 ki je Semion Zvodovski. Vseh vseh komputacij, nekaj je zelo, je zelo vseh P.H. B.R.A.M.I. V tej vseh vseh nekaj je zelo, kaj je Kadi Kapust, Boris Mikarlovič v Stadi MHD v Bangoriju Nr. in Efid Semach v Nr. Reščenih Stadi Vse reče pravno vsoje je skonviko počkala vzvoljnjenja postavljenja vsega ili zvedanje, da je vzvoljnja vzvoljnja zvedanje, in v sej novi reče. V sej novi reče prijevno zvedanje se večstva zvrkaj nama počkala vzvoljnja zvedanje, nama počkala vzvoljnja izvokovitjo in dučne vzvoljnje, potem vzvoljnje in dučne uvrkaj. ta je vse konstrukcije. In tako, sezelo zelo je, da je povedalo radioizotopov za medikalne aplikacije. Vse zelo je zelo svorila, da je zelo vsezelo. To je jedna od nekaj stadi, kaj smo prišli vzajno vzajno vzajno radioizotob. In nekaj metod vzajno radioizotob je, da je vzajno vzajno vzajno vzajno vzajno, ta taj prišla, da je srednit, pa je vzajno prišla proton. If, for example, the target is filled with lithium as a liquid metal, then during the irradiation the berylu will be produced, which is a source of neutron for medical applications, boron neutron therapy. A typical target contains liquid metal placed in a container, which is covered by thin metallic foil. Now, due to the interaction between the protons and liquid metal, the material inside the target becomes radioactive, and the kinetic energy of protons is converted into heat, which is absorbed in the target. The total heat release, total kinetic energy of protons is 20 kilowatt, and this, all these flux of protons is concentrated in a rather small area, such that the total heat flux for square meter is 200 megawatt, which is very high. This high flux generates high temperature at local point, close to the center of the target, and far away from this point the temperature is close to the ambient temperature, so the high temperature gradients are generated inside the target. These high gradients may cause damage to the target itself. For example, it may create cracks in the foil that cover the target. High heat flux removal is a very important problem to solve before this application will be used. Now, the liquid metal becomes radioactive, so one cannot apply any mechanical means to enhance heat transfer inside the target. Also, natural convection is very insufficient, it actually doesn't do much. So the heat distribution remains with very high gradients, if nothing special can be done with it. So we suggest to use rotating magnetic field to produce force convection. Rotating magnetic field will be not intrusive, so it would be very safe and reliable. Now, the typical target looks like you see in this picture. Here the beam of proton, and here the target placed inside. This is schematic diagram of the target. It's not in correct scales. Maximum length is about 9 cm, but the width is just 1.5 mm. It's a very narrow target. So the flow should be very weak here. That's what happens. That's the main reason why natural convection doesn't help here. Now, we suggest to place inductor of magnetic field at the bottom size of the target and generate the rotating magnetic field to produce circulation and force convection. This is one of the possible inductors with six poles. Magnetic field, sorry, the electrical current and three phase current will generate rotating magnetic field. And we hope to remove the heat more efficiently with such mixing. Now, analytical solution for such complicated geometry of course doesn't exist. So we use number simplifying assumptions. The most simplifying assumption is that magnetic Reynolds number is small and that's really the fact because magnetic diffusivity is in order of 1 and the size of the target is small. So the induced magnetic field is much smaller than the field generated in the inductor. So we have to compute only the field generated in the inductor and then use it as a part of the Lorentz force in Navier-Stokes equation. Now, the geometry is cylindrical. We assume that inductor currents are localized at the surface attached to the target base. So that, of course, is not accurate representation but as we will see later it produces rather good results. So instead of all the coils in the inductor we have just surface currents at the back side of the target. Now, due to harmonic dependence of the current, electrical current the solution is also harmonically time and angular dependent. So that is another very important simplification. Now I will show you that there is approximate solution for the magnetic field and this solution will be used first of all it clarifies the physical phenomenon then it can be used as in a parametric study to quickly find the optimal or near optimal parameters for the all apparatus. Now the analytical model. First of all we defined vector potential because the magnetic field is non-divergent, solenoidal so vector potential can be defined and with column calibration divergence of vector potential is equal to zero. Then, of course, induction equation can be derived from Maxwell equation and ohm's law for moving conducting media. So that's basic equation for the magnetic field generated by inductor. Now, if you find, if you solve this equation find vector potential then we can compute currents inside the liquid metal just by using ampere slow and another simplification is that due to the cylindrical geometry and time dependence, harmonic time dependence actually we have to find only radial component of the vector potential and then using continuity equation we can find azimutal component and radial component of current inside the liquid metal and vector component of magnetic field. So with radial component of the current density and vertical component of magnetic field we compute the main force which is azimutal in this case J cross B that generates the circulation of liquid metal. So first we have to specify boundary condition for radial component of the vector potential. They are rather simple. The vector potential is non-singular in the center of the target. Then it is equal to zero at the outer external radius of the target because the current doesn't lift the target. There is no conductive material outside of the target so that gives us another boundary condition that far away from the target magnetic field tends to zero and the additional boundary condition is at the current surface where we specify surface currents and using ampere low and taking into account that the inductor core is made of ferromagnetic material with magnetic probability very high. We can use this boundary condition to find boundary condition for magnetic field or for A at z equal to zero. Now, first we make non-dimensionalization of the parameter. There is interesting parameter which is non-dimensional frequency. It is the frequency of the AC current multiplied by characteristic time of magnetic field diffusion over the widths of the target. So that's non-dimensional parameter which is quite important. Now, because the A vector potential is solenoidal we can find dependence between radial component and azimutal component and we assume that all components of vector potential depend on the time and the azimutal angle in the harmonic wave. So, after substitution of all these components in the divergence of A equal to zero we find that azimutal component and radial component differ on angle phi zero which is pi over two. This is very important because as you can see when we write the equation for the radial component of vector potential we see that the left side of this equation is real while the right side is imaginary. So, both of them are equal to zero. That is a strong simplification which allows me to solve the Red-Head-Sun equation in a simple way by just separation of variables r and z and the solution will be presented as a series of Bessel function of first kind and exponential decay with characteristic scale of the decay as a eigenvalue of the characteristic equation. Now, gamma here are just roots of Bessel function due to boundary condition which I explained on the previous slide. So, that's a solution. C and D are coefficients which can be found easily by integrating the currents at the base of the target and here you see the results in graphic form. So, also it's not in the scale the vertical scale is non-dimensionalized by the width of the target and it is rather small and the horizontal is dimensionalized by the radius. This is magnetic field near the pole. So, we see that our assumption that it will be mostly in the vertical direction. So, the main component the vertical component is really reproduced itself and this is quite an accurate solution in the area close to the pole. Outside of the pole, of course far away from it the solution is not accurate because geometry is very simplified relative to real geometry. So, real geometry you see on the following slide. So, one cannot solve the magnetic field with this real geometry so the numerical calculation were used in self by means of answers Maxwell code. In numerical conversion rather time computing so we used parallel computing computer with highly efficient computer. Now, the analytical solution for flow can be found but it doesn't make sense because it's a laminar solution and the real flow is turbulent very much turbulent. So, I don't show you the analytical solution for the flow and just go to the numerical solution. Now, we use answers fluent to solve the Navier-Stokes equation with Lorentz force J cross B magnetic induction method was used for solving the MSE equation and I explained here shortly what it means and for turbulence computation we use a large simulation option with Smogorinski viscosity but because the number of elements is very high and we have more than 3 million elements most of them in the horizontal so because of the very small size of elements the hydrodynamic Reynolds number is inside the element is smaller than 1 so LES doesn't help much it's basically direct numerical simulation but also the magnetic field doesn't affect the turbulent viscosity also because of the same reason the characteristic time of turbulence inside the element is very short relative to the time of characteristic time of magnetic field so we check that the simulation doesn't depend on the number of elements above this number we increase it but the result remains the same here you see the typical snapshot actually it's not snapshot it's mean velocity at time when the flow reached steady state we shifted the inductor from the center to the area the heating is the most intense in order to generate highest velocity in this region now I will show you short video of velocity field so what you can see here that there is vortex generated by magnetic field and the interesting that there is a counter rotating vortex attached to this one one vortex in this direction another one in this direction and also the flow is three dimensional so there is varticity also in the vertical so it looks like we can generate quite efficient mixing using the rotating magnetic field flow is very turbulent and we expect significant improvement of heat transfer we will see it on the next slide so on the left figure here there is a temperature distribution temperature contour without magnetic field so you see high temperature spot in the center in temperature reaching I forgot the number 800 above and 100 degrees Kelvin when we applied magnetic field hot spot significantly decreased it didn't disappear completely but it is much less hot now the difference between these two is shown on this slide so you see in the center of the target the reduction of the temperature is highest and there is also increase of the temperature in some adjacent areas so the gradient of the temperature in the horizontal plane and also in the vertical will decrease you can see the movie of the effect the rotating magnetic field of the temperature I will explain first what you will see at the beginning you see the heating of the target by the beam of protons when the temperature reaches steady state then we apply magnetic field and you will see how the temperature decreases so let's start the movie you see that the high temperature spot in the center just disappears so it's a very effective method non-intrusive very reliable and easy to apply to significantly increase heat transfer now I come to the conclusions so we have the analytical solution it's not bad for the magnetic field it's not accurate for hydrodynamic field which is very, very clear because the flow is turbulent but with analytical solution we can find the area of parameters where we have to look for and use this area of parameters for optimization using more advanced numerical tools now the numerical simulation proved that the application of rotating magnetic field allows one to achieve the main goal of the project which is to decrease the gradients and decrease the highest temperature by at least 60 degrees degrees carrying we saw in certain areas temperature drop up to 150 degrees so the goal was achieved also from analytical solution we easily see that the effect of magnetic field is proportional to frequency of the electrical current but it is correct only until the certain value of the frequency above this value there is a optimal value of the frequency above which increasing the frequency of the electrical field would not help even work in the opposite way because there is increase in reactive resistivity of the coils and some other effects that make it not worse to go above certain optimal frequency of the electrical field now it's ongoing project the next stage that I didn't show yet is to use different geometry of the inductor and different connection of the inductor poles in order to generate running magnetic field instead of the rotating field and orienting in such way that that will be jet which will sweep the high temperature area outside of the area of the high heating and with think suppose that it will be more effective even than the use of rotating field but this experiment is not finished now in addition to numerical simulation we also made laboratory experiment and it will be presented in another talk by another graduate student Tzahesho Kroon during this conference so please come visit him and ask the questions thank you very much