Using magnetic levitation to produce cryogenic targets for inertial fusion energy: experiment and theory 6 1 0 2 D. Chatain and V. S. Nikolayev1 n a CEA/DSM/SBT/ESEME, CEA Grenoble, 17, rue des Martyrs, 38054, Grenoble J Cedex 9, France 5 2 ] t Abstract f o s . We present experimental and theoretical studies of magnetic levitation of hydrogen t a gas bubble surrounded by liquid hydrogen confined in a semi-transparent spherical m shell of 3 mm internal diameter. Such shells are used as targets for the Inertial Con- - d finement Fusion (ICF), for which a homogeneous (within a few per-cent) layer of a n hydrogen isotope should be deposited on the internal walls of the shells. The grav- o ity does not allow the hydrogen layer thickness to be homogeneous. To compensate c [ this gravity effect, we have used a non-homogeneous magnetic field created by a 10 T superconductive solenoid. Our experiments show that the magnetic levitation 1 v homogenizes the thickness of liquid hydrogen layer. However, the variation of the 5 layer thickness is very difficult to measure experimentally. Our theoretical model 7 allows the exact shape of the layer to be predicted. The model takes into account 2 6 the surface tension, gravity, van der Waals, and magnetic forces. The numerical 0 calculation shows that the homogeneity of the layer thickness is satisfactory for the . 2 ICF purposes. 0 6 1 Key words: ICF, IFE target, magnetic levitation : v i X r a 1 Introduction Several concepts have been proposed for the design of a commercial power plant based on Inertial Fusion Energy production [1,2]. Targets are direct or indirect drive targets but must be at cryogenic temperature [3]. They must be injected in the vacuum chamber of the reactor at a rate of about 5 Hz and a speed depending on the temperature and the residual pressure of the vacuum vessel [4]. The targets are then tracked and hit on-the-fly with laser or heavy ion beams [5]. 1 Mailing address: ESEME-CEA, Institut de Chimie de la Mati`ere Condens´ee de Bordeaux, CNRS, Avenue du Dr. Schweitzer, 33608 Pessac Cedex, France; e-mail: [email protected] Preprint submitted to Elsevier 22 February 2016 The targets are hollow spherical shells made of beryllium or polystyrene. Their diameter ranges from 2 to 5 mm. Their internal wall must be covered with a solid layer of deuterium or a mixture of deuterium and tritium of several hundred microns in thickness. The thickness of the layer must be uniform within a few percent. If tritium is present in the mixture, the beta energy produced by tritium naturally drives the solid to a uniform thickness layer covering the internal walls of the shell [6,7]. If tritium is not present in the liquid layer, it stays at the bottom because of gravity. In this paper, we describe how, by using the diamagnetic properties of the hydrogen, we can compensate this gravity effect and obtain a homogenous thickness of liquid layer inside the sphere before freezing it. 2 Forces that act on hydrogen molecules In order to obtain the homogeneous thickness of the liquid layer on the inner walls of a hollow spherical shell, one needs to satisfy simultaneously two conditions: • the shape of the gas bubble inside the liquid should be spherical, • the gas bubble should levitate in the middle of the shell. We show now how the various forces influence the satisfaction of these conditions. 2.1 Surface tension The force of the surface tension tends to minimize the interface area. Therefore, the surface tension helps to maintain the spherical shape of the gas bubble. Obviously, we need to look for the conditions where the value σ of the surface tension is large. As a matter of fact, σ is the decreasing function of the temperature T and goes to zero at the critical temperature T , which c is about 33 K for hydrogen. The working temperature should thus be at least several degrees smaller than T . c The contribution of the surface tension (i.e. the Laplace pressure) is inversely proportional to the radius of curvature of the interface. For the thicker liquid layer the radius of the gas bubble is smaller and the Laplace pressure is thus larger. Therefore, the thickness of the thicker liquid layer is a priori more homogeneous (provided that the gas bubble is levitated in the middle of the shell) than the thickness of the thinner liquid layer. As a consequence, we need to analyze the homogeneity only for the thinnest layer under consideration, which is 200 µm. If for this case the homogeneity 1% criterion is satisfied, it will be satisfied for all larger thicknesses. 2.2 Van der Waals forces Since the hydrogen completely wets the solid shell (zero contact angle), the van der Waals force manifests itself as an attraction between the hydrogen molecules and the solid wall [12]. It thus tends to create the layer of the densest (liquid) phase at the shell wall, leaving the less 2 dense phase (gas) in the middle of the shell. However, the value of this force is very small in comparison with the surface tension. While the van der Waals force influences strongly [12] the shape of the layers of microscopic (of the order of 1 µm) thickness, we do not expect a strong effect for the case of the thick liquid layers (average thickness larger than 200 µm), which we analyze in this report. We carry out all our calculations for the non-retarded van der Waals interactions instead of the more suitable (because of the large layer thickness) retarded expression. The reason is that the retarded interactions are weaker [12] and would result in even smaller contribution. 2.3 Gravity and magnetic forces: magnetic levitation The gravitational force per unit volume f = ρg is proportional to the mass density ρ of g hydrogen, g being the gravitational acceleration. Since the liquid mass density ρ is larger than L ~ the gas mass density ρ , the resulting Archimedes force ∆f = (ρ −ρ )~g that acts on the gas G g G L bubble tends to increase the thickness of the liquid layer on the bottom of the shell by lifting the bubble upwards. Gravitation thus needs to be compensated. The gravity compensation by means of the static magnetic field is based on the expression for ~ 2 the magnetic force per unit volume fm = χ∇(B )/2µ0 that acts on a body with the magnetic susceptibility χ, where B is the magnetic induction that would be created by the same solenoid in the free space and µ0 is the magnetic permeability of free space. The magnetic susceptibility is proportional to the mass density ρ, χ/ρ = α (see Table 1). The magnetic force that acts on Table 1 Parameters of hydrogen at T = 20 K [11]. Description Notation Value Units −8 3 Magnetic susceptibility/mass density α −2.51·10 m /kg Surface tension σ 0.002 N/m 3 Mass density of liquid phase ρ 71.41 kg/m L 3 Mass density of gas phase ρ 1.19 kg/m G the bubble is thus (see Appendix) ~ 2 ∆fm = (ρG −ρL)α∇(B )/2µ0 (1) ~ ~ ~ and the condition of the bubble levitation is ∆f = ∆f +∆f = 0. m g 2 ~ Note that ∇(B )/2 = B dB dz along the solenoid axis (axis z) and the curve ∆f(z) exhibits z z a maximum (see Fig. 1) at some value of z = z (I) that is almost independent of the solenoid m current I. All magnetic field calculations reported in this article were performed with the code BOBOZ translated to the C programming language. As an additional input to this code, the solenoid current in amperes multiplied by the number of coils is required. 3 0,2 I +1% min 0 2m/s) -0,2 Imin ρ )(g -0,4 ρ-l I -1% +f)/(g -0,6 min m-0,8 (f z z (I ) -1 1 m min -1,2 -10 -5 0 5 10 z/R Fig. 1. The effective force per unit mass that acts on a small gas bubble in presence of the magnetic fieldofthesolenoidversusz-coordinatealongthesolenoidaxiscalculated fortreevaluesofthesolenoid current: I , I ·1.01, and I ·0.99. min min min Exact gravity compensation |α| dB z g = B (2) z µ0 dz can be achieved when I ≥ I , where I is a minimum current at which the exact com- min min pensation is possible at all. In the experiments (see sec. 3), the current I is 60 A. In our min calculations we choose the point z (I ) as a zero reference point. Its position is 8.502 cm m min 6 abovethesolenoidcenter.Accordingtoourcalculations,I correspondsto1.8300·10 ampere- min coils. The unknown (the solenoid documentation is not available) number of coils is obtained by division of this number by I . min Fig. 1 shows that when I < I , no compensation is possible at all, the force is non-zero min everywhere. When I > Imin, there are two points of compensation z1 and z2 such as z1 < zm < z2. However, it can be shown that levitation in z2 is not stable, so that the compensation can be achieved in only one point z1 < zm. Everywhere else, the hydrogen molecules are exposed to the 2 residual acceleration~γ = |α|/(2µ0) ∇B −~g the vertical (z) and radial (r) components of which are shown in Fig. 2. As we can see on these graphs, the radial component of the acceleration 5 5 -0.1 -0.05 4 4 0 3 3 0 2 2 1 -0.1 1 0.025 H, H, 0 mm 0 mm -1 -0.2 -1 0 -2 -0.3 -0.3 -2 -3 -3 -0.05 -4 -4 -0.1 -0.4 -0.4 -5 -5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 Radial distance, mm Radial distance, mm (a) (b) Fig. 2. The iso-acceleration curves for (a) γ and (b) γ . The position of the sphere is indicated by the z r −2 dotted lines. The acceleration values are given in m s . in the vicinity of the center of the sphere is more important than the axial component. Inside 4 −2 a 3 mm diameter sphere, the maximum radial acceleration is about 0.125 m s , while the vertical acceleration is ten times lower. The radial acceleration is directed toward the vertical axis. This means that the interface can deviate from the spherical shape because it follows the variation of ~γ. It is thus important to know if the homogeneous thickness of the liquid layer can in principle be achieved with the magnetic field created by the superconductive solenoid. 3 Experimental 3.1 Test facility The test facility (Figs. 3) consists of a superconductive solenoid immersed into a helium bath. The vacuum vessel containing the sphere is introduced into the core of the solenoid. The sphere Light Camera Vacuum vessel Hydrogen Hollow sphere Light guide Camera endoscope Heater Thermometer Vacuum vessel Thermal valve with its heater Helium bath (T=2.2K) Hydrogen capillary Sphere Coil Indium seal Thermal resistance Thermal valve Hydrogen capillary (a) (b) Fig. 3. (a) A scheme of the cryostat. (b) A photograph of the bottom of the cryostat showing the sphere. is illuminated from the top of the cryostat with a light guide. Observation is performed with an endoscope and a CCD camera. The bottom of the cryostat with the sphere is shown in Fig. 3b. The sphere center is placed at 15 mm from the top of the solenoid i.e. into the calculated point z (I ). The hydrogen is introduced into the sphere by capillary. The characteristics of the m min solenoid are listed in Table 2. The following operations are performed in order to condense and levitate the hydrogen in the sphere: −6 • The vacuum vessel is pumped out to about 10 mb and the capillary is pumped out to about 0.1 mb. • The solenoid and the vacuum vessel are cooled down to 2.2 K. • The thermal valve is heated (a power of 300 mW is necessary) • The sphere is heated to about 20 K. • Hydrogen is slowly introduced into the capillary. 5 Table 2 Parameters of the superconductive solenoid. Description Value Units Inner radius 48 mm Outer radius 93 mm Total height 20 cm B at 4.2 K and I = 53 A 8 T B at 2.17 K and I = 67 A 10 T Critical current at 2.17 K 72 A D=3.5mm D=3 mm Glue 3 mm D=0.6 mm D=0.2mm D=1.5 mm 5 mm D=2.3 mm (a) (b) Fig. 4. (a) Design of the transparent hollow sphere. (b) A photograph of the sphere. • When a sufficient quantity is condensed inside the sphere, the heater of the valve is cut so that an ice plug clogs up the capillary. • The current in the solenoid is increased to about 60 A to compensate the gravity. • The temperature of the sphere can be decreased below the triple point (13 K) to freeze the liquid. 3.2 Sphere The sphere was made of Plexiglas (PMMA) machined in two hemispheres according to the design shown in Figs. 4. The two hemispheres were glued with Epoxy 501 stick. 3.3 Effect of the magnetic field As one can see in Figs. 5, it was possible to condense the hydrogen in the sphere. The optical imperfections of the sphere did not allow a high image quality to be obtained. Consequently, a precise measurement of the homogeneity of the layer thickness was not possible. Nevertheless, 6 we can see in Fig. 5d that the gas-liquid interface deformation due to the residual gravity force does not seem to be large. The figures below show the effect of the magnetic field on the shape of the vapor bubble. (a) (b) (c) (d) Vapor Liquid Fig. 5. The sphere half filled with liquid hydrogen at a temperature close to the triple point (14 K)for the different values of the current I in the solenoid that correspond to the different amplitudes of the magnetic field: (a) I = 0, (b) I = 30 A, (c) I = 50 A, (d) I = 60 A. The gravity is completely compensated by the magnetic force. The gas bubble is stable and well centered. 3.4 Effect of the liquid/vapor volume ratio The images in Figs. 6 show the effect of the liquid/gas volume ratio for a gravity completely compensated by the magnetic field. This ratio controls the thickness of the liquid layer. (a) (b) (c) (d) Fig.6.SameasinFig. 5dforthegravity compensation fieldandfor thedifferentliquidlayer thickness: (a) 520 µm, (b) 350 µm, (c) 200 µm, (d) It becomes difficult to measure the liquid layer thickness. 4 Numerical modeling 4.1 Mathematical formulation The equilibrium shape of the interface can be found by two different approaches. One of them consists in direct numerical minimization of the free energy of the system. In this work we adopt another, variational approach, in which the minimization itself is performed analytically. It results in a variational equation that should be solved numerically to obtain the interface shape. The general form of this variational equation is the Laplace equation Kσ = ∆p, (3) 7 where K is the local curvature of the interface, and ∆p is the difference between the forces per unit area that act on the interface from the liquid and gas sides. There are several contributions to ∆p: ∆p = ∆p +∆p +∆p +λ (4) g m w that correspond to gravitation (∆p ), magnetic field (∆p ), and the vander Waals force (∆p ). g m w The constant λ is a Lagrange multiplier that appears as a result of the constrained gas volume V . λ can be viewed otherwise as an unknown a priori difference of pressures inside the liquid G and gas phases. The mathematical expression for the curvature K depends on the choice of the reference system and the independent variable. It is convenient to use the cylindrical (r,z) coordinate system because the solenoid (z) axis is vertical and ∆p = ∆p(r,z) is thus cylindrically symmetric. In this reference system, the interface is fully defined by its half-contour for which r > 0. None of the variables r,z can be chosen as independent because for the closed interface contour (at least) two values of z exists for each value of r and vise-versa. We choose as an independent variable the curvilinear coordinate l that varies along the interface contour counter-clockwise with l = 0 at thepoint onthe symmetry axis where r = 0. Using this parameterization, z = z(l) and r = r(l). Since l measures the running length along the interface contour, the following equation is valid ′2 ′2 z +r = 1, (5) where prime means the derivative d/dl. The expression for the local curvature then takes the form ′ ′′ ′′ ′ ′ K = r z −r z +z /r. (6) By introducing an auxiliary function u(l), one can reduce (3) (with K given by (6)) to the set of the first-order ordinary differential equations: ′ u = ∆p(r,z)/σ −sinu/r  r′ = cosu (7) ′ z = sinu  The physical meaning of the variable u can be found out by dividing two last equations: u is the angle between the r axis and the tangent to the interface contour. The gas bubble volume V is fixed. This condition allows λ to be determined from the equation G L 2 V = π r sinu dl, (8) G Z0 8 where r = r(l) and u = u(l) are the solutions of the set (7) and L is the unknown a priori half-length of the interface contour. Fourboundaryconditions fortheset (7)shouldbespecified atthepointsl = 0andl = L.Three of them serve to determine the integration constants in (7) and the fourth serves to determine L. There are two possible types of the boundary conditions. The first type corresponds to the case of the continuous liquid layer [12] and reflects the symmetry of the contour: u(0) = 0,  ru((0L))==0π,, (9) r(L) = 0. The boundary conditions of the second type should be specified when the liquid layer is discon- tinuous whenthevander Waalsforcesareneglected inthecalculationandthepoint l = Listhe triple (gas-liquid-wall) contact point. Since the contact angle is zero, the boundary conditions take the form u(0) = 0,  rr((0L))==0R, sinu(L), (10) z(L) = −Rcosu(L), where R is the internal radius of the shell. The mathematical problem is now complete. However, it is difficult to solve because of the moving boundary conditions specified at the unknown upper boundary L. This problem can be reduced to the simpler problem with the fixed boundary conditions by the following mathemat- ical trick. We introduce new independent variable ξ = l/L and two more dependent variables L and λ. The set (7) then reduces to ′ u = L∆p(r,z)−sinu/r  ′ zr′ ==LL csionsuu (11) ′ L = 0 λ′ = 0 where u,r,z,L,λ are supposed to be the functions of ξ, prime means now the derivatives d/dξ, and from now on we express r, z, and L in the units of R and ∆p in the units σ/R. Five 9 unknown integration constants for these five equations should be found from the condition 1 2 V = πL r sinu dξ, (12) G Z0 and from the four fixed boundary conditions (specified at ξ = 0,1) for the problem of the first type (continuous liquid layer) u(0) = 0,  ru((01))==0π,, (13) r(1) = 0, or of the second type (discontinuous liquid layer, i.e. direct wall-gas contact) u(0) = 0,  rr((01)) == 0si,nu(1), (14) z(1) = −cosu(1). The set of equations (4,11,12) with the boundary conditions (13) or (14) provide two fully defined fixed boundary mathematical problems if the functional forms of ∆p (r,z), ∆p (r,z), m g and ∆p (r,z) are known. The magnetic induction B that enters the first of them (A.5) is w calculated using the code BOBOZ. The second is given by ∆p (r,z) = Bo·z, where Bo is the g non-dimensional Bond number 2 Bo = (ρ −ρ )gR /σ. (15) L V The expression for the van der Waals contribution was calculated in [12]. For the non-retarded interaction 3 2 −3 −3 ∆p (r,z) = C [R (R −d) −(1−d) ], w w e e 2 2 where d = r +z , R is the external shell radius (1.75 mm) expressed in the units R and e 2 4π b N ρ C = (ρ −ρ ) HS A S (16) w 3 L V σR2m m S H is the non-dimensional number that reflects the strength of the van der Waals forces relative to −78 6 the surface tension, b ≈ 4·10 Jm is the London constant for the interaction of the shell HS and hydrogen molecules, N is the Avogadro number, ρ and m is the mass density and the A S S molecular weight of the shell material, and m is the molecular weight of hydrogen. H 10