The realistic boundary condition of the surface temperature varying with time is used in the heat transfer analysis of the solidification of a binary eutectic alloy system. The system occupies a semi-infinite region and consists of a solid zone, a freezing zone, and a liquid zone. The heat released in the freezing zone is treated as a discontinuous heal generation. The solid-fraction distribution in the freezing zone, given as a function of temperature, is solved simultaneously with the temperature distribution, which is given as a function of distance within the freezing zone. Using the &quot;heat balance integral&quot; technique, the method for the solution of the temperature distribution and the thickness of each region is given. A s an example, the results are shown in graphical form for an Al-5 percent Cu alloy

G E Geiger

R H Tien

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R. H . T I E N 
Senior Scientist o f E d g a r C. Bain 
L a b o r a t o r y f o r Fundamen ta l Research, 
U. S. S tee l C o r p o r a t i o n , M o n r o e v i l l e , Pa. 
G. E. G E I G E R 
Assoc ia te Professor , D e p a r t m e n t o f 
M e c h a n i c a l Eng ineer ing , Univers i ty o f 
Pittsburgh,. Pi t tsburgh, Pa. 
The Unidimensional Solidification of a 
Binary Eutectic System Willi a Time-
Dependent Surface Temperature1 
The realistic boundary condition of the surface temperature varying with time is used 
in the heat transfer analysis of the solidification of a binary eutectic alloy system. The 
system occupies a semi-infinite region and consists of a solid zone, a freezing zone, 
and a liquid zone. The heat released in the freezing zone is treated as a discontinuous 
heal generation. The solid-fraction distribution in the freezing zone, given as a func-
tion of temperature, is solved simultaneously with the temperature distribution, which 
is given as a function of distance within the freezing zone. Using the "heat balance 
integral" technique, the method for the solution of the temperature distribution and the 
thickness of each region is given. A s an example, the results are shown in graphical 
form for an Al-5 percent Cu alloy. 
In t roduct ion 
THE change from the liquid to the solid phase during 
the solidification of an alloy does not take place isothermally, as 
is the case for a pure metal. The liquid starts to freeze at the 
liquidus temperature, and freezes partially and gradually until 
its temperature drops to a eutectic temperature; then the re-
maining liquid freezes isothermally at that temperature. The 
physical representation of this process can therefore be con-
structed b y separating the liquid and the solid b y a "freezing-
zone" which is formed b y two isothermal surfaces at the liquidus 
and the solidus temperatures, respectively. The mathematical 
model consists of the Fourier's conduction equation and the 
solid-fraction temperature relationship within the freezing zone. 
A general account of the derivation of the relationship between 
the temperature and the solid fraction within the freezing zone is 
given b y Brody [ l ] 1 and Pfann [2], 
The one-dimensional solidification of a binary eutectic system 
with a constant surface temperature, and assumed solid fraction 
distribution within the freezing zone, was analyzed previously 
b y the authors [3] from the heat transfer point of view. A more 
realistic model is one in which the cooling temperature at the 
1 The material presented in this paper consists of a portion of a 
PhD dissertation submitted by R. H. Tien to the Department of 
Mechanical Engineering, University of Pittsburgh, Pittsburgh, Pa. 
2 Numbers in brackets designate References at end of paper. 
Contributed by the Heat Transfer Division of T H E A M E R I C A N 
S O C I E T Y OF M E C H A N I C A L E N G I N E E R S and presented at the ASME-
AICliE Heat Transfer Conference and Exhibit, Seattle, Wash., 
August 6-9, 1967. Manuscript received by Heat Transfer Division, 
February 21, 1967. Paper No. 67—HT-43. 
surface is time-dependent and the solid fraction distribution and 
the temperature distribution are solved simultaneous]}' f rom 
Fourier's conduction equation and a specific relationship be-
tween the temperature and the solid fraction. 
The time-dependent boundary condition prohibits an error 
function form of solution. Applying Goodman's [4] approximate 
method of the heat balance integral to both the solidified part 
and the freezing zone reduces the system from a set of two second-
order partial differential equations to a set of two first-order 
ordinary differential equations, with the thickness of the solidi-
fied part and the freezing zone as dependent variables. The 
latter then is solved by the Runge-Kutta method, a simple but 
reliable numerical technique. 
The temperature distribution within the freezing zone depends 
on the heat generation arising from freezing, and is dependent 
upon the solid fraction distribution. If this temperature dis-
tribution can be expressed only in terms of a dimensionless 
space variable, x, such that the functional relationship is inde-
pendent of the form of an assumed solid fraction distribution, 
then the combination of this temperature distribution with a 
specific solid-fraction temperature relationship is the actual solid-
fraction distribution in terms of the same dimensionless space 
variable. 
Sta tement of the P r o b l e m 
The liquid, at liquidus temperature, initially occupies a semi-
infinite region which extends over the positive x-axis. At time 
zero, the surface (x = 0) is at solidus temperature. At later 
times the temperature is allowed to decrease in an arbitrary 
^ N o m e n c l a t u r e -
.4 = internal heat generation, 
Btu / f t 3 -hr 
c = specific heat, B t u / l b cleg F 
Co = initial concentration of the 
alloy 
e = position of solidus front, f t 
At = thickness of freezing zone, 
f t 
f , = solid fraction 
f,u = eutectic solid fraction 
k = equilibrium distribution co-
efficient 
K = thermal conductivity, B t u / 
ft-hr-deg F 
L = latent heat of fusion, B t u / 
lb 
M i = magnitude of the slope of 
the liquidus line, deg F 
a = thermal diffusivity, f t 2 /hr 
p = density, l b / f t 3 
R = rate of temperature change 
at cooling surface, deg 
F / h r 
t = time, hr 
T = temperature, deg F 
To = equilibrium temperature 
for zero concentration, 
deg F 
T , = solidus temperature, deg F 
= liquidus temperature, deg 
F 
Tc = surface temperature, deg F 
AT = temperature difference be-
tween liquidus and soli-
dus, deg F 
x = distance, ft 
subscript 1 = solidified part 
subscript 2 = freezing zone 
Journal of H e a t Transfer F E B R U A R Y 1 9 6 8 / 2 7 
Copyright © 1968 by ASME
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L IQU ID 
S O L I D AND LIQUID 
S O L I D 
Y 
X = 0 x = € X=e+A€ 
F i g . 1 S c h e m a t i c r e p r e s e n t a t i o n 
The coefficients C,'s are chosen such that 
f,(x = 0) = / , „ i.e., Ci s are all finite 
and 
f.(S = 1) = 0 i.e., £ Ct = 1 
i = l 
Combining equations (4) and (5) provides 
pL ( de _ dAe\ 
A = Ae\-cU+Slu E , i = i 
where .%- = 
Ae 
(6) 
manner. A schematic representation of this problem is shown 
in Fig. 1. The rate of change of the solid fraction with respect 
to time within the freezing zone provides the latent heat effect. 
A solid-fraction temperature relationship can be written as, 
/ . = 1 -
M,C 0 
r 
i 
i - i t 
(i) 
d-T, 1 dT, 
ds2 dt 
for 0 < a < e (2) 
d=T2 A 
dx* K, 
1 M\ 
a2 dt 
for e < x < e + Ae (3) 
where Ti and Ti are relative to the solidus temperature, Ts. 
The boundary conditions are: 
1 r,(s = 0) = -(T, - Tc) 
2 Ti(x = e) = 0 
3 T2(x = e) = 0 
4 T2(x = e + Ae) = AT 
dT, 
5 — (x = 6 + Ae) = 0 
ox 
6 Iu ^ (x = e) = Kt ^ {x = e) + pL( 1 - / , „ ) ^ 
OS 03: «< 
The initial conditions are: 
7 £(i = 0) = 0 8 Ae(i = 0) = 0 
The first five boundary conditions specify the temperature or 
the temperature gradient at the respective positions. The sixth 
boundary condition represents the conservation of energy at the 
eutectic front. The second term on the right side of the equation 
represents the latent heat effect clue to freezing of the remaining 
liquid at the eutectic temperature. 
The internal heat generation term, A, associated with the 
latent heat effect, can be written as 
A = pL •Ml 
dt 
(4) 
where the solid fraction, f„ can be represented generally b y a 
ii th degree polynomial. 
fs J si 1 - E C{x* 
i = l 
(5) 
S o l u t i o n 
I t will be assumed that the "thermal layer" which has been 
defined by Goodman [4] in the heat-balance-integral method is 
the en the solidified part of the system in which equation (2) 
is applicable. Then integrating equation (2) with respect to 
x f rom x = 0 to x = e results in 
For the normal nonequilibrium mode of freezing, the derivation 
of equation (1) is given b y Pfann [2]. 
Assuming constant physical properties, one-dimensional tem-
perature variation, conduction as the only heat transfer mechan-
ism, and the heat of fusion the only internal heat generation, 
Fourier's conduction equation for the solidified part and the 
freezing zone can be written as: 
dT! dT7! 
— 3 = 0 - — ( 3 = 0, t) = 
OX ox 
1 re dTi 
« J o &f 
dx (7) 
B y defining 
J o 
Ti(x, t)dx 
the right side of equation (7) can be reduced to 
2 
a i 
d&i „ , de 
~di + ™ = « ¥ (8) 
b y using Leibniz's formula. 
B y combining equations (7), (8) and the boundary conditions 
(2) and (6), equation (2) becomes 
IU dl\ 
— — (x = e, t.) 
Iu ox 
PzL 
Ki 
( 1 / ) -JSUJ i , at 
f — It w 
ox at 
Next the "thermal layer" for equation (3) is defined as the 
entire freezing zone. Then integrating equation (3) with respect 
to x f rom x = e to x = e + Ae yields 
dT2 dT, 
— (x = e + Ae, t) - —" (x = e, t) 
i r e + A e i r - A e dJj 
dt 
dx (10) 
On the left side of equation (10), the first term is eliminated 
b y applying boundary condition (5), the second term is replaced 
b y boundary condition (6), while the last term can be integrated 
with the aid of equation (6). The right side of equation (10) 
can be reduced in a similar manner to 
1_ d9i 
dt 
+ AT 
de dAe\ 
~dt ~dt ) _ 
where 
j: 6-1-Ac Ts(x, t)dx 
With the aid of all the foregoing conditions, equation (3) is 
reduced to the following final form: 
2 8 / F E B R U A R Y 1 9 6 8 T r a n s a c t i o n s o f t h e A S M E 
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\C2 / df _ C2 V i t i i 4 - J 7 
C{ + AT 
dAe 
dt 
iu ar. rfft 
— (a = e, J) = — (11) 
pc2 ox dt 
N o w a quadratic temperature distribution in both the solidi-
fied part and freezing zone is assumed, 
T,{x, t) = + A, (e - a ) + A s (e - x¥ 
T2(x, t) = A, + A& - e) + - e)2 
(12) 
A 4 = 
2AT 
Ae 
AT 
A > = " A e 2 
= 
1 
1 
(ft + ft) 
where 
A2 = - - [(? ' , - r . ) - ( f t + ft)] 
A i Ae c i 
ft = ^ft* L + 2 - (1 C2 • / . u ) ( r . - r . ) 
Hence the temperature distribution in the two regions is: 
r , ( « , o = - - ( f t + ft)(e - « ) 
- - r [(r, - Tc) - ( f t + ft)](e - a ; ) 2 
2A71 A T 
T&, t ) - — (x - e) - — (x - e)2 
Equation (19) can also be represented b} ' an ;ith degree 
polynomial using a modified least-squares method as shown in 
equation (5). However, the coefficients now are not assumed 
but calculated from equation (19). 
Wi th the temperature distributions Ti and T*_ as known func-
tions of x, e, and Ae as shown in equations (16) and (17), the 
parameters 61 and d-2 are calculated according to their definitions. 
Upon substituting these into equations (9) and (11), two first-
order ordinary differential equations of e and Ae are obtained 
with respect to time, t, as independent variables as: 
where the are independent of x, and can be determined by 
the prescribed boundary conditions. 
From the boundary conditions (2) and (3), Ao and A3 are zero. 
B y using the boundary conditions (4) and (5), A t and A i are 
determined as: 
de 
dt 
dAe 
dt 
NZM, - AWs 
NiMi 
A W s 
M i 
N3M1 
ATiMt - N«Mi 
= Me, Ae, t) 
= Me, Ae, I) 
(20) 
(13) 
where 
iVi = + - (1 - / . „ ) Ci 
Ae 1 + S 
Ai and A i must be solved by the use of boundary condition 
(1) and the combination of boundary conditions (2) and (6), 
and equation (2). The boundary condition (6) alone is not 
suitable for the present case. Since — appears in boundary 
at 
condition (6), this will make equation (9) a second-order differ-
ential equation in e which cannot be solved because only one 
initial condition for e is available. 
1 dT 
N* = ^ e ~ $2 - a, ( A, + 2A2e + „ , , 
3 dt \ Ki Ae / 
K* 2AT\ 
Mi = — + AT 
Ci 
M, = - / . „ £ 7 
Ci \ * — C.A + - AT t i i + 1 7 3 
(14) 
K l a 
Mi = — — Aioct 
K 2 
ft + ft + 2 { T , - Tc) + ^ AT ± 
Kx Ae 
1 L , „ 1 
2̂  c7 ft 
(15) and Ah Ai, ft, and ft are given in equations (14) and (15). 
T o generalize the solution, the following dimensionless varia-
bles are defined: 
e Ae la, 
1 = D ' A ' = D ] l = ^ T = 
T, 
AT 
(16) 
(17) 
_ IU d _ a-z 
A = — ; c = — ; a = ; 
K i Ci ai 
L = 
c\AT' 
where D is a characteristic length. Then 
A, - <lr) = T ( f t + ft) 
B y defining x = 
Ae 
-, equation (17) can also be written as 
Tt(x) = 2 A T x - ATx2 (18) 
The functional relationship between T 2 and x as shown in 
equation (18) appears to be independent of the distribution of 
the solid fraction f„ however e and Ae which appear in x are 
functions o f / , . Hence, substituting equation (18) into equation 
(1) will give the true solid fraction distribution in terms of x, and 
is shown as: 
/ = i _ r m a - p 9) 
L ( 2 ' „ - T.) - AT(2x - * 2 ) J U J ; 
Journal of Heat Transfer 
D"- 1 
A, = A2 ( — ) = - IT. + ( f t + ft)] 
- i T - * * - 1 1 1 - ' " * 
AT 
yft2 2 L — (1 - U T C 
= 
AT 
ft + ft - 2 Tc + K 
$ , = $1 = 1 + ^ ( 1 - / , „ ) i 
/ 0 2 
1 + J ft 
F E B R U A R Y 1 9 6 8 / 2 8 
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M2 L . _ 
M l = — = - / , „ E 
ji/3 = 
AT 
M, 
\i+ 1 
a 1 r (ft + ft) 
Ar> = ^ = + i d - / .„> 
N3 = iYa 1 _ — e — -
3 dt 
A1 + 2,4,1 + 
2K 
" " ^ A T o , 
Therefore equation (20) can also be written as: 
de N3M2 - NsM3 
dl NiMi - NiMi 
= Uh Ae, 1) 
dAe NUM3 - N,Mi - , 
6n+i = 6„ + (»o + 2?i! + 2ns + "3) 
6 
i tr . i l = Ae„ + — (»!o + 2>m + 2rm + Viz) 
o 
where 
wo = MM* h, M.) 
m0 = Alft(ln, e„, AeJ 
1 1 1 
>h = A Ift (?„ + -— At, e„ + — no, Aen + — mo 
nil = Alft Z„ + — At, e„ + - Vo, Ae„ + 77 ?»o 
112 = Alft ( k + \ A/:, 
1 1 1 
+ A'en + V '»1 
1 
To = 1220 deg F 
71, = 101S deg F 
AT = 173.1 deg F 
This fs ~ :v curve is then fitted b y a tenth degree polynomial 
with the maximum difference less than 7 X 1 0 - 5 through a 
modified least-squares method. Thus, 
!, = f.u 1 
9 
E c (s (23) 
where 
(21) 
c, = 1.9516656 X 10" 1 
c. = 2.6572437 X 10" 1 
c3 = 1.940S31 X 10" 1 
Ci = 3.52208 
C5 = - 2 . 3 7 9 3 6 4 X 101 
CO = 9.227445 X 101 
c , = - 1 . 9 9 7 9 6 7 4 X 10'2 
Cs = 2,4912653 X 102 
C'S = — 1.6538519 X 102 
C10 = 4.4397551 X 101 
The Runge-Kutta [5] method is used to solve equation (21) 
with the initial conditions (7) and (8) for e and Ae. With a 
uniform interval At, the (11 + l ) th values of e and Ae can then 
be calculated in terms of nth values of l and Ae as: 
(22) 
With a solid fraction distribution as in equation (23) and a 
linearly decreasing surface temperature such as 
TC = Ta — R-t; or Tc = -Rl (24) 
/ D 2 \ 
where R = — — E. the positions of those two isothermal 
\ « i A T) 
fronts e and Ae are calculated b y equation (22) with a carefully 
selected time interval, A?, to insure convergence. The following 
physical properties are necessary to proceed with the calculation: 
IU = 114 (Btu/ f t -hr deg F ) 
Ko = 105 K = 0.921 
c, = 0.25 (Btu / lb deg F ) c = 1.2 
c2 = 0.3 (Btu / lb deg F ) 5 = 0.7675 
p = 170 ( lb / f t 3 ) or L = 3.92S 
L = 170 ( B t u / l b ) 
/ . . = 0.903 
T1 = 1191.1 (deg F ) 
T. = 101S (deg F ) 
ftu = 0.903 
E = 5.75 
m, = Alft [i„ + - At, e„ + - mi , Aen + - m i 
1H = Alft(l„ + At, en + 712, Aen + mi) 
m3 = Atft(]n + At, en + m2, Ag„ + mi) 
with l0 = Ae0 = 0 at ~t0 = 0. 
N u m e r i c a l Example and Discussion 
The distribution of the solid fraction within the freezing zone 
in terms of x is calculated f rom equation (19) b y use of the fol-
lowing numerical data for 5 percent Cu-aluminum alloy. 
k = 0.16S7 
M i = 5.78 
Co = 5.0 percent 
The results which include the growth of e and Ae, and their 
• • & dAe . , . „ time derivatives, — and , against time are shown in I'ig. 2. 
dt dt 
The corresponding temperature distributions within the freezing 
zone as well as in the solidified part are calculated by equations 
(16) and (17) for several values of I, and are shown in Fig. 3. 
The heat flux at the surface, x = 0, and the temperature, Tc, 
defined in equation (24) are plotted against, time in Fig. 4. 
As seen from Fig. 2 the rate of movement of the solidus front, 
— i n c r e a s e s rapidly at first, reaches a maximum, then decreases 
dl 
dAe 
gradually. On the other hand, —— for the liquidus front de-
af 
creases with freezing time. If the surface temperature remains 
constant during freezing, as shown in the previous study [3], both 
de dAe 
— and —— for solidus and liquidus fronts decrease with freezing 
dl 
time 
dl 
3 0 / February 1 9 6 8 T r a> Is a c t i o n s of t h e A S M E 
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de dAe 
Fig. 2 G r o w t h of s o l i d i f i e d a n d f r e e z i n g zones a n d the i r rates 
Variations in the heat flux and temperature at the surface with 
freezing time are shown in Fig. 4. At the initial stage of solidifi-
cation the heat flux at the surface decreases rapidly with time, 
then becomes essentially constant even though the temperature, 
Tc, decreases at a constant rate. This is consistent with the 
de dAe . . . . . 
fact that — and change rapidly only during the initial stage 
dt dt 
of solidification. 
References 
1 Brody, H. D., "Solute Redistribution in Dendritic Solidifica-
tion," ScD thesis, MIT , 1965. 
2 Pfann, W. Ci., "Principles of Zone-Melting," Trans. AIME, 
Vol. 135, July 1952, p. 85. 
3 Tien, R. II., and Geiger, G. E., " A Heat Transfer Analysis of 
the Solidification of a Binary Eutectic System," ASME Paper No. 
66—WA/IIT-23. To lie published in Journal of Heat Transfer. 
4 Goodman, T. R., "The Heat-Balance Integral and Its Applica-
tion to Problems Involving a Change of Phase," TRANS. ASME, Vol. 
30, 1958, p. 335. 
5 Hildebrand, F. B., Introduction to Numerical Analysis, McGraw-
Hill, New York, 1956, p. 233. 
Fig. 3 T e m p e r a t u r e d i s t r i b u t i o n 
Fig. 4 Heat f l u x a n d t e m p e r a t u r e at su r f ace 
Journa l of H e a t T rans fe r February i 9 6 s / 31 
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The Unidimensional Solidification of a Binary Eutectic System Willi a Time- Dependent Surface Temperature 1

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The Unidimensional Solidification of a Binary Eutectic System Willi a Time- Dependent Surface Temperature 1

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