Simultaneously developing duct

Attention was then turned to developing duct flows. A numerical solution for thermally developing flow in a pipe was first considered. Attention was then turned to plane duct flow when both the velocity and temperature fields are simultaneously developing. An approximate solution based on the use of the boundary layer integral equations was discussed. 

Santos, C.A.C., Medeiros, M.J., Cotta, RM., and Kaka9, S. (1998), Theoretical Analysis of Transient Laminar Forced Convection in Simultaneous Developing Flow in Parallel-Plate Channel, 7 AIAA/ASME Joint Themophysics and Heat Transfer Conference, AIAA Paper 97-2678, Albuquerque, New Mexico, June. Cheroto, S., Mikhailov, M.D., Kaka , S., and Cotta, R.M, (1999), Periodic Laminar Forced Convection -Solution via Symbolic Computation and Integral Trmsforms, Int. J. Thermal Sciences, V.38, no.7, pp.613-621. Kaka, S., Santos, C.A.C., Avelino, MR., and Cotta, R.M. (2001), Computational Solutions and Experimental Analysis of Transient Forced Convection in Ducts, Invited Paper, Int. J. of Transport Phenomena, V.3, pp. 1-17. 

As a result of the development of the hydrodynamic and thermal boundary layers, four types of laminar flows occur in ducts, namely, fully developed, hydrodynamically developing, thermally developing (hydrodynamically developed and thermally developing), and simultaneously developing (hydrodynamically and thermally developing). In this chapter, the term fully developed flow refers to fluid flow in which both the velocity profile and temperature profile are fully developed (i.e., hydrodynamically and thermally developed flow). In such cases, the velocity profile and dimensionless temperature profile are constant along the flow direction. The friction factor and Nusselt number are also constant. 

Simultaneously developing flow is fluid flow in which both the velocity and the temperature profiles are developing. The hydrodynamic and thermal boundary layers are developing in the entrance region of the duct. Both the friction factor and Nusselt number vary in the flow direction. Detailed descriptions of fully developed, hydrodynamically developing, thermally developing, and simultaneously developing flows can be found in Shah and London [1] and Shah and Bhatti [2],... 

FIGURE 5.4 Local Nusselt number NutT for simultaneously developing flow in a circular duct [1]. 

Heat Transfer on the Walls With Uniform Heat Flux. The solutions for simultaneously developing flow in circular ducts with uniform wall heat flux are reviewed by Shah and London [1], Recently, a new integral or boundary layer solution has been obtained by Al-Ali and Selim [33] for the same problem. However, the most accurate results for the local Nusselt numbers [1] are presented in Table 5.6. 

TABLE 5.6 Local Nusselt Number Nu, T3 for Simultaneously Developing Flow in a Circular Duct [1]... 

Simultaneously Developing Flow. The local Nusselt numbers obtained theoretically by Deissler [92] for simultaneously developing velocity and temperature fields in a smooth circular duct subject to uniform wall temperature and the uniform heat flux for Pr = 0.73 are plotted in Fig. 5.12. It can be seen from this figure that the Nusselt numbers for two different thermal boundary conditions are identical for xlDh > 8. 

It is worth noting that the duct entrance configuration affects simultaneously developing flow [98,99]. The local Nusselt number is different for each duct entrance configuration. For practice usage, Bhatti and Shah [45] suggest the following formula for the calculation of the mean Nusselt number. 

For liquid metals (Pr < 0.03), Chen and Chiou [95] have obtained the correlations for simultaneously developing flow in a smooth circular duct with a uniform velocity profile at the inlet. These follow ... 

FIGURE 5.12 Local Nusselt numbers Nu, T and Nu, H for simultaneously developing turbulent flow in a smooth circular duct for Pr = 0.73 [92],... 

TABLE 5.21 Fundamental Solution of the First Kind for Simultaneously Developing Flow in Concentric Annular Ducts for Pr = 0.7 [104]... 

Simultaneously Developing Flow. Little information is available on simultaneously developing turbulent flow in concentric annular ducts. However, the theoretical and experimental studies by Roberts and Barrow [118] indicate that the Nusselt numbers for simultaneously developing flow are not significantly different from those for thermally developing flow. 

Simultaneously Developing Flow. The results for simultaneously developing flow in parallel plate ducts are provided for the following thermal boundary conditions. 

Equal and Uniform Temperatures at Both Walls. For simultaneously developing flow in a parallel plate duct with fluids of 0.1 < Pr < 1000, the following equations are recommended for the computation of the local and mean Nusselt numbers [2,136,137] ... 

When one duct wall is insulated and the other is at a uniform temperature, the local and mean Nusselt numbers for simultaneously developing flow have been obtained for fluids of 0.1 < Pr < 10. These follow [1] ... 

FIGURE 5.23 Local and mean Nusselt numbers for simultaneously developing flow in a flat duct with uniform heat flux at one wall and the other wall insulated [34,140]. 

In this section, the friction factors and Nusselt numbers for fully developed, hydrodynami-cally developing, thermally developing, and simultaneously developing laminar flows in rectangular ducts are presented. 

Simultaneously Developing Flow Table 5.35 presents the results for simultaneously developing flow in rectangular ducts these were obtained by Wibulswas [160] for the and boundary conditions for air (Pr = 0.72). Transverse velocity is neglected in this analysis. However, Chandrupatla and Sastri [163] include transverse velocity in their analysis for a square duct with the boundary condition. 

TABLE 5.35 Local and Mean Nusselt Numbers for Simultaneously Developing Flow in Rectangular Ducts With the and Boundary Conditions [160]... 

Thermally and Simultaneously Developing Flows. Hydrodynamically developing laminar flow in triangular ducts has been solved by different investigators as is reviewed by Shah and London [1]. Wibulswas [160] obtained a numerical solution for the problem of simultaneously... 

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. 

V. Javeri, Simultaneous Development of the Laminar Velocity and Temperature Fields in a Circular Duct for the Temperature Boundary Condition of the Third Kind, Int. J. Heat Mass Transfer, (19) 943-949,1976. 

The previous section was concerned with a flow in which only the temperature field was developing, the velocity field having reached the fully developed state before the heating began. In general, however, both the velocity and temperature fields develop simultaneously [24],[25]. In order to illustrate the nature of such flows, developing two-dimensional flow in a plane duct will be considered here. The flow situation considered is shown in Fig. 7.11. 

Stage ni The septal stage displays periportal and bridging necrosis as well as an increasing loss of bile ducts with a simultaneous decline in portal inflammatory infiltrations. Dense concentric fibrosis develops around... 

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