Spalding number

A commonly used expression for this number is Sh = 2.0 - - 0.55Re/ Scp where Rcp = d u —u /i/ is the particle Reynolds number and Scp is the fuel vapor Schmidt number. In this definition v is the carrier phase kinematic viscosity. In Eq. (8.2), one important parameter is the Spalding number Bm = Ypx f)/(1 — Yfx) where Ypx is the fuel mass fraction at the droplet surface, calculated from the fuel vapor partial pressure at the interface ppx which is evaluated from the Clausius-Clapeyron relation ... 

Developed for a single wet particle, taking into account the resistance of the liquid vapors around the particle to the heat transfer by Spalding number, B. Cp denotes the heat capacity of the liquid vapors in the gas phase and Hfg is the latent heat of evaporation for the fluid. 

The terms appearing as argument of the logarithm in Eq. (17.69) are the molar fractions of the species in the gas phase at a sufficiently far distance Xy and the molar fractions at saturation. These terms are also used in a more compact form and referenced as the Spalding number B ... 

Spalding number Adimensionless number, B, used in liquid droplet evaporation studies. It relates the sensible heat and latent heat of the evaporated material ... 

When the concentration profile is fully developed, the mass-transfer rate becomes independent of the transfer length. Spalding (S20a) has given a theory of turbulent convective transfer based on the hypothesis that profiles of velocity, total (molecular plus eddy) viscosity, and total diffusivity possess a universal character. In that case the transfer rate k + can be written in terms of a single universal function of the transfer length L and fluid properties (expressed as a molecular and a turbulent Schmidt number) ... 

The dimensionless group in the log term is called the Spalding B number, named after Professor Brian D. Spalding who demonstrated its early use [5], From Equation (9.39), the surface conditions give... 

Both correlating parameters are in the form of Peclet numbers, and the air-fuel ratio dependence is in Figure 4.58 shows the excellent correlation of data by the above expression developed from the Spalding analysis. Indeed, the power dependence of d with respect to blowoff velocity can be developed from the slopes of the lines in Fig. 4.58. Notice that the slope is 2 for values... 

There are now three fundamental diffusion equations one for the fuel, one for the oxidizer, and one for the heat. Equation (6.74) is then written as two equations one in terms of m and the other in terms of m0. Equation (6.73) remains the same for the consideration here. As seen in the case of the evaporation, the solution to the equations becomes quite simple when written in terms of the b variable, which led to the Spalding transfer number B. As noted in this case the b variable was obtained from the boundary condition. Indeed, another b variable could have been obtained for the energy equation [Eq. (6.73)]—a variable having the form... 

Comprehensive treatment of the evaporation constant X is given by Spalding 109), and a comprehensive examination of Nusselt numbers for droplets is presented by Ranz and Marshall 102). The first work in the field is that of Froessling 29), after whom the appropriate equations are named. According to the combined treatments of Spalding and of Ranz and Marshall, when diffusion is controlling at relatively low vaporization rates... 

The parameters (2 + 0.60 Re1/2 Sc1/3) and (2 -f 0.60 ReI/2Pr1/3) represent the Nusselt numbers for mass and heat transfer, respectively (87). Equation 3 derives from Froessling directly, while Spalding substituted thermal diffusivity for molecular diffusivity to establish the basis for Equation 4. 

The transfer number B in Equations 3 and 4 is Spalding s contribution. It is the driving force for mass transfer in dimensionless form. With diffusion controlling (Equation 3) ... 

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). 

The case of combustion of an entire spherical surface with forced convection has not yet been solved. Frossling (4) originally proposed a semi-empirical relation for the low-temperature evaporation of droplets in motion. Spalding (60) has applied the equation to his heterogeneous combustion data with some success by including the term containing the transfer number ... 

Spalding (51, 55, 60) presents, in dimensionless form, data on natural convection for kerosine, gas oil, petrol, and heavy naphtha burning from a 1.5-inch sphere. He suggests the following empirical relation (see Equation 8) for a range of transfer numbers, B, of 0.25 to 3 ... 

Spalding obtained additional information with forced air convection past 0.5-, 0.75-, and 1-inch spheres, with kerosine, petrol, ethyl alcohol, and benzene as fuels. For the range B = 0.6 to 5 and Reynolds number (based on sphere diameter and ambient gas conditions) ranging from 800 to 4000 he found the following to apply (see Equations 9 and 10) ... 

As was the case for the First Law, one may provide a parable that illuminates the Second Law of Thermodynamics. Here we follow the example provided by Spalding and Cole7. Entropy change is to be likened to a difference in elevation between two points A and B, AS, on rough mountainous terrain. A mountain climber undertakes to measure AS in terms of the number of steps nf, each one-foot in height, which he must take to get from level A to level B. If the climber elects to take a circuitous route along a firm path, then AS — nf. If he... 

Spalding developed a successful eddy-breakup model for describing rates of flame spread in high-intensity flows of this type [143]-[146]. There are a number of difYerent versions of the model one employs... 

Here B = boo — and is generally referred to as the Spalding transfer number. The mass consumption rate per unit area G = where = p v Anr, ... 

However, the sorption of a number of pesticide compounds to some soils has been found to be more accurately described by the nonlinear Freundlich isotherm (e.g., Hamaker and Thompson, 1972 Widmer and Spalding, 1996), i.e.. 

The modeled transport equations for z differ mainly in the diffusion and secondary source term. Launder and Spalding (1972) and Chambers and Wilcox (1977) discuss the differences and similarities in more detail. The variable, z = e is generally preferred since it does not require a secondary source, and a simple gradient diffusion hypothesis is fairly good for the diffusion (Launder and Spalding, 1974 Rodi, 1984). The turbulent Prandtl number for s has a reasonable value of 1.3, which fits the experimental data for the spread of various quantities at locations far from the walls, without modification of any constants. Because of these factors, the k-s model of turbulence has been the most extensively studied and used and is recommended as a baseline model for typical internal flows encountered by reactor engineers. 

This equation contains three new terms, namely flux of scalar variance, production of variance and dissipation of scalar variance, which require further modeling to close the equation. The flux terms are usually closed by invoking the gradient diffusion model (with turbulent Schmidt number, aj, of about 0.7). This modeled form is already incorporated in Eq. (5.21). The variance production term is modeled by invoking an analogy with turbulence energy production (Spalding, 1971) ... 

The scalar length scale, Ls, is assumed to be equal to in the above expression. For systems with low values of Schmidt number, Corrsin s model reduces to that of Spalding, albeit with higher coefficient (0.5). Corrsin s model is found to be useful... 

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