Particle mass transfer coefficients

For the case where there is a mass transfer resistance in the fluid external to the particle (mass transfer coefficient hn), express the mass transfer rate in terms of the bulk concentration C , rather than the concentration Cts at the surface of the particle. 

Figure 2.2 Experimental gas-particle mass transfer coefficients. Adapted from Kunii, D. and Levenspiel, O., Fluidization engineering, 1991, with permission from Elsevier.

Table 2.2 Example calculation of gas-particle mass transfer coefficients.

K Langmuir constant (m moL ) k reaction rate constant (m moL s ) k overall adsorption rate parameter (s ) k intra-particle mass transfer coefficient (s )... 

Figure 5.3-6. Correlation of fluid/particle mass transfer coefficients in MSSR (after [10]).

Since the liquid is saturated with hydrogen, only the liquid-to-particle mass transfer coefficient and the intrinsic rate constant will be significant. In the case that the reaction is fast, the reaction rate will depend only on the liquid-sold mass transfer resistance. Since the particle are very small (10 micrometers), and the loading is moderate (0.8% mass), the Sherwood number will be that of lonely spheres, so Sh = 2. For this case we can take Sh = =4 [50], rather safely. 

The extraction of toluene and 1,2 dichlorobenzene from shallow packed beds of porous particles was studied both experimentally and theoretically at various operating conditions. Mathematical extraction models, based on the shrinking core concept, were developed for three different particle geometries. These models contain three adjustable parameters an effective diffusivity, a volumetric fluid-to-particle mass transfer coefficient, and an equilibrium solubility or partition coefficient. K as well as Kq were first determined from initial extraction rates. Then, by fitting experimental extraction data, values of the effective diffusivity were obtained. Model predictions compare well with experimental data and the respective value of the tortuosity factor around 2.5 is in excellent agreement with related literature data. 

The unknown parameters are the characteristic particle diameter, dp, the gas-to-particle mass transfer coefficient, k and the solid hold-up e,. 

Actually Sato et al. expressed their particle mass-transfer coefficients in terms of an enhancement factor representing the ratio of with two-phase flow to ks at the same liquid flow rate in single-phase flow. For pulsing and dispersed bubble flow this enhancement factor was found to be inversely proportional to liquid holdup j3, which in turn is a function of the two-phase parameter A or A (see Section IV,A,3,a). For comparison, the data for single-phase liquid flow are best represented by an equation of the same form as Eq. (115) but with a constant of 0.8. 

Pp is the density of the particles of packing material, kfp is the fluid to particle mass transfer coefficient, is the adsorption equilibrium constant of compoimd i. 

Boltzmann constant corrective growth crowding factor coordination number for an aggregate particle mass-transfer coefficient... 

Breakthrough curves obtained in a fixed-bed ion exchange process by Pansini et al. (1996) were interpreted by means of the model, the only parameters of which are a solid-liquid and an intra-particle mass transfer coefficient. The experimental results were obtained using a... 

Liquid particle mass transfer coefficients k are usually correlated as particle Sherwood number, depending on Ihe 0.5-0.7th power of the liquid Reynolds number and the 1/3 power of the liquid Schmidt number and in case of bubble flow additionally on a power ratio of gas to liquid Reynolds number. Also here, the dependence of k on the operating variables varies (see Fig. 14) with the flow pattern [26,9]. In trickle flow and bubble flow operation the Sherwood number is comparable to single phase flow, but increases rapidly in pulsing flow. At high gas velocities surface tension of liquid becomes important and under these conditions also Weber number is a significant correlation parameter [27]. 

The ratio of l> j /hbio represents an effective mass transfer coefficient, Thoms [5] report 38 measiuements in freshwater systems for which this ratio can be calculated. The average value is 130 cm/year and about 60% of the measurements exceed 1 cm/year and 90% exceed 0.1 cm/year. In shallow waters such as would be observed nearshore, Boudreau [7] provided correlations between biotiubation and depth that suggest a 2.54 and 29.5 cm/year effective particle mass transfer coefficient. A value of 1 cm/year appears to be a reasonably conservative estimate of the effective particle bioturbation mass transfer coefficient in a clean shallow system, such as the top of a nearshore sediment cap. A coefficient of the order of 1 cm/year or more has also been observed by Thibodeaux [8]. 

Particle Mass Transfer Coefficients. It is well known that the advantages of supercritical fluids in extractions are based on their optimal property values, which lie between those of gases and liquids. Thus, with high diffiisivity and low viscosity values, they are normally higher tlian those of liquids. Wakao and Kaguei s [45] correlation for mass transfer from particles is a good starting point. It has been successfully applied by a... 

Fluid to particle mass transfer coefficient ki is determined from fluid dynamic conditions as well as diffusion property of the fluid. 

Obviously this model is suitable for those cases where fluid-to-particle mass transfer is a dominant mass transfer resistance. In such cases Kra, can be replaced by fluid-to-particle mass transfer coefficient, ktoy. 

In packed beds, the main parameters of transport of adsorbates are the axial dispersion coefficient and the fluid-to-particle mass transfer coefficient. The other important parameter, the intraparticle diffusion coefficient, is not dependent on type of adsorption contactor and the treatment described in Chapter 4 can be applied. 

While these simplifications ignore disengagement of small particles, all with terminal settling velocities > U, particle recyle from the cyclones and non-isothermal behaviour, the model allows some estimation of the extent of the freeboard region contribution. Some profiles of gas concentration derived from Equation (41) are given in Figure 12 with typical values of f 0.2(2), C=0.4(74), Q /A from Equation (17) with Y=0.7, and gas-to-particle mass transfer coefficients, gp> from ref.(8). 

An equation identical in form to Equation 4.16a also arises if we describe the diffusion process by an ordinary differential equation (ODE) model using an equivalent (and empirical) "particle mass transfer coefficient" k. We have, for either uptake or release. 

Relation between Particle Mass Transfer Coefficient and Diffusivity... 

This correlation applies to the trickle-flow regime much higher values can be obtained at higher liquid and gas rates. The trickle-flow regime is usually for Gl < 3 X 10 g/cm h. A j factor correlation (Dharwadkar and Sylvester 1977) in the trickle-flow regime for the liquid-particle mass transfer coefficient kc is ... 

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