Nusselt and Sherwood numbers

There is apparently an inherent anomaly in the heat and mass transfer results in that, at low Reynolds numbers, the Nusselt and Sherwood numbers (Figures. 6.30 and 6.27) are very low, and substantially below the theoretical minimum value of 2 for transfer by thermal conduction or molecular diffusion to a spherical particle when the driving force is spread over an infinite distance (Volume 1, Chapter 9). The most probable explanation is that at low Reynolds numbers there is appreciable back-mixing of gas associated with the circulation of the solids. If this is represented as a diffusional type of process with a longitudinal diffusivity of DL, the basic equation for the heat transfer process is ... 

The expressions given by Eqns. (3.4-9) correspond to the theory of the boundary layer. Similar expressions are obtained with different theories. In practical work, expressions of the type given below are used for different arrangements. Generally, the exponent of the Reynolds number is less than unity, while the exponent of the Schmidt and Prandtl numbers has been kept as 1/3. The usual expressions for the Nusselt and Sherwood numbers are ... 

So far we have considered an infinite value of the gas-to-particle heat and mass transfer coefficients. One may encounter, however, an imperfect access of heat and mass by convection to the outer geometrical surface of a catalyst. Stated in other terms, the surface conditions differ from those in the bulk flow because external temperature and concentration gradients are established. In consequence, the multiple steady-state phenomena as well as oscillatory activity depend also on the Sherwood and Nusselt numbers. The magnitudes of the Nusselt and Sherwood numbers for some strongly exothermic reactions are reported in Table III (77). We may infer from this table that the range of Sh/Nu is roughly Sh/Nu (1.0, 104). 

To assess the influence of the Nusselt and Sherwood number on multiplicity explicit condition (5) may be consulted (11) ... 

Considering these Biot numbers, we can observe that they are similar to the Nusselt and Sherwood numbers. The only difference between these dimensionless numbers is the transfer coefficient property characterizing the Biot numbers transfer kinetics for the external phase (a x heat transfer coefficient for the external phase, k ex- mass transfer coefficient for the external phase). We can conclude that the Biot number is an index of the transfer resistances of the contacting phases. 

The model was developed to show that if the proper boundary conditions are used,one should not expect at low Reynolds numbers that the Nusselt and Sherwood numbers approach the limiting value of two, which is valid for a sphere in an infinite static medium. Since the particles are members of an assemblage, they assume in their model that there is a concentric spherical shell of radius... 

To this end, experimental heat and mass transfer coefficients were determined in a fluidized bed. Nusselt and Sherwood numbers were obtained in terms of Reynolds number and aspect ratios dp/L and dp/D. The results are also analyzed in terms of the Kato and Wen(5) and Nelson and Galloway(6) models. 

In natural convection mass transfer, the analogy betsveen the Nusselt and Sherwood numbers still holds, and thus Sh =y(Gr, Sc). But the Grashof number in this case should be determined directly from... 

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. 

The first five correlations predict limiting values of the Sherwood or Nusselt number as 2 and are mainly valid for single-particle or very dilute flows. The last two correlations predict Nusselt and Sherwood numbers lower than 2 for dense flows and should be used to simulate dense granular flows. 

The heat- and mass-transfer coefficients were then calculated from the definitions of the Nusselt and Sherwood numbers. 

If the Reynolds number is increased in a packed bed, the Nusselt and Sherwood numbers increase corresponding to the left branch of the curve in Fig. 3.40. 

Once the upward flowing fluid has reached the minimum fluidisation velocity wml and with that the Reynolds number the value Remi = w dp/i/, point a in Fig. 3.40, a fluidised bed is formed. The heat and mass transfer coefficients hardly change with increasing fluid velocity The Nusselt and Sherwood numbers are only weakly dependent on the Reynolds number, corresponding to the slightly upwardly arched line a b in Fig. 3.40. After a certain fluid velocity has been reached, indicated here by point b in Fig. 3.40, the particles are carried upwards. At point b the heat and mass transfer coefficients are about the same as those for flow around a single sphere of diameter dP. 

The process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

M. M. Yovanovich, New Nusselt and Sherwood Numbers for Arbitrary Isopotential Geometries at Near Zero Peclet and Rayleigh Numbers, AIAA-87-1643, AIAA 22nd Thermophysics Conference, Honolulu, HI, 1987. 

The effective media properties D , Df, D , and Ds, which include both the molecular (i.e., conductive) and the hydrodynamic dispersion components, are also modeled. Due to the lack of any predictive correlations for the nonequilibrium transport, local thermal equilibrium conditions are used. For the interfacial convective transport, the local Nusselt and Sherwood numbers are prescribed. The effect of the solid particles geometry must also be addressed [160],... 

In Equations 10.1 through 10.3, the coefficients of convective heat and mass transfer, h and ho, are evaluated in terms of the appropriate Nusselt and Sherwood numbers for evaporating droplet [22]. 

Fig. 4.3-1 Nusselt and Sherwood number depending on the inlet characteristic RePr- d/L respectively Re- Sc- d/L for several flow regimes...

Abstract Evaporation of multi-component liquid droplets is reviewed, and modeling approaches of various degrees of sophistication are discussed. First, the evaporation of a single droplet is considered from a general point of view by means of the conservation equations for mass, species and energy of the liquid and gas phases. Subsequently, additional assumptions and simplifications are discussed which lead to simpler evaporation models suitable for use in CFD spray calculations. In particular, the heat and mass transfer for forced and non-forced convection is expressed in terms of the Nusselt and Sherwood numbers. Finally, an evaporation model for sprays that is widely used in today s CFD codes is presented. 

For a drop in a quiescent atmosphere, that is, a drop under non-forced convection, exact expressions for the Nusselt and Sherwood numbers can be derived tmder the assumption that the problem is spherically symmetric. The initial derivation for a single species drop is due to Spalding [26, 27]. An insightful presentation of this approach can be found in Kuo [15]. Essentially the same methodology can also be apphed to multi-species droplets and the resulting expressions for the Sherwood and Nusselt numbers are formally the same as for a single species drop. For more details see Gradinger [12]. 

The asymptotic Nusselt and Sherwood numbers depend on the geometry of the channel as summarized in Table 5.1. 

Re, Pr Reynolds and Prandtl nos. of gas phase Nu, Sh Nusselt and Sherwood numbers... 

Figure 10.11 Variation in Nusselt and Sherwood number profiles with inlet flow velocity, Uoi other conditions are the same as those in Figure 10.10. The inlet Pe varies from 14 to 115 in the curves shown in this figure.

Hayes RE, Kolaczkowski ST. A study of Nusselt and Sherwood numbers in a monolith reactor. Catalysis Today 1999 47 295-303. 

The semi-empirical global transport correlation proposed by Hawthorn [15] is the most commonly used for the definition of local Nusselt and Sherwood numbers, applicable to laminar flows in square ducts ... 

A model developed earlier (4, ) used the collocation method to solve the equations for heat, mass and momentum transfer In a single, adiabatic channel of the monolith. The basic model Is the one described as Model II-A(5) a square duct with axial conduction of heat longitudinally In the solid walls, but with Infinitely fast conduction peripherally around the square, and Including the diffusion of heat and mass In the transfer direction In the fluid (See for a discussion of the Importance of Including this effec.) Nusselt and Sherwood numbers are not assigned priori, but are derived from the solution. The reaction rate expression P2 In (3) with a basic form... 

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