Catalyst Project: Identification of Effective Heat Conductivity Coefficient of Particulate Two-phase Materials

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Catalyst Projects provide support for Historically Black Colleges and Universities to work towards establishing research capacity of faculty to strengthen science, technology, engineering and mathematics undergraduate education and research. It is expected that the award will further the faculty member's research capability, improve research and teaching at the institution, and involve undergraduate students in research experiences. This project at Savannah State University focuses on mathematical modeling for the identification of heat conductivity coefficients of two-phase materials and provides an opportunity for undergraduate students to enhance their education through research experiences in mathematical modeling. The researcher has established a strong collaboration with faculty at Purdue University. One of the central issues of the contemporary mechanics of continuous media is the problem of identification of the mechanical properties of particulate two-phase materials and flows based on the respective properties of the constituents. This project will investigate the problem of identifying one such mechanical property, namely, effective heat conductivity. The particulate material (or suspension of particles in a fluid, such as a liquid-solid mixture) can be conceptualized as a particulate "phase" comprised of randomly dispersed particles throughout a continuous "phase". The presence of suspended particles strongly influences the overall mechanical properties. This phenomenon remains a major challenge in the field of fluid mechanics especially while estimating the effective transport coefficients, such as the heat conductivity coefficient or the viscosity coefficient, of suspensions. The long-term goal is the estimation of the effective heat conductivity coefficient for suspension by developing an in-depth mathematical model. This model will be based on a system of stochastic differential equations. This system will be recovered from the steady and unsteady heat diffusion equations by applying a method called random point approximations - also, expanding the temperature field by using a suitable stochastic functional series. In this process, the model will be fully capable of countering the problems associated with the randomness of sizes and positions of the suspended particles in a continuous media. Furthermore, it will also be able to tackle the significant issue of inhomogeneity interactions which is strongly validated by experimental data. Once developed, this model will lead to a mathematically rigorous formula for effective heat conductivity coefficient of suspension, improving Maxwell's formula for terms up to the square of concentration of the suspension. The expected outcomes will also improve our understanding of the mechanism and theory that underlies this phenomenon which will subsequently extend design options in manufacturing devices and developing processes applicable to these mechanisms, therefore opening the door to possibilities of novel processes and devices that use heat transfer and fluid flow.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.