Shah and London

The investigation shows agreement between the standard laminar incompressible flow predictions and the measured results for water. Based on these observations the predictions based on the analytical results of Shah and London (1978) can be used to predict the pressure drop for water in channels with as small as 24.9 pm. This investigation shows also that it is insufficient to assume that the friction factor for laminar compressible flow can be determined by means of the well-known analytical predictions for its incompressible counterpart. In fact, the experimental and numerical results both show that the friction factor increases for compressible flows as Re is increased for a given channel with air. 

In general, the axial heat conduction in the channel wall, for conventional size channels, can be neglected because the wall is usually very thin compared to the diameter. Shah and London (1978) found that the Nusselt number for developed laminar flow in a circular tube fell between 4.36 and 3.66, corresponding to values for constant heat flux and constant temperature boundary conditions, respectively. 

The data on pressure drop in irregular channels are presented by Shah and London (1978) and White (1994). Analytical solutions for the drag in micro-channels with a wide variety of shapes of the duct cross-section were obtained by Ma and Peterson (1997). Numerical values of the Poiseuille number for irregular microchannels are tabulated by Sharp et al. (2001). It is possible to formulate the general features of Poiseuille flow as follows ... 

The values of the Nusselt and the Poiseuille numbers for heat transfer and friction for fully developed laminar flows through specifled channels are presented in Table 7.1 (Shah and London 1978). 

In their studies of frichon factors in channels with a number of different cross-sechonal geometries, Shah [103] and Shah and London [102] also computed heat transfer properties. A few characterishc cross-sechons for which Nusselt numbers were obtained are displayed on the left side of Figure 2.17. Their results include both the Nusselt numbers for fixed temperature and fixed heat flux wall boundary condihons and are given as tabulated values for different geometric parameters. 

Shah and London [40] have compiled the heat-transfer and fluid-friction information for fully developed laminar flow in ducts with a variety of flow cross sections as shown in Table 6-1. In this table the following nomenclature applies ... 

For parallel plates, Shah and London [14] propose the following law for the PoiseuiUe number which takes the entrance length into account ... 

These heat-transfer mechanisms have been modeled both analytically and numerically for many flow geometries. Most numerical methods require a computer to obtain an analytical answer, and because of the simphflcations and assumptions required to obtain a solution, they are usually more accurate than the empirical methods discussed in this chapter. Good introductions to these methods are provided by Hewitt [1], Patankar [10], and Shah and London [11]. 

Prediction of the heat-transfer coefficient for this problem is complicated by the fact that the local coefficient is very high at the start of heat transfer and decreases as the conduction into or out of the fluid builds up an adverse temperature gradient. A number of analytical and numerical solutions for this and other constant cross-sectional geometries are given by Shah and London [11], but most... 

The single-phase heat transfer coefficients in Eq. (17) were determined by asymptotic interpolation of standard correlations, i.e. Shah and London (SL) for Re < 2,300 and Gnielinski (G) for Re > 2,300. Thus,... 

Simultaneously developing flow is fluid flow in which both the velocity and the temperature profiles are developing. The hydrodynamic and thermal boundary layers are developing in the entrance region of the duct. Both the friction factor and Nusselt number vary in the flow direction. Detailed descriptions of fully developed, hydrodynamically developing, thermally developing, and simultaneously developing flows can be found in Shah and London [1] and Shah and Bhatti [2],... 

Shah and London [1] have obtained the Nusselt number for -51.36 maximum error of 3 percent ... 

Solutions for Very Large Reynolds Number Flows. For very large Reynolds number flow, boundary layer theory simplifications are involved in the solutions. The numerical solution found by Hornbeck [10] is the most accurate of the various solutions reviewed by Shah and London [1]. The dimensionless axial velocity and pressure drop obtained by Hornbeck [10] are presented in Table 5.2. 

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 effects of fluid axial conduction on the Graetz solution have been reviewed extensively by Shah and London [1]. Furthermore, Laohakul et al. [21], Ebadian and Zhang [22],... 

Other extended Graetz problems in which the effect of viscous dissipation, inlet velocity, and temperature profiles are considered are reviewed in detail by Shah and London [1]. 

The effects of viscous dissipation on the thermal entrance problem with the uniform wall heat flux boundary condition can be found in Brinkman [27], Tyagi [6], Ou and Cheng [28], and Basu and Roy [29]. Other effects, such as inlet temperature, internal heat source, and wall heat flux variation, are reviewed by Shah and London [1] in detail. 

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 ... 

Heat Transfer on the Walls With Uniform Heat Flux. The solutions for simultaneously developing flow in circular ducts with uniform wall heat flux are reviewed by Shah and London [1], Recently, a new integral or boundary layer solution has been obtained by Al-Ali and Selim [33] for the same problem. However, the most accurate results for the local Nusselt numbers [1] are presented in Table 5.6. 

Uniform Temperature at Both Walls. When 7) Tot the problem is designated as la, and the fully developed Nusselt numbers at the two walls are designated as Nufully developed Nusselt numbers at the two walls are designated as Nu)u,) and Nufi4). These are presented in Fig. 5.13. Tabulated values for these and the subsequent solutions are available in Shah and London [1]. 

Uniform Temperature at One Wall and Uniform Heat Flux at the Other. The subscripts 1 and 2 refer to either the inside or the outside wall. When T, T2, the problem is known as 4a, and when T, = T2 it is known as 4b. It has been shown by Shah and London [1] that... 

Hydrodynamically Developing Flow. Shah and London [1] summarize the solutions for the hydrodynamic development of laminar flow in concentric annuli. The apparent friction factor in the hydrodynamic entrance region, derived by Shah [103], is expressed as ... 

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. 

TABLE 5.16 Fundamental Solutions of the First Kind for Thermally Developing Flow in Concentric Annular Ducts (compiled from Shah and London [1])... 

Effects of Eccentricity. In practice, a perfect concentric annular duct cannot be achieved because of manufacturer tolerances, installation, and so forth. Therefore, eccentric annular ducts are frequently encountered. The velocity profile for fully developed flow in an eccentric annulus has been analyzed by Piercy et al. [105]. Based on Piercy s solution, Shah and London [1] have derived the friction factor formula, as follows ... 

Hydro dynamically Developing Flow. For hydrodynamically developing flow in parallel plate ducts, Shah and London [1] obtained the apparent friction factor/app and Chen [11] has obtained Lhyf K oo) as the function of x+ and Re, respectively, as shown in the following equations ... 

Equal and Uniform Temperatures on Both Walls. The local and mean Nusselt numbers for parallel plate ducts with equal and uniform temperatures on both walls can be computed from Nusselt s [131] solution, which is displayed in Fig. 5.21. The tabulated values for Fig. 5.21 are available in Shah and London [1]. 

It is also suggested that the following set of empirical equations proposed by Shah and London [1] be used for the practical calculation of the local Nusselt number ... 

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 ... 

Convective Boundary Condition at Both Walls or One Wall The solutions for the convective boundary condition on both walls or one wall are reviewed in Shah and London [1], where more detailed descriptions are available. 

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