Laminar duct flow thermally-developing

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

Aparecido, J., and Cotta, R. (1990) Thermally Developing Laminar Flow inside Rectangular Ducts, Int. J. Heat andMass Transfer. Vol. 33, pp. 341-347. 

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. 

For a circular duct with a diameter of 2a, the characteristics of laminar flow and heat transfer for four kinds of flows, namely, fully developed, hydrodynamically developing, thermally developing, and simultaneously developing, are outlined in the following sections. 

Heat Transfer on Walls With Uniform Temperature. For this boundary condition, denoted as , temperature distribution in a circular duct for fully developed laminar flow in the absence of flow work, thermal energy sources, and fluid axial conduction has been solved by Bhatti [3] and presented by Shah and Bhatti [2], as follows ... 

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. 

In this section, the friction factors and Nusselt numbers for fully developed, hydrodynami-cally developing, thermally developing, and simultaneously developing laminar flows in rectangular ducts are presented. 

Thermally and Simultaneously Developing Flows. Hydrodynamically developing laminar flow in triangular ducts has been solved by different investigators as is reviewed by Shah and London [1]. Wibulswas [160] obtained a numerical solution for the problem of simultaneously... 

In subsequent research [235], thermally developing flow in curved elliptic ducts is analyzed for different a and Prandtl numbers. The local Nusselt numbers along the flow direction are shown in graph form, and the asymptotic values of the Nusselt numbers have been obtained, as is shown in Table 5.46. In a related study, the effects of buoyancy on laminar flow in curved elliptic ducts are discussed by Dong and Ebadian [236]. 

Dong and Ebadian [293]. An analysis of thermally developing laminar flow in cusped ducts can be found in Dong et al. [292]. 

T. V. Nguyen, Laminar Heat Transfer for Thermally Developing Flow in Ducts, Int. J. Heat Mass Transfer, (35) 1733-1741,1992. 

The Rectangular Duct Thermal-Entry Length, with Hydrodynamically Fully Developed Laminar Flow... 

Santos, C.A.C., Medeiros, M.J., Cotta, RM., and Kaka9, S. (1998), Theoretical Analysis of Transient Laminar Forced Convection in Simultaneous Developing Flow in Parallel-Plate Channel, 7 AIAA/ASME Joint Themophysics and Heat Transfer Conference, AIAA Paper 97-2678, Albuquerque, New Mexico, June. Cheroto, S., Mikhailov, M.D., Kaka , S., and Cotta, R.M, (1999), Periodic Laminar Forced Convection -Solution via Symbolic Computation and Integral Trmsforms, Int. J. Thermal Sciences, V.38, no.7, pp.613-621. Kaka, S., Santos, C.A.C., Avelino, MR., and Cotta, R.M. (2001), Computational Solutions and Experimental Analysis of Transient Forced Convection in Ducts, Invited Paper, Int. J. of Transport Phenomena, V.3, pp. 1-17. 

Uniform Temperature at One Wall and Uniform Heat Flux at the Other. When the two walls of a parallel plate duct are subject to a thermal boundary condition such as uniform temperature at one wall and uniform heat flux at the other, the Nusselt numbers for fully developed laminar flow for qZ = 0 and q"w 0 are determined to be ... 

TABLE 5.41 Local and Mean Nusselt Numbers for Thermally and Simultaneously Developing Laminar Flows and Equilateral Triangular Ducts [160]... 

An elliptical duct with four internal longitudinal fins mounted on the major and minor axes, as shown in Fig. 5.48, has been analyzed by Dong and Ebadian [275] for fully developed laminar flow and heat transfer. In this analysis, the fins are considered to have zero thickness. The thermal boundary condition is applied to the duct wall, and / is defined as a ratio of Ha a = Hbib. The friction factors and Nusselt numbers for fully developed laminar flow are given in Table 5.52. 

A cardioid duct is shown in Fig. 5.63. Fully developed laminar flow and heat transfer under the boundary condition have been analyzed by Tyagi [294]. The/Re and NuHj values derived from this analysis are 5.675 and 4.208, respectively. The Nusselt number for the thermal boundary condition was found to be 4.097 [1]. 

Turbulent Flow. The thermal entry length solutions for smooth ducts for several cross-sectional geometries have been summarized [46]. As for laminar flow, the Nusselt numbers in the thermal region are higher than those in the fully developed region. However, unlike laminar flow, Nu,x and NuxHi are very nearly the same for turbulent flow. The local and mean Nusselt numbers for a circular tube with and boundary conditions are [46] ... 

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