1. Background

Multilayer flows are often encountered in the industry during co-extrusion, film coating, and lubricated pipeline flow. These processes are often restricted by interfacial instability and fouling problems. Therefore, methods to suppress this instability are of prime importance [1].

Lubricated pipelining is a method in which a high viscous oil core is surrounded by an annular low viscous liquid (water) film, thereby preventing its contact with the wall. As a consequence, the total pressure drop during flow reduces drastically to almost the pressure drop encountered during the flow of water with the same mixture velocity through the same pipe. This configuration commonly termed as Core-annular Flow (CAF) has been explored for energy-efficient transportation of high viscous crude oil since 1904 [2], but has not gained much popularity in practical applications primarily due to the challenges associated with the establishment and re-establishment of stable CAF. A few researchers have proposed the use of viscoplastic fluid as the annular fluid to alleviate this problem. These studies have performed linear and nonlinear analysis to stabilize the CAF [1] [3][4]. The viscoplastic fluid with yield stress ensures a plug flow zone at the interface where the shear stress is less than the yield stress, and viscous flow occurs closer to the wall, when the shear stress exceeds the yield value. This suppresses interfacial disturbances and stabilizes CAF configuration.

In this study, we present a mathematical analysis of CAF configuration for high viscous oil core with viscoplastic lubrication and discuss the effect of the rheological properties on pressure drop reduction factor. The analysis reduces to ordinary CAF (both liquids Newtonian) when yield stress of the outer fluid approaches zero. This simplified analytical approach can be used in lieu of complex numerical simulation [3] to comprehend the influence of rheological parameters on the thickness of annulus film and its corresponding pressure drop.

2. Analysis

The flow geometry (Fig. 1) considers the axisymmetric core-annular flow of two liquids through a circular pipe of uniform radius (R) and length (L). It is noted from experiments that the high viscous oils mostly exhibit Newtonian characteristics. Accordingly, the analysis considers the inner fluid (Fluid 1 in Fig. 1) to be Newtonian (with density ρ₁ and viscosity µ₁) and the outer fluid (Fluid 2 in Fig. 1) to exhibit Bingham plastic (B.P) characteristics given by the following constitutive relation, Eqn. 1. (a, b):

The pressure drop reduction factor (ΔP_r) is the ratio of pressure drop under CAF configuration (P_2𝞥) to the pressure drop for the flow of oil only through the pipe (P_1𝞥). In both cases, the volumetric flow rate of the oil is same. ΔP_r is related to other parameters of the process by Eqn. 2, where the radius terms are labeled in Fig. 1. Evidently a low value of ΔP_r suggests a more energy-efficient transportation.

To solve Eqn. (2a), the velocity profiles for the inner and outer fluids are required, which are obtained from the Navier-Stokes (N-S) equation along with the appropriate constitutive relationships. Navier-Stokes (N-S) equations are integrated considering ‘no slip’ and ‘velocity continuity’ boundary conditions to give the following equations Eqn. 3. (a, b, c):

The analysis reduces to the case of two-phase Newtonian flow for τ_y = 0, which is validated by matching the obtained pressure drop with the simulation work of Gupta et al.,2016 [5]. P_2𝞥 predicted from this analysis is within ± 0.6% for the Newtonian – Newtonian case reported by them.

3. Results and Discussion

The velocity profiles for Newtonian oil core and B.P. annulus along with the dimensions for the zone of core, and annular (plug and viscous) region depend on the following input parameters: pipe radius (R), oil density (ρ₁) and viscosity (µ₁), B.P. fluid density (ρ₂), viscosity (µ₂), and yield stress (Ï„_y). Upon solving the flow rate equations (obtained by integrating the velocity profiles for individual regions) for both the core and annular region, we can obtain the value of core radius (r_i), radius at the end of the plug flow region of the annulus (r_y) and the corresponding pressure drop for the two-phase flow of oil and water (P_2𝞥). These values define the velocity profile of the liquids as a function of ‘r’.

3.1 Parametric variation

It is observed from the analytical expressions developed, that the core radius (r_i) is influenced by both the core and annular fluid, and this in turn influences the two-phase pressure drop and the pressure drop reduction factor.

Numerical simulation (CFD) of the system was carried out using COMSOL for conduit radius (R = 0.025m), in which core oil (ρ₁ = 886.8 kg/m³, µ₁= 0.314 Pa. s) and an annular B.P. liquid (ρ₂ = 1037.35 kg/m³, Ï„_y = 5.331 Pa, µ₂ = 0. 0039Pa.s) flowed at 0.0000878 and 0.000391 m³/s respectively. The simulation (not presented here) has shown stable CAF. The parametric effects of the rheological properties of the core and annular fluid on the CAF geometry and pressure drop reduction factor are carried out by varying Ï„_y (0 ≤ Ï„_y ≤12), µ₂ (0.001 ≤ µ₂ ≤ 0.0064) and µ₁ (0.05 ≤ µ₁ ≤ 0.7).

Introducing yield stress (Ï„_y) changes the annular liquid from Newtonian to a B.P. liquid. Increase in Ï„_y does not affect the core radius (r_i) but makes the plug flow region (r_y - r_i) of annulus thicker at the cost of the viscous flow zone thickness (R – r_y). The plug flow at higher Ï„_y sets in at a closer distance from the wall (higher r_y), where shear stress value is higher. An increase in Ï„_y increases ΔP_r almost linearly (Fig. 2). This suggests that if a Newtonian annular fluid (viscosity µ₂), is converted to a B.P. liquid (with same µ₂ and Ï„_y > 0), using an appropriate additive, the effect will be higher pressure drop compared to the Newtonian annular liquid case.

It is observed that in order to reduce the pressure drop (P_2𝞥), the annular liquid viscosity should be lowered. In fact, increasing µ₂ from 0.001 to 0.0064 Pa. s, did raise the pressure drop reduction factor (ΔP_r) by a small amount (from 2.5 to 2.73), almost linearly. An increase in µ₂ increases only the thickness of the shear region of annular flow, thus bringing the plug zone away from the wall.

Effect of increasing the viscosity (µ₁) of core oil is found to decrease the pressure ratio (ΔP_r), clearly showing the lubrication to be more effective for higher viscosity core oils. Also, the shear region of the annulus gets thicker at the cost of its plug flow region.

4. Conclusion

The results show that this CAF configuration of flow for pressure drop reduction effects are more prominent with higher viscosity of the core oil. If a B.P. annular liquid is used for stabilizing the CAF, the pressure drop will be more compared to a Newtonian liquid with same viscosity.

Pipeline pressure drop increases with increasing yield stress of an annular BP liquid. The beneficial effect of stabilizing CAF as reported by Moyers-Gonzalez et al. [3] needs to be weighed against the increased pressure drop incurred due to the presence of yield stress of the annular liquid.

5. References

[1] S. Hormozi, K. Wielage-Burchard, and I. A. Frigaard, “Multi-layer channel flows with yield stress fluids,” J. Nonnewton. Fluid Mech., vol. 166, no. 5–6, pp. 262–278, 2011.

[2] D. D. Joseph, R. Bai, and K. P. Chen, “Core-Annular Flows,” pp. 65–90, 1997.

[3] M. A. Moyers-Gonzalez, I. A. Frigaard, and C. Nouar, “Nonlinear stability of a visco-plastically lubricated viscous shear flow,” J. Fluid Mech., vol. 506, pp. 117–146, 2004.

[4] S. Garmeh, A. Dolatabadi, and I. Karimfazli, “A novel suspension transport method: Viscoplastic lubrication of high-density fluids,” J. Nonnewton. Fluid Mech., vol. 287, no. November 2020, p. 104449, 2021.

[5] R. Gupta, C. K. Turangan, and R. Manica, “Oil-water core-annular flow in vertical pipes: A CFD study,” Can. J. Chem. Eng., vol. 94, no. 5, pp. 980–987, 2016.