TAILIEUCHUNG - Mechanical Engineers Handbook Episode 10

Tham khảo tài liệu 'mechanical engineers handbook episode 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 528 Principles of Heat Transfer where u0 is the maximum velocity in the axis . The problem then reduces to finding u0. It is obvious that u y z thus obtained does not identically match the true solution to the problem. The maximum value of the velocity results by imposing that verifies integrated averaged over the duct cross section namely abdP m dx 2 I - 92u d2u t t . -1V d dzdy 2 2 x Consequently ab - 6 muo b v dx 3 a b and using the definition of the average 2 Ud f 1 ab udz dy __a 2 yields u0 4 U. Finally this result produces the following expression for the friction factor f a 21. where Dh ZaL a b Re Dh a b Heat Transfer HEAT TRANSFER IN THE FULLY DEVELOPED REGION The key problem of heat transfer in duct flows is to determine the relationship between the wall-to-stream temperature drop and its associated heat transfer rate. For flow in a circular duct of radius r0 and with average longitudinal velocity U the mass flow rate is m p-pr U Fig. 10 . The heat transfer rate from the wall to the stream should equal the change in the enthalpy of the stream. To verify this we write the first principle for a control volume of length dx q 2pr0dx m hx dx - hx . In the perfect gas limit of the fluid dh cPdTm or in the limit of an incompressible flow and under a negligible pressure variation dh cPdTm this balance equation implies dTm 2 q m -hĩ dx r0 pcU where for incompressible flows cP was replaced by c. The temperature Tm that appears in the energy balance written for the control volume is the average temperature of the stream and it indicates that the actual tempera- 3. Convection Heat Transfer 529 Figure Heat transfer in the fully developed region for flow in a cylindrical duct. ture field in the stream is not uniform. Its definition results from the first principle applied to a stream tube that is q 2nr0dx d JApucPTdA as T - pcpũA. pucPTdA. A For ducts with uniform-temperature walls this means Tm

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