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.
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Abbreviations
- b :
-
parameter in velocity distribution function (Eq. (28))
- c K :
-
factor in Kolmogoroff spectrum law (Eq. (20))
- D t(r p ) m2/s:
-
characteristic dispersivity at particle radius scale (Eq. (13))
- E(k, t) m3/s2 :
-
energy spectrum as function of k and t (Eq. (16))
- E K (k) m3/s2 :
-
energy spectrum as function of k in Kolmogoroff-region (Eq. (20))
- E p :
-
dimensionless mean kinetic energy of a colliding particle (Eq. (36))
- 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))
- 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))
- g m/s2 :
-
gravitational acceleration
- I cp :
-
dimensionless collision intensity per particle (Eq. (38))
- I cv :
-
dimensionless volumetric collision intensity (Eq. (39))
- k l/m:
-
reciprocal of length scale of velocity fluctuations (Eq. (17))
- K :
-
dimensionless viscosity (Eq. (13))
- n(2):
-
dimensionless particle collision rate (Eq. (12))
- n′(r) l/s:
-
particle exchange rate as function of distance from observatory particle center (Eq. (7))
- r m:
-
vector describing position relative to observatory particle center (Eq. (2))
- r m:
-
scalar distance to observatory particle center (Eq. (3))
- r pm:
-
particle radius (Eq. (1))
- s :
-
dimensionless time (Eq. (10))
- SC kg/ms3 :
-
Severity of collision (Eq. (1))
- t s:
-
time (Eq. (2))
- u(r, t) m/s:
-
velocity vector as function of position vector and time (Eq. (2))
- u(r, t) m/s:
-
magnitude of velocity vector as function of position vector and time (Eq. (3))
- u r(r, t) m/s:
-
radial component of velocity vector as function of position vector and time (Eq. (3))
- u r (r, t) m/s:
-
magnitude of radial component of velocity vector as function of position vector and time (Eq. (3))
- u ϕ (r, t) m/s:
-
latitudinal component of velocity vector as function of position vector and time (Eq. (3))
- u ϕ (r, t) m/s:
-
magnitude of latitudinal component of velocity vector as function of position vector and time (Eq. (3))
- u ψ (r, t) m/s:
-
longitudinal component of velocity vector as function of position vector and time (Eq. (3))
- u ψ (r, t) m/s:
-
magnitude of longitudinal component of velocity vector as function of position vector and time (Eq. (3))
- u gsm/s:
-
superficial gas velocity
- u′(r) m/s:
-
root mean square velocity as function of distance from observatory particle center (Eq. (3))
- u′r(r) m/s:
-
root mean square radial velocity component as function of distance from observatory particle center (Eq. (4))
- u′ϕ (r) m/s:
-
root mean square latitudinal velocity component as function of distance from observatory particle center (Eq. (4))
- u′ψ (r) m/s:
-
Root mean square longitudinal velocity component as function of distance from observatory particle center (Eq. (4))
- w′(x) :
-
dimensionless root mean square velocity as function of dimensionless distance from observatory particle center (Eq. (11))
- V pm3 :
-
particle volume (Eq. (36))
- w′(2):
-
dimensionless root mean square collision velocity (Eq. (34))
- w * :
-
parameter in boundary layer velocity equation (Eq. (24))
- x :
-
dimensionless distance to particle center (Eq. (9))
- x * :
-
value of x where G Band G K-curves touch (Eq. (32))
- x K :
-
dimensionless micro-scale (Kolmogoroff-scale) of turbulence (Eq. (15))
- α :
-
volumetric particle hold-up
- ε m2/s3 :
-
energy dissipation per unit of mass
- ν m2/s:
-
kinematic viscosity
- ρ kg/m3 :
-
density
- Φ(r) m3/s:
-
fluid-exchange rate as function of distance to observatory particle center
- φ :
-
Latitudinal co-ordinate (Eq. (5))
- ψ :
-
Longitudinal co-ordinate (Eq. (5))
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Beverloo, W.A., Tramper, J. Intensity of microcarrier collisions in turbulent flow. Bioprocess Engineering 11, 177–184 (1994). https://doi.org/10.1007/BF00369627
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DOI: https://doi.org/10.1007/BF00369627