Entrance length, thermal

As a rough guide for plate heat exchangers, the rate of the hydrodynamic entrance length to the corresponding thermal entrance length is given by... 

Reference slip length Thermal entrance length... 

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. 

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. 

Corresponding to the hydrodynamic entrance length, the thermal entrance length L,h is defined as the axial distance needed to achieve a value of the local Nusselt number Nu which is 1.05 times the fully developed Nusselt number value. The dimensionless thermal entrance length is expressed as L+,h = L,hl(Dh Pe). 

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 ... 

It has been confirmed that the effect of fluid axial conduction can be neglected for Pe > 50 [24]. However, the thermal entrance length L T varies with the Peclet number. Nguyen [23]... 

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

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 ... 

However, the thermal entrance lengths for uniform wall temperature and uniform wall heat flux are much different. These are shown in Figs. 5.10 and 5.11, respectively. 

FIGURE 5.10 Thermal entrance lengths for the turbulent Graetz problem for a smooth circular duct [80],... 

The thermal entrance length for liquid metals has been found by Genin et al. [97] ... 

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 Temperature at One Wall and Uniform Heat Flux at the Other. In this case, the subscripts 1 and 2 refer to either the inside or the outside wall. The thermal entrance length solution to this problem is expressible in terms of the following equations [1] ... 

From these results, the dimensionless thermal entrance length can be determined as follows ... 

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. 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. 

FIGURE 533 Thermal entrance lengths for thermally developing and hydrodynamically developing turbulent airflow (Pr = 0.7) in a smooth equilateral triangular duct [176]. 

The fully developed Nusselt numbers for the thermal boundary conditions of uniform wall temperature and axial uniform wall heat flux with circumferential uniform temperature obtained by Prakash and Liu [266] are given in Table 5.49, along with the corresponding thermal entrance lengths. The term n in Table 5.49 denotes the number of fins, whereas / represents the relative length of the fins. 

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. 

L dimensionless hydrodynamic entrance length=LkyIDh Re Ln thermal entrance length, m... 

The dimensionless thermal entrance length L,/d can be estimated using the following expression ... 

FIGURE 10.15 Thermal entrance lengths for the T and H boundary conditions for power-law fluids. 

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],... 

FIGURE 10.27 Thermal entrance length for drag-reducing viscoelastic fluids. Dimensionless wall temperature versus dimensionless axial distance [93]. 

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. 

Hasson et al. [179] studied laminar film condensation on falling jets and sheets using a Graetz-type thermal entrance length solution. For a sheet of thickness 25, the local Nusselt number is... 

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