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Computational Fluid Mechanics and Heat Transfer Third Edition_9
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Tham khảo tài liệu 'computational fluid mechanics and heat transfer third edition_9', kỹ thuật - công nghệ, điện - điện tử phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 6.5 Heat transfer coefficient for laminar incompressible flow over a flat surface 301 or d dx St u T - T dy qw 0 pcp 6.47 Equation 6.47 expresses the conservation of thermal energy in integrated form. It shows that the rate thermal energy is carried away by the b.l. flow is matched by the rate heat is transferred in at the wall. Predicting the temperature distribution in the laminar thermal boundary layer We can continue to paraphrase the development of the velocity profile in the laminar b.l. from the preceding section. We previously guessed the velocity profile in such a way as to make it match what we know to be true. We also know certain things to be true of the temperature profile. The temperatures at the wall and at the outer edge of the b.l. are known. Furthermore the temperature distribution should be smooth as it blends into T for y 5t. This condition is imposed by setting dT dy equal to zero at y 5t .A fourth condition is obtained by writing eqn. 6.40 at the wall where u v 0. This gives d2T dy2 y 0 0. These four conditions take the following dimensionless form T T 1 .ỉ X n ự 1 at y St 0 J-w T T T . ự 0 at y St 1 Tw T d T T Tw T _ n -------d .xW-------- 0 at y St 1 d y St d 2 T T Tw T _n ------- w------------ 0 at y St 0 d y St 2-------------J 6.48 Equations 6.48 provide enough information to approximate the temperature profile with a cubic function. T- T Tw T - z X 2 z X 3 a by 4y 4y Ot Ot St 6.49 Substituting eqn. 6.49 into eqns. 6.48 we get a 1 1 b c d 0 b 2c 3d 0 2c 302 Laminar and turbulent boundary layers 6.5 which gives a 1 b -3 c 0 d 2 so the temperature profile is T - I. 3 y 1 f y 3 Tw - T. 2 St 2 St 6.50 Predicting the heat flux in the laminar boundary layer Equation 6.47 contains an as-yet-unknown quantity the thermal b.l. thickness St. To calculate St we substitute the temperature profile eqn. 6.50 and the velocity profile eqn. 6.29 in the integral form of the energy equation 6.47 which we first express as u. Tw T. dx u 7 T - T. d i y 0 u. Tw