Explicit calculation of compressible flow

The exact method for calculating compressible flow summarized in Section 6.4 and illustrated in Section 6.7 requires the iterative solution of a set of highly nonlinear simultaneous equations. However, it is shown in Appendix 2 that it is possible to use the results produced by the exact method as the basis for an approximate method of calculation that will 

Here fipa is the generalized, long-pipe approximation function  

A set of polynomial approximating functions are derived in Appendix 2 to be used with the explicit equations listed above. The critical pressure ratio just inside the pipe outlet is approximated by the polynomial  

Similarly the two terms needed to calculate the correction factor, fro, for a specified value of specific heat ratio, y, take the form of a point value, fro(ATr, 0.95)  

The value of fro at any given pair of values (Kj, Pi/pi) is given by an equation that is linear in Pi/pi  

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Appendix 2 Explicit calculation of compressible flow using approximating functions... 

The numerical jet model [9-11] is based on the numerical solution of the time-dependent, compressible flow conservation equations for total mass, energy, momentum, and chemical species number densities, with appropriate in-flow/outfiow open-boundary conditions and an ideal gas equation of state. In the reactive simulations, multispecies temperature-dependent diffusion and thermal conduction processes [11, 12] are calculated explicitly using central difference approximations and coupled to chemical kinetics and convection using timestep-splitting techniques [13]. Global models for hydrogen [14] and propane chemistry [15] have been used in the 3D, time-dependent reactive jet simulations. Extensive comparisons with laboratory experiments have been reported for non-reactive jets [9, 16] validation of the reactive/diffusive models is discussed in [14]. 

The remarks made in Section 8.1 translate across to the gas flow cases more or less word for word, except that the methods of Chapter 9 must now be allied to those of Chapter 6 in order to calculate gas flow through line and valve. But the more complicated equations for both line flow and valve flow render explicit solutions to the full set of equations impossible. Two implicit methods, the Velocity-Head Implicit Method (VHIM) and the Smoothed Velocity-Head Implicit Method (SVHIM), will be presented, where the solution process has been reduced to iteration on a single variable. The SVHIM is judged to be more accurate because it deals with the compressible-flow valve equations at all times. 

The method given in Sections 6.2 to 6.4 of Chapter 6 allows an iterative calculation of steady-state, compressible flow. However, the control engineer is likely to need to calculate the flow of a gas or vapour under a changing set of conditions, implying iteration at each timestep. An explicit method is obviously to be preferred, and this Appendix shows how it is possible to use the established basis of the exact but implicit method to construct an approximate but explicit method of calculation which preserves most of the accuracy. 

In this section, an R-field formulation for representing the convection of a two-gas mixture is given [12, 13, 15], This is an extension of the bond graph formulation for the forced convection of a compressible ideal gas given in [6], The details of the sub-model for modelling the convection of a two-component gas mixture are given in Fig. 10.3. The most important element in the expanded model of the MR element is the RS-field element (see Fig. 10.3). This element receives the downstream side temperature and the information of the valve position (x), the upstream side chemical potentials and temperature, and the downstream side chemical potentials to calculate the mass and entropy flow rates. Note that all these variables are inputs to the MR element. To maintain the clarity of the figure, the connections needed to explicitly show these modulations are not drawn. 

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