14 Numerical Methods for Coupled Population Balance Systems …
477
the conservation of mass for the fluid flow, which are modeled by the incompressible
Navier–Stokes equations
∂ t u − ∇ ·
ρ
η
∇u
+ (u · ∇) u + ∇p = f in (0, t end ) × Ω,
∇ · u = 0 in (0, t end ) × Ω.
(1.1)
In (1.1), t end (s) is a final time, Ω ⊂ R
3 is a bounded domain, which is assumed to be
constant in the whole time interval, u ( m /s) is the velocity field, p (Pa) is the pressure,
f ( m /s
2
) represents forces acting on the fluid, ρ ( kg /m
3
) is the density of the fluid, and
μ ( kg /s m) is the dynamic viscosity of the fluid. Often, the body forces possess the
form f = (0, 0, g)
T with g ( m /s
2
) being the gravitational acceleration.
The other equations of a PBS are usually coupled. These are equations for the
energy balance, where the unknown quantity is the temperature T (K), for the balance
of the molar concentration c ( mol /m
3
) of dissolved species, and for the balance of the
particle population density f ( 1 /kg m
3
) (the unit is for a particle population density
with the only internal coordinate mass, it is different in other situations).
The energy balance of the PBS has the form
∂ t T − D T ΔT + u · ∇T = F ener,growth (c, T , f ) in (0, t end ) × Ω,
(1.2)
where D T ( m
2
/s) is a diffusion coefficient, u is the velocity from (1.1), and the righthand side F ener,growth (c, T , f ) ( K /s) models the energy consumption or production in
the growth process of the crystals. Since the velocity is divergence-free, it holds that
u · ∇T = ∇ · (T u).
In a crystallization process, the dissolved material in the fluid is used in the growth
process of the crystals. The corresponding balance equation has the form
∂ t c − D c Δc + u · ∇c = F conc,growth (c, T , f ) in (0, t end ) × Ω.
(1.3)
Here, D c ( m
2
/s) is again a diffusion coefficient and F conc,growth (c, T , f ) ( mol /s m
3
) represents the consumption or production of dissolved material. We like to mention
that there are PBSs with a coupled system of equations of type (1.3) for several
concentrations, like in the modeling of precipitation processes, e.g., see [36].
The final part of a PBS is an equation for the particle population density. Assuming
that the number of internal or property coordinates is d int ≥ 1, then this equation
might read as follows
∂ t f + (u + u sed ) · ∇f + ∇ int · (G(c, T )f )
= F agg (u, c, T , f ) + F break (u, c, T , f ) in (0, t end ) × Ω × Ω int .
(1.4)
Here, Ω int is the d int -dimensional domain for the internal coordinates and u sed ( m /s)
is the sedimentation velocity, which is assumed to be divergence-free. The growth
term is assumed to be linear with the growth rate G(c, T ) ( kg /s), and ∇ int is the
477
the conservation of mass for the fluid flow, which are modeled by the incompressible
Navier–Stokes equations
∂ t u − ∇ ·
ρ
η
∇u
+ (u · ∇) u + ∇p = f in (0, t end ) × Ω,
∇ · u = 0 in (0, t end ) × Ω.
(1.1)
In (1.1), t end (s) is a final time, Ω ⊂ R
3 is a bounded domain, which is assumed to be
constant in the whole time interval, u ( m /s) is the velocity field, p (Pa) is the pressure,
f ( m /s
2
) represents forces acting on the fluid, ρ ( kg /m
3
) is the density of the fluid, and
μ ( kg /s m) is the dynamic viscosity of the fluid. Often, the body forces possess the
form f = (0, 0, g)
T with g ( m /s
2
) being the gravitational acceleration.
The other equations of a PBS are usually coupled. These are equations for the
energy balance, where the unknown quantity is the temperature T (K), for the balance
of the molar concentration c ( mol /m
3
) of dissolved species, and for the balance of the
particle population density f ( 1 /kg m
3
) (the unit is for a particle population density
with the only internal coordinate mass, it is different in other situations).
The energy balance of the PBS has the form
∂ t T − D T ΔT + u · ∇T = F ener,growth (c, T , f ) in (0, t end ) × Ω,
(1.2)
where D T ( m
2
/s) is a diffusion coefficient, u is the velocity from (1.1), and the righthand side F ener,growth (c, T , f ) ( K /s) models the energy consumption or production in
the growth process of the crystals. Since the velocity is divergence-free, it holds that
u · ∇T = ∇ · (T u).
In a crystallization process, the dissolved material in the fluid is used in the growth
process of the crystals. The corresponding balance equation has the form
∂ t c − D c Δc + u · ∇c = F conc,growth (c, T , f ) in (0, t end ) × Ω.
(1.3)
Here, D c ( m
2
/s) is again a diffusion coefficient and F conc,growth (c, T , f ) ( mol /s m
3
) represents the consumption or production of dissolved material. We like to mention
that there are PBSs with a coupled system of equations of type (1.3) for several
concentrations, like in the modeling of precipitation processes, e.g., see [36].
The final part of a PBS is an equation for the particle population density. Assuming
that the number of internal or property coordinates is d int ≥ 1, then this equation
might read as follows
∂ t f + (u + u sed ) · ∇f + ∇ int · (G(c, T )f )
= F agg (u, c, T , f ) + F break (u, c, T , f ) in (0, t end ) × Ω × Ω int .
(1.4)
Here, Ω int is the d int -dimensional domain for the internal coordinates and u sed ( m /s)
is the sedimentation velocity, which is assumed to be divergence-free. The growth
term is assumed to be linear with the growth rate G(c, T ) ( kg /s), and ∇ int is the