Analysis and Optimization of Kelvin's Generator

The goal of the project is the analysis of the Kelvin's generator and the optimization of the induction ring position for optimizing the electric field and therefore the voltage difference in the whole system.

Kelvin's Generator

Schematic operation

Kelvin's generator configuration is showed above. Due to naturally fluctuations, droplets from one of the injectors will be altered electrically and their charge will vary. This positive or negative charge will generate a positive or negative behaviour of the tanks as well as the induction rings. As the water keeps flowing from the injectors the voltage difference will keep growing as a feedback loop.

Electrical Induction

The model followed for the analysis will consist in the electric field generated by an induction ring and the water jet will be considered as a rod that will be positively or negatively charged. The interaction between both components will be the subject of analysis.

Electric field for different charge density and ring size

Electric potential for different charge densities and ring size

Conducting rod model for different meshing

The results for charge density in the rod, consider as the water jet, is based on the potential equation that after simplifying neglecting the rod potential when its ground connected the whole integrand can be represented as the equation to the right. The results implemented in MATLAB are shown below.

Rod charge density results

Charge density of the rod for different meshings, the one of the left is from a model implemented by Janisic and Prosen and on the right the model from the equation above for 700 grid points

Sensitivity Analysis

A sensitivity analysis was made to establish the most influent parameters when determining the optimal position of the ring to maximize density charge in the rod. The dimensionless parameters considered were:

Rod thickness (𝜂)
Charge density (𝜎𝑜)
Distance from the ring (𝛽)
Relation between ring radii and rod length (𝜆)

Default values for parameters

Charge density for different 𝛽

Charge density for different initial 𝜎𝑜

Charge density for different 𝜆

Charge density for different initial 𝜂

Discretization Analysis

In order to add more accuracy to the meshing used in the model a new variable was proposed to account for this factor

Ring's optimal position

The total charge in the rod for different ring positions is shown in the figure to the left. A discrete variable was proposed to define the different points along the rod going from 1 to 640 representing each of rod's limits.

Rod's optimal radii

A jet breakup study was made to stablish the droplet size with respect to the jet's profile velocity and the atomizer opening. Considering the studies from Hanalein, Ohnesorge and Reynolds that were first adapted by Rayleigh, this last one's theory was applied with the relationship D = 1.89d, being D the size of the droplet and d the injector's opening size. The main concept was to maintain a laminar flow to not reach high-Reynold numbers breakup mechanisms.

Numerical Analysis

A numerical analysis was performed to study the different oscillations and instabilities of the model proposed. The idea here is that the conducting rod, in a realistic model, the charge collected by the ends of the rod would tend to infinte. By applying the discrete variable giving more complexity in the central region and leaving high discrete values on the edges the main analysis was developed in this central area (where the ring position matters) neglecting oscillations in both ends. By increasing the discrete variable towards this ends the instabilities can be seen like shown in the figures below.

Charge density for n=1 and 𝜂 = 0.07

Charge density for n=10 and 𝜂 = 0.085

Conclusions

During the research, analysis and optimization it can be concluded that an optimal performance of the Kelvin's generator can be designed when considering the different parameter of the system.

The different parameters were analyzed and optimized together as well as the discrete values included that allowed the whole research.

The process requires a specific distance between atomizer and induction ring so the droplets can be charged, the conducting rod simplifies the process and the ring must be positioned immediately after the rod.

Design and Assembly

A preliminary design in Catia V5 was made to attempt a performance analysis based on the conclusions from the research project. Altough the design was consistent and solid the accuracy required by the theretical model was presumed to be milimetric and specific tools were needed to accomplish this task. However the design was able to generate a potential difference between the rings.