Thermal Conductivity of Liquid Metals and Alloys D. S. VISWANATH

AND

B. C. MATHUR

The t h e r m a l conductivity data of liquid m e t a l s have b e e n c o r r e l a t e d b a s e d on the c e l l t h e o r y of l i q u i d s . The t e m p e r a t u r e dependence of t h e r m a l conductivity is p r e d i c t e d in the c a s e of t h i r t e e n m e t a l s for which e x p e r i m e n t a l data a r e a v a i l a b l e . The o v e r a l l a v e r a g e absolute d e v i a t i o n of p r e d i c t e d v a l u e s is 4.25 pct. The method has b e e n extended to a l l o y s .

LIQUID

m e t a l s make good heat t r a n s f e r m e d i a b e c a u s e of t h e i r h i g h heat c a p a c i t y , good t h e r m a l c o n d u c t i o n , and high heat t r a n s f e r c o e f f i c i e n t s . The vapor p r e s s u r e of liquid m e t a l s b e i n g v e r y low allows heat t r a n s f e r o p e r a t i o n s to be c a r r i e d out at n o r m a l p r e s s u r e s even at high t e m p e r a t u r e s . The p r e d i c t i o n of heat t r a n s f e r c o e f f i c i e n t s is far f r o m s a t i s f a c t o r y b e c a u s e of the tack of data and d i s c r e p a n c i e s in the a v a i l a b l e data. T h e r e has been so far no s y s t e m a t i c a n a l y s i s of the data and p r a c t i c a l l y no method to p r e d i c t t h e r m a l c o n duetivity of liquid m e t a l s except the c o r r e l a t i o n of Ewing 1 et al. L i t e r a t u r e s u r v e y shows that work on the c o r r e l a t i o n of t h e r m a l conductivity of liquid m e t a l s s t a r t e d as e a r l y as 1920. Rao 2 has shown that the d e c r e a s e in t h e r m a l conductivity of liquid m e t a l s o b s e r v e d n e a r the m e l t i n g point could be explained on the a s s u m p t i o n that the f r e q u e n c y of v i b r a t i o n of m o l e c u l e s c h a n g e s n e a r the m e l t i n g point. He a t t r i b u t e s the d e c r e a s e in t h e r m a l c o n d u c t i v i t y in the liquid phase to the d i s o r d e r e d state in the liquid. P o w e l l 3 has c o m p i l e d and d i s c u s s e d the work of Konno, 4 Bidwell, s and Brown. 6 As the c o e f f i c i e n t s of t h e r m a l conductivity of g a s e s and liquid alloys of m e t a l s a r e of the s a m e o r d e r of m a g n i t u d e , it can be concluded that the m e c h a n i s m s of heat t r a n s f e r in liquid alloys and g a s e s a r e s i m i l a r . Ewing 1 el al. have given a g e n e r a l c o r r e l a t i o n for m e t a l s - s o l i d , liquid, and alloys b a s e d on the concept of e l e c t r o n i c and m o l e c u l a r conduction, a r e l a t i o n s i m i l a r to W i e d e m a n n - F r a n z r e l a t i o n . T h i s r e l a t i o n as s t a t e d by the a u t h o r s p r e d i c t s t h e r m a l conductivity of liquid m e t a l s with • pet d e v i a t i o n . T h e i r r e l a t i o n is K = 2.61 x 10 -8 ( r / p )

- 2 x 10 -'7 ( T / p ) 2 / C p d

+ 97 Cp(d)2/MT

[1]

The u s e of hole theory of liquids in obtaining a s a t i s f a c t o r y c o r r e l a t i o n for t h e r m a l conductivity of o r g a n i c liquids is outlined by V i s w a n a t h ] The author also a p plied the r e s u l t s to p r e d i c t the t h e r m a l c o n d u c t i v i t y of liquids m e t a l s . T h i s method p r e d i c t e d the r e s u l t s with an e r r o r of • pet. H o r r o c k s and M c L a u g h l i n 8 developed a r e l a t i o n for t h e r m a l conductivity of liquids b a s e d on the a s s u m p t i o n that the e n e r g y t r a n s p o r t in liquids is due e n t i r e l y to v i b r a t i o n a l c o n t r i b u t i o n . The v i b r a t i o n a l c o n t r i b u t i o n is e v a l u a t e d a s s u m i n g the liquid to p o s s e s s a q u a s i c r y s t a l l i n e l a t t i c e s t r u c t u r e i n t e r a c t i n g a c c o r d i n g to L e n n a r d - J o n e s 12-6 p o t e n t i a l . The a u t h o r s however, use the concept of f r e e volume f r o m E y r i n g ' s r a t e p r o c e s s theory in e v a l u a t i n g the

c o n v e c t i v e p a r t . T h e i r e v a l u a t i o n of the c o n v e c t i v e p a r t shows that it i s n e g l i g i b l e c o m p a r e d to the v i b r a tional part. Although this t h e o r y is developed for s i m p l e m o l e c u l e s in c o n d e n s e d phase, it should be p o s s i b l e to e x tend the a r g u m e n t s to liquid m e t a l s and a l l o y s , as m e t a l s and alloys can be grouped as s i m p l e m o l e c u l e s and t h e i r c r i t i c a l t e m p e r a t u r e s a r e v e r y m u c h higher c o m p a r e d to t e m p e r a t u r e s of i n t e r e s t . The v i b r a t i o n a l p a r t can be e v a l u a t e d if it is a s s u m e d that the e n e r g y t r a n s p o r t due to a t e m p e r a t u r e g r a d i e n t , dT/dx takes place b e t w e e n the a d j a c e n t p l a n e s which a r e d i s p l a c e d by a d i s t a n c e 1. The r a t e of flow of heat dQ/dt can be e v a l u a t e d and the coefficient of t h e r m a l conductivity is then given by

K = 2P~n 1Cv and the t e m p e r a t u r e dependence by

(1/K)(dK/dT) = - ~ [ ( 1 / 3 ) - (51n .*/6In V)p]

METALLURGICALTRANSACTIONS

[3]

T h i s has been extended by Viswanath and Rao 9 to p r e d i c t t h e r m a l conductivity of o r g a n i c l i q u i d s . Eq. [3] can be w r i t t e n as

(1/K)(dK/dT) = -N (1/V)(dV/dT)

[4]

w h e r e N = [(1/3) - (Sin ~/51n V)p]. F o r o r d i n a r y p r e s s u r e s e s p e c i a l l y in the e a s e of liquid m e t a l s , we can a s s u m e that volume is a function of t e m p e r a t u r e only and that it is d i r e c t l y p r o p o r t i o n a l to t e m p e r a t u r e . T h e r e f o r e Eq. [4] b e c o m e s

(K/no) = ( T / T o ) - N

[5]

F i g . 1 shows the validity of Eq. [5]. The value of Ko u s e d is the value a v a i l a b l e at the lowest t e m p e r a t u r e . Some v a l u e s as can be s e e n f r o m F i g . 1 do not fit the equation to a g r e a t e r m e a s u r e of p r e c i s i o n and this m a y be b e c a u s e of the i n a c c u r a c i e s i n the e x p e r i m e n t a l v a l u e s . T a b l e I gives the e x p e r i m e n t a l and c a l c u l a t e d v a l u e s together with the p e r c e n t e r r o r . The value of

~O'O8 tf

?o.o b o 0L

9

A

Sn

LEGEND o Na 9 Cd

Zn o Bi

-o-

K

e"

Pb x

o

9

AI

9

Sb Rb ~"

~ 0.9 _~_O.121

D. S. VISWANATHand B, C. MATHUR are Professor of Chemical Engineeringand Research Scholar, respectively, Indian Institute of Science, Bangalore, India. Manuscript submitted March 8, 1971.

[2]

L

O

j

o

L*

__

0.1 0.2 0.5 Reduced Temperclure. log T/To

Fig. 1--Relation between reduced temperature and reduced thermal conductivity of liquid metals. VOLUME 3, JULY 1972-1769

0.14

Table I. Predictionof Thermal Conductivity of Liquid Metals Experimental and Calculated Values of Thermal Conductivityof Liquid Metals

~" 0.12

S1 No. 1

2

3

4

4

Liquid Metal Tin ~~

Zinc w

Bismuth m

Lead m

Potassium ~~

Thermal Conductivity Temp, cal/cm-sec, K ~

K/Ko K/Ko T/To

Exptl

cal

K, cal

Pct Error*

513 565 690 771

0.080 0.081 0.079 0.078

1.000 1.000 1.000 0.080 1.101 1.013 1.021 0.082 1.345 0.988 0.937 0.075 1.502 0.975 0.9]5 0.073

1.25 5.18 6.]5

778 873 973

0.138 0.136 0.135

1.000 1.000 1.129 0.985 1.258 0.978

1.47 2.96

1.000 0.138 0.974 0.134 0.951 0.131

573 673 773 873 973

0.041 0.037 0.037 0.037 0.037

1.000 1.000 1.000 1.174 0.902 0.965 1.349 0.902 0.936 1.523 0.902 0.912 1.698 0.902 0.890

0.041 0.040 0.038 0.037 0.037

8.11 2.70 -

603 673 773 873 973

0.039 0.038 0.037 0.037 0.036

1.000 1.110 1.281 1.447 1.613

1.000 0.974 0.948 0.948 0.923

0.039 0.038 0.037 0.036 0.035

2.70 2.78

473 573 673 773 873

0.107 0.10l 0.096 0.089 0.085

1.000 1.211 1.422 1.634 1.845

1.000 1.000 0.107 0.944 0.959 0.103 0.891 0.925 0.099 0.837 0.897 0.096 0.788 0.874 0298

1.48 3.84 7.23 15.48

1.000 1.000 0.205 0.947 0.949 0.195 0.880 0.910 0.]87 0.828 0.878 0.181 0.777 0.852 0.175

0.16 3.31 6.40 9.64

1.000 0.976 0.947 0.922 0.900

6

Sodium m

373 473 573 673 773

0.2055 0.1947 0.1809 0.1701 0.1596

1.000 1.268 1.536 1.804 2.072

7

Cadmium m

628 631 653 708

0.1060 0.1050 0.1050 0.1190

1.000 1.000 1.000 0.106 1.004 0.991 0.992 0.105 1.039 0.991 0.992 0.105 1.127 1.122 0.974 0.103

13.40

8

Aluminum ~~

700 973 1063

0.2470 0.2470 0.2900

1.000 1.000 1.390 1.000 1.518 1.174

7.00 20.40

903 913 1003 1063

0.0520 0.0520 0.0510 0.0500 0.0500

1.000 1.000 1.000 0.052 1.011 1.000 0.999 0.052 1.077 0.980 0.984 0.051 1.110 0.962 0.978 0.051 1.177 0.962 0.965 0.050

312 323

0.0700 0.0750

1.000 1.035

9

Antimony 1~

973

10

Rubidium m

1.000 1.071

1.000 0.247 0.930 0.230 0.912 0.230

-

x Hg 9 Mg

040

A ~

0.08

,I

0

8.00 4.25

x

9

0.04

I

"~ 0.02 a2

030

0.12

Recluced Temperature, log T/To Fig. 2 - - R e l a t i o n b e t w e e n r e d u c e d t e m p e r a t u r e and r e d u c e d t h e r m a l c o n d u c t i v i t y of l i q u i d m e t a l s (Ga, M g , Hg).

Table II, Experimental and Calculated Values of Thermal Conductivity of Liquid Metals

SI No.

Liquid Metal

Thermal Conductivity Temp, cal/cm-sec, K ~

K/Ko, K/Ko, T/To

Exptl

cal

K, cal

Pct Error*

I

Gallium ~

350 400 450 500 550

0.0672 0.0754 0.0813 0.0865 0.0915

1.000 1.000 1.142 1.122 1.285 1.200 1.428 1.280 1.571 1.360

1.00 1.10 1.21 1.31 1.40

0.067 0.075 0.081 0.088 0.095

0.53 0.37 1.74 3.20

2

Mercury 1~

333 393 433 473 493

0.0231 0.0261 0.0279 0.0295 0.0303

1.000 1.000 1.180 1.120 1.300 1.200 1.420 1.277 1.480 1.310

1.00 1.13 1.22 1.30 1.34

0.023 0.026 0.028 0.030 0.031

0.39 0.72 2.00 2.30

3

Magnesium

920 950 1000 1050 1100 1150 1200

0.1861 0.1935 0.2050 0.2080 0.3175 0.2241 0.2340

1.000 1.031 1.088 1.141 1.195 1.250 1.305

1.00 1.02 1.06 1.10 1.14 1.18 1.22

0.186 0.]90 0.200 0.200 0.210 0.220 0.230

1.85 2.44 3.90 3.50 1.80 1.70

Average absolute error:

1.88

2.00 -

Average absolute error:

x

~

9-e 0.06 E

-

1.000 0.070 0.993 0.069

~ ~

1.000 1.040 1.100 1.118 1.169 1.205 1.255

Superscript in liquid metal column refers to reference number. * [(Kcalc - Kexp)/Kexptl] X 100.

Superscript in liquid metal column refers to reference number. * [(Kcalc - Kexp)/Kexptl] X 100. o

N is found to be 0.23 by the l e a s t - s q u a r e s technique. Although t h e r m a l conductivity of m e t a l s g e n e r a l l y d e c r e a s e s with t e m p e r a t u r e , it is found to be the opposite in the c a s e of c e r t a i n m e t a l s like Hg, Ga, and Mg as shown in F i g . 2 (data in Table IIL The relation obtained for the liquid m e t a l s excepting Hg, Ga, and Mg is (K/Ko) (T/To) -~ [6] =

which c o n f i r m s the relationship given by Eq. [5]. In the c a s e of s u b s t a n c e s like Hg, Ga, and Mg for which t h e r m a l conductivity i n c r e a s e s with t e m p e r a t u r e , a relation s i m i l a r to Eq. [6] holds and t a k e s the f o r m

(K /Ko) (T / To)-~ .98 : (T / To)-O .z3

[7]

The experimental data 6'I~ considered here on the thermal conductivity of alloys reveals that I) the 1 7 7 0 - V O L U M E 3, J U L Y 1972

I

~ 004

LEGEN______DD Na-K ( 5 6 0 %

No] o

•

;

~

0.02

CL/~

t

l

o

o.I

o,2

Reduced

Temperature, log T/To

Fig. 3 - - R e l a t i o n b e t w e e n r e d u c e d t e m p e r a t u r e t h e r m a l c o n d u c t i v i t y of N a - K a l l o y s .

[

o5 and reduced

METALLURGICAL TRANSACTIONS

Table III. Experimental and Calculated Values of Thermal Conductivity of Liquid Metals

Table IV. Experimental and Calculated Values of Thermal Conductivity of Alloys Thernral

Thermal Conductivity S1 No.

1

2

3

4

Conductivity

K/Ko, K/Ko,

Temp, K

cal/cm.sec, ~

433 473 513 573 593

0.0220 0.0230 0.0240 0.0260 0.0270

1.323 1.369

509 512 583 609 693

0.0553 0.0533 0.0610 0.0615 0.0710

1.000 1,000 1.00 0.055 1.005 0.964 1.00 0.055 1.145 1,103 1.10 0.061 1.196 1.I12 1.14 0.063 1.361 1.283 1.24 0.068

2.50 4.20

Pb-Sb 6 (87pctPb)

589 598 645 655

0.0386 0.0396 0.0455 0.0457

1.000 1.015 1.095 1.112

1.000 1,025 1.178 1,183

0.038 0.039 0.041 0.042

1.51 9.90 8.i0

Sn-Zn 6 (92 pct Sn)

488 616 706

0.0570 0.0731 0.0878

1.000 1.262 1.446

1.000 1.00 0.057 1.282 1.19 0.068 1.540 1.31 0.075

6.50 14.50

Average absolute error:

5.53

Alloys Pb-Bi6 (44.6 pct Pb)

Sn-Pb s (62 pctSn)

T/To

Exptl

cal

K, cal

Pct Error*

1.000 1.092

1.000 1.045 1.090 1.181 1.227

1.00 1.07 1.14 1.23 1.27

0.022 0.024 0.025 0.027 0.028

-4.34 4.16 3.80 3.70

t.184

1.00 1.01 1.07 1.08

SI No.

Alloys

K/Ko,

Temp, K

cal/cm'sec, ~

T/To

Exptl

K/Ko, cal K, cal

Pct Error*

1

Na-K t~ (56 pct Na)

373 473 573 673 773

0.0617 0.0633 0.0647 0.0662 0.0672

1.000 1.268 1,538 1,804 2.072

1.000 1.025 1.050 1.073 1.094

0.061 0.064 0,065 0,066 0.067

1.10 0.46 0.30 0.30

2

Na-K m (22 pct Na)

373 673

0.0583 0.0636

1.000 1.800

1.000 1.00 0.058 1.090 1.07 0.063

0.90

Average absolute error:

0.61

3.20

1.00 1.02 1.05 1.06 1.09

Additional Data: 3

Na-K 12 (56.5 pct K)

473 573 673 773

0.0595 0.0626 0.0642 0.0647

1.000 1.211 1.422 1.634

1.000 1,052 1.076 1.087

1.00 1.02 1.04 1.06

0.059 0,061 0.062 0.063

2.50 3.42 2.60

4

Na-K n (77.7 pct K)

373 423 473 573 673 773 873 973

0.0583 0.0568 0.0592 0.0619 0.0626 0.0626 0.0619 0.0610

1.000 1.134 1.268 1.536 1.804 2.072 2.340 2.608

1,000 0.974 1.015 1,061 1.073 1.073 1.061 1.045

1.00 1.01 1.03 1.05 1.09 1.09 1.11 1.12

0,058 0.056 0.060 0.062 0.064 0,064 0,065 0.066

5.26 1.35 0.10 1.45 1.45 5.00 7.37

Average absolute error:

3.05

Superscript in alloys column r e a r s to re~rence number. *[(Kcalc-K~xp)/Ke• X 100,

Superscript in alloys column refers to reference number.

*[(Kcalc- Kexp)/Kexptl] • ff

0.18

9~

0.14

8 E

LEGEND Pb-Bi (44.6 % Pb) e Sn-Pb(62.0%Sn) x Pb-Sb(87.0 % Pb) Sn-Zn(92.0 %Sn) *

.

like organic l i q u i d s , f o l l o w the g e n e r a l r e l a t i o n (K/I%) = A ( T / T o ) N

w i t h A in all the e a s e s t e s t e d so far being one. T h e r e is a good e a s e and urgent need to e s t a b l i s h a c o r r e lation to evaluate Ko or a c o r r e l a t i o n w h i e h can do away with -fro. F u r t h e r it is n e c e s s a r y to find a c o r r e l a t i o n w h i c h is m o r e g e n e r a l than the one given in this p a p e r . It is hoped that this paper w i l l i n t e r e s t others in this direetion.

o.~ 0.06

-0.02 I 0

I

I

0.05

0.1

Reduced

[10 ]

9

0.02

g

100.

I 0.15

0.18

Temperoture, log T/To

between reduced temperature t h e r m a l c o n d u c t i v i t y of d i f f e r e n t a l l o y s , Fig. 4--Relation

NOMENC L A T U R E

and reduced

t h e r m a l conductivity of these a l l o y s i n c r e a s e s w i t h t e m p e r a t u r e even if the t e m p e r a t u r e d e p e n d e n c e of the constituent m e t a l s i s o t h e r w i s e , and 2) the l i m i t s of t h e r m a l conductivity of a l l o y s need not f a l l b e t w e e n the t h e r m a l conductivity of the c o n s t i t u e n t m e t a l s . With the e x c e p t i o n of N a - K a l l o y s , other a l l o y s t e s t e d f o l l o w e d the r e l a t i o n

CV

c o n s t a n t v o l u m e heat c a p a c i t y , c a l p e r g - m o l e , ~

d

d e n s i t y , p e r cu c m

Ko

t h e r m a l c o n d u c t i v i t y at l o w e s t t e m p e r a t u r e , To, cal per sec, cm per ~ K=Ko

at

T = To

t h e r m a l c o n d u c t i v i t y , cal p e r s e c , c m p e r ~

l

distance between nearest neighbors

M

molecular weight

n

n u m b e r of m o l e c u l e s p er unit area

g i v e n in found after t e s t e d and

N

t h e r m a l conductivity index

p

p r o b a b i l i t y that e n e r g y is t r a n s f e r r e d in each collision

alloys,

T

absolute t e m p e r a t u r e , K

Table III shows the alloys tested and the percent deviation. N a - K alloys followed the relation

(K/Ko) = ( T / t o ) o.z~

METALLURGICAL TRANSACTIONS

Specific heat, cal p e r g, ~

K

[8]

(K/Ko) : (T,/To) ~

T h i s r e l a t i o n w a s e s t a b l i s h e d with the data T a b l e IV. S o m e additional data w h i c h w e r e the c o r r e l a t i o n w a s e s t a b l i s h e d , w e r e a l s o are shown in T a b l e IV. It can be concluded that liquid m e t a l s and

Cp

[9 ]

V O L U M E 3, J U L Y 1 9 7 2 - 1 7 7 1

V

molal

volume,

t

time,

sea.

cu cm per

Greek (3/

P

coefficient

of v o l u m e

vibrational

frequency,

electrical

resistivity,

g-mole

Symbols expansion,

K -~

cm ohm per

cm

REFERENCES 1. C. T. Ewing, B. E. Walker, and J. A. Grand: Liquid Metal Technology Part 1. Chem. Eng. Progr. Syrup. Ser., 1957, vol. 53, no. 20, p. 19. 2. M. Rama Rao: Indian J. Phys., 1942, vol. 16, pp. 155.59. 3. R. W. Powell: Z lronSteellnst., 1949, vol. 162, p. 315. 4. S. Konno: Sci. Rep. Tohoku Imp. Univ., 1919, vol. 8, p. 169. 5. C. C. Bidwell and C. L. Hougan: J. AppL Phys., 1947, vol. 18, p. 776. 6. W. B. Brown: Phys. Rev., 1923, vol. 22, p. 171. 7. D. S. V~swanath: A.LCh.E. ,L, 1967, vol. 13, pp. 850-58.

1 7 7 2 - V O L U M E 3, J'ULY 1972

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