Non-Steady-State Rates of Dissolution of Iron and Cobalt in Liquid Copper under Static Conditions

REIICHI OHNO

The horizontal bottom face of cylindrical iron or cobalt is exposed to liquid copper, and their dissolution rates are determined at 1473 to 1476 (-+10) K and 1573 to 1576 (+10) K for the Cu-Fe system and at 1473 to 1475 (-+ 10) K for the Cu-Co system. The decrease in height of the cylinder, z (m), is proportional to the square root of time, Vt-(sV2), as is expected in mass transfer controlled by non-steady-state diffusion. The observed value of c~ (m �9 s 1/2) in the equation z = c ~ / t is 1.95 x 10 -6 ( m �9 s-1/2) in the lower temperature range and 3.53 x 10 -6 ( m �9 S -1/2) in the higher temperature range for the Cu-Fe system, and 2.66 x 10 6 (m �9 s- 1/2) for the Cu-Co system. Under fixed experimental conditions, the value of c~ calculated on the basis of the expression that the rate of diffusion in liquid is proportional to the activity gradient of a diffusing substance is in agreement with the observed value, but that calculated on the basis of Fick's first law is 1.9 to 2.7 times as great as the observed value. For this expression to be valid it is necessary that the ratio of the phenomenological coefficient defined by irreversible thermodynamics to the activity is independent of the concentration. An estimate of the density gradients caused by dissolution suggests that no natural convection occurs in the vicinity of the solid-liquid interface.

I. I N T R O D U C T I O N

THE liquid Cu-Fe and Cu-Co systems exhibit great positive departures from ideality. In view of this fact, studies have been conducted on the dissolution rates of iron and cobalt cylinders in liquid copper under natural and forced con- vections. 1'2'3 Under natural convection, the steady-state rates of dissolution of iron and cobalt cylinders in liquid copper follow an equation in which the dissolution rate is proportional to the difference between the activities of each dissolving substance at saturation and in the bulk, whereas under forced convection, their dissolution rates do not.3 This fact suggests that the diffusion rates of iron and cobalt in liquid copper are proportional to their respective activity gradients when their concentration gradients are small.

The present study has been conducted under the condi- tions that mass transfer is supposed to proceed almost by diffusion. Under these conditions the dissolution rates of iron and cobalt in liquid copper have been determined. Although under natural or forced convection it is difficult to calculate the dissolution rate, the dissolution rate controlled by diffusion in the absence of fluid flow can be calculated. In the present study, the dissolution rates of iron and cobalt are calculated on the basis of two different expressions for the rate of diffusion in the liquid: (1) it is proportional to the concentration gradient of a diffusing substance, i.e., Fick's first law, and (2) it is proportional to the activity gradient of a diffusing substance. The results of this calculation are compared with those of the determination. The second ex- pression for the rate of diffusion is discussed briefly from the viewpoint of irreversible thermodynamics. In addition, the density gradients caused by dissolution are estimated and the possibility of natural convection is discussed.

REIICHI OHNO is Associate Professor with The Research Institute for Iron, Steel and Other Metals, Tohoku University, Katahira, Sendai 980, Japan.

Manuscript submitted November 22, 1982.

II. EXPERIMENTAL

A. Apparatus and Procedure

A molybdenum resistance furnace described elsewhere 1 was used. A cylindrical alundum crucible (5.0 cm in OD, 3.8 cm in ID, and 11.3 cm in depth) was charged with 490 to 520 g of deoxidized copper and placed in the furnace. As shown in Figure 1, a cylindrical iron or cobalt specimen (8 mm in diameter and 10 mm in height) was fixed to the quartz holder made of two transparent quartz tubes having different inside diameters to limit dissolution to the bottom face of the cylinder, and the holder was connected to the steel shaft by a molybdenum rod having a molybdenum cylinder at its lower end. A ceramic fiber was packed softly between the holder and the specimen, and the specimen was fixed so that its bottom face was horizontal. The lower quartz tube of the holder is from 8.0 to 8.1 mm in inside diameter and from 28 to 33 mm in height. A narrow gap between the upper and lower quartz tubes of the holder was filled with alumina powder.

The furnace was heated to 1323 to 1353 K under vacuum, and then heated to the desired temperature under a mixture of argon at 81 kPa and hydrogen at 20 kPa. In some experi- ments on the Cu-Co system at 1573 to 1578 ( -10 ) K, pure argon at 101.3 kPa or a mixture of argon at 94.2 kPa and hydrogen at 7.1 kPa was used. For the purpose of pre- heating the cylindrical specimen, after the quartz holder was held for two to five minutes above the liquid copper, the lower end of the holder was immersed in the liquid for three to six minutes and then the holder was immersed to a depth of about 20 mm for the same periods. After this preheating the bottom face of the specimen was kept in contact with the liquid copper, as shown in Figure 1, for a predetermined time. As will be discussed later, it is supposed that this arrangement of the solid specimen and the liquid hardly produces natural convection. The dissolution experiments were conducted at 1473 to 1476 (-+ 10) K and 1573 to 1578 (-+ 10) K for both the Cu-Fe and Cu-Co systems. The time

METALLURGICAL TRANSACTIONS B VOLUME 14B, DECEMBER 1983--667

8

I

I

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g

I

9

9

9

q

I

11 =

12 4

I

I

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i H , ' 1

o o l . !

�9 L �9 , !

; - I �9 o . I "' ~ L______ 3 �9 .', �9

�9 i

~ �9 i

"~ ~ ".! �9

o | �9 !

�9 o |

�9 !

~ I - - - - ~ �9 i - - ~

�9 1 - - �9

�9 i - - "

. ~ _ _ " ~

" . I : " �9 �9 : . .

o , �9 , - - - - - - ~ 1 7 6 �9

. ' i _ y _ _ ." I i �9

�9 ~ �9 �9 i - - .

~ - - - -

�9 ~ I - - - ' .

" . . . . . . . - : .. : . . . - : . : : . . ' ~

Fig. 1 - -The solid-liquid assembly: (1) steel shaft, (2) molybdenum rod, (3) molybdenum cylinder, (4) cylindrical iron or cobalt, (5) ceramic fiber, (6) quartz holder, (7) thermocouple, (8) thermocouple protection tube, (9) molybdenum heat coil, (10) alumina crucible, (11) alumina tube, (12) liquid copper.

of immersion was 0.6, 1.8, and 3.6 ks in the lower tem- perature range and was 0.6, 1.2, and 1.8 ks in the higher temperature range. The melt temperature was measured by means of a Pt/Pt-13 pct Rh thermocouple in an alumina sheath immersed in the melt and was kept constant within _+ 10 K during dissolution. After a measured time of dis- solution, the specimen was raised to a position over the furnace tube, being cooled. In the course of this operation, in most of the experiments of the lower temperature range, the melt inside the lower quartz tube flowed out but in any of the experiments of the higher temperature range it did not. Copper alloy adhering to the iron cylinders after dissolution was dissolved in nitric acid (60 wt pct). That ad- hering to the cobalt cylinder was filed off because cobalt dissolves in the nitric acid; in practice, it is difficult to file off only copper alloy at the interface between cobalt and cop- per alloy, and hence this filing was carried out until the area of cobalt became one-half to two-thirds of the total filed

area so that the position of a surface obtained might be as close as possible to that of the interface, i.e., the dissolu- tion surface. In order to determine the decrease in height of the cylindrical specimen, its height was measured by a micrometer before and after dissolution. The standard devia- tion of the measurements ranged from 0 to 3.5 • 10 -~ mm.

B. Materials

Commercial electrolytic copper (99.99 wt pct), iron (99.9 wt pct), and cobalt (99.9 wt pct) were used as basic raw materials. The deoxidation of these materials was car- fled out as follows: by use of an induction fumace, the copper was vacuum-melted in a graphite crucible, and the iron or cobalt was vacuum-melted in an alundum crucible after addition of a small amount of graphite. The iron or cobalt melt was slowly solidified in the crucible under vacuum in order to promote grain growth. These solidified speci- mens were machined to the cylinders. The iron specimens contained carbon of 0.035 to 0.053 wt pct and aluminum of 0.016 to 0.020 wt pct, and the cobalt specimen contained carbon of 0.064 wt pct and aluminum of 0.035 wt pct. The bottom face of each cylinder was polished with emery papers just before use. The purities of commercial argon and hydrogen used were 99.999 and 99.9995 wt pct, respectively.

III. RESULTS AND DISCUSSION

The observed decrease in height of the iron and cobalt specimens was plotted against the square root of time in each temperature range in Figures 2(a) and (b), and 3. In all the experiments on the Cu-Co system at 1573 to 1578 (---10) K, many cavities are formed on the side of copper solidified inside the lower tube of the quartz holder and the measured values of decrease in height of the cobalt speci- men scattered largely. The formation of these many cavities probably produces fluid flow which affects the dissolution rate. For this reason, the experimental data under this con- dition are not shown.

The dissolution rates for the present investigation were calculated on the basis of the mathematical treatment for diffusion in a system with a moving boundary given by Lommel and Chalmers 4 and the expression that the rate of diffusion in liquid is proportional to the activity gradient of a diffusing substance. The validity of this expression for diffusion in solid metals has been already discussed. 5

On the basis of a solution of Fick's second law for dif- fusion in the liquid phase, the displacement of the solid- liquid interface, z(m), is given by the equation: 4

z : 2 , ~ x / b L t [1]

where /gL is the interdiffusion coefficient in the liquid (m 2 �9 s-~), being assumed to be independent of the concen- tration of a diffusing substance, t is the reaction time (s), and h is constant because at the interface the concentration of the diffusing substance (liquid phase) is constant (saturation), and hence the value of the error function is constant.

When the interdiffusion coefficient in the solid is much smaller than DL, on the basis of Fick's laws, the value of h

668--VOLUME 14B, DECEMBER 1983 METALLURGICAL TRANSACTIONS B

0 .15 I

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I I

I I

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0 14 7 3 - 1 4 7 6 ( •

1473

1473

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0 10 20 30 40 50 60

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(a)

O O

T P r I I I r / i I I I i r } I 1 I I I I

015 /

r i : II C o.

�9 "~ 0.I0

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T

"6

o.o5 0 1573- 1576 (•

r-~ / 1573

/ 1573 0 U I I L ~ i 1 1 ~ 1 1 1 i 1 1 1 1 [ I I

10 20 30 40 ,/T ( s '~)

(b) Fig. 2 - - Dissolution of iron cylinder in liquid copper vs square root of time at temperatures in the ranges: (a) 1473 to 1476 (-+ 10) K and (b) 1573 to 1576 (+-10) K. O experimental; --- calculated from Eqs. [1] and [2];--calculated from Eqs. [1] and [7].

is given by the equation: 4

F(A) = X/'~A(exp A 2) (1 + erf A)

Cs - Co C:- Q

k, Ns - koNo - [ 2 ]

kiN: - ksUs

where C, is the molar concentration of the solute i, i.e., the component of a pure solid, at saturation in the liquid (real �9 m-3), Co and C:are the initial molar concentrations of i in the liquid and the solid (mol �9 m-a), respectively, Ns is the mole fraction of i at saturation in the liquid, No and N/ are the initial mole fractions of i in the liquid and the solid,

METALLURGICAL TRANSACTIONS B

E E

o . e l O

( _ 1

O

f - 0 3

-r-

r

Q

0 . 1 5

0 .10

0.05

I I I I I I I I I I I . L / C)

- /

_ / O /

_ /

/ - /

/ /

/ /

/ /

/ - /

/ - /

/

/ / / / / O

/ / Temperature(K)

014731473-1475(+-10)1473

0 I t ~ ~ r ~ ~ ~ ~ ~ I r 0 10 20 30 40 50 60

25- (s,,'2) Fig. 3--Dissolution of cobalt cylinder in liquid copper vs square root of time at temperatures in the range of 1473 to J475 (-+ 10) K. O experi- mental; - -- calculated from Eqs. [1] and [2]; - - calculated from Eqs. Ill and [7].

respectively, ks and ko are C,/N, and Co~No, respectively, approximating to p~/Mj when N, ~ 1, where Pt and Mj are the density of the pure liquid (kg �9 m -3) and the molar mass of its component, j (kg �9 mol-~), respectively, and ks is CJN: which approximates to pi/Mi when Nr - 1, where Pi and Mi are the density of the pure solid (kg �9 m -3) and the molar mass of i (kg �9 mol-l) , respectively.

Now, instead of Fick's first law, it is assumed that the diffusion flux of i in the x-direction in the liquid referred to a cell-fixed frame, J (mol �9 m -2 �9 s l), is expressed by

- Oa J = - D L ~ x [31

where a is the activity of i, the standard state of which is chosen so that a approaches the molar cgncentration of i, C (real �9 m-3), as C approaches zero, and DL is independent of the concentration of i, being defined by the following re- lation at sufficiently low concentration of i:

J = -bL(OC/Ox) = -bL(aa/Ox)

Fick's second law and Eq. [3] give

Ox \ Ox /

o ( ,oq Ox \ Ox /

[4]

o {b aa oc) \ oa

(b oc, oc \ Ox/

_ 02a DL-S-~x~

[51 OC

Ot

VOLUME 14B, DECEMBER 1983--669

Hence,

Oa 02a - - = / ) L - - [6] Ot OX 2

In the present study, the x-direction is perpendicular to the bottom face of a cylinder and the value of x at the solid- liquid interface is denoted by z, which is given by Eq. [1]. On the basis of Eqs. [3] and [6], and the mathematical treatment with Fick 's laws, 4 A is given by the equation:

F(A) = ~ A ( e x p A 2) (1 + erf A)

as - ao

G-C~ 1

y---o [k as(N) - k a0(N)] = [7]

kiU~- ksUs

where as and a0 are the activities of i in the liquid at satura- tion and at t = 0, respectively, being defined so as to ap- proach the molar concentration of i at infinite dilution; as(N) and ao(N) are the activities of i in the liquid at saturation and at t = 0, respectively, being referred to pure solid or liquid i; y~ is the activity coefficient of i at infinite dilution in the liquid, being referred to the same state as as(N) and ao(N), and k is 1 /N (mol �9 m -3) at C = 1 mol �9 m 3, where N is the mole fraction of i in the liquid, approximating to pt/Mj.

The relations between z and ~ - calculated from Eqs. [1] and [7] as well as Eqs. [1] and [2] were drawn in Fig- ures 2(a) and (b), and 3. Numerical values used for this cal- culation are presented in Table I. The values of as(N) in the Cu-Fe system are based on the activities of iron in copper- saturated y-iron, which were calculated from the measured activities of copper by Arita et al. 6 The values of y~ in the liquid Cu-Fe system referred to pure liquid iron were calcu- lated from the relative partial molar enthalpy and the relative partial molar excess entropy of iron at NFe = 0, which were recommended by Kulkarni, 7 i.e., AHFe = 70.96 kJ �9 mol 1 and A S ~ ( = ASve + R l n N v o ) = 19.3 J ' m o l ~ ' K 1 where the reference state of iron is pure liquid, ASF~ is the relative partial molar entropy of iron, and R is the gas constant (J �9 mol -j �9 K- l ) ; the values of y~ referred to y-iron were obtained from those calculated above by use of the Gibbs free energy changes for Fe(y) = Fe(l) s where 3' and I denote y-iron and liquid iron, respectively. The values of as(N) and y~ in the Cu-Co system are those determined electrochemically by Oishi et al. 9 The values of Ns in the Cu-Fe system were taken from published phase diagram,I~ and the value of N~ in the Cu-Co system is based on the study by Taskinen. 11 The values of/)L are those determined at very low concentration of iron or cobalt by Ejima and Kameda. 12 The values of the densities of liquid copper and y-iron used are 7.8 x 10313 and 7.5 • 103 k g " m -314, respectively.

The density of cobalt at 1473 K was est imated to be 8.0 • 103 kg �9 m -3 from those at 293 K is and 1823 K 13 on the assumption tha t it decreases linearly with temperature from 293 K up to 1823 K.

As may be seen from Figures 2(a) and (b) and 3, in each temperature range, the observed decrease in height of the iron or cobalt specimen increases linearly with the square root of t ime in accordance with Eq. [1], and the experi- mental points are close to the solid line which was drawn on the basis of Eqs. [1] and [7]. Equation [1] is rewritten as

z = ~W~- [81

where a (m �9 s - I n ) i s the proportionality constant, corre- sponding to 2h~v//)t. Table II shows the values of a observed and calculated on the basis of Eqs. [2] and [7]. In Table II, under fixed experimental conditions the value of a based on Eq. [7] is in good agreement with the observed value of a , but that based on Eq. [2] is 1.9 to 2.7 times as great as the observed value. This fact indicates that the dissolution rates of iron and cobalt in liquid copper under the present experimental conditions are controlled by non- steady-state diffusion and demonstrates that they can be expressed by Eqs. [1] and [7].

Now, diffusion rate is discussed briefly from the view- point of irreversible thermodynamics. If the diffusion flux of i, Jv (mol �9 m -2 �9 s -~) in a binary solution is measured with respect to a plane across which no net transfer of volume occurs, Jv is given by 16

0C Jv = - b , . - - [9]

Ox

where Dv ( m Z ' s 1) is the interdiffusion coefficient. Ac- cording to irreversible thermodynamics, Jv is expressed as 16

Jv = - L 0_~_~ [10] Ox

where/x is the chemical potential of i (J �9 mol ~) and L is the phenomenological coefficient. The experimental inter- diffusion coefficients determined by means of the normal techniques can be identified with D~.16 Then the diffusion flux of i, J (mol �9 m -2 �9 s-l) , expressed by this experimental interdiffusion coefficient is given by

0/z J = - L - -

Ox

L Oa : - R T - - - - [11]

a Ox

As may be seen from Eq. [11], for J to be proportional to Oa/Ox with constant coefficient in accordance with Eq. [3] which expresses the diffusion flux of i in a cell-fixed frame, it is necessary that L / a is independent of the concentra- tion of i.

Table I. Thermodynamic Properties, Solubilities, and Interdiffusion Coefficients in the Liquid Cu-Fe and Cu-Co Systems

System Temperature (K) as(N) y~ Ns DL (m 2 �9 s-1)

Cu-Fe 1473 0.938 40.2 0.060 5.25 x 10 9 Cu-Fe 1573 0.936 25.6 0.105 6.92 x 10 9 Cu-Co 1473 0.873 22.9 0.072 4.76 X 10 -9

Note: a~(N) and 3,~ refer to pure solid iron or cobalt.

670 VOLUME 14B, DECEMBER 1983 METALLURGICAL TRANSACTIONS B

Table II. Values of Proportionality Constant (a)

Temperature (K)

System Experimental Calculated Observed

OL X 10 6 (m �9 s ,/2)

Based on Eq. [7] Based on Eq. [2]

Cu-Fe 1473 to 1476 (+ 10) 1473 1573 to 1576 (-+ 10) 1573

Cu-Co 1473 to 1475 (+ 10) 1473

1.95 1.83 4.59 3.53 3.39 9.38 2.66 2.65 4.91

The possibility that natural convection may be caused by dissolution is discussed below. The densities of Cu-Fe alloy at 1823 and 1873 K and of Cu-Co alloy at 1823, 1873, and 1923 K decrease almost linearly with NFe and Nco up to those at 0.4, respectively, 17 and hence they are expressed by

P = P, + (Op~ N [12] \ON/T

where p is the density of liquid Cu-Fe or Cu-Co alloy (kg �9 m - 3 ) , Pl is the density of pure liquid copper (kg �9 m-3), N is the mole fraction of iron or cobalt, and (Op/ON)r is almost independent of temperature, being - 1 . 0 x 10 3

kg �9 m 3 for the Cu-Fe system and - 0 . 3 x 10 3 kg �9 m -3 for the Cu-Co system. Hence, the buoyancy coeff icient , Ap = (p, - p~)/p~ where p, is the density of liquid alloy at saturation (kg �9 m-3), is expressed by

Ap ~ l ( O p ] N~ [133 Pt \ ON/r

The values of Ap estimated from Eq. [13] on the assump- tion that (Op/ON)r is constant down to 1473 K are in the range of - 3 x 10 3 to - 1 x 10 -2. They are negative and indicate the upward flow of the liquid. The lighter liquid is located at the higher position and hence natural convection hardly occurs under the present arrangement of solid and liquid. Recently, it has been demonstrated that the dissolu- tion of antimony placed in top position in molten bismuth is governed by diffusion. ~8

The dissolution rate of i, Jz ( m o l - m -2 . s-l) , is ex- pressed as

p~ dz Jz -

M~ dt

__ P i 1 a t 1/2 [14] Mi 2

The dissolution rate, Jz, equals the rate of diffusion of i at the solid-liquid interface. Therefore, on the basis of Eq. [3] Jz is expressed also by

r k /Oa(N)~

r k [Oa(N)~ /ON~ =

= - D L ~ [ % + Ns(-~N),] (~x)~=~ [15]

where a(N) is the activity of i in the liquid defined by TN, being referred to pure solid or liquid i, y is the activity coefficient of i at N in the liquid referred to the same state as a(N), 3', is y at N,, (0T/0N), is the concentration de- pendence of y at N,, and (ON/Ox)~=~ i s the concentration

gradient of i at the interface. Equations [14] and [15] give

P_L 1 at -1/2

The density gradient at the interface corresponding to this concentration gradient, (Op/Ox)r .... (kg �9 m-4), is given by

ON

The values of (Op/OX)r .... were estimated from those of (ON/Ox)x=z at Is, 1.8, and 3.6 ks evaluated from Eq. [16] and those of (Op/ON)r mentioned above on the basis of Eq. [17], being 2.1 x l 0 4 (2.1 x 10 ~) to 2.3 x 106 k g " m -4 (2.3 x 10 g ' c m 4) for the Cu-Fe system and 8.8 x 103 (8.8 • 10 2) to 5.3 x 105 k g . m -4 (5.3 g - c m -4) for the Cu-Co system. The values of , / u sed for the calculation of (0y/0N), in Eq. [16] are those calculated on the basis of the Wilson equation for the Cu-Fe system 3 and those determined by Oishi et al. 9 for the Cu-Co system.

The rate at which heat is formed per unit area in the dissolution of solid i in liquid at the interface, dQ/dt (J �9 m -2 �9 s-l), is given by

dQ dz A H i

dt - Pi--dtt Mi

1 A H i - - piOLt -1/2 [18]

2 Mi

where AHi is the relative partial molar enthalpy of i in a liquid alloy at the saturation concentration referred to pure solid (J �9 mol ~). In the present alloy systems, AHFe = 48 kJ �9 mol ~ at NFe = 0.06, A H F e = 44 kJ �9 mo1-1 at NFe =

0.105 andAHco = 64 kJ �9 mol -~ at Nco = 0.072; the val- ues of AHFe w e r e calculated from those referred to liquid iron ~9_and enthalpy for the fusion of y-iron, 2~ and the value of AHco was calculated from that referred to liquid cobalt 2~ and enthalpy for the fusion of /3-cobalt. 22 The values of dQ/dt calculated from Eq. [18] range from - 1 0 to - 0 . 1 0 kJ �9 m -2 �9 s ~ at ls to 3.6 ks for the Cu-Fe system and from - 1 2 to - 0 . 1 9 kJ �9 m -2 �9 s -~ in the same range of time for the Cu-Co system, expressing endothermic rate.

The thermal conductivities of liquid copper at the experi- mental temperatures are supposed to be close to those of solid copper at temperatures near its melting point, e.g., 0.31 kJ �9 m -~ �9 K -l �9 s -~ at 1323 K . 23 The temperature gradient, OT/Ox, estimated f rom the above mentioned values of dQ/dt on the basis of Fourier 's law by assuming heat transfer only from liquid phase and steady-state heat conduction ranges from 0.32 to 34 K �9 m -L for the Cu-Fe system and from 0.61 to 37 K �9 m ~ for the Cu-Co system,

METALLURGICAL TRANSACTIONS B VOLUME 14B, DECEMBER 1983--671

being negligibly small. The density gradient of liquid cop- per, Op~/Ox (kg �9 m-4), corresponding to this temperature gradient is given by

Opl_ Opt OT Ox OT Ox

[19]

where Opt/OT (kg . m - 3 - K -~) is the temperature de- pendence of the density of liquid copper, being -0.801 kg �9 m -3 �9 K - I . 24 The values of Opz/Ox calculated from the maximum values of OT/Ox on the basis of Eq. [19] are - 2 7 k g - m -4 ( - 2 . 7 x 10-4g �9 cm -4) for the Cu-Fe system and - 3 0 kg �9 m -4 ( - 3 . 0 x 10 -4 g �9 cm -4) for the Cu-Co system; their absolute values are very small.

The values of (Op/Ox)~ . . . . and Opt/Ox are opposite in sign, but the former values are much greater than the abso- lute values of the latter. This discussion suggests that no natural convection is caused by the heat of dissolution in the vicinity of the interface. In practice, however, there may be slight temperature gradients (liquid phase) depending on heating condition and a slight deviation of the bottom face of the solid specimen from the horizontal position may arise; these will produce fluid flow though its velocity is probably very small.

IV. C O N C L U S I O N S

The observed decrease in height of iron or cobalt cylinder is proportional to the square root of time in agreement with the model in which mass transfer is controlled by diffusion in the absence of fluid flow and is in good agreement with the decrease calculated on the basis of the equation which expresses that the rate of diffusion in liquid is proportional to the activity gradient of a diffusing substance.

A C K N O W L E D G M E N T S

The author wishes to thank Messrs. S. Yamada, K. Enami, and T. Takahashi for carrying out vacuum melting of the basic raw materials, Messrs. T. Nakazawa, G. Ihara, and K. Saito for preparing the cylindrical specimens, Messrs. K. Shoji, Y. Oyama, M. Moriya, and Y. Waga for preparing the quartz holders, and Mr. S. Takeyama and his associates for the elemental analyses.

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Steelmaking, Addison-Wesley Publishing Co., Reading, MA, 1963, vol. 2, p. 626.

15. J.F. Elliott and M. Gleiser: Thermochemistry for Steelmaking, Addison-Wesley Publishing Co., Reading, MA, 1960, vol. 1, p. 6.

16. H.J .V. Tyrrell: Diffusion Processes, J.N. Sherwood, A.V. Chadwick, W. M. Muir, and F. L. Swinton, eds., Gordon and Breach Science Publishers, London, 1971, vol. 1, pp. 67-86.

17. S. Watanabe and T. Saito: Nippon Kinzoku Gakkaishi, 1971, vol. 35, pp. 554-60.

18. Y. Shoji and S. Uchida: Metall. Trans. A, 1981, vol. 12A, pp. 1681-86.

19. R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, and K.K. Kelley: Selected Values of the Thermodynamic Properties of Binary Alloys, ASM, Metals Park, OH, 1973, p. 740.

20. R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelley, and D. D. Wagman: Selected Values of the Thermodynamic Properties of the Elements, ASM, Metals Park, OH, 1973, p. 185.

21. Y. Iguchi, Y. Tozaki, M. Kakizaki, S. Ban-ya, and T. Fuwa: Tetsu- to-Hagand, 1977, vol. 63, pp. 953-61.

22. R. Hultgren, P. D. Desai, D.T. Hawkins, M. Gleiser, K. K. Kelley, and D. D. Wagman: Selected Values of the Thermodynamic Properties of the Elements, ASM, Metals Park, OH, 1973, p. 130.

23. G.H. Geiger and D. R. Poirier: Transport Phenomena in Metallurgy, Addison-Wesley Publishing Co., Reading, MA, 1973, p. 192.

24. C.J. Smithels and E.A. Brandes: Metals Reference Book, 5th ed., Butterworth & Co., London, 1976, p. 945.

672--VOLUME 14B, DECEMBER 1983 METALLURGICAL TRANSACTIONS B