can calculate .in the Figure 13 the variations of (ionic temperature) versus time for different values of is shown.

Also fuel energy gain of D-T fusion reaction is given by:

In the Table.1 numerical values of different parameters for calculation energy gain is given.

In the figure 14, the variation of energy gain versus temperature for different values of f_{T} is shown. From this figure we conclude that, by increasing temperature energy gain is increased because by enhancement temperature the fusion cross section increases therefore the number of fusion reactions are increased also is increased .Finally the total energy gain is given by:

By assuming that η_{c}≈η_{ig} and E_{dig}<< E_{d we will have:.GDT≈ ηcGF In the figure 15 the three dimensional variations of total energy gain in terms of fT and temperature is shown.}

ρR=1.2g/cm^{2},η_{c} = 0.3

Also in Table2, the numerical values of total energy gain versus at available physical conditions is given. Our calculations show that by increasing ρR and constant f_{T} the total energy is increased and maximized at ρR=1.2g/cm^{2} .In Table 2 the maximum values of total energy gain in the interval[ρR_{1},ρR_{2}] at different values of f_{T} are given. Table2: the maximum values of total energy gain in the interval[ρR_{1},ρR_{2}] at different values of f_{T} and η_{c}=0.3

From this table we see that at all different values of f_{T} there is an interval of[_{1},ρR_{2}] such that in this interval the total energy gain is greater than 200.

Notice that in our calculations we use the data inside the Table3.

### Conclusion

In this paper, by solving the balance equations for deuterium-tritium fusion reaction in the presence of deuterium-tritium source by utilization of DCI design, for the first time, we obtained the time dependent particle density. Then we determined the total energy of the system. We see that by reducing f_{T} the energy density is decreased but their amount are still considerable. If we analyze the system from economical point of view we conclude that decreasing the is f_{T} an important point because tritium is rare ,radioactive and expensive fuel . Also we saw that by decreasing f_{T} the total energy gain of the system is decreased but even with selecting low f_{T} and making a system with ρR in the suggested interval[ρR_{1},ρR_{2}] we can get the energy gain more than 200 . Therefore from this work we conclude that using of DCI design the energy gain is increased respect to cone guided ignition . Therefore, it is hoped that the use of DCI design in the future can lead us to more energy efficiency, but in the meantime, more research is needed on the fabrication of such fuel pellets and the challenges that govern it.

- Doran TJ (1982) Fusion research.prtgamon press,Newyork.
- Miyamoto k (1989) plasma of nuclear fusion ,2nd edn.MITPress, Cambridge, Mass.
- Taller E(ed.) fusion. Academicpress, London (1981).
- Keishiro, Nuclearfusion, Tokyo Institue of Technology English Edition(1989)
- Rand McNally,R.and et al.Fusion Reactivity,OakRidge Nat.Lab(1979)
- Lawson JD (1989) some criteria for a power production thermonuclear reactor, Proceedings of the physical society.
- Artsimovitsh LA Tokamak device. Nuclear Fusion 215 (1972)
- Conn RW Magnetic fusion reactor. Fusion(ed.E.Teller) Vol.1,AcademicPress,Newyork(1981).
- Gross J. Fusion energy. Willey, NewYork (1984).
- Zhang J, Wang WM, Yang XH, Wu D, Ma YY, Jiao JL,Zhang Z, Wu FY, Yuan XH, Li YT, Zhu JQ. Double-cone ignition scheme for inertial confinement fusion. Phil. Trans. R. Soc. A 378: 20200015,( 2020).
- Norreys P et al. Fast electron energy transport in solid density and compressed plasma. Nucl. Fusion 54, 054004. (2014).
- Jarrott LC et al. Visualizing fast electron energy transport into laser-compressed highdensity fast-ignition targets. Nature Phys. 12, 499-504. (2016).
- Theobald W et al. Initial cone-in-shell fast-ignition experiments on OMEGA. Phys. Plasmas 18, 056305. (2011).
- Nora R et al. Gigabar spherical shock generation on the OMEGA Laser. Phys. Rev. Lett. 114, 045001. (2015).
- Fujioka S et al. Kilotesla magnetic field due to a capacitor-coil target driven by high power laser. Sci. Rep. 3, 1170.(2013).
- Zhu BJ et al. Strong magnetic fields generated with a simple open-ended coil irradiated by high power laser pulses. Appl. Phys. Lett. 107, 261903 (2015).
- Wang W-M, Gibbon P, Sheng Z-M, Li Y-T. Magnetically assisted fast ignition. Phys. Rev. Lett. 114, 015001 (2015)
- Bailly-Grandvaux M et al. Guiding of relativistic electron beams in dense matter by laser-driven magnetostatic fields.Nat. Commun. 9, 102. (2018).
- Sakata S et al. Magnetized fast isochoric laser heating for efficient creation of ultra-highenergy-density states. Nat.Commun 9, 3937. (2018).
- Kidder RE. Theory of homogeneous isentropic compression and its application to laser fusion. Nucl. Fusion 14, 53. (1974).
- Henestroza E, Grant Logan B. Progress towards a highgain and robust target design for heavy ion fusion. Phys. Plasmas 19, 072706. (2012) 22. Wu D, Yu W, Fritzsche S, He XT. 2019 High-order implicit particle-in-cell method for plasma simulations at solid densities. Phys. Rev. E. 100, 013207. (2019).
- Cui Y-Q, Wang W-M, Sheng Z-M, Li Y-T, Zhang J. Laser absorption and hot electron temperature scalings in laser-plasma interactions. Plasma Phys. Control. Fusion 55, 085008.( 2013).

## Tables at a glance

## Figures at a glance