#### Date of Award

Spring 2012

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

#### Department

Mathematical Sciences

#### First Advisor

John Kenneth Bechtold

#### Second Advisor

Shahriar Afkhami

#### Third Advisor

Michael R. Booty

#### Fourth Advisor

Linda Jane Cummings

#### Fifth Advisor

Hong Geun Im

#### Abstract

In this dissertation, we develop new mathematical theories of flame propagation that are valid at elevated, or extreme, pressures. Of particular interest is the regime of burning in which the pressure exceeds the critical pressure of the species undergoing chemical reaction. Fluids and flames are known to behave differently under these extreme conditions as opposed to atmospheric pressure. The focus of this dissertation is to investigate these differences by deriving reduced models that contain the unique features.

In the first part of this dissertation, we analyze the structure of laminar diffusion flames at high pressure in the limit of large activation energy for the particular configuration of a steady flame in counterflow. We consider a dense fluid in which normal Fickian diffusion of the fuel is limited, and thermal diffusion, i.e., the Soret effect, is the dominant mechanism for fuel mass transport. Temperature and species profiles, as well as flame temperature and location, are determined as a function of Damköhler number and Soret diffusion coefficient. In particular, we find that oxidant is entirely consumed by the flame, while some fuel leaks through. For light fuels, the fuel profile is found to have a local peak on the oxidant side as a result of thermal diffusion. Our analysis includes a description of extinction phenomenon, including explicit criteria in terms of the Soret diffusion coefficient, ratio of temperature of the two streams, and the Damköhler number at extinction.

In the second part of this dissertation, we derive an asymptotic theory of laminar premixed flames in high density fluids in the limit of large activation energy. The model is intended to provide insights into the structure and dynamics of deflagration waves in high pressure, dense fluids where normal Fickian diffusion is limited. In such cases, particularly under conditions exceeding the thermodynamic critical point of the fluid, the primary mode of species transport is through thermal diffusion, i.e., the Soret effect. Such a model for diffusive transport is considered, and we derive a model with an explicit dependence on the Soret effect for a one-step overall reaction. The density is assumed sufficiently high to adopt a constant density formulation. The local reaction-diffusion structure is found to be fundamentally different from that of an ideal gas with Fickian diffusion, which results in new conditions relating the equations for thermal and mass transport in the bulk flow. The model is used to investigate the basic structure of planar flames, as well as their stability. Stability boundaries are identified that mark the transition from planar to either steady, spatially periodic structures, or time-dependent modes of propagation. The combined effects of the Soret diffusion coefficient and Lewis number are discussed. Furthermore, a weakly nonlinear analysis of the derived model is carried out, resulting in a modified Kuramoto-Sivashinsky (K-S) equation, accounting for effects of Soret Diffusion. Linear stability analysis shows that the flame front is unstable with respect to long-waves in a range of Soret diffusion coefficient that corresponds to no and weak Soret effect. However, there exists a range of Soret diffusion coefficient for which a flame front is unconditionally stable.

#### Recommended Citation

Fong, Daniel, "Mathematical models of combustion at high pressure" (2012). *Dissertations*. 320.

https://digitalcommons.njit.edu/dissertations/320