(233b) Population Balance Equation for Calculation of the Inlet Distribution for Oil and Water Droplets in Continuous Gravity Separators

In this work, we use a population balance equation (PBE) to describe the distribution of oil and water droplets in the respective water- and oil-continuous phases in the turbulent inlet section of a continuous gravity separation unit. The volumes of the oil- and water-continuous layers in the inlet section are considered as ideal stirred tanks, thus the PBE for each layer can be expressed as df(r)/dt = C₊(r) – C_–(r) + B₊(r) – B_–(r), where the right-hand side expressions describe the birth (index +) and death (index -) integrals for both coalescence (C) and breakage (B). We use the particle radius as inner coordinate and formulate the PBE as a dimensionless equation (Grimes, 2012). We do not regard existence and uniqueness of solutions for the above PBE in this work, however, our studies indicate that the PBE converges rapidly to a unique solution for a distinct set of parameters, but independent of the initial condition.

Thus, the equilibrium solution to the dynamic PBE could represent the inlet (boundary) condition for a simple dynamic gravity separator model as introduced in (Backi & Skogestad, 2017). The model consists of three dynamic states, namely the levels of water- and overall liquid (oil + water) as well as the gas pressure. These dynamic states are controlled to nominal values by PI controllers via the respective outflows. In addition, simple steady-state droplet balance calculations solely based in Stokes’ Law are applied to the initial distribution of oil and water droplets. Hence, birth and death due to coalescence and breakage are neglected as the droplets move along the length of the separator towards the outlets; this will be subject to future work. The droplet size classes are static inside the separator and thus only sedimentation and creaming will be regarded.

Further assumptions include

• There are no liquid droplets in the gas phase and no gas droplets in the liquid phases
• No emulsion layer (dense-packed layer) is considered between the continuous water and oil phases; the droplets are regarded to directly leave into their respective bulk phases at the interface. This means that the rate of interfacial coalescence is smaller than the flux of droplets to the interface (reasonable if an effective demulsifier is properly dosed to the separator inlet stream)
• The horizontal velocity of the droplets inside the gravity separator is equal to the velocity of the respective phase (volumetric inflow divided by the respective cross sectional area); hence, no slip between dispersed droplets and continuous phase is assumed
• The velocity of the droplets in vertical direction is determined by Stokes’ Law without regarding a correction factor as e.g. described in (Richardson & Zaki, 1954)

By comparing horizontal and vertical velocities of each (constant) particle class, we can calculate the total volumes of particles going into their respective bulk phases or leaving the separator through the dispersed phase’s outlet.

Backi, C. J., & Skogestad, S. (2017). A Simple Dynamic Gravity Separator Model for Separation Efficiency Evaluation Incorporating Level and Pressure Control. Proceedings of the 2017 American Control Conference.Seattle: IEEE Control Systems Society.

Grimes, B. A. (2012). Population Balance Model for Batch Gravity Separation of Crude Oil and Water Emulsions. Part I: Model Formulation. Journal of Dispersion Science and Technology, 33, 578-590.

Richardson, J. F., & Zaki, W. N. (1954). The sedimentation of a suspension of uniform spheres under conditions of viscous flow. Chemical Engineering Science, 8, 65-73.