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New one-dimensional hydrodynamics of circulating fluidized bed risers

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Abstract

A one-dimensional analysis of the hydrodynamics of circulating fluidized bed risers is performed following brand new mass and momentum balances for a gas–solids flow that the author has recently proposed. This analysis also includes a characteristic slip velocity between the phases derived from classical particle mechanics (the single particle equation). The predictions of the pressure drop and the solids holdup axial profiles, and the dense bed height are successfully compared with a rather limited amount of data.

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Abbreviations

\(A_p \) :

Projected area of the particle

\(C_D \) :

Drag coefficient

\(d_p \) :

Particle diameter

\(d_{ps}\) :

Mean surface diameter of the particles population

\(F_{D,p}\) :

Drag force on a single particle \(\gamma v^{2}\)

\(F_D^{\prime }\) :

Drag force per unit volume

\(F_{D,bed} \) :

Total drag force along the bed

g :

Gravitational constant

G :

Gas mass flux

\(G_s\) :

Solids mass flux

\(H_d\) :

Height of the dense bed

\(m_p\) :

Mass of the particle

\(N_p\) :

Number of particles per unit volume

p :

Gas pressure

r :

Radial coordinate with origin the bed vertical axis

Re :

Reynolds number \(\frac{\rho vd_p }{\mu }\)

u :

Gas velocity \(\frac{G}{\rho \left( {1-\varepsilon _s } \right) }\)

\(u_p\) :

Particle velocity i.e., characteristic one dimensional velocity of the population of particles

\(u_s \left( {r,z} \right) \) :

Local solids velocity

\(u_t\) :

Terminal velocity (for the mean surface diameter \(d_{ps} )\)

v :

Slip velocity i.e., characteristic one dimensional slip velocity of the population of particles derived from one dimensional Newton’s laws of motion, \(v=u-u_p \)

\(v_\gamma \) :

‘Friction’ velocity \(\sqrt{\frac{m_p \cdot g}{\gamma \left( {v,d_p } \right) }}\)

z :

Vertical coordinate (height), positive upwards and origin the gas distributor

\(\varepsilon _s\) :

Cross-sectional average of solids void fraction

\(\gamma \) :

Friction coefficient between gas and particles, \(\gamma =\gamma \left( {v,d_p } \right) \)

\(\rho \) :

Gas density

\(\rho _s\) :

Solids mass density

i :

Inlet

d :

Dense bed surface

o :

Outlet

\(\mathrm{R}\) :

Return line

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Authors and Affiliations

  1. Mechanical Engineering Department, Universidad de Zaragoza, Maria de Luna 3, 50018, Zaragoza, Spain

    Francisco J. Collado

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  1. Francisco J. Collado
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Correspondence to Francisco J. Collado.

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Collado, F.J. New one-dimensional hydrodynamics of circulating fluidized bed risers. Granular Matter 18, 78 (2016). https://doi.org/10.1007/s10035-016-0674-5

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  • DOI: https://doi.org/10.1007/s10035-016-0674-5

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