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Publicly Available Published by De Gruyter February 7, 2015

A Comparison Between the Burn Condition of Deuterium–Tritium and Deuterium–Helium-3 Reaction and Stability Limits

  • Seyed Mohammad Motevalli EMAIL logo and Fereshteh Fadaei

Abstract

The nuclear reaction of deuterium–tritium (D–T) fusion by the usual magnetic or inertial confinement suffers from a number of difficulties and problems caused by tritium handling, neutron damage to materials and neutron-induced radioactivity, etc. The study of the nuclear synthesis reaction of deuterium–helium-3 (D–3He) at low collision energies (below 1 keV) is of interest for its applications in nuclear physics and astrophysics. Spherical tokamak (ST) reactors have a low aspect ratio and can confine plasma with β≈1. These capabilities of ST reactors are due to the use of the alternative D–3He reaction. In this work, the burn condition of D–3He reaction was calculated by using zero-dimensional particles and power equations, and, with the use of the parameters of the ST reactor, the stability limit of D–3He reaction was calculated and then the results were compared with those of D–T reaction. The obtained results show that the burn conditions of D–3He reaction required a higher temperature and had a much more limited temperature range in comparison to those of D–T reaction.

1 Introduction

Deuterium–tritium (D–T) fuel for a fusion reactor is the dominant fuel in use in today’s fusion research project [1–3]. The D–T fuel cycle has the largest cross section of all the fusion reactions, and it burns at the lowest temperature. In the D–T fuel cycle, 80% of the fusion energy is in the form of 14-MeV neutrons. These neutrons activate the reactor wall and induce radiation damage and radioactivity in the reactor structure. Furthermore, a breeding and storage system for a radioactive substance, tritium, that needs extra complexity, cost and radial space for a lithium blanket is required for the D–T reactor. The D–3He reaction generates no neutrons, so a fusion reactor with this fuel is very attractive from this viewpoint. Neutron activation levels and radiation damage rates by a factor of 17–20 relative to the D–T fuel cycle are reduced by the low neutron production of the D–3He fuel cycle [4]. In the D–3He reaction, if the D–D reactions are disregarded, all fusion energy is released in charged particles that can be directly converted to electricity at high efficiency in some configurations. The other advantages of D–3He over D–T are full lifetime materials, easier maintenance and proliferation resistance [5–9]. But this reaction has many problems. The D–3He fusion cross section is lower than that of D–T fuel. Consequently, D–3He requires a density-confinement time product that is 50 times higher and a fusion power density in the plasma that is 80 times higher than that for D–T [5, 6]. Burning D–3He fuel thus requires substantial, continued progress in plasma physics. There has been only relatively small progress beyond the progress already accomplished in the historically well-funded tokamak program. The crucial physics issues for advanced fusion configuration are the confinement and control of the resulting fusion ash buildup [10]. In innovative confinement concepts such as the spherical tokamak (ST), the key physics issues have been identified, but resources to test issues adequately have not been available [10]. The fact that the terrestrial supply of the helium isotope 3He is severely limited is another impediment to competitive D–3He fusion, while the hydrogen isotope deuterium is found in plentiful supply in nature. Fusion can be applied to space propulsion. Research projects address the application of fusion to generate electrical power in space, as well as propulsion. As mentioned previously, the standard D–T fusion reaction has some fundamental limitations; moreover, the lack of 3He on the earth is the main problem for the D–3He fusion reaction, but 3He is found all throughout the universe and thus the reactor could be refueled at any number of locations such as the moon. For theses reasons, it seems that D–3He fusion reaction is an appropriate fuel for space propulsion. The first study on D–3He fusion reactors for space propulsion was published in 1962, and one of the noted studies to use D–3He fusion reaction for propulsion was done by the British Interplanetary Society called Daedalus [11].

As mentioned previously, a spherical tokamak is capable of operating with the alternative D–3He fuel because the fact that ST reactors can confine plasma with β≈1 allows plasma to be confined at the same pressure as in classical tokamak reactors but at a significantly lower toroidal magnetic field [12]. The physics key to the attractiveness of the ST approach is in the order unity β values expected to be achieved by the combination of low aspect ratio. A low aspect ratio makes it possible to create a relatively compact ST reactor, while a high β value allows the reactor to operate with the alternative D–3He reaction [12]. The advantages of the ST approach have been discussed for many years [13, 14]. In recent years, interest in the ST approach has grown rapidly [15, 16]. The low cost of the ST approach in comparison with other conventional tokamaks allows ST reactors to continue to be considered as a potential option for producing thermonuclear burn. The possibility of D–3He fueled ST reactors has been indicated already [17].

The aim of this paper was to study the burn condition of D–3He nuclear fusion reaction in comparison to that of D–T fusion reaction. In these reactions, investigation of the burn criterion is very important. Several attempts have been made to extend the original burn criterion. By using τR, the time scale for reaction kinetics, which replaced τE, the energy confinement time, Maglich and Miller [18] introduced burn criterion in 1975. This approach was questioned by Chen et al. [19] in 1977. Burn criterion was defined by Chen et al. [19] as the product of plasma density and energy confinement time in which all loss processes can be included realistically in the calculation of the energy confinement time. In this work, τE has been applied in the calculation of the triple product neτET for the D–3He reaction. This parameter was obtained from the work of Rebhan and Oost [20] on D–T reaction, and it was also calculated in this work for the D–3He reaction and for comparison with the D–T reaction.

Zero-dimensional particles and power equations have been used for nuclear fusion reactions. In this model, ions and electrons are assumed to have Maxwellian distributions and share the same temperature at all time, Ti=Te. Using the global power equation and scaling ansatz brings about the closed ignition curves for D–3He nuclear fusion reaction, which are compared with D–T reaction results [21], then, with the use of energy confinement scaling and the parameters of the ST reactor, plasma instability limits were obtained and compared with D–T reaction results [20]. In this work, in addition to D–3He nuclear fusion reaction, the two branches of D–D reaction and D–T reaction were taken into consideration in our calculations. Synchrotron radiation power was also taken into account in the D–3He reaction, while it was ignored in the D–T reaction.

2 Theoretical Calculation

The main fusion reactions in this work are

(1)D+T4He (3.5 MeV)+n (14.1 MeV) (1)

and for the D–3He cycle

(2)D+3He4He (3.7 MeV)+p (14.7 MeV) (2)
(3)D+DT (1.01 MeV)+p (3.02 MeV) (3)
(4)D+D3He (0.82 MeV)+n (2.449 MeV) (4)
(5)D+T4He (3.5 MeV)+n (14.1 MeV) (5)

all the aforementioend reactions were applied in this work. The reactions (D–D, D–3He) for the D–T reaction were ignored due to their small fusion cross section.

The equations to be solved in this paper were the general particle balance and general global plasma power balance equations. The particle balance equation, which describes the diffusion loss, fuel injection, production and consumption of a given species, is of the form

(6)dnkdt=nkτpk+Sk+i,jninjδijσvijininkσvik (6)

where i and j represent the appropriate species for reaction i+ jk. The term nkτpk is the diffusion loss term, and Sk is the external particle supply of nucleus k. 〈σνij is the fusion reaction rate for all particle production and consumption (i, j, k= D, T, p, 3He, α), and for all reactions, 〈σνij was taken from [22]. For like particles, δij1/2 when i= j; otherwise, δij1 when ij.

In this work, the particle confinement time τp was coupled to the energy confinement time τE by this relation [20],

(7)ρ=τpτE (7)

The general global plasma power balance equation is

(8)dWdt=Pf+PauxPbrPsynPie+PohWτE (8)

where Pf is the fusion power density, which is calculated by considering all fusion reactions, which are given by

(9)Pf=ijninjσvijδijEfij (9)

where Efij is the energy yielded by a fusion reaction i+ jk, and for like particles, δij1/2 when i= j; otherwise, δij1 when ij.

Paux is the auxiliary power density; it is useful to express the auxiliary power in multiples of fusion power by

(10)Paux=fPf (10)

Pbr is the bremsstrahlung radiation power loss and is defined as [23]

(11)Pbr=AbZeff2ne2T1/2 (11)

Zeff is the effective charge of the plasma ions and is defined as

(12)Zeff=iZi2nine. (12)

and Ab is the bremsstrahlung radiation coefficient [24]. The electron density is governed by the neutrality condition.

(13)ne=izini (13)

Psyn is the synchrotron radiation loss and is defined as [23]

(14)Psyn=6.214×1023neTB2ΦT (14)

where

(15)ΦT=5.198×103Λ1/2T1.5(1+22.61AT1/2)1/2(1Rw)1/2 (15)
(16)Λ1/2=7.78×109(neaB) (16)

where A is the aspect ratio, which is R/a; B is the toroidal magnetic field on axis; and Rw is the reflection factor of the radiation from the vacuum wall. The synchrotron power depends on the plasma parameters, i.e., aspect ratio, β value, magnetic field, minor radius of plasma, wall reflectivity and density. In this work, it is used by the parameters of the ST for the D–3He reaction [25]. Synchrotron radiation is important at high temperature; at low temperature, however, bremsstrahlung radiation dominates the losses, so, in this work, synchrotron radiation was ignored in the D–T reaction.

Pie is the equilibration power between ions and electrons

(17)Pie=2.4×1041CielnΛiene2ZeffTiTeTe1.5 (17)

where

(18)Cie=(1+αn)2(2αn0.5αT+1)(1+αT)(1/2) (18)

The parameters αn, αT represent the parabolic profile exponents. In this work, it was assumed that Ti=Te, so Pie=0.

Poh is the ohmic power density, which, for steady-state operations, is negligible. The last parameter in (8) is the power density of the transport losses, where W is the plasma thermal energy content defined as

(19)W=32ntotkT (19)

where ntot=ni + ne and ni is the ion density.

Now, the neτET parameter of the ideal ignition and the breakeven values of the D–T and D–3He reaction should be calculated. To do this, in steady state, the d/dt term equals zero, so, in this state, we shall neglect the presence of impurities as well as the helium and proton ash [20]. For breakeven, the f= 1 in (10) and f= 0 for ignition were set. By solving (8), considering (9), (10), (11) and (14), the ideal fusion product neτET for the D–T and D–3He reaction in the state of ignition and breakeven was obtained [20]. Its temperature dependence for both of the states and two reactions is shown in Figure 1. Comparison of these two graphs shows that the minimum temperature required for ignition in the D–T reaction was 14 keV and for breakeven was 13 keV. The minimum for D–3He reaction was 65 and60 keV for ignition and breakeven, respectively. The graph shows that the minimum value of neτET for the D–3He reaction was 26 and 68 times larger than that for the D–T reaction for ignition and breakeven, respectively.

Figure 1 Comparison of the effect of D–3He and D–T reactions on neτET for ideal ignition (f= 0) (solid lines) and ideal breakeven (f= 1) (dashed lines) vs. temperature.
Figure 1

Comparison of the effect of D–3He and D–T reactions on neτET for ideal ignition (f= 0) (solid lines) and ideal breakeven (f= 1) (dashed lines) vs. temperature.

We shall consider the influence of ash particle and calculate the neτET parameter, but this time we need the particle balance equations with regard to (6) and the scaling ansatz (7) [21, 26]. One can derive an equation for neτET where it is a function of T and ρ and can be put into the form ρ(neτET, T) [21]. The dependence of this parameter on temperature in the D–T and D–3He reactions is shown in Figure 2. This time it is a function of ρ and T.

Figure 2 Curves of neτET for non-ideal ignited equilibria D–3He reaction compared to that for D–T reaction.
Figure 2

Curves of neτET for non-ideal ignited equilibria D–3He reaction compared to that for D–T reaction.

Figure 2 shows the ignition curves (ρ = const) for the D–T and D–3He reactions. This figure shows the closed ignition curves that exist for the D–T reaction for ρ < 15 [20]; these curves exist for D–3He fuel for ρ = 5.5 and ρ = 6, and for larger values of ρ they disappear. The largest value of ρ in the D–3He reaction was 6.21. It is obvious from Figure 2 that the burn condition of the D–3He reaction in comparison to that of the D–T reaction is at a higher temperature. The minimum neτET for the state ρ = 6 in the D–3He reaction in comparison with ρ = 6 in the D–T reaction was approximately 40 times larger.

The energy confinement time is a quantity of central importance in the design of next-generation tokamak reactors, which is equal to the plasma stored energy divided by the total power loss from the plasma in equilibrium. Because tokamak transport is problematic, it is very difficult to predict the energy confinement time theoretically. Empirical scaling laws are needed. Empirical scaling has been derived from databases for global plasma parameters collected from different devices. The energy confinement time depends on the plasma parameters. In this work, the energy confinement scaling used for the D–T reaction is ITER 89-P [27].

This scale with plasma parameters is

(20)τE=0.048fHM0.5IP0.85Bt0.2R1.2a0.3κ0.5P0.5ne0.1 (20)

There are several confinement regimes; the H-mode is the most important regime for reactor applications. The currently recommended H-mode scaling is known as IPB98 (y,2), which is used for the D–3He reaction

(21)τE=0.144HhBt0.15IP0.93κ0.78ne0.41a0.58R1.39M0.19P0.69 (21)

where P, the net plasma heating power in megawatts, is given by

(22)P=Pf+PauxPbrPsyn (22)

In the steady state, it can be replaced by

(23)P=WτE (23)

Hh is the enhancement factor; the recent NSTX experiments show that an enhancement factor of Hh>1.5 is possible. However, we conservatively set Hh=1.0. At this point, by using the parameters in Table 1 for D–T reaction and those in Table 2 for D–3He reaction, the ignition contours can be translated into an ne, T plane.

Table 1

Parameters of ITER [20].

ParameterUnitValue
Toroidal current, IPMA22
Magnetic field, BtT4.85
Elongation, κ2.2
Major radius, Rm6
Minor radius, am2.15
Plasma volume, Vm31100
Troyon coefficient, gf2.5
Table 2

Parameters of the D–3He ST Reactor [17].

ParameterUnitValue
Toroidal current, IPMA128
Magnetic field, BtT2.7
Elongation, κ3
Major radius, Rm8
Minor radius, am6.15
Plasma volume, Vm31.792×104
Wall reflection, Rw1
Troyon coefficient, gf4.2

By substituting the energy confinement time in the equation neτET, this equation can be transformed to an ne, T plane. Using this transformation, we obtained the previous ignition curves for ρ =0, 3, 5 for the D–T reaction and ρ =5, 5.5, 6 for the D–3He reaction in the ne, T diagram shown in Figure 3. In this figure, the ne, T plane for the D–3He reaction in comparison to that for the D–T reaction [20] is shown. The fact that the minimum temperature range of 6 keV in the D–T reaction shifted to 32 keV in the D–3He reaction was obtained from the comparison of the ignition curve in the state ρ =5 in Figure 3. The other finding of this comparison is that the minimum of electron density for the D–T reaction at a temperature of about 14 keV was 4.7×1019 (m–3) for the ignition curve in the state ρ =5, while the temperature for the D–3He reaction was about 50 keV and 1×1020 (m–3).

Figure 3 Ignition curve of the ne, T plane for ρ = 0,3,5 in the D–T reaction and ρ = 5,5.5,6 in the D–3He reaction compared to that of the D–T reaction.
Figure 3

Ignition curve of the ne, T plane for ρ = 0,3,5 in the D–T reaction and ρ = 5,5.5,6 in the D–3He reaction compared to that of the D–T reaction.

And using the β parameter as

(24)β=ntotkT/(B2/2μ0) (24)

the ignition curves can be transformed to the β, T plane.

The normalised βN is given by

(25)βN=β/(Ip/aBt) (25)

The following operation limits were mainly implied from the tokamak operation window. The β limit is expressed as

(26)β<gfIpaBt (26)

where gf is the Troyon coefficient and identified in Tables 1 and 2 for the D–T reaction and D–3He reaction, respectively. The Greenwald density limit is

(27)ne=kGIpπa02 (27)

where kG=0.85 for the D–3He reaction [25]. In Figure 4, the βN, T plane for the D–3He reaction is shown in comparison to that for the D–T reaction [20]. The importance of representing this ignition curve as ne(T) or β(T) is that the impact of plasma stability limits like the β limit and the density limit can be seen immediately.

Figure 4 Ignition curve of the βN, T plane for ρ = 0,3,5 in the D–T reaction and ρ = 5,5.5,6 in the D–3He reaction compared to that of the D–T reaction.
Figure 4

Ignition curve of the βN, T plane for ρ = 0,3,5 in the D–T reaction and ρ = 5,5.5,6 in the D–3He reaction compared to that of the D–T reaction.

3 Conclusions

In this work, using zero-dimensional particles and power equations for D–T and D–3He reactions, the parameter neτET was first obtained in ideal conditions where the presence of impurities as well as of helium and proton ash was ignored and then its temperature dependence was shown for two states. Then, with regard to particle equations, the temperature dependence of this parameter was both calculated and shown again.

The results show that the burn conditions of the D–3He reaction required a higher temperature in comparison that of the D–T reaction. In addition, the burn condition of the D–3He reaction in comparison to that of the D–T reaction has a much more limited temperature range. Afterwards, using and substitding the energy confinement time and plasma parameter for the D–3He and D–T reactions in the equation neτET, the equation can be transformed to the ne, T plane. It is worth noting that the parameters of ST reactor were used for the D–3He reaction, which can be seen in Table 2.

Regarding the present progress in building reactors, it seems unlikely that D–T will be replaced by D–3He reaction as fuel for fusion reactors. Because D–3He reaction requires a much higher plasma temperature, density and neτET parameter than a comparable D–T reactor, it seems challenging and unlikely to consider D–3He reaction as a suitable fuel in a fusion reactor. Lack of a reliable and suitable source of 3He is another problem in D–3He reaction. The results show that the burn conditions of D–3He reaction required a higher temperature in comparison to those of D–T reaction. In addition, the burn condition of D–3He reaction in comparison to that of D–T reaction has a much more limited temperature range.


Corresponding author: Seyed Mohammad Motevalli, Department of Physics, Faculty of Science, University of Mazandaran, P.O. Box 47415-416, Babolsar, Iran, E-mail:

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Received: 2014-4-29
Accepted: 2015-1-7
Published Online: 2015-2-7
Published in Print: 2015-2-1

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