Thermal entrance flow

The plot of growth rate in Figure 8a shows that even without buoyancy-driven secondary flows, a considerable variation in the growth rate in the transverse direction exists. The decrease in the axial velocity near the side walls leads to both a shorter thermal entrance length and a greater depletion near the walls compared with the behavior in the middle of the reactor. These perturbations from two-dimensional behavior induced by the side walls extend away from the side walls to a distance about equal to the reactor height. Thus, two-dimensional models may not be sufficient to predict CVD reactor performance even in the absence of buoyancy-driven rolls. 

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 ... 

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. 

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

Hartnett, J. P. Experimental Determination of the Thermal Entrance Length for the Flow of Water and of Oil in Circular Pipes, Trans. ASME, vol. 77, p. 1211, 1955. 

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... 

The average Nusselt number for the thermal entrance region of flow between isothermal parallel plates of length L is expressed as (Edwards 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 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 ... 

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 External Convection. Figure 5.3 presents the results obtained by Hsu [30] for the thermal entrance problem with the convective duct wall boundary condition without consideration of viscous dissipation, fluid axial conduction, flow work, or internal heat sources. As limiting cases of the boundary condition, the curves corresponding to Bi = 0 and Bi = °° are identical to Nu H and Nu T, respectively. Significant viscous dissipation effects have been found by Lin et al. [31] for larger Bi values. 

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 ... 

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

TABLE 5.20 Thermal Entrance Lengths for Thermally Developing Flows in Concentric Annular Ducts (Shah and London [1])... 

Uniform and Equal Heat Flux at Both Walls. Thermally developing flow in a parallel plate duct with uniform and equal heat flux at both walls has been investigated by Cess and Shaffer [132] and Sparrow et al. [133] in terms of a series format for the local and mean Nus-selt numbers. The dimensionless thermal entrance length for this problem has been found by Shah and London [1] to be as follows ... 

Thermally Developing Flow. Wibulswas [160] and Aparecido and Cotta [161] have solved the thermal entrance problem for rectangular ducts with the thermal boundary condition of uniform temperature and uniform heat flux at four walls. However, the effects of viscous dissipation, fluid axial conduction, and thermal energy sources in the fluid are neglected in their analyses. The local and mean Nusselt numbers Nu j, Num T, and Nu hi and Num Hi obtained by Wibulswas [160] are presented in Tables 5.32 and 5.33. 

Thermally Developing Flow. Altemani and Sparrow [176] have conducted experimental measurements of the thermally developing flow of air (Pr = 0.7) in an equilateral triangular duct with the boundary condition on two walls and the third wall insulated. The local Nusselt numbers Nu,Ml and the thermal entrance lengths from their results are given in Figs. 5.32 and 5.33, respectively. 

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]

Hong and Bergles [284] have analyzed the thermal entrance solution of heat transfer for a circular segment duct with 20 = 180° (i.e., a semicircular duct). Two kinds of thermal boundary conditions are used (1) a constant wall heat flux along the axial flow direction with a constant wall temperature along the duct circumference, and (2) a constant wall heat flux along the axial flow direction and a constant wall temperature along the semicircular arc, with zero heat... 

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. 

J. W. Ou, and K. C. Cheng, Viscous Dissipation Effects on Thermal Entrance Heat Transfer in Laminar and Turbulent Pipe Flows with Uniform Wall Temperature, AlAA, paper no. 74-743 or ASME paper no. 74-HT-50,1974. 

T. F. Lin, K. H. Hawks, and W. Leidenfrost, Analysis of Viscous Dissipation Effect on Thermal Entrance Heat Transfer in Laminar Pipe Flows with Convective Boundary Conditions, Wiirme-und Stoffubertragung, (17) 97-105,1983. 

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. 

The thermal entrance lengths for purely viscous nonnewtonian fluids in turbulent pipe flow are on the order of 10 to 15 pipe diameters, the same order of magnitude as for newto-nian fluids [74],... 

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] ... 

In the case of turbulent channel flow of purely viscous power-law fluids, the hydrodynamic and thermal entrance lengths can be taken as the same as the corresponding values for a new-tonian fluid. 

L, thermal entrance length in duct flow m, ft Ni first normal stress difference N/m2, lb,/ft2 N2 second normal stress difference N/m2, lbf/ft2 Nu mean Nusselt number = h lk,... 

S. S. Yoo and J. P. Hartnett, Thermal Entrance Lengths for non-Newtonian Fluid in Turbulent Pipe Flow, Lett. Heat Mass Transfer (2) 189,1975. 

Length Effect. The heat transfer coefficient can vary significantly in the entrance region of the laminar flow. For hydrodynamically developed and thermally developing flow, the local and mean heat transfer coefficients h, and h, for a circular tube or parallel plates are related as [19]... 

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