Four Fundamental Thermal Boundary Conditions

As shown in Fig. 5.13, there are two walls in concentric annular ducts. Either or both of them can be involved in heat transfer to a flowing fluid in the annulus. Four fundamental thermal boundary conditions, which follow, can be used to define any other desired boundary condition. Correspondingly, the solutions for these four fundamental boundary conditions can be adopted to obtain the solutions for other boundary conditions using superposition techniques. 

The four fundamental thermal boundary conditions are as follows ... 

Heat Transfer. Fundamental solutions for boundary conditions of the first, second, and third kinds for fully developed flow in concentric annular ducts are given in Table 5.14. The nomenclature used in describing the corresponding solutions can best be explained with reference to the specific heat transfer parameters G) and 0 which are the dimensionless duct wall and fluid bulk mean temperature, respectively. The superscript k denotes the type of the fundamental solution according to the four types of boundary conditions described in the section entitled Four Fundamental Thermal Boundary Conditions. Thus, k = 1,2, 3, or 4. The subscript l in Gj 1 refers to the particular wall at which the temperature is evaluated / = i or o when the temperature is evaluated at the inner or the outer wall. The subscript j in G) 1 refers... 

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. 

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. 

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]. 

Similar to the four fundamental thermal boundary conditions for concentric annuli, the four kinds of fundamental conditions for parallel plate ducts are shown in Fig. 5.20. The fully developed Nusselt numbers for the four boundary conditions follow [1] ... 

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. 

Cheng and Hwang [108] analyzed the heat transfer problem in eccentric annular ducts. The Nusselt numbers for fully developed flow in eccentric annular ducts with the and thermal boundary conditions are given in Table 5.26. For eccentric annular ducts with boundary conditions different from the four described in the section entitled Four Fundamental... 

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... 

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