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Tham khảo tài liệu 'heat conduction basic research part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Energy Transfer in Pyroelectric Material 239 J ttủids j pbiuidv J c jkiUk iUi j - ekljEkui j - Yijduij dv j pủtủtdV 18 which can be of the form Ị tịủids ỊpbiUidv Ị we ekijEkUi j Yij 9u jdv Ị Kdv 19 where 1 a We cijkluk lui j 2 at cijkluk lui j which represents the rate of mechanical potential energy density. _ . _ 1 a K piiiiu i 2 at piu- iii i which is the rate of kinetic energy density. II. Multiplying ty by the time derivative of Equation 3 integrating the resulting expression over volume n and using the identity equation Dkty k Dk kty Dk ty k and Gaussian Theorem we have J tyDknkds J EkDkdV 0 where nk is the unit outward normal of dS. Substitution the constitutive equation Equation 7 2 into the above equation yields Ị tyìdknkdS Ị Ek ekijè ij ẢikẺ i ĩk é dv 0 which is of the form f tyDknkds Ị ekij ijEk WE ỉkéE dv 0 20 an Q where the rate of electric energy density is defined as 1a WE 2 at ựikEiEk The addition of Equation 19 and Equation 20 yields J t íi ds j pb u dv j 7 jeu jdv J tyD knkdS J Ek ĩk édv J We K dv an n n an n n 21 III. Taking the time differential on Equation 7 3 and using Equation 5 we get T0 Y jỀ ij T0ĨiE i pCỒ qt t 240 Heat Conduction - Basic Research Applying the operator L on both sides of this equation and using Equation 6 yields Kij 9 ij - L T0 Yij Ề. ij T0 Ị iE i pC9 0 22 Multiplying Equation 22 by 9 and apply volume integral on this expression we obtain J K j 0fij 9dV -J T0 YjL ẻ 9dV -J T0 Ị L E 9dV - Ị pCL è 9dV 0 23 n n n n Using the identity 99 i j 9 j0 i 99 ij and Gaussian Theorem then we have J Kij9 ij9dV J Kij 99 i j 9 j9 i dV J Kijnj99pdS J Kij9rl9 dV Inserting this relation into Equation 23 and expanding the result by using the entropy equation Equation 7 3 we get ly Kij9 i9njdS - T I Kij9 j9 idV Ị Yijảij9dV Ị Ị Ei9dV Ị TC99dV TI TdV 0 an 0 n n n n n 24 Thus the rate of thermal energy density w9 can be expressed as w9 PC 9 9 1 C P-C 92 9 T0 2 di T0 J Combining Equation 21 and Equation 24 by eliminating J YijỀij 9dV finally we .
