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1 Rev Rum Sci Tecn Électrtecn et Énerg Vl 6,, pp 54 58, Bucarest, 07 CYLINRICAL INUCTION HEATER THE QUALITY FACTOR ROMAN KIEROŃSKI Key wrs: Electreat Inuctin eating Parameters f inuctin furnaces an eaters Crucible furnace erive frmulas n quality factr cylinrical furnaces an inuctin eaters In rer t simplify te frmulas intruce witut imensinal relative energy absrptin cefficient f te electrmagnetic fiel relative t cpper at 0 C Te values f tis cefficient are summarize in te table as a functin f temperature fr te cmmnly use metals Receive simple frmulas are aitinally useful fr rapi implementatin f initial engineering jects an fr teacing Wavefrms pltte efficiency quality factr an cmpare wit te values appearing in te literature INTROUCTION eterminatin f parameters f inuctin eaters an furnaces base n pysical ata is ften necessary wen ealing wit te subject f inuctin eating Te article erive simple frmulas fr ften use cylinrical inuctin eater esigns can als be applie t cylinrical crucible inuctin furnaces ETERMINATION OF THE PARAMETERS ON THE BASIS OF THE PHYSICAL On te basis f te pysical ata f te inuctin eater tat is its gemetric imensins an material perties can calculate te varius parameters eg resistance reactance an terefre te pwer factr quality factr Te article erive simple esigns an are mre accurate tan tse given in te literature [] an [] a) b) Fig Cnstructin (a) an an electric equivalent circuit (b) f te cylinrical inuctin eater inuctr-carge system in iameters, an eigts H,, wit te: z cils inuctr an te penetratin ept t carge; μ, μ 0 magnetic permeability, ρ, ρ resistivity, equivalent resistance R an equivalent reactance X Cnsiere cylinrical inuctin eater as a gemetrical imensins an material perties f inuctr an carge as inicate in Fig clarity were it was pssible t intruce simple witut inex ces (like Liwiński in [3]) in suc a way tat te large letters refer t te inuctr an small t carge Equivalent resistance R an reactance X (fig ) als epen n te frequency f Te eater is supplie by generatr values, i, u frequency f, as swn in scematic blck-iagram f Fig Fig Te blck-iagram sceme f te supply inuctin eater wit generatr f frequency f Te mst useful is t calculate te electrical parameters f te eater resistance R an reactance X t etermine te equivalent circuit (Fig b) an te quality factr Q X R Q () Te teretical calculatin is very muc epenent frmulas erive frm nly ne parameter quality factr Q because tey fcus n temselves at te same time ave tw parameters R an X Electrtermal literature [3 0] are ften serve frmulas n R an X Terefre, tis article is erive frmulas fr te Q factr Te mst cmmn frmula is given in te literature n resistance R (Fig ): Rw ρ K r R R + Rw πz () η η were: R resistance f te inuctr; R w resistance f te carge; F r carge cylinrical sape cefficient (Fig 3); K r crrectin cefficient (Fig 4); η electrical efficiency f te inuctin eating system (4) Reactance X (Fig an Fig ): X ω ρ μ πz K + N K x F xk x (3) 4 H were: F x carge cylinrical sape cefficient (Fig 3); K x crrectin cefficient (Fig 4); K N Nagaka cefficient (Fig 5); ω πf te pulsatin Te cefficients fr te carge cylinrical given in frmulas () an (3), respectively sape F r, F x, crrectin K r, K x an Nagaka K N are rawings in Figs 3 t 5 Te sape cefficients F r, F x, f te cylinrical carge is AGH Akaemia Górnicz-Hutnicza w akwie, Plan,

2 Cylinrical inuctin eater te quality factr 55 pltte as a functin (base n te literature [3, 5, 9]) in Fig 3 Fig 5 Nagaka cefficient K N fr inuctr cylinrical n te basis f [5] 3 EFFICIENCY Fig 3 Te sape cefficients F r an F x f te cylinrical carge as a functin Crrectin cefficients K r, K x f carge as a functin f varius parameters etermine by [3] are swn in Fig 4 Electrical efficiency η f te inuctin eating system accring t [5 9] sw a frmula: η, H ρ H ρ + (4) + FKk μρ FKk k r r w r r r w p were: k p ρμr is te cefficient f te energy absrptin f te electrmagnetic fiel, k w te fill cefficient f te wining Te literature is rare esignatin k p ρμr calle energy absrptin cefficient f te electrmagnetic fiel accring t [9], because it as a imensin an is uncmfrtable range f values Terefre intruce witut imensinal esignatin k ρrμr relative energy absrptin cefficient f te electrmagnetic fiel ρ by intrucing te relative resistivity ρ r ρcu0 (pse in relatin t cpper at 0 C) Te values f tis cefficient k fr te mst cmmnly use metals are calculate in Table by [] Table Relative cefficient k f electrmagnetic fiel energy absrptin as a functin f temperature fr varius metals wit respect t te cpper at 0 C Temperature Cefficient k fr varius metals Fig 4 Crrectin cefficients K r, K x f carge as a functin f varius parameters Nagaka cefficient K N as a functin by [5] is swn H in Fig 5 C Cu Al Ag Au Sn Steel * Steel * μ r 0 μ r

3 3 Rman Kierński 56 (table cntinuate) Melting pint C ~ 500 ~ 500 * Steel wit lw carbn cntent Table sws tat te steel f curse as te largest value f te cefficient ue μ >> Hwever, abve te Curie pint abve apx 750 C accring t [0] after te lss f te magnetic perties f ter metals (particularly in a liqui state) absrb energy nearly as steel ata inicate in bl: fr metals in liqui an fr steel wen te lss f magnetic perties Take int accunt tat te inuctr is a little warmer even tug it is cle an as up t 60 C Ten exessin in te frmula (4) is a giving te wining fill cefficient H a frmula efficiency: ρ 7 μ r k an by ρ 8 8 w 0 η + FK 8 7 r r k FK 07 k r r k an assuming (5) ωμ ρ KN Kx+ FxKx 4 H Q ρ FK r r η ωμ ρ ( KN Kx) + FK x x 4 ρ FK r r η On te basis f te cnversin frmula (5) ave te relatinsip + η 0 7 k (6) (7) After inserting te exessin (7) t (6) is btaine by te frmula ( ) ωμ ρ KN Kx + FK x x Q 4 ρ FK r r + 07 k ωμμ r ( KN Kx) + FK x x 4ρμr + 07 k After inserting btaine: ρ ω μ μ r (8) int te frmula (8) was wic wavefrms are rawing in Fig 6 as a functin f varius values parameter k, assuming values F r 09 (Fig 3) an 4 r Q K N K x + F x K x μ r FK r r + 07 k (9) K x K x + Fx μr K N K N Q K F r r + k 07 K N (0) Fig 6 Efficiency f eater as a functin f varius values k, assuming values F r 09 an 4 4 QUALITY FACTOR Q After insertin f () an (3) t () an cntinues t H is erive frmula n Q factr inuctin eater: Fig 7 Wavefrms te Q factr f eater as a functin f varius parameters k, μ r,

4 4 Cylinrical inuctin eater te quality factr 57 5 VERIFICATION OF THE RESULTS Te calculate Q factr are cmpare in te graps (Fig 7 8) wit te calculate examples in te literature [ 5] (Table ) base n measurements f electrical quantities (n mecanical imensins) Pints 5 in Figs 7 8 results sw g agreement wit te given literature Table Results verificatin f te ata an te imensins f figures an Fig 8 Wavefrms te Q factr f eater as a functin f varius parameters k, μ r, an a rug calculatins by [] can be cefficients wit frmula (0) apximate: exp exp K N Kx 09 K N () Wavefrms example f te frmulas (9) an (0) are rawing in Fig 7 as a functin f varius values parameters k, μ r, It is eferre t perate wit lw values f tse swn in Fig 7 Wavefrms te Q factr as a functin f varius values k, μ r, an pltte in Fig 8 m Fig 8 can be seen tat te values little cange fr te analyze range an tat te smallest quality factr Q are esent at ig values μ r 5 Often tere is a cance tat tis frmulas can be simplifie wen F r F x fr > 0 (Fig 3) an after inserting frmulas () t equatin (0) was btaine: Q μr exp + 07 k + exp () m frmula () furter sws tat eater quality factr Q ecreases wit te increase cefficient μ an increases wit increasing an an epening n te carge size fr te analyze range ( 0, ) r Carge f H Q Pint Fig 7 8 khz mm mm mm nr Literature Cu [] Cu [3] Stal [3] Stal [4] Stal [5] 6 SPECIAL CASES Wen tere is n carge in te inuctr tat is btaine by Q (3) were penetratin ept f te wave t te inuctr Tis is te basic frmula fr Q factr eater witut la It als fllws frm te imary literature [3 0] Wen te la is a ferrmagnetic carge an at te best size fr eating:, 0, Q factr epens mainly n carge an frmula (0) simplifies t: Fx K x Fx Q (4) K r Tis frmula is as in basic literature [3 0] t te Q factr f an electrmagnetic wave incient n te cnuctive material Verificatin f te results sws tat in actice te mst cmmn ccurs > 0 Ten fr carges f ferrmagnetic tis is ften true tat wen inserte int equatin μr () gives te result Q exp + exp 07 k (5) Especially fr tin ferrmagnetic carges ( > 5 ) can

5 5 Rman Kierński 58 ccur: an ten equatin (5) is te easiest: 07 k Q cs (6) Special cases are simple frmulas (3 6) an (3 4) are te same as in te electreat imary literature [3 0] 7 CONCLUSIONS Te erive frmulas are simpler tan tse available in te literature (fr example [, 6 ] an algritms in [, 3]) Sme f tem are smewat simplifie but crrect t make quick calculatins an eliminary engineering jects In aitin ave t te value f teacing fr teacers an stuents because f te frmula can be seen irectly as efficiency an quality factr epens n te imensins an cefficients inuctin eater wit Fig an te frequency f its supply by Fig Cllecte values efficiencies an quality factr generally agree wit tse aministere in te literature cncerning inuctin eating (eg [5, 9]) Verificatin f results an special cases cnfirm te crrect erivatin f frmulas fr Q factr Receive n May 5, 06 REFERENCES R Kierński, Uszczenie wzrów na parametry cylinrycznyc pieców i nagrzewnic inukcyjnyc zasilanyc z falwnika Elektrtecnika i Elektrnika, 6,, pp (007) R Kierński, Nagrzewnica inukcyjna cylinryczna, wzry na parametr tłumienia i brć, Przeglą Elektrtecniczny, 9, 7, pp 0 05 (05) 3 W Liwiński, Nagrzewnice inukcyjne, WNT, J Hauser, Elektrtecnika, Pstawy elektrtermii i tecniki świetlnej, Wy, Plitecniki Pznańskiej, M Hering, Pstawy elektrtermii cz, II, WNT, M Hering, Pranik inżyniera elektryka T Elektrtermia, WNT, P Nvak, Elektricke Tepl, Vyala Equilibria, 00 8 T Racki, A Kanyba, Urzązenia elektrtermiczne, Wyawnictw Plitecniki Śląskiej, 00 9 C Sajak, E Samek, Nagrzewanie inukcyjne, Wy, Śląsk, J Turwski, Elektrynamika tecniczna, WNT, 968 R Kierński, Nagrzewnica inukcyjna stpałścienna Sawnść wzry na parametr tłumienia i brć, Przeglą Elektrtecniczny, 9, 8, pp (05) C Sajak, J Barglik, L Pieszczek, R Przyłucki, Wspmaganie kmputerwe jektwania pieców inukcyjnyc tyglwyc tpienia metali nieżelaznyc i stpów żelaza, Knferencja Wspmaganie kmputerwe jektwania urzązeń elektrtermicznyc, Plitecnika Śląska, Szczyrk, 994, pp R Kierński, Analiza acy i spsób sterwania wułącznikweg falwnika napięcia nagrzewania inukcyjneg, ktrat, aków, 0 4 B Grzesik, Z Kaczmarczyk, M Kaszak, Falwniki tranzystrwe częsttliwści pwyżej 00 khz nagrzewania inukcyjneg, Knferencja Naukw-Tecniczna Energelektrnika Energszczęnść, Plitecnika Śląska, Gliwice, A Skała, Falwnik wcz klasy E zastswania w cesac elektrtermicznyc, ktrat, aków, 04 6 K Kurek, M Niklewicz, Ocena zyatnści nagrzewania inukcyjneg w cesie artwania kół zębatyc, Przeglą Elektrtecniczny, 7, pp (008) 7 K Kurek, A Smalcerz, Baanie ekranów wielwarstwwyc tłumiącyc ple magnetyczne wkół nagrzewnic inukcyjnyc Przeglą Elektrtecniczny, 84, 7, pp 89 9 (008) 8 R Przyłucki, A Kacel, Mel bliczeniwy inukcyjneg ukłau grzejneg z uwzglęnieniem raiacyjnej wymiany ciepła, Przeglą Elektrtecniczny, 84, 7, pp (008) 9 M Wciślik, T Kwaśniewski, Inukcyjnść cewki jenwarstwwej Wzry analityczne, Przeglą Elektrtecniczny, 84, 7, pp (008) 0 J Barglik, T Wieczrek, A Smalcerz Meling f te surface inuctin arening cess, Przeglą Elektrtecniczny, 90,, pp 4 (04) M Wesłwski, R Niebała, wuwzbunikwa nagrzewnica inukcyjna krystalizacji metą PVT kncepcja i symulacja Przeglą Elektrtecniczny, 90,, pp (04) T Leuca, Ş Nagy, N Trip, H Silagi, C Mic-Vancea, Optimal esign fr inuctin eating using genetic algritms, Rev Rum Sci Tecn Électrtecn et Énerg, 60,, pp 33 4 (05) 3 T Leuca, Ş Nagy, N Trip, H Silagi, A Burca, esign f experiments apac fr inuctin eating ptimizatin, Rev Rum Sci Tecn Électrtecn et Énerg, 6,, pp 69 7 (06)