Developing flow simultaneously

Simultaneously Developing Flow. Simultaneously developing flow usually occurs when the fluid exhibits a moderate Prandtl number. In such a flow, the velocity and the temperature profiles develop simultaneously along the flow direction. Therefore, the heat transfer rate strongly depends on the Prandtl number of the fluid and the thermal boundary condition. 

Two-phase flow in parallel micro-channels, feeding from a common manifold shows that different flow patterns occur simultaneously in different microchannels. The probability of appearance of different flow patterns should be taken into account for developing flow pattern maps. 

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. 

Heat transfer to a laminar flow in an annulus is complicated by the fact that both the velocity and thermal profiles are simultaneously developing near the entrance and, often, over the length of the heated channel. Natural convection may also be a factor. It is usually conservative (i.e., predicted heat-transfer coefficients are lower than those experienced) to use equations for the fully developed flow. 

Under very rapid mechanical actions or in observations with characteristic time t < to, the substance behaves as an ideal elastic medium. For t to the developing flow becomes stronger than the elastic deformation, and the substance can be treated as a simple Newtonian fluid. It is only if t is of the same order of magnitude as to that the elastic and viscous effects act simultaneously, and the complex nature of the deformation displays itself. 

The earliest studies related to thermophysieal property variation in tube flow conducted by Deissler [51] and Oskay and Kakac [52], who studied the variation of viscosity with temperature in a tube in macroscale flow. The concept seems to be well-understood for the macroscale heat transfer problem, but how it affects microscale heat transfer is an ongoing research area. Experimental and numerical studies point out to the non-negligible effects of the variation of especially viscosity with temperature. For example, Nusselt numbers may differ up to 30% as a result of thermophysieal property variation in microchannels [53]. Variable property effects have been analyzed with the traditional no-slip/no-temperature jump boundary conditions in microchannels for three-dimensional thermally-developing flow [22] and two-dimensional simultaneously developing flow [23, 26], where the effect of viscous dissipation was neglected. Another study includes the viscous dissipation effect and suggests a correlation for the Nusselt number and the variation of properties [24]. In contrast to the abovementioned studies, the slip velocity boundary condition was considered only recently, where variable viscosity and viscous dissipation effects on pressure drop and the friction factor were analyzed in microchannels [25]. 

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],... 

The term x, denoting thermally and simultaneously developing flows, is expressed as ... 

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

The thermal entrance lengths for simultaneously developing flow with the thermal boundary condition of uniform wall temperature provided by Shah and London [1] are as follows ... 

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. 

The axial diffusions of heat and momentum were considered by Pagliarini [35] for simultaneously developing flow, whereas the viscous dissipation effect has been taken into account by Barletta [36]. 

Heat Transfer With the Convective Boundary Condition. The solution for simultaneously developing flow with the convective boundary condition has been obtained by Javeri [37]. The results are listed in Table 5.6. It should be noted that when Bi = < , Nu T3 is identical to No,, - When Bi = 0, Na,n is the same as Nu H. 

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 ... 

Simultaneously Developing Flow. For the four fundamental thermal boundary conditions, the solutions to simultaneously developing velocity and temperature fields in concentric annuli with r = 0.1,0.25,0.50, and 1.0 and Pr = 0.01,0.7, and 10 have been obtained by Kakaij and Yiicel [104]. Presented in Tables 5.21 to 5.23 are the results for Pr = 0.7. The results for Pr = 0.01 and Pr = 10 have also been tabulated in Kaka and Yiicel [104]. 

Unlike thermally developing flow, the superposition method cannot be applied directly to the simultaneously developing flow because of the dependence of the velocity profile on the axial locations. However, certain influence coefficients are introduced to determine the local Nusselt number for simultaneous developing flow in concentric annuli with thermal boundary conditions that are different from the four fundamental conditions the influence coefficients 0 through 0 2, determined by Kakacj and Yiicel [104] are listed in Tables 5.24 and 5.25. 

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. 

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