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Thermal conductivities of solid and liquid phases for pure Al, pure Sn and their binary alloys
Fluid Phase Equilibria 298 (2010) 97–105
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Thermal conductivities of solid and liquid phases for pure Al, pure Sn and their binary alloys Fatma Meydaneri a,∗ , Buket Saatc¸i b , Mehmet Özdemir c a b c
Department of Metallurgy and Materials Science Engineering, Faculty of Engineering, Karabük University, 78050 Karabük, Turkey Faculty Arts and Sciences, Department of Physics, Erciyes University, 38039 Kayseri, Turkey Faculty Arts and Sciences, Department of Chemistry, Erciyes University, 38039 Kayseri, Turkey
a r t i c l e
i n f o
Article history: Received 5 April 2010 Received in revised form 11 June 2010 Accepted 14 July 2010 Available online 27 July 2010 Keywords: Thermal conductivity Radial heat flow Directional solidification Sn–Al alloy Thermal coefficient
a b s t r a c t The variations of thermal conductivities of solid phases versus temperature for pure Sn, pure Al and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al binary alloys were measured with a radial heat flow apparatus. From thermal conductivity variations versus temperature, the thermal conductivities of the solid phases at their melting temperature and temperature coefficients for same materials were also found to be 60.60 ± 0.06, 208.80 ± 0.22, 69.70 ± 0.07, 80.30 ± 0.08, 112.30 ± 0.12, 142.00 ± 0.15, 188.50 ± 0.20 W/K m and 0.00098, 0.00062, 0.00127, 0.00114, 0.00112, 0.00150, 0.00116 K−1 , respectively. The thermal conductivity ratios of liquid phase to solid phase for the pure Sn, pure Al and eutectic Sn–0.5 wt.% Al alloy at their melting temperature are found to be 1.11, 1.13, 1.06 with a Bridgman type directional solidification apparatus, respectively. Thus the thermal conductivities of liquid phases for pure Sn, pure Al and eutectic Sn–0.5 wt.% Al binary alloy at their melting temperature were evaluated to be 67.26, 235.94 and 73.88 W/K m, respectively by using the values of solid phase thermal conductivities and the thermal conductivity ratios of liquid phase to solid phase. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Recently, Sn-based alloys have been investigated intensively as potential alternative anode materials for lithium ion batteries because of its higher capacity storage than that of the carbonaceous anodes [1]. Al-based alloys, which consist of Al–Pb and Al–Sn, are widely used for sliding bearing applications due to their good load carrying capacity, fatigue resistance, wear resistance and sliding properties. However, environmental legislation has driven manufacturers to eliminate Pb from bearing alloys. Thus, focus has being concentrated on Sn–Al alloys with high content of Sn. Al–Sn-based alloys are widely used in sliding bearings in the aerospace, combustion engine pistons, cylinder liners and automobile industries, because they offer good balance between load bearing capacity and fatigue resistance, their high wear resistance, together with favorable conformability and compatibility [2–5]. Sliding bearing is applied broadly in modern industry and its quality directly affects the precision and life-span of machines. Friction and wear performance of sliding bearing is of great interest, especially with environmental and health concerns in recent years [6]. In this binary alloy system, tin is a necessary soft phase in the aluminium matrix. Due to the excellent anti-welding characteristics with iron,
∗ Corresponding author. Tel.: +90 370 433 82 00; fax: +90 370 433 82 04. E-mail address: [email protected] (F. Meydaneri). 0378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2010.07.015
its low modulus and low strength, tin can provide suitable shear surface during sliding. Such alloys are also recommended as anodic protective coating for anti-corrosive protection of steel and these alloys provide a good combination of strength, corrosion resistance and surface properties [2,7,8]. For pure Al, pure Sn and their binary alloys do not satisfy some technologic parameters as working temperature, for example. The knowledge of the thermodynamics of materials provides fundamental information about the stability of phases and about the driving forces for chemical reactions and diffusion processes. Also, the data are of great importance for the development of electronic materials, interconnection technologies, sliding properties, especially in microelectronics and modern industry. Hence, the purpose of this study is to determine thermal coefficients and thermal conductivities of solid and liquid phases in order to estimate some of thermodynamic properties such as Gibbs–Thomson coefficient ( ), solid–liquid interfacial energy ( SL ) and grain boundary energy ( GB ). The thermal conductivity, , is one of the main fundamental properties of materials such as density, melting point, entropy, resistance, and crystal structure parameters and it plays a critical role in controlling the performance and stability of materials. The investigation of thermal conductivity is a valuable tool for the study of transport mechanisms of alloys. Although the value of for pure materials was obtained theoretically and experimentally, there are not enough information and data available about the ther-
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Fig. 1. The phase diagram of Sn–Al system [26].
Fig. 2. Block diagram of radial heat flow system.
mal conductivity of alloys. The values of for alloys change, as in pure materials, not only with temperature but also it changes by compositions of the materials. In the experimental determination of the thermal conductivity of solids, a number of different methods of measurements are used for different ranges of temperature and for various classes of materials having different ranges of thermal conductivity values. Many attempts have been made to determine the thermal conductivity values of solid and liquid phases in various materials by using different methods [9–23]. One of the common techniques for measuring the thermal conductivity of solids is the radial heat flow method. There are several different types of apparatus all employing radial heat flow. The classification is mainly based upon specimen geometry, i.e. cylindrical, spherical, ellipsoidal, concentric sphere, concentric cylinder, and plate methods. In the present work, radial heat flow method was used. The radial heat flow method uses a specimen in the form of a right circular cylinder with a coaxial central hole which contains either a heater or a heat sink, depending on whether the described heat flow direction is to be radially outward or inward. Temperatures within the specimen are measured by thermocouples. This method was first used for measuring the thermal conductivity of solids for pure materials by Callender and Nicolson [9] then this method was used by Niven [10] and Powell [11]. A review of radial heat flow methods was presented by McElroy and Moore [12] and this method was widely used for measuring the thermal conductivity of solids for various materials. [13–23]. Woodcraft [24] described a method of predicting the thermal conductivity of any aluminium alloy between the superconducting transition temperature (approximately 1 K) and room temperature, based on a measurement of the thermal conductivity or electrical resistivity at a single temperature. Smontara et al. [25] also calculated using the Wiedemann–Franz law and the measured electrical resistivity (T) data. The phase diagram of Sn–Al system is given in Fig. 1 [26]. In the present work, firstly, the dependence of thermal conductivity of solid phases on temperature for the pure materials and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al binary alloys have been investigated. Secondly, the temperature coefficients of the pure Sn, pure Al and their binary alloys have been determined from the graph of the thermal conductivity versus temperature. Finally, the thermal conductivity ratios of liquid phases to solid phases at their melting temperatures have been determined to obtain the thermal conductivities of liquid phases for same materials.
2. Experimental procedure 2.1. Measurement of thermal conductivity of solid phases In the present work, the radial heat flow apparatus was chosen to determine the thermal conductivity of solids, because of its symmetrical characteristics. A radial heat flow apparatus, originally designed by Gündüz and Hunt [13,14] and modified by Maras¸lı and Hunt [15] were used to experimentally determine the thermal conductivity of solid phases. More details of the apparatus are described in Refs. [13–23] as shown in Fig. 2. Consider a cylindrical specimen heated by a heating element along the axis at the centre of the specimen. At the steady-state conditions, thermal conductivity of solid can be determined by using appropriate boundary conditions with Fourier’s law for the radial heat flow, and the temperature gradients are given as:
dT dr
S
=−
Q 2rS
(1)
where Q is the total input power from the centre of the specimen, the length of the heating element, S the thermal conductivity of the solid phase, and r is the distance from the centre. Integration of Eq. (1) for the radial heat flow gives: S =
Q ln (r2 /r1 ) 2 (T1 − T2 )
(2)
or S = a0
Q T1 − T2
(3)
where a0 = ln (r2 − r1 )/2 is an experimental constant, the length of the heating element, r1 and r2 are fixed distances from the centre of the sample (r2 > r1 ), and T1 , T2 are the temperatures at the fixed positions r1 and r2 , respectively. If the values of Q, r1 , r2 , , T1 and T2 can be accurately measured for the well-characterized sample, then reliable S values can be evaluated provided that the vertical temperature variations are minimum or zero [18]. The crucible was made from high purity graphite as symmetrical as possible to ensure that the isotherms were almost parallel to the central axis, as shown in Fig. 3a. The crucible consisted of three parts, a 120 mm length of cylindrical tube, the top and the bottom lids, as shown in Fig. 3b. The lids were pushed tightly into the cylindrical tube. The cylindrical tube was 30 mm inner diameter
F. Meydaneri et al. / Fluid Phase Equilibria 298 (2010) 97–105
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Fig. 3. (a) Transverse section of the sample, (b) graphite crucible, and (c) longitudinal section of a part of the sample at the central of the specimen.
(ID) × 40 mm outer diameter (OD) × 120 mm long. The top and bottom lids were made as symmetrical as possible. The top lid had four air or feeding holes, one vertical thermocouple hole and one central hole for the alumina tube. The bottom lid had five holes, three holes for the fixed thermocouples (one of them for the control unit, two others for measurement thermocouples), one for the vertical thermocouple and one for the central alumina tube. The control unit thermocouple and one of the measurement thermocouples were placed 0.5–3 mm away from the central alumina tube, the control and the measurement thermocouple holes were drilled at 85.5◦ to the ends of the cylinder tube. The vertical (moveable) and one of the measurements thermocouple holes were drilled 11–13 mm away from the centre, as shown in Fig. 3c. Sufficient amount of materials to produce an ingot of approximately 120 mm long and 30 mm inner diameter were melted in a graphite crucible using the vacuum melting furnace by using 99.9% pure Sn and 99.99% pure Al supplied by Alfa Aesar. The specific amount of metal was melted under the vacuum approximately 50 K above the melting point of the alloys. Then, molten metal was
stirred with a graphite plunger. The molten metal was poured into the graphite crucible held in a specially constructed casting furnace at approximately 50 K above the melting temperature of the alloys. After that, it was directionally solidified from the bottom to the top to ensure that the crucible was completely filled. The sample was taken out from the casting furnace and after thermocouples were placed into alumina tube holes drilled to bottom lid and placed into the radial heat flow apparatus. The specimen was heated from the center by using a single heating wire (∼155 mm lengths and 1.7 mm in diameter, Kanthal A-1) in steps of 20–50 K up to 10 K below the melting temperature and the outside of the specimen was kept cool with the water cooling jacket to get a radial temperature gradient. The length of the central heating wire was chosen to be slightly longer than the length of the specimen to make the vertical isotherms parallel to the axis. The gap between the cooling jacket and the specimen was filled with free running sand to get a large radial temperature gradient on the specimen. The larger radial temperature gradient is desired to increase the experimental sensitivity for the solid phase thermal conductivity
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Fig. 4. Typical vertical temperature variations in the specimen at different setting temperatures for Sn.
Table 1 Experimental data for the thermal conductivity determination of solid pure Sn. T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
373 393 413 433 453 473
17.79 23.78 27.22 31.26 38.65 42.79
371.70 392.41 412.69 432.31 452.88 473.13
371.25 391.79 411.96 431.45 451.83 471.93
0.45 0.62 0.73 0.86 1.05 1.20
67.20 65.20 63.30 61.70 62.50 60.60
r1 = 2.47 × 10−3 m; r2 = 13.61 × 10−3 m; l = 0.155 m; a0 = 1.75 m−1 . Table 2 Experimental data for the thermal conductivity determination of solid pure Al. T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
423 473 523 573 623 673 723 773 823 873 923
35.80 74.70 81.07 95.20 137.90 162.47 197.56 221.28 279.18 322.77 372.08
422.69 472.43 522.02 572.30 622.11 672.40 722.10 771.88 822.77 873.82 920.32
422.56 472.14 521.70 571.90 621.52 671.67 721.20 770.86 821.42 872.25 918.36
0.13 0.29 0.32 0.40 0.59 0.73 0.90 1.02 1.35 1.57 1.96
302.90 288.30 278.60 261.80 255.70 244.80 241.40 238.60 227.40 226.10 208.80
r1 = 3.77 × 10−3 m; r2 = 11.68 × 10−3 m; l = 0.157 m; a0 = 1.14 m−1 .
measurements. The temperature of the specimen was controlled with a Euroterm 9706 type controller and the temperature in the specimen was stable to ±0.01–0.02 K for short periods (e.g. half an hour, 1 h). At the steady state, the total input power and the temperatures of the stationary thermocouples were recorded with Hewlett-Packard 34401 type multimeter and Pico TC-08 model data logger. The vertical isotherm for each setting was made parallel to the axis at the measurement region by moving the central heater up and down, as shown in Fig. 4. After all the desired power setting and the temperature measurements had been completed during the heating procedure, the specimen was started in same steps cool to the room temperature. Then the sample was moved from the furnace and cut transversely near the temperature measurement point, after that the specimen was ground and polished for the measurements of the r1 and r2 . The positions of the thermocouples were then photographed with digital camera placed in conjunction with an Olympus BH2 optical microscope and a graticule (100 × 0.01 = 1 mm) was also photographed using the same objective. The photographs of the positions of the thermocouples and the graticule were superimposed on one another using Adobe PhotoShop CS2 version software,
Table 3 Experimental data for the thermal conductivity determination of solid Sn–0.5 wt.% Al alloy. T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
363 383 403 423 443 463 483 493
26.38 32.24 36.91 38.43 49.59 55.47 51.93 59.10
364.79 383.59 403.96 424.54 444.83 464.31 484.15 492.16
364.19 382.82 403.07 423.61 443.62 462.93 482.85 490.55
0.60 0.77 0.89 0.93 1.21 1.38 1.30 1.61
83.50 79.50 78.70 78.50 77.80 76.30 75.80 69.70
r1 = 2.509 × 10−3 m; r2 = 16.885 × 10−3 m; l = 0.156 m; a0 = 1.94 m−1 .
Table 4 Experimental data for the thermal conductivity determination of solid Sn–2.2 wt.% Al alloy. T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
323 343 363 383 403 423 443 463 483 493
5.39 12.40 23.75 30.35 39.69 47.74 62.54 73.83 86.48 97.19
322.56 342.50 363.97 382.81 403.07 420.32 443.36 463.34 483.69 493.58
322.50 342.36 366.69 382.45 402.60 419.74 442.58 462.40 482.54 492.25
0.06 0.14 0.28 0.36 0.47 0.58 0.78 0.94 1.15 1.33
98.80 97.40 93.30 92.70 92.90 90.50 88.20 86.30 82.70 80.30
r1 = 4.24 × 10−3 m; r2 = 13.55 × 10−3 m; l = 0.155 m; a0 = 1.19 m−1 .
so that accurate measurement of the distances of stationary thermocouples could be made to an accuracy of ±10 m. The transverse and longitudinal sections of the specimen were examined for the porosity, crack and casting defects to make sure that these would not introduce any uncertains to the measurements. Experimental data for the thermal conductivity determination of solid phases for pure Sn, pure Al and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al binary alloys are given in Tables 1–7. The variations of solid phase thermal conductivity versus temperature for same binary alloys are also shown in Fig. 5. 2.2. Determination of the temperature coefficient For a given composition, the dependence of the thermal conductivity of solid phase on temperature can be expressed as: S = S0 [1 + ˛(T − T0 )]
(4)
F. Meydaneri et al. / Fluid Phase Equilibria 298 (2010) 97–105 Table 5 Experimental data for the thermal conductivity determination of solid Sn–25 wt.% Al alloy.
101
Table 8 Temperature coefficient (˛) obtained from the graph of thermal conductivity versus temperature.
T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
System
S0 (W/K m)
S (W/K m)
T0 (K)
T (K)
˛ (K−1 )
323 343 363 383 403 423 443 463 483 493
4.64 10.26 16.99 22.05 37.00 43.92 54.76 67.86 86.46 101.07
321.72 341.74 366.00 382.35 402.69 419.99 442.97 463.09 483.25 493.11
121.68 341.65 365.85 382.15 402.35 419.57 442.44 462.40 482.36 491.03
0.04 0.09 0.15 0.20 0.34 0.42 0.53 0.69 0.89 1.08
139.30 136.80 135.90 132.30 130.50 125.40 123.90 118.00 116.50 112.30
Pure Sn[PW] Pure Al[PW] Sn–0.5 wt.% Al[PW] Sn–2.2 wt.% Al[PW] Sn–25 wt.% Al[PW] Sn–50 wt.% Al[PW] Sn–75 wt.% Al[PW] Pure Sn [34] Pure Sn [35] Pb–95 wt.% Sn [36]
67.200 302.90 83.500 99.800 139.00 191.00 235.00 54.27 61.90 –
60.600 208.80 69.700 80.300 112.30 142.00 188.50 34.27 55.20 58.3
373 423 363 323 323 323 323 300 300 –
473 923 493 493 493 493 493 500 500 456
0.00098 0.00062 0.00127 0.00114 0.00112 0.00150 0.00116 0.00816 0.000541 0.00094
r1 = 3.83 × 10−3 m; r2 = 13.02 × 10−3 m; l = 0.155 m; a0 = 1.25 m−1 . Table 6 Experimental data for the thermal conductivity determination of solid Sn–50 wt.% Al alloy. T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
323 343 363 383 403 423 443 463 483 493
8.39 17.38 30.49 38.72 50.22 57.58 68.75 87.07 103.06 102.10
322.52 342.72 364.17 383.00 403.66 421.42 443.45 463.38 483.70 493.68
322.45 342.57 363.90 382.65 403.20 420.88 442.79 462.51 482.62 492.53
0.07 0.15 0.27 0.35 0.46 0.54 0.66 0.87 1.08 1.15
191.70 185.30 180.60 177.00 174.60 170.60 166.60 160.10 152.60 142.00
r1 = 2.57 × 10−3 m; r2 = 13.30 × 10−3 m; l = 0.155 m; a0 = 1.68 m−1 . Table 7 Experimental data for the thermal conductivity determination of solid Sn–75 wt.% Al alloy. T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
323 343 363 383 403 423 443 463 483 493
9.70 16.50 25.51 36.96 48.09 62.72 70.91 77.93 80.81 92.06
323.72 343.87 364.21 383.69 403.43 419.78 442.62 464.69 483.84 493.04
323.65 343.75 364.02 383.41 403.06 419.29 442.05 464.05 483.14 492.21
0.07 0.12 0.19 0.28 0.37 0.49 0.57 0.64 0.70 0.83
235.50 233.70 228.20 224.40 220.90 217.60 211.40 207.00 196.20 188.50
−3
r1 = 2.67 × 10
−3
m; r2 = 15.39 × 10
−1
m; l = 0.155 m; a0 = 1.79 m
.
where S is the thermal conductivity of the solid phase at the temperature T, S0 is the thermal conductivity at the initial temperature and ˛ is the temperature coefficient. From Eq. (4), the temperature coefficient, ˛ is written as: ˛=
=
1 S0 T
(5)
This means that the temperature coefficient, ˛ can be obtained from the graph of thermal conductivity versus temperature. The values used in the determination of the temperature coefficient, ˛ of solid phases are given in Table 8. 2.3. Thermal conductivity ratio of liquid phase to solid phase It is not possible to measure the thermal conductivity of liquid phase with the radial heat flow apparatus since a thick liquid layer (10 mm) is required. A layer of this size would certainly have led to convection. If the ratio of thermal conductivity of the liquid phase to solid phase is known and the thermal conductivity of the solid phase is measured at the melting temperature (or eutectic), the thermal conductivity of the liquid phase can then be evaluated. The thermal conductivity ratio can be obtained during directional growth with the Bridgman type growth apparatus. The heat flow away from the interface through the solid phase must balance that liquid phase plus the latent heat generated at the interface, i.e. [29]: VL = S GS − L GL
(6)
where V is the growth rate, L is the latent heat, GS and GL are the temperature gradients in the solid and liquid, respectively and S and L are the thermal conductivities of the solid and the liquid phases, respectively. For very low growth rates VL S GS , so that the conductivity ratio, R is given by: R=
Fig. 5. Thermal conductivities of pure Sn [27], pure Al [28] and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al binary alloys versus temperature.
S − S0 S0 (T − T0 )
L GS = S GL
(7)
If the right hand side of Eq. (7) is obtained and the value of S is measured, then the liquid thermal conductivity can be evaluated from Eq. (7) [13–23]. In the present work, the thermal conductivity ratio of liquid phase to solid phase, R, was obtained in a directional growth apparatus (Bridgman type) [30]. A directional growth apparatus which was first constructed by McCartney [31] was used to determine the thermal conductivity ratio R = L /S . A thin walled graphite crucible was produced by drilling out a graphite rod of 6.35 mm outer diameter (OD) and 220 mm total length to a depth of 180 mm with a 4 mm inner diameter (ID). As mentioned above, the sufficient amount of materials (pure Sn, pure Al and their binary alloys) was melted in a vacuum furnace. After stirring, the molten metal was poured into thin walled graphite crucibles and the molten alloy was then directionally frozen from bottom to top to ensure that the crucible was completely full. Then, the sample was positioned in a Bridgman type furnace in a graphite rod of 6.35 mm
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Fig. 7. Temperature versus time for pure Al.
Fig. 6. Temperature versus time for pure Sn.
outer diameter (OD) and 220 mm total length to a depth of 180 mm with a 4 mm inner diameter (ID) and it was heated to 50 K over the melting temperature. The specimen was then left to reach the thermal equilibrium for at least 2 h. The temperature in the specimen was measured with an insulated 0.5 mm K-type thermocouple. 1.2 mm outer diameter (OD) × 0.8 mm inner diameter (ID) alumina tube was used to insulate the thermocouple from the melt and the thermocouple was placed perpendicular to heat flow. The temperature on the sample was controlled to an accuracy of ±0.5 K with a Eurotherm 815 type controller and the temperature of fluid in the reservoir was approximately 288–293 K (15–20 ◦ C). When the specimen temperature stabilized, the directional growth begun by turning on the motor to observe a change in the slope of the cooling rate for liquid and solid phases, controlled-cooling with time and the temperature change with time was recorded by a data logger via computer. While the solid–liquid interface was crossing the thermocouple, a change in the slope of the temperature change with time was observed. In the present measurements, the growth rate was 66 m/s. After the temperature reading is 20–30 K below the melting temperature the growth was stopped and the samples were quenched by rapidly pulling it down into the fluid reservoir. The conductivity ratio was evaluated from the change in the slope of the temperature versus time curves. From the temperature versus time curves, the slope of the liquid and solid phases can be written as:
dT dt
L
=
dT dx dx
L
dt
L
= GL VL
(8)
and
dT dt
S
=
dT dx dx
S
dt
S
= GS VS
(9)
VL = VS then from Eqs. (8) and (9), the thermal conductivity ratio can be written as: R=
(dT/dt)S GS L = = S GL (dT/dt)L
(10)
where the values of (dT/dt)S and (dT/dt)L for pure Sn, pure Al and eutectic Sn–0.5 wt.% Al alloy were directly measured from the temperature versus time curve as shown in Figs. 6–8, respectively.
Fig. 8. Temperature versus time for eutectic Sn–0.5 wt.% Al alloy.
3. Results and discussions 3.1. Thermal conductivity and temperature coefficient of solid phase The thermal conductivities of the solid phase, S , versus temperature for pure Sn, pure Al and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al alloys are shown in Fig. 5. The thermal conductivities of solid phases for same alloys decreases with increasing temperature and the values of S for same systems lie between the values of S for pure Al and pure Sn as shown in Fig. 5. The estimated experimental uncertainty in the measurement of S is the sum of the fractional uncertainty of the measurements of power, temperature difference, length of heating wire and thermocouples positions which can be expressed as:
S = Q + T ∗ + + r1 + r2 S Q T r1 ln r2 /r1 r2 ln r2 /r1
(11)
F. Meydaneri et al. / Fluid Phase Equilibria 298 (2010) 97–105
103
Fig. 9. Schematic-drawing of calibration experimental setup.
3.1.1. Fractional uncertainty in the power measurement The input power is expressed as: Q = Vh I
(12)
where Vh is the potential difference between the ends of the central heating wire and I is the current outgoing from the central heating wire. The fractional uncertainty in the power measurement can be expressed as:
Q Vh I Q = V + I
(13)
h
The potential difference between the ends of the central heating wire was measured with a Hewlett-Packard 34401 multimeter to an accuracy of ±1%. Current outgoing from the central heating wire was measured with Clampmeter to accuracy of ±1%. Thus the total fractional uncertainty in power measurement is about 2%. 3.1.2. The fractional uncertainty in the measurement of heating wire’s length, , and the fixed distances (r1 , r2 ) As can be seen from Tables 1–7, the average lengths of heating wire was 155 mm and measured to an accuracy of ±0.5 mm. The fractional uncertainty in the measurement of heating wire’s length is about 0.3%. The fixed distances of thermocouples from centre of the specimen (r1 , r2 ) were measured using Adobe PhotoShop CS2 version software from the photographs of the thermocouple’s positions to an accuracy of ±10 m. The fixed distances from the center of specimen are about 3 mm and 14 mm. The fractional uncertainty in the measurements of the fixed distances is between 0.2% and 0.5%. Therefore the total fractional uncertainty for measuring the heating wire’s length and the fixed distances is 1%. 3.1.3. Fractional uncertainty in the measurement of temperature difference between two thermocouples, T = T1 − T2 at the setting temperature The difference of the thermocouples readings, T* at the same point of specimen with a setting temperature must be known or measured to determine the uncertainty of temperature measurement. To determine the difference of thermocouples readings, the thermocouples were calibrated by detecting the melting point of
Fig. 10. Thermocouples calibration by detecting melting temperature.
metallic material as shown in Fig. 9. It can be seen from Fig. 10, the difference between two thermocouples readings, T*, at the melting temperature of metallic material was measured to be an accuracy of ±0.1–0.2 K. As can be seen for Sn–Al binary alloys from Tables 3–7, the temperature difference between two fixed thermocouples, T = T1 − T2 at 493 K changes between 1.61 K and 0.83 K. Thus the uncertainty in the temperature measurement at 493 K is about 8%. Therefore the total fractional uncertainty in the measurements of thermal conductivity of solid phases is about 11%. The thermal conductivities of solid phases for pure Sn, pure Al and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al alloys at approximately their melting temperature are found to be 60.60 ± 0.06, 208.80 ± 0.22, 69.70 ± 0.07, 80.30 ± 0.08, 112.30 ± 0.12, 142.00 ± 0.15, 188.00 ± 0.20 W/K m, respectively and the temperature coefficients for same systems were found to be 0.00098, 0.00062, 0.00127, 0.00114, 0.00112, 0.00150, 0.00116 K−1 , respectively from Fig. 5.
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Table 9 Thermal conductivities of solid and liquid phase at their melting temperature for pure Sn, pure Al and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al binary alloys. System
Phase
Temperature (K)
Liquid Sn Solid Sn
504 473 300 500 300 500
67.26 60.60
Liquid Al Solid Al
933 923
235.94 208.80
Sn–0.5 wt.% Al (eutectic composition)
(Liquid phase)Sn–0.5 wt.% Al (Solid phase) Sn–0.5 wt.% Al
501 493
73.88 69.70
Sn–2.2 wt.% Al
(Liquid phase)Sn–0.5 wt.% Al (Solid phase) Sn–2.2 wt.% Al
501 493
73.88 80.30
– 81.20 [32]
0.91
Sn–25 wt.% Al
(Liquid phase)Sn–0.5 wt.% Al (Solid phase) Sn–25 wt.% Al
501 493
73.88 112.30
– 114.20 [32]
0.65
Sn–50 wt.% Al
(Liquid phase)Sn–0.5 wt.% Al (Solid phase) Sn–50 wt.% Al
501 493
73.88 142.00
– 139.30 [33]
0.51
Sn–75 wt.% Al
(Liquid phase)Sn–0.5 wt.% Al (Solid phase) Sn–75 wt.% Al
501 493
73.88 188.50
– 173.20 [33]
0.39
Pure Sn
Pure Al
(W/K m) [PW]
(W/K m) [literature]
R = L /S
– 59.80 [27] 54.27 [34] 34.27 [34] 61.90 [35] 55.20 [35]
1.11
– 213.00 [28] 240.00 [25] 210.00 [24]
1.13
–
1.06
3.2. Thermal conductivity ratio of liquid phase to solid phase
4. Conclusions
R is ratio of the solid phase temperature gradient to the liquid phase temperature gradient or ratio of the liquid phase thermal conductivity to the solid phase thermal conductivity (Eq. (10)). As can be seen from Figs. 6–8, the value of R can be evaluated from the ratio of the solid phase cooling rate to the liquid phase cooling rate. The values of R for pure Sn, pure Al and eutectic Sn–0.5 wt.% Al alloy at their melting temperatures were found to be 1.11, 1.13 and 1.06, respectively by Bridgman type directional solidification apparatus and the results are given in Table 9. The thermal conductivities of solid phases for the same systems at their melting temperature are measured to be 60.60 ± 0.06, 208.80 ± 0.22, 69.70 ± 0.07 W/K m, respectively with radial heat flow apparatus. Thus the thermal conductivities of liquid phases, L for pure Sn, pure Al and eutectic Sn–0.5 wt.% Al alloy at their melting temperature were determined to be 67.26, 235.94 and 73.88 W/K m, respectively from Eq. (10) using the values of S and R and the evaluated values are also given in Table 9. The values of R for Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al alloys at their melting temperature were found from R = L (eutectic liquid)/S (solid). L (eutectic liquid) was found to be 73.88 W/K m. The thermal conductivities of solid phases for same systems at approximately their melting temperature are found to be 80.30 ± 0.08, 112.30 ± 0.12, 142.00 ± 0.15, 188.00 ± 0.20 W/K m, respectively with radial heat flow apparatus. Thus the values of R for same binary alloys at their melting temperatures were found to be 0.91, 0.65, 0.51 and 0.39, respectively. The values of thermal conductivities used in the calculations are given in Table 9. A comparison of our results with values of S found in the literature is also given in Table 9. As can bee seen from Table 9 and Fig. 5, the values measured in the present work for pure Sn and pure Al are in a good agreement with the values obtained by Touloukian [27,28], Arı and Saatc¸i [34,35]. In the literature, there were little data for Sn–Al binary alloys to make comparison but the values of S for above Sn–Al binary alloys are between the S values of pure Sn and pure Al measured in present work.
Thermal conductivities of solid and liquid phases for pure Sn, pure Al and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al alloys were surveyed as functions of temperature. The results are summarized as follows: (a) The variations of thermal conductivities of solid phase versus temperature for pure Sn, pure Al and same systems have been measured with ± 11% experimental uncertainty by using radial heat flow apparatus. (b) The thermal conductivities of solid phases at approximately their melting temperature and the temperature coefficients for same materials have been determined from the graphs of the solid phase thermal conductivity versus temperature. (c) The thermal conductivity ratios of liquid phase to solid phase for pure Sn, pure Al and eutectic Sn–0.5 wt.% Al alloy at their melting temperature have been determined with a Bridgeman type directional solidification apparatus. (d) The values of R for Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al alloys at their melting temperature were found from R = L (eutectic liquid)/S (solid). (e) The thermal conductivity of liquid phases for pure Sn, pure Al and eutectic Sn–0.5 wt.% Al alloy at their melting temperature has been evaluated by using the values of solid phase thermal conductivities and the thermal conductivity ratios of liquid phase to solid phase. Acknowledgments This project was supported by Erciyes University Scientific Research Project Unit under Contract No: FBD-09-846. Authors would like to thank to Erciyes University Scientific Research Project Unit for their financial supports. References [1] M. Winter, J.O. Besenhard, Electrochim. Acta 45 (1999) 31–50.
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Contents lists available at ScienceDirect
Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid
Thermal conductivities of solid and liquid phases for pure Al, pure Sn and their binary alloys Fatma Meydaneri a,∗ , Buket Saatc¸i b , Mehmet Özdemir c a b c
Department of Metallurgy and Materials Science Engineering, Faculty of Engineering, Karabük University, 78050 Karabük, Turkey Faculty Arts and Sciences, Department of Physics, Erciyes University, 38039 Kayseri, Turkey Faculty Arts and Sciences, Department of Chemistry, Erciyes University, 38039 Kayseri, Turkey
a r t i c l e
i n f o
Article history: Received 5 April 2010 Received in revised form 11 June 2010 Accepted 14 July 2010 Available online 27 July 2010 Keywords: Thermal conductivity Radial heat flow Directional solidification Sn–Al alloy Thermal coefficient
a b s t r a c t The variations of thermal conductivities of solid phases versus temperature for pure Sn, pure Al and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al binary alloys were measured with a radial heat flow apparatus. From thermal conductivity variations versus temperature, the thermal conductivities of the solid phases at their melting temperature and temperature coefficients for same materials were also found to be 60.60 ± 0.06, 208.80 ± 0.22, 69.70 ± 0.07, 80.30 ± 0.08, 112.30 ± 0.12, 142.00 ± 0.15, 188.50 ± 0.20 W/K m and 0.00098, 0.00062, 0.00127, 0.00114, 0.00112, 0.00150, 0.00116 K−1 , respectively. The thermal conductivity ratios of liquid phase to solid phase for the pure Sn, pure Al and eutectic Sn–0.5 wt.% Al alloy at their melting temperature are found to be 1.11, 1.13, 1.06 with a Bridgman type directional solidification apparatus, respectively. Thus the thermal conductivities of liquid phases for pure Sn, pure Al and eutectic Sn–0.5 wt.% Al binary alloy at their melting temperature were evaluated to be 67.26, 235.94 and 73.88 W/K m, respectively by using the values of solid phase thermal conductivities and the thermal conductivity ratios of liquid phase to solid phase. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Recently, Sn-based alloys have been investigated intensively as potential alternative anode materials for lithium ion batteries because of its higher capacity storage than that of the carbonaceous anodes [1]. Al-based alloys, which consist of Al–Pb and Al–Sn, are widely used for sliding bearing applications due to their good load carrying capacity, fatigue resistance, wear resistance and sliding properties. However, environmental legislation has driven manufacturers to eliminate Pb from bearing alloys. Thus, focus has being concentrated on Sn–Al alloys with high content of Sn. Al–Sn-based alloys are widely used in sliding bearings in the aerospace, combustion engine pistons, cylinder liners and automobile industries, because they offer good balance between load bearing capacity and fatigue resistance, their high wear resistance, together with favorable conformability and compatibility [2–5]. Sliding bearing is applied broadly in modern industry and its quality directly affects the precision and life-span of machines. Friction and wear performance of sliding bearing is of great interest, especially with environmental and health concerns in recent years [6]. In this binary alloy system, tin is a necessary soft phase in the aluminium matrix. Due to the excellent anti-welding characteristics with iron,
∗ Corresponding author. Tel.: +90 370 433 82 00; fax: +90 370 433 82 04. E-mail address: [email protected] (F. Meydaneri). 0378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2010.07.015
its low modulus and low strength, tin can provide suitable shear surface during sliding. Such alloys are also recommended as anodic protective coating for anti-corrosive protection of steel and these alloys provide a good combination of strength, corrosion resistance and surface properties [2,7,8]. For pure Al, pure Sn and their binary alloys do not satisfy some technologic parameters as working temperature, for example. The knowledge of the thermodynamics of materials provides fundamental information about the stability of phases and about the driving forces for chemical reactions and diffusion processes. Also, the data are of great importance for the development of electronic materials, interconnection technologies, sliding properties, especially in microelectronics and modern industry. Hence, the purpose of this study is to determine thermal coefficients and thermal conductivities of solid and liquid phases in order to estimate some of thermodynamic properties such as Gibbs–Thomson coefficient ( ), solid–liquid interfacial energy ( SL ) and grain boundary energy ( GB ). The thermal conductivity, , is one of the main fundamental properties of materials such as density, melting point, entropy, resistance, and crystal structure parameters and it plays a critical role in controlling the performance and stability of materials. The investigation of thermal conductivity is a valuable tool for the study of transport mechanisms of alloys. Although the value of for pure materials was obtained theoretically and experimentally, there are not enough information and data available about the ther-
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Fig. 1. The phase diagram of Sn–Al system [26].
Fig. 2. Block diagram of radial heat flow system.
mal conductivity of alloys. The values of for alloys change, as in pure materials, not only with temperature but also it changes by compositions of the materials. In the experimental determination of the thermal conductivity of solids, a number of different methods of measurements are used for different ranges of temperature and for various classes of materials having different ranges of thermal conductivity values. Many attempts have been made to determine the thermal conductivity values of solid and liquid phases in various materials by using different methods [9–23]. One of the common techniques for measuring the thermal conductivity of solids is the radial heat flow method. There are several different types of apparatus all employing radial heat flow. The classification is mainly based upon specimen geometry, i.e. cylindrical, spherical, ellipsoidal, concentric sphere, concentric cylinder, and plate methods. In the present work, radial heat flow method was used. The radial heat flow method uses a specimen in the form of a right circular cylinder with a coaxial central hole which contains either a heater or a heat sink, depending on whether the described heat flow direction is to be radially outward or inward. Temperatures within the specimen are measured by thermocouples. This method was first used for measuring the thermal conductivity of solids for pure materials by Callender and Nicolson [9] then this method was used by Niven [10] and Powell [11]. A review of radial heat flow methods was presented by McElroy and Moore [12] and this method was widely used for measuring the thermal conductivity of solids for various materials. [13–23]. Woodcraft [24] described a method of predicting the thermal conductivity of any aluminium alloy between the superconducting transition temperature (approximately 1 K) and room temperature, based on a measurement of the thermal conductivity or electrical resistivity at a single temperature. Smontara et al. [25] also calculated using the Wiedemann–Franz law and the measured electrical resistivity (T) data. The phase diagram of Sn–Al system is given in Fig. 1 [26]. In the present work, firstly, the dependence of thermal conductivity of solid phases on temperature for the pure materials and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al binary alloys have been investigated. Secondly, the temperature coefficients of the pure Sn, pure Al and their binary alloys have been determined from the graph of the thermal conductivity versus temperature. Finally, the thermal conductivity ratios of liquid phases to solid phases at their melting temperatures have been determined to obtain the thermal conductivities of liquid phases for same materials.
2. Experimental procedure 2.1. Measurement of thermal conductivity of solid phases In the present work, the radial heat flow apparatus was chosen to determine the thermal conductivity of solids, because of its symmetrical characteristics. A radial heat flow apparatus, originally designed by Gündüz and Hunt [13,14] and modified by Maras¸lı and Hunt [15] were used to experimentally determine the thermal conductivity of solid phases. More details of the apparatus are described in Refs. [13–23] as shown in Fig. 2. Consider a cylindrical specimen heated by a heating element along the axis at the centre of the specimen. At the steady-state conditions, thermal conductivity of solid can be determined by using appropriate boundary conditions with Fourier’s law for the radial heat flow, and the temperature gradients are given as:
dT dr
S
=−
Q 2rS
(1)
where Q is the total input power from the centre of the specimen, the length of the heating element, S the thermal conductivity of the solid phase, and r is the distance from the centre. Integration of Eq. (1) for the radial heat flow gives: S =
Q ln (r2 /r1 ) 2 (T1 − T2 )
(2)
or S = a0
Q T1 − T2
(3)
where a0 = ln (r2 − r1 )/2 is an experimental constant, the length of the heating element, r1 and r2 are fixed distances from the centre of the sample (r2 > r1 ), and T1 , T2 are the temperatures at the fixed positions r1 and r2 , respectively. If the values of Q, r1 , r2 , , T1 and T2 can be accurately measured for the well-characterized sample, then reliable S values can be evaluated provided that the vertical temperature variations are minimum or zero [18]. The crucible was made from high purity graphite as symmetrical as possible to ensure that the isotherms were almost parallel to the central axis, as shown in Fig. 3a. The crucible consisted of three parts, a 120 mm length of cylindrical tube, the top and the bottom lids, as shown in Fig. 3b. The lids were pushed tightly into the cylindrical tube. The cylindrical tube was 30 mm inner diameter
F. Meydaneri et al. / Fluid Phase Equilibria 298 (2010) 97–105
99
Fig. 3. (a) Transverse section of the sample, (b) graphite crucible, and (c) longitudinal section of a part of the sample at the central of the specimen.
(ID) × 40 mm outer diameter (OD) × 120 mm long. The top and bottom lids were made as symmetrical as possible. The top lid had four air or feeding holes, one vertical thermocouple hole and one central hole for the alumina tube. The bottom lid had five holes, three holes for the fixed thermocouples (one of them for the control unit, two others for measurement thermocouples), one for the vertical thermocouple and one for the central alumina tube. The control unit thermocouple and one of the measurement thermocouples were placed 0.5–3 mm away from the central alumina tube, the control and the measurement thermocouple holes were drilled at 85.5◦ to the ends of the cylinder tube. The vertical (moveable) and one of the measurements thermocouple holes were drilled 11–13 mm away from the centre, as shown in Fig. 3c. Sufficient amount of materials to produce an ingot of approximately 120 mm long and 30 mm inner diameter were melted in a graphite crucible using the vacuum melting furnace by using 99.9% pure Sn and 99.99% pure Al supplied by Alfa Aesar. The specific amount of metal was melted under the vacuum approximately 50 K above the melting point of the alloys. Then, molten metal was
stirred with a graphite plunger. The molten metal was poured into the graphite crucible held in a specially constructed casting furnace at approximately 50 K above the melting temperature of the alloys. After that, it was directionally solidified from the bottom to the top to ensure that the crucible was completely filled. The sample was taken out from the casting furnace and after thermocouples were placed into alumina tube holes drilled to bottom lid and placed into the radial heat flow apparatus. The specimen was heated from the center by using a single heating wire (∼155 mm lengths and 1.7 mm in diameter, Kanthal A-1) in steps of 20–50 K up to 10 K below the melting temperature and the outside of the specimen was kept cool with the water cooling jacket to get a radial temperature gradient. The length of the central heating wire was chosen to be slightly longer than the length of the specimen to make the vertical isotherms parallel to the axis. The gap between the cooling jacket and the specimen was filled with free running sand to get a large radial temperature gradient on the specimen. The larger radial temperature gradient is desired to increase the experimental sensitivity for the solid phase thermal conductivity
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Fig. 4. Typical vertical temperature variations in the specimen at different setting temperatures for Sn.
Table 1 Experimental data for the thermal conductivity determination of solid pure Sn. T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
373 393 413 433 453 473
17.79 23.78 27.22 31.26 38.65 42.79
371.70 392.41 412.69 432.31 452.88 473.13
371.25 391.79 411.96 431.45 451.83 471.93
0.45 0.62 0.73 0.86 1.05 1.20
67.20 65.20 63.30 61.70 62.50 60.60
r1 = 2.47 × 10−3 m; r2 = 13.61 × 10−3 m; l = 0.155 m; a0 = 1.75 m−1 . Table 2 Experimental data for the thermal conductivity determination of solid pure Al. T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
423 473 523 573 623 673 723 773 823 873 923
35.80 74.70 81.07 95.20 137.90 162.47 197.56 221.28 279.18 322.77 372.08
422.69 472.43 522.02 572.30 622.11 672.40 722.10 771.88 822.77 873.82 920.32
422.56 472.14 521.70 571.90 621.52 671.67 721.20 770.86 821.42 872.25 918.36
0.13 0.29 0.32 0.40 0.59 0.73 0.90 1.02 1.35 1.57 1.96
302.90 288.30 278.60 261.80 255.70 244.80 241.40 238.60 227.40 226.10 208.80
r1 = 3.77 × 10−3 m; r2 = 11.68 × 10−3 m; l = 0.157 m; a0 = 1.14 m−1 .
measurements. The temperature of the specimen was controlled with a Euroterm 9706 type controller and the temperature in the specimen was stable to ±0.01–0.02 K for short periods (e.g. half an hour, 1 h). At the steady state, the total input power and the temperatures of the stationary thermocouples were recorded with Hewlett-Packard 34401 type multimeter and Pico TC-08 model data logger. The vertical isotherm for each setting was made parallel to the axis at the measurement region by moving the central heater up and down, as shown in Fig. 4. After all the desired power setting and the temperature measurements had been completed during the heating procedure, the specimen was started in same steps cool to the room temperature. Then the sample was moved from the furnace and cut transversely near the temperature measurement point, after that the specimen was ground and polished for the measurements of the r1 and r2 . The positions of the thermocouples were then photographed with digital camera placed in conjunction with an Olympus BH2 optical microscope and a graticule (100 × 0.01 = 1 mm) was also photographed using the same objective. The photographs of the positions of the thermocouples and the graticule were superimposed on one another using Adobe PhotoShop CS2 version software,
Table 3 Experimental data for the thermal conductivity determination of solid Sn–0.5 wt.% Al alloy. T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
363 383 403 423 443 463 483 493
26.38 32.24 36.91 38.43 49.59 55.47 51.93 59.10
364.79 383.59 403.96 424.54 444.83 464.31 484.15 492.16
364.19 382.82 403.07 423.61 443.62 462.93 482.85 490.55
0.60 0.77 0.89 0.93 1.21 1.38 1.30 1.61
83.50 79.50 78.70 78.50 77.80 76.30 75.80 69.70
r1 = 2.509 × 10−3 m; r2 = 16.885 × 10−3 m; l = 0.156 m; a0 = 1.94 m−1 .
Table 4 Experimental data for the thermal conductivity determination of solid Sn–2.2 wt.% Al alloy. T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
323 343 363 383 403 423 443 463 483 493
5.39 12.40 23.75 30.35 39.69 47.74 62.54 73.83 86.48 97.19
322.56 342.50 363.97 382.81 403.07 420.32 443.36 463.34 483.69 493.58
322.50 342.36 366.69 382.45 402.60 419.74 442.58 462.40 482.54 492.25
0.06 0.14 0.28 0.36 0.47 0.58 0.78 0.94 1.15 1.33
98.80 97.40 93.30 92.70 92.90 90.50 88.20 86.30 82.70 80.30
r1 = 4.24 × 10−3 m; r2 = 13.55 × 10−3 m; l = 0.155 m; a0 = 1.19 m−1 .
so that accurate measurement of the distances of stationary thermocouples could be made to an accuracy of ±10 m. The transverse and longitudinal sections of the specimen were examined for the porosity, crack and casting defects to make sure that these would not introduce any uncertains to the measurements. Experimental data for the thermal conductivity determination of solid phases for pure Sn, pure Al and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al binary alloys are given in Tables 1–7. The variations of solid phase thermal conductivity versus temperature for same binary alloys are also shown in Fig. 5. 2.2. Determination of the temperature coefficient For a given composition, the dependence of the thermal conductivity of solid phase on temperature can be expressed as: S = S0 [1 + ˛(T − T0 )]
(4)
F. Meydaneri et al. / Fluid Phase Equilibria 298 (2010) 97–105 Table 5 Experimental data for the thermal conductivity determination of solid Sn–25 wt.% Al alloy.
101
Table 8 Temperature coefficient (˛) obtained from the graph of thermal conductivity versus temperature.
T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
System
S0 (W/K m)
S (W/K m)
T0 (K)
T (K)
˛ (K−1 )
323 343 363 383 403 423 443 463 483 493
4.64 10.26 16.99 22.05 37.00 43.92 54.76 67.86 86.46 101.07
321.72 341.74 366.00 382.35 402.69 419.99 442.97 463.09 483.25 493.11
121.68 341.65 365.85 382.15 402.35 419.57 442.44 462.40 482.36 491.03
0.04 0.09 0.15 0.20 0.34 0.42 0.53 0.69 0.89 1.08
139.30 136.80 135.90 132.30 130.50 125.40 123.90 118.00 116.50 112.30
Pure Sn[PW] Pure Al[PW] Sn–0.5 wt.% Al[PW] Sn–2.2 wt.% Al[PW] Sn–25 wt.% Al[PW] Sn–50 wt.% Al[PW] Sn–75 wt.% Al[PW] Pure Sn [34] Pure Sn [35] Pb–95 wt.% Sn [36]
67.200 302.90 83.500 99.800 139.00 191.00 235.00 54.27 61.90 –
60.600 208.80 69.700 80.300 112.30 142.00 188.50 34.27 55.20 58.3
373 423 363 323 323 323 323 300 300 –
473 923 493 493 493 493 493 500 500 456
0.00098 0.00062 0.00127 0.00114 0.00112 0.00150 0.00116 0.00816 0.000541 0.00094
r1 = 3.83 × 10−3 m; r2 = 13.02 × 10−3 m; l = 0.155 m; a0 = 1.25 m−1 . Table 6 Experimental data for the thermal conductivity determination of solid Sn–50 wt.% Al alloy. T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
323 343 363 383 403 423 443 463 483 493
8.39 17.38 30.49 38.72 50.22 57.58 68.75 87.07 103.06 102.10
322.52 342.72 364.17 383.00 403.66 421.42 443.45 463.38 483.70 493.68
322.45 342.57 363.90 382.65 403.20 420.88 442.79 462.51 482.62 492.53
0.07 0.15 0.27 0.35 0.46 0.54 0.66 0.87 1.08 1.15
191.70 185.30 180.60 177.00 174.60 170.60 166.60 160.10 152.60 142.00
r1 = 2.57 × 10−3 m; r2 = 13.30 × 10−3 m; l = 0.155 m; a0 = 1.68 m−1 . Table 7 Experimental data for the thermal conductivity determination of solid Sn–75 wt.% Al alloy. T (K)
Q (W)
T1 (K)
T2 (K)
T = T1 − T2 (K)
S (W/K m)
323 343 363 383 403 423 443 463 483 493
9.70 16.50 25.51 36.96 48.09 62.72 70.91 77.93 80.81 92.06
323.72 343.87 364.21 383.69 403.43 419.78 442.62 464.69 483.84 493.04
323.65 343.75 364.02 383.41 403.06 419.29 442.05 464.05 483.14 492.21
0.07 0.12 0.19 0.28 0.37 0.49 0.57 0.64 0.70 0.83
235.50 233.70 228.20 224.40 220.90 217.60 211.40 207.00 196.20 188.50
−3
r1 = 2.67 × 10
−3
m; r2 = 15.39 × 10
−1
m; l = 0.155 m; a0 = 1.79 m
.
where S is the thermal conductivity of the solid phase at the temperature T, S0 is the thermal conductivity at the initial temperature and ˛ is the temperature coefficient. From Eq. (4), the temperature coefficient, ˛ is written as: ˛=
=
1 S0 T
(5)
This means that the temperature coefficient, ˛ can be obtained from the graph of thermal conductivity versus temperature. The values used in the determination of the temperature coefficient, ˛ of solid phases are given in Table 8. 2.3. Thermal conductivity ratio of liquid phase to solid phase It is not possible to measure the thermal conductivity of liquid phase with the radial heat flow apparatus since a thick liquid layer (10 mm) is required. A layer of this size would certainly have led to convection. If the ratio of thermal conductivity of the liquid phase to solid phase is known and the thermal conductivity of the solid phase is measured at the melting temperature (or eutectic), the thermal conductivity of the liquid phase can then be evaluated. The thermal conductivity ratio can be obtained during directional growth with the Bridgman type growth apparatus. The heat flow away from the interface through the solid phase must balance that liquid phase plus the latent heat generated at the interface, i.e. [29]: VL = S GS − L GL
(6)
where V is the growth rate, L is the latent heat, GS and GL are the temperature gradients in the solid and liquid, respectively and S and L are the thermal conductivities of the solid and the liquid phases, respectively. For very low growth rates VL S GS , so that the conductivity ratio, R is given by: R=
Fig. 5. Thermal conductivities of pure Sn [27], pure Al [28] and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al binary alloys versus temperature.
S − S0 S0 (T − T0 )
L GS = S GL
(7)
If the right hand side of Eq. (7) is obtained and the value of S is measured, then the liquid thermal conductivity can be evaluated from Eq. (7) [13–23]. In the present work, the thermal conductivity ratio of liquid phase to solid phase, R, was obtained in a directional growth apparatus (Bridgman type) [30]. A directional growth apparatus which was first constructed by McCartney [31] was used to determine the thermal conductivity ratio R = L /S . A thin walled graphite crucible was produced by drilling out a graphite rod of 6.35 mm outer diameter (OD) and 220 mm total length to a depth of 180 mm with a 4 mm inner diameter (ID). As mentioned above, the sufficient amount of materials (pure Sn, pure Al and their binary alloys) was melted in a vacuum furnace. After stirring, the molten metal was poured into thin walled graphite crucibles and the molten alloy was then directionally frozen from bottom to top to ensure that the crucible was completely full. Then, the sample was positioned in a Bridgman type furnace in a graphite rod of 6.35 mm
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F. Meydaneri et al. / Fluid Phase Equilibria 298 (2010) 97–105
Fig. 7. Temperature versus time for pure Al.
Fig. 6. Temperature versus time for pure Sn.
outer diameter (OD) and 220 mm total length to a depth of 180 mm with a 4 mm inner diameter (ID) and it was heated to 50 K over the melting temperature. The specimen was then left to reach the thermal equilibrium for at least 2 h. The temperature in the specimen was measured with an insulated 0.5 mm K-type thermocouple. 1.2 mm outer diameter (OD) × 0.8 mm inner diameter (ID) alumina tube was used to insulate the thermocouple from the melt and the thermocouple was placed perpendicular to heat flow. The temperature on the sample was controlled to an accuracy of ±0.5 K with a Eurotherm 815 type controller and the temperature of fluid in the reservoir was approximately 288–293 K (15–20 ◦ C). When the specimen temperature stabilized, the directional growth begun by turning on the motor to observe a change in the slope of the cooling rate for liquid and solid phases, controlled-cooling with time and the temperature change with time was recorded by a data logger via computer. While the solid–liquid interface was crossing the thermocouple, a change in the slope of the temperature change with time was observed. In the present measurements, the growth rate was 66 m/s. After the temperature reading is 20–30 K below the melting temperature the growth was stopped and the samples were quenched by rapidly pulling it down into the fluid reservoir. The conductivity ratio was evaluated from the change in the slope of the temperature versus time curves. From the temperature versus time curves, the slope of the liquid and solid phases can be written as:
dT dt
L
=
dT dx dx
L
dt
L
= GL VL
(8)
and
dT dt
S
=
dT dx dx
S
dt
S
= GS VS
(9)
VL = VS then from Eqs. (8) and (9), the thermal conductivity ratio can be written as: R=
(dT/dt)S GS L = = S GL (dT/dt)L
(10)
where the values of (dT/dt)S and (dT/dt)L for pure Sn, pure Al and eutectic Sn–0.5 wt.% Al alloy were directly measured from the temperature versus time curve as shown in Figs. 6–8, respectively.
Fig. 8. Temperature versus time for eutectic Sn–0.5 wt.% Al alloy.
3. Results and discussions 3.1. Thermal conductivity and temperature coefficient of solid phase The thermal conductivities of the solid phase, S , versus temperature for pure Sn, pure Al and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al alloys are shown in Fig. 5. The thermal conductivities of solid phases for same alloys decreases with increasing temperature and the values of S for same systems lie between the values of S for pure Al and pure Sn as shown in Fig. 5. The estimated experimental uncertainty in the measurement of S is the sum of the fractional uncertainty of the measurements of power, temperature difference, length of heating wire and thermocouples positions which can be expressed as:
S = Q + T ∗ + + r1 + r2 S Q T r1 ln r2 /r1 r2 ln r2 /r1
(11)
F. Meydaneri et al. / Fluid Phase Equilibria 298 (2010) 97–105
103
Fig. 9. Schematic-drawing of calibration experimental setup.
3.1.1. Fractional uncertainty in the power measurement The input power is expressed as: Q = Vh I
(12)
where Vh is the potential difference between the ends of the central heating wire and I is the current outgoing from the central heating wire. The fractional uncertainty in the power measurement can be expressed as:
Q Vh I Q = V + I
(13)
h
The potential difference between the ends of the central heating wire was measured with a Hewlett-Packard 34401 multimeter to an accuracy of ±1%. Current outgoing from the central heating wire was measured with Clampmeter to accuracy of ±1%. Thus the total fractional uncertainty in power measurement is about 2%. 3.1.2. The fractional uncertainty in the measurement of heating wire’s length, , and the fixed distances (r1 , r2 ) As can be seen from Tables 1–7, the average lengths of heating wire was 155 mm and measured to an accuracy of ±0.5 mm. The fractional uncertainty in the measurement of heating wire’s length is about 0.3%. The fixed distances of thermocouples from centre of the specimen (r1 , r2 ) were measured using Adobe PhotoShop CS2 version software from the photographs of the thermocouple’s positions to an accuracy of ±10 m. The fixed distances from the center of specimen are about 3 mm and 14 mm. The fractional uncertainty in the measurements of the fixed distances is between 0.2% and 0.5%. Therefore the total fractional uncertainty for measuring the heating wire’s length and the fixed distances is 1%. 3.1.3. Fractional uncertainty in the measurement of temperature difference between two thermocouples, T = T1 − T2 at the setting temperature The difference of the thermocouples readings, T* at the same point of specimen with a setting temperature must be known or measured to determine the uncertainty of temperature measurement. To determine the difference of thermocouples readings, the thermocouples were calibrated by detecting the melting point of
Fig. 10. Thermocouples calibration by detecting melting temperature.
metallic material as shown in Fig. 9. It can be seen from Fig. 10, the difference between two thermocouples readings, T*, at the melting temperature of metallic material was measured to be an accuracy of ±0.1–0.2 K. As can be seen for Sn–Al binary alloys from Tables 3–7, the temperature difference between two fixed thermocouples, T = T1 − T2 at 493 K changes between 1.61 K and 0.83 K. Thus the uncertainty in the temperature measurement at 493 K is about 8%. Therefore the total fractional uncertainty in the measurements of thermal conductivity of solid phases is about 11%. The thermal conductivities of solid phases for pure Sn, pure Al and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al alloys at approximately their melting temperature are found to be 60.60 ± 0.06, 208.80 ± 0.22, 69.70 ± 0.07, 80.30 ± 0.08, 112.30 ± 0.12, 142.00 ± 0.15, 188.00 ± 0.20 W/K m, respectively and the temperature coefficients for same systems were found to be 0.00098, 0.00062, 0.00127, 0.00114, 0.00112, 0.00150, 0.00116 K−1 , respectively from Fig. 5.
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Table 9 Thermal conductivities of solid and liquid phase at their melting temperature for pure Sn, pure Al and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al binary alloys. System
Phase
Temperature (K)
Liquid Sn Solid Sn
504 473 300 500 300 500
67.26 60.60
Liquid Al Solid Al
933 923
235.94 208.80
Sn–0.5 wt.% Al (eutectic composition)
(Liquid phase)Sn–0.5 wt.% Al (Solid phase) Sn–0.5 wt.% Al
501 493
73.88 69.70
Sn–2.2 wt.% Al
(Liquid phase)Sn–0.5 wt.% Al (Solid phase) Sn–2.2 wt.% Al
501 493
73.88 80.30
– 81.20 [32]
0.91
Sn–25 wt.% Al
(Liquid phase)Sn–0.5 wt.% Al (Solid phase) Sn–25 wt.% Al
501 493
73.88 112.30
– 114.20 [32]
0.65
Sn–50 wt.% Al
(Liquid phase)Sn–0.5 wt.% Al (Solid phase) Sn–50 wt.% Al
501 493
73.88 142.00
– 139.30 [33]
0.51
Sn–75 wt.% Al
(Liquid phase)Sn–0.5 wt.% Al (Solid phase) Sn–75 wt.% Al
501 493
73.88 188.50
– 173.20 [33]
0.39
Pure Sn
Pure Al
(W/K m) [PW]
(W/K m) [literature]
R = L /S
– 59.80 [27] 54.27 [34] 34.27 [34] 61.90 [35] 55.20 [35]
1.11
– 213.00 [28] 240.00 [25] 210.00 [24]
1.13
–
1.06
3.2. Thermal conductivity ratio of liquid phase to solid phase
4. Conclusions
R is ratio of the solid phase temperature gradient to the liquid phase temperature gradient or ratio of the liquid phase thermal conductivity to the solid phase thermal conductivity (Eq. (10)). As can be seen from Figs. 6–8, the value of R can be evaluated from the ratio of the solid phase cooling rate to the liquid phase cooling rate. The values of R for pure Sn, pure Al and eutectic Sn–0.5 wt.% Al alloy at their melting temperatures were found to be 1.11, 1.13 and 1.06, respectively by Bridgman type directional solidification apparatus and the results are given in Table 9. The thermal conductivities of solid phases for the same systems at their melting temperature are measured to be 60.60 ± 0.06, 208.80 ± 0.22, 69.70 ± 0.07 W/K m, respectively with radial heat flow apparatus. Thus the thermal conductivities of liquid phases, L for pure Sn, pure Al and eutectic Sn–0.5 wt.% Al alloy at their melting temperature were determined to be 67.26, 235.94 and 73.88 W/K m, respectively from Eq. (10) using the values of S and R and the evaluated values are also given in Table 9. The values of R for Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al alloys at their melting temperature were found from R = L (eutectic liquid)/S (solid). L (eutectic liquid) was found to be 73.88 W/K m. The thermal conductivities of solid phases for same systems at approximately their melting temperature are found to be 80.30 ± 0.08, 112.30 ± 0.12, 142.00 ± 0.15, 188.00 ± 0.20 W/K m, respectively with radial heat flow apparatus. Thus the values of R for same binary alloys at their melting temperatures were found to be 0.91, 0.65, 0.51 and 0.39, respectively. The values of thermal conductivities used in the calculations are given in Table 9. A comparison of our results with values of S found in the literature is also given in Table 9. As can bee seen from Table 9 and Fig. 5, the values measured in the present work for pure Sn and pure Al are in a good agreement with the values obtained by Touloukian [27,28], Arı and Saatc¸i [34,35]. In the literature, there were little data for Sn–Al binary alloys to make comparison but the values of S for above Sn–Al binary alloys are between the S values of pure Sn and pure Al measured in present work.
Thermal conductivities of solid and liquid phases for pure Sn, pure Al and Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al alloys were surveyed as functions of temperature. The results are summarized as follows: (a) The variations of thermal conductivities of solid phase versus temperature for pure Sn, pure Al and same systems have been measured with ± 11% experimental uncertainty by using radial heat flow apparatus. (b) The thermal conductivities of solid phases at approximately their melting temperature and the temperature coefficients for same materials have been determined from the graphs of the solid phase thermal conductivity versus temperature. (c) The thermal conductivity ratios of liquid phase to solid phase for pure Sn, pure Al and eutectic Sn–0.5 wt.% Al alloy at their melting temperature have been determined with a Bridgeman type directional solidification apparatus. (d) The values of R for Sn–0.5 wt.% Al, Sn–2.2 wt.% Al, Sn–25 wt.% Al, Sn–50 wt.% Al, Sn–75 wt.% Al alloys at their melting temperature were found from R = L (eutectic liquid)/S (solid). (e) The thermal conductivity of liquid phases for pure Sn, pure Al and eutectic Sn–0.5 wt.% Al alloy at their melting temperature has been evaluated by using the values of solid phase thermal conductivities and the thermal conductivity ratios of liquid phase to solid phase. Acknowledgments This project was supported by Erciyes University Scientific Research Project Unit under Contract No: FBD-09-846. Authors would like to thank to Erciyes University Scientific Research Project Unit for their financial supports. References [1] M. Winter, J.O. Besenhard, Electrochim. Acta 45 (1999) 31–50.
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