J. Mater. Sci. Technol. ›› 2022, Vol. 97: 123-133.DOI: 10.1016/j.jmst.2021.04.036
• Research Article • Previous Articles Next Articles
Shiyi Wena, Yong Dua,*(), Jing Tana, Yuling Liua, Peng Zhoub, Jianzhan Longc, George Kaptayd,*()
Received:
2021-02-23
Revised:
2021-04-08
Accepted:
2021-04-08
Published:
2021-06-17
Online:
2021-06-17
Contact:
Yong Du,George Kaptay
About author:
kaptay@hotmail.com (G. Kaptay).Shiyi Wen, Yong Du, Jing Tan, Yuling Liu, Peng Zhou, Jianzhan Long, George Kaptay. A new model for thermal conductivity of “continuous matrix / dispersed and separated 3D-particles” type composite materials and its application to WC-M (M = Co, Ag) systems[J]. J. Mater. Sci. Technol., 2022, 97: 123-133.
Physical model | Factors | Refs. |
---|---|---|
Series | Phase fraction | [ |
Parallel | Phase fraction | [ |
Maxwell-Eucken (two-forms) | Phase fraction | [ |
Effective Medium Theory (EMT) | Phase fraction | [ |
Hasselman-Johnson | Phase fraction, Grain size | [ |
Bruggeman | Phase fraction, Grain size | [ |
Kapitza | Grain size | [ |
Table 1 Summary of the main factors for thermal conductivity considered in each model.
Physical model | Factors | Refs. |
---|---|---|
Series | Phase fraction | [ |
Parallel | Phase fraction | [ |
Maxwell-Eucken (two-forms) | Phase fraction | [ |
Effective Medium Theory (EMT) | Phase fraction | [ |
Hasselman-Johnson | Phase fraction, Grain size | [ |
Bruggeman | Phase fraction, Grain size | [ |
Kapitza | Grain size | [ |
Fig. 1. The dependence of thermal conductivity on the grain size of the WC particles for WC-10Co hard metal at room temperature. The dotted line is calculated by Eq. (1) of Vornborger et al. [33], while the solid line is calculated by our new Eq. (9,9a), using parameters of Table 4 applied for T = 298 K and 10 wt% Co: kM = 97.3 W/(m K), kWC = 203.3 W/(m K), RM/WC = 1.48 × 10-8 m2 K/W, a = 7.76 W/(m K)), b = 169.52 W/(m K) and c = 5.67 × 10-7 m.
No. | Grades | Mass fraction of Co | Mole fraction of Co | Grain size of WC (µm) |
---|---|---|---|---|
1 | YG6B | 0.06 | 0.1750 | 1.6 |
2 | YG6C | 0.06 | 0.1750 | 2.4 |
3 | YG10B | 0.10 | 0.2697 | 1.6 |
4 | YG10C | 0.10 | 0.2697 | 2.4 |
5 | YG15B | 0.15 | 0.3697 | 1.6 |
6 | YG15C | 0.15 | 0.3697 | 2.4 |
7 | YG20B | 0.20 | 0.4538 | 1.6 |
8 | YG20C | 0.20 | 0.4538 | 2.4 |
Table 2 Compositions and grain sizes for the WC-Co samples prepared in the present work.
No. | Grades | Mass fraction of Co | Mole fraction of Co | Grain size of WC (µm) |
---|---|---|---|---|
1 | YG6B | 0.06 | 0.1750 | 1.6 |
2 | YG6C | 0.06 | 0.1750 | 2.4 |
3 | YG10B | 0.10 | 0.2697 | 1.6 |
4 | YG10C | 0.10 | 0.2697 | 2.4 |
5 | YG15B | 0.15 | 0.3697 | 1.6 |
6 | YG15C | 0.15 | 0.3697 | 2.4 |
7 | YG20B | 0.20 | 0.4538 | 1.6 |
8 | YG20C | 0.20 | 0.4538 | 2.4 |
Fig. 2. (a) One representative microstructure of YB10B by SEM; (b) Schematic of the structure of the material if looked perpendicular to its xy plane. The 3D particles of different shapes of average effective side length dX = dWC are separated from each other by the binder metal M (porosity is not shown).
No. | Grain size (µm) | Temperature (K) | Volume Fraction of WC | Density (g/cm3) | Heat Capacity (J/(g K)) | Thermal diffusivity (mm2/s) | Thermal conductivity (W/(m K)) |
---|---|---|---|---|---|---|---|
1 | 1.6 | 300 | 0.8982 | 14.725 | 0.197 | 45.277 | 131.3 |
600 | 0.8974 | 14.646 | 0.250 | 26.993 | 98.6 | ||
900 | 0.8963 | 14.568 | 0.273 | 20.650 | 82.0 | ||
1200 | 0.8953 | 14.490 | 0.301 | 16.946 | 73.9 | ||
2 | 2.4 | 300 | 0.8982 | 14.856 | 0.197 | 55.503 | 162.4 |
600 | 0.8974 | 14.777 | 0.250 | 29.641 | 109.3 | ||
900 | 0.8963 | 14.698 | 0.273 | 21.366 | 85.6 | ||
1200 | 0.8953 | 14.619 | 0.301 | 16.189 | 71.2 | ||
3 | 1.6 | 300 | 0.8352 | 14.121 | 0.207 | 43.088 | 125.9 |
600 | 0.834 | 14.036 | 0.260 | 25.787 | 94.1 | ||
900 | 0.8323 | 13.951 | 0.287 | 19.565 | 78.2 | ||
1200 | 0.8309 | 13.867 | 0.326 | 15.118 | 68.3 | ||
4 | 2.4 | 300 | 0.8352 | 14.137 | 0.207 | 48.021 | 140.5 |
600 | 0.834 | 14.052 | 0.260 | 27.737 | 101.4 | ||
900 | 0.8323 | 13.967 | 0.287 | 20.284 | 81.2 | ||
1200 | 0.8309 | 13.883 | 0.326 | 15.569 | 70.4 | ||
5 | 1.6 | 300 | 0.7614 | 13.811 | 0.219 | 40.447 | 122.4 |
600 | 0.7598 | 13.717 | 0.273 | 24.530 | 92.0 | ||
900 | 0.7576 | 13.624 | 0.304 | 18.364 | 76.1 | ||
1200 | 0.7557 | 13.530 | 0.356 | 14.247 | 68.7 | ||
6 | 2.4 | 300 | 0.7614 | 13.608 | 0.219 | 45.270 | 135.0 |
600 | 0.7598 | 13.515 | 0.273 | 25.906 | 95.7 | ||
900 | 0.7576 | 13.423 | 0.304 | 19.587 | 79.9 | ||
1200 | 0.7557 | 13.331 | 0.356 | 15.434 | 73.3 | ||
7 | 1.6 | 300 | 0.6925 | 13.309 | 0.232 | 36.847 | 113.6 |
600 | 0.6907 | 13.209 | 0.287 | 23.064 | 87.3 | ||
900 | 0.6881 | 13.109 | 0.321 | 17.452 | 73.5 | ||
1200 | 0.6858 | 13.010 | 0.387 | 13.456 | 67.7 | ||
8 | 2.4 | 300 | 0.6925 | 13.221 | 0.232 | 40.183 | 123.0 |
600 | 0.6907 | 13.121 | 0.287 | 24.033 | 90.4 | ||
900 | 0.6881 | 13.022 | 0.321 | 17.998 | 75.3 | ||
1200 | 0.6858 | 12.924 | 0.387 | 14.117 | 70.6 |
Table 3 The density, volume fraction, heat capacity, thermal diffusivity and thermal conductivity for the WC-Co samples at 300, 600, 900 and 1200 K, respectively. *
No. | Grain size (µm) | Temperature (K) | Volume Fraction of WC | Density (g/cm3) | Heat Capacity (J/(g K)) | Thermal diffusivity (mm2/s) | Thermal conductivity (W/(m K)) |
---|---|---|---|---|---|---|---|
1 | 1.6 | 300 | 0.8982 | 14.725 | 0.197 | 45.277 | 131.3 |
600 | 0.8974 | 14.646 | 0.250 | 26.993 | 98.6 | ||
900 | 0.8963 | 14.568 | 0.273 | 20.650 | 82.0 | ||
1200 | 0.8953 | 14.490 | 0.301 | 16.946 | 73.9 | ||
2 | 2.4 | 300 | 0.8982 | 14.856 | 0.197 | 55.503 | 162.4 |
600 | 0.8974 | 14.777 | 0.250 | 29.641 | 109.3 | ||
900 | 0.8963 | 14.698 | 0.273 | 21.366 | 85.6 | ||
1200 | 0.8953 | 14.619 | 0.301 | 16.189 | 71.2 | ||
3 | 1.6 | 300 | 0.8352 | 14.121 | 0.207 | 43.088 | 125.9 |
600 | 0.834 | 14.036 | 0.260 | 25.787 | 94.1 | ||
900 | 0.8323 | 13.951 | 0.287 | 19.565 | 78.2 | ||
1200 | 0.8309 | 13.867 | 0.326 | 15.118 | 68.3 | ||
4 | 2.4 | 300 | 0.8352 | 14.137 | 0.207 | 48.021 | 140.5 |
600 | 0.834 | 14.052 | 0.260 | 27.737 | 101.4 | ||
900 | 0.8323 | 13.967 | 0.287 | 20.284 | 81.2 | ||
1200 | 0.8309 | 13.883 | 0.326 | 15.569 | 70.4 | ||
5 | 1.6 | 300 | 0.7614 | 13.811 | 0.219 | 40.447 | 122.4 |
600 | 0.7598 | 13.717 | 0.273 | 24.530 | 92.0 | ||
900 | 0.7576 | 13.624 | 0.304 | 18.364 | 76.1 | ||
1200 | 0.7557 | 13.530 | 0.356 | 14.247 | 68.7 | ||
6 | 2.4 | 300 | 0.7614 | 13.608 | 0.219 | 45.270 | 135.0 |
600 | 0.7598 | 13.515 | 0.273 | 25.906 | 95.7 | ||
900 | 0.7576 | 13.423 | 0.304 | 19.587 | 79.9 | ||
1200 | 0.7557 | 13.331 | 0.356 | 15.434 | 73.3 | ||
7 | 1.6 | 300 | 0.6925 | 13.309 | 0.232 | 36.847 | 113.6 |
600 | 0.6907 | 13.209 | 0.287 | 23.064 | 87.3 | ||
900 | 0.6881 | 13.109 | 0.321 | 17.452 | 73.5 | ||
1200 | 0.6858 | 13.010 | 0.387 | 13.456 | 67.7 | ||
8 | 2.4 | 300 | 0.6925 | 13.221 | 0.232 | 40.183 | 123.0 |
600 | 0.6907 | 13.121 | 0.287 | 24.033 | 90.4 | ||
900 | 0.6881 | 13.022 | 0.321 | 17.998 | 75.3 | ||
1200 | 0.6858 | 12.924 | 0.387 | 14.117 | 70.6 |
Parameter, unit | T-dependence (T in K) | T-range (K) | Refs. |
---|---|---|---|
kWC (W/(m K) | 8052·T-0.645 | 300-1273 | This work |
kCo (W/(m K) | 97.75-0.0836·T+7284/T | 300-695 | [ |
kCo (W/(m K) | 46.21-0.00918·T+12705/T | 695-1388 | [ |
kAg (W/(m K) | 844·T-0.117 | 300-1200 | [ |
RWC/WC (m2 K/W) | (5±1)·10-9 | 300-1073 | This work |
RCo/WC (m2 K/W) | 4·10-8·T-0.53 | 300-695 | This work |
RCo/WC (m2 K/W) | 6.96·10-3·T-2.3 | 695-1273 | This work |
RAg/WC (m2 K/W) | 1.59·10-9·e0.0029·T | 300-1200 | This work |
Table 4 The optimized values of parameters kM, kWC, RM/WC and RWC/WC (M = Co, Ag)
Parameter, unit | T-dependence (T in K) | T-range (K) | Refs. |
---|---|---|---|
kWC (W/(m K) | 8052·T-0.645 | 300-1273 | This work |
kCo (W/(m K) | 97.75-0.0836·T+7284/T | 300-695 | [ |
kCo (W/(m K) | 46.21-0.00918·T+12705/T | 695-1388 | [ |
kAg (W/(m K) | 844·T-0.117 | 300-1200 | [ |
RWC/WC (m2 K/W) | (5±1)·10-9 | 300-1073 | This work |
RCo/WC (m2 K/W) | 4·10-8·T-0.53 | 300-695 | This work |
RCo/WC (m2 K/W) | 6.96·10-3·T-2.3 | 695-1273 | This work |
RAg/WC (m2 K/W) | 1.59·10-9·e0.0029·T | 300-1200 | This work |
Fig. 6. Modelled thermal conductivities versus temperature and grain size of WC-Co samples at different compositions i.e. YG6, YG10, YG15 and YG20. In order to make a comparison in one diagram, the calculated values are added with M = 200, 400, 600 for YG10, YG15 and YG20, respectively.
Fig. 8. Temperature dependence of the yAg ratio for the WC-Ag system calculated by Eq. (12) for 3 different particle sizes of WC (see the 3 sub-figures) and for 3 different phase fractions of WC particles in each figure.
Fig. 9. Temperature dependence of the yCo ratio for the WC-Co system calculated by Eq. (12) for 3 different particle sizes of WC (see the 3 figures) and for 3 different phase fractions of WC particles in each figure.
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