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In this paper a multidisciplinary simulation of a capacitive droplet sensor based on an open plate capacitor as transducing element is presented. The numerical simulations are based on the finite volume method (FVM), including calculations of an electric field which changes according to the presence of a liquid droplet. The volume of fluid (VOF) method is applied for the simulation of the ejection process of a liquid droplet out of a dispenser nozzle. The simulations were realised using the computational fluid dynamic (CFD) software CFD ACE+. The investigated capacitive sensing principle enables to determine the volume of a micro droplet passing the sensor capacitor due to the induced change in capacity. It could be found that single droplets in the considered volume range of 5 nL < V_{drop} < 100 nL lead to a linear change of the capacity up to ΔQ < 30 fC. The sensitivity of the focused capacitor geometry was evaluated to be S_{i} = 0.3 fC/nL. The simulation results are validated by experiments which exhibit good agreement.
An increasing demand in the field of micro dispensing is the control of the delivered quantities of liquids. Especially noncontact dispensing applications in the pico to the microliter range entail the need for novel measurement techniques to evaluate droplet volumes and the stability of the process in time. E.g., pharmaceutical research, based on quantification of several thousands of drug like substances or the imprint of quantitative readable lateral flow tests require precise information about the applied amount of liquids to improve the accuracy of the analysis [
The simulation study presented in this paper is based on measurement results achieved with a capacitive sensor prototype which was applied to detect single droplets in flight [
A dispensing process comprises two distinct phases which are the droplet's growth until its tearoff from the dispenser nozzle and the free flight of the droplet after it has detached from the nozzle. It turns out that these two situations result in different boundary conditions for the electrical problem of the charged capacitor. Both situations can be described by two different electrical equivalent circuits like follows: in first consideration, a growing droplet is connected to the liquid inside the nozzle, which stays in contact to the aluminum housing of the used dispenser unit via the dispensing piston, like described in detail in [
The numerical multiphysics model to study the electrical field of the droplet sensor is based on a structured 3D grid consisting of the droplet generator, implemented by a liquid column, and the capacitor electrodes embedded in the sensor support material (bulk material). The computational domain, shown in
The simulation of hydrodynamic fluid flows using CFD is based on the iterative solution of the discretized set of equations consisting of the basic physical conservation principles of mass, momentum and energy, given by the Navier Stokes Equations (NSE). The following sections introduce the basic equations required for the numerical calculation of the coupled simulation, considering the fluid dynamics of droplet ejection as well as the electrical interaction of a droplet and a capacitor.
The numerical simulation is based on the application of the finite volume method (FVM). The FVM is a numerical method for the solution of partial differential equations that calculates the values of the conserved variables across a considered volume. This, so called control volume, is defined by the discretization using a computational mesh. In case of incompressible Newtonian fluids considering the properties of an external electric field, which is the most appropriate for solving the described problem, the Navier Stokes momentum equation can be written as follows:
The electrostatic force
A further method that is required for simulation of the droplet generation is the volume of fluid method (VOF). The VOF method enables the simulation of two phase flows of immiscible fluids, which are in the considered case water and ambient air. The approach is based on the introduction of an additional field variable, the volume fraction
Since this method determines the phase distribution in terms of the
In order to model the surface tension of the liquid the pure VOF method like briefly described above is not sufficient. Furthermore, the surface tension has to be considered when dealing with capillary liquid flows like droplet ejection processes. The implementation of the surface tension to the numerical calculation is based on a surface reconstruction method to determine the curvature of the interface between the two phases. Here, the Piecewise Linear Interface Construction (PLIC) of 2nd order [
In addition to the fluid dynamics the electrical field of the capacitive transducer is modeled by the application of the electric module, provided by the CFD ACE+ tool [
The electrostatic potential is then used as a source for coupling force at the interface of the droplet through the virtual force calculated by the CFD Ace+ software:
The calculation of the total charge on an electrode by the electric module in a discrete way is implemented by:
The numerical study of the presented capacitive measurement method requires the implementation of a droplet ejection process which generates droplets with realistic properties in terms of shape, volume and velocity. To keep the focus on the electrostatic interaction, a simple model of a droplet ejection process was realised, based on a liquid flow boundary condition driving the droplet ejection. Therefore, a liquid flow of a constant velocity (v_{flow}) of 2.5 m/s was set as boundary condition at the top inlet of the cylindrical liquid column (cf. Section 3.1). The flow was active for 70 μs and then stopped instantly to initiate the droplet tear off. The applied parameters led to a droplet ejection process like shown in
The boundary condition for the electric field in between the capacitor electrodes is given by constant electrical potentials on the two opposite electrodes. The electrode on the left side (see
The applied solution technique follows the finite volume method, like described in Section 3.2, to solve the partial differential equations in the computational domain. The whole domain consists of 540,000 cells, whereas the area along the flight path of the droplet consists of a grid of smaller cells in comparison to the surrounding cells (cell size∼3:1). The planar geometry of the considered setup enabled to set a symmetry condition, which allowed for the calculation of only half of the real geometry, to save computation time. The simulation required transient conditions at a defined time step of Δt = 1 μs and a convergence criteria of 0.0001. To evaluate the accuracy of the numerical calculations, a brief grid refinement study was accomplished to investigate the influence of the cell size. To estimate the discretisation error the side length of all cells was decreased in all dimensions by a factor 2 and 4, as well as increased by a factor 2 respectively. Simulations were performed with the different grids by stationary simulation of an empty capacitor as well as for a capacitor with a droplet introduced in the middle of the electrodes at identical conditions like described above for the dynamic simulation model. The value of interest here was the change of the charge on the measurement electrode for the empty capacitor compared to the droplet filled one. The results are given in
It can be seen that the charge, depicted on the yaxis, declines for higher grid definition (xaxis). The polynomial fit converges to a value of about 1 fC representing the “real” physical condition. Based on this grid study the error of the transient simulations to be presented below was estimated to be about +29% if a grid of scaling 1 is applied. Though, this error is quite considerable, a grid or scaling 1 was applied for all further studies to hold the computational time within a reasonable time frame. Obviously, the presented grid study exhibits a convergence from larger to smaller values. However, in the general case it cannot be assumed that this is the case for any initial or boundary condition. Therefore, the estimated error should be assumed to be symmetric about the simulated values. Nevertheless, it turns out in all considered cases, like presented below, that the simulated results tend to overestimate the experimental or analytical findings by about approximately this error estimate of 29%.
To investigate the feasibility to solve the described multidisciplinary problem the presented computational model was used in a simplified setup. Simulations were performed with defined spherical droplets of various volumes in the range from 5 to 100 nL neglecting the described droplet generation model. The droplets were defined as initial conditions in the model passing the capacitor with a constant shape at a defined velocity of v = 1 m/s. The charge characteristic on the measurement electrode was extracted as a function of time and is given in
The simulation of the full model as described in Section 3 results in different signal characteristics. Following the explanation given in Section 2, two separate situations can be distinguished, like depicted in
To confirm the influence of the capacitive coupling effect in a quantitative manner the height of the liquid column was varied, to increase the displacement of the ground potential at the end of the column, see
The parameter of major interest is the effect of variable droplet volumes to the generated signals. A first proof of this influence was given in Section 4.1,
It can be seen that the maximum change of the charge decreases with lower droplet volumes also in the full model simulation. A noticeable detail is the waved signal characteristics exhibited by the signals generated especially by smaller droplets (V = 22 nL and V = 17 nL, see
To validate the established simulation model by experiment the maximum change in charge values taken from the simulation of spherical droplets were fed into an electrical network simulation representing the electrical amplification circuit of the capacitive sensor as described in [
It has been shown that a multidisciplinary simulation, comprising the simulation of a droplet ejection process and the interaction of the droplet with the electromagnetic field in the electrostatic limit can be successfully applied to quantitatively model the considered capacitive droplet sensor. The fundamental reason for the specific negative signal characteristics, known from previous experiments, could be explained by the effect of capacitive coupling. It was found that this effect can be used to identify the point of droplet tearoff from the nozzle, indicated by the negative signal peak. The study of variable droplet volumes has shown, that the change of the charge caused by the measured droplets follows a linear relation resulting in a sensitivity of S_{i} = 0.3 fC/nL. The verification of the CFD simulation results by comparison to the experimental results, considering the grid refinement study, confirmed that the presented models are correctly describing the considered problem. A final validation proved that the combination of the accomplished CFD simulations with an electrical network model enabled a complete numerical description of the experimental sensor prototype, including the electronic amplification circuit. In summary the established models and results can contribute significantly to the explanation and optimization of the capacitive measurement method. The presented capacitive droplet sensor can be considered as valuable contribution to the list of process control methods in the field of noncontact dispensing. In particular, the linear dependency of the change in charge on the volume of the causative droplet provides the basis for novel quantitative droplet sensors that are able to determine the volume of droplets with high accuracy in a noncontact manner. The application of such sensors can simplify the calibration and characterization of droplet generating devices as well as improve the quality of products, requiring small liquid quantities, by the application of online process control systems.
The authors are thankful to the Ministry of Science, Research and Art of the Federal State of BadenWürttemberg, Germany for the financial support.
Previously achieved experimental signal characteristics correlated to droplet positions while a droplet is passing the capacitor.
Two electrical equivalent circuits occurring during droplet ejection (
Computational domain as used for the CFD simulations.
Simulated droplet ejection process applying the grid as described in Section 3.1. A flow (2.5 m/s) towards the nozzle is instantly stopped after 70 μs which initializes the droplet ejection.
Results of the accomplished grid study to evaluate the accuracy of the used computational grid.
Change in charge as a function of time caused by spherically shaped droplets of different volumes passing the capacitor as shown in the illustration for a spherical droplet of V = 50 nL at three specific points in time.
Simulated field distribution and droplet flight through the capacitor, including the droplet ejection process.
Investigation of the influence of the capacitive coupling effect by variations of the liquid column height.
Charge alternation caused by dispensed droplets of different volumes and velocities.
Correlation of droplet volume to the corresponding maximum change in charge for the spherical droplets as well as for the dispensed droplets.
Comparison of the voltage signals obtained from the experiments and the results of the network simulation.
Material properties of the fluids used for the presented simulations.
dynamic viscosity [mPas]  1.0  0.0185 
permittivity  80.1 (f < GHz)  1 
conductivity [1/Ωm]  5.5 × 10^{−6}  1 × 10^{−4} 
Physical property  DI water  air 
density [kg/m^{3}]  1,000  1.161 
Surface tension [N/m]  0.0725  – 
Droplet dispensing process actuation parameters and resulting droplet properties.

 

pulse duration [μs]  flow velocity [m/s]  volume [nL]  droplet velocity [m/s] 
70  2.8  33  1.4 
70  2.5  29  1.2 
70  2.2  22  1.1 
50  2.5  17  0.9 