Mass transfer coefficient single particle

Parameters a and b are related to the diffusion coefficient of solutes in the mobile phase, bed porosity, and mass transfer coefficients. They can be determined from the knowledge of two chromatograms obtained at different velocities. If H is unknown, b can be estimated as 3 to 5 times of the mean particle size, where a is highly dependent on the packing and solutes. Then, the parameters can be derived from a single analytical chromatogram. 

For heat and mass transfer through a stationary or streamline fluid to a single spherical particle, it has been shown in Volume 1, Chapter 9, that the heat and mass transfer coefficients reach limiting low values given by ... 

Table 2.2 gives examples of mass transfer coefficients determined from both the single particle and fixed bed models for the evaporation of water from particles of the same diameter and density as in Table 2.1, assuming the diffusivity of water in air to be 3 x 10 m s h Once... 

The double lines in Figure 3.44 represent the Sh number based on the mass transfer coefficient, in the case of a single-particle fall in water, for three different particle densities (Harriot, 1962). This value is considered to be the minimum mass-transfer coefficient in liquid-solid films in agitated vessels. Taking into account the fact that the actual Sh value in an agitated vessel is 1.5 -8 times its minimum value, it is apparent that the mass transfer coefficients are much higher in the case of agitated vessels. 

Single Sphere Model II (Equations 4, 5, 8, 9 and 10 in reference 6) In this model allowance is made for the resistance to mass transfer offered by the surface film surrounding the herb particles. The mass transfer coefficient kf was obtained from correlations proposed by Catchpole et al (8, 9) for mass transfer and diffusion into near-critical fluids. An average of the binary diffusivities of the major essential oil components present was used in calculating kf (these diffusivities were all rather similar because of their similar structures). 

In the limiting case of mass transfer from a single sphere resting in an infinite stagnant liquid, a simple film-theory analysis122 indicates that the liquid-solid mass-transfer coefficient R s is equal to 2D/JV, where D is the molecular diffusivity of the solute in the liquid phase and d is the particle diameter. In dimensionless form, the Sherwood number... 

Term 1 is a function of temperature and pressure only. The diffusivity always increases with increasing temperature for both gas and liquid systems. However, the kiinematic viscosity v increases with temperature (v oc T 3/2 for gases and decreases exponentially with temperature for liquids. Term 2 is a function of flow conditions and particle size. Consequently, to increase k and thus the overall rate of reaction per unit surface area, one may either decrease the particle size or increase the velocity of the fluid flowing past the particle. For this particullar case of flow past a single sphere , we see that if the velocity is doubled, the mass transfer coefficient and consequently the rate of reaction is increased by a factor of... 

In order to assess transport mechanisms due to convection various correlation for heat and mass transfer coefficients in a packed bed have been derived. For the present application the transfer coefficient in the bed is related to the transfer coefficient of a single particle in a gas flow according to [15]. Due to the outflow of the gases during pyrolysis and char conversion the calculated transfer coefficient is decreased, thus Stefan correction is included to calculate the transfer coefficient at a finite flow over the boundary. 

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. 

Heat and mass transfer coefficients in a fluidised bed lie between the values for a packed bed and those for a single particle. The fundamental pattern of the Nusselt or Sherwood number as functions of the Reynolds number is illustrated in Fig. 3.40. In this the Nusselt number Nu = adP/A or the Sherwood number Sh = l3dP/rj and the Reynolds number Re = wmdP/v are all formed with the particle diameter, which for non-spherical particles is the same as the equivalent sphere diameter according to (3.273). In the Reynolds number wm is the mean velocity in the imaginary empty packing. 

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. 

Figures 16.23a to d compare experimental profiles of mixtures of the enantiomers of 1-indanol on cellulose tribenzoate with those calculated with the GMS-GRM model of these authors [57]. For the numerical calculations, they assmned that surface diffusion plays the dominant role in mass transfer across the particles and neglected the contribution of pore diffusion to the fluxes. Unfortunately, it was impossible independently to measure or even estimate the surface diffusion parameters. So, the numerical values of the surface diffusion coefficients needed for the calculation were estimated by minimizing the discrepancies between the measured and the calculated band profiles i.e., by parameter adjustment). Yet, it is impressive that, using a unique set of diffusion coefficients, it was possible to calculate band profiles of single components of binary mixtures in the whole range of relative composition, for loading factors between 0 and 10%.

In principle, the mass transfer coefficient for a single liquid spherical droplet in an immiscible liquid flowing with velocity z/ll past the spherical droplet can be calculated from a Froessling-type equation, which was originally derived for a solid particle (see Atiemo-Obeng, Penney, and ArmenanteJ ) ... 

Maximum stable drop diameter, m Impeller diameter, m Diffusivity of dissolved component or reactant in liquid, m /s Gravitational acceleration, m/s Height of liquid in vessel, m Mass transfer coefficient, m/s Mass transfer coefficient for a single spherical droplet immersed in a liquid flowing at constant velocity past the droplet, m/s Mass of liquid, kg Rate of mass transfer of solute or reactant, kg/s Impeller speed, rotations/s Minimum speed to just suspend solid particles in vessel, rotations/s Minimum impeller speed to completely incorporate dispersed phase into continuous phase in liquid-liquid systems, rotations/s Power dissipation, W Time, s... 

The mass transfer coefficient to a single solid spherical particle immersed in a liquid flowing with velocity rsL past the particle can be calculated from ... 

The Mass Transfer Coefficient 771 Ma.ss Transfer Coefficient 773 Correlations for the Mass Traiurfer Coefficient Mass Transfer to a Single Particle 776 Ma.ss Transfer-Umited Reactions in Packed Beds 780 Robert the Worrier 783... 

Wesselingh, J. A. Bollen, A. M. Single Particles, Bubbles and Drops Their Velocities and Mass Transfer Coefficients, Trans J ChemE, Vol 77, Part A, March (1999) 89-96... 

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]. 

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