Experimental Nusselt number

Figure 4 Comparison of experimental Nusselt number from several contributors [33].

Figure 5.37 compares Eq. 5.272 with some of the available theoretical predictions [204,205] and experimental Nusselt number data [190,198]. The figure indicates a fairly good agreement between the correlation and most of the available data. 

FIGURE 5.36 A comparison of the recommended design correlation (Eq. 5.268, solid line) with theoretical and experimental Nusselt numbers for the boundary condition [201]. 

Figure 5 shows a ratio of the experimental Nusselt number to the predicted Nusselt number (based upon the above definition of Reynolds number) plotted as a function of x t. The data points represent local conditions at four axial stations near the midportion of the small diameter Inconel test section. Such a selection of axial stations obviated the effects of end conditions. 

When Mam > 0.3 the hypothesis of incompressibility no longer holds and the gas acceleration leads to changes in the velocity profile not only in magnitude but also in shape. The magnitude increments produce additional pressure drop while the shape changes alter the friction factor at the walls. The continuous variation in shape of the velocity profile means that neither fully developed nor locally fully developed flows occur. This fact influences the convective heat transfer coefficient since no fully developed temperature profiles can occur if the flow is developing and the experimental Nusselt numbers differ by the theoretical values for fully developed flow. It is possible to demonstrate numerically that when the Mach number increases the Nusselt number decreases along the... 

Fig. 4.3a-c Experimental results for smooth circular tubes, (a) Dependence of the Nusselt number on non-dimensional axial distance rfin = 125.4, 300 and 500 pm, Re = 95—774. Reprinted from Lelea et al. (2004) with permission, (b) d =... 

Reynolds number. It should be stressed that the heat transfer coefficient depends on the character of the wall temperature and the bulk fluid temperature variation along the heated tube wall. It is well known that under certain conditions the use of mean wall and fluid temperatures to calculate the heat transfer coefficient may lead to peculiar behavior of the Nusselt number (see Eckert and Weise 1941 Petukhov 1967 Kays and Crawford 1993). The experimental results of Hetsroni et al. (2004) showed that the use of the heat transfer model based on the assumption of constant heat flux, and linear variation of the bulk temperature of the fluid at low Reynolds number, yield an apparent growth of the Nusselt number with an increase in the Reynolds number, as well as underestimation of this number. 

Adams et al. (1998) investigated turbulent, single-phase forced convection of water in circular micro-channels with diameters of 0.76 and 1.09 mm. The Nusselt numbers determined experimentally were higher than those predicted by traditional Nusselt number correlations such as the Gnielinski correlation (1976). The data suggest that the extent of enhancement (deviation) increases as the channel diameter decreases. Owhaib and Palm (2004) investigated the heat transfer characteristics... 

Qu et al. (2000) carried out experiments on heat transfer for water flow at 100 < Re < 1,450 in trapezoidal silicon micro-channels, with the hydraulic diameter ranging from 62.3 to 168.9pm. The dimensions are presented in Table 4.5. A numerical analysis was also carried out by solving a conjugate heat transfer problem involving simultaneous determination of the temperature field in both the solid and fluid regions. It was found that the experimentally determined Nusselt number in micro-channels is lower than that predicted by numerical analysis. A roughness-viscosity model was applied to interpret the experimental results. 

Experimental and numerical analyses were performed on the heat transfer characteristics of water flowing through triangular silicon micro-channels with hydraulic diameter of 160 pm in the range of Reynolds number Re = 3.2—84 (Tiselj et al. 2004). It was shown that dissipation effects can be neglected and the heat transfer may be described by conventional Navier-Stokes and energy equations as a common basis. Experiments carried out by Hetsroni et al. (2004) in a pipe of inner diameter of 1.07 mm also did not show effect of the Brinkman number on the Nusselt number in the range Re = 10—100. 

Equation (4.12) indicates the effect of viscous dissipation on heat transfer in micro-channels. In the case when the inlet fluid temperature, To, exceeds the wall temperature, viscous dissipation leads to an increase in the Nusselt number. In contrast, when To < Tv, viscous dissipation leads to a decrease in the temperature gradient on the wall. Equation (4.12) corresponds to a relatively small amount of heat released due to viscous dissipation. Taking this into account, we estimate the lower boundary of the Brinkman number at which the effect of viscous dissipation may be observed experimentally. Assuming that (Nu-Nuo)/Nuo > 10 the follow-... 

Number of chemical components Number of tanks in series Molar flow rate of component A Moles initially present Moles of component A Number of experimental data Rotational velocity of impeller Total moles in the system Nusselt number... 

In the articles cited above, the studies were restricted to steady-state flows, and steady-state solutions could be determined for the range of Reynolds numbers considered. Experimental work on flow and heat transfer in sinusoidally curved channels was conducted by Rush et al. [121]. Their results indicate heat-transfer enhancement and do not show evidence of a Nusselt number reduction in any range... 

The parametric effect of system pressure on the heat transfer coefficient was studied by Wirth (1995). They obtained experimental measurements of the heat transfer Nusselt number for fast fluidized beds... 

The experimental results obtained for a wide range of systems(96-99) are correlated by equation 6.58, in terms of the Nusselt number (Nu = hd/k) for the particle expressed as a function of the Reynolds number (Re c = ucdp/fx) for the particle, the Prandtl number Pr for the liquid, and the voidage of the bed. This takes the form ... 

As Re increases further and vortices are shed, the local rate of mass transfer aft of separation should oscillate. Although no measurements have been made for spheres, mass transfer oscillations at the shedding frequency have been observed for cylinders (B9, D6, SI2). At higher Re the forward portion of the sphere approaches boundary layer flow while aft of separation the flow is complex as discussed above. Figure 5.17 shows experimental values of the local Nusselt number Nuj c for heat transfer to air at high Re. The vertical lines on each curve indicate the values of the separation angle. It is clear that the transfer rate at the rear of the sphere increases more rapidly than that at the front and that even at very high Re the minimum Nuj. occurs aft of separation. Also shown in Fig. 5.17 is the thin concentration boundary layer... 

Fig. 5.17 Local Nusselt number for heat transfer from a sphere to air (Pr = 0.71). Experimental results of Galloway and Sage (Gl). Dashed lines are predictions of boundary layer theory by Lee and Barrow (LIO).

Experimental data are available for large particles at Re greater than that required for wake shedding. Turbulence increases the rate of transfer at all Reynolds numbers. Early experimental work on cylinders (VI) disclosed an effect of turbulence scale with a particular scale being optimal, i.e., for a given turbulence intensity the Nusselt number achieved a maximum value for a certain ratio of scale to diameter. This led to speculation on the existence of a similar effect for spheres. However, more recent work (Rl, R2) has failed to support the existence of an optimal scale for either cylinders or spheres. A weak scale effect has been found for spheres (R2) amounting to less than a 2% increase in Nusselt number as the ratio of sphere diameter to turbulence macroscale increased from zero to five. There has also been some indication (M15, S21) that the spectral distribution of the turbulence affects the transfer rate, but additional data are required to confirm this. The major variable is the intensity of turbulence. Early experimental work has been reviewed by several authors (G3, G4, K3). 

While the basic equations for Nusselt number have not been proved for small droplets, there is ample demonstration that Equations 3 and 4 apply to small droplets when Reynolds number is zero and Nusselt number is 2. Such discrepancies as arise may be attributed to uncertainties in the necessary physical property values and in the particular experimental data. Pertinent investigations are those for burning droplets performed by Godsave (37-39), Goldsmith and Penner (42), and Graves (44), with analysis by Spalding (109). 

For gas-solid fluidized beds, Wen and Fane (1982) suggested that the determination of the bed-to-surface mass transfer coefficient can be conducted by using the corresponding heat transfer correlations, replacing the Nusselt number with the Sherwood number, and replacing the Prandtl number by Sc(cpp)/(cpp)/(l — a). Few experimental results on bed-to-surface mass transfer are available, especially for gas-solid fluidized beds operated at relatively high gas velocities. 

Hence, in experimental heat transfer, the number of variables to be studied is significantly reduced. The Nusselt number or heat transfer coefficient is correlated to only two dimensionless numbers. 

Comparison between predicted and experimental mean Nusselt numbers. 

These expressions give results that are in reasonably good agreement with experimental results, a comparison of some measurements of mean Nusselt number with the values predicted by Eq. (3.51) being shown in Fig. 3.7. 

Dimensionless numbers have proved useful for analyzing relationships between heat transfer and boundary layer thickness for leaves. In particular, the Nusselt number increases as the Reynolds number increases for example, Nu experimentally equals 0.97 Re0-5 for flat leaves (Fig. 7-9). By Equations 7.18 and 7.19, d/8bl is then equal to 0.97 (vd/v)V2y so for air temperatures in the boundary layer of 20 to 25°C, we have... 

Several correlations, all based on experimental data, have been proposed for the average Nusselt number for cross flow over tube banks. Mote recently, Zukauskas has proposed correlations whose general form is... 

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