Gas dynamics

In vacuum technology, some calculations involve the steady flow of fluid through duct of changing circular cross-section (diffusers, jets, nozzles, etc.) and the methods of gas dynamics can be applied to calculate pressures, velocities and temperatures. (A criterion for the applicability of gas dynamics is that Kn 0.01 although, according to ref. (d), the methods can be applied even at Kn values up to 0.3.) 

A very useful equation to deal with phenomena associated with the flow of fluids is the Bernoulli equation. It can be used to analyse fluid flow along a streamline from a point 1 to a point 2 assuming that the flow is steady, the process is adiabatic and that frictional forces between the fluid and the tube are negligible. Various forms of the equation appear in textbooks on fluid mechanics and physics. A statement in differential form can be obtained  

For adiabatic changes in the state of an ideal gas, the following relationships exist  

Further, the equation of state for an ideal gas can be rearranged to yield  

It can be shown that if a stagnant fluid (vj = 0, T = Tx) is expanded through a jet, from px to p2, the flow velocity in the streamline is given by  

---

Torgunakov V.G. et al. Two-level system for thermographic monitoring of industrial thermal units. Proc. of VTI Intern. S-T conference. Cherepovets, Russia, pp. 45-46, 1997. 2. Solovyov A.V., Solovyova Ye.V. et al. The method of Dirichlet cells for solution of gas-dynamic equations in cylindrical coordinates, M., 1986, 32 p. 

The resulting overall energy balance for the plant at nominal load conditions is shown in Table 3. The primary combustor operates at 760 kPa (7.5 atm) pressure the equivalence ratio is 0.9 the heat loss is about 3.5%. The channel operates in the subsonic mode, in a peak magnetic field of 6 T. AH critical electrical and gas dynamic operating parameters of the channel are within prescribed constraints the magnetic field and electrical loading are tailored to limit the maximum axial electrical field to 2 kV/m, the transverse current density to 0.9 A/cm , and the Hall parameter to 4. The diffuser pressure recovery factor is 0.6. 

G. A. Bird, Molecular Gas Dynamics, Oxford University Press, Oxford, U.K., 1976. 

The shape of the converging section is a smooth trumpet shape similar to the simple converging nozzle. However, special shapes of the diverging section are required to produce the maximum supersonic-exit velocity. Shocks result if the divergence is too rapid and excessive boundary layer friction occurs if the divergence is too shallow. See Liepmann and Roshko (Elements of Gas Dynamic.s, Wiley, New York, 1957, p. 284). If the nozzle is to be used as a thrust device, the diverg-... 

Additional theoretical bac-kground can be obtained from Preiswerk, Application of the Methods of Gas Dynamics to Water Flows with Free Suiface, part 1 Flows with No Energy Dissipation, NACA Tech. Mem. 934, 1940 part 11 Flows with Momentum Discontinuities Hydraulic Jumps), NACA Tech. Mem. 935, 1940. 

J.A. Owczarek, Fundamentals of Gas Dynamics, International Textbook, Scranton, PA, 1964. 

The process gas of ethylene plants and methyl tertiary butyl ether plants is normally a hydrogen/ methane mixture. The molecular weight of the gas in such processes ranges from 3.5 to 14. The tliermodynamic behavior of hydrogen/methane mixtures has been and continues to be extensively researched. The gas dynamic design of turboexpanders, which are extensively used in such plants, depends on the equations of state of the process gas. Optimum performance of the turboexpander and associated equipment demands accurate thermodynamic properties for a wide range of process gas conditions. 

The determination of the first bending critical speed is well established however, there is also concern with regard to the rotor support system s sensitivity to exciting forces. These come from unbalance and/or gas dynamic forces arising during operation in service. Operation with dirty corrosive gas will soon cause rotor unbalance. The rotor dynamics verification test is concerned with synchronous excitaticm, namely unbalance. The test must also verify that the separation margins are to specification. 

Another potential problem is due to rotor instability caused by gas dynamic forces. The frequency of this occurrence is non-synchronous. This has been described as aerodynamic forces set up within an impeller when the rotational axis is not coincident with the geometric axis. The verification of a compressor train requires a test at full pressure and speed. Aerodynamic cross-coupling, the interaction of the rotor mechanically with the gas flow in the compressor, can be predicted. A caution flag should be raised at this point because the full-pressure full-speed tests as normally conducted are not Class IASME performance tests. This means the staging probably is mismatched and can lead to other problems [22], It might also be appropriate to caution the reader this test is expensive. 

A detonation shock wave is an abrupt gas dynamic discontinuity across which properties such as gas pressure, density, temperature, and local flow velocities change discontinnonsly. Shockwaves are always characterized by the observation that the wave travels with a velocity that is faster than the local speed of sound in the undisturbed mixtnre ahead of the wave front. The ratio of the wave velocity to the speed of sound is called the Mach number. 

In the surrounding atmosphere, a blast wave is experienced as a transient change in gas-dynamic-state parameters pressure, density, and particle velocity. Generally, these parameters increase rapidly, then decrease less rapidly to sub-ambient values (i.e., develop a negative phase). Subsequently, parameters slowly return to atmospheric values (Figure 3.7). The shape of a blast wave is highly dependent on the nature of the explosion process. 

If the combustion process within a gas explosion is relatively slow, then expansion is slow, and the blast consists of a low-amplitude pressure wave that is characterized by a gradual increase in gas-dynamic-state variables (Figure 3.7a). If, on the other hand, combustion is rapid, the blast is characterized by a sudden increase in the gas-dynamic-state variables a shock (Figure 3.7b). The shape of a blast wave changes during propagation because the propagation mechanism is nonlinear. Initial pressure waves tend to steepen to shock waves in the far field, and wave durations tend to increase. 

Making a detailed estimate of the full loading of an object by a blast wave is only possible by use of multidimensional gas-dynamic codes such as BLAST (Van den Berg 1990). However, if the problem is sufficiently simplified, analytic methods may do as well. For such methods, it is sufficient to describe the blast wave somewhere in the field in terms of the side-on peak overpressure and the positive-phase duration. Blast models used for vapor cloud explosion blast modeling (Section 4.3) give the distribution of these blast parameters in the explosion s vicinity. 

Experimental research has shown that a vapor cloud explosion can be described as a process of combustion-driven expansion flow with the turbulent structure of the flow acting as a positive feedback mechanism. Combustion, turbulence, and gas dynamics in this complicated process are closely interrelated. Computational research has explored the theoretical relations among burning speed, flame speed, combustion rates, geometry, and gas dynamics in gas explosions. 

Analytical methods relate the gas dynamics of the expansion flow field to an energy addition that is fully prescribed. A first step in this approach is to examine spherical geometry as the simplest in which a gas explosion manifests itself. The gas dynamics of a spherical flow field is described by the conservation equations for mass, momentum, and energy ... 

This section describes how this set of equations can be solved analytically by the introduction of various simplifications. First, gas dynamics is linearized, thus permitting an acoustic approach. Next, a class of solutions based on the similarity principle is presented. The simplest and most tractable results are obtained from the most extensive simplifications. 

The similarity solution for a flow field in front of a steady piston is a special case from a much larger class of similarity solutions in which certain well-defined variations in piston speed are allowed (Guirguis et al. 1983). The similarity postulate for variable piston speed solutions, however, sets stringent conditions for the gas-dynamic state of the ambient medium. These conditions are unrealistic within the scope of these guidelines, so discussion is confined to constant-velocity solutions. 

Solving the gas dynamics expressions of Kuhl et al. (1973) requires numerical integration of ordinary differential equations. Hence, the Kuhl et al. paper was soon followed by various papers in which Kuhl s numerical exact solution was approximated by analytical expressions. 

The flow field in front of an expanding piston is characterized by a leading gas-dynamic discontinuity, namely, a shock followed by a monotonic increase in gas-dynamic variables toward the piston. If both shock and piston are regarded as boundary conditions, the intermediate flow field may be treated as isentropic. Therefore, the gas dynamics can be described by only two dependent variables. Moreover, the assumption of similarity reduces the number of independent variables to one, which makes it possible to recast the conservation equations for mass and momentum into a set of two simultaneous ordinary differential equations ... 

Now the distribution of the gas-dynamic variables can be computed from the isen-tropic relations ... 

Once the piston-driven flow field is known, the flame-driven flow field is found by fitting in a steady flame front, with the condition that the medium behind it is quiescent. This may be accomplished by employing the jump conditions which relate the gas-dynamic states on either side of a flame front. The condition that the reaction products behind the flame are at rest enables the derivation of expressions for the density ratio, pressure ratio, and heat addition... 

If the values of the gas dynamic variables are known, these expressions may be evaluated for any position throughout the flow field. The location of the flame front is found where Q matches the heat of combustion of the fuel-air mixture in question. If the coordinate of the front X, is known, the burning velocity Mach number can be computed from... 

Gas Dynamics Resulting from a Prescribed Energy Addition... 

Generally speaking, tbe flow field induced by a gas explosion is characterized by two different gas-dynamic discontinuities ... 

In general, discontinuities constitute a problem for numerical methods. Numerical simulation of a blast flow field by conventional, finite-difference schemes results in a solution that becomes increasingly inaccurate. To overcome such problems and to achieve a proper description of gas dynamic discontinuities, extra computational effort is required. Two approaches to this problem are found in the literature on vapor cloud explosions. These approaches differ mainly in the way in which the extra computational effort is spent. 

As a consequence of implicit mass conservation, the gas-dynamic conservation equations, expressed in Lagrangean form, can describe contact discontinuities. To prevent oscillating behavior in places where shock phenomena are resolved in the... 

Finite-difference schemes used to solve Lagrangean gas dynamics have been described many times (Richtmyer and Morton 1967 Brode 1955, 1959 Oppenheim 1973 Luckritz 1977 MacKenzie and Martin 1982 Van Wingerden 1984 and Van den Berg 1984). 

Whereas Fishbum was mainly interested in the detonative mode of explosion, Luckritz (1977) and Strehlow et al. (1979) focused on the simulation of generation and decay of blast from deflagrative gas explosions. For this purpose, they employed a similar code provided with a comparable heat-addition routine. Strehlow et al. (1979), however, realized that perfect-gas behavior, which is the basis in the numerical scheme for the solution of the gas-dynamic conservation equations, is an idealization which does not reflect realistic behavior in the large temperature range considered. 

To overcome this problem, they proposed a working-fluid heat-addition model. This model implies that the gas dynamics are not computed on the basis of real values for heat of combustion and specific heat ratio of the combustion products, but on the basis of effective values. Effective values for the heat addition and product specific heat ratios were determined for six different stoichiometric fuel-air mixtures. Using this numerical model, Luckritz (1977) and Strehlow et al. (1979) systematically registered the properties of blast generated by spherical, constant-velocity deflagrations over a large range of flame speeds. 

Fishbum et al. (1981) used the HEMP-code of Giroux (1971) to simulate gas dynamics resulting from a large cylindrical detonation in a large, flat, fuel-air cloud containing 5000 kg of kerosene. Blast effects were compared with those produced by a 100,000-kg TNT charge detonated on the ground. 

The solid lines in Figure 4.5 represent extrapolations of experimental data to full-scale vessel bursts on the basis of dimensional arguments. Attendant overpressures were computed by the similarity solution for the gas dynamics generated by steady flames according to Kuhl et al. (1973). Overpressure effects in the environment were determined assuming acoustic decay. The dimensional arguments used to scale up the turbulent flame speed, based on an expression by Damkohler (1940), are, however, questionable. 

Hjertager, B. H. 1985. Computer simulation of turbulent reactive gas dynamics. Modeling, Identification and Control. 5(4) 211-236. 

Istratov, A. G., and V. B. Librovich. 1969. On the stability of gas-dynamic discontinuities associated with chemical reactions. The case of a spherical flame. Astronautica Acta 14 453-467. 

Van den Berg, A. C. 1989. RE AG AS—a code for numerical simulation of 2-D reactive gas dynamics in gas explosions. TNO Prins Maurits Laboratory report no. PML1989-IN48. 

Liepmann, H. W., and A. Roshko. 1967. Elements of Gas Dynamics. New York John Wiley and Sons. 

---