Thermally developing flows

Fig. 4.5d-f Rectangular micro-channels, (d) dh = 750 jin. Test section used by Warrier et al. (2002) (schematic view) 1 upper aluminum plate, 2 down aluminum plate, 3 micro-channel, 4 heater, (e) rfh = 200—2,000 pm. Test section used by Gao et al. (2002) (schematic view) 1 brass block, 2 micro-channel, 3 heater, (f) Thermally developing flow in rectangular micro-channel (Lee et al. 2005) (schematic view) 1 cover plate, 2 micro-channel, 3 copper block, 4 heater. Reprinted from Peng and Peterson (1996), Harms et al. (1999), Warrier et al. (2002), Qu and Mudawar (2002a), Gao et al. (2002), and Lee et al. (2005) with permission... 

Nguyen TV (1992) Laminar heat transfer for thermal developing flow in ducts. Int J Heat Mass Transfer 35 1733-1741... 

As far as heat transfer is considered, Fenner [27] made a detailed comparison of the thermally fully developed flow and thermally developing flow. He indicated that the thermally developed flow will not be achieved when heat conduction effects become significant [34]. Bruker et al. [35] experimentally verified that the thermally developing flow analysis provided a more accurate description of the flow in the extruder. 

Attention will in this section be given to thermally developing flow in a pipe. In this case, the velocity profile, which is not changing with z, is given, as discussed in Section 4.2, by ... 

FIGURE 4.15 Nusselt number variation in thermally developing flow in a pipe. 

Next, consider thermally developing flow in the case where the wall heat flux, qw, rather than the wall temperature is uniform. In this case, the following dimensionless temperature is used ... 

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. 

Consider thermally developing flow in a smooth 60-mm diameter pipe. Air, at an initial temperature of 10°C, flows through this pipe, the mean air velocity being 30m/s. The first portion of the pipe is unheated and the velocity profile becomes fully developed in this portion of the pipe. The second portion of the pipe, which has a length of 2 m, is heated to a uniform temperature of 50°C. Determine how the wall heat transfer rate in W/m2 varies with distance along the heated portion of the pipe. 

For electro-osmotic flow only the limiting Nusselt numbers for thermally fully-developed flow in parallel plate channel and circular tube are obtained as a special case from the solution for thermally developing flow. 

Fig. 3.34 Velocity and temperature profiles of a hydrodynamically and thermally developing flow...

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. 

FIGURE 5.1 Local and mean Nusselt numbers Nu, T and Num T for thermal developing flow in a circular duct. 

The local Nusselt number and mean Nusselt number computed from Eqs. 5.36 and 5.37 are shown in Fig. 5.1. The data corresponding to this figure can be found in Shah and London [1], The thermal entrance length for thermally developing flow in circular ducts can be obtained using the following expression ... 

Heat Transfer on Walls With Uniform Heat Flux. The temperature profile and the local and mean Nusselt numbers for thermally developing flow in a circular duct with uniform wall heat flux are provided by Siegel et al. [25] as follows ... 

The thermal entrance length for thermally developing flow under the uniform wall heat flux boundary condition is equal to the following ... 

Heat Transfer on Walls With Radiation. The local Nusselt numbers normalized with respect to Nu H have been obtained by Kadaner et al. [8] for thermally developing flow with the radioactive duct wall boundary condition . This is expressed as ... 

Thermally Developing Flow. Numerous investigators [80, 89-94] have carried out the investigation of turbulent thermally developing flow in a smooth circular duct with uniform wall temperature and uniform wall heat flux boundary conditions. It has been found that the dimensionless temperature and the Nusselt number for thermally developing turbulent flow have the same formats as those for laminar thermally developing flow (i.e., Eqs. 5.34-5.37 and Eqs. 5.50-5.53). The only differences are the eigenvalues and constants in the equations. 

The mean Nusselt number Num for thermally developing flow with uniform wall temperature or uniform wall heat flux conditions can be calculated using Al-Arabi s [95] correlation ... 

Thermally Developing Flow. The solutions for thermally developing flow in concentric annular ducts under each of the four fundamental thermal boundary conditions are tabulated in Tables 5.16, 5.17,5.18, and 5.19. These results have been taken from Shah and London [1]. Additional quantities can be determined from the correlations listed at the bottom of each table using the data presented. 

TABLE 5.16 Fundamental Solutions of the First Kind for Thermally Developing Flow in Concentric Annular Ducts (compiled from Shah and London [1])... 

The thermal entrance lengths for thermally developing flow with these four fundamental thermal boundary conditions are given in Table 5.20. 

Using the four fundamental solutions presented in Tables 5.16-5.19, thermally developing flow with thermal boundary conditions different from the fundamental conditions presented in the section entitled Four Fundamental Thermal Boundary Conditions can be obtained by the superposition method. Three examples are detailed in the following sections. 

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. 

Thermally Developing Flow. Kays and Leung [111] present experimental results for thermally developing turbulent flow in four concentric annular ducts, r = 0.192,0.255,0.376, and 0.500, with the boundary condition of one wall at uniform heat flux and the other insulated, that is, the fundamental solution of the second kind. In accordance with this solution, the local Nusselt numbers Nu and Nu at the outer and inner walls are expressed as... 

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. 

Few results that can be used in practice are available for thermally developing flow and simultaneously developing flow in eccentric annuli. According to the discussion in Bhatti and Shah [45], the Nusselt numbers may be estimated from the corresponding results for concentric annuli (e = 0). 

Thermally Developing Flow,. The results for thermally developing flow in parallel plate ducts are presented for the following practical thermal boundary conditions of interest. 

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