Intensity of microcarrier collisions in turbulent flow |
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Authors: | W A Beverloo J Tramper |
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Institution: | (1) Department of Food Science and Technology, Food and Bioprocess Engineering Group, Wageningen Agricultural University, P.O. Box 8129, 6700 EV Wageningen, The Netherlands |
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Abstract: | A model is developed, allowing estimation of the share of inelastic interparticle collisions in total energy dissipation for stirred suspensions. The model is restricted to equal-sized, rigid, spherical particles of the same density as the surrounding Newtonian fluid. A number of simplifying assumptions had to be made in developing the model. According to the developed model, the share of collisions in energy dissipation is small.List of Symbols
b
parameter in velocity distribution function (Eq. (28))
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c
K
factor in Kolmogoroff spectrum law (Eq. (20))
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D
t(r
p
) m2/s
characteristic dispersivity at particle radius scale (Eq. (13))
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E(k, t) m3/s2
energy spectrum as function of k and t (Eq. (16))
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E
K
(k) m3/s2
energy spectrum as function of k in Kolmogoroff-region (Eq. (20))
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E
p
dimensionless mean kinetic energy of a colliding particle (Eq. (36))
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E
cp
dimensionless kinetic energy exchange in a collision (Eq. (37))
-
G(x, s)
dimensionless energy spectrum as function of x and s (Eq. (16))
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G
B(x)
dimensionless energy spectrum as function of x for boundary region (Eq. (29))
-
G
K(x)
dimensionless energy spectrum as function of x for Kolmogoroff-region (Eq. (21))
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g m/s2
gravitational acceleration
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I
cp
dimensionless collision intensity per particle (Eq. (38))
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I
cv
dimensionless volumetric collision intensity (Eq. (39))
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k l/m
reciprocal of length scale of velocity fluctuations (Eq. (17))
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K
dimensionless viscosity (Eq. (13))
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n(2)
dimensionless particle collision rate (Eq. (12))
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n(r) l/s
particle exchange rate as function of distance from observatory particle center (Eq. (7))
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r m
vector describing position relative to observatory particle center (Eq. (2))
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r m
scalar distance to observatory particle center (Eq. (3))
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r
pm
particle radius (Eq. (1))
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s
dimensionless time (Eq. (10))
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SC kg/ms3
Severity of collision (Eq. (1))
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t s
time (Eq. (2))
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u(r, t) m/s
velocity vector as function of position vector and time (Eq. (2))
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u(r, t) m/s
magnitude of velocity vector as function of position vector and time (Eq. (3))
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u
r(r, t) m/s
radial component of velocity vector as function of position vector and time (Eq. (3))
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u
r
(r, t) m/s
magnitude of radial component of velocity vector as function of position vector and time (Eq. (3))
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u
(r, t) m/s
latitudinal component of velocity vector as function of position vector and time (Eq. (3))
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u
(r, t) m/s
magnitude of latitudinal component of velocity vector as function of position vector and time (Eq. (3))
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u
(r, t) m/s
longitudinal component of velocity vector as function of position vector and time (Eq. (3))
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u
(r, t) m/s
magnitude of longitudinal component of velocity vector as function of position vector and time (Eq. (3))
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u
gsm/s
superficial gas velocity
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u(r) m/s
root mean square velocity as function of distance from observatory particle center (Eq. (3))
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ur(r) m/s
root mean square radial velocity component as function of distance from observatory particle center (Eq. (4))
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u (r) m/s
root mean square latitudinal velocity component as function of distance from observatory particle center (Eq. (4))
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u (r) m/s
Root mean square longitudinal velocity component as function of distance from observatory particle center (Eq. (4))
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w(x)
dimensionless root mean square velocity as function of dimensionless distance from observatory particle center (Eq. (11))
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V
pm3
particle volume (Eq. (36))
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w(2)
dimensionless root mean square collision velocity (Eq. (34))
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w
*
parameter in boundary layer velocity equation (Eq. (24))
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x
dimensionless distance to particle center (Eq. (9))
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x
*
value of x where G
Band G
K-curves touch (Eq. (32))
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x
K
dimensionless micro-scale (Kolmogoroff-scale) of turbulence (Eq. (15))
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volumetric particle hold-up
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m2/s3
energy dissipation per unit of mass
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m2/s
kinematic viscosity
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kg/m3
density
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(r) m3/s
fluid-exchange rate as function of distance to observatory particle center
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Latitudinal co-ordinate (Eq. (5))
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Longitudinal co-ordinate (Eq. (5)) |
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Keywords: | |
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