The transport of charged particles in a field reversed configuration (FRC) was previously conside... more The transport of charged particles in a field reversed configuration (FRC) was previously considered to be turbulent because it is much faster than classical predictions. Classical transport has mainly been developed for plasmas in which the gyroradii of particles are small compared to the scale lengths of the variation of the density and magnetic field. This assumption is quite inappropriate for an FRC where the magnetic field vanishes on a surface within the plasma. Classical theory has been extended to include large ion gyroradii. A classical loss-cone process is revealed that is consistent with the transport experiments in which the ion gyroradii were comparable in size to the plasma radius.
In an FRC the magnetic field vanishes on a null surface within the plasma. The usual assumptions ... more In an FRC the magnetic field vanishes on a null surface within the plasma. The usual assumptions of classical diffusion are invalid. The two main types of orbits are betatron orbits that circulate around the null surface and drift orbits. Cumulative small-angle scattering does not change the orbit type. However, large-angle scattering from neighboring particles can change the orbit topology: i.e., from betatron orbit to drift orbit(A. Qerushi, phParticle and Energy Transport in a Field Reversed Configuration), Ph.D. Thesis, University of Florida, Gainesville (2000). Containment requires that the orbit rotates in the diamagnetic direction. In a conventional FRC the nabla B × B drift orbit rotates in the counterdiamagnetic direction so that it would be lost on a fast time scale. The relatively infrequent large-angle scatterings from close collisions in all previous considerations of plasma are less important than the cumulative small-angle scatterings from distant particles. In the present case the close collisions result in a betatron orbit changing to a drift orbit which is not contained. This process is similar to the loss-cone phenomena and leads to a loss rate that is faster than conventional diffusion by an order of magnitude. This process provides an explanation of the experimental loss rate measurements of FRC's that are usually about a factor of 10 faster than predicted by classical diffusion.(M. Tuszewski, phNuclear Fusion), ph28, 2033 (1988) These results were previously attributed to anomalous transport.
This work has been part of a collaboration between the University of Florida and the University o... more This work has been part of a collaboration between the University of Florida and the University of California, Irvine, aimed at building a fusion reactor which is compact, environmentally friendly, and easy to maintain. The specific work of this dissertation concerns theoretical issues about equilibrium and particle transport in such a reactor, which is based on magnetic configurations known as Field Reversed Configurations (FRC). The equilibrium study shows how to obtain solutions for various physical quantities of interest in the case of fusion reactions with one or many types of ions. The main attribute that comes out of this study is the existence of mixed confinement states in a Colliding Beam Fusion Reactor (CBFR). In these mixed confinement states ions are confined magnetically while the electrons are confined electrostatically. A whole range of electric fields of different strengths can be accessed by tuning the externally applied magnetic field. A strong electric field that is confining for electrons can avoid their anomalous transport. The next part of this work concerns the effect of collisions on particle orbits in a CBFR. Simulations of particle orbits show that there are two main types of orbits: (1) betatron orbits, which can be thought of as sine waves propagating along a circle, and (2) drift orbits, which can be thought of as small circles rolling over larger circles. It is shown that large-angle collisions between ions can change a betatron orbit to a drift orbit. The direction of rotation of the drift orbit is in the diamagnetic direction in all cases where the electric field is confining for electrons and the E⃗xB⃗ drift dominates over the gradient drift B⃗x1B . This is an important finding for the ion transport in a CBFR. Simulations show also that small angle collisions between electrons and ions do not change the topology of betatron orbits, but only increase the amplitude of their radial oscillations with time. The last part of this work is related to the development of a formal diffusion theory that enables the calculation of the diffusion rates of betatron orbits and drift orbits due to small angle collisions. This diffusion theory is based on test particle methods used in kinetic theory of plasmas. It has the merit of being applicable to particle orbits of any size and rapidly varying magnetic fields that pass through zero.
We present a two-dimensional equilibrium model for FRCs with rotation. This is a multi-fluid trea... more We present a two-dimensional equilibrium model for FRCs with rotation. This is a multi-fluid treatment that uses rigid rotors (B. Spivey phConfinement of Non-adiabatic Vlasov-Maxwellian Plasmas), PhD Thesis, University of California, Irvine, 1992., the only drifted Maxwellian solutions of Vlasov's equation for systems with cylindrical symmetry. A complete description of equilibria for FRCs with rotation is provided by a Grad-Shafranov-like equation for the flux function. We solve this fundamental equation with Green's function methods. The original partial differential equation is converted to an equivalent integral equation involving Green's function. The integral equation is solvable by iteration, i.e., an algorithm exists that converges to a solution of the fundamental equation starting from any initial approximation. All computations are done with phMathematica 4.0 on a PC: the flux function is obtained with high accuracy in several hours of computing time. In addition to rapid convergence for a complex non-linear problem the Green's function method guaranties that the boundary conditions are satisfied for every iteration.
The recirculating power for fusion reactors has previously been calculated and published(Todd H. ... more The recirculating power for fusion reactors has previously been calculated and published(Todd H. Rider phPhys. Plasmas) 4, 1039, (1997); also 1998 APS/DPP General Meeting, invited paper R8M3-7, New Orleans Nov. 19, (1998). for many types of reactors and fuels. The calculations are generic. They involve assuming distribution functions of the form [ f(v)= nK . exp[-(v-v_0)^2v_t^2]+exp[-(v+v_0)^2v_t^2]. and determining from the Fokker-Planck equation the power Pc required to maintain f(v). Here v=√v_x^2+v_y^2+v_z^2 n, K, v0 and vt are adjustable parameters. For example results were given for p-B^11 of P_cP_f=9100, 350, 52, and 33 depending on the choice of the parameters. Pf is the fusion power. We have carried out the calculation for a Colliding Beam FusionReactor(N. Rostoker and M. Binderbauer phJ. Plasma Phys.) 56, 451, (1996); also APS/DPP General Meeting invited paper R8M3-4, (1998). with p-B^11 fuel and obtained P_cP_f=0.5. A distribution function like f(v) does not apply to the CBFR, nor to any reactor concept that we are aware of.
The Colliding Beam Fusion Reactor Space Propulsion System, CBFR-SPS, is an aneutronic, magnetic-f... more The Colliding Beam Fusion Reactor Space Propulsion System, CBFR-SPS, is an aneutronic, magnetic-field-reversed configuration, fueled by an energetic-ion mixture of hydrogen and boron11 (H-B11). Particle confinement and transport in the CBFR-SPS are classical, hence the system is scaleable. Fusion products are helium ions, α-particles, expelled axially out of the system. α-particles flowing in one direction are decelerated and their energy recovered to ``power'' the system; particles expelled in the opposite direction provide thrust. Since the fusion products are charged particles, the system does not require the use of a massive-radiation shield. This paper describes a 100 MW CBFR-SPS design, including estimates for the propulsion-system parameters and masses. Specific emphasis is placed on the design of a closed-cycle, Brayton-heat engine, consisting of heat-exchangers, turbo-alternator, compressor, and finned radiators.
ABSTRACT A system and method for containing plasma and forming a Field Reversed Configuration (FR... more ABSTRACT A system and method for containing plasma and forming a Field Reversed Configuration (FRC) magnetic topology are described in which plasma ions are contained magnetically in stable, non-adiabatic orbits in the FRC. Further, the electrons are contained electrostatically in a deep energy well, created by tuning an externally applied magnetic field. The simultaneous electrostatic confinement of electrons and magnetic confinement of ions avoids anomalous transport and facilitates classical containment of both electrons and ions. In this configuration, ions and electrons may have adequate density and temperature so that upon collisions they are fused together by nuclear force, thus releasing fusion energy. Moreover, the fusion fuel plasmas that can be used with the present confinement system and method are not limited to neutronic fuels only, but also advantageously include advanced fuels.
A two-dimensional equilibrium model for field reversed configurations (FRCs) with rotation is pre... more A two-dimensional equilibrium model for field reversed configurations (FRCs) with rotation is presented. In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] it was shown that a complete description of equilibria for FRCs with rotation is provided by a generalized Grad-Shafranov equation for the plasma flux function. In this paper it is shown how to solve that fundamental equation for the case of two space dimensions and one type of ion. Periodic boundary conditions and a Green's function are used to convert the original differential equation to an equivalent integral equation. The integral equation is solved by iteration. An iteration algorithm is described which converges to a solution of the generalized Grad-Shafranov equation starting with a one-dimensional trial function. Analytic one-dimensional solutions are shown to be a limiting case of two-dimensional solutions when the applied magnetic field is constant. In addition to rapid convergence for a complex nonlinear problem, the Green's function method guarantees that the boundary conditions are satisfied in every iteration.
In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] a generalized Grad... more In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] a generalized Grad-Shafranov equation for the plasma flux function was derived which provides a complete description of equilibria of field reversed configurations with rotation. In this paper this fundamental equation is solved for two space dimensions and many ion species. The following fusion fuels are considered: D-T, D-He3, and p-B11. Using periodic boundary conditions the original differential equation is converted to an equivalent integral equation which involves a Green's function. The integral equation is solved by iteration. Approximate solutions are found for all the fusion fuels considered using a two-dimensional equilibrium model for one type of ion [A. Qerushi and N. Rostoker, Phys. Plasmas 9, 5001 (2002)]. They are used as starting trial functions of the iterations. They turn out to be so close to the real solutions that only a few iterations are needed.
In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] it was shown that ... more In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] it was shown that a complete description of equilibria of field reversed configurations with rotation can be obtained by solving a generalized Grad-Shafranov equation for the flux function. In this paper we show how to solve this equation in the case of one space dimension and many ion species. The following fusion fuels are considered: D-T, D-He3, and p-B11. Using a Green's function the generalized Grad-Shafranov equation is converted to an equivalent integral equation. The integral equation can be solved by iteration. Approximate analytic solutions for a plasma with many ion species are found. They are used as starting trial functions of the iterations. They turn out to be so close to the true solutions that only a few iterations are needed.
A two-dimensional equilibrium model for field reversed configurations (FRCs) with rotation is pre... more A two-dimensional equilibrium model for field reversed configurations (FRCs) with rotation is presented. In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] it was shown that a complete description of equilibria for FRCs with rotation is provided by a generalized Grad-Shafranov equation for the plasma flux function. In this paper it is shown how to solve that fundamental equation for the case of two space dimensions and one type of ion. Periodic boundary conditions and a Green's function are used to convert the original differential equation to an equivalent integral equation. The integral equation is solved by iteration. An iteration algorithm is described which converges to a solution of the generalized Grad-Shafranov equation starting with a one-dimensional trial function. Analytic one-dimensional solutions are shown to be a limiting case of two-dimensional solutions when the applied magnetic field is constant. In addition to rapid convergence for a complex nonlinear problem, the Green's function method guarantees that the boundary conditions are satisfied in every iteration.
The recirculating power for virtually all types of fusion reactors has previously been calculated... more The recirculating power for virtually all types of fusion reactors has previously been calculated [1] with the Fokker–Planck equation. The reactors involve non-Maxwellian plasmas. The calculations are generic in that they do not relate to specific confinement devices. In all cases except for a Tokamak with D–T fuel the recirculating power was found to exceed the fusion power by a large factor. In this paper we criticize the generality claimed for this calculation. The ratio of circulating power to fusion power is calculated for the Colliding Beam Reactor with fuels D–T, D–He3 and p–B11. The results are respectively, 0.070, 0.141 and 0.493.
The transport of charged particles in a field reversed configuration (FRC) was previously conside... more The transport of charged particles in a field reversed configuration (FRC) was previously considered to be turbulent because it is much faster than classical predictions. Classical transport has mainly been developed for plasmas in which the gyroradii of particles are small compared to the scale lengths of the variation of the density and magnetic field. This assumption is quite inappropriate for an FRC where the magnetic field vanishes on a surface within the plasma. Classical theory has been extended to include large ion gyroradii. A classical loss-cone process is revealed that is consistent with the transport experiments in which the ion gyroradii were comparable in size to the plasma radius.
In an FRC the magnetic field vanishes on a null surface within the plasma. The usual assumptions ... more In an FRC the magnetic field vanishes on a null surface within the plasma. The usual assumptions of classical diffusion are invalid. The two main types of orbits are betatron orbits that circulate around the null surface and drift orbits. Cumulative small-angle scattering does not change the orbit type. However, large-angle scattering from neighboring particles can change the orbit topology: i.e., from betatron orbit to drift orbit(A. Qerushi, phParticle and Energy Transport in a Field Reversed Configuration), Ph.D. Thesis, University of Florida, Gainesville (2000). Containment requires that the orbit rotates in the diamagnetic direction. In a conventional FRC the nabla B × B drift orbit rotates in the counterdiamagnetic direction so that it would be lost on a fast time scale. The relatively infrequent large-angle scatterings from close collisions in all previous considerations of plasma are less important than the cumulative small-angle scatterings from distant particles. In the present case the close collisions result in a betatron orbit changing to a drift orbit which is not contained. This process is similar to the loss-cone phenomena and leads to a loss rate that is faster than conventional diffusion by an order of magnitude. This process provides an explanation of the experimental loss rate measurements of FRC's that are usually about a factor of 10 faster than predicted by classical diffusion.(M. Tuszewski, phNuclear Fusion), ph28, 2033 (1988) These results were previously attributed to anomalous transport.
This work has been part of a collaboration between the University of Florida and the University o... more This work has been part of a collaboration between the University of Florida and the University of California, Irvine, aimed at building a fusion reactor which is compact, environmentally friendly, and easy to maintain. The specific work of this dissertation concerns theoretical issues about equilibrium and particle transport in such a reactor, which is based on magnetic configurations known as Field Reversed Configurations (FRC). The equilibrium study shows how to obtain solutions for various physical quantities of interest in the case of fusion reactions with one or many types of ions. The main attribute that comes out of this study is the existence of mixed confinement states in a Colliding Beam Fusion Reactor (CBFR). In these mixed confinement states ions are confined magnetically while the electrons are confined electrostatically. A whole range of electric fields of different strengths can be accessed by tuning the externally applied magnetic field. A strong electric field that is confining for electrons can avoid their anomalous transport. The next part of this work concerns the effect of collisions on particle orbits in a CBFR. Simulations of particle orbits show that there are two main types of orbits: (1) betatron orbits, which can be thought of as sine waves propagating along a circle, and (2) drift orbits, which can be thought of as small circles rolling over larger circles. It is shown that large-angle collisions between ions can change a betatron orbit to a drift orbit. The direction of rotation of the drift orbit is in the diamagnetic direction in all cases where the electric field is confining for electrons and the E⃗xB⃗ drift dominates over the gradient drift B⃗x1B . This is an important finding for the ion transport in a CBFR. Simulations show also that small angle collisions between electrons and ions do not change the topology of betatron orbits, but only increase the amplitude of their radial oscillations with time. The last part of this work is related to the development of a formal diffusion theory that enables the calculation of the diffusion rates of betatron orbits and drift orbits due to small angle collisions. This diffusion theory is based on test particle methods used in kinetic theory of plasmas. It has the merit of being applicable to particle orbits of any size and rapidly varying magnetic fields that pass through zero.
We present a two-dimensional equilibrium model for FRCs with rotation. This is a multi-fluid trea... more We present a two-dimensional equilibrium model for FRCs with rotation. This is a multi-fluid treatment that uses rigid rotors (B. Spivey phConfinement of Non-adiabatic Vlasov-Maxwellian Plasmas), PhD Thesis, University of California, Irvine, 1992., the only drifted Maxwellian solutions of Vlasov's equation for systems with cylindrical symmetry. A complete description of equilibria for FRCs with rotation is provided by a Grad-Shafranov-like equation for the flux function. We solve this fundamental equation with Green's function methods. The original partial differential equation is converted to an equivalent integral equation involving Green's function. The integral equation is solvable by iteration, i.e., an algorithm exists that converges to a solution of the fundamental equation starting from any initial approximation. All computations are done with phMathematica 4.0 on a PC: the flux function is obtained with high accuracy in several hours of computing time. In addition to rapid convergence for a complex non-linear problem the Green's function method guaranties that the boundary conditions are satisfied for every iteration.
The recirculating power for fusion reactors has previously been calculated and published(Todd H. ... more The recirculating power for fusion reactors has previously been calculated and published(Todd H. Rider phPhys. Plasmas) 4, 1039, (1997); also 1998 APS/DPP General Meeting, invited paper R8M3-7, New Orleans Nov. 19, (1998). for many types of reactors and fuels. The calculations are generic. They involve assuming distribution functions of the form [ f(v)= nK . exp[-(v-v_0)^2v_t^2]+exp[-(v+v_0)^2v_t^2]. and determining from the Fokker-Planck equation the power Pc required to maintain f(v). Here v=√v_x^2+v_y^2+v_z^2 n, K, v0 and vt are adjustable parameters. For example results were given for p-B^11 of P_cP_f=9100, 350, 52, and 33 depending on the choice of the parameters. Pf is the fusion power. We have carried out the calculation for a Colliding Beam FusionReactor(N. Rostoker and M. Binderbauer phJ. Plasma Phys.) 56, 451, (1996); also APS/DPP General Meeting invited paper R8M3-4, (1998). with p-B^11 fuel and obtained P_cP_f=0.5. A distribution function like f(v) does not apply to the CBFR, nor to any reactor concept that we are aware of.
The Colliding Beam Fusion Reactor Space Propulsion System, CBFR-SPS, is an aneutronic, magnetic-f... more The Colliding Beam Fusion Reactor Space Propulsion System, CBFR-SPS, is an aneutronic, magnetic-field-reversed configuration, fueled by an energetic-ion mixture of hydrogen and boron11 (H-B11). Particle confinement and transport in the CBFR-SPS are classical, hence the system is scaleable. Fusion products are helium ions, α-particles, expelled axially out of the system. α-particles flowing in one direction are decelerated and their energy recovered to ``power'' the system; particles expelled in the opposite direction provide thrust. Since the fusion products are charged particles, the system does not require the use of a massive-radiation shield. This paper describes a 100 MW CBFR-SPS design, including estimates for the propulsion-system parameters and masses. Specific emphasis is placed on the design of a closed-cycle, Brayton-heat engine, consisting of heat-exchangers, turbo-alternator, compressor, and finned radiators.
ABSTRACT A system and method for containing plasma and forming a Field Reversed Configuration (FR... more ABSTRACT A system and method for containing plasma and forming a Field Reversed Configuration (FRC) magnetic topology are described in which plasma ions are contained magnetically in stable, non-adiabatic orbits in the FRC. Further, the electrons are contained electrostatically in a deep energy well, created by tuning an externally applied magnetic field. The simultaneous electrostatic confinement of electrons and magnetic confinement of ions avoids anomalous transport and facilitates classical containment of both electrons and ions. In this configuration, ions and electrons may have adequate density and temperature so that upon collisions they are fused together by nuclear force, thus releasing fusion energy. Moreover, the fusion fuel plasmas that can be used with the present confinement system and method are not limited to neutronic fuels only, but also advantageously include advanced fuels.
A two-dimensional equilibrium model for field reversed configurations (FRCs) with rotation is pre... more A two-dimensional equilibrium model for field reversed configurations (FRCs) with rotation is presented. In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] it was shown that a complete description of equilibria for FRCs with rotation is provided by a generalized Grad-Shafranov equation for the plasma flux function. In this paper it is shown how to solve that fundamental equation for the case of two space dimensions and one type of ion. Periodic boundary conditions and a Green's function are used to convert the original differential equation to an equivalent integral equation. The integral equation is solved by iteration. An iteration algorithm is described which converges to a solution of the generalized Grad-Shafranov equation starting with a one-dimensional trial function. Analytic one-dimensional solutions are shown to be a limiting case of two-dimensional solutions when the applied magnetic field is constant. In addition to rapid convergence for a complex nonlinear problem, the Green's function method guarantees that the boundary conditions are satisfied in every iteration.
In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] a generalized Grad... more In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] a generalized Grad-Shafranov equation for the plasma flux function was derived which provides a complete description of equilibria of field reversed configurations with rotation. In this paper this fundamental equation is solved for two space dimensions and many ion species. The following fusion fuels are considered: D-T, D-He3, and p-B11. Using periodic boundary conditions the original differential equation is converted to an equivalent integral equation which involves a Green's function. The integral equation is solved by iteration. Approximate solutions are found for all the fusion fuels considered using a two-dimensional equilibrium model for one type of ion [A. Qerushi and N. Rostoker, Phys. Plasmas 9, 5001 (2002)]. They are used as starting trial functions of the iterations. They turn out to be so close to the real solutions that only a few iterations are needed.
In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] it was shown that ... more In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] it was shown that a complete description of equilibria of field reversed configurations with rotation can be obtained by solving a generalized Grad-Shafranov equation for the flux function. In this paper we show how to solve this equation in the case of one space dimension and many ion species. The following fusion fuels are considered: D-T, D-He3, and p-B11. Using a Green's function the generalized Grad-Shafranov equation is converted to an equivalent integral equation. The integral equation can be solved by iteration. Approximate analytic solutions for a plasma with many ion species are found. They are used as starting trial functions of the iterations. They turn out to be so close to the true solutions that only a few iterations are needed.
A two-dimensional equilibrium model for field reversed configurations (FRCs) with rotation is pre... more A two-dimensional equilibrium model for field reversed configurations (FRCs) with rotation is presented. In a previous paper [N. Rostoker and A. Qerushi, Phys. Plasmas 9, 3057 (2002)] it was shown that a complete description of equilibria for FRCs with rotation is provided by a generalized Grad-Shafranov equation for the plasma flux function. In this paper it is shown how to solve that fundamental equation for the case of two space dimensions and one type of ion. Periodic boundary conditions and a Green's function are used to convert the original differential equation to an equivalent integral equation. The integral equation is solved by iteration. An iteration algorithm is described which converges to a solution of the generalized Grad-Shafranov equation starting with a one-dimensional trial function. Analytic one-dimensional solutions are shown to be a limiting case of two-dimensional solutions when the applied magnetic field is constant. In addition to rapid convergence for a complex nonlinear problem, the Green's function method guarantees that the boundary conditions are satisfied in every iteration.
The recirculating power for virtually all types of fusion reactors has previously been calculated... more The recirculating power for virtually all types of fusion reactors has previously been calculated [1] with the Fokker–Planck equation. The reactors involve non-Maxwellian plasmas. The calculations are generic in that they do not relate to specific confinement devices. In all cases except for a Tokamak with D–T fuel the recirculating power was found to exceed the fusion power by a large factor. In this paper we criticize the generality claimed for this calculation. The ratio of circulating power to fusion power is calculated for the Colliding Beam Reactor with fuels D–T, D–He3 and p–B11. The results are respectively, 0.070, 0.141 and 0.493.
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