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The hydrodynamic and thermal characterization of a yield stress fluid in stirred tanks equipped with simple helical ribbons with two stages

  • Simulation, Optimization & Identification
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Abstract

The objective of this paper is to characterize the hydrodynamic and thermal behaviors of yield stress fluids within a cylindrical agitated vessel equipped with simple helical ribbon stirrers (SHR) (one and two-stages) by means of the numerical simulation approach. For this purpose, a computational fluid dynamic simulation using the 3D finite volume technique has been carried out to solve the continuity, momentum and thermal energy equations. In this study, we have analyzed the Oldroyd and Reynolds numbers effects on the hydrodynamic and thermal behaviors for the two mentioned stirrers types. In addition, the influence of the impeller width and its clearance from the vessel wall on the velocity and thermal fields has been investigated. Velocity and thermal fields’ visualization has been presented in different (r–z) and (r–θ) planes. Moreover, the power constant and Nusselt number are correlated by a relationship relating the physical properties and the geometric ratios defining the SHR.

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Abbreviations

\({\text{a}} = \frac{{\uplambda }}{{{\uprho{\text{ C}}}_{\text{p}} }}\) :

Thermal diffusivity (m s−2)

C = (D − da)/D:

Dimensionless clearance

\({\text{C}}_{\text{p}}\) :

Specific heat capacity (J Kg−1 K−1)

D:

Vessel diameter (m)

da :

Impeller diameter (m)

dr :

Rotor diameter (m)

g:

Acceleration of gravity (m s−2)

H:

Liquid height in the tank (m)

h:

Heat transfer coefficient (W/m2K)

hi :

Impeller height (m)

µ:

Viscosity of Bingham fluid (Pa s)

µa :

Apparent viscosity (Pa s)

N:

Rotation speed of the agitator (tr/s)

Nr :

Number of impeller ribbon

\({\text{R}} = \frac{{{\text{d}}_{\text{a}} }}{2}\) :

Impeller radius (m)

P1 :

Pressure (Pa)

Pw :

Power (w)

p:

Pitch of the ribbon (m)

ts :

Time (s)

\({\text{V}}_{\text{c}} = 2{\uppi{\text{ NR}}}\) :

Characteristic velocity (m s−1)

\({\text{V}}^{\prime}\) :

Vessel volume

w:

Ribbon width (m)

\({\bar{\uptheta }}\) :

Average temperature (K)

\({\uptheta }_{\text{i}}\) :

Initial internal temperature (K)

\({\uptheta }_{\text{w}}\) :

The vessel wall temperature (K)

\({\uptau }\) :

Shear stress (Pa)

τ0 :

Yield stress (Pa)

\({\dot{\upgamma }}\) :

Flow shear rate (s−1)

λ:

Thermal conductivity (W/mK)

ρ:

Fluid density (Kg m−3)

A = Po * Re:

Power constant

\({\text{dv}} = {\text{rdrd}}\uptheta {\text{dz}}\) :

Element volume

\({\text{F}}_{\text{o}} = \frac{2}{{{\uppi {\text{Pe}}}}}{\text{t}}\left( {\frac{{{\text{d}}_{\text{a}} }}{\text{D}}} \right)^{2}\) :

Fourier number

\({\text{Fr}} = \frac{{ 2\left( {{\uppi {\text{N}}}} \right)^{2} {\text{D}}}}{\text{g}}\) :

Froude number

G:

Viscous dissipation function

m:

Regularization parameter

\({\text{Nu}} = \frac{\text{hD}}{{\uplambda }}\) :

Nusselt number

\({\text{Od}} = \frac{{{\uptau }_{0} {\text{D}}}}{{2 {\upmu {\text{V}}}_{\text{c}} }}\) :

Oldroyd number

\({\text{Pe}} = {\text{RePr}} = \frac{{{\text{Nd}}_{\text{a}}^{2} }}{\text{a}}\) :

Peclet number

\({\text{P}}_{\text{o}} = \frac{{{\text{P}}_{\text{w}} }}{{{\uprho \text{N}}^{3} ({\text{d}}_{\text{a}} )^{5} }}\) :

Power number

\({ \Pr } = \frac{{{{\upmu \text{C}}}_{\text{p}} }}{{\uplambda }}\) :

Prandtl number

\({\text{P}}_{\text{re}} = \frac{{{\text{P}}_{1} }}{{{{\uprho }}\left( {{{\uppi \text{ND}}}} \right)^{2} }}\) :

Dimensionless pressure

(r, \({\uptheta }\), z):

Dimensionless co-ordinates (dimensional coordinates divided by D/2)

\({\text{Re}} = \frac{{{{\uprho N d}}_{\text{a}} ^{2} }}{\upmu }\) :

Reynolds number

\({\bar{\text{T}}} = \frac{{{\bar{\uptheta }} - {\uptheta }_{\text{i}} }}{{{\uptheta }_{\text{w}} - {\uptheta }_{\text{i}} }}\) :

Average dimensionless temperature

\({\text{t}} = 2{\uppi \text{Nt}}_{\text{s}}\) :

Dimensionless time

\({\text{V}}_{\text{i}}\) :

Viscosity correction factor

\({\vec{\text{V}}}({\text{U}}, {\text{V}}, {\text{W}})\) :

Dimensionless velocity vector (dimensional velocity divided by \(2{\uppi{\text{ NR}}}\))

\({\upeta }_{\text{a}} = \frac{{\upmu }_{\text{a}}}{\upmu }\) :

Dimensionless apparent viscosity

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Authors and Affiliations

  1. National Engineering School of Sfax, B.P. W3038, Sfax, Tunisia

    A. Gammoudi, A. Ayadi & M. Baccar

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  1. A. Gammoudi
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  2. A. Ayadi
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  3. M. Baccar
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Correspondence to A. Gammoudi.

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Gammoudi, A., Ayadi, A. & Baccar, M. The hydrodynamic and thermal characterization of a yield stress fluid in stirred tanks equipped with simple helical ribbons with two stages. Meccanica 52, 1743–1766 (2017). https://doi.org/10.1007/s11012-016-0506-z

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