The case of creeping flow or Stokes flow implies Rer 1. Because of the relationship Per Rer×Pr, unless Pr for the sphere is very large, as is the case of some organic oils, this condition also implies that Per
1 or at least Per 1. Convection is comprised of two parts, conduction and advection and, in this case, the conduction part dominates in the convection process, while the advection part is insignificant.
Therefore, studies that implicitly or explicitly assume that Pe 1 or Rer 1 essentially neglect the effects of the advection parts of the process and are treating the conduction part of the process alone. The steady-state solution of the conduction equation yields the following expression for the Nusselt number:
Nu 2 (1.200)
Acrivos and Taylor (1962) conducted a study on the heat transfer from a sphere, analogous to the study by Proudman and Pierson (1956) for the equation of motion. They implicitly assumed a Stokesian flow around a sphere and derived an asymptotic heat transfer solution, which is valid at higher order of Per. With the corrections in the coefficients (Acrivos, 1980; Leal, 1992) their expression for the steady-state Nusselt number, which is applicable in the ranges of the parameters Rer 1 and Per 1, is as follows:
Nu 2 Per2ln 0.2073Per2 Per3ln (1.201) In the same study, Acrivos and Taylor (1962) proved that the functional relationship Nu(Pe) as obtained in Stokesian flow is less sensitive to an increase of Re than the corresponding functional relation for the drag coefficient, CD(Re). Therefore, it is generally accepted that Eq. (1.201) is valid not only under creeping flow conditions, but also when Reris finite but small. Acrivos and Goddard (1965) also derived an asymptotic solution for high Pe assuming a Stokesian velocity distribution. Their expression, which is valid for Pe 5, may be written as follows:
Nu 1.249冢 冣1/3 0.922 (1.202)
In the case of viscous spheres, with viscosity ratio λ, Levich (1962) provided an asymptotic first-order solution for a liquid sphere at very large Perunder the condition of creeping flow (Rer 1):
Nu冪莦 (1.203)
More recently, Feng and Michaelides (2000b), under the assumption of a Stokesian velocity profile around a viscous sphere (drop or bubble), solved numerically the energy equation. They derived the fol-lowing correlation for the heat transfer coefficients from a sphere in terms of Perand λ:
Nu 1.49Per0.322 0.113/(0.361λ1) (1.204)
It must be pointed out that the implicit conditions for the use of the last three expressions (1.202), (1.203), and (1.204) are Rer 1, or at least Rer 1, and Per 1. These conditions are satisfied only for spheres with Pr 1. Several organic liquids, including gasoline and engine oil satisfy these conditions.
1.4.4.2 Reynolds Number Effects
Among the correlations of experimental data that have been used for the steady-state heat transfer from a solid sphere without mass transfer on the surface, e.g., in the absence of evaporation, sublimation, or chemical reactions, are the expressions derived by Ranz and Marshal (1952) and Whitacker (1972) which may be written, respectively, as follows:
Nu 2 0.6Rer0.5Pr0.33 (1.205)
and
Nu 2 (0.4Rer1/2 0.06Rer2/3)Pr0.4 (1.206) These correlations are valid in the case of solid spheres only and may be used up to Rer 104.
Feng and Michaelides (2000a, 2001b) conducted two numerical studies on the subject of heat transfer from viscous spheres without mass transfer and derived useful correlations, with Rerand Per as inde-pendent variables. The first study pertains to high Rerand any Per(Feng and Michaelides, 2000a) and the second to any values of Rerand Per(Feng and Michaelides, 2001b). Their results in correlation form may be summarized as follows:
1. At small but finite values of Rer(0 Rer 1) and Per 10, the general expression for the Nusselt number is as follows:
Nu(λ, Per, Rer)冢 Per1/2 Per1/3冣冤1α(Rer)冥冢 冣(1.207)
where the function α(Rer) may be written as follows:
α(Rer) 0.032 (1.208)
2. As with the case of the expression of the drag coefficient in Section 1.4.3.2, for higher Rer, the analysis of the data revealed that the best correlations of the numerical data are obtained when the general expression for the Nusselt number is given in terms of the following three functions, which pertain to specific values of the viscosity ratio λ:
A. The correlation for an inviscid sphere (λ 0), which is given by the following expression:
Nu(0, Per, Rer) 0.651 Per1/2冢1.032 冣冢1.60 冣 (1.209)
B. The Nusselt number expression for a solid sphere (λ ):
Nu(, Per, Rer) 0.852 Per1/3(1 0.233 Rer0.287) 1.3 0.182 Rer0.355 (1.210) C. The corresponding function for a sphere with viscosity ratio equal to 2, which was derived from
the numerical results and may be written as follows:
Nu(2, Per, Rer) 0.64 Per0.43(1 0.233 Rer0.287) 1.41 0.15 Rer0.287 (1.211) Hence, the final correlations for the heat transfer coefficients are given by the following expressions in the two ranges of the viscosity ratio, 0λ 2 and 2 λ :
Nu(Per, Rer,λ) Nu(Per, Rer, 0) Nu(Per, Rer,2) (1.212) for 0λ 2, 10 Per 1000
64λλ 2λ
2
0.61 Rer
Rer21 0.61 Rer
Rer21 0.61Rer
Rer21
1λλ 1.65(1α(Rer))
10.95λ
0.991λ
1λ 0.651
10.95λ
and
Nu(Per, Rer,λ) Nu(Per, Rer, 2) Nu(Per, Rer,) (1.213) for 2λ , 10 Per 1000
In the case of smaller values of Per 10, it was not possible to obtain a simple correlation of the numer-ical results, Nu(Per,Rer,λ), with any satisfactory degree of accuracy. For this reason, for applications in the range 0 Per 10, it is recommended that one uses the numerical results in the original publication (Feng and Michaelides, 2001b). These results are given in tabular form and their accuracy is only limited by the numerical accuracy of the method used.
As in the case of the hydrodynamic force on a viscous sphere, it was found that, for a fixed value of Rer and viscosity ratio,λ, the variations of the density ratio,ρs/ρf, have only a minimal effect on the external flow field. When one considers the governing equation for the heat or mass transfer processes and the per-tinent boundary conditions, one will conclude that the density ratio (or equivalently the internal Reynolds number, Rei) would not affect the corresponding transport coefficients, h or hMand, consequently, the Nusselt or Sherwood numbers. This was verified numerically by Feng and Michaelides (2001a, 2001b) in extensive numerical computations for both the hydrodynamic force and for the rate of heat transfer. The results of the computations show conclusively that the influence of the density ratio on the heat transfer coefficient is less than 0.1%, a number that is of the same order of magnitude as the numerical uncertainty of the computations and much lower than the required accuracy for any engineering calculations.
1.4.4.3 Blowing Effects
Blowing effects are important for burning droplets when the timescale of burning (mass transfer from the droplet) is of equal or lesser order of magnitude than the timescale for energy transfer. As in the case of momentum transfer, corrections to the heat transfer coefficient have been developed. These correc-tions take into account the change of the properties of the gaseous boundary layer and the phase change on the surface of the sphere. The two dimensionless numbers, called blowing factors or transfer numbers, BHand BM, which were defined in Eqs. (1.138) and (1.140), account for any heat and mass transfer effects on the surface of the sphere. These factors are used in corrections for the empirical or analytical correla-tions on heat and mass transfer from the surface of a constant volume sphere.
Since the origin of the two blowing factors is the radial mass transfer from the surface of the sphere to the carrier fluid, it is evident that the two are not independent. Abramzon and Sirignano (1989) con-ducted an analytical study on the evaporation of drops and derived expressions for these blowing factors to be used in engineering calculations. In the case of a fuel droplet that burns in air, they derived the fol-lowing relationship between the two bfol-lowing factors:
BH (1 BM)n1 (1.214)
where the exponent n is given by the ratio
n (1.215)
In the last equation, cpFis the specific heat of the fuel vapor, cpfis the specific heat of the carrier gas, Le is the dimensionless Lewis number,
Le kf/(ρfcpfᑞfp) Sc/Pr (1.216) k is an empirical coefficient equal to 0.848, and F is a function of the corresponding blowing factor,
F(B) (1 B)0.7log(1B) (1.217)
Under the conditions of thermodynamic equilibrium and properties that satisfy the equalities Pr Sc 1, the two transfer coefficients are equal: BH BM (Sirignano, 1999).
Abramzon and Sirignano (1989) obtained semianalytical expressions for the Nusselt and Sherwood numbers for a drop with mass transfer on its surface. Later, Chiang et al. (1992) improved on that study by relaxing some of the most restrictive assumptions and by conducting numerical computations on the vaporization of drops. They derived more general and, very likely, more accurate correlations for the heat and mass transfer coefficients, which may be written as follows:
Nu 1.275(1 BH)0.678Rem0.438Prm0.619 (1.218) and
Sh 1.224(1 BM)0.568Rem0.385Scm0.492 (1.219) The Schmidt number, Sc is the dimensionless group of the fluid properties, Sc (µf/ρfᑞ). Both the Reynolds and Schmidt numbers, Remand Scm, must be calculated using the mean-film transport coeffi-cients, which are defined by Eq. (1.136) and, in the case of Rem, the free-stream gas density,ρf∞. A few use-ful details, on the definition of the film properties and their usage are given in Section 1.4.3.4, and the corresponding correction for the steady-state drag coefficient is given by Eq. (1.138).
One may combine the above correlations and derive the following expressions for the rate of heat and mass transfer of vapor from the surface of a spherical drop:
Q.
1.275πkfd(Ts T)(1 BH)0.678Rem0.438Prm0.619
m. 1.224πρfdᑞ (Ys Y)BM(1 BM)0.568Rem0.385Scm0.492 (1.220) An alternatively way is to use the empirical relationship derived from the experimental results by Renksizbulut and Yuen (1983). They correlated their experimental data on the heat transfer from a sphere with mass transfer at its surface by the following expression:
Nu (1.221)
Equation (1.221) yields the following expressions for the rates of heat and mass transfer from the surface of the spherical drop:
Q.
πkfd(TsT) (1.222)
and
m. πdρfᑞ(YsY) (1.223)
All the fluid properties in the last four equations that define the Nusselt, Prandtl, and Schmidt numbers, are the film properties defined by Eq. (1.136). The relative Reynolds and Peclet numbers Remand Pemare defined in terms of the gas-film viscosity as in (1.136) and the free-stream gas density,ρf. It is evident from an inspection of these equations that their functional form and coefficients are very similar. It has been confirmed (Sirignano, 1999) that the results of the two expressions do not differ substantially.
Correlations such as (1.220), (1.222), and (1.223) are frequently used in engineering computations in order to provide the necessary transport coefficients in models for the gas-phase flow and to determine the details of the processes of droplet motion, heating, and vaporization or burning.