Equations for thermal conduction

Transcription

1 Equations for therma conduction Heat conduction and heat conductivity q - T q heat fux heat conductivity T temperature Incropera & De Witt 1990

2 Therma conductivities for some materias Materia / Part λ [W/mK] Copper 370 Auminium 240 Casted iron 58 Stee (0.1 % C) 52 Stainess stee Laminated core (radia direction) Laminated core (axia direction) 1 4 NeFeB magnets 10 Ename coating of conductors 0.20 Sot insuation Air at 50 C Water at 20 C 0.60 Typicay, a good eectrica conductor is aso a good therma conductor, and vice versa. Exceptions: Superconductors are perfect eectrica conductors but poor therma conductors. Diamonds are eectric insuators but they conduct heat better than copper 2000 W/mK

3 Temperature dependence of therma conductivities Incropera & De Witt 1990

4 Heat equation Power baance of a voume eement V T, p h r, c p q n p h power or oss density n surface norma vector r density c p specific heat capacity Power conducted into the voume P c - S q ds Power generation inside the voume Power absorbed in heat capacity P ge V p dv h T Phc rcp dv Ł t ł V

5 Power baance Power baance T rcp dv - q ds + phdv Ł t ł V S V Gauss s aw (Divergence theorem) q ds qdv S V Power baance V T rcp + q - p dv Ł t ł h 0 T, p h r, c p q n

6 Power baance Vaid for any voume > rc p T + q - p t h 0 Substituting the equation of heat conduction: Biot-Fourier equation T rcp - ( T) p t h

7 Boundary conditions G 1 Dirichet s condition on boundary G 1 W T T G 1 Cauchy condition on boundary G 2 G G1 G2 G 2 qn - T n at + g ( T T ) a G - a

8 Probem: Heat transfer from soid to fuid v What is veocity distribution? Turbuent or aminar fow? x Conduction or convection? T m v T a What is therma distribution? T > Semi-empirica heat transfer coefficient a c for the surface x ( ) q a T - T c c m a

9 Heat transfer between two soid bodies Incompete contact between to soids > A semi-empirica heat-transfer coefficient a c is defined for the boundary ( ) q a T - T c c m a Contact α [W/m²K] Auminium frame Stator core Cast iron frame Stator core Rotor bar Rotor core

10 Soution of heat equation using finite eements Static heat equation Boundary conditions ( T) ph 0 + T T G 1 Method of weighted residuas ( ) hø W 0 R wøº T + p ßd W (- ) n a( - ) T T T a ( ) - ( ) + ( ) w T dw w T dw w T dw W W W ( ) ( ) - w T dw + w T ndg W G

11 Soution by finite eements II [ ] ( ) n ( ) R - w T - wp dw + w T dg + w T ndg h W G G 1 2 G 1 ( ) w T ndg 0 Choice of weight functions G ( ) n - ( - ) w T dg wa T T dg 2 2 G a Boundary condition ( ) ( ) ø ( ) R غ - w T + wp ßdW - wa T - T dg W h a 0 G 2

12 Soution by finite eements III N f T N( x, y, z) t() t j 1 j Gaerkin s method w N ( xyz,, ), i 1,... N i j f Ø Nf ø Nf ri Œ- ( Ni) Njt j + NiphœdW - Nia Njt j-ta dg Wº Ł j1 ł ß G j1 2 Ł ł N f f j1 N j1 ( )( ) - N N dw+ an N dg t + N pdw + NaTdG H t + f 0 ij j i i j i j j i h i a W G W G 2 2

13 Soution by finite eements IV We obtain a matrix equation Hτ + f 0 where the matrix and vector entries are ( ) ( ) H N N dw+ anndg ij i j i j W G f N p dw + NaTdG i i h i a W G 2 2

14 Temperature rise of the PM rotor Tota oss in PM rotor: P rt 320 W Heat transfer coeff. on rotor surface: a 350 W/Km 2 Temperature in air-gap: T amp 60 C

15 Therma resistance A conductor having a constant cross-sectiona area P I q, U A P Power, q Temperature difference Equations for the heat fow and eectric current (p h 0) P I A r q r da A - r q T da A r r r U J da - s f da s A A A U R e q R R R e A s A

16 1-D exampe p h A T 1 x T 2 ( T ) - ph constant; p h constant 2 d T 2 dx - ph > T p 2 h 2 ( x) - x + c1x + c2 T ( 0) () T T T 2 1 p 2 > h T ( x) x( - x) + T1 + T2 - x x

17 Average temperature of a conductor Knowedge of the average temperature is often sufficient p h A T 1 x T 2 1 Ø ph - x x ave ( ) 1 2 Œ ø T x x T T œ dx º ß ph T + T 2 12 ( ) 1 2

18 s i Ø, W/mm ø º ß Visibe ight T 1100 K T 1300 K T 900 K T 700 K Therma radiation Wave ength [mm] Panck s radiation aw i, s 5 i radiation intensity c speed of ight h Pank s constant k Botzmann s constant Maximum intensity at max 2 2πch ch kt e -1 Ł ł 2898 T Source: Water Wagner, Lämmönsiirto mmk

19 Heat transfer by radiation Stefan-Botzmann aw T 1 T 0 ( 4 4 ) Q se A T - T s r m W 2 4 K Radiation: T K, T K > q 992 W/m 2 Natura convection: a c 14 W/m 2 K, DT 100 K > q 1400 W/m 2

20 Literature Incropera F.P., De Witt D.P, Fundamentas of heat and mass transfer. Third edition, John Wiey & Sons, New York p + Appendices. Pyrhönen J., Jokinen T., Hrabovcova V., Design of Rotating Eectrica Machines. Second edition, John Wiey & Sons, p. Wagner W., Lämmönsiirto, Painatuskeskus, Hesinki 1994.