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Fluid Delivery Modeling

This research is a partnership of:
Santa Clara University (SCU)
Hewlett Packard-Aguadilla (HP-Aguadilla)
University of Puerto Rico at Mayaguez (UPRM)
with support from the National Science Foundation (NSF)


Study of the pressure behavior and fluid tension effect in small fluid reservoirs


Closed fluid reservoirs resembling cartridges are modeled and validated experimentally.

Reservoir and Fluid Delivery Modeling

Related work also developed under the collaboration between UPRM and HP and co-funded by the NSF concerns an experimental and theoretical investigation of the fluid storage and discharging mechanisms within TIJs. The model used for this discharging process is at once both a generalization of the Navier-Stokes equations and the Darcy’s law usually used for flow in porous regions. Figure 1 shows the physical representation of the problem. The fluid, ink in this case, is stored here in a chamber containing a foam material. The porous material provides enough surface area to hold the fluid by capillarity effects. A discharging hole is included at the bottom from which the fluid is extracted at a constant rate.



Figure 1: Discharging process in three and two-dimensional representation and experimental setup for the discharging process

The computational fluid dynamics tool used to solve the resulting set of conservation equations was CFX, version 4.4. The model used by CFX to take into consideration the capillarity effect is based on the Continuum Surface Force model of Brackbill. This approach models the surface tension force as a force which exists throughout the flow based upon derivatives of volume fraction, which has the same overall effect as the surface force, even when the surface is smeared. It was also considered that the two fluids are ink and air, separated by an interface at time t.

An experimental analysis was conducted with the twofold objective of understanding this discharging process and to validate the CFX simulations. The pressures drops for three different foams (A, B and C) at three different flows were studied. The appropriate combinations between the foam and flow resulted in a total of nine experiments. The three flow rates considered were: 0.75 ml/min, 1.5 ml/min, and 3 ml/min.

The experimental set-up resembled an actual discharging process in TIJ’s and consisted of a lexan chamber with capacity for single Robinhood foam (Foam A), Robinhood Cyan ink, a pressure transducer, a data acquisition system, an infusion syringe pump and a computer. The chamber built for this case was designed to fit a Robinhood foam. This second experiment consisted in filling the foam sample in the chamber with Robinhood Cyan ink until saturation and extraction of the ink was completed using the pump or until air penetrated into the line. When the ink was extracted, the foam provided capillarity lift or negative pressure that increases as the amount of ink decreases. This is known as the backpressure of the system. The pressure measurement decreases as the ink was drawn out of the foam. When the foam suction had the ability to defeat the pump suction, air would enter the extraction point provoking an increase in the pressure reading. This point was considered the end of the test.

From the simulations and a first set of experiments, it was shown that the pressure dropped linearly through the porous media. It was noticed that the hydraulic conductivity exerted the main influence in this magnitude. These results suggest that it is possible to reduce the magnitude of the pressure drop by changing the material properties of the porous medium. The accuracy and efficiency of the numerical solution was very good when compared with the experimental results. The average deviation found between experimental and numerical results was +1.076%.

The numerical solution for the discharging process used an homogeneous approach to treat the free surface because it works in a flow under gravity, where the phases have completely stratified, which was the case in this work. It was found in the simulation process that the addition of the body forces, i.e. Darcy’s flow and surface tension effects in the momentum equations, cause a degrading effect in the convergence of the program. Figure 2(a) shows the fringe plot for the volume fractions. It can be noted in this figure the presence of an interface between the ink and the air. This interface had an irregular shape and it is responsible for the surface tensions effects. This result revealed that there is a surface tension force that can have considerable effect on the final surface shape. Physically, this force acts directly on the surface.




Figure 2: (a) Visualization of the free surface by CFX-4 and (b) the backpressure as function of time

(FR=2.75, Q=24ml/h, σ = 0.029N/m, µ=3.6E-03Pa.s, T=300°C).

Figure 2(b) is a comparison between the numerical results and the experimental data for the discharging process, showing good agreement. The experimental results validate the linear behavior of the backpressure. When the ink is extracted, the foam provides a resistance that increases as the amount of ink decreases. This is known as the backpressure of the system. This response was basically an effect of the capillarity pressure. The average error between experimental and numerical results resulted in 0.1447 %. The efficiency of the extraction rate obtained by experimental data was 30.76923 %; meanwhile by simulation was 46.15%. The difference between these results could be due to the fact that in the simulation the porous media was an approximate to an arrangement of small tubes; but in actual conditions the porous medium had channels not interconnected that store small quantities of ink and air making a non-continuous flow through the channel.