Nusselt number model

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. 

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. 

The heat transfer coefficient calculated numerically using an exact model with regard to the heat transferred through the solid substrate represents the correct variation of the Nusselt number with respect to the Reynolds number. 

The results of calculations of the Nusselt number are presented in Fig. 10.19. Here also the data of the calculated heat transfer by the quasi-one-dimensional model by Khrustalev and Faghri (1996) is shown. The comparison of the results related to one and two-dimensional model shows that for relatively small values of wall superheat the agreement between the one and two-dimensional model is good enough (difference about 3%), whereas at large At the difference achieves 30%. 

The simulations of fluid flow and heat transfer in such microstructured geometries were carried out with an FVM solver. Air with an inlet temperature of 100 °C was considered as a fluid, and the channel walls were modeled as isothermal with a temperature of 0 °C. The streamline pattern is characterized by recirculation zones which develop behind the fins at comparatively high Reynolds numbers. The results of the heat transfer simulations are summarized in Figure 2.34, which shows the Nusselt number as a fimction of Reynolds number. For... 

The second approach assigns thermal resistance to a gaseous boundary layer at the heat transfer surface. The enhancement of heat transfer found in fluidized beds is then attributed to the scouring action of solid particles on the gas film, decreasing the effective film thickness. The early works of Leva et al. (1949), Dow and Jacob (1951), and Levenspiel and Walton (1954) utilized this approach. Models following this approach generally attempt to correlate a heat transfer Nusselt number in terms of the fluid Prandtl number and a modified Reynolds number with either the particle diameter or the tube diameter as the characteristic length scale. Examples are ... 

Here d is the model particle diameter Nu is the Nusselt number. The Nusselt number for the th particle is determined as... 

Very little work has been reported on vaporization under conditions of turbulent gas flow. Ingebo (61), for example, took pains to minimize approach stream turbulence. Two exceptions are the investigations of Maisel and Sherwood (83) and Fledderman and Hanson (27). Neither went so far in analysis as insertion into the Nusselt number equations of allowance for the additional relative velocity between droplet and air stream occasioned by turbulence. In the case of Maisel and Sherwood s investigation with model droplets at fixed positions, the effect would not be expected to be extreme, because at all times there was appreciable relative velocity, discounting turbulence. However, in Fledderman and Hanson s experiments the relative velocity, discounting turbulence, fell away as the droplets accelerated up to stream velocity. Thus turbulence would eventually provide the only appreciable relative velocity. The results indicate a substantial increase in vaporization rate because of the turbulence and provide some basis for gross engineering estimates. 

At higher Reynolds numbers, i.e., between points A and B in Fig. 9.22, the effect of the buoyancy forces on the turbulence quantities is the dominant effect and the Nusselt number is, therefore, decreased below its forced convective value. At the lower Reynolds number, i.e., between B and C in Fig. 9.22, the direct effect of the buoyancy forces on the mean momentum balance becomes the dominant effect and the Nusselt number rises above its forced convective value. The changes are displayed by the numerical results shown in Fig. 9.23. These results were obtained using a more advanced turbulence model than that discussed here. 

It is worth noting that one of the uses to which computers can be put is to derive continuum models for highly structured systems. Thus in the case of drilling muds, simulations on the scale of clay platelets can be used to provide rheological models for use in finite element models for flow mechanics, which are then used to derive Nusselt number estimates for uniaxial mean temperature predictions in oil wells. The range of length scales goes from 1 nm to 1000 m ... 

Now the dimensionless ratio hu/K is known as the Nusselt number Nu(ro), and for systems with convection it takes values of about 5 if the flow is not turbulent. (In the absence of convection /i, the heat transfer at the walls is determined by the temperature gradient at the walls, which in turn is proportional to K/tq,) It is interesting to note that the simple model which permits laminar convection gives values of 8c of about the order of 0, which is reasonably close to the value of 3.32 calculated for pure conduction. 

Kato and Wen (5) found, for the case of packed beds,that there was a dependency of the Sherwood and Nusselt numbers with the ratio dp/L. They proposed that the fall of the heat and mass transfer coefficients at low Reynolds numbers is due to an overlapping of the boundary layers surrounding the particles which produces a reduction of the available effective area for transfer of mass and heat. Nelson and Galloway W proposed a new model in terms of the Frossling number, to explain the fall of the heat and mass transfer coefficients in the zone of low Reynolds numbers. 

Because pressure drop measurements are much faster and cheaper than mass transfer or heat transfer measurements, it is tempting to try to relate the Sherwood and Nusselt numbers to the friction factor. A relation that has proved successful for smooth circular tubes is obtained from a plausible assumption that is known as the film layer model. The assumption is that for turbulent flow the lateral velocity, temperature, and concentration gradients are located in thin films at the wall of the channel the thickness of the films is indicated with 8/, 87, and 8., respectively. According to the film model, the lateral velocity gradient at the channel surface equals (m)/8/, the lateral temperature gradient equals (T/, - rj/87 and the lateral concentration gradient equals (c. /, - C , )/8,.. From these assumptions, and the theoretical knowledge that 8//8r Pr and 8//8e Sc (for... 

Compressible two-dimensional fluid flow and heat transfer characferistics of a gas flowing between two parallel plates with both uniform temperature and uniform heat flux boundary conditions were solved in [22]. They compared their results with the experimental results of [t7]. The shp flow model agreed well with these experiments. They observed an increase in the entrance length and a decrease in the Nusselt number as Kn takes higher values. It was found that the effect of compressibiUty and rarefaction is a function of Re. Compressibihty is significant for high Re and rarefaction is significant for low Re. 

Here the particle Nusselt number is Nup, where Nup = hfdp/kc, and kc is the thermal conductivity of the gas [W/(m-K)]. They stated that the heat-transfer rates predicted by this procedure were much larger than those measured on an industrial cooler, which is probably due to the particles on the inside of the cascades not e3q>eriencing the full gas velocity. Kamke and Wilson (1986) used a similar approach to model the drying of wood chips, but used the Ranz-Marsha (1952) equation to predict the heat-transfer coefficient ... 

The type of relationship between the Nusselt number and the other characteristic numbers, or the form of the functions in (1.46) and (1.47), has to be determined either through theory, the development of a suitable model or on the basis of experiments. It must also be noted that it varies from problem to problem. In the case of flow in a tube, with L0 = d, the tube diameter we get... 

The incorporation of solutal convection into morphological stability theory is here illustrated on the basis of a very simple model. It is assumed that the interfacial solute gradient Gc in the liquid is enhanced by a Nusselt number Nu given by... 

In the simple model equation (14) the onset of convection corresponds to the growth velocity at which Nu exceeds unity. At larger growth velocities Gc is not modified by convection and Nu — 1. Because the convection cannot transport more solute than is available at the interface, there is an upper limit for the Nusselt number at low V. Solute conservation implies the bound... 

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