DEVELOPMENT OF A NUMERICAL METHOD FOR SIMULATING TRANSONIC MULTIPHASE FLOWS F. Put, H.W.M. Hoeijmakers, P.H. Kelleners Engineering Fluid Dynamics Department of Mechanical Engineering University of Twente, The Netherlands P.O. Box 217, 7500 AE Enschede
[email protected]
F.A. Lammers Twister B.V. P.O. Box 60, 2288 AB Rijswijk The Netherlands
[email protected]
Abstract
A generally applicable numerical method has been developed to simulate the flow of a mixture of condensing real gases. The flow is described by the Euler equations of gasdynamics, whereas the thermodynamically non–equilibrium process of condensation is modeled by an integral description method, i.e. Hill’s method. The fluid considered, be in gaseous or liquid phase, may be a mixture of several different inert or condensing components. The development of an equation of state that accurately describes the behavior of a real gas (i.e. a gas at high pressure) is important, e.g. for natural gas applications. The developed equation of state satisfies Maxwell’s thermodynamic relations. The numerical method is a general 3-dimensional node-centered finite-volume method for unstructured tetrahedral meshes. A fractional, explicit, time-stepping method is used to solve the stiff system of partial differential equations describing the condensation process. So far only steady-flow cases have been considered. The first test case is the validation case of supercritical flow of an air/water mixture through a nozzle with very small cross-sectional area divergence downstream of the throat. The second test case is the transonic swirling flow of a natural gas, consisting of 20 components.
Keywords:
condensation, real gas, numerical method, vortical flow
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Introduction Condensation in flows of gas mixtures at transonic speed has been investigated by, amongst others, Hill [1], Wegener [2], Schnerr et al. [3] [4] [5] and van Dongen et al. [6] [7]. Expansion in nozzles of gases to supersonic speeds has often been used to investigate the physics of condensation. Also condensation in the flow around airfoil sections and in steam turbines has been investigated extensively. In this paper the implementation is described of models for condensing gas flows in a generally applicable numerical simulation method for three-dimensional flows of real gases.
1.
Governing Equations
The system of governing equations, containing the Euler equations (continuity, momentum and the energy equation) of gasdynamics and Hill’s moment equations (extended to multi-component condensing mixtures) describing the condensation process, can be cast in the following form [8] [9] ∂ ∂t
Z V
U dV +
Z
F~ · ~n dS =
∂V
Z
W dV.
(1)
V
The system of equations is closed by an equation of state for a real gas, since for the cases with high stagnation pressures the ideal gas law is no longer valid. High pressure effects can be incorporated in the perfect–gas law, by including a compressibility factor z p = zρRT.
(2)
For given R, the compressibility factor and the temperature are both functions of the thermodynamic state of the gas, i.e. z = z(ρ, e) and T = T (ρ, e). The Maxwell relations, derived from the first law of thermodynamics, provide a relation between the admissible forms for z(ρ, e) and T (ρ, e), see [9] for more details. Weighted least–squares fits are used to determine the coefficients in fully Maxwell-compatible fits for z(ρ, e) and T (ρ, e), within a specified range of p and T , employing a user-specified data-set (p, T, ρ, e). As an example, the real-gas model has been determined from a data-set describing the behavior of a mixture containing 20 components, with the (p, T ) values ranging from 15-100 bar and 180-300 K, respectively. Along with the main components of the mixture CH4 (≈ 81%) and N2 (≈ 14%) the mixture contains H2 O and further light and heavy hydrocarbons. From the differences between the fits and the original dataset for a typical 3D configuration, we observed that the maximum deviation is about 0.4%, while the average deviation
Development of a Numerical Method for Simulating Transonic Multi-phase Flows 3
is about 0.1%. For the current applications this is quite an acceptable fitting procedure.
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Numerical Method
The flow solver developed is based on a node-centered finite-volume formulation for tetrahedral meshes. The volumes are formed by the median dual mesh volumes around the vertices. The flux terms are computed based on upwind differencing. Both Van Leer’s flux vector splitting method for real gases [10] and Liou’s AUSM+ method [11] have been implemented for this purpose. Higherorder accuracy of the spatial differencing is achieved by linear extrapolation towards the cell-face, with the gradient of the conserved quantities calculated at the cell-center using a Green-Gauss formulation. A Van Albeda type limiter is used to achieve second-order monotonic behavior. The magnitude and variation of the source terms results in stiffness of the set of equations. This restricts the maximum allowable time-step for the present explicit time integration scheme. Therefore time integration is carried out in two stages, using a different method for each stage. First, an explicit multi-stage Runge-Kutta scheme is used to integrate the homogeneous system (i.e. with omitted source term) in time. Second, the source terms are calculated based on the homogeneous solution, and, with the flux terms omitted, a fractional time-stepping method is used to advance the solution in time. To improve the convergence characteristics of the numerical scheme, local time-stepping is used, restricting the method to steady-flow solutions.
3.
Results
Two cases computed with the 3D flow method are considered. Emphasis in these cases is on the condensation process. The first one is a case at atmospheric pressure, the second one is a case at high stagnation pressure.
Supercritical flow in A1–Nozzle The A1–nozzle is a nozzle with almost no divergence downstream of the throat section, see [5] for a description. The transonic flow in the nozzle is therefore very sensitive to variations in back pressure and inlet vapour fractions. The nozzle geometry and the flow in this nozzle have been extensively investigated and described by Schnerr and Adam [5]. These geometric features make this case very suitable for validation and verification of the numerical method. As it turns out, the results of the present numerical method and the numerical and experimental results presented by Adam [5], agree very well, see [8] and [9] for a full account of results.
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Twister configuration The Twister configuration consists of a tube in which swirl is generated by a wing–like geometry. fig. 1. The flowing medium is a natural gas such as described in the section Governing Equations.
Figure 1.
Layout of the Twister configuration
A nozzle is located upstream of the vortex generator. In the nozzle the flow expands to low pressure and temperature at a high cooling rate, resulting in homogeneous condensation. Since the outflow of this nozzle is supersonic, it can be treated separately. A quasi 1D version of the current flow solver was used to simulate the flow in the nozzle. The inflow conditions of the nozzle are p0 = 98 bar, T0 = 293 K. The flow conditions at the outlet of the nozzle were used as inflow conditions for the Twister configuration. The results are presented in fig. 2. All plots show the flow-field quantities on the upper-side of the delta wing and in a number of cross-flow planes normal to the tube-axis. As can be seen in fig. 2a, the flow is supersonic everywhere in the tube. Fig. 2a also shows that the Mach number in the vortex core increases further, leading to temperatures as low as 180K. The vortex core can easily be detected in fig. 2b, in the plot of the total pressure. The total liquid mass fraction is shown in fig. 2c. As can be seen, the highest liquid mass fractions occur in the vortex core. In figs. 2d and 2e the liquid mass fractions of C3 H8 and C9 H20 are shown. These components present the relatively light and heavy hydrocarbon components present in the mixture, respectively. As can be seen from these figures, the lighter hydrocarbons tend to condensate less than the heavier ones. Note that the liquid mass fractions of the individual components are shown as fraction of the total mass fraction of that component.
Development of a Numerical Method for Simulating Transonic Multi-phase Flows 5
M 1.65 1.35 1.05
a) ptot [bar] 100 80 60
b) g [g/kg] 14 9 4
c) C3H8 [g/gmax] 0.17 0.12 0.07 0.02
d) C9H20 [g/gmax] 1 0.95 0.9 0.85
e)
Figure 2. Flow in Twister configuration: a) Mach number; b) Total pressure; c) Total liquid mass fraction; d) Liquid mass fraction C3 H8 e) Liquid mass fraction C9 H20 . Flow is from left to right.
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4.
Conclusions
An unstructured-grid numerical method for the flow of transonic high total pressure mixtures of real gases with homogeneous condensation has been developed. It is based on the Euler equations of gasdynamics, Hill’s method of moments for the condensation process, extended to multi-component mixtures and a real-gas equation of state consistent with Maxwell’s thermodynamic relations. The results of the method have been validated with results for the transonic moist-air flow in a supercritical nozzle. The method has been applied to the flow in a high total pressure Twister configuration, which demonstrates the potential of the method to numerically simulate the transonic flow of such gas mixtures in complex configurations.
References [1] Hill, P.G., Condensation of Water Vapour during Supersonic Expansion in Nozzles, Journal of Fluid Mechanics, Vol. 25, part 3, pp. 593–620, 1966. [2] Wegener, P.P., Nonequilibrium Flows, Part 1, Marcel Dekker, New York and London, 1969. [3] Dohrmann, U., Ein numerisches Verfahren zur Berechnung station a¨ rer transsonischer Str¨omungen mit Energiezufuhr durch homogene Kondensation, PhD Thesis, Universit¨at Karlsruhe (TH), Germany, 1989. [4] Mundinger, G., Numerische Simulation instationa¨ rer Lavald¨usenstr¨omungen mit Energiezufuhr durch homogene Kondensation, PhD Thesis, Universit¨at Karlsruhe (TH), Germany, 1994. [5] Adam, S., Numerische und experimentelle Untersuchung instation a¨ rer D¨usenstr¨omungen mit Energiezufuhr durch homogene Kondensation, PhD Thesis, Universit¨at Karlsruhe (TH), Germany, 1996. [6] Luijten, C.C.M., Nucleation and Droplet Growth at High Pressure, PhD Thesis, Eindhoven University of Technology, the Netherlands, ISBN 90-386-0747-4, 1998. [7] Lamanna, G., On Nucleation and Droplet Growth in Condensing Nozzle Flows, PhD Thesis, Eindhoven University of Technology, the Netherlands, ISBN 90-386-1649-X, 2000. [8] Put, F., Kelleners, P.H., A Three Dimensional Unstructured Grid Method For Flows With Condensation, Presented at the ECCOMAS Computational Fluid Dynamics Conference 2001, Swansea, Wales, UK, September 4-7, 2001. [9] Kelleners, P.H., Put, F., Development of a Numerical Method for the Simulation of Condensing Real Gas Flows, Presented as paper 152 at the 4th International Conference On Multiphase Flow, New Orleans, Louisiana, May 27th - June 1st, 2001 [10] Shuen, J.S., Liou, M.S., van Leer, B., Inviscid Flux-Splitting Algorithms for Real Gases with Non-Equilibrium Chemistry, Journal of Computational Physics 90, pp. 371-395, 1990 [11] Liou, M.S., A Sequel to AUSM: AUSM+, Journal of Computational Physics 129, pp. 364382, 1996.
