A Comparison of numerical simulation models for predicting temperature in solidification analysis with reference to air gap formation J. Kron, Michel Bellet, Andreas Ludwig, Bjorn Pustal, Joachim Wendt, Hasse Fredriksson To cite this version: J. Kron, Michel Bellet, Andreas Ludwig, Bjorn Pustal, Joachim Wendt, et al.. A Comparison of numerical simulation models for predicting temperature in solidification analysis with reference to air gap formation. International Journal of Cast Metals Research, 2004, 17 (5), pp.pages 295-310. <10.1179/136404604225020669>. <hal-00576280> HAL Id: hal-00576280 https://hal-mines-paristech.archives-ouvertes.fr/hal-00576280 Submitted on 14 Mar 2011 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
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
! c p E fs h "*%!#!%&?/, 0 0 @ C,+ $"?@ A $+'!%" $!#'%!?0@ &'"#'% ##!%!?7@?+ 0< 0 @ I m!!-" '?0@ p = 'σ *'""'?@ 2 "'!0'""!!A!-!.?0@ s = σ + pi A! '!%"'""" '?@ T T S T L α h f ε ε tr el ε pl ε vp ε th ε ρ? T = L @ ρ? T ρ? T L @ S +*''? 'D7@ " $!"+*''? 'D7@ $!1!"+*''? 'D7@ &'+$$!'.*"! % ##!%!? 0 @ "*%!#!%$& ##"!?/, 0 @ @ < 5 "&'!/,'! 8'$!AA $+%&,"" %!)!&& $ $!1!0" $!'"!!?0@ "'!'" '?" 0 @ $"!%*' #&"'!'" '?" 0 @ *$"!%*' #&"'!'" '?" 0 @ A!"% *$"!%*' #&"'!'" '?" 0 @ &'+$*' #&"'!'" '?" 0 @ pl < eq 2 ε pl ij pl ij = ε ε A!"" 1!A$*$"!%? ' A!"% *$"!%8 vp!" # pl@ < eq 2 ε ij ij "'!0'?" 0 @ = ε ε A!""1!A$"'!0'?" 0 @ pl = t pl eq ε 5 eq ε dt A!""1!A$*$"!%? 'A!"% *$"!%8vp!" #pl@ "'!?0@ <
ε dt A!""1!A$"'!?0@ = t eq ε 5 eq λ ν ρ σ &'+$% %!A!-?+ 0 0 @!"" % ##!%!?0@ "*%!#!%+""?/,+ 02 @ 7%&-"'""" '?@ σ = 2 < s s A!""1!A$"'""?@ eq ij ij σy "!% -!$ "'""?@?*$"!% &'"& $8!# eq σy +'!$!"$"!%@ σ < & 2
! "#!!',*# '+!!"!+* '*' ($+! +-%"!,*' %""""!%!&" $',!#$% &&'"#'()%"!,+ $ '/ &% %! () A'-!,&'"#'% ##!%!&# '+! #!',*&"(*'"!A'! " '!%$"( + $%"!,E0:F&,*# '+! &"(# "')&&" $! +$"&$$!""',,& '"!"&*'""'#' +&$!1!+$E4F&&'"#' #' +&+$ &+ $!" +!(-% %! (# '!',*&"# '+& +%' "% *!%!',*!"# '+ &% %!!"'% && '"#'% $( "%'!("&"+ #&% %! &'!! A'&"$!EGF&%" # $+!!+(" $$ -"&&'"#' '!!!"!"!,!#!% E=F 7 A%! &" (*' A &A,$!,!($##%E5F&,'$.*$! #&!',*# '+!!"&"&'!/, #&%"!,.*"! #&+ $ %%' "'""""'!" #' +&'+$,'!"%''!,&%"!,*' %""E50;F +&+!%$ + $$!, # %"!, *' %"", A'!, 1! " &'+ *&-"!%$ *' *'!" ( '-%!! " ' ' + "*'%!%$%"!,*' %""" & &'+$ ( '- %!! " '!##!%$ %! & ',&$- # ' $$ *!" # & + $H+$!'#%&'" # '&!"!"&&&'"#'%A'-% "!'($-)!&!++*''$,&#% #&%"!,!$$&"( '-%!! "%!& '"$" #&'+$")$$"&'+ +%&!%$+ $$!, #&*' %"" &*'") '/.*'!+"&A( )&'+*''"!"*$%+" #& %"!,&+ $&A(+"''!," $!!#!%! *' %""&, +'- # &"0*&"(%& " ( #"!+*$%-$!'!%$"&*! '' +/!* ""!($ % & '"#'% ##!%!#' + & +*'' +"'+" &!" A$!# ' & )& $( '- #&%"!, &!+ #&*'") '/!" + $&.*'!+"%'!(( A+'!%$$-)!&& %$%$&'"#'% ##!%!"( '-%!! &!##'&'+ +%&!%$ **' %&""(-# '!##'"!+$! *','+""& $("!&+ $$!,) '/ &'"$" # & + $$!,*'!%! "','!, &" $!!#!%! *' %"" ")$$ " &!"*$%+"# '+!,!',*"& $(% +*' %& &' &.*'!+$ '"$" &"!+$! *','+" "&')' 702 E:F E4F 7EGF7E=F &$$ -"%& "# '&.*'!+")'%!%$023!? 0+!#!@$043!0 523,?,'!'#!@!+$! ) '/&"'$!'(!&!"'E <F (".*'!+"*'# '+ '$!'(- %&+! A"" E<5F )A'! &!' ) '/ $- & $!',* )",!A & &'+$.*"! # & + $ )"!"!,!"&#' +&% '%! #&%"!, $%&'&!&("&!! "&#( "'!" #%"!,.*'!+"&A(*'# '+!%-$!'!%$+ $)!&% '!!" %'&&!,& #&+ $)"55++ &.'$!+'" #&% ' & >
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h )" '!A#' +&/ )$, #&#$."!&A!%!!- #&!'#%&+&+!%$.*'""! " "!+h#' +&+"'+*''"!",!A"8 λ casting T r casting T = h? Ti casting Ti mould@ = λmould?@ r mould ;
)&' λcasting T i casting λmould T i mould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
' & +'!%$ "!+$! # & %-$!'.*'!+" # ' +'!$,' *" )' % "!'8!@ % '!!@ +$?%"!,@J?!!!@ + $?!A@!"$! ' & &'+$ "%'!*! # & *' %"" & # $$ )!, +'!$ *' *'!" & (,' 8 & % %!A!- λ?t@ "*%!#!%+"" ρ?t@ "*%!#!%& c p?t@!! $$- &$& ##"! h f &A $! #" $!#'%! f s?t@ &$!1!"" $!"+*''" T L T S &A (/ )# '&%"+$ '&+$&+ $),' +*'' **' *'!" )&'"# ' &% ' &!"$! ) "% " " &!#&'$" #&$&!"+ $$A! h f f s?t@&a$" # c p?t@"& $ %!-% '!(! #&$& $&,& & % '!" %" )!& 1'I (!!" + $$!'$- (- "!, " *' *'!" ' & +%&!%$% +*! &# $$ )!, +'!$*' *'!")'% +*!$8 C,K" + $" E?T@!"" K"'! ν?t@ &'+$.*"! % ##!%! α?t@ -!$"'"" σ y?t@# &$!'&'+$.*"! % ##!%! α?t@&'$!aa $+'!% " $!!#!%! "&'!/, ε tr )'%#' +&*'!+'-ρ?t@ &!##'**' %&""" %!)!&&!##'% "*'"!%! <<!!" ""+&&+'!$(&A"$"!%$$-!#&A!""1!A$"'""σeq!""+$$' &&-!$"'"" σy# σeq.%" σ y!!"""+& *!, &+ $ & +'!$(&A"!&'*$"!%$$- 'A!"% 0*$"!%$$-8 σ σ eq eq < σ $"!%(&A! ' y > σ $"!% 0*$"!% y '$"!%0A!"% 0*$"!%(&A! '?<@ "%'!(&"'""H"'!%'A( A&-!$"'""!)"""+&&$"!%0*$"!% +(',0", "'""0"'! + $ E<2 <>F%( **$! $&,& & # '+! # $!1! "+!0" $! &!,& +*''" $!0" +$$!% $$ -"!" (A! "$-"'!0' * &!"%&!%&"(! '' (($ % +*'&!##'+'!%$ % "(-"!,% ++ % "!!A+ $&&# '+'!%$% ""!&!""- %'""&"!&*$"!%',!+"'""?'#'%@"'!!"'$(- σ eq σy n ε eq = + n # 'εeq ε y?2@ ε y ' ε = σ H E!" & +.!++ $"!% "'! (A! "$- & '#'% "'! εeq y y '*'"" & $"'! &!" &"+ # & $"!% &?A!"% 0@*$"!%"'! &!+"! $""*'+'n?t@(&a"$!/&!a'" #"'!&'!,% ##!%! 'n N 1?2@'"! /K"$)"& )" $-$"!%(&A! ' )&'"# 'nn & +'!$**' %&"!$*$"!%!-? σeq!"%$ " σ y )&A'&A$ # ε eq@"&!##'+'!%$% ""!##'"'""0"'!+ $" &% "!!A!# '+!,!A $$+'!$"*' *'!""!&!""-%(+A!$($(-&& '" 4
(- 1?2@ +" ( '"#''! & + $ " (- & % ''"*!, *','+" & *' %')!$$("%'!(,&')!&&"%'!*! #%&+'!%$% <>!!$%!! " & +6 ' (6%!A" # &!") '/ ' &"!+$! # &,* &# '+"() & %-$!'!%$%"!,&+ $(-"!,!##'**' %&" % +*'&*'!%! )!&.*'!+$#!!," %! A'$ & "/(-"!+$!, & + $ #!$$!,&"!!"""+ "$'-!!%!%! &&*' %"""'"(-% $!, # +$)!&!# '+ +*''!"'!(!! *'& ""+($- # & + $ & % '&!"$! &A!,&"+!!!$+*'' <; '-%!! " " '&'+$( '-%!! )""+% "7 #h out N<5+ 0< 0 ()+ $!"$! "''!,&+*'' #&"''!,)"""+ (T out N<5D7 ) %& *!' # +'!$" #' + % ' +$ + $!"$! ) ""+ &.!"% #&'"#'% ##!%!& "7")'%& "!&' (% "!+ * '* -,*)&!%&# '+"()&+'!$" &7()*'% '!"""+ (% " h castinghcore N<555+ 0< 0 % ("& )&&!"""+*! &",$!,!($!#$% &'"$""!%&% ''*!$- '%&"&+*'' #&+$ &&.%&,% ##!%!)!&!"$! &"(#!. '(!''-A$ h OHinsulation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
7! '' *'!%"'"""!" '! "%"(-!& +, "% $!," $, " "1% ##!$")!& +*'''"$"!" A!$($ 2%( " % +*$$-!* # 7 +-*'!%"'"""!" '! "# ' - +'!$ %$""""!'J+ $ % ' '%"!, '# '$$&' )A'!#) +'!$"'%& " )&!%& '!!!$$-!% % 2% "!'" & ) +'!$" (!*'+ % %&'# ' $-&# '+! #&%"!,&"("!+$ &%$%$! #"'""" # '+! "!"(" * &'+ 0$"!%0*$"!% + $ &1! "''!A#' +&'$0""1! "')'!"# $$ )"8 el pl th ε = ε + ε + ε el + ν ν + ε = σ σ I + T ν T ν '? @ σ '? σ@ I E E T E T E f = σeq σy 5 2 ε pl eq pl ε = s!# f = 5s8 σ 5 ε = 5 < σeq th ε = αti &')!"?>@ & "% 1! & ) $" '+" %% # ' & +*'' *%- # $"!%!-% ##!%!"Eν&&!'1!!"&%!!!+* "(-&A!"" *$"!%!-%'!'! &-!$"'""σ y %A'-)!&&*$"!%# '+! ε $!'"'!&'!,+ $&# '&1!!"&A!""K*$"!%#$ )'$!" # pl eq %% '!, ' &# ' +'!$,' *" + $ %"!, % '!"$! " &'0!+"! $,'!% A'!, 1'' # &#$$ + $)" "!, "!, G2:5&.&' 55G *&' <24> " &% ''"*!,%&'%'!"!%"!I # & $+"!" ( >;++ '&&'+$"!+$! % "!+"* tn5;")""&"("1 % *$"!+$! #"'"""!" '! ")'*'# '+)!& tn" &"&'!/,!""%'!((-+!#!&'+$.*"! % ##!%!&" $!!#!%!!'A$ &A$ #α!&# '&1! #?>@!"%& ""%&&8 L T S tr 2α? T @ = ε?;@ )&' tr ε <<<8!"&"&'!/,'! "" %!)!&&" $!!#!%! P!"20% ++'%!$"!+$! " #)'(" &#!!!##'% +&!"% +* " #"A'$+ $"# '!##'*'* ""$!/%$%$! " #"'"""!" '! " +%' ",',! +!%' "'%'*'!%! "# '.+*$&'!"$" ("A!$($!&*','+ =
$%&! "'""!" & + $ " (- # ' %$%$! " # "'"""!" '! "# '+! " $!!#!%! *' %"" &*','+ +/" &'+$%$%$! "#!'" "" &%$%$ +*''!"'!(! "!0# ' "'"" & &'+$ &&'+ +%&!%$%$%$! "% $ (% *$ '&"'"""!+$! &'+ 0$"!%0*$"!%+ $!"% "!'!"(" &"+ 1! " "?>@ & # $$ )!, +'!$ *' *'!" ' A!$($! & (" # 8C,K"+ $"!"" K"% ##!%! &'+$.*"! % ##!%! -!$ "'""% ##!%!" #"'!&'!, $%&! # &"'""A'"! >5)""!&!") '/ '&# '+'!$,' *"#!J + $ %"!, % '!"$! &#$$, +'!%$+ $)"!"%'!")!&,'! "!,:5:;<GA $+$+"=5>5 #&"A $+")'"# '&%"!,?A', %&'%'!"!%"!I <> ++@ ' & &'+$"!+$! +!%!+"*% ' $$!,!" **$!!% % *$ +*''0"'"" %$%$! " ' * ""!($! & +*''!"'!(! )" %$%$ #!'" # $$ ) (- %$%$! # "'"""!" '! " &!""("1"!+$! )"*'# '+)!& +.!++!+"* # <5 "% "&"&'!/,)""%'!((-+!#!&'+$.*"! % ##!%! $!/! 7028"%! << & *','+ ",!A & * ""!(!$!- %$%$!"*$%+"! ) % % +'!$""!+$ "$-&'# '&%$%$! #!',*& (+!) "*"J )&'&!"*$%+ #&%"!,)"%$%$ )&'&!"*$%+ # &+ $)"%$%$ <<27 7!"% ++'%!$#!!$+" #)'*%/,# '%"!,"!+$! & %!" A$ *(- &% +*- E<:F % "!"" # (" + $ "A *! $ + $" & " $% "!"" # " $!!#!%! " $A')!&*' * " *' %"" '"&!! $+ $"'"&!, $! '""!!!%' "'%' $%' +,!% A'" '%! & + $ %% " # ' '"! 0$!' 2 & % %! & % A%! '!! )!&*&"%&," ' "!, &$*- # '+$! & &.%&, ()#%!, +!"%(')!&%!%! ' 0%!%!+"&" '"#'"'""%$%$! "'% *$&'+$% %()*'"!"% "!' (-+" ###%!A7 h eff )&!%&!"#!" 5
h eff = )!& + h h + h 5 air rad h air λ = g air?:@ λ air (!,&% %!A!- #!'?I' # 'A%+@ g&,*)!& h rad &'!! & '"#'% ##!%!&% ##!%!h 5!"&#$&'"#'% ##!%!)&,*.!""8h 5!"* ""!($-'$ &*'"% #%!,&!'#%!"A$(!,&,!A(-&'! #&%!,+'!$% %!A!- A'&%!,&!%/"""','" +%&!%$% %()!##' +!"!!" +,(-,+,', +$!*$!'# '+$! E<4F &"'""+ $"A!$($!7!%$&'+ 0$"!%0A!"% *$"!%+ $ #& 'I--*E<GF &'+ 0$"!%0*$"!%+ $?"!+!$' 1?>@@$"!%+ $# ' &%"*'")$$"'!,!( -+ $# '+ $+'!$"E<=F$$+'!$*' *'!" %(%& "+*''* '% "&!"% +*'!" &'!,! $!'$"!% &'+ 0$"!%0*$"!%+ $")'" " ' *!% /!+!%&'!, + $" ' A!$($ '!" ' *!%&'!, $!' # '+$! * )'$)# '+$! 'A!$($8 eq y pl eq σ = σ + Hε '"*%!A$-σ eq βε = σ55 +? σy σ55@ e?4@ H!"&*$"!%+ $"σ55&$!+"'"" β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pl eq
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mn&!"%" KN 2η)!&η&-+!%A!"% "!- #&$!1!$$ - vp th vp 2 vp m th tr ε = ε + ε ε =? εeq@ s ε = ( αt + f 2 s ε )I?G@ < K vp eq m σeq = K?ε @?=@ $ )T C!##'&'+ 0$"!%0A!"% *$"!%+ $"%("&!""- &+ $ "%'!((- 1?5@&"(%& " 1!?5(@-!$" &&-* $"!% /Q" $) )&!$?5%@,!A" &.*'""! # & A!"% *$"!% "'! '?&.*'""! () ('%/" '%" I' )&,!A8 )& σ eq!" $ )' & & *$"!% &'"& $ vp eq n σ y + H?ε @ A!"% *$"!%# '+! %%'"@ el vp th ε = ε + ε + ε el + ν ν + ε = σ σ I + T ν '? @ σ T E E T E vp n m eq y H vp 2 σ σ? εeq@ ε = s < σeq K th tr ε = ( αt + f 2 s ε ) I ν '? σ@ I T E?5@ ' +&!"" #" '1! " & 0!+"! $$)%(.'%8 <
σ eq y vp eq n vp eq m = σ + H? ε @ + K? ε @?@ & $!1!0" $! % *$!,!" ' " # $$ )" & " $!0-* #!! $+" ',',!?! +"& A$ %!- 1$" +'!$ A$ %!-@ )&'" )& $!1! &- ' $'!0,',!?+"& A$ %!-!"%$%$!*$- # +'!$ A$ %!-@ &!" *'A" +"&,'%-!%" # &'+$% A%! $$ )" & +"&( '- # $$ )&A $! #&#'"'#% #&'+!!,$!1!* $& + $ * "&'!/,E2< 2>F!('! # / A, # &"-++'- # &*' ($+ &#!! $+ +"&"&A( '"'!%,$'"% ' #<;,'"&+"&"!I"!&*')"( < 2;++!&*' 2++!&+ $,'!,% "!!A1! "! '' % +*')!&& &'% " $-&" $!0$!/% "!!A+ $&"("?1?5@0?@@)!&% ##!%! K(!, $$! '' **' %& & +(',0", "'!0'!*$"!%0*$"!%+ $8*'+'"H(T)n(T)&A(!!#! #!1?2@ &)& $+*'' +! 7 & &'+$ $!'.*"! % ##!%!!" % +!%$$-#' + & *'!+'-ρ?t@,!a*!)!" '!"%!#!!$+ α!1?g@ '?5@!" #!"8 α = 2? T t+ t T t @ ρ? T t+ t ρ? T @ ρ? T t @ t @?<@ )&'&"*%!#!%+""!"!'* $!&($ρ?t@%"&"&'!/,'! ε tr!" " &% ##!%!α!"" I'!&" $!!#!%!!'A$ +!%!+"*% ' $&"(**$!# '&'+$&'+ 0+%&!%$"!+$! " )!&+.!++ #;" /%&#(""!#! )! ".*$!!%!!&!"#!'""* &%"!,&+ $'% "!''!,! &!#$% # &!',* && '"#'( '-%!! "!"!, ' &'# ' $-*'& '"#'!"% "!' )!&% " 7" &!" %+!%*'$!+!'-"* "'A""% +*'!" ()&&'"#'" $A'" #&!##'+'!%$% " 2$043!0523,$$ - "* %("!!,'; & &'"$" (!)!& &!##'"!+$! % " ' %$ " 2
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
" $!!#!%! &"+),' *" #'"$"%(!!#! )A' )&!$7 7"& ) A'-"!+!$''"$" "& ) + ''*!% $!, & & &' &' % " &'# ' *','""!A$-!##' #' + 7 & +.!++ +*''!##'%)!&&+"'+"!"( =5D7?!,:@ )! &"% "* #&% +*! " &" $!!#-!,*'&+ $# '+ &'+ +%&!%$ % +*!!" %''! & " ) +!" & % ' &!"$! '""+'!,!)*'+ $!+*7%#' +.*'!+$!"% "!' ".*$!! %!!" A$ )&!%&!" ""+!# '+ &)& $!'#%!"!'%$-%#' +&.*'!+$+"'+" &'!" ##%!A% *$!,#' + & +%&!%$%$%$! )'" && '"#' %$%$! $- & % *$!, #' + & &'+$ %$%$! )'" & +%&!%$ %$%$!!"/! %% &',&&+*''*%- #&*'+ $ % "!!A*'+'" %" #&!" 0)-% *$!, &"+'""# '"*%( ("'A# '& % $!,%'A"*'"!!,'"4G,'!,*'+ $!"*$%+"!)!$$ ("&&''(!,!##'%"()&!##'% " 2<$043!0523,$$ - "*< %("!,' 4 & &*'!% +*'' A $! " '!,,'+ )!&&.*'!+$+"'+"&!"!"1! '+$"!%!&!"%"&% +*! " &A " & 7"!'%$- %#' +.*'!+$ +*'' +"'+" & " % ##!%!"&A %'",*# '+!?!, >@8% "1$- &% $!, # & *'!""$ )'&!"* &&!, #&+ $!"$" "$ ) )#'&!!!! #&!',* ' 55"&+*''A $! "%$%$(-&!##' % "'A'-"!+!$' "# '"*,'!, & +%&!%$"! # &% +*! "!,' 4"& )" & A $! # &!"*$%+ #&*'+ $"'#%"&*'H+ $!'#%?'N4;++@+!0 &!,& #&%"!,&.*'!+$%'A"'&"+ ""!!,'2&-"& ) &.*"! #&+ $ )&!%&"(!$!"">55"&(,!!, #&% $!, &!".*"!!"# $$ )(-&*'"'#% * 55")&!%&!"&!',*# '+!!+ &+ $.*"! )&!%&!", A'(-&'+ 0$"!%!-!"% ''%$-*'!%(-& ) % "&)'($ /!! %% 87"& $( & &%$%$! " )!& 702 7 ""+ '!,! + $&%" #7 &" #)'% '!+*7"< A'"! #&% +< )""# '"*< (*'+!!, $-'!,!+ $ % +*! " &% +*'!" " #&!##'%'A"# '!"*$%+ #&*'"& )&,!"*'"! %(" & 702 7?<@"& )!++! % '%! # & $$ - ( & +*$! &!+ A $! # &!" # '+!!" $$-!##' 8!'+!" A'-"+$$)!& )&'" & ;
A $! '!,&#!'":5"!""!+!$')!&7027?<@ ( &(% +"% +*$$-!##'8 G55" &,**'!%(- 702.%" &.*'!+$ (- ;5 3?' =55 µ+!" #:55 µ+@ &,* *'!%(-7!"&'!+",''?' G55µ+@7"& )"+-( &("? '$"(T@'"$ )!&!!!$.*"! #&$$ - # $$ )!,&+ $ ( )!& '$-,*# '+! (-% +*'!" )!&&.*'!+,*"!I)!%(!,,' &&+"' 8G55" <55µ+!" #:55µ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
!##' &(% +" 1!"!+!$'# '&!,&'*' %""!+" & A$" '&!,&$- A'"!+)!&'"*% &+"'+"&% +*! ""& )!"*$%+ $$ # & *'!$ 255 " & % '%! )!& A$" & ' % +*'($ & +"' " '"$" '"& )# ' 702(%" & +%&!%$% +*! #&% #!$!&!"%" )* & &!' $""* # &% +*'!" &'+ +%&!%$% +*! "&A( %''!!)&!%&&A$ #&7()*'+ $*"$ %$$- &"!I # &!',* $!/"* < & +%&!%$%$%$! &" )!#$% && '"#' %$%$! $- ) % " 7 7 *'+! "%& #$$- % *$&'+ +%&!%$% +*! " "!,7 &7!"'+!%% '!, 1?:@ "!, 7 &,* "!I *%- # & 7 &" ( "($!"&! & # $$ )!,+'' +&+"',*"!IA $! g?t@?!,2@#' +&+"' 7 h?t@?!, >@!)"* ""!($ % &!'!"!%,*"!I *%- # & 7 h?g@&!"'$! )"!' %"*)!"!&#!$ #&% &" $A!,&& '"#'*' ($+ &$ %$,*"!I!"/ )?#' +&+%&!%$+ $@ &'# '& **' *'!$ %$A$ #h!"%#' +&($h?g@(-$!'!'* $! 22$043!0523,$$ - "*2 %("!!,'= 7&&*'!%% $!, #&*'!"+%&"$ )'!&!" %"&" $!!#!%!!+*'!%(-7!"**'.!+$-)!%"$',"& +"' )&!$" 7"& )" A'-"$ )% $!, )8 &*'!""!$$ " $!!#! #' 555" %% '!,$- &&!, # & + $?!, =7@!"% "!'($- '%"% +*' "*<# '.+*$&%" #&!"(&A! '%(#!&.+!! #&!"*$%+%'A"?!,=@ &% "*'!%,*"!I&!",'' &&+"'?+%&,''!&%" #7@% "1$-&A$" #h' A'"!+ $!, '"!+&#$."&!'#%()*' + $' +&+ $"! )%"&!".*"!!"$!+!% +*' "*< % +*! " (%" #'%&!, "# '"*< &"+'" ".*$!&"'"$" 22<%!%$023!$$ - "*2 &!"%"& +%&!%$ + $ # 7#!$ *'# '+ %$%$!!,' 5 $-7'"$"',!A "& )!,&"+'")!&&$043!0523, $$ - +,-. '' A'% + & $!+!! "!%%'%!"% +!,#' + & " # $"!%0 *$"!% % "!!A 1!?" &!"%""!! %! 2<@ % +*$+'- 4
% +*! "&A(%''! )!& 7" #)'! &%" # $043!0 523,$$ -""%'!(!%! <<> $!1!0$!/" $!0$!/% "!!A1! " '!"!,!"&""!+$ "$-!&"!+$! ( A&+*''T C &' %& "1$ ;:=D7?% ''"*!, &%!%+*''f s N544@&+'!$!" % "!' " A!"% *$"!%?)! A'T L @ )&'"! (-" $"!%0A!"% *$"!% + $($ )T C " #1?5%@ +$!*$!%!A0-*.*'""! &"(%& "# ' &A!"% *$"!%"'!0'" '8 m vp 2 σeq σy ε = s?2@ < σ vp n K? ε @ eq eq -!$!,8 σ vp n vp m eq σy + K? εeq@? εeq@ =?>@ & +*'' **'+'" # &"% "!!A 1! " ',!A! ($ < $$ &''&"+7 +*'!,!,'!,'=!%("&&'"$" '!(','+)!&&.*'!+$+"'+"% +*'!" ()!,'" = "*%!$$- "& )" & & +! # & %"!,!" (' '*'" )!& 7)& "!, & $"!%0A!"% *$"!%% "!!A 1! " & +.!++ '' ' &*'!"*$%+!"( 5;++!&!"%" )&'"!!""&!,&"5:++"!, $" 0*$"!%!- &!" + ''$!"!%*'!"*$%+-!$" '" ($-,,*"!I *'!%!?!,'@ )&!%&!'$" ('"!+! #&&!, #&+ $?!,7 )&!%&%(% +*'!,'=7@ #&% $!, #&*'?!, (% +*')!&!,' =@!!!!"!'"!, % +*' &"&* # &*' &!" (! "!, & ) **' %&"8 "!,' < "!, & #!'" $"!%0*$"!% **' %& &*'**'" "&'!/'&'!# '+$- )!&$',!',*#$#' "'#% & * ' *!! &!"% +"#' + & " # $"!%0*$"!%% "!!A 1! &!" "!## + $&#$ ) #$!1!+"&-+'!$7 A'"$- "!, &$!1!H" $!**' %& &,*"$'-"!!"+%&"+$$'&#'"'#%"& )" %'A#'"'#%)&!%&!"!1$!!A,'+)!&&.*'!+ (#! &!" **' & "!+$! " *'# '+ (- # '!##' +'!%$ % " &A ( % +*')!&'#'%.*'!+$ +"'+"# ' ) $+!!+ $$ -%"!," )!&&"+'!,0-*, +'-&+!% %$"! " #&!") '/'&# $$ )!," &'"* "" #&!##'" #)'!&%" #*'&'"#'% +*! &" (# 1!"!+!$'.%*!&%" #&%!%$$ -# ')&!%&&"%'!, # &'"$"!"!+* ','!,&A$! #&(!$!- *'!%!',*# '+! "!,% *$&'+ 0 +%&!%$% +*! "!)"!##!%$ *'# '+% +*$% +*'!" () G
% " +!$-# ') '" "!'" +,&# '% "% "!' $-) )'($ '""#$$-% *$&'+ 0+%&!%$*' ($+!)&!%&&!',*# '+! $ %$$- +!#!" && '"#' %!)"!##!%$ '+! % "!!A 1! &% $(%%*(-$$# '% " &'!""!$$ % "!'($ $%/ #/ )$,','!, & +%&!%$% "!!A 1! " &!'"" %!*'+'" ("!" $!!#!%! %!! "&!" (A! "$-"#'&') '/&+!+ &$!+!! " #&" # A'"!+*$!#! (&A! ' $)""%& " $" 0*$"!%!- A' )! +*''!'A$"!%$!, $!1!+"&-"" &A(+ "' &'"$"#' +&!") '/$",!A'" $ /+ '& ',&$-! &+'!%$ '+ # " $!!#!%! "&'!/, $$ # ' *','+" % "!' & "&'!/, % '!( &!',*# '+! J!&'(-!,'+ &&'+$% '%!! &.*'""! # ' &'+$ "'! ' (- 6"!, & &'+$.*"! % ##!%! )A' &'"$"#' +"*<!&!") '/"& )&)!&&" #&&'+$ #' +&".*'!+" &*'!%! #&!',* )!&&% +*'*','+"!" "!"#% '-&'!"$".*'!+$"** '" )&-&" $!!#!%! "&'!/, "& $('$.(-&+$&'# '% '!( %A!-!& * #&%"!, '* ' "!- '&' & ##%!, &!',*.+*$ # "%&.*'!+$ ("'A!!"&* "!!A",',! &!" ("'A'&"'#% )'"&% $ "! #%"!, "& )!,&'$.! #&"&'!/,!"%$$-/!,*$%& % %$"! +"(&&+ $# '"%'!(!,& $"'!!" $!!#-!,+'!$ " (#!+ '%'#$$- 0 1(&"&& &!"**'!" % + #&0) '/R!%' "'%'$,!'!,- $!!#!%! ' %""!,?@S "(,' * > R&'+$"'""" "'!" '!," $!!#!%! S & & '")!"&.*'""&!','!# '#!%!$"** '(-&' *7 ++!""! =
&&&& EF$$- S&&'+$!'#%()%"!,"+ $"S AFSTrans. =4: 42;0 4>> E<F &$/ S$0+ $!'#%!$&'"#'S Metall.andMat.Trans.B =G; ;G;0;=> E2F &$/ S'"! +& "# ' '+!! # +$0+ $!'#%!$ &'"#'S AFSTrans. =G2 :G=0:=G E>F "% - &'+ SB'!! " #& '"#'% ##!%!" '!, " $!!#!%! #%"!,"!+$$!%+ $"S TheBritishFoundryman =G; >:;0>:G E;F'!##!&" S&&'"#'% ##!%!'!,&!!'%! $" $!!#!%! # $0!$$ -%"!,S Metall.andMat.Trans.B === 25 >420>G< E:F '(& 7+*($$ SA"!,! #%"!,H%&!$$!'#%!$ & '"#' '!," $!!#!%! #$03!$$ -(-!A'"+ $$!,'$!+U0'-!+,!,S Int.J.CastMetalsRes. === < 240>2 E4F!' &$/ SB $+'!% "&'!/,,* # '+! '!, " $!!#!%! #% **'0("$$ -"S AFSTrans. =G2 G0GG EGF &$/ S%&!"+" #& '"#' +$0+ $!'#%, AFS Trans. =G> ;G40;=G E=FI$A'! S*# '+!!*'++ $%"!,"S Trans.ofthe Americanfoundrymen ssociety =:5 :G 242024= E5FC!"&! ' ",$' S&!',*# '+! *' %""&%"!,0+ $!'#%&&'"#'+%&!"+&',&&,*S Metall.andMat.Trans.B =G: G220G>> EF$$ %$!. 9V - 7A!$$ 07& %&+! A"" S&'+ +%&!%" # & % $!, ",! %"!, *' %"""8 &'0!+"! $#!!$+$-"!".*'!+$A$!! S,Metall.andMat.Trans.B ==: <4 G0== E<F%$!. 9V - 7A!$$ %&+! A"" $$ R&'+ +%&!%$ + $!,! %"!, )!&.*'!+$ A$!! S Modelling in Welding, Hot Powder Forming and Casting '$""?@ '! $ ==4 %&*5 <=022 E2FW!+C7&! S&+ A+ #&% %A%"!,"'#% '!, +"&-0-*" $!!#!%!!" ##% && '"#'% ##!%!S Metall. and Mat.Trans.B ==G <= 5;05;: E>F ' A ',-' * $ " S&!+*$+! # +&+!%$ + $ %&'%'!I+ $+$!'#%##%"!+$%"!,S,CanadianMetall.Quarterly ==G 24 G;0=: <5
E;F!%&+! S& '- #&'+$"'"""!'0,*# '+! '!,& '$-"," #" $!!#!%!!'%,$'+ $S J.Mech.Solids =4 = <420<G> E:F "%&!""!(%& S!%' %' "!+$! # 7"!, ' %"""S %' " # > X+ % $ Q99 7 '1' $$" '% ==G *($!"&(-!! "7 ==G %&*7G 0>< E4F&*8HH)))+,+" #% + EGF&*8HH)))%$% +%&H' %"H' %"&+$ E=F7*'"! )))'"A$ '% +H)))"%% "$"% +H E<5F %&+! Heat transfer and air gap formation in permanent mould casting of aluminium alloys & &"!" '!0%05;> & -$ "! # %& $,- %/& $+ ) ==> E<F' "" ''!/"" S!',*# '+! '!," $!!#!%!! %-$!'!%$%"!," #*'$+!!+%!%$0!S Int.J.CastMetalsRes. <55< > <4;0<G; E<<F' ''!/"" R!',*# '+! '!," $!!#!%! #$+!!+(" $$ -"! *'+ + $S ' % '!$" $! " 7 #'%.* "!! <55< % ('405 7 $+(" &! *($!"&(-'! $ &! <55< E<2F +(', ", R'+!! # '""0"'! 7'A" (- &' '+'"S %&!%$ ;52! $A!" '-7 ++! '!%7?=>@ E<>F", R'""0"'!# '+$"S J.Aeronaut.Sci =>: 2 >;0>G E<;F &&"!" %&!%$!A'"!-?@ %& '+- ==> E<:F&*8HH)))"0" #)'% +H E<4F7!+ '" R,+,',!'+ #7 %' ($+" A $A!,'!%! S Comput.Struct. ==< >< =40: E<GF'I- R+$' ($+"!B!"% *$"!%!-S Adv.Appl.Mech. =:: = <>20 244 E<=F + " W W&, R7 *$ &'+$0#$!"0"'"" $-"!" # 7"!,"S Proc. 9 th Int.Conf.onModelingofCasting,WeldingandAdvancedSolidificationProcesses %&?'+-@ <555 &+ " 7 $-?"@ &/' B'$, %& <555 G50G4 E25F Modélisation tridimensionnelle par éléments finis pour l analyse thermomécanique du refroidissement des pièces coulées?three5dimensional finite element modelling for thermomechanical analysis of the cooling of cast parts@ & &"!"?! #'%&@ % $"!"'!" ==G E2F $$ R +'!%$ +%&!%$% *$!, $, '!&+# ' # '+($ (!"8 **$!%! *'H+ $!'%!! %"!, *' %""S Proc.8 th Int.Conf.on <
Modelling of Casting, Welding and Advanced Solidification Processes!,?7 @ ==G & +" 7 %/'+?"@ &!'$" $" '!$" %!- ==G 42=04>: E2<F$$ R!!$+**' %& #&'+ +%&!%" #" $!!#!%! *' %"""S Proc. Int. Conf. On Cutting Edge of Computer Simulation of Solidification and Casting "/?*@ === &/C"?"@ &' $"! # *?@ === 420=5 E22F$$ 7$!, R!!$+"# '&'+ +%&!%$ #$-"!" # " $!!#!%! *' %"""S Proc.9 th Int.Conf.onModelingofCasting,Weldingand Advanced Solidification Processes %&?'+-@ <555 &+ " 7 $-?"@ &/'B'$, %& <555 504 E2>F$$!'$ R0**' %& &&'+ +%&!%" # " $!!#!%! *' %""" )!& **$!%! & *'!%! # *!* "&'!/,S ( *($!"&!Int.J.Num.Meth.forHeat&FluidFlow <<
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
($87 "A$"# '&&'"#'% ##!%! ()*'+ $# '"*% +*! " $$ - h castinghmould $023! $043!0523, 2<5+ 0< 0 G=G+ 0< 0 <>
($<8& $,!%$*' *'!" #$043!0523,"!7% +*! ' % #!!$!-'" " %$ A$"% (,!A -*!%$ A$" ',!A &!'* $! ()&,!AA$""& $ (""+$!' $" 0A!"% *$"!%!- ($ )%'!!%$+*''T C N;:=D7 +*'' ED7F 7 "!"%-K EF '!' ""!!A!- % ##!%!me0f '!, % ##!%!ne0f!%-!$ "'""σ y EF C,+ $" EEF <5 <555 555; 5 >; :G555 <55 <25 55 54 >5 :;555 >55 << 554 5: >; >2555 ;><?NT S @ ; 5<; 5 5 2555 ;:= 5< 52 55 5 255 B!"% *$"!%!- A'%'!!%$+*''T C N;:=D7 +*'' ED7F 7 "!"%-K EF '!' ""!!A!- % ##!%!me0f ;:= 5< 52 :2?NT L @ 2x5 02 5 G55 2x5 02 5 <;
LVDT:s INSULATING GRANULES THERMO- COUPLES CORE CASTING MOULD LVDT- POSITION 24 INSULATION 150 250 LVDT:s!,'8 +'- #&.*'!+$"0* <:
800 700 Measured temperatures for unmodified Al-Si 800 700 A 600 200 100 0 100 200 600 600 200 100 0 100 200 600!,'<87 $!,%'A"#' +&.*'!+"8%!%$$ -$023!?+"'+ r N 45; ++! & %"!, r N 44 ++! & + $@ 8 $043!0523,?+"'+rN:=++!&%"!,rN44++!&+ $@ B <4
Displacement [m] 2 x 10-4 1 0-1 mould casting Displacement [m] x 10-4 1 0-1 -2-3 mould casting A -2 0 100 200 600 B -4 0 100 200 600!,'28"''!$!"*$%+"!.*'!+"8%!%$$ -$023! 8$0 43!0523, ( & %&'" &!' "'#% # & + $ &" * "!!A '!$!"*$%+)&!$&"'#% #&%"!,&"#!'"* "!!A # $$ )!,&+ $ +! &,!A " $!!#!%! "&'!/, <G
Heat transfer coefficient [W m -2 K -1 ] Heat transfer coefficient [W m -2 K -1 ]!,' >8 7$%$& '"#'% ##!%!"# ' &.*'!+" 8 %!% $$ - $0 23! 8$043!0523, <=
700 600 200 100 Casts Magma Procast Thercast 8+*''!% ''NG++ 800 800 700 700 600 600 Casts Magma Procast Thercast Experiment Casts Magma Procast Thercast 8+*''!*''N>;++ 78+*''!*''N:=++ 200 200 100 Experiment Casts Magma Procast Thercast 8+*''!+ $'N44++ 100 Casts Magma Procast Thercast 8+*''!+ $'NG;++!,';8"$"# '"*$043!0523,'&'"#'%$%$! ")!&% " &'"#'% ##!%!" 25
700 600 200 100 Casts Magma Procast Thercast 8+*''!% ''NG++ 800 800 700 700 600 600 Casts Magma Procast Thercast Experiment Casts Magma Procast Thercast 8+*''!*''N>;++ 78+*''!*''N45;++ 200 200 100 Experiment Casts Magma Procast Thercast 8+*''!+ $'N44++ 100 Casts Magma Procast Thercast 8+*''!+ $'NG;++!,':8 "$"# '"* $023! '& '"#'%$%$! ")!&% "& '"#'% ##!%!" 2
700 600 200 100 Casts Magma Procast Thercast (R2sol) 8+*''!% ''NG++ 800 700 600 200 100 Experiment Magma Thercast (R2sol) Casts Procast Experiment Magma Thercast (R2sol) Casts Procast 8+*''!*''N:=++ 78+*''!+ $'N44++ 0,0003 Radial displacement [m] 0,0000-0,0003-0,0006-0,0009-0,0012-0,0015-0,0018 Experiment part Experiment mould Casts part Magma part Magma mould Procast part Procast mould Thercast (R2sol) part 8*'+ $!"*$%+"!,' 48 "$"# '"* < $043!0523, &'+ +%&!%$%$%$! ")!&!+ *&'"#'% ##!%!" 2<
700 600 200 100 Casts Magma Procast Thercast (R2sol) 8+*''!% ''NG++ 800 700 600 Experiment Casts Magma Procast Thercast (R2sol) 8+*''!*''N45;++ 200 100 Experiment Casts Magma Procast Thercast (R2sol) 78+*''!+ $'N44++ 0,0003 Radial displacement [m] 0,0000-0,0003-0,0006-0,0009-0,0012 Experiment part Experiment mould Magma part Magma mould Procast part Procast mould Thercast (R2sol) part 8*'+ $!"*$%+"!,'G8"$"# '"*<$023!&'+ +%&!%$%$%$! ")!&!+* &'"#'% ##!%!" 22
700 600 200 100 Procast Thercast 8+*''!% ''NG++ 800 700 600 200 100 Experiment Procast Thercast Experiment Procast Thercast 8+*''!*''N:=++ 78+*''!+ $'N44++ 0,0002 Radial displacement [m] 0,0000-0,0002-0,0004-0,0006-0,0008-0,0010-0,0012-0,0014 Experiment part Experiment mould Procast part Procast mould Thercast part Thercast mould 8*'+ $!"*$%+"!,' =8 "$"# '"* 2 $043!0523, &'+ +%&!%$%$%$! ")!&,* *&'"#'% ##!%!" 2>
800 700 600 200 100 Thercast 8+*''!% ''NG++ 800 700 600 200 100 Experiment Thercast Experiment Thercast 8+*''!*''N45;++ 78+*''!+ $'N44++ 0,0002 Radial displacement [m] 0,0000-0,0002-0,0004-0,0006-0,0008-0,0010 Experiment part Experiment mould Thercast part Thercast mould 8*'+ $!"*$%+"!,'58"$"# '"*2$023!&'+ +%&!%$%$%$! ")!&,** &'"#'% ##!%!" 2;
700 600 200 100 Thercast 8+*''!% ''NG++ 800 700 600 200 100 Experiment Thercast Experiment Thercast 8+*''!*''N:=++ 78+*''!+ $'N44++ 0,0008 Radial displacement [m] 0,0002 0,0000-0,0002-0,0004 Air gap [m] 0,0006 0,0004 0,0002-0,0006 Experiment part Experiment mould Thercast part Thercast mould 8*'+ $!"*$%+" 0,0000 Experiment Thercast 8"" %!!',*A $!!,' 8 "$"# '"* 2 $043!0523, &'+ +%&!%$%$%$! ")!&,* *&'"#'% ##!%!" "!,$!1!0" $!% "!!A+ $!7 2:
!,' <8 "$"# '"* 2 $043!0, -"+% #!,'! 555" % +*)!& 7 #8 +(',0", $" 0*$"!% % "!!A + $!,&8!"!%! () #$!0$!/ % "!!A + $?)!HA!"% *$"!%@ " $!0$!/ % "!!A + $?$"!%0A!"% *$"!%@ 24