Volume of Fluid Simulations of Heat Transfer in Liquid-Liquid Segmented Flows in Microfluidic channel

This study presents the investigations on the effect of heat transfer on droplet formation in T-type microfluidic channel. Mineral oil acts as a continuous phase, and water acts as a dispersed phase. The Volume of Fluid model is used to investigate the formation of droplets of water in oil in the microchannel. The physical properties of both fluids change with an increase in temperature of dispersed phase, of which the dominant properties are viscosities of fluids and interfacial tension. The parameters such as droplet diameter, distance between two consecutive droplets and detachment time were studied with respect to temperature in the range of 300 K to 325 K. These properties are also studied at different flow rate ratios 0.125, 0.25, and 0.5. The distance between two consecutive droplets was found to decrease with the increase in temperature due to the rapid motion of droplets in the temperature zone. The droplet detachment time also decreases, due to the decrease in interfacial forces which hold the droplets in the dispersed phase. The droplet diameter decreases initially with the increase in temperature, but it was observed that after 320 K the diameter of the droplet increases due to the coalescence of two consecutive droplets downstream in the microchannel. This temperature is reported as a limiting condition for thermally controlled water-in-oil microemulsions.


Introduction
In the modern era, the field of micro-fluidics experienced tremendous growth. Many researchers and industries show their interest in microchannels, because of their compact dimensions and high functionality. The field of microfluidics lies at the intersection between different fields i.e. engineering, chemistry, and biology with an aim to develop lab-ona-chip systems. Technology of micro-fluidics achieve goals like investigation of single-cell and biophysics of a single molecule, efficiency, portability of chemical assays, cost, and material savings of minimal reagent use (Zimmerman 2006). With the expansion of Microfluidics, progress in the fabrication of Micro and Nanodevices has led to inventive changes in our capability to manipulate minute volumes of fluids or micro and nanoparticles. Consequently, it led towards numerous applications such as chemical, biological and particulate separation analysis, developing sensors, capture and counting of cells, micro pumps, actuators, and system integration. The chemical and biological analysis usually deals with molecules and bioparticles having small dimensions (Kolev 2005). Microdroplet formation of uniform size in microchannels is a complicated phenomenon, however, plays an important role as it increases the surface area which increases the contact between the fluids (Iqbal et al. 2020). Microfluidic systems may be used to control chemical reactions in micro-droplets at the rate of micro to nanosecond scale. These microdroplets provide a large surface-to-volume ratio, therefore reactions inside these micro-droplets become efficient and fast, as well as results in a high quality yield (Nguyen and Wereley 2002).
Numerous methods have been developed based on different physical devices; designated by the flow-field topology in the surrounding area of the droplet formation zone. The devices commonly used to produce droplets are based on T-junction and flow-focusing configurations. In both cases, to be dispersed phase is compelled into a microchannel; where it comes across the immiscible carrier fluid and shears off into the continuous stream of droplets. The formation of droplets has been performed by varying flow conditions, wettability, viscosity and channel geometry (Baroud et al. 2010;Bashir et al. 2011Bashir et al. , 2014Soleymani et al. 2008;Guo and Chen 2009;Garstecki et al. 2006). However, the effects of temperature on the droplet formation mechanism have not been given much attention which can result in coalescence of droplets and disruption of uniform flow patterns in microchannels. The merging of droplets is also important phenomena where droplets should touch each other and overwhelmed the force that stabilizes it caused by surface tension which results in commencing the chemical reaction of the reagents enclosed within these droplets (Gu et al. 2011). Murshed and his co-workers studied experimentally the phenomena of droplet formation of water and Nanofluids in heat-induced T-shaped microchannel. They studied the temperature effect on droplet size with microchannel depth. It was found that with the increase of temperature the size of droplets increases, the depth of the microchannel also plays a significant role in the droplet formation, the smaller the depth of the channel larger the increase in droplet size with the increase in temperature (Murshed et al. 2009). Yap et al. studied experimentally thermal mediated control of liquid micro-droplet at a bifurcation using T-junction. It was observed that at room temperature droplets were equally divided. However after varying the temperature, resistance to the fluid is reduced at the side where the temperature is provided, which increases the tendency of the droplets to flow in the branch with the high temperature (Yap et al. 2009). Fujiu et al. experimentally studied the effect of temperature on oil-water emulsions in a microchannel. They studied the variation of droplet size and frequency of droplet formation with temperature for the range of 10 °C to 70 °C at different ranges for velocities. It was observed that droplet diameter depends on temperature. The production of droplets is increased as the temperature is increased due to a decrease in viscosity of both fluids (Fujiu et al. 2011). Jafari et al. studied the phenomenon of re-coalescence of droplets during emulsification at high energy and addressed the factors affecting the re-coalescence phenomena (Jafari et al. 2008). Wong et al. studied numerically and experimentally the control of droplet breakup process via inducing thermal surface tension gradient. A 2D model in T-type geometry was used to predict the thermally mediated passive breakup mechanism. It was determined that at normal temperature of 27 °C, identical size of droplets were formed on both sides of the junction. However, as temperature was increased the droplet moves towards the heated zone and large size of droplets were formed in heated zone. The droplet sorting takes place when the temperature reaches up to 40 °C (Ting et al. 2006). Despite of all such existing studies, there is no particular detailed numerical study performed so far in order to investigate the temperature effects on droplet formation in microfluidic channel, in order to understand the phenomena and disruption of uniformity of dispersed phase droplet stream. This research aims to investigate the droplet breakup mechanism in a two dimensional T-junction microchannel under the effect of induced temperature gradient using Volume of fluid (VOF) model. The parameters like droplet diameter, detachment time, and distance between two consecutive droplets were quantitatively investigated to optimize the microfluidic device. Furthermore, these parameters will be studied at different flow rate ratios for the range of temperatures, to investigate the effect of temperature with flow rates ratios. This work is a valuable addition for the validation of experimental work and provides an understanding of temperature mediated microfluidic droplet dynamics.

Computational Methodology
The schematic diagram of the two dimensional T-junction microchanel geometry has been shown in Fig. 1.
The dispersed phase flow (i.e. water) is carried into the main channel perpendicular to the continuous phase flow (i.e. mineral oil). The width "w" of the main channel and for both flow inlets are 100 μm. However, the width of the diverging section is double than that of the main channel width. The flow rate ratio of the dispersed phase to continuous phase "Q" is studied for three different cases Q = 0.125, Q = 0.25 and Q = 0.5 by changing the flow rates of oil phase in accordance with our previous work (Bashir et al. 2014). The flow rate Q = , where subscripts d and c denote the dispersed phase and continues phase velocities respectively. The computations were carried out using VOF method. The two dimensional approximation works very well in microchannels having large aspect ratio. The same has been reported in previous studies with aspect ratio ≥ 20 (Bashir et al. 2011;Brown et al. 2006). The governing equations for continuity, volume fraction, and momentum are given by: Here α q ,ρ q , V q, m pq , m qp , S q are volume fraction of q th phase, density fraction of q th phase, velocity of phase q th phase, mass transfer from the p th to the q th phase, mass transfer from the q th to the p th phase and source term of q th phase. There is no source term and volume fraction equation α q for the primary phase will be calculated based on the following equation: Momentum equation solve the resulting velocity field shared between the phases. This equation is dependent on the volume fractions of all phases, viscosity and density.
(  Here; ρ, p, g, F are density shared by all phases, pressure shared by all phases, body force and sum of all external forces.
To study the dynamics of droplets with the variation of temperature energy equation is solved along with continuity and momentum equation. Energy equation is shared among the phases and is given by: The VOF model gives temperature T and energy E as massaveraged variables, therefore: where E q for each of the phase is based on its specific heat and shared temperature. The properties ρ and k eff are shared by the phases. The S h is heat source term which contains contributions from all volumetric heat sources.

Correlation Functions
The materials used in our case are mineral oil and water, we have to study the effect of temperature of dispersed phase on droplet formation. As temperature varies the material properties like density, thermal conductivity, viscosity and specific heat of materials vary, to account for these variations user-defined functions were used. Interfacial tension is a property that allows the liquid to resist an external force. Interfacial tension is a property at the surface of a liquid that allows it to resist an external force. Interfacial tension is temperature-dependent property. Equation (6) shows the co-relation for the interfacial tension between mineral oil and water (Yap et al. 2009).
The rest of the correlations of properties of water and oil which are dependent on temperature are separately mentioned below (Coker 2007):

The Variation of Properties of Water with Temperature
The density (kg/m 3 ) variations with respect to temperature for water is given by The correlation of specific heat (J/kg.K) with respect to temperature for water is given by: where A = -10.2158, B = 1.7925 × 10 3 , C = 1.7730 × 10 -2 and D = -1.2631 × 10 -5 .

Variation of Properties of Mineral Oil with Temperature
The correlation of density with respect to temperature for mineral oil is given by: The correlation of specific heat (J/kg.K) with respect to temperature for mineral oil is given by:  The correlation of thermal conductivity (W/m.K) with respect to temperature for mineral oil is given by: The correlation of thermal conductivity (Kg/m.sec) with respect to temperature for mineral oil is given by:

VOF Model and Solver
VOF model is selected for the tracking of an interface between the continuous fluid (oil) and the dispersed fluid (water). As flow is incompressible therefore an implicit pressure-based solver was used. Explicit scheme was used and was set to be 0.25 to avoid instability of convergence. Surface tension varies with temperature therefore inserted as user-defined function for the formation of droplets in T-Junction microchannel. The operating pressure was set as 101,325 Pa. As we are dealing with micro-channel, for this gravity effects are negligible therefore we acquaint this within simulation. Bashir et al. (2014) studied experimentally wetting properties in T-type microchannel. They added surfactants, due to which the changes take place in the value of both interfacial surface tension as well as contact angle between two phases. When the value of surfactant concentration was 1% percent, the contact angle was almost approached to ideal value of 180. Therefore, at wall contact angle was set as 180 from phase 2 i.e. water to mineral oil. Volume fraction of both phases was also set with reference to phase 2 which was water liquid. A simple scheme is used for the pressure-velocity coupling. Green-Gauss cell based scheme for gradient and Presto for Pressure. First order upwind scheme for momentum and energy equation with under relaxation factor 0.7 and 1. Geo-Reconstruction for Volume fraction of both phases. The geometry is not complex; therefore, simple schemes are used as compared to complex ones. Time step was set as 0.0001to gave stability in results and gave convergence within given iteration per time step for generation of water droplets in oil.

Mesh Independent Study
Mesh counts per 100 µm were studied to find the optimum mesh counts in order to simulate our problem. Two perimeters mixture velocity and pressure were studied at these counts to find optimized mesh counts.  Table 1 represents maximum mixture velocity and counts/100 µm. The graph between interval count and mixture velocity was plotted as shown in Fig. 3. The mesh count from which results become stabilized is 30, same results were obtained at higher mesh counts. For more accurate results 50 counts/100 µm was selected as the final mesh for simulation of droplet formation.

Results and Discussion
The numerical simulations of water-in-oil droplet formation were performed using VOF method for different sets of conditions. Simulations were performed for three different flow rate ratios Q = 0.125, Q = 0.25 and Q = 0.5 by altering the flow rates of oil phase. With the variation of flow rate ratios, the formation of spherical droplets in a divergent section has been analyzed. Generally, the droplets of larger diameters were formed as we increased the flow rate ratios. Droplet formation depends on different parameters such as viscosity and velocity of both phases, interfacial surface tension between two phases as well as geometry. The droplet dynamic was studied at different flow rate ratios for the temperature range of 300 K to 325 K.

Validation of Simulation Scheme
Verification is essential while working with CFD simulation, therefore, simulations were performed and compared with the previous computational and experimental work. For the verification of computational work, the result was compared with the experimental work performed by Bashir et al. (2014). Table 2 shows that the error percentage of less than 2% with experimental work, which is acceptable. Figure 4 shows the comparison of contours for volume fraction with the previous computational result.

Effect of Temperature on Droplet Formation Process
The effect of temperature was studied for three different flow rate ratios. The velocity of the dispersed phase (water) was kept constant in all simulations while the velocity of continuous phase is varied. The droplet detachment time, droplet diameter, and distance between two consecutive droplets in the diverging were studied for the temperature range of 300 K to 325 K.

Detachment Time
The time in which droplet detaches from the dispersed phase to the continuous phase is known as detachment time. This phenomenon is known as droplet fission. A force that holds the droplet in the dispersed phase is interfacial tension which was balanced by pressure force and viscous force exerted by the continuous phase. With the increase of temperature interfacial force and viscous forces decrease, but interfacial force decreases more rapidly as compared to the viscous force due to which droplets detached rapidly as we increase temperature. For the flow rate ratio of 0.25 and 0.5 it was observed that the detachment time decreases linearly with the increase in temperature as shown in Fig. 5, but for the flow rate ratios of 0.125, it was observed that no changes occurs in the detachment time of the droplet. Therefore, the velocity of the dispersed phase increased four times due to which the droplet didn't have enough time to encounter changes with temperature.

Droplet Diameter
The velocity of the continuous phase is increase which results in the decreases in detachment time. Due to this slight amount of water droplet enters into the continuous phase, which causes the reduction of droplet diameter. Figure 6 shows changes with temperature. As the detachment time decreases with the increase of temperature. The slight amount of water droplet enters in the continuous phase channel as we increase temperature as shown in Fig. 6(a). It was predicted that further increase in temperature results in droplet coalescence (Binks and Rocher 2009). The phenomenon is also known as droplet fusion as shown in Fig. 6(b), (c). Coalescence occurs because of the rapid movement of droplets in the temperature zone, as temperature increases viscous effects decreases which cause rapid motion of droplets.

Distance Between Consecutive Droplets
Distance between droplets increases as we increase the flow velocity of the continuous phase as shown in Fig. 7. It was observed that as velocity increases droplet diameter decreases due to which droplets move rapidly which increases the distance between consecutive droplets. The distance between two consecutive droplets decreases linearly with increase of temperature but changes rapidly after 320 K as the diameter of droplet decrease with increase of temperature as well as decrease in viscosities of continuous and dispersed phase. Due to droplet coalescence, droplet of larger diameter moves slowly as compared with the droplet of small diameter therefore distance decreases rapidly.

Conclusions
The droplet dynamics was studied numerically with respect to temperature using the VOF model. Droplet diameter, distance between two consecutive droplets, and detachment time was studied with respect to temperature for different flow rate ratios. Following conclusions have been made from numerical simulation.
1. It was found that droplet detachment time decreases as we increase temperature, due to a decrease in interfacial forces which hold the droplets in the dispersed phase. Droplet detachment time increases as flow rate ratios increase. 2. Distance between two consecutive droplets decrease with an increase in temperature due to rapid motion of droplets in the temperature zone. With the increase of flow rate ratios, this distance decreases because the size of droplets reduces due to which distance between droplets increases. 3. Droplet Diameter decreases initially with the increase in temperature, it was observed that at certain temperatures the diameter of droplet increases due to coalescence of droplets. When the flow rate ratio was reduced to 0.125 coalescence didn't take place, due to high flow rate of continuous phase droplets didn't get enough time to encounter changes takes place due to temperature.

Declarations
Ethics Approval Not Applicable.

Conflicts of Interest
The authors declare that they have no competing interests, or other interests that might be perceived to infuence the results and/or discussion reported in this paper.