THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

2.4 Fluid Flow Equations 19which is usually established experimentally. The implication of the equation ofstate is that only two of the variables are independent; this is to say if the valuesof two of the independent thermodynamic variables are given for a fluid, thespecific value of any other thermodynamic property is automatically established.The equation of state for an ideal gas,p= RT (2.5)ρcan be derived from simple kinetic theory. Here,R = gas constant, energy per unit mass per degreeR =R/MR=universal gas constant, energy per mole per degree= 8.314.3 kJ/kg mol K = 1545.5 ft lb f /lb mol R= 1.986 Btu/lb m mol RM = molecular weight of gas, kg/kg mol or lb m /lb m molEach kilogram-mole of the gas contains N 0 = 6.02 × 10 26 molecules. N 0 constitutesAvogadro’s number for the MKS system of dimensional units. With η ⌢ representingthe mass of a single-gas molecule, M = N 0 η, the number of molecules⌢per unit volume is N = ρ/ η. ⌢ The equation of state for the ideal gas can now berewritten as:p = N R N0T = NkTwhere k is the Boltzmann constant =R/N 0 = 1.38 ×10 –26 kJ/K.In liquids and gases under extreme pressures, the relationships between thethermodynamic variables p, T , ρ, X (here X is the quality or the fractional massof gas comprising a saturated liquid–gas mixture, e.g., X = 1.00 represents a fullysaturated gaseous state and X = 0 represents the fully saturated liquid state) arenot so simple, but the fact remains that these parameters are fully dependent uponeach other, and specifying two thermodynamic parameters (including enthalpy,entropy, etc.) will fully specify the thermodynamic state of the fluid.2.4 Fluid Flow EquationsIn the Eulerian description of fluid mechanics the field variables, such as pressure,density, momenta, and energy, are considered to be continuous functions of thespatial coordinates x, y, z and of time t. Because velocity has three componentsin three-dimensional space and only two independent thermodynamic variablesneed to be selected to fix the thermodynamic state of the fluid (we chose p andρ), we have a total of five field variables for which we need five independentequations. We can take advantage of conservation laws to establish these equations,namely the conservation of mass, which supplies one equation; the conservation ofmomentum along each of the three principal axes, which provides three equations;