Nusselt number estimation

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 ... 

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-... 

The maximum possible heat flux, which corresponds to the maximum allowed wall temperatures is estimated. This maximum wall heat flux is determined by the difference between the permissible wall temperature and the vapor temperature in the outlet cross-section, which is a function of the Reynolds and Nusselt numbers. 

The heat transfer coefficient to the vessel wall can be estimated using the correlations for forced convection in conduits, such as equation 12.11. The fluid velocity and the path length can be calculated from the geometry of the jacket arrangement. The hydraulic mean diameter (equivalent diameter, de) of the channel or half-pipe should be used as the characteristic dimension in the Reynolds and Nusselt numbers see Section 12.8.1. 

Now let us consider the mixing time, t. This will be estimated by an order of magnitude estimate for diffusion to occur across the boundary layer thickness, <5Bl- If we have turbulent natural conditions, it is common to represent the heat transfer in terms of the Nusselt number for a vertical plate of height, , as... 

In this correlation, the material properties are evaluated at the melting temperature. The left hand side of the correlation is the dimensionless minimum melt superheat. The right hand side of the correlation is also dimensionless, and represents a combination of the Prandtl number, Euler number, Reynolds number and Nusselt number, as well as temperature and length ratios TJTG and l0/d0. The correlation is accurate within 10%. In addition, considering the effects of the surface roughness of nozzle wall, the pre-basal coefficient in the regression expression has been increased by 25% in order to predict a safe estimate of the minimum melt superheat. 

The designer now needs to make some estimates of mass transfer. These properties are generally well known for commercially available adsorbents, so the job is not difficult. We need to re-introduce the adsorber cross-section area and the gas velocity in order to make the required estimates of the external film contribution to the overall mass transfer. For spherical beads or pellets we can generally employ Eq. (7.12) or (7.15) of Ruthven s text to obtain the Sherwood number. That correlation is the mass transfer analog to the Nusselt number formulation in heat transfer ... 

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. 

For a constant wall temperature, a fully developed laminar velocity profile, and a developing thermal profile, the average Nusselt number is estimated by [Hausen, Mlg. Waermetech., 9, 75 (1959)]... 

The average Nusselt number for laminar flow over an isothermal flat plate of length x is estimated from [Churchill and Ozoe, J. Heat Transfer, 95, 416 (1973)]... 

Estimates of the film thicknesses, 8j, needed to determine the gas-phase temperature and weight fraction profiles are based on the empirical Nusselt number correlations developed by Ranz and Marshall (23) for... 

In turbulent flow, one can estimate the Nusselt number using the analogy between heat and momentum transfer (Colburn analogy). This analogy relates the Nusselt number to llie coefficient of friction, Cj, as (a) Nu = 0.5 Cy Re Pr (h) Nu = 0.5 C>Rc Pr ... 

The corresponding Nusselt number (Nu) is calculated as Nu = hspDh/ki, where Iq is the liquid thermal conductivity. The measurement uncertainty in the single-phase heat transfer coefficients were estimated to about 15% - 25% close to the channel inlet and about 10% - 15% further downstream. 

Although further diseussion of heat transfer correlations is no doubt worthwhile, it will not help us to determine the mass transfer coefficieni and the mass flux from the bulk fluid to the external pellet smface. However, the preceding discussion on heat transfer was not entirely futile, because, for similar geometries, the heat and mass transfer correlafions are analogous. If a heat transfer coirrelation for the Nusselt number exists, the mass transfer coefficient can be estimated by replacing the Nusselt and Prandtl numbers in this correlation by the Sherwood and Schmidt numbers, respectively ... 

According to Stein and Schmidt a few adaptations make it possible to use correlations for horizontal tube coils to estimate heat transfer in a vessel with a halfpipe jacket. Necessary adaptations include using the thermic diameter dth = Tt/2)di instead of the tube inner diameter d to calculate the Reynolds and Nusselt numbers, and replacing the bending ratio (dhjDb) with the ratio of d-J2 T + 26 ) " ... 

For the empty tube the heat transfer coefficient can be estimated by the traditional correlations. If d is the diameter of the pipe, the Nusselt number Ud/k is commonly related to the Prandtl and Reynolds numbers by the Dittus-Boelter relation... 

A large number of empirical correlations is available for estimating the Nus-selt number in both packed- and fluidized beds. A Nusselt number correlation proposed by Gunn [60] was used. The Nusselt number parameterization represents a functional fit to experimental data for Reynolds number up to 10 in the porosity range 0.35 — 1 ... 

The limit Pe 0 yields the pure conduction heat transfer case. However, for a fluid in motion, we find that the pure conduction limit is not a uniformly valid first approximation to the heat transfer process for Pe 1, but breaks down far from a heated or cooled body in a flow. We discuss this in the context of the Whitehead paradox for heat transfer from a sphere in a uniform flow and then show how the problem of forced convection heat transfer from a body in a flow can be understood in the context of a singular-perturbation analysis. This leads to an estimate for the first correction to the Nusselt number for small but finite Pe - this is the first small effect of convection on the correlation between Nu and Pe for a heated (or cooled) sphere in a uniform flow. 

In the intermediate domain (2200 < Re < 4000) at moderate Prandtl numbers, the Nusselt number can be estimated using the formula [254]... 

TABLE 3. Results for the local Nusselt number, Nu(Z), for the parallel-plates case with Bi = CO, =0.1 and fS = h Comparison of exact results. Ref. [6], and recovered results from estimated parameters. 

This is obtained from the average Nusselt number Nu defined in Fig. 4.6. A rough estimate of Nu can be obtained for Ra > 105 by using the equations in the section on vertical flat plates with uniform Tw and = 90°, with AT replaced by AT. For higher accuracy, and for convenience since Ra is used in place of Ra, use the following equation set ... 

Stratified Medium. For a long horizontal circular cylinder in a thermally stratified environment in which the temperature increases linearly with height (see Fig. 4.16h for nomenclature), AT is the temperature difference at the mid-height of the cylinder. First calculate the laminar isothermal Nusselt number Nu, from Eq. 4.45 with AT = AT and rename it NUf iS0 the corresponding calculated heat flow is qis0. The laminar Nusselt number Nu, corrected to account for the stratification is then estimated from... 

From the Nusselt number for pure natural convection, Nu, and that for pure forced convection, Nu,., a rough estimate of the actual Nusselt number for a given problem is... 

Horizontal Flow. For laminar flow over the upper surface of a horizontal heated plate (or over the bottom surface of a cooled plate), the center of the mixed convection regime can again be estimated by equating the forced convection Nusselt number from Eq. 4.154 to that for natural convection from Eq. 4.39c (for detached turbulent convection). This results in... 

Equating the Nusselt numbers for pure natural convection and pure forced convection provides a good estimate of the Ra-Re curve along which mixed convection effects are most important, as already discussed. After a careful study of available data, Morgan [198] proposed the following equation for forced convection heat transfer from a cylinder for cross flow in a low-turbulence airstream ... 

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