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Non-classical Behavior of BZT Gas in Isentropic Quasi-One-Dimensional Flow

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A thermodynamic property of gases called the fundamental derivative was first proposed by Bethe(1942) and later defined as the dimensionless quantity $\Gamma=\dfrac{c^4}{2v^3}\left(\dfrac{\partial^2 v}{\partial p^2}\right)_s$. The sign of $\Gamma$ reflects the sign of the curvature of the isentrope in the pressure-specific volume plane. The value of $\Gamma$ significantly affects the gas behavior and flow properties. Gases at relatively low pressure away from the critical pressure levels usually have values of $\Gamma$ above 1.0. For an ideal gas, $\Gamma = \dfrac{\gamma +1}{2}$, where $\gamma$ is the ratio of specific heats. Previous studies identified flow behaviors of gases with $\Gamma <0$ that are qualitatively opposite to classical gas dynamic theories based on perfect gas laws. For example, a divergent channel accelerates a subsonic flow and expansion shocks exist for gases with negative $\Gamma$. Although no experimental evidence has yet been found to confirm such non-classical gas flow behaviors, present interests in the use of super-critical heavy gases as well as pure academic curiosity call for more in-depth and definitive studies of such gas flows. A dense gas called $MDM$ is selected as the working fluid in the present work. A region of negative fundamental derivative is found near the critical point using the Van der Waals real gas Equation of State (EoS) for this heavy gas. Contrary to previous studies, the present work considers $\Gamma$ as a local thermodynamic variable instead of a constant in an isentropic flow or across a shock wave. Formulas of the relation of the fundamental derivative to other thermodynamic variables are given. To compare with the ideal gas model, the thermodynamic properties of this dense gas and the gas dynamic behaviors near its critical point are investigated. The conservation laws have been applied to develop the ordinary differential equation system for the quasi-one-dimensional isentropic flow. Since analytical solutions as in the classical theory are no longer possible for the non-ideal gas, numerical simulations are obtained for different upstream conditions. Various seemingly counter-classical gas dynamics flow behaviors are demonstrated. For example, a divergent-convergent nozzle is needed for transonic flow when the gas is within the negative fundamental derivative range. These unconventional gas behaviors are vitally interrelated in a flow of such non-ideal gas as it expands from high pressure to low pressure going through regions of $\Gamma >1$, $0<\Gamma <1$, and $\Gamma <0$ due to changes of its thermodynamic properties in the isentropic expansion process. Specific counter-classical behaviors are identified and discussed in this thesis.

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