Simulation of non-Newtonian fluid flow in a horizontal pipe

Microchannel reactors exhibit distinct gas-liquid two-phase flow behaviors compared to macro-scale channels, offering advantages like efficient heat and mass transfer, compact size, and low energy consumption. Sodium carboxymethyl cellulose (CMC) emerges as a potent stabilizer, enhancing mixing, homogenization and pipeline transport. Its judicious application reduces equipment strain, amplifies homogenization efficiency and finds diverse utility in food processing and beyond. However, incorrect employment bears the risk of compromised outcomes, potentially leading to product wastage. Consequently, investigating N2-CMC solution flow in microchannels holds paramount significance for elevating efficacy, minimizing usage, refining product quality and bolstering yield. In this study, we simulated the N2-CMC solution flow dynamics within an interleaved T-shaped microchannel using FLUENT which is a numerical simulation software. The simulation delved into alterations in gas-liquid two-phase flow patterns and pressure drops. We examined the impacts of liquid concentration and gas-liquid flow rate ratio on flow patterns, pressure drops and bubble lengths. The simulation results exhibited congruence with experimental data. Notably, elevated liquid concentrations correlated with higher pressure drops and elongated bubble lengths. Conversely, augmenting the gas-liquid flow rate ratio led to diminished pressure drops while elongating bubble lengths. These findings furnish insights into flow patterns and pressure drop behaviors for gas-non-Newtonian fluids within microchannels, forming a pivotal reference for microfluidic system design.


Introduction
Microchannels are conduits characterized by micrometer or submicrometer dimensions, typically spanning from 10 to 1,000 micrometers.These diminutive passageways host distinctive gas-liquid twophase flow dynamics, marked by specific properties and behaviors.The microchannel environment boasts an exceptional surface area to volume ratio, markedly augmenting the efficiency of heat and mass transfer within the gas-liquid two-phase flow.This milieu's small dimensions and optimal interfacial contact facilitate rapid intermixing or separation of gas and liquid phases, rendering it a versatile candidate in fields such as chemicals and pharmaceuticals [1][2][3].
The profound utility of microchannels is exemplified in various applications.They serve as integral components for microdroplet generation systems, enabling biochemical reactions, swift reagent mixing and micro-particle synthesis.This technology, distinguished by its cost-effectiveness, automation and high throughput, bears relevance in controlled drug release, virus detection, and catalyst and particulate material synthesis.Through microfluidic approaches hinged on microchannel architectures, meticulous manipulation of microdroplet size, structure, morphology and functionality is realized, signifying a pivotal platform in the microdroplet domain.

Physical model
T-type microchannels constitute a prevalent microscale channel configuration, encompassing both staggered and symmetric variants.Notably, staggered T-type microchannels stand out for their remarkable attributes: exceptional uniformity, precise sizing and meticulous control.Given these advantages, our study adopts the staggered T-type microchannel [6] as the designated physical model.The schematic representation of the microchannel geometry employed in this investigation is illustrated in Figure 1.The channel's total length spans 50 mm.However, to mitigate the influence of inlet effects on flow patterns, we opt for an observation length of 30 mm.This section of the structure positioned at a vertical distance of 2 mm from inlet 2 attains gas equilibrium in both experimental and simulation settings.The microchannel takes on a rectangular cross-section, featuring a width and depth of 300 μm each.
Within this setup, the CMC solution [7] is introduced horizontally through inlet 1, while the gas is injected vertically via inlet 2. The point of confluence between the two streams gives rise to microdroplets and microbubbles, eventually exiting through the outlet.
The three-dimensional geometric model was established by using Fluent Meshing software.The specific structure of the three-dimensional coordinate system and geometric model is shown in Figure

Mesh division
To simulate the gas-liquid two-phase flow within the microchannel, the microchannel's geometry is discretized by using ANSYS-Meshing.The geometry is intentionally kept straightforward, leading to the selection of uniform meshing to enhance mesh quality.Through meticulous experimentation, an optimal mesh count of 220,000 is determined, resulting in a commendable grid cell quality of 0.92 across the entire computational domain.By conducting a mesh-independence test, the mesh exhibits distinct boundaries.More importantly, it reduces the computational time for the specified parameters.Figure 3 illustrates the microchannel meshing configuration adopted in this study.

Control equations
The VOF (volume of fluid) model [12] simulates two or more immiscible fluids by solving a single momentum equation and tracking the volume fraction of a single-phase fluid in the region which can accurately compute and track the kinematic changes at the gas-liquid phase interface.Since the role of surface tension is obvious in microchannels, the continuum surface force (CSF) model is also used to add surface tension as a source term to the momentum equation so that the simulation results are closer to the real flow conditions.In the study of this paper, the transfer process of phase change between gas and liquid is not considered and the governing equations are as follows: The continuity equation [8] is: Here, αg is the volume fraction of fluid in phase g; ρg is the density of phase g; vg is the velocity of phase g. αl is the volume fraction of fluid in phase l; ρl is the density of phase l; vl is the velocity of phase l and t is time.
The momentum equation [9] is: (4) Here, ρ is the volume-weighted average density; v is the mixing velocity; p is the pressure; μ is the volume-weighted average viscosity; g is the gravitational acceleration; F is the surface tension.

Boundary condition
The physical parameters of the solutions used for the simulations (0. 1% CMC, 0. 2% CMC and 0. 3% CMC solutions) were set to be the same as in the experimental case (density ρ, surface tension σ, fluid consistency coefficient k and flow characteristic index n), as shown in Table 1.The density of nitrogen was 1.25 kg/m 3 and the dynamic viscosity was 17.9 × 10-6 Pa-s.
The microchannel contains two inlets (inlet1 and inlet2) and one outlet.The CMC solution and nitrogen gas flow in from the two inlets at different flow rates respectively and start to form bubbles at the meeting point, which flow out through the mixing channel.In FLUENT software, in order to reduce the influence of the inlet boundary, the inlet fluid is set as a laminar flow, which is convenient for improving the convergence speed.The VOF model [10] is chosen to calculate and capture the gas-liquid phase interface.The continuous surface tension model is used to add the surface tension as a source term to the momentum equation so that the simulation results are closer to the real flow state.

Solver settings
Based on the instability of the two-phase flow state, the choice of solver, parameter settings and solution method are selected based on the pressure solver for transient simulation with the addition of gravity term.Choice of solver, parameter settings and solution method are selected based on the pressure solver for transient simulation with the addition of gravity.The liquid is set as an incompressible fluid and the gas phase is selected as nitrogen and set as a compressible fluid with no energy or mass transfer.The solution method is based on the PISO pressure-velocity coupling algorithm.In order to improve the convergence speed, the gradient term is selected based on the least squares format.The pressure difference is selected from the PRESTO format term.The volume fraction [11] difference is selected from the geometric reconstruction format and the simulation is carried out for different working conditions by changing the apparent velocity of the gas-liquid phase.The outlet is set as the pressure outlet.The contact angle is .The simulation parameters are shown in Table 2.The microchannel's gas-liquid phase interface is meticulously monitored and reconstructed through the implementation of the Volume of Fluid (VOF) model.Within the FLUENT software framework, the VOF model orchestrates the simulation of gas-liquid two-phase flow by autonomously solving the momentum equations while concurrently managing the volume fractions attributed to distinct fluids traversing the system.
Functioning on the principle of fractionally defining fluid volumes within each control volume, the VOF model embodies the cornerstone concept underpinning the simulation of two-phase flows.Notably, the sum of volume fractions for gas and liquid phases within every computational cell remains invariant at 1. To encapsulate the volume fraction of the liquid phase within the control volume, an "F" function is introduced accompanied by variables crafted to augment the model's representational capacity. (5)

Effect of flow rate ratio on flow pattern
Figure 4 illustrates a characteristic flow pattern, notably the segmented plug flow, which is impeccably simulated within a CMC solution at a concentration of 0.2%.The gas ingress velocity stands at 0.3 m/s, mirroring the liquid's inlet velocity.Notably, all simulations consistently manifest the emergence of nitrogen-CMC solution two-phase flow adopting a segmented plug flow configuration within the microchannel, encompassing the conditions scrutinized in this study.Notably absent are manifestations of vesicular flow or alternate flow patterns.This predilection can be attributed to the heightened viscosity inherent to non-Newtonian fluids.It is noteworthy that, operating under identical experimental conditions, the interaction between the CMC solution and the channel wall eclipses that of water, leading to pronounced elongation of bubbles.As a result, the formation of vesicular flow is hindered.
Three distinct scenarios of gas-liquid two-phase flow are meticulously simulated, featuring liquid phase mass fractions of 0.1%, 0.2%, and 0.3% within the CMC solution.Nitrogen takes the role of the gas phase.The resultant flow patterns are vividly portrayed in Figure 5. Remarkably, the microchannels subjected to the varying concentrations exhibit a transition in the flow pattern, wherein the elongated segmented plug flow morphs into a shorter segmented plug flow as the gas-liquid flow rate ratio diminishes.This observation concurs harmoniously with experimental findings [7] and acutely captures the nuances inherent to gas-liquid two-phase flow within horizontal tubes.As depicted in Figure 5, when confronted with a notable gas-liquid flow rate, the microchannel exhibits a heightened prevalence of gas bubbles, resulting in the formation of extended bubbles within the channel's confines.Consequently, this configuration aligns with a long-segment plug flow pattern.Conversely, as the gas phase flow rate diminishes, gas distribution manifests through small bubbles dispersed within the liquid phase.This restructuring of gas distribution corresponds to a reduction in bubble content within the microchannel.The typical bubble diameter registers slightly below the tube's diameter.Consequently, this shift in conditions yields a transition to a short-segment plug flow pattern.

Effect of concentration ratio on flow pattern
Illustrated in Figure 5, it is evident that distinct liquid concentrations do not elicit alterations in the flow pattern.Remarkably, this outcome resonates with the findings from Yuan Xigang et al. simulation study [13] involving T-shaped structured microchannels.In their investigation, varying water and ethanol solutions with a 10% mass fraction and pure ethanol liquids, it was observed that the flow pattern remained essentially constant.The critical insight garnered here is that surface tension exerts substantial influence over the flow pattern.Notably, even solutions featuring different viscosities showcased minimal impact on the vacuole's shape, this underscores the limited influence of viscosity in comparison to the dominant role played by surface tension in shaping the flow pattern.

Solver settings
Due to the inhomogeneity of the gas-liquid two-phase flow distribution in the horizontal microchannel, the average length of the bubble length was calculated as: Here, i = 1, 2, 3...; is the length of each bubble in the observation region in the steady state of the model.
The mean values of bubble length for the segmented plug flow for different concentrations of CMC solution and different gas-liquid flow rate ratios are shown in Figure 6.As evident from Figure 6, the length of bubbles within the microchannel exhibits a steady rise in correspondence with escalating gasliquid flow rate ratios.Simultaneously, a comparable trend is observed with increasing CMC solution concentrations.The length of bubbles swells even under identical flow rate ratios.This phenomenon can be attributed to the heightened viscosity resulting from augmented CMC solution concentrations.This heightened viscosity, in turn, intensifies the interaction between the bubbles and the channel's inner wall.During the bubbles' movement, the influence of viscous forces surmounts that of inertial forces, engendering a phenomenon where the bubbles become elongated to a certain degree.

Two-phase flow pressure drop
The pressure drop in two-phase flow consists of three main components, which are friction drop, gravity drop and acceleration drop.
(7) Since the gas mass is very small in the microchannel and the liquid flows horizontally, the effect of gravity on the pressure drop in the two-phase flow is negligible compared to the frictional pressure drop and the acceleration pressure drop is similar.Therefore, only the frictional pressure drop needs to be taken into account when calculating the total pressure drop to simplify the calculation and the pressure drop is: (8) To ascertain the fidelity of the simulations, a comparative analysis was performed, aligning the simulation outcomes with experimental findings conducted by Gao Yuhang [7].The experimentally obtained and simulated pressure drop values within the two-phase flow context are presented in Figure 7.This comparison encapsulates flow ratios spanning from 0.25 to 3, along with CMC liquid concentrations of 0.1%, 0.2% and 0.3%.Remarkably, the examination reveals that the disparity between the experimental and simulated results remains confined within the range of ± 10%, thereby signifying a heightened level of concordance and validation.Figure 8 illustrates the pressure drop's response to alterations in the gas-liquid flow rate ratio across diverse concentrations of CMC solutions.Notably, within the horizontal tube microchannel, an elevated liquid concentration correlates with a proportionally higher pressure drop.Moreover, the impact of concentration augmentation becomes increasingly conspicuous as the flow rate escalates.Interestingly, the pressure drop of the CMC solution exhibits gradual attenuation with the progressive increase in the gas-liquid flow rate ratio, primarily owing to the influence of the shear thinning effect.As the pressure drop within the gas is almost zero, it decreases as the proportion of gas increases.When the gas velocity is certain, as the gas-liquid flow rate ratio increases (i.e., the liquid velocity is relatively reduced), the liquid phase interface will be sheared by the high-speed gas so that small liquid droplets appear in the pipeline.Due to the effect of droplet entrainment, the gas-liquid phase interface will become relatively rough, which leads to the gas-liquid two-phase flow and the friction between the fluid and the pipe wall to increase the friction loss through the gas to make the microchannel of the lowspeed liquid flow rate increases.It is more obvious that the pressure drop decreases more slowly.

Conclusions
In this paper, the flow pattern and pressure drop of gas-liquid two-phase flow in a microchannel were simulated by taking nitrogen-CMC solution two-phase flow as an example.The effects of gas-liquid flow rate ratio and liquid concentration on pressure drop and bubble length were analyzed and the main conclusions obtained were as follows.
(1) In different concentrations of CMC solutions, varying the gas-liquid flow rate ratio, only one flow pattern with segmented plug flow was observed in all of them.
(2) As the gas-liquid flow rate ratio increases, the length of the bubble increases.As the concentration of the liquid increases, the length of the bubble is also gradually elongated.
(3) In nitrogen-CMC solution two-phase flow, the pressure drop of all three concentrations of CMC solution tended to decrease with the increase of the gas-liquid flow rate ratio.When the concentration of the solution increased, the pressure drop increased as well.
The pressure drop and flow pattern are pivotal parameters governing gas-liquid two-phase flow within microchannels, which is crucial for microchannel system design.This study's numerical simulations delve into the pressure drop and bubble length within gas-non-Newtonian fluid two-phase flow across microchannels.These analyses are correlated with liquid properties and gas-liquid flow rate ratios, thereby furnishing indispensable theoretical insights instrumental in the design of microfluidic systems.

Figure 1 .
Figure 1.Structure and size of microchannel.

Figure 5 .
Figure 5. Flow pattern of nitrogen-CMC solution in the middle position within the observation length.

Figure 6 .
Figure 6.Average bubble length at different concentrations of CMC solution and different gas-liquid flow rate ratios.

Figure 7 .
Figure 7. Experimental and simulated values of pressure drop in two-phase flow at different gasliquid flow ratios and concentrations.

Figure 8 .
Figure 8. Variation of pressure drop with gas-liquid flow rate ratio.

Table 1 .
Physical parameters of experimental solution