Ranz-Marshall correlation for

For a reaction represented by A(g) + bB(s) —> produc1(g), derive the relation between time (t) of reaction and fraction of B converted (/B), if the particle is spherical with an initial radius R0, and the Ranz-Marshall correlation for kAg(R) is valid, where R is the radius at t. Other assumptions are given above. 

Assume that the particle is spherical and isothermal, that both gas-film mass transfer resistance and reaction resistance are significant, and that the Ranz-Marshall correlation for k g is applicable. Do not make an assumption about particle size, but assume the reaction is first-order. 

FIGURE 22.11 Particle Sherwood number (Sh) versus particle Reynolds number (Re). Comparison of results in gas-flowing solids-flxed bed contactor and Ranz-Marshall correlation for single sphere. 

In these equations, Tj is the temperature at coordinates (r, t) in the macrograin, pcp is the average value of the heat capacity per unit volume of the macroparticle, is the effective thermal diffiisivity in the macrograin and (—Affr) the enthalpy of polymerization. In Equation 2.141, the parameter hp is the average convective heat transfer coefficient, usually calculated from a Nusselt number correlation. Early works tended to use the well-known Ranz-Marshall correlation for evaporation from a droplet however, it has been... 

Experimental mass transfer data at low Reynolds numbers show a great deal of scatter, but most of the reported values fall somewhat bi ow the limit of 2.0 predicted from the Ranz-Marshall correlation. For examii>le, expressed in terms of Sherwood number, the correlation of Petrovic and iTiodos for gases is equivalent to... 

Ranz-Marshall correlation for spheres. Assuming a constant Prandtl number for air, the most relevant parameter is the Reynolds number, which is mainly determined by the relative velocity between the dispersed and the continuous phase. The transfer of heat between the phases is given by the product of Nusselt number and the local, driving temperature difference of droplet and gas. 

For a free-falling spherical particle of radius R moving with velocity u relative to a fluid of density p and viscosity p, and in which the molecular diffusion coefficient (for species A) is DA, the Ranz-Marshall correlation relates the Sherwood number (Sh), which incorporates kAg, to the Schmidt number (Sc) and the Reynolds number (Re) ... 

Correlations for heat transfer coefficient between a single sphere and surrounding gas have been proposed by many researchers (Table 5.2), for example, Whitaker,1584 and Ranz and Marshall,15051 among others. The correlation recommended by Whitaker is accurate to within 30% for the range of parameter values listed. All properties except jus should be evaluated at Tm. For freely falling liquid droplets, the Ranz-Marshall correlation 505 is often used. The correlations may be applied to mass transfer processes simply by replacing Nu and Pr with Sh and Sc, respectively, where Sh and Sc are the Sherwood number and Schmidt number, respectively. Modifications to the Ranz-Marshall correlation have been made by researchers to account... 

It is important to remind the reader that U is the velocity of the fluid phase seen by the particle, U - U is the slip velocity, dp is the particle diameter, and Vf is the kinematic viscosity of the fluid phase. Note that Eq. (5.33) depends on the particle velocity U and is valid in the zero-Stokes-number limit where U = U so that particles follow the fluid. The correlation in Eq. (5.31) is valid only for RCp < 1 and Sc > 200. For larger particle Reynolds numbers the following correlations can be used Sh = 2 -i- 0.724Rep Sc, which is valid for 100 < RCp < 2000, and Sh = 2 -i- 0.425RCp Sc, which is valid for 2000 < RCp <10. Among the other correlations available, it is important to cite the one proposed by Ranz Marshall (1952) for macroparticles Sh = 2.0 -i- O.bReJ Sc. These expressions assume that the fluid velocity U is known. For micron-sized (or smaller) particles moving in turbulent fluids for which only the ensemble-mean fluid velocity (Uf) is known, it is instead better to employ the mesoscale model derived by Armenante Kirwan (1989) Sh = 2.0 -i- 0.52(Re ) Sc, where Re = is the modi-... 

Thus, the Ranz and Marshall correlations for heat transfer may be written... 

Rate-process parameters estimation of kAg for spherical particle. The three rate-process parameters in the expressions for t(fB) (kAg,De, and jkAj),may each require experimental measurement for a particular situation. However, we consider one correlation for estimating kA for spherical particles given by Ranz and Marshall... 

Most of the data available to substantiate Equations 1 to 4 pertain to single droplets in still air. Therefore, in most cases, the Nusselt number is 2, and Froessling s correlation for the Nusselt number in terms of the Reynolds and Schmidt numbers is neither verified nor cast in doubt. Data of Ranz and Marshall (102) do afford substantiation in particular, they verify Equation 4. 

O. For liquids and gases, Ranz and Marshall correlation Nsh = - = 2.0 + 0.eNgNg AT dpVt uperP i-yRe R [E] Based on freely falling, evaporating spheres (see 5-20-C). Has been applied to packed beds, prediction is low compared to experimental data. Limit of 2.0 at low is too high. Not corrected for axial dispersion. [121][128] p. 214 [155] [110]... 

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... 

A standard correlation for heat transfer to a sphere is given by (Ranz and Marshall, 1952)... 

Bt = Cp(Tg,p — 7 )/(A/tv), where FFg.p is the fuel vapor mass fraction interpolated to the droplet location. For 7 > Jb, By is set equal to Bj. The Clausius-Qapeyron equilibrium vapor-pressure relationship is used to compute the fuel mass fraction at the droplet surface. In addition, convective correction actors (based on Ranz and Marshall correlations) are applied to obtain spray evaporation rates at high Reynolds numbers. Liquid properties are evaluated using the one third rule for reference mass fractions [28]. Advanced models for droplet evaporation accounting for nonequilibrium effects can also be incorporated in the above framework by altering the timescales associated with the droplet lifetime and the convective heating. 

The overall rate of mass transfer of adsorbate in the bed is affected by external transport from the bulk of the gas to the external surfaces of the adsorbent particles, axial dispersion and backmixing in the gas phase, and internal transport within the pores. External transport can be correlated by equations similar to those used for mass transfer in packed absorption columns, such as the Ranz-Marshall (1952) equation ... 

In a supersonic gas flow, the convective heat transfer coefficient is not only a function of the Reynolds and Prandtl numbers, but also depends on the droplet surface temperature and the Mach number (compressibility of gas). 154 156 However, the effects of the surface temperature and the Mach number may be substantially eliminated if all properties are evaluated at a film temperature defined in Ref. 623. Thus, the convective heat transfer coefficient may still be estimated using the experimental correlation proposed by Ranz and Marshall 505 with appropriate modifications to account for various effects such as turbulence,[587] droplet oscillation and distortion,[5851 and droplet vaporization and mass transfer. 555 It has been demonstrated 1561 that using the modified Newton s law of cooling and evaluating the heat transfer coefficient at the film temperature allow numerical calculations of droplet cooling and solidification histories in both subsonic and supersonic gas flows in the spray. 

Although some controversy exists on the correlation of Sherwood Number applicable to fluidized beds, well-defined combustion experiments support the use of the Ranz and Marshall (35) or Frossling (36) correlation with an approximate correction of mf to allow for the obstruction to diffusion by the inert particles surrounding the burning char particles (37). Thus... 

For spray dryers, the popular equation of Ranz and Marshall [132] is presented in Table 4.9 (Equations T9.13 and T9.14). They correlated data obtained for suspended drops evaporating in air. 

There are various expressions for the Sherwood and Nusselt numbers if the relative drop-gas velocity is nonzero, i.e., for forced convection. Widely used correlations are those by Ranz and Marshall [20]. These were obtained from experiments of vaporizing single-component drops at atmospheric pressure and moderate ambient temperatures with low transfer rates, that is, when B = B kQ and, therefore, Nmo = 5/jo = 2. These correlations are given by... 

On the other hand, for very small particles (Rep<< lor 0), mass transfer can be depicted by diffusion in an infinite quiescent medium. For such a situation, the limit of Shp=2 implicit in Ranz and Marshall s correlation suggests that with decreasing particle diameter, must increase in a hnearly reciprocal manner. 

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