Nusselt flow model

Two flow models are used to estimate the mean condensation coefficient in horizontal tubes stratified flow, Figure 12.45a, and annular flow, Figure 12.45. The stratified flow model represents the limiting condition at low condensate and vapour rates, and the annular model the condition at high vapour and low condensate rates. For the stratified flow model, the condensate film coefficient can be estimated from the Nusselt equation, applying a suitable correction for the reduction in the coefficient caused by... 

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. 

Annular. When the vapor velocity is high enough (j > 1.5), gravitational effects can be neglected, and the condensate collects as a thin annular film around the inside of the tube walls, with no stratification. A significant portion of most condensers operate in this flow regime. There are numerous predictive models described in the literature for annular flow. Laminar flow models predict heat transfer coefficients that are too low, and turbulent models must be used. The most commonly used models are listed in Table 14.1. All models have a form for the local Nusselt number... 

While the fluid dynamics of the actual film-flow process is dauntingly complex, a very approximate interim flow model may be based on Nusselt s (1916) treatment of the flow of a condensate film. This model assumes that there is no shear at the gas-liquid interface, that the film is ripple-free and that there is no tangential slip... 

Guo and Wu (31) have pointed out that the effects of acceleration are more important than the effects of friction at high Mach numbers. They conclude that the assumption of a fully developed or locally fully developed velocity profile does not apply to near-sonic flow in a microtube. Because the velocity profile becomes flatter, higher velocity gradients are expected near the wall. The results of their two-dimensional flow model and their experimental findings indicate that the Reynolds number multiplied by the Darcy friction factor can be as high as 80 as compared with 64 from the standard literature (22). Accordingly, they also found a Nusselt number that was about 25% higher than that predicted by conventional correlations. 

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 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 basic equations for filmwise condensation were derived by Nusselt (1916), and his equations form the basis for practical condenser design. The basic Nusselt equations are derived in Volume 1, Chapter 9. In the Nusselt model of condensation laminar flow is assumed in the film, and heat transfer is assumed to take place entirely by conduction through the film. In practical condensers the Nusselt model will strictly only apply at low liquid and vapour rates, and where the flowing condensate film is undisturbed. Turbulence can be induced in the liquid film at high liquid rates, and by shear at high vapour rates. This will generally increase the rate of heat transfer over that predicted using the Nusselt model. The effect of vapour shear and film turbulence are discussed in Volume 1, Chapter 9, see also Butterworth (1978) and Taborek (1974). 

In a bank of tubes the condensate from the upper rows of tubes will add to that condensing on the lower tubes. If there are Nr tubes in a vertical row and the condensate is assumed to flow smoothly from row to row, Figure 12.42a, and if the flow remains laminar, the mean coefficient predicted by the Nusselt model is related to that for the top tube by ... 

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. 

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. 

Considerations along the above lines lead to analogous correlations for the Sherwood number for the description of mass transfer in a single channel. The application of the rather simple Nusselt and Sherwood number concept for monolith reactor modeling implies that the laminar flow through the channel can be approached as plug flow, but it is always limited to cases in which homogeneous gas-phase reactions are absent and catalytic reactions in the washcoat prevail. If not, a model description via distributed flow is necessary. 

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... 

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... 

Nusselt provided a simple model for laminar liquid flow down an inclined plane. This assumed that the liquid had reached fully developed conditions in which drag due to viscous shear exactly balanced the weight of the film. Under these conditions, Nusselt showed that for a Newtonian fluid of kinematic viscosity, v, film thickness, /, could be written in terms of the liquid flow rate, Q, moving over a vertically inclined surface of width, w, under a gravitational acceleration, g, using the following relationship ... 

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