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Tham khảo tài liệu 'nonlinear continua part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 4.5 Energy balance First Law of Thermodynamics 111 tp u tdy Í ầ dV - J v Di J v L.ip V q dV tr Mr . 4.49e The above equation is the integral form of the Eulerian formulation for the energy conservation principle. Since the above equation has to be fulfilled for any control volume that we define inside the continuum we obtain the localized form of the energy conservation principle in the Eulerian formulation t pp u t Í T i t i 1 M V a r . q 4.50 Example 4.15.J JJJJ Following from the topic discussed in Examples 4.5 and 4.8 we consider a fluid moving with a velocity field v and a moving control volume with a velocity field w. In this example we are going to analyze the energy balance inside the moving control volume. Using the generalized Reynolds transport theorem Eq. 4.11 for the fluid internal energy we get tp u w d f u w p u n M dS Di L t p L t dt X t p nw From the generalized Gauss theorem Eq. 4.5 ft t t t JT r I t ft t t i o y p u v dl n p u v d5 . JQ i Jơ t Subtracting the above equation from the previous one L lp u fdV L Jn t d fp u dt L J ơ t p u n V p u v w v d5 . fdV Using the result in Example 4.2 and Eq. 4.19 continuity equation nrw tp u dV lp Cu v V tu dV Dt L t p L t p I dt v - J p u n w v d5 . J ơ t On the r.h.s. of the above equation the term between brackets in the first 112 Nonlinear continua integral is the material derivative of the fluid internal energy. Hence using Eq. 4.50 L l d 2 tp s MV Ị Dt JtQ Ja t - V 4q tp tr 4dV tp 4u n 4v t w tdS . _JJJJJ 4.5.2 Lagrangian material formulation of the energy balance Let us call 4U the internal energy per unit mass in the t-configuration but referred to the reference configuration. Obviously 4U X1 t 4u txi t . 4.51a From Eqs. 3.10 and 3.20c y M 4dV and using Eq. 2.76a we define Q tQi A SA S J v U M. a V SA 4.51b 4.51c M dV For the heat sources per unit mass we can also define in the reference configuration t-fí far1 fa r tri A 4 51d n I X t I I I X I . l .5Kl .