Question

Results of calculations for four different designs of a fusion reactor using D-D reaction are given below. Which of these is most promising based on Lawson criterion? (A) deuteron density \(=2.0 \times 10^{12} \mathrm{~cm}^{-3}\), confinement time \(=5.0 \times 10^{-3} \mathrm{~s}\) (B) deuteron density \(=8.0 \times 10^{14} \mathrm{~cm}^{-3}\), confinement time \(=9.0 \times 10^{-1} \mathrm{~s}\) (C) deuteron density \(=4.0 \times 10^{23} \mathrm{~cm}^{-3}\), confinement time \(=1.0 \times 10^{-11} \mathrm{~s}\) (D) deuteron density \(=1.0 \times 10^{24} \mathrm{~cm}^{-3}\), confinement time \(=4.0 \times 10^{-12} \mathrm{~s}\)

Short answer

Based on the Lawson criterion, Design B with a product of deuteron density and confinement time equal to \(7.2 \times 10^{14} \mathrm{~cm}^{-3} \cdot \mathrm{s}\) is the most promising fusion reactor design.

Step-by-step solution

Understanding the Lawson Criterion

The Lawson criterion provides a method for evaluating whether a fusion reactor can achieve net energy gain. It defines the product of the plasma density (deuterons in this case) and the energy confinement time that needs to be exceeded for a fusion reactor to work effectively. To identify the most promising reactor design based on Lawson criterion, calculate the product of the deuteron density and the confinement time for each option.

Calculating the Lawson Criterion for Design A

For design A, multiply the deuteron density by the confinement time:Design A: Lawson Criterion = Deuteron Density \times Confinement Time\[ = 2.0 \times 10^{12} \mathrm{~cm}^{-3} \times 5.0 \times 10^{-3} \mathrm{~s} \]\[ = 1.0 \times 10^{10} \mathrm{~cm}^{-3} \cdot \mathrm{s} \]

Calculating the Lawson Criterion for Design B

For design B, multiply the deuteron density by the confinement time:Design B: Lawson Criterion = Deuteron Density \times Confinement Time\[ = 8.0 \times 10^{14} \mathrm{~cm}^{-3} \times 9.0 \times 10^{-1} \mathrm{~s} \]\[ = 7.2 \times 10^{14} \mathrm{~cm}^{-3} \cdot \mathrm{s} \]

Calculating the Lawson Criterion for Design C

For design C, multiply the deuteron density by the confinement time:Design C: Lawson Criterion = Deuteron Density \times Confinement Time\[ = 4.0 \times 10^{23} \mathrm{~cm}^{-3} \times 1.0 \times 10^{-11} \mathrm{~s} \]\[ = 4.0 \times 10^{12} \mathrm{~cm}^{-3} \cdot \mathrm{s} \]

Calculating the Lawson Criterion for Design D

For design D, multiply the deuteron density by the confinement time:Design D: Lawson Criterion = Deuteron Density \times Confinement Time\[ = 1.0 \times 10^{24} \mathrm{~cm}^{-3} \times 4.0 \times 10^{-12} \mathrm{~s} \]\[ = 4.0 \times 10^{12} \mathrm{~cm}^{-3} \cdot \mathrm{s} \]

Comparing the Results

Compare the products obtained for each design to determine which is the most promising based on the Lawson criterion:Design A: \(1.0 \times 10^{10}\) Design B: \(7.2 \times 10^{14}\) Design C: \(4.0 \times 10^{12}\) Design D: \(4.0 \times 10^{12}\)It is clear that Design B with a product of \(7.2 \times 10^{14} \mathrm{~cm}^{-3} \cdot \mathrm{s}\) has the highest value, and thus is the most promising according to the Lawson criterion.

Key concepts

Fusion Reactor Design

The design of a fusion reactor is crucial for harnessing energy from nuclear fusion, the process that powers our sun. Fusion reactors aim to replicate this process on Earth by combining light nuclei like deuterium, or heavy isotopes of hydrogen, to form helium and release substantial amounts of energy.
In designing a viable fusion reactor, multiple factors must be considered. These factors include the choice of fuel, methods to achieve and maintain the extreme conditions for fusion, magnetic and inertial confinement systems to contain the hot plasma, and materials that can withstand the harsh environment inside the reactor.

Key Components of Fusion Reactor Design

To achieve fusion, reactors must have a method to heat the plasma to millions of degrees Celsius, a means to compress the plasma to a sufficient density, and a way to confine the plasma long enough for fusion to occur.

• Magnetic confinement uses powerful magnetic fields to contain the plasma in a toroidal shape, such as in tokamaks or stellarators.
• Inertial confinement uses lasers or ion beams to compress a small fuel pellet to achieve the necessary conditions.
• Plasma heating and stabilization are achieved through a combination of external heating methods and internal current drive.
• Materials technology is critical because the reactor must handle extreme heat and neutron flux without degrading.

The ultimate goal in fusion reactor design is to reach and surpass the break-even point where the energy produced by the fusion reaction is greater than the energy put into the system to maintain the fusion conditions, leading to net energy gain.

Net Energy Gain in Fusion

Net energy gain in fusion refers to the point where the amount of energy produced from the fusion process exceeds the energy used to initiate and sustain the fusion reaction. For a fusion power plant to be practical, it must produce more energy than it consumes, which is still a significant challenge.

Importance of Net Energy Gain

The Lawson criterion is central to understanding net energy gain. It sets out the minimum conditions that plasma must achieve to obtain a net output of energy. To satisfy this criterion, the product of the plasma density, the confinement time, and the temperature must exceed a specific threshold.

• Plasma density indicates how many fusionable particles are within a given volume.
• Confinement time represents how long the plasma can be held at fusion temperatures and density.
• Temperature must be high enough to overcome repulsive forces between nuclei, allowing them to fuse.

Achieving net energy gain is not just about hitting these parameters. The efficiency of heat conversion to electricity, thermal insulation, and minimizing energy losses are also critical. Current research in fusion technology is focused on improving each aspect to achieve a sustainable and economically viable net energy gain.

Plasma Density in Fusion

Plasma density is a key factor in achieving fusion and is one of the parameters of the Lawson criterion. In fusion reactors, the plasma consists of positively charged nuclei and negatively charged electrons, with density reflecting the number of particles per unit volume.

Understanding Plasma Density

High plasma density increases the likelihood of collision between fusion particles, which in turn increases the probability of fusion reactions. However, achieving a high plasma density is not simple. It requires sophisticated technology to confine a hot plasma without it coming into contact with the reactor walls and cooling down or damaging the equipment.
Plasma density is intricately linked with temperature and confinement time. These three parameters interact in a way that makes balancing them a delicate task. For instance, a very high density could shorten the confinement time if the plasma becomes unstable. Conversely, lower densities may require longer confinement times and higher temperatures to achieve net energy gain.
The designs presented in the exercise reflect various approaches to optimizing plasma density. Design B, which according to the Lawson criterion appears the most promising, strikes a balance by offering a high plasma density combined with a reasonably long confinement time, suggesting an improved likelihood of achieving sustainable fusion reactions within the reactor.