Proceedings of the 9th World Congress on Momentum, Heat and Mass Transfer (MHMT'23)
London, United kingdom – April 11 – 13, 2024
Paper No. ENFHT 330
DOI: 10.11159/enfht24.330

Correlation for the Mass Transfer Coefficient between Liquid andParticles in Multiparticle Systems
Ziming Wang1, Charalampos Christodoulou2, Luca Mazzei1*
1

University College London, Department of Chemical Engineering
Torrington Place, London WC1E 7JE, UK
uceczw4@ucl.ac.uk; l.mazzei@ucl.ac.uk (*corresponding author)
2
GlaxoSmithKline (GSK), Gunnels Wood Road, Stevenage SG1 2NY, UK
charalampos.x.christodoulou@gsk.com

Extended Abstract
To assess the mass transfer rate between liquid and particles, correlations involving the mass transfer coefficient 𝑘 or its
dimensionless counterpart, the Sherwood number Sh, are commonly employed [1-7]. In the context of multiparticle systems,
the presence of other particles affects the mass transfer of a specific particle, and therefore correlations must account for thiseffect. The available correlations lack rigorous theoretical support and do not always agree with experimental data [2-7]. In
this work, we propose a theoretically robust approach based on scaling and order of magnitude analysis. Using the definitionof 𝑘 and Fick's law, we estimate the order of magnitude of 𝑘 and Sh as follows:𝑘∆𝐶 ~ 𝐷 ∆𝐶 ⁄𝛿𝑐 → 𝑘~ 𝐷⁄𝛿𝑐 → Sh~ 𝑑𝑝 ⁄𝛿𝑐(1)
where 𝐷 is the solute molecular diffusivity, ∆𝐶 is the concentration difference between the particle surface and the bulk ofthe liquid, 𝛿𝑐 is the length scale for significant concentration changes near the particle surface, and 𝑑𝑝 is the particle diameter.
Due to the large Schmidt number (Sc) in the liquid, when the Reynolds number (Re) is not extremely small, the Peclet number(Pe) is much larger than unity. In such cases, a thin concentration boundary layer forms around the particle, 𝛿𝑐 representing
its thickness [8]. Given that 𝛿𝑐 is very small, we can simplify the mass balance equations for the liquid mixture and the solute
within this boundary layer. Scaling these simplified equations yields a relationship between 𝛿𝑐 and the velocity scale 𝑢𝑐 at
the outer edge of the concentration boundary layer; this relationship reads:
1/2
𝛿𝑐𝐷~
()
𝑑𝑝𝑢𝑐 𝑑𝑝(2)

Now, we relate 𝛿𝑐 and 𝑢𝑐 to the length and velocity scales of the velocity field around the particles, denoted as 𝛿𝑣 and𝑢𝑣 , respectively. Due to the large Sc and to Pe being far larger than Re, we can assume that 𝛿𝑐 is far smaller than 𝛿𝑣 .
Employing the scaling method, we can write:𝛿𝑐𝑢~ 𝑐𝛿𝑣 𝑢 𝑣

𝛿𝐷𝛿

𝑝𝑣 𝑝𝑝

1/3

→ 𝑑𝑐 ~ (𝑢 𝑑 𝑑𝑣 )(3)

To estimate 𝛿𝑣 and 𝑢𝑣 , we employ the order of magnitude analysis applied to the drag force acting on a particle (𝐹𝑝 ).
For a uniform suspension at equilibrium, 𝐹𝑝 is related to the unhindered terminal velocity (𝑢𝑡 ) through the equation [9]:𝜋1

𝐹𝑝 = ( 4 𝑑𝑝2 ) (2 𝜌𝑒 𝑢𝑡2 ) 𝐶𝐷𝑡(4)

where 𝜌𝑒 is the liquid density and 𝐶𝐷𝑡 is the particle drag coefficient characterized by 𝑢𝑡 . To calculate 𝐶𝐷𝑡 , we employ thecorrelation of Dallavalle [10]:2−1/2𝐶𝐷𝑡 = (0.63 + 4.8Ret ) , Ret ≡ 𝜌𝑒 𝑢𝑡 𝑑𝑝 ⁄𝜇𝑒
where 𝜇𝑒 is the liquid viscosity. 𝑢𝑡 can be related to the superficial velocity (𝑢) by the equation [11]:𝑢 = 𝑢𝑡 𝜀 𝑛 with 𝑛 =(5)3/4

4.8+2.4∙0.175Ret
3/41+0.175Ret

Moreover, 𝐹𝑝 can be estimated as:

ENFHT 330-1(6)𝑢

𝐹𝑝 ~ (𝜇𝑒 𝛿 𝑣 ) 𝜋𝑑𝑝2𝑣(7)

For multiparticle systems, assuming 𝑢𝑣 has the same order of magnitude as the interstitial velocity 𝑢⁄𝜀 and using Eqs. (2) to(7), we obtain:
1/3
𝛿𝑐𝐷
𝜀𝑛~2(), Pe ≫ 1𝑡𝑑𝑝
𝑢𝑑𝑝 Ret 𝐶𝐷(8)Given the application of concentration boundary layer theory, Pe must be significantly larger than unity. Finally, utilizing
Eq. (1) and introducing a constant 𝐶 (expected to be of unit order of magnitude), the correlation for Sh reads:
𝐶2/3

Sh = 𝜀 −2𝑛/3 (0.63Re + 4.8Re1/2 𝜀 𝑛/2 ) Sc1/3 , Re ≡ 𝜌𝑒 𝑢𝑑𝑝 ⁄𝜇𝑒 , Pe ≫ 1(9)2
The constant 𝐶 is estimated by matching Eq. (9) with experimental data from packed and fluidized beds [12-19]. The
obtained value indeed has unit order of magnitude, and the correlation aligns well with experimental data, errors being lessthan 30%. These results affirm the applicability of the newly proposed approach.

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ENFHT 330-2