A mathematical description of the separation process of the watered oil-containing mixture has been proposed. The mathematical model is based on the determination of the speed of movement of oil globules in a constrained flow. The formula for determining the velocity of the globules is derived from theoretical provisions and experimental data. The key issue of simulation is determining the separation time. This characteristic is calculated as a time period from the moment of arrival of the portion of oil-containing mixture to the first chamber of the separator to the moment of formation of the continuous phase layer of the given thickness on the surface. The mathematical model enables us to determine the separation time taking into account geometric characteristics of the separation device at the high level of water content. Thus, a topical problem of improving the performance of the separation device without sacrificing the quality of the commercial oil is solved.
Computer Modeling / V.A. Zelenskiy, A.A. Sushin, A.I. Shchodro
indirect measuring of density the mixture watercut value is obtained [
1) Separation time is a period of time from the moment the first chamber of the separator is filled (up to a given level, not exceeding the overflow level) until full decomposition of the oil-containing mixture. This approach did not find practical use as it does not take into account the real velocity of demulsification process. 2) Separation time is a period of time from the moment of arrival of the portion of oil-containing mixture to the first chamber of oil and gas separator to the moment of its decomposition to the state determined by GOST P 51858-2002. This approach is more progressive than the first one, but does not take into account the continuous nature of OTCS operation. 3) Separation time is a period of time from the moment of arrival of the proportion of oil-containing mixture to the first chamber of the separator to the moment of formation of the continuous phase layer of the given thickness on the surface.
This is the preferred method to be used in the framework of this research.
There is no strict mathematical description of the stratification of the oil-containing mixture as Navier-Stokes equation is valid
just for laminar flow and some special cases [
6 where Δρ is the difference between dispersed phase and dispersed medium particles' density, g is the gravitational acceleration, d is the particle's diameter. Resistance force of the continuous medium:
Fc о d42 2о с , where ξ0 is the coefficient of hydraulic resistance of the continuous medium to the movement of a single particle in it, ω0 is the velocity of a single particle relative to the medium, ρc is the density of the continuous medium. Let us assume that the temperature at all points of a separation device chamber is the same. Then there is no thermal convection. If the particle's velocity in the medium is constant: ΔF = Fc the Reynolds criterion is determined by the following ratio: Reo 0d c , c where μc is the dynamic viscosity of the continuous medium. Archimedes' criterion is as follows:
Ar d 3 g д с c2 с , where vc is the kinematic viscosity of continuous medium, ρd is the density of the dispersed phase. Taking these criteria into account, we can derive the following equality: од о (1 ) n ,
Reд (1 ) Reо . where ξod is the coefficient of hydraulic resistance for a continuous medium for a single particle under the constrained flow conditions, φ is the volume fraction of the dispersed phase within the system. It would be useful to define the type of function f (φ) for the small and large Reynolds criterion values. Experimental studies showed that the velocities of particles depositing are connected by the following relation: where ω0d is the depositing rate of the particle relative to the continuous medium in constrained flow conditions, ω0d is the rate of free depositing of the particle, and n index is to be determined. Using the obtained parameter called the volume fraction of the dispersed phase, we have:
Computer Modeling / V.A. Zelenskiy, A.A. Sushin, A.I. Shchodro
It is experimentally shown [
For large values of Reynolds coefficient Re the following expression is valid: f ( ) (1 )n f ( ) (1 )2n .
It is experimentally shown [
f ( ) (1 )4.72 .
Taking into account the expressions derived, it is obtained that the ratio of the particle depositing rate relative to the continuous medium under the constrained flow conditions to the particle's free depositing rate is equal to: оd /0 (1 )4.72 .
There exist empirical formulas which enable us to account for the influence of constrainity [
оd /o (1 )2101.82 .
For φ>0,3 the following formula is applied: оd / o 0.123 (1 )3 .
Table 1 shows comparative data of the calculation of the velocity of the oil globules rising for the known relations and calculated according to the obtained formula. Numerical values are reduced to the velocity of oil globules freely rising to t he surface, i.e. relative ones.
Data for the watercut degree from 70% to 90% have not been found. The velocities of oil globules rising distributed by
fractions are of practical interest. The known relations are obtained for a case when the dispersed phase is water. In our case, the
dispersed phase is oil. In paper [
Computer Modeling / V.A. Zelenskiy, A.A. Sushin, A.I. Shchodro where pd, pc is the density of dispersed and continuous medium, kg/m3; μс is the viscosity of the continuous medium, Pa s; d is the diameter of the globule; g is the gravitational acceleration, m/s2. While rising, globules of different sizes are moving at different speeds. It is proposed to use the following expression to describe the calculation of the constrained rising of globules: 4.72 (w0d / w0 )i ( pd pc )d 2 g 1 B ,
18 c 1 B 1 (di / dmax )2
The equation obtained enables us to calculate the spectrum of velocities of constrained rising of the oil globules, taking into account the change of watercut of emulsion according to the height of the partition wall between the chambers of the separation device. In Table 2 there are given the data for the watercut values of the studied range from 70% to 90%. In accordance with table 2 and geometric characteristics of the separation device, it is possible to determine the separation time for oil-containing mixture, taking into account the composition of the mixture and watercut value. 100 150 200
The velocity of the oil globules rising, cm/s
The obtained relations enable us to mathematically describe the separation process through the velocity of the globules in the case of a high watercut value for oil-containing mixture, which demonstrates the scientific novelty. Using the mathematical model data and knowing the geometric characteristics of the separation device, separation time can be calculated. The precise definition of separation time improves the performance of the separation device without sacrificing the quality of the commercial oil, which is of great practical importance.
The authors express their gratitude for LLC Coordination (Ufa), LLC "GIRS", "Neftestroy" Educational Centre (Samara) for the fruitful cooperation and experimental data provided.