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Australasian Universities Power Engineering Conference (AUPEC 2004) 26-29 September 2004, Brisbane, Australia THERMAL ANALYSIS OF NEODYMIUM IRON BORON (NDFEB) MAGNET IN THE LINEAR GENERATOR DESIGN W. N. L. Wan Mahadi, S. R. Adi and K. M. Nor Department of Electrical Engineering University of Malaya Abstract This paper presents a study of thermal design aspect of Neodymium Iron Boron (NdFeB) magnet in the free piston linear generator design. The generator is coupled with the internal combustion engine which tends to produce heat. Due to the high temperature from the combustion engine, the performance of NdFeB magnet used in the generator is examined. In this paper, heat sources from the exhaust of the internal combustion engine, current carrying conductors and friction are studied. The heat transfer mechanism such as convection and radiation are also explored. Finite element method is used to generate and analyze a thermal model of the NdFeB magnet in the linear generator. 1. INTRODUCTION Fundamental knowledge of magnet and linear generator in various aspects such as electrical and thermal A linear generator is developed in the University of properties has been explored. This paper is written with Malaya, Malaysia for automotive and stand-alone the objective to investigate and analyze thermal (standby or remote) generator applications. It is driven properties and simulation of NdFeB magnet in the linear by an internal combustion engine and converts generator design. FEM simulation plays an important mechanical power into electrical power. It consists of a role to evaluate thermal performance of the magnet and stator and a linear translator, which carries axial field generator. Although the exhaust of the combustion permanent magnets. The main difference of linear engine acts as the main contributor in producing generator to a conventional generator is that the motion maximum heat, others various heat sources produced in of the rotor is linear. The linear system gains the generator have also been investigated. improvement in efficiency and volume reduction as the crankshaft is removed. 2. PERMANENT MAGNET FOR LINEAR GENERATOR DESIGN In order to select the appropriate permanent magnets for the linear generator, four major factors are emphasized; a) magnetic properties, b) corrosion resistance, c) material cost and d) maximum operating temperature. Fig. 1: Linear generator with a free-piston engine A rare earth permanent magnet NdFeB, is chosen based on a number of considerations especially in terms of The development and investigation of linear generator producing high energy product. The physical, have been continuing for a number of years. In order to mechanical and magnetic properties of the magnet such design a better performance of linear generator as thermal conductivity, specific heat, electrical prototype, previous research has been reviewed. Based resistivity and magnetic flux density which are affected on past research at West Virginia University, Sandia by temperature are investigated. Since NdFeB magnet is National Laboratories in California, Sunpower Inc. and brittle, a tin coating is used as a protective layer for the The Royal Institute of Technology in Sweden, basic magnet to overcome corrosion resistance. ideas in designing free-piston generator have been studied [5,6,10]. Fig. 1 illustrates the principle of the linear generator with a free-piston combustion engine. 1 2.1 Magnetic properties A detailed study on thermal analysis of linear generator is furthered by designing a thermal model using FEM. The intensity of magnetization and the coercive force The temperature difference at certain regions in the are elements which determine the performance of linear generator is then can be estimated. For thermal permanent magnets. For a better performance, NdFeB analysis in the linear generator, various kinds of heat magnet is a good choice for a linear generator sources and heat transfer mechanisms are studied. Heat application. It offers the highest energy product sources in the linear generator include heat from the compared to other permanent magnets. The strongest exhaust of the combustion engine, current carrying magnet has the highest maximum energy product conductors and friction are investigated. In this paper, (BHmax). For the linear generator, NdFeB magnet with our analysis concentrates on heat sources from the grade N30EH is used. internal combustion engine and the electromagnetic field Important parameters for NdFeB permanent magnet part. The heat transfer mechanisms are also investigated. with grade N30EH: This analysis can be classified as one of multi-physics a) Remanance flux density, Br = 1.114 T problems which involve conduction, convection, b) Coercive force, Hcb = 871 kA/m radiation, electromagnetic field and friction. c) Intrinsic force, Hcj = 2411 kA/m d) Maximum energy product, BHmax = 241 kJ/m3 3.1 Governing equations In any applications, before thermal model is designed, numerical computation should be considered and the load applied to the model should be represented by the governing equations. For linear generator, the net function of heat, which is considered to be generated can be represented as ∑ f (Q) = f (Qt + Qe + Q f ) (1) where Qt is the heat from the combustion engine, Qe is the heat from electromagnetic field and Qf is the heat produced by piston and sleeve bearing through friction. In general, the heat equation is derived from the conservation of energy principle, which states that the net heat conducted out is equal to the summation of heat Fig. 2: Demagnetization curves as a function of generated and change in energy stored within the temperature system. Mathematically, this expression can be expressed as [2] Fig. 2 which is taken from reference [12] shows the v ∂e demagnetization curves for N30EH NdFeB magnet at ∇•q = Q− (2) different temperatures. ∂t where q is the heat conduction, Q is the heat generated 3. THERMAL ANALYSIS within the system and ∂ e/ ∂ t is the change in energy stored. q is described by Fourier’s Law of heat Thermal analysis is very important in generator conduction and is given by application to evaluate the effect of temperature in the v system. The focus of our analysis is to develop a thermal q = −κ∇T (3) model and to study the heat distribution in linear Therefore, the heat transfer in a solid material is generator. Compared to other materials used, the expressed by a partial difference equation as follows [2] permanent magnet temperature is one of the key items to v v ∂T be emphasized in generator design. A simulation is ∇ • (κ∇T ) + Q − ρC p =0 (4) ∂t developed to ensure that the magnet can be operated at where κ is the thermal conductivity, Cp is the specific certain temperature conditions due to the limited heat capacity, ρ is the mass density, Q is the heat maximum operating temperature within NdFeB magnet. generation rate per unit volume and T is the unknown temperature distribution that is to be determined. 2 3.2 Heat transfer mechanisms Then, the total heat loss, Qcr by convection and radiation in the inner part of linear generator is [5] To study the behavior of materials at certain Qcr = (hc + hr ) A(T − T A ) (8) temperature, a basic knowledge of heat transfer mechanism should be studied. Several properties of Radiation is usually significant relative to conduction or materials change with temperature are also can be natural convection, but negligible relative to forced predicted and investigated with the aid of heat convection. Thus, radiation in forced convection distribution estimation from FEM simulation. Heat can applications is normally disregarded, especially when be transferred in three different ways: conduction, the surfaces involved have high emissivities and low convection and radiation. temperatures [11]. For this reason, the radiation effect is negligible in our simulation because of the high In order to transfer heat from a solid surface to a cooling emissivities of copper and silicon steel materials in the medium such as air, it is necessary for the surface to be generator’s stator, which are 0.63 and 0.7 respectively. hotter than the surrounding medium. The heat transfer coefficient is a constant, which depends upon the surface material, the surrounding medium and the relative 3.3 Electro-thermal analysis velocity of the surrounding medium [1]. For convection, the heat transfer is complicated due to the fact that it Electro-thermal simulation has become an essential part involves fluid motion as well as heat conduction. of many engineering applications. Thermal analysis for Newton determined that the heat transfer through a the linear generator is quite complex due to the presence surface area is proportional to the fluid solid temperature of heat source from electromagnetic field part. It difference. The temperature difference usually occurs requires a solution of coupled field analysis. In the across a boundary layer of fluid adjacent to the solid coupled field analysis, heat generation inside the coils is surface. The Newton’s equation can be represented as considered in order to get a prescribed temperature field. Qc = hc A(T − T A ) (5) To generate a model that takes into account both electrical and thermal effects still remains a challenge. where, hc is the convection heat transfer coefficient, A is The relationship between the electromagnetic field and the heat transfer surface area, T is the temperature of temperature are studied to analyze the overall system. the surface area and TA is the ambient temperature. For a natural convection of air, the range of the heat transfer In our design, we consider heat produced by the coefficients is between 2 to 25 W/m2.0C and for a forced electromagnetic part as a heat generation. Heat convection, the range is between 25-250 W/m2.0C [11]. generation in linear generator refers to resistance heating The temperature effect and heat loss at the outer part of in copper wire where electrical energy is converted to the linear generator is caused by a natural convection heat. Heat generation is usually expressed per unit while the inner part is caused by forced convection and volume of the medium and it can be represented as radiation. I 2 Rcoil Qe = (9) The radiation heat transfer is more difficult to calculate, π ( r02 L) − π (ri 2 L) as radiation loss depends on the forth power of the absolute material temperature. In the inner part of linear where Rcoil is coil resistance, I is current capacity of wire generator, the radiation heat transfer coefficient can be and L is coil thickness. From calculation, heat generated expressed as [6] in copper coil is 374082.6773 W/m3. σ t ε (T 4 − T A ) hr = (6) 3.4 Friction (T − T A ) where ε is emissivity of the material and T is the Friction is associated with bodies in motion. When two material temperature. The radiation heat transfer to or bodies in contact are forced to move relative to each from surface surrounded by an air occurs parallel to other, a friction force that opposes the motion develops convection between the surface and the air. Thus, the at the interface of these bodies. The energy supplied as total heat transfer is determined by adding the work is eventually converted to heat during the process contributions of both heat transfer mechanisms. The and is transferred to the bodies in contact, as evidenced radiation heat transfer can be represented as [11] by a temperature rise at the interface [11]. This Qr = εσ t A(T 4 − TA ) 4 (7) phenomenon is occurred in linear generator while it is 3 operating. Piston and sleeve bearing in a cylinder are ∂T ∂N 1 ∂N 2 ∂x ∂x ... example of objects, which produce heat through friction. ∂T ∂N ∂x 1 ∂N 2 (16) Heat generated through friction can be calculated as = ... {T } = [ B ]{T } ∂y ∂y ∂y Q f = ηFν (10) ∂ T ∂ N 1 ∂ N 2 ... ∂z ∂z ∂z where η is friction coefficient, F is frictional force and ν is sliding velocity. where {T} is the temperature at nodes, [N] is a matrix of shape functions and [B] is a matrix for temperature 4. SIMULATION ANALYSIS gradients interpolation. By using Galerkin method, the equation (11) can be written as [3] 4.1 Complete partial differential equations (PDE) ∂qx ∂qy ∂qz ∂T for thermal model ∫ ∂x + ∂y + ∂z V − Q + ρCp Ni dV = 0 ∂t (17) For a basic equation of heat transfer, the equation (5) can be represented as [3] By applying the divergence theorem to the first three terms, the equation can be expressed as [4] ∂q x ∂q y ∂q z ∂T −( + + ) + Q = ρC p (11) ∂T ∂N i ∂N i ∂N i ∂x ∂y ∂z ∂t ∫ ∫ ρC p ∂t N i dV − V ∂x ∂y ∂z {q}dV V where qx, qy and qz are components of heat flow through (18) ∫ = QN dV − { q } { n } N dS ∫ T the unit area and Q = Q(x, y, z, t) is the inner heat i i generation rate per unit volume. It is assumed that the V S boundary conditions can be of the following types [4] {q}T = [qx qy qz ] and {n}T = [nx ny nz ] a) Specified temperature Ts = T1 ( x, y, z, t ) on S1 (12) where {n} is an outer normal to the surface of the body. b) Specified heat flow After insertion of boundary conditions equations (12) to q x n x + q y n y + q z n z = −q s on S2 (13) (15) into equation (18), then the equations for the overall system is [4] c) Convection boundary conditions q x n x + q y n y + q z n z = h(Ts − Te ) (14) ∂T ∂N ∂N i ∂N i ∫ ρC V p ∂t N i dV − ∫ [ i V ∂x ∂y ∂z ]{q}dV where Ts is an unknown surface temperature, Te is a (19) known environment temperature and h is the convection = ∫ QN i dV − ∫ {q}T {n}N i dS + ∫ q S N i dS − V S1 S2 coefficient. ∫ h(T − T ) N dS − ∫ (σεT − αq r ) N i dS 4 d) Radiation e i q x nx + q y n y + q z nz = σεT − αqr 4 S3 S4 s (15) where α is the surrounding absorption coefficient and qr 4.2 Finite element model simulations is incoming heat flow per unit surface area. For transient problems, it is necessary to specify a Finite element analysis is a tool used in engineering temperature field for a body at the time t = 0. A domain application to determine the physical effects of a given V is divided into finite elements connected at nodes. set of boundary conditions will have on a part. The finite Global equations for the domain can be assembled from element method is the most useful technique and finite element equations using connectivity information. flexible tool to determine the unknown temperature in Shape functions Ni are used for interpolation of most industrial applications. The geometry being temperature and temperature gradients inside a finite modeled will always be divided up into smaller element [4]. This can be expressed as follows divisions known as elements and the elements are connected together to form the finite element mesh. T = [ N ]{ T } Each element contains nodes, which are points and the [ N ] = [ N 1 N 2 .....] elements are mathematically connected to one another. It is actually an approximate mathematical method for {T } = {T1 T 2 .....} solving differential equations problems. 4 Theoretical analysis, heat transfer mechanisms, simulation, the heat source is largely governed by the electromagnetic-thermal and friction have been heat conduction from the exhaust of the combustion discussed. The integral finite element equation that is engine. referred to the equation (19) is solved using 2D Finite Element Analysis. For this purpose, ANSYS 7.1 The heat source from the combustion engine is software is used as a simulation tool. Since the linear considered about 7270C (1000K). The heat generation generator is symmetric about the centerline of the shaft, from the copper coil, which is governed by equation (9); it is only necessary to model half of the geometry in and natural convection at the enclosure of the generator ANSYS. Therefore, 2D axis-symmetry expansion is are also applied as thermal load of the linear generator. used to design and simulate a full model of the linear For our simulation, it is noticed that the radiation effect, generator. forced convection and friction are not considered. The linear generator dimensions used is given in Table 1 Radiation effect is almost negligible because the below, existence of the air gap between the stator and translator Table 1: Linear generator dimensions is very narrow. The effects for these three mechanisms will be further investigated in future work. Components Dimension Stator pole pitch 76 mm Stator slot pitch 76 mm Stator slot depth 30 mm Stator tooth width 8 mm Stator back iron thickness 6.5 mm Magnet outer diameter 55 mm Magnet inner diameter 25 mm Magnet thickness 56 mm Magnet spacer 20 mm Air gap 1 mm Fig. 3: 2D linear generator model Starting from a sketch of the geometry, closed regions have to be identified. Determination of properties of materials such as specific heat, thermal conductivity and Fig. 4: Heat distribution in linear generator without density at different temperatures are important for cooling fins and permanent magnet part thermal problem solving. In order to get a final solution, thermal load as boundary conditions of the model are Fig. 4 and Fig. 5 show the heat distribution in linear defined properly. Fig. 3 shows the 2D linear generator generator as well as permanent magnet. Temperature model. region around the permanent magnet has also been analyzed to ensure that it is below than its maximum 4.3 Simulation results and discussion operating temperature. From mathematical analysis of thermal model, the heat sources boundary conditions are applied. In this 5 6. REFERENCES [1] Design of Machines and Drives – Generalised Machine Designed Concept. http://www.staff. ncl.ac.uk/barrie.mecrow /dmd Intro.pdf. [2] Efunda (Engineering Fundamentals), Conduction: General Theory. http://www.efunda.com/formulae/heat_transfer/ conduction/overview_cond.cfm. [3] G.P.Nikishov, “Introduction to the Finite Element Method”, University of Aizu, Japan, 1998. [4] Marc T.Thomson, “Electrodynamic Magnetic Suspension – Models, Scaling Laws, and Fig. 5: Heat distribution in linear generator with cooling Experimental Results”, IEEE Transactions on fins and permanent magnet part Education, Vol. 43, No. 3, 2000. [5] N. W. Lane and W.T. Beale, “A 5 kW Electric From thermal simulation of both linear generators with Free-Piston Stirling Engine”, Proceedings of 7th and without cooling fins, it is observed that the range of International Conference on Stirling Cycle temperature distribution in permanent magnet is Machines, Tokyo, Japan, 1995. between 43.9010C to 133.7480C (without cooling fins) [6] P. V. Blarigan, “Advanced Internal and 40.820C to 113.8480C (with cooling fins). This Combustion Electrical Generator”, Proceedings range can be accounted as a safety margin for the of U.S DOE Hydrogen Program Review, permanent magnet which has maximum operating NREL/CP-610-32405, Sandia National temperature of 2000C. Laboratories, Livermore, Ca 94550, 2002. [7] S.R. Trout, “Understanding Permanent Magnet From the results, the amount of heat to be removed will Materials: An Attempt at Universal Magnetic be determined in order to maintain an optimum Literacy”, Magnequench International, Inc. operating temperature. To reduce as much heat within http://www.magnequench.com/techresources/ the generator, the design of the finned surfaces is still in techcenter/reference/papers/CoilWinding2000. the process of improvement. The use of the fins in this pdf. application is very effective in order to reduce the [8] S.R. Trout and Yuriy Zhilichev, “Effective Use temperature rise in the linear generator. As a of Neodymium Iron Boron Magnets, Case continuation, a cooling mechanism design such as Studies”, Proceedings of Electric forced cooling will be investigated and simulated. Manufacturing and Coil Winding ’99 Conference, 1999. 5. CONCLUSION [9] S.R. Trout, “Material Selection of Permanent Magnets, Considering Thermal Properties Thermal model from FEM analysis has been studied. Correctly”, Proceedings of Electric The simulation results are required to observe the Manufacturing and Coil Winding Conference, temperature rise within the permanent magnet. The Ohio, USA, 2001. temperature distribution within the permanent magnet [10] W. R. Cawthorne, “Optimization of a Brushless has been determined. To improve the simulation, Permanent Magnet Linear Alternator for Use coupled field analysis, cooling mechanism, radiation, with a Linear Internal Combustion Engine”, forced convection and friction effects will be Ph.D. dissertation, Dept. Computer Science and investigated further. An analysis of thermal, heat Elect. Eng., Univ. West Virginia, Morgantown, transfer and cooling mechanisms will aid in the 1999. estimation of heat distribution in the linear generator. [11] Y. A. Cengel, “Introduction of Thermodynamics For further investigation, the data that have been and Heat Transfer”, the Mac-Graw Hill obtained from the simulation is then used in cooling Companies Inc., United States of America, mechanism design by using better fins and forced 1997. cooling. [12] Electron Energy Corp., EEC Magnetic Properties, http://www.Electroenergy.com/prod _demag_curves.html. 6

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