Application of 2-D population balance equation for modeling of an industrial spiral-type gravity coalescer

ABSTRACT

The main purpose of this article is to model an industrial spiral-type gravitational coalescer by applying population balance as the governing equation. In this model, considering the two-dimensional movement of crude oil flow in the coalescer, the mechanisms of differential sedimentation, laminar shear, and Brownian motion are considered to determine the collision rate of droplets. Then, the population balance equation was solved to determine the droplet size distribution, dehydration, and desalination efficiency using the class method. The simulation results were in good match with the gathered industrial data. Finally, the influence of interfacial tension of water and oil, the pressure drop of the mixing valve, oil temperature, oil density, and washing water flow rate on the process efficiency was analyzed. The modeling results indicated that increasing the interfacial tension of water and oil from 30 to 36 mN/m and the oil density from 800 to 900 kg/m³ will decrease the water removal efficiency by 4.35% and 14.18%, respectively. Also, increasing the temperature of oil from 300 to 340 K and the volume fraction of washing water from 3% to 15% improves the efficiency of separating water from crude oil by 9.55% and 9.95%, respectively.

KEYWORDS:

• Modeling
• mixing valve
• population balance
• gravitational coalescer
• coalescence
• collision

Statement of novelty

The process of separating brine from crude oil in an industrial gravity coalescer is modeled using the two-dimensional population balance equation under steady state conditions. In the proposed model, the effects of the mechanisms of differential sedimentation, Brownian motion and laminar shear on the collision and coalescence of droplets were considered. To confirm the accuracy of the presented model, the modeling results were compared with the data obtained from the NISOC desalination unit. Finally, the effect of some parameters such as interfacial tension of water and crude oil and density of crude oil on dehydration and desalination of crude oil was investigated.

Nomenclature

A	=	Hamker coefficient [J]
b	=	Coalescence rate empirical constants [-]
B	=	Interfacial force constant [N m2]
Ca	=	Capillary number [-]
d	=	Droplet diameter [m]
eij	=	Collision efficiency [-]
f (v,w)	=	Daughter drop size distribution function [-]
g	=	acceleration Gravity [m s−2]
g (d)	=	Breakage frequency [s−1]
K	=	Adjustable parameter [-]
kB	=	Boltzmann’s constant [J K−1]
m	=	The number of droplets formed in each break of a larger droplet [-]
n(v,x)	=	Continuous number density [m−3]
Ni	=	Discrete number density [m−3]
P	=	Pressure [Pa]
Pe	=	Peclet number [-]
Re	=	Reynolds number [-]
Stk	=	Stokes number [-]
tres	=	Resident time [s]
T	=	Temperature [K]
u	=	Velocity [m s−1]
v	=	Droplet volume [m3]
w	=	Droplet volume [m3]
We	=	Weber number [-]
x	=	Length [m]
y	=	Length [m]
Greek symbols	=
$\mathit{\beta }$	=	Coalescence rate [m3 s−1]
$\dot{\gamma }$	=	Shear rate [s−1]
$\mathit{\theta }$	=	Collision frequency [s−1]
$\mathit{\lambda }$	=	Kolmogorov length scale [m]
$\mathit{\mu }$	=	Dynamic viscosity [Pa s]
$\mathit{v}$	=	Kinematic viscosity [m2 s−1]
$\mathit{\xi }$	=	Rate of energy dissipation [m2 s−3]
$\mathit{\rho }$	=	Density [kg m−3]
$\mathit{\sigma }$	=	Interfacial tension [N m−1]
$\mathit{\tau }$	=	Contact time [s]
$\mathit{\phi }$	=	Water fraction in emulsion [-]
Superscript	=
Br	=	Brownian force
Ds	=	Differential sedimentation
Ls	=	Laminar shear
T	=	Turbulent force
Subscripts	=
i, j, k	=	Class number index
$\mathrm{c}$	=	Continuous phase
crit	=	Critical
d	=	Dispersed phase
eq	=	Equivalent
s	=	Settling

Acknowledgments

The research of the corresponding author is supported by a grant from Ferdowsi University of Mashhad (N. 3/57428)

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors reported there is no funding associated with the work featured in this article.