OPTIMIZATION OF INDUCTION HARDFACING
OF THIN DISCS, ALLOWING FOR THERMAL
AND ELECTROMAGNETIC SHIELDING
O .M . S H A B L Y 1, C h .V . P U L K A1 and A .S . P IS M E N N Y2
I. P u ly u j T crnopil S ta te Technical U n iv ersity , T e m o p il, U k rain e
2E .O . P a to n E lec tric W eld in g In stitu te , NASU, K yiv, U k rain e
A m ath em atical m odel has been developed to determ ine th e tem p eratu re field in th e process o f disc h ard facin g u sin g a tw o -tu rn rin g in d u c to r w ith allo w an ce for th erm al and electrom agnetic shielding. T his m odel enables o p tim izin g th e m en tio n ed te m p e ra tu re in th e zone o f disc h ardfacing depending on th e param eters o f th e in d u cto r, disc, electro m ag n etic and th erm al shields, as w ell as e le c tric c u rre n t.
K e y w o r d s : in d u ctio n h a r d fa c in g , th in discs, tw o -tu rn ring in d u cto r, o p tim izin g th e in d u c to r p aram eters, tem perature f i e l d , th e rm a l a n d elec tro m a g n etic sh ie ld in g , in vestig a tio n s, ca lc u la tio n s
In work [ 1 ] studies have been performed t o optim ize th e design parameters of a tw o-tu rn ring inductor for hardfacing thin round and shaped discs o f an arbitrary diam eter and w idth o f hardfacing zone, taking into account sh ield in g o f ju st the electrom agnetic field, in order to obtain a specified power distribution over th e hardfacing zone w id th . D im ensions o f the tw o- turn ring inductor are selected , depending on disc diam eter and hardfacing zone w id th , as w ell as al lo w in g for the values o f th e coefficien ts o f electro m agnetic field sh ield in g ( K sh = 1; K sh = 0 .2 5 , K & = = 0 — full sh ield in g ) [1 ].
It is o f interest to study the temperature field in th e hardfacing zone w ith developm ent o f a mathe m atical model to determ ine th e tem perature in the disc through the parameters o f a tw o-turn ring induc tor, w hich is used to perform heating, allow in g for electrom agnetic and therm al shield in g sim ultaneously [2 ], w hich significan tly affect th e temperature distri bution in the hardfacing zone. The developed model w ill allow designin g th e heating system (inductor, thermal and electrom agnetic shields, part) for hard- facing thin round and shaped discs.
Let a round disc (F igure 1) of 2h thickness and
r2 radius be heated, using a tw o-turn ring inductor.
In this case the specific power o f the heat sources, which emerge in the disc area under th e im pact o f an electrom agnetic field , has the form o f [1]
(
1)
oco2Ho
W -- --- 5“ X
1287E-/2
x [Al]A2a] + U \ a \ B 2 + Aha^C^e'2^ ' n A],
where A 2, B 2, C 2 are the functions o f radius r [3], geometrical dimensions o f the inductor and disc h\,
2h, a \ t #2» r 2 (Figure 2 ), as well as physical parameters of the electromagnetic field; A = V 2/(a)jia) is the depth of penetration o f the electromagnetic field into disc material (co is the circular frequency of the electromag netic field; |i is the magnetic permeability; a is the electric conductivity); no = 4n*10”7 H / m .
Temperature field in th e disc satisfies the equation of heat condu ctivity [4]:
c ^ t i a r _ 2T _ 1 5*1 = _ E
dr2 r dr a Bt X*
2 a
m = t t ,
(
2)
where X is the coefficient o f heat condu ctivity o f disc material; a is the coefficien t o f heat transfer to theF ig u re 1. C ross-section of th e h e a tin g system : 1 — p a rt; 2 — ch arge; ) - tw o -tu rn rin g in d u cto r; </, 5 - th erm al and electro- nugm*t*c sh ield s, respectively
F ig u re 2. Schem atic of a disc w ith th e in d u cto r: / — u p p e r tu rn of th e in d u cto r; 2 — low er tu rn
en viron m en t In th e absence o f a shield;
T
■
T\
-
f cn( T \ %
^ th e tem perature o f th e disc and environm ent, resp e c tiv e ly ); boundary con d ition s are as fol low s: Y " = 0 at r = 0; or dT* X — + a T = 0 at r = r>. or (3) (4)
A nalysis o f calcu lation s, g iven in [1 ], show s that electrom agn etic sh ield in g o f the specific power o f heat sources at disc edge sig n ifica n tly influences the uni
form ity o f its distribution along the radius, particu larly over the w idth o f th e hardfacing zone O2 - rg ® = 1 0 -5 0 m m ). If thermal sh ield in g is also im ple m ented at disc edge (see Figure 1), the heat flow through the edge w ill be sign ifican tly reduced or stopp ed com p letely, w hich w ill strongly influence the level o f tem perature distribution in the hardfacing zone, and th e heat losses through the edge w ill also be reduced.
In the case o f electrom agn etic shieldin g o f disc edge formula ( 1 ) may be expressed as follow s:
2 2
crarw
w = --- — x 1 28tc /i
x t&I2A2ci2 + &l\a2B2 + K s\i\ha\l2C2e{~2(r- “r>)
(5)
K sh = exp _ 2 — 1
Ashj*
(
6)
w here d s h is the sh ield thickness; Ash = V2/(o)|xshCTsh) is th e depth o f electrom agn etic field penetration into th e shield; a sh is th e electric conduc tiv ity o f th e sh ield m aterial.A ssum ing sh ield thickness d 9h = Ash, th e pow er of heat sources on disc edge w ill decrease e tim es, if rfsh = 2Ash» it w ill decrease e4 tim es, at d sh = 4Ash it w ill decrease e8 tim es (i.e . it w ill be p ractically zero). At K sh = 0 w e achieve full electrom agn etic sh ield in g, at /Csh = 1 th e sh ield in g is absent.
W hen a therm al shield is also in stalled at disc edge, boundary condition ( 4 ) has the fo llo w in g form:
\ 4 L + K .a T = 0
d r at r = r2, (7)
where K t is th e co efficien t o f therm al sh ield in g [6 ], w hich varies in th e range o f 0 < K t < 1. At K t = 0 w e have full therm al sh ield in g , at = 1 the sh ield in g is absent. C o efficien t o f therm al sh ield in g , w hen using
a
sh ield o f th ick n ess d t, w ill be found from th e following relationship [6]:
K ta « £ ; K t «
d t 1 d ta
(
8)
where A, / d t is th e co effic ien t o f heat c o n d u ctiv ity o f the shield.
I f X / d t = a , then K t = 1, i.e. sh ield in g is ab sent,
an inten sive co n v ectiv e heat exch an ge w ith th e en vi ronment is in place and boundary con d ition ( 7 ) has th e form o f equation ( 4 ) .
Let us assume th a t at th e in itial m om ent o f tim e the disc tem perature is equal to th at o f th e environ m ent. Then, initial co n d itio n for equation ( 2 ) w ill be expressed in the fo llo w in g form:
T = 0 at t = 0. (9)
Solu tion o f eq uation ( 2 ) at boundary con d ition ( 7 ) and in itial co n d itio n ( 9 ) in th e case, w hen th e specific pow er is determ ined from form ula ( 5 ) , has the fo llo w in g form:
' Jie(r, (*/„(/„, r)rdr - ^ J 2 _ ---0 r)rdr
(
10)
J o(^v» r),where /Csh* the coefficien t o f electrom agnetic sh ield in g for a sh ield , located close to the disc [5 ], has the fo llo w in g form:
where /J = Xv - m 2; Xy = V/J + m 2 \ S&Qy* r ) is th e B es sel function o f the first kind of zeroth order o f the real argument; a is th e therm al d iffu siv ity; /v are th e roots of th e characteristic equation
X/v y ,(/v, r2) + a K t / 0(/v, r2) = 0.
(
11)
Thus, a m athem atical m odel is obtained o f deter m ination o f tem perature in th e disc through th e source o f its induction h eatin g, using electrom agn etic and thermal sh ield in g o f disc ed ge. T his a llo w s determ i nation and optim izing th e above tem perature in the zone of disc hardfacing, d ep end ing on the parameters o f the inductor, disc, electrom agn etic and therm al sh ield s, as w ell as electric current.
To optim ize th e param eters o f th e inductor and electric current, flo w in g through it, it is necessary to optim ize th e fo llo w in g functional:
T r 7
0 = 1 \ ( T - ThA)2 rd rd t.
(
12)
w’here Th.d is th e tem perature at w h ich sound hard- facing o f a p ow d er-lik e hard a llo y on th e w ork in g surface o f th e disc is perform ed; T is th e tem perature, w hich is determ ined from form ula ( 1 0 ) .
•c
F ig u r e 3. Temperature distributions around disc radius: r2 =
= 0 .1 2 5 m for different moments in tim e: / — 22; 2 — 19; 3 —
15; 4 - I t; 5 — 5 s
R esults o f temperature calculation (*C) in different m o m e n t of tim e, depending on disc radius
f t. nt T 5 11 15 19 22 0 .0 1 0 129.82 347.82 543.63 792.95 1023.71 0.020 131.79 352.20 550.40 802.74 1036.20 0.030 133.33 357.25 558.69 815.69 1052.42 0.040 137.38 367.11 573.66 836.58 1079.77 0.050 140.55 376.76 589.08 859.19 1108.96 0.060 145.98 390.55 610.06 889.09 1146.96 0.070 150.74 402.31 627.75 914.10 1178.64 0.075 155.25 412.48 624.39 934.13 1199.76 0.080 154.59 410.35 639.98 931.55 1202.21 0.090 154.01 411.21 641.67 934.36 1204.68 0 .0 9 5 152.86 411.99 641.96 935.54 1205.64 0.1 0 0 154.90 413.08 644.25 937.72 1205.64 0 .1 0 5 154.57 412.16 642.79 935.57 1205.93 0 .1 1 0 154.62 411.99 641.96 934.56 1201.72 0 .1 2 0 156.55 4 13.50 642.76 933.56 1201.99 0.124 157.89 414.78 643.65 933.87 1201.74 0 .1 2 5 157.67 414.10 642.54 932.22 1199.59
entire working surface of a thin round and shaped
disc.
As an example of calculation of the inductor, allow
ing for the design features and optimization capabilities,
let us assign the geometrical dimensions of the heating
system: radius of the inductor external turn#j = 0.131 m
(see Figures 1 and 2), criterion B, = 0.27 for a thermal
insulation material on the external contour of the disc
of asbestos sheeting of thickness
d t
= 0.004 m, radius
r
3= 0.075 m (see Figure 1), hardfacing time x = 22 s,
X » 0.35 W /(m -#C).
Experimental data confirms that hardfacing tem
perature
T\
= 1220 #C. Then, at
Tcn
= 20 °C we have
T'h.d = 1200 °C. If the electromagnetic shield is made
of copper, at circular frequency co = 2n-440 kHz the
depth of the electromagnetic field penetration into
the shield is A, = 0.1 mm.
Let us assume #
2» ^
2»
A , K e
to be the optimi
zation parameters. Performing the procedure of func
tional minimizing by these parameters, we obtain their
values:
= 0.0945 m;
A
= 165.20;
K e
= 0.655;
h i
=
= 0.0315 m;
a\ -
0.131 m;
h\
= 0.01 m;
r$
= 0.075 m.
Results of temperature calculations in the disc area
at these values in different moments of time are given in
the Table, and their graphic representation — in Figure 3.
It can be seen from the Figure that the temperature is
almost the same across the width of the hardfacing zone
(in this case 5 = r
2- r
3= 0.125 - 0.075 = 0.05 m),
deviation from the specified temperature being 0.5 %,
and it is equal to 1200 °C over time x = 22 s (which
is highly important at induction hardfacing). As the
hard alloy (for instance, PG-S1) melts from the sur
face of the base metal, the thickness of the deposited
metal is uniform over the entire working surface.
It follows from the conducted analysis that the
final (at x = 22 s) temperature in the hardfacing zone
in this case deviates from the required one by not
more than 0.5%. For implementation of the found
optimum coefficient of electromagnetic shielding/C<, =
= 0.655 it is sufficient, according to formula (6), to
use a copper plate of thickness
d e
= 0.021 mm, i.e.
spraying of copper pow’der onto the thermal shield or
pasting of copper foil of the same thickness on it can
be performed in practice.
The following data are assumed in calculations:
• for disc: 2
h
= 3 mm; c = 846 J /(k g -#C); X =
= 40 W /(m °C);
y =
5969.2 k g /m 3; a = 1.25-1 O’*
l/(O hm -m ); r
2= 0.125 m; a = 455 W / ( m 2-°C);
r$
=
= 0.075 m; x = 22 s; 7Vd = 1200 °C. Base metal is
steel St3, deposited layer is of alloy PG-S1, thickness
of deposited metal is 0.8-1.5 mm;
• for inductor (copper): n = 2.75^o>
£0=
= 8.854-l0“12 F /m ; n0 = 4n-10"7 H /m ; o =
= 2.763-10"6 Hz;
C\
= 5.0 mm;
c-i
= 8.0 mm; p =
= 0.17-10'7 Ohm*m;
• for electromagnetic shield (copper):
0,1 = P = b . ' ^ K T 7 = 5 8 l 8 ' 1C)6
l/(O h m m); ne =
= HHo = 1-4J110'7 = 12.5610'7 H /m ;
K c
= 0.655.
,
t * e l
i
(
___________________ __ 9/2003
f ' ' SCIENTIFIC AND TECHNICAL V
For thermal shield (asbestos
s h e e tin g ) : d { * • 0004
m ; X -0.35 W
(n v * C );a
«455
W / ( m 2-°C );K%
* X W ta ) * 0.192.
C O N C L U S I O N S 1 T h e d e r iv e d m a th e m a ti c a l m o d e l fo r d e te r m in a tio nof
t h e t e m p e r a t u r e in t h e d is c th r o u g h th e p a ra m e te rsof
a t w o - t u r n r in g i n d u c to r , w h ic h is u sed to p e rfo rm h e a t i n g w ith e le c tr o m a g n e tic a n d th e rm a l s h ie ld in g , a llo w s o p tim iz in g t h e a b o v e t e m p e r a t u r e in th e zo n eo f
d is c h a r d f a c i n g , d e p e n d in g o n t h e p a r a m e te r s o f t h e i n d u c to r , d is c , e le c tr o m a g n e tic a n d th e r m a l s h ie ld s , a s w e ll as e le c tr ic c u r r e n t in th e in d u c to r . 2. D e v e lo p e d a l g o r ith m a ls o a llo w s d e s ig n in g t h e h e a t i n g s y s te m ( i n d u c t o r , th e r m a l a n d e le c tr o m a g n e tic s h ie ld s , p a r t ) , w h ic h p r o v id e s t h e r e q u ir e d co n d i t i o n s fo r p e rf o rm a n c e o f t h e te c h n o lo g ic a l p ro c e ss o f h a r d f a c in g . 3 . T h e d e v e lo p e d h e a ti n g s y s te m e n s u r e s th e re q u ir e d te m p e r a t u r e in t h e h a r d f a c in g z o n e w ith th e a c c u r a c y o f u p t o 0 .5 %. 4. A p ro c e d u re h as b e en d e v e lo p e d fo r fin d in g th e c o e ffic ie n ts o f th e e le c tr o m a g n e tic a n d th e rm a l s h ie ld s , w h ic h a re u sed fo r te m p e r a tu r e re g u la tio n a c ro ss t h e w id th o f th e h a r d f a c in g z o n e w ith a c o m p le x g e o m e tric a l s h a p e o f th e s u rfa c e . 1. S h a b ly , O .M ., P u lk a , C h .V ., P ism e n n y , A .S . (2 0 0 2 ) O p t i m isatio n o f in d u c to r p a ra m e te rs fo r u n ifo rm h e a tin g o f d iscs across th e w id th o f th e h a rd fa c in g zone, a llo w in g fo r sc re e n in g . T h e P a to n W e ld in g J ., 11, 2 3 -2 5 .2. S h a b ly , O .N ., P u lk a , C h .V ., M ik h a jlis h in , M .S . e t a l. D e- v ic e fo r p o w e r c o n tr o l in zo n e o f h a r d fa c in g . P o s itiv e d e c i sio n on issu in g o f p a te n t o n a p p lic a tio n 200 2 1 1 9 4 9 1 . F ile d 2 8 .1 1 .0 2 .
3 . S h a b ly , O .N ., P u lk a , C h .V ., P ism e n n y , A .S . (1 9 9 7 ) O p t i m isa tio n o f d e sig n p a ra m e te rs o f in d u c to r fo r in d u c tio n h a rd fa c in g o f th in ste e l d isc s. A v to m a tic h . S v a r k a , 6, 1 7 - 20.
4. S h a b ly , O .N ., P u lk a , C h .V ., B u d z a n , B .P . (1 9 8 7 ) O p tim i s a tio n o f p o w e r c o n s u m p tio n d u r in g in d u c tio n h a rd fa c in g o f th in -w a lle d d iscs. I b i d . , 1, 3 6 - 3 9 .
5 . S em y o n o v , N .A . (1 9 7 3 ) T e c h n ic a l e le c tr o d y n a m ic s . M o s cow': Svyaz.
