Mass transfer from spherical particles in tandem arrangement in Visco-Plastic Fluids S. Al-Najjar 1,2, N. Nirmalkar 1 and M. Barigou1

1School of Chemical Engineering, University of Birmingham, UK 2Department of Chemical Engineering, Nahrain University, Baghdad, Iraq

shahadgzs@gmail.com

n =

1.8

n

= 0

.3

This work was funded by Ministry of higher education and

scientific research in Iraq (MOHESR) and Nahrain

University

Acknowledgement Twin vortices are observed in between the two spheres (Fig. 2)

Yielded zone expands with increasing Reynolds number (Fig. 3)

Concentration field shrinks at higher Reynolds numbers (Fig. 4)

Classical inverse dependence of drag coefficient with modified Reynolds number is observed (Fig. 5)

Shear-thinning promotes mass transfer whereas shear-thickening impedes it (Figs 6 & 7)

Fig. 5 Variation of drag coefficient with modified Reynolds number (Re**) at Bingham number (Bn) = 10

Objectives

Down stream cylinder

Up stream cylinder

Computational Domain

Line of symmetry

𝑫∞

𝑳

𝒛

𝒓

shear-thinning yield-stress

fluid

g 𝑈. 𝑛 = 0, 𝑐 = 1

𝑈𝑧 = 𝑈𝑎𝑣𝑔 & 𝑐 = 0

𝑝 = 0, 𝑛. 𝛻𝑐 = 0

Fig. 1 Flow and computational domain

Governing Equations, Boundary Conditions & Dimensionless Numbers

Fig. 3 Yielded and unyielded regions at Bn = 10 Fig. 2 Typical streamline contours at Bn = 10 Fig. 4 Typical concentration contours at Bn = 10

Results

Conclusions

• Continuity

𝛻. 𝑈 = 0

• Equations of motion

𝑈. 𝛻 𝑈 = −𝛻𝑝 +1

𝑅𝑒𝛻. 𝜎 where; 𝜎 = 𝜎𝑜 + 𝑘𝛾 𝑛

• Mass transport equation 𝑈. 𝛻 c =1

𝑅𝑒 × 𝑆𝑐 𝛻2𝑐

• Reynolds number : 𝑅𝑒 = 𝜌 𝑈𝑎𝑣𝑔

2−𝑛 𝑑𝑛

𝑘

• Bingham number : 𝐵𝑛 =𝜎°

𝑘

𝑑

𝑈𝑎𝑣𝑔

𝑛−1

• Schmidt number : 𝑆𝑐 = 𝑘

𝜌 𝐷 𝑈𝑎𝑣𝑔

𝑑

𝑛−1

• Modified Re number : 𝑅𝑒∗∗ = 𝑅𝑒

1+𝐵𝑛

• Sherwood number : 𝑆ℎ𝑎𝑣𝑔 = 1

𝑆 𝑆ℎ𝐿 𝑑𝑆𝑆

Mass transfer from two spherical particles settling in visco-plastic fluids placed

in tandem arrangement has been investigated over a wide range of conditions:

1≤ Re ≤150; 1≤ Sc ≤100; 0≤ Bn ≤10; 0.3≤ n ≤1.8

To investigate flow and mass transfer from two spherical particles in a shear-

thinning yield-stress flow stream.

To visualize streamlines and mass transfer contours.

To report reliable values of drag-coefficient as a function of pertinent

dimensionless numbers, i.e., modified Reynolds number (Re**) and Flow

behaviour index (n).

To report the average Sherwood number as a function of modified Reynolds

number (Re**), flow behaviour index (n) and Schmidt number (Sc).

Fig. 6 Variation of average Sherwood number (Shavg) with modified Reynolds number (Re**) at Schmidt number (Sc) = 1

n =

1.8

n

= 0

.3

n =

1.8

n

= 0

.3

Fig. 7 Variation of average Sherwood number (Shavg) with modified Reynolds number (Re**) at Schmidt number (Sc) = 100

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