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Application of inductive heating to granular media: temperature distribution in a granular bed.

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(1)Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.. This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID : 20114. To link to this article: DOI: 10.1016/S0017-9310(05)80130-9 URL: http://dx.doi.org/10.1016/S0017-9310(05)80130-9. To cite this version : Duquenne, Philippe and Deltour, Alain and Lacoste, Germain Application of inductive heating to granular media: temperature distribution in a granular bed. (1993) International Journal of Heat and Mass Transfer, vol. 36 (n° 9). pp. 2473 -2477. ISSN 0017-9310. Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr.

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(111)      @jg V“ 6“ Uj‚}qN‹‚“ zx“ „hV“ ‚NtV“ gN}h“ V‰}V€jtVx„Nq“ „Vt V‰}VS„VU “„iV“VxVg‹“j‚“‚}VNU“V‡Vxq‹“jx“„iV“VNS„z“NxU“„hN„“ VNSi“zW j„‚“}N€„jSqV‚“iN‚“„hV“‚NtV“QVhN‡jz €“„zˆNU‚“jxU S„j‡V“ }VN„ V“}zXlqV‚“ jx“„iV“QVU“ ˆj„hz „“ IqV‰jgqN‚“ ˆNqq‚“ Szu iVN„jxg“N‚“Nx‹“z„hV“zxV“= „“„ij‚“ShVVW†q“€V‚ q„“SNx“Nq‚z“QV“ }N€VU“ ˆj„i“ }VUjS„jzx‚“ gj‡Vx“ Q‹“ „iV“ }q gbzˆ“ ˆj„h“ N‰jNq“ U V“ „z“ NUjNq“ Uj‚}V€‚jzx“ qVNUjxg“ „z“ N“ „Vt}VN„ €V“izuz Uj‚}V‚jzx“ uzUVq-“ JhV“ V]VS„j‡V“ b jU}hN‚V“ „hVvNq“ Szx U S„j‡’„‹“ ˆN‚“‚V„“V~ Nq“„z“0n‘“K01“03 %“ gVxjŒN„jzx“jx“Nx‹“gj‡Vx“‚VS„jzx“zW“„hV“QVU“ JhV“‚NtV“tVN‚ VtVx„‚“ˆV€V“}VY{uVU“ˆijqV“„hV“VNS„z€“ ?Š}VjtVx„Nq“‡Nq V‚“N„“N“gj‡Vx“Uj‚„NxSV“ezt“„iV“€VNS„z“ ˆN‚“V~ j}}VU“ˆj„i“„iV“„ˆz“SzxSVx„jS“IqV‰jgqN‚“„ QV‚“jx

(112) “ˆj„h“ NS„j‡V“ŒzxV“Vx„NxSV“NV“ Nx“N‡VNgV“zW“„hV“„Vt}VN„ €V‚“N„“ €V‚ q„‚“gj‡Vx“ jx“ Ajg“ 4;“‚jxSV“IqV‰jgqN‚“ˆNqq‚“jx“„iV“€VNS„z“ „hV“‚NtV“Uj‚„NxSV“Z|“„hV“„hVV“€NUjNq“}z‚j„jzx‚“V‰}qzVU&“<‚“ ‚„z}“iVN„“„Nx‚WV€‚“ˆiN„V‡V“„hVj“xN„ V“„hVtNq“€NUjN„jzx“ xz“ju}z€„Nx„“UV‡jN„jzx“V‰j‚„‚“QV„ˆVVx“„hV“„ˆz“‚V„‚“zW“}zjx„‚ “ j‚“xVgqjgjRqV“jx“„ij‚“€NxgV“zW“„Vv}VN„ €V “VNSi“zW“„iV“„hVV“ ˆV“ SzxSq UV“ „hN„“ „iV“ i‹}z„iV‚j‚“ zW“N“ }q gbzˆ“ ˆj„h“ N‰jNq“ UVqjtj„VU“ŒzxV‚“j‚“N‚‚ wVU“„O“ QV“ jxUV}VxUVx„“ jx“„V€v‚“zW“ Uj‚}V‚jzx“}hVxzvVxN“j‚“o ‚„j`VU'“ ‚zqjU.s jU“„Nx‚WV€‚ “JhV“`g €V“SqVNq‹“‚izˆ‚“„iN„“„hV“„h€VV“ Fz€Vz‡V€“„iV“„Vt}V€N„ €V‚“}z`qV‚“ˆV“iN‡V“VxSz x„V€VU“ ŒzxV‚“VSVj‡V“„iV“‚NtV“}zˆV“UVx‚j„‹!“>zt}N€j‚zx“QV„ˆVVx“ „i€z ghz „“„iV‚V“V‰}VjtVx„‚“Nkq“‚izˆ“~ j“„ V“N“gzzU“qjxVNj„‹“ Ajg‚"“ 2“ NxU“ 5“ ˆijSi“ ‚izˆ“ Nquz‚„“ „hV“ ‚NtV“ „Vw}V€N„ €V“ qVNUjxg“ ‚“„z“QVqjV‡V“„hN„“„hV“V^VS„‚“zW“N‰jNq“Uj‚}V‚jzx“Sz qU“ }€z`qV‚“ˆj„hjx“„iV“QVU“W|“„iV“‚„VNU‹‚„N„V“€VgmtV “Nq‚z“jxUjŽ hN‡V“QVVx“xVgqVS„VU“L7“9M(“ SN„V‚“ „hN„“ IqV‰jgqN‚“ ˆNqq‚“ hN‡V“ xz“ jxb VxSV“ zx“ „€Nx‚}z€„“ }iVxztVxN“ ˆijSi“ SNx“QV“ V‰}qNjxVU“Q‹“ˆVNp“ jxb VxSV“ zW“     €NUjNq“Uj‚}V‚jzx#“ >NqS qN„jzx‚“ ˆV€V“tNUV“„z“UV`xV“„hV“VxVgV„jS“V_SjVxS‹“ H„hV“ V‰}V€jtVx„‚“ ˆV€V“ }V€duVU“ ˆj„i“ xz“ b jU“ bzˆ“ zW“ „iV“ jxU S„j‡V“ iVN„V€)“ Jij‚“ V_SjVxS‹“ j‚“ „iV“ N„jz“ zW“ „hV“ „h€z gh“„iV“gNx qN€“QVU“‚ztV“ ‚jxg“N“1“uuhjgh“ 6“ut VxV€g‹“ „€Nx‚WV€€VU“ „z“ „iV“ b jU“ }hNƒV“ SNqS qN„VU“ [zt“ „iV“ UjNtV„V“ V‰}qz€V“ Szj“}q ggVU“„z“„iV“z‚Sjqqz‚Sz}V“jx“z€UV€“ b jU“ szˆ“ NxU“ „hV“ jxqV„“ NxU“ z „qV„“ „Vu}VN„ V‚“ „z“ „hN„“ „z“tVN‚ V“rzSNq“uNgxV„jS“jxU S„jzx“`VqU“N„“Uj]V€Vx„“}qNSV‚“ UVqj‡V€VU“Q‹“„iV“gVxVN„z“UV„VujxVU“ˆj„h“„hV“z‚Sjqqz‚Sz}V“ jx“ „iV“ €VNS„z“ z„iV‚“ ˆj„i“ Nx“ :“ tt“ ‚„VVq“ QNqq“ jx“ ˆhjSh“ N“ Ajg €V“7“‚hzˆ‚“„hj‚“V_SjVxS‹“NxU“j„‚“‡NjN„jzx‚“Z|“Uj]VVx„“ „hV€uzSz }qV“ ˆN‚“ ‚V„,“ JhV‹“ Nq“ Szxat“„hV“\S„“„iN„“QP„h“ fV~ VxSjV‚ “ jxU S„jzx“NxU“j„‚“V]VS„‚“N€V“ xjdt“jx“„iV“RVU$“ Dx“„iN„“‚VjV‚“zW“V‰}VjtVx„‚“„hV“c jU“bzˆ“€VuNjx‚“Szx. . . # #. #. . #. . . .  . .  .

(113) . .  . .   . #.  . . . AEC*“5+“Aq jU“„Vw}V€N„ €V“jx“„iV“RVU“ˆj„i“SzxTVx„€jS“ˆNqq‚+“. .

(114) .   

(115)  .   .  

(116). . 3 5 . . 3 : . 3< . TYV0 >0  U€•~j‡}f‘k ’k‡k‰f’•‰kE k˜‡k‰~kƒ’‘2‡€•| w — —~’} f˜~f€ j~‘‡k‰‘~ ƒ  jk€ h ‡f‰~‘ ƒ0 . ‘’fƒ’ fƒj ’}k |kƒk‰f’ ‰ okˆ•kƒhš ~‘  ‘k’  ’  j~rk‰kƒ’  –f€•k‘  –f‰š~ƒ| mŽ  =  ’  C X›* Qhh ‰j~ƒ| ’   —}f’  ~‘ h  ƒ€š  g‘k‰–kj fg •’ f }  |kƒk •‘ € fj ~ƒ fƒ ~ƒj•h’~–k }kf’k‰

(117)   –f€•k‘  l  ksh~kƒhš  f‰k  ˆ•~’k  }~|}  fƒj  ’kƒj  ’   ‰~‘k  —~’}  ~ƒh‰kf‘~ƒ| mŽkˆ•kƒh~k‘N z  4+A? h ‰‰k‘‡ ƒj~ƒ| ’  = X›  ~’  h€~g‘  ‰f‡~j€š ’  40C  f’  C  X›,  ck  hfƒ  ‘kk  f‘  p†‰  }  œ |kƒk •‘  € fj‘  d?e ’}f’  ’}k‰k  ~‘ f h‰~’~hf€  {kˆ•kƒhš gkš ƒj  —}~h}  ~’  ~‘  ƒ ’  — ‰’}  — ‰~ƒ|  y‰  fƒ  ~ƒh‰kf‘k  m ’}~‘  {kˆ•kƒhš  ‡‰ –~jk‘  ƒ  ‘~|„~vhfƒ’  gk’’kkƒ’  m kth~kƒhš.  Zƒ  •‰ hf‘k ’}~‘ mŽkˆ•kƒhš fš gk k‘’~f’kj f’  fg •’ @  ‰  B X›0  5(16-)! 6%#0+1.46 _ƒk }š‡ ’}k‘~‘ —f‘ fjk f‘‘•~ƒ| ’}f’ “ O 7!9Š  [ƒ  ‰jk‰ ’  fk ‘•‰k  m~’‘ –f€~j~’š —k }f–k ‡k‰y‰kj ‘ k  hf€h•€f’~ ƒ‘  —~’}  ’}k‰  jk€‘  ‡‰kj~h’~ƒ|  –f€•k‘  m \“  ~ƒ  ‡fhkj gkj‘ 67 6;e- `}k ‰k‘•€’ ~‘ ’}f’ ƒ  ~ƒw•kƒhk  n f˜~f€  j~‘‡k‰‘~ ƒ hfƒ gk ‘kkƒ ~ƒ ’}k j f~ƒ —k ‘’•j~kj fhh ‰j~ƒ|  ’  —}f’ hfƒ gk €kf‰ƒ’ mŽ  ’}k €~’k‰f’•‰k @ AeF ’}~‘ ~ƒw•kƒhk  hfƒƒ ’  gk jk’kh’kj y‰  –f€•k‘  m  € —k‰  ’}fƒ 74 c "  ]#.  `}•‘ ~ƒ ’}k k™‡k‰~kƒ’f€ j f~ƒ —k ~ƒ–k‘’~|f’kj —k hfƒ  k˜‡‰k‘‘  ’}k ’k‡k‰f’•‰k ‡‰ v€k ~ƒ ’}k ‰kfh’ ‰ —~’}  f ‡€•|œ x — —~’} •’  j~‘‡k‰‘~ ƒ   jk1  Q‘‘•~ƒ| ’}~‘ —k }f–k ~ƒ  ‘’kfjš$‘’f’k h ƒj~’~ ƒ‘G . ž y‰ ’}k w•~j ‡}f‘kH. ž fƒj y‰ ’}k ‘ €~jG. —~’} ’}k y€€ —~ƒ| g •ƒjf‰š h ƒj~’~ ƒ I  a‹M O 4 P  b   R ƒ‘~jk‰~ƒ| ’}f’ ~ƒ f ‘’kfjš%‘’f’k ‰k|~‚k ’}k —} €k kƒk‰|š  ~ƒ‡•’ ~‘ ’‰fƒ‘lk‰‰kj ’  ’}k w•~j —k f‰k f€€ —kj ’  —‰~’kJ   &2*1  6$6

(118)  6 ck hfƒ ’}kƒ k˜‡‰k‘‘ ’}k ’k‡k‰f’•‰k ‡‰ v€k fi ‰j~ƒ| ’   ’}k ‡€•| w —  jk€K  aŒL  . . ''  .%#6%63')"#. 6 .6 .) #,)6 %#/6 0 /%#6 `}k ~ƒ’k‰‰•‡’~ ƒ  l  }kf’~ƒ|  fš  gk k˜‡‰k‘‘kj  f‘ f  ‘’k‡  k’} j y‰ ’}k }kf’ ’‰fƒ‘q‰ k–f€•f’~ ƒ+ Q ’}k‰f€ gf€fƒhk  gk’—kkƒ ’}k –f‰~f’~ ƒ  m ’}k y‰hkj w — kƒ’}f€‡š ~ƒ’k|‰f’kj . 83  Ÿ&&&&&&&&'&&&&&&&(&&&&&&&&&&&&) 

(119)  . 3D . . .     

(120)  . . . . . . T”W0 @/ Sƒk‰|k’~h kuh~kƒhšG ~ƒw•kƒhk  m pkˆ•kƒhš0 . . .

(121)

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