Flow, adiabatic thermally fully developed

This says that the temperature increase at the wall in a thermal fully developed flow changes with the length x in the same way as the difference between the wall and the adiabatic mixing temperature. The temperature profile that satisfies this condition is of the general form... 

For X+ oo, the known end value of 3.6568 for the Nusselt number for thermal fully developed flow is obtained. The calculation of the Nusselt number for small values of the length X+ = L/(dPe) requires many terms of the series for the adiabatic mixing temperature (3.249). Therefore the exact solution has been approximated by empirical equations. According to Stephan [3.30] through... 

The development of the thermal and dynamic boundary layers can be either coupled or uncoupled. When the thermal and dynamic boundary layers develop together, the flow is said to be a simultaneously developing flow. When only the dynamic boundary layer develops, with the fluid in thermal equilibrium with the walls, the flow is said to be a hydrodynamically developing flow. When the flow is hydrodynamically fully developed but the thermal boundary layer develops, the flow is said to be a hydrodynamically fully developed and thermally developing flow, this situation can be obtained in a microchannel with adiabatic walls in the region near the entrance (adiabatic preparation) and where heating at the walls starts beyond the hydrodynamic entrance region. 

The micro-channels utilized in engineering systems are frequently connected with inlet and outlet manifolds. In this case the thermal boundary condition at the inlet and outlet of the tube is not adiabatic. Heat transfer in a micro-tube under these conditions was studied by Hetsroni et al. (2004). They measured heat transfer to water flowing in a pipe of inner diameter 1.07 mm, outer diameter 1.5 mm, and 0.600 m in length, as shown in Fig. 4.2b. The pipe was divided into two sections. The development section of Lj = 0.245 m was used to obtain fully developed flow and thermal fields. The test section proper, of heating length Lh = 0.335 m, was used for collecting the experimental data. 

Simultaneously developing flow in annular sector ducts for air (Pr = 0.7) has been analyzed by Renzoni and Prakash [287]. In their analysis, the outer curved wall is treated as adiabatic, and the boundary condition is imposed on the inner curved wall as well as on the two straight walls of the sector. The fully developed friction factors, incremental pressure drop numbers, hydrodynamic entrance lengths, and thermal entrance lengths are presented in Table 5.62. The term L y used in Table 5.62 is defined as the dimensionless axial distance at which /app Re = 1.05/ Re. The fully developed Nusselt numbers are represented by Nu/< in order not to confuse the reader since the thermal boundary condition applied in Renzoni and Prakash [287] is different from those defined in the section. 

The velocity profile is fully developed in the thermal entrance region and it remains fixed while the temperature profile develops. In this case, the flow is designated as hydrodynami-cally fully developed and thermally developing flow this situation can be obtained in a microchannel with adiabatic walls in the region near the entrance adiabatic preparation) (see Fig. 3). 

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