Thermal entrance region

Fully developed velocity profile with developing temperature profile (i.e. the thermal entrance region) or... 

The thermal entrance region in a hydrodynamically fully developed flow in a rectangular duct may be studied by the use of the integral method. In this section, the uniform wall temperature and the uniform wall heat flux cases are discussed. The physical model is based on the following assumptions ... 

In order to illustrate the type of solution obtained in the thermal entrance region, attention will be given first to the case where the wall temperature of the pipe is kept constant in this thermal entrance region, i.e., to the case where ... 

Discuss how the computer program for flow in the thermal entrance region to a plane duct must be modified in order to apply to the case where slug flow exists, i.e., for the situation in which the velocity can be assumed to be constant across the pipe and equal, of course, to the mean velocity in the pipe. 

Discuss the modifications required to the computer program given for flow in the thermal entrance region of a plane duct to deal with the case where there is a uniform heat flux at one wall of the duct and where there is a uniform specified temperature at the other wall of the duct. 

Variation of Nusselt number with Z in thermal entrance region for Pr = 0.7 for various values of ReD. 

Sparrow, E.M., Hallman, T.M., and Siegel, R., Turbulent Heat Transfer in the Thermal Entrance Region of a Pipe with Uniform Heat Rux , Appl. Sci. Rest. Section A, Vol. 7, p. 37. 1957. 

Wang, M., Tsuji, T., and Nagano, Y., Mixed Convection with Row Reversal in the Thermal Entrance Region of Horizontal and Vertical Pipes , Int. J. Heat Mass Transfer, Vol. 37, pp. 2305-2319,1994. 

Chou, Fu Chu, Laminar Mixed Convection in the Thermal Entrance Region of Horizontal Rectangular Channels with Uniform Heat Input Axially and Uniform Wall Temperature Circumferentially , Canadian J. Chem. Eng., Vol. 68. pp. 577-584.1990. 

Fig. 5-5 Local and average Nusselt numbers tor circular tube thermal entrance regions in fully developed laminar flow.

Th regioii of flow over which the thermal boundary layer develops and re.iches (he tube center i.s called the thermal entrance region, and the length of this region is called the thermal entry length L,. Flow in the thermal... 

For a circular tube of length L subjected to constant surface temperature, the average Nussell number for the thermal entrance region can be determined from (Ed vards et al., 1979)... 

Joule heat generated by the ionic current is removed through cooling jackets mounted on the transverse walls. This creates a transverse temperature gradient which drives a stable and usually deleterious convective flow (3). Furthermore, the axial temperature gradient in the thermal entrance region (10,11) near the tips of the electrodes may drive a buoyancy instability when the thermal gradient exceeds a critical value (4,12). Apart from the physical properties of the carrier fluid, the critical temperature... 

The mean Nusselt number based on hm in the thermal entrance region is defined as... 

FIGURE 5.21 Local and mean Nusselt numbers in the thermal entrance region of a parallel plate duct with the and (8) boundary conditions [1]. 

TABLE 5.32 Local and Mean Nusselt Numbers in the Thermal Entrance Region of Rectangular Ducts With the Boundary Condition [160]... 

For equilateral triangular ducts having rounded corners with a ratio of the corner radius of curvature to the hydraulic diameter of 0.15, Campbell and Perkins [180] have measured the local friction factor and heat transfer coefficients with the boundary condition on all three walls over the range 6000 < Re < 4 x 104. The results are reported in terms of the hydrodynamically developed flow friction factor in the thermal entrance region with the local wall (Tw) to fluid bulk mean (Tm) temperature ratio in the range 1.1 < TJTm < 2.11, 6000 < Re < 4 x 10 and 7.45 [Pg.382]

Nu, a. mean Nusselt number for the thermal entrance region for the specified thermal boundary condition... 

J. W. Ou, and K. C. Cheng, Viscous Dissipation Effects on Thermal Entrance Region Heat Transfer in Pipes with Uniform Wall Heat Flux, Appl. Sci. Res., (28) 289-301,1973. 

A. Roberts and H. Barrow, Turbulent Heat Transfer in the Thermal Entrance Region of an Internally Heated Annulus, Proc. Inst. Mech. Engrs., (182/3H) 268-276,1967. 

The Nusselt number for power-law fluids for constant wall heat flux reduces to the newto-nian value of 4.36 when n = 1 and to 8.0 when n = 0. Equation 10.47 is applicable to the laminar flow of nonnewtonian fluids, both purely viscous and viscoelastic, for the constant wall heat flux boundary condition for values of xId beyond the thermal entrance region. The laminar heat transfer results for the constant wall temperature boundary condition were also obtained by the separation of variables using the fully developed velocity profile. The values of the Nusselt number for n = 1.0, Vi, and A calculated by Lyche and Bird [40] are 3.66, 3.95, and 4.18, respectively, while the value for n = 0 is 5.80. These values are equally valid for purely viscous and viscoelastic fluids for the constant wall temperature case provided that the thermal conditions are fully established. 

Laminar Heat Transfer in the Thermal Entrance Region... 

The prediction of the local laminar heat transfer coefficient for a power-law fluid in the thermal entrance region of a circular tube was reported by Bird and colleagues [41]. Both the constant wall heat flux and the constant wall temperature boundary condition have been studied. The results can be expressed by the following relationships [42-48]. 

It is interesting to note that the nonnewtonian effect has been taken into account by simply multiplying the corresponding newtonian result by [(3n + l)/4n]1/3. Equations 10.48 and 10.49 may be used to predict the local heat transfer coefficient of purely viscous and viscoelastic fluids in the thermal entrance region of a circular tube. Figure 10.6 shows a typical comparison of the measured local heat transfer coefficient of a viscoelastic fluid with the prediction for a power-law fluid. The good agreement provides evidence to support the applicability of Eq. 10.48 in the case of the constant heat flux boundary condition. 

Values of the asymptotic heat transfer factors jH in the thermal entrance region are reported for concentrated aqueous solutions of polyacrylamide and polyethylene oxide. The results are shown in Fig. 10.30, as a function of the Reynolds number Re . These values were measured in tubes of 0.98,1.30, and 2.25 cm (0.386,0.512, and 0.886 in) inside diameter in a recirculating-flow loop. The asymptotic turbulent heat transfer data in the thermal entrance region are seen to be a function of the Reynolds number Re and of the axial position xld. The following empirical correlation is derived from the data [35,37] ... 

FIGURE 10 JO Experimental results of turbulent heat transfer for concentrated solutions of polyethylene oxide and polyacrylamide in the thermal entrance region. 

Proportional to the ratio of thermal energy convected to the fluid to thermal energy conducted axially within the fluid the inverse of Pe indicates relative importance of fluid axial heat conduction Useful in describing the thermal entrance region heat transfer results... 

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