1. Stokes' Law in Centrifugal Acceleration Fields
In standard gravity separation, droplet rising velocity is governed by terrestrial gravity (\(g = 9.81\ \text{m/s}^2\)). Inside a deoiling hydrocyclone, tangential feed injection converts available fluid pressure head into a high-speed rotational vortex, generating radial centrifugal accelerations \(G = \omega^2 r / g\) between \(1,000\ g\) and \(2,500\ g\). The radial terminal velocity \(v_r\) of an oil droplet of diameter \(d_p\) is:
\[ v_r = \frac{d_p^2 \left( \rho_w - \rho_o \right) \cdot (G \cdot g)}{18 \mu_w} \]
where \(\rho_w\) is produced water brine density, \(\rho_o\) is oil/condensate density, and \(\mu_w\) is water dynamic viscosity. Lighter oil droplets migrate inward to the low-pressure axial core and exit through the upstream reject orifice, while de-oiled water discharges out the downstream underflow.
2. Colman & Thew Hydrocyclone Scaling & Separation Efficiency
The cumulative grade separation efficiency \(\eta_{\text{hc}}\) for a given droplet size distribution is evaluated using the Rosin-Rammler / Weibull probability function:
\[ \eta_{\text{hc}} = 1 - \exp\left[ -0.693 \left( \frac{d_{50}}{d_{50,\text{cut}}} \right)^m \right] \]
where \(d_{50,\text{cut}}\) is the characteristic cut size (\(10\text{--}15\ \mu\text{m}\)) of the liner geometry, and \(m \approx 1.6\text{--}1.8\) is the separation sharpness index. The required total active liner count \(N_{\text{liners}}\) is:
\[ N_{\text{liners}} = \left\lceil \frac{Q_{\text{water}}}{q_{\text{liner}}} \right\rceil \]
3. Pressure Drop Ratio (PDR) & Reject Split Ratio Optimization
Hydrocyclones operate without moving parts; their separation efficiency depends strictly on maintaining the correct balance between the reject pressure drop and underflow pressure drop, defined by the Pressure Drop Ratio (PDR):
\[ \text{PDR} = \frac{P_{\text{inlet}} - P_{\text{reject}}}{P_{\text{inlet}} - P_{\text{underflow}}} \approx 1.5\text{ to }2.2 \]
To avoid oily core re-entrainment into the clean water underflow, the reject split ratio \(R_{\text{split}}\) must be controlled between \(1.5\%\) and \(3.0\%\) of the total volumetric feed flow.
4. Secondary Dissolved Gas Flotation (DGF / IGF) Flotation Cell Sizing
To eliminate fine dispersed droplets (\(5\text{--}20\ \mu\text{m}\)) leaving the hydrocyclone underflow, a secondary flotation cell introduces microscopic gas bubbles (\(30\text{--}70\ \mu\text{m}\)). The bubbles collide with and attach to hydrophobic oil droplets, lowering effective particle density and accelerating flotation. Flotation vessel volume is sized from hydraulic retention time (\(\text{HRT} = 15\text{--}25\ \text{min}\)):
\[ V_{\text{active}} = Q_{\text{underflow}} \cdot \left( \frac{\text{HRT}}{60} \right) \quad [\text{m}^3] \]
The final polished water effluent concentration is:
\[ \text{OIW}_{\text{final}} = \text{OIW}_{\text{hc,out}} \cdot \left( 1 - \eta_{\text{dgf}} \right) \quad [\text{ppm}] \]