52
principal groups: models that use characteristic flow parameters for up- or downscaling of a median diameter and models that calculate a complete droplet size
distribution through mechanistic modelling of the flow. Both groups differ widely
with regard to the computational effort but also with regard to the level of detail of
the result.
4.3.1 Scaling-Based Models Using Dimensionless Numbers
Probably the most widely used scaling approach in the oil spill community is the
modified Weber number scaling by Johansen et al. (2013). Based on the model of
Wang and Calabrese (1986) for stirred-tank reactors, they proposed the implicit
scaling law for the volume median diameter d v50
d
D
A
v
We
50
3 5
= ⋅
∗− /
(4.7)
with the modified Weber number We
*
We
We
Vi
v
∗
=
+ ⋅ ⋅
1
50
1
3
B
d
D
(4.8)
where We
l
l
l
=
⋅ ⋅
D
u
ρ
σ
2
is the Weber number,Vi
We
= Re
the viscosity number and A
and B are empirical coefficients. Those coefficients were calculated using the dataset of Brandvik et al. (2013) to be A = 15 and B = 0.8; a later work based on a larger
dataset updates these to A = 24.8 and B = 0.08 (Socolofsky et al. 2015). The data
used to calibrate the model span a range of approximately 10
3
≤ We ≤ 10
4
and jets
with and without additional dispersant injection. The authors claim the model to be
applicable to multiphase discharges of oil and gas as well by adjusting the exit
velocity u l to account for the gas phase and its buoyancy.
A second model, called the unified droplet size model, by Li et al. (2017) scales
the volume median diameter d v50 with a combination of Weber and Ohnesorge number in an explicit equation:
d
D
r
p
q
v
Oh We
50
1 10
= ⋅ + ⋅
(
) ⋅
(4.9)
where r = 14.05, p = 0.460 and q = − 0.518 are empirically derived coefficients for
a liquid-liquid jet. The model is proposed for determining both the d v50 of a jet and
of wave entrainment at the surface. The coefficients p and q were determined based
K. Malone et al.
principal groups: models that use characteristic flow parameters for up- or downscaling of a median diameter and models that calculate a complete droplet size
distribution through mechanistic modelling of the flow. Both groups differ widely
with regard to the computational effort but also with regard to the level of detail of
the result.
4.3.1 Scaling-Based Models Using Dimensionless Numbers
Probably the most widely used scaling approach in the oil spill community is the
modified Weber number scaling by Johansen et al. (2013). Based on the model of
Wang and Calabrese (1986) for stirred-tank reactors, they proposed the implicit
scaling law for the volume median diameter d v50
d
D
A
v
We
50
3 5
= ⋅
∗− /
(4.7)
with the modified Weber number We
*
We
We
Vi
v
∗
=
+ ⋅ ⋅
1
50
1
3
B
d
D
(4.8)
where We
l
l
l
=
⋅ ⋅
D
u
ρ
σ
2
is the Weber number,Vi
We
= Re
the viscosity number and A
and B are empirical coefficients. Those coefficients were calculated using the dataset of Brandvik et al. (2013) to be A = 15 and B = 0.8; a later work based on a larger
dataset updates these to A = 24.8 and B = 0.08 (Socolofsky et al. 2015). The data
used to calibrate the model span a range of approximately 10
3
≤ We ≤ 10
4
and jets
with and without additional dispersant injection. The authors claim the model to be
applicable to multiphase discharges of oil and gas as well by adjusting the exit
velocity u l to account for the gas phase and its buoyancy.
A second model, called the unified droplet size model, by Li et al. (2017) scales
the volume median diameter d v50 with a combination of Weber and Ohnesorge number in an explicit equation:
d
D
r
p
q
v
Oh We
50
1 10
= ⋅ + ⋅
(
) ⋅
(4.9)
where r = 14.05, p = 0.460 and q = − 0.518 are empirically derived coefficients for
a liquid-liquid jet. The model is proposed for determining both the d v50 of a jet and
of wave entrainment at the surface. The coefficients p and q were determined based
K. Malone et al.