An Application of Roe s High Resolution Scheme to Transonic Two-Phase Flow through Nozzles

Transcription

1 An Application of Roe s High Resolution Scheme to Transonic Two-Phase Flow through Nozzles M. J. Kermani 1 Energy Laboratory, Department of Mechanical Engineering Amirkabir University of Technology (Tehran Polytechnic) Tehran, A. G. Gerber Department of Mechanical Engineering University of New Brunswick Fredericton, New Brunswick, Canada, E3B 5A3 J. M. Stockie Department of Mathematics Simon Fraser University Burnaby, British Columbia, Canada, V5A 1S6 A high resolution flux difference splitting scheme of Roe has been extended to transonic twophase liquid and vapor mixture through nozzles. To discretize the convective terms in dry regions, the pressure (p), temperature (T ), and velocity (u) are extrapolated to cell faces by the MUSCL approach using a third-order upwind-biased scheme, while in wet regions T, u and the quality (χ) are used for extrapolation purposes. Pressure in moist regions is, therefore, equated to the saturated pressure at the local temperature. The Roe-averaged values at the cell faces are obtained based on mixture (liquid + vapor) properties at the sides of the cell faces. Similar to solely dry cases, the van Albada flux limiter is used to prevent spurious numerical oscillations. The present model can also be used as a base-ground if an analytical solution for isentropic expansion of two-phase steam flow is sought. The results are validated with experimental data of Moore et al. Keyword: Roe scheme, high resolution MUSCL, moisture prediction, thermodynamic equilibrium 1 Introduction Condensing flow occurs in a wide range of industrial applications, from very low speed flows such as in fuel cells [1, 2, 3], pure steam flow through the components of steam power plant turbines [4, 5], to high speed cases such as flow of moist air mixtures in aeronautical applications [6]. In all of these situations, an accurate prediction of moisture level is a vital issue, and so the choice of an appropriate numerical scheme is essential. The high resolution upwind scheme of Roe [7] is particularly useful for compressible flows; especially for Mach numbers greater than 0.5, because of its non-diffusive nature in these flow 1 Corresponding Author, mkermani@aut.ac.ir 1

2 regimes. It is also an Approximate Riemann Solver (ARS) and so is computationally much more efficient compared the Exact Riemann Solvers (ERS) used in a Godunov scheme. These features of the Roe scheme have made it one of the most popular density-based schemes for compressible flows. The purpose of the present work is to extend Roe s high resolution Flux Difference Splitting scheme to compressible two-phase (liquid + vapor) flows. Roe scheme has initially been developed for solely single phase (dry) flows. A common approach for dry flows, when determining left and right values at the cell faces, is to extrapolate the primitive variables (for example, pressure p, temperature T, and velocity, u) to cell faces using a high resolution MUSCL approach [8]. These primitive variable are then used to determine the so-called Roe-averaged parameters corresponding to density (ˆρ), velocity (û) and total enthalpy (Ĥ). Within wet regions, however, the primitive variables p and T are not independent and so a suitable substitute must be chosen to replace either p or T. On the other hand, speed of sound, which are used in the eigenvalues of the Jacobian flux matrix and determines the slope of the characteristic lines, exhibits a jump across the saturation line owing to discontinuities in the partial derivative of the pressure across the saturation line (see Eqn. 18). Various formulae are given in the literature for the speed of sound in the two-phase regions [9, 10], with the simplest being the frozen speed of sound, which is determined based solely on the dryphase parameters of the mixture (i.e., the vapor part only). A different choice for speed of sound can only affect the solution in the transient region and the convergence rate to the steady-state. A sensitivity study of the steady-state solution to the frozen value of speed of sound ± 20% have been performed, and identical answers for the steady solution have been obtained in all cases (see Sec. 2.7). In the present study, p is replaced by the mixture quality χ, so that in wet regions, T, u and χ are used for the third order MUSCL extrapolation. The pressure within wet regions is simply evaluated using the saturated pressure at the local temperature. To determine a Roe-averaged value in the twophase region, the density and total enthalpy for the mixture (liquid + vapor) are used at the left and right sides of the cell faces. The numerical method is spatially third-order accurate, with an explicit time-marching method that gives second-order accuracy in time. Any spurious numerical oscillations in the present high resolution computations have been damped by using the van Albada flux limiter [11]. The expansion shocks have been avoided using the entropy correction formula given by Kermani and Plett [12]. This method is applied to quasi one-dimensional condensing flows through converging-diverging nozzles through an isentropic process, as shown in Fig. 1. However, further generalization to multi-dimensions is straight forward. The present model can also be used as a base-ground if an analytical solution for isentropic expansion of the two-phase steam is sought [13]. The numerical results of the present study are validated with experiments of Moore et al. [14]. 2 Governing Equations The governing equations of fluid motion for quasi one-dimensional, unsteady, compressible flow in full conservative form with no body force can be written as (see Ref. [15]): F (SQ) + t x H s = 0, where x and t are the space and time coordinates, S is the cross-sectional area of the duct. The quantities Q, F, and H s are, respectively, the conservative vector, the flux vector and the source (1) 2

3 term, and are given by: ρ Q = ρu, ρe t F = S ρu ρu 2 + p ρuh, H s = ds dx Here, ρ is the mixture (vapor + liquid) density, u is the velocity, e t the total mixture energy, p the pressure, and H the total mixture enthalpy. For the present study of low pressure steam flows of below one atmosphere, it is sufficient to use the ideal gas equation of state p = ρ g RT wherein ρ g is the gas density and R is the gas constant (= J/kg.K for vapor). In dry regions, ρ g is identical to the mixture density, while in wet regions ρ g = ρχ, where χ is the quality of the mixture (i.e., the mass fraction of the vapor to that of the mixture). In the present study, the volume of the liquid is assumed to be much smaller than that of the vapor and hence is ignored. The enthalpy of evaporation h fg is obtained as follows: h fg = e fg + RT where e fg is the change in internal energy from the saturated liquid state to that of the saturated vapor. A second-order polynomial can accurately represent the relationship between e fg and T, and the coefficients are provided in Appendix A. The internal energy of the vapor, e g, is determined by assuming a constant value for the specific heat at constant volume; that is, e g = C v T, where C v = R/(γ 1), with γ = 1.32 for vapor. Equation 4 provides the saturated vapor internal energy with e g = C v T sat, where can be used to obtain the saturated liquid internal energy: e f = e g (T sat ) e fg. In the present study, entropy is used only for post-processing purposes, and its calculation is addressed in Appendix A. 2.1 Time Discretization For the time discretization, an explicit, two-step scheme belonging to the Lax-Wendroff family of predictor-corrector methods has been used to step the solution from time level n to n + 1 (see for example [16]). The predictor step determines the flow conditions at an intermediate step (time level n + 1/2): 1 [ (SQ) n+1/2 (SQ) n] + 1 [ F n E t/2 x F ] n W H n s = 0, (6) where F n and F n are the numerical fluxes evaluated at the east (E) and west (W) faces of the E W control volume, the evaluation of which are addressed in Section 2.3. The predictor step is followed by a corrector step, which involves a central difference approximation in time around n + 1/2 implemented as follows: 1 [ (SQ) n+1 (SQ) n] + 1 [ F n+1/2 E t x 0 p 0. (2) (3) (4) (5) ] F n+1/2 W H n+1/2 s = 0. (7) 3

4 2.2 Space Discretization The spatial discretization for the numerical scheme is determined by the formulae used to compute numerical fluxes on cell faces. A third order upwind-biased algorithm with a MUSCL extrapolation strategy [8] is applied in this study to obtain left (L) and right (R) values for the flow conditions at the cell faces. In this approach, the nodal values are assumed to be known and extrapolation provides values on both sides of each cell face. In the present study, the L and R values are used to obtain the Roe-averaged conditions, [7], on the cell faces. Various parameters are used in the literature for extrapolation purposes, [17], including the elements of the conservative vector Q, the primitive variables p, ρ, T, u, etc., or even the components of the flux vector F. We have chosen to use the primitive variables in the present study. For the dry region, values for the primitive variables p, T and u at cell faces uniquely fix the thermodynamic state point. However, for the computation of wet cell faces, pressure and temperature are not independent parameters, and so the quality (χ) is extrapolated to the cell face instead of pressure. Therefore, local pressure is equated to the saturated pressure at the local mixture temperature. We use the following third-order extrapolation scheme: q L E = q j [(1 κ) W q + (1 + κ) E q], q R E = q j [(1 κ) EE q + (1 + κ) E q], (8) where j corresponds to any arbitrary nodal point, and q represents either of p, u, or T if the flow is dry at the cell faces; otherwise q represents T, u, or χ. In these formulae, a subscript E or W refers to the east or west face of a control volume respectively, and W q = q j q j 1, E q = q j+1 q j, and EE q = q j+2 q j+1. The third-order upwind biased scheme corresponds to a choice of κ = 1/ The Numerical Flux Roe The numerical flux Roe, used in the present study, is obtained from (for more detail the reader is referred to either of the Ref. [20], [18] or [16]): F E = 1 ( ) F L 2 E + FE R 1 3 ˆλ (k) E δw (k) (k) E ˆT E (9) 2 k=1 where λ is the eigenvalue of the Jacobian flux matrix, T represents the corresponding eigenvector, and δw is the wave amplitude vector. In Eqn. 9, k corresponds to each wave propagating in the x t domain. An analogous formula can be written for the flux vector F W on the west face. 2.4 Roe s Averaging for Wet Flow Computation The numerical flux for Roe s scheme is calculated at the so-called Roe-averaged value obtained from the L and R states on both sides of a cell face. For dry flow conditions, this has been well explained in several texts (see, for example, [16] or [18]). For wet flow conditions, it should be noted that gas properties are taken to be those of the mixture; for example, density and total enthalpy at the Roe-averaged condition are obtained for the east face of a control volume by: ˆρ E = ρ L E ρr E (10) Ĥ E = ρ L E HL E + ρ R E HR E ρ L E + ρ R E (11) 4

5 where all the properties in wet regions correspond to the mixture (liquid+vapor) values. For example, ρ L E = (ρ g) L E / (χ)l E (12) where ρ g = p sat /(RT) for wet flow, and similarly: H L E = (h f) L E + (χ)l E (h fg) L E }{{} mixture enthalpy [ ] (u) L 2. (13) E As noted in Eqn. 13, h f +χh fg represents the mixture (liquid+vapor) static enthalpy, shown by h m in Fig. 7 (Left). It is also noted that for an adiabatic, steady and two-phase flow, likewise single phase cases, the stagnation enthalpy (h 0 ) remains constant throughout the flow field, as shown in Fig. 7 (Left). 2.5 Flux Limiter Spurious numerical oscillations are unavoidable in these high resolution computations, but can be prevented using the van Albada flux limiter [11]. This limiter has been reported and implemented in various forms in the literature (see [17] and [19]). The following form of van Albada s limiter has been used in the present work (and in previously reported results [20]), and has been shown to minimize spurious numerical oscillations and give better convergence [21, 22]: q L E = q j + φ j 4 [(1 κ) W q + (1 + κ) E q], q R E = q j+1 φ j+1 4 [(1 κ) EE q + (1 + κ) E q]. (14) The limiter function φ depends on forward and backward differences according to: φ j 2( W q)( E q) + ε ( W q) 2 + ( E q) 2 + ε, (15) where ε is a small parameter that prevents indeterminacy in regions of zero gradients, i.e. where ( W q) = ( E q) = Entropy Correction Roe s scheme incorrectly captures expansion shocks in the regions where eigenvalues of the Jacobian matrix of flux vector vanish. These regions are sonic regions or stagnation points. To avoid expansion shocks from appearing, several entropy correction formulae were reported in the literature, with one of the most popular being that of Harten and Hyman [23]. A modified version of this formula which totally eliminates expansion shocks from sonic expansion regions without affecting the rest of the domain has been previously developed, [12], and used here: (ˆλ 2 + ɛ 2 )/2 ɛ, if λ < ɛ, ˆλ new = ˆλ if λ ɛ (16) 5

6 where ɛ = 4.0 max [ 0, (ˆλ λ L ), ( λ R ˆλ )], (17) and ˆλ is the eigenvalue of the Jacobian flux matrix determined at Roe-averaged condition, and λ L and λ R are the eigenvalues determined at L and R conditions, respectively. 2.7 Speed of Sound An accurate estimate of the speed of sound is essential for time-varying upwind algorithms such as the one we employ here. This is because the wave speeds and correspondingly the eigenvalues of the Jacobian matrix of the flux vector, λ 1,2,3 = u, u + a, and u a, determine the directions along which information propagates within the x t plane. For solely dry (or single-phase) flows the speed of sound is a well-defined thermodynamic property, given by: ( ) p a 2 =, (18) ρ s where s is the entropy. However, the speed of sound is not so well-determined in wet flows, and in general it cannot be treated as a thermodynamic property if thermodynamic equilibrium is not prevailed, [9]. On the other hand, even for thermodynamically equilibrium cases computing Eqn. 18 is a cumbersome task, and therefore using it as a part of an iterative algorithm is not suggested. Alternatively, Guha (1995) has proposed the following algebraic equation, as an approximation for Eqn. 18, to conveniently obtain the speed of sound for only wet steam flows in equilibrium thermodynamic models (see [9]): a 2 χγrt = γ [1 RT/h fg (2 CT/h fg )], (19) where C = C p + (1 χ)c l /χ. The quantities C p and C l are the specific heat values at saturated vapor and liquid states, respectively. Speed of sound predictions by Eqns. 18 and 19 are compared and shown in Fig. 2, illustrating a very good agreement in wet regions. However as shown in this figure, Eqn. 18 exhibits a discontinues behavior across the saturation line, due to discontinuities in the partial pressure derivatives. This may lead to some instabilities when numerical solutions are sought. Another approach to calculating the speed of sound is using the so-called frozen value, which can be derived from Eqn. 18 solely based on the gas side (dry side) parameters and is valid only for under isentropic conditions. This approach leads to: a 2 = γrt. Figure 2 shows a plot of the speed of sound obtained along an isobar process. As shown in this figure, the frozen value of the speed of sound matches very well with that obtained by Eqn. 18 in the dry region. Furthermore, the sound speed provided by Guha (Eqn. 19) is indistinguishable from that of Eqn. 18 in the wet region. There is no single formulation capturing the speed of sound in both dry and wet regions, and the formula of Eqn. 18 is not a computationally efficient choice. We have performed a sensitivity study of the steady-state solution to the frozen value of speed of sound ± 20%, and identical answers for the steady solution have been obtained in all cases. Consequently, for the present computation containing both dry and wet regions with our interest being only on the steady-state solution, the frozen value of speed of sound (i.e. Eqn. 20) has been used. (20) 6

7 2.8 Moisture Evaluation As the solution marches in time, the conservative vector Q is obtained at each time level. Q provides values for mixture density and total internal energy. Knowing the value of velocity at the same time level, a static value of the internal energy can be obtained. The thermodynamic state point can then be fixed based on current values of mixture density and internal energy. The moisture content (if present) for this equilibrium state can then be determined from: χ = e m e f e fg, where e m is the mixture (liquid+vapor) internal energy. (21) 2.9 Boundary Conditions The inflow is assumed subsonic and dry, where the stagnation pressure (P 0in ) and temperature (T 0in ) are specified (see Fig. 1). At the exit plane for which supersonic conditions prevail, all flow properties are extrapolated from the interior domain. 3 Results and Discussions 3.1 Numerical Validation To validate the results of the present computation, two nozzle geometries are chosen, labeled as nozzles (A) and (D) in Fig. 3. The computed pressure distributions along the nozzle axis are compared with those from experiments of Ref. [14] using nozzles (A) and (D). Nozzle (A) has the highest expansion rate of these series of nozzles (i.e. the largest exit to throat area ratio) and nozzle (D) has the lowest expansion rate. The inflow stagnation conditions are: P 0in = 25 kpa, taken the same for all nozzles geometries, T 0inA = K, and T 0inD = K (see Fig. 1). The outflow conditions for all cases are supersonic. Figure 4 compares the computed pressure distributions along the nozzle with the experimental values obtained for nozzles (A) and (D). As shown in these figures, the computed and experimental results agree well, with differences between them owing to two reasons: (1) different choked mass flows through the nozzles, and (2) irreversibility introduced by a condensation shock causing a slight rise in the pressure. These issues are further explained below: 1. Rapid expansion through a nozzle forces the flow to experience non-equilibrium conditions, and therefore condensation does not ensue until a significant amount of supercooling is achieved. This rapid expansion delays the moisture generation seen in experiment to a post-throat location. Simulated results from several different models are consistent with the dry nozzle throat conditions seen in experiments, including: (i) a non-equilibrium computation [4], (ii) a totally dry flow computation with no condensation used in the present study, and (iii) using Eqn. 23 given in Sec All of these results indicate a value of ṁ dry = kg/s for nozzle A (to within 4 decimal digits of accuracy in all cases). However, according to the equilibrium thermodynamic model, moisture is formed upstream of the throat as soon as the saturation line is crossed (represented as Point S in Fig. 1). In the computations reported here which assume moisture formation under equilibrium conditions, 7

8 the flow is moist at the nozzle throat (i.e. at X = 0 in Fig. 6) which causes the choked mass flow rate to differ slightly from experiments. This moist-throat mass flow for nozzle A is computed in Sec. 3.2 as ṁ wet = kg/s. To adjust the mass flow of our equilibrium model case to that of the experiment, we computed nozzle A with an enlarged throat area in such a way that the mass flow kg/s is met for the wet case. The corrected (modified) pressure distribution is shown in Fig The slight rise in pressure exhibited by the experimental data in Fig. 4-(Top) is attributed to the irreversibility caused by a condensation shock, which the present model is not capable of capturing because of our assumption of equilibrium thermodynamic conditions. A condensation shock, similar to an oblique shock wave in single phase flow, possesses supersonic post shock conditions. Both types of shock (condensation and oblique shock waves) give rise to so-called aerodynamic losses; in addition, condensation shocks generate thermodynamic losses. A complete study of aerodynamic and thermodynamic losses, and condensation shocks in the context of non-equilibrium steam flow was performed earlier and reported in Ref. [4]. 3.2 Results Stagnant flow introduced at an imaginary reservoir upstream of the inflow plane is accelerated to the inlet boundary. This flow is continuously accelerated along the nozzle and crosses the saturation line, say at point S upstream of the throat (see Figs. 1 and 6). At the converged state, a uniform profile for the mixture mass flow rate (ṁ = ṁ l + ṁ v ), mixture total enthalpy (h 0 = h m + u 2 /2, where h m = h f + χh fg ) and mixture entropy (s m = s f + χs fg ) has been obtained along the nozzle. These profiles are shown in Fig. 7 for h 0 and s m. As shown in Fig. 7-(Left) total enthalpy (denoted by h 0 ) is uniform along the nozzle axis at the converged state, showing that the energy equation, with adiabatic flow condition, is well satisfied. Also shown in this figure is the mixture enthalpy, h m, which decreases as the flow accelerates along the nozzle with increasing kinetic energy. Mixture entropy (s m ) is also uniform, as shown in Fig. 7-(Right), representing an isentropic process through the nozzle. The mixture mass flow profile along the nozzle is also perfectly uniform (not shown here) with a relative value of inflow and outflow differences as: ṁ out ṁ in < = 0.008%. (22) ṁ in As mentioned in Sec. 3.1, according to the equilibrium model, moisture is generated as soon as the saturation line is crossed. Unlike the dry flow case, where the inflow boundary condition and the throat area determine the maximum mass flow through a nozzle, in the wet flow computation the amount of moisture at the throat is also important and reduces the choked mass flow. This is because the latent heat released from the condensate flows toward the vapor phase and increases the vapor side stagnation temperature, [13], reducing the choked mass flow. Using the present twophase equilibrium model, the mass flow rate in nozzle (A) with p 0 in = 25 kpa and T 0in = K is computed as kg/s. However, the choked mass flow rate under dry throat conditions, as is the case from the experiment, can be obtained by: ṁ dry = p 0A ( ) (γ+1)/(γ 1) 2 γ (23) RT0 γ + 1 8

9 (see Moran and Shapiro [24], for example). Equation 23 yields a mass flow value of kg/s which is an overestimate of approximately 3.5% to the wet-throat computation using equilibrium model. In Eqn. 23, A represents the sonic throat area. The Mach number profile is also shown in Fig. 8, which is obtained based on the frozen value of speed of sound (Eqn. 20). As shown in this figure, the Mach number at the nozzle throat is not perfectly obtained as unity, because flow is wet at the throat and therefore the speed of sound differs from its frozen value at throat. 4 Conclusion The highlights of the present paper are as follows: A method for the computation of moisture content in subsonic-supersonic steam flow has been described based on an equilibrium thermodynamic model, and implemented using Roe s high resolution scheme. The method is third order accurate in space with a second order accuracy in time. In this method the primitive variables p, T and u are extrapolated to the cell faces in dry regions, while in wet regions T, u and χ are used. Pressure in moist regions is, therefore, equated to saturated pressure at the local temperature. Therefore, Roe-averaged values are obtained based on mixture properties (liquid + vapor) at the cell faces. The physical features of the isentropically expanding two-phase steam flow are correctly captured with the present numerical model, and described. Three various formulae representing the speed of sound in two-phase flows under equilibrium conditions were reviewed. The results compare fairly well versus experimental results on non-equilibrium condensation where the effect of throat moisture levels on choked mass flow is considered. The choked mass flow through a nozzle, unlike the dry flow case, cannot be determined directly based only on inflow stagnation conditions, nozzle geometry and fluid properties (Eqn. 23). In addition, the moisture content at the throat also affects the mass flow rate through the duct relative to a non-equilibrium prediction when moisture formation is delayed until a location beyond the throat. Acknowledgments Financial support was provided by Amirkabir University of Technology and the MITACS Network of Centres of Excellence. References [1] Weber, A.Z. and Newman, J., Modeling Transport in Polymer-Electrolyte Fuel Cells, Chem. Rev., Vol. 104, No. 10, pp (2004). 9

10 [2] Wang, C.Y., Fundamental Models for Fuel Cell Engineering, Chem. Rev., Vol. 104, No. 10, pp (2004). [3] Kermani, M.J., Stockie, J.M. and Gerber, A.G., Condensation in the Cathode of a PEM Fuel Cell, Proceedings of the 11th Annual Conference of the CFD Society of Canada, Vancouver, BC, Canada (2003). [4] Kermani, M.J. and Gerber, A.G., A General Formula for the Evaluation of Thermodynamic and Aerodynamic Losses in Nucleating Steam Flow, Int. J. Heat Mass Transfer, Vol. 46, pp ). [5] Gerber, A.G. and Kermani, M.J., A Pressure Based Eulerian-Eulerian Multi-Phase Model for Non-Equilibrium Condensation in Transonic Steam Flow, Int. J. Heat Mass Transfer, Vol. 47, pp (2004). [6] Yamamoto, S., Computation of Practical Flow Problems with Release of Latent Heat, Energy, Vol. 30, pp (2005). [7] Roe, P.L., Approximate Riemann Solvers, Parameter Vectors and Difference Schemes, J. Comput. Phys., Vol. 43, pp (1981). [8] van Leer, B., Towards the Ultimate Conservation Difference Scheme, V, A Second Order Sequel to Godunov s Method, J. Comput. Phys., Vol. 32, pp (1979). [9] Guha, A., Two-Phase Flows with Phase Transition, von Karman Institute Lecture Series , May 29-June 1 (1995). [10] Paillere, H., Kumbaro, A., Viozat, C., Broquet, A. and Corre, C. A Comparison of Roe, VFFC and AUSM+ Schemes for Two-Phase Water/Steam Flows, Proceedings of Godunov Methods, Theory and Applications Conference, Oxford, UK, Oct. (1999). [11] van Albada, G. D., van Leer, B. and Roberts, W. W., A Comparative Study of Computational Methods in Cosmic Gas Dynamics, Astron. Astrophys., Vol. 108, pp (1982). [12] Kermani, M. J. and Plett, E. G., Modified Entropy Correction Formula for the Roe Scheme, AIAA Paper # (2001) [13] Zayernouri, M., and Kermani, M. J., Development of an Analytical Solution for Compressible Two-Phase Steam Flow, Canadian Journal of Mechanical Engineering: Transaction of the CSME (Accepted), (2006). [14] Moore, M.J., Walters, P.T., Crane, R.I. and Davidson, B.J., Predicting the Fog Drop Size in Wet Steam Turbines, Inst. of Mechanical Engineers (UK), Wet Steam 4 Conference, University of Warwick, paper C37/73 (1973). [15] Hoffmann, K.A. and Chiang, S.T., Computational Fluid Dynamics for Engineers, Vol. II, Engineering Education Systems, Wichita, Kansas, USA (1993). [16] Tannehill, J.C., Anderson, D.A. and Pletcher, R.H., Computational Fluid Mechanics and Heat Transfer, Second Edition (1997). 10

11 [17] Thomas, J.L. and Walters, R. W., Upwind Relaxation Algorithms for Navier-Stokes Equations, AIAA J., Vol. 25, pp (1987). [18] Hirsch, C., Numerical Computation of Internal and External Flows, John Wiley & Sons, Vol. 2 (1990). [19] Amaladas, J.R., Implicit, multigrid and Local-Preconditioning Procedures for Euler and Navier-Stokes Computations with Upwind Schemes, Department of Aerospace Engineering, Indian Institute of Technology, Bangalore, India, June (1995). [20] Kermani, M.J., Development and Assessment of Upwind Schemes with Application to Inviscid and Viscous Flows on Structured Meshes, Ph.D. thesis, Department of Mechanical & Aerospace Engineering, Carleton University, Canada (2001). [21] Amaladas, J.R., private communication, Indian Institute of Technology, Bangalore, India, March (2000). [22] Thomas, J.L., private communication, NASA Langley, March (2000). [23] Harten, A. and Hyman, J.M., Self-Adjusting Grid Methods for One-Dimensional Hyperbolic Conservation Laws, J. Comput. Phys., Vol. 50, pp (1983). [24] Moran, M.J. and Shapiro, H.N., Fundamentals of Engineering Thermodynamics, 4 th Edition, John Wiley & Sons (1998). Nomenclature a speed of sound (m/s) (Eqns ) A sonic throat area (m 2 ) (Eqn. 23) A 0 A 5 constants of the curve-fit for the p sat polynomial (Eqns. 24 and 25) C l liquid specific heat(j/kg.k) (Eqn. 19) C p vapor isobaric specific heat (J/kg.K) C v vapor isochoric specific heat (J/kg.K) e internal energy (J/kg) E 0 E 2 constants of the curve-fit for the internal energy of evaporation (Eqns. 26 and 27) F flux vector = S [ρu,ρu 2 + p,ρuh] t h fg enthalpy of evaporation (J/kg) H total mixture enthalpy (J/kg) H s source term k wave propagation number (Eqn. 9) ṁ mass flow rate (kg/s) n time level (Eqns. 6 and 7) p pressure (Pa) q a representative of primitive variable (q {p,t,u} for dry case, or q {T,u,χ} for wet case) Q conservative vector = [ρ,ρu,ρe t ] t R gas constant ( = J/kg.K for vapor) 11

12 s entropy (J/kg.K) S nozzle s cross sectional area (m 2 ) t time coordinate (s) T temperature (K), and Eigenvectors (Eqn. 9) u velocity (m/s) x space coordinate (m) Greek symbols γ specific heat ratio ( = 1.32 for vapor) ɛ entropy correction band (Eqn. 17) ε a specified small parameter in the size of O(10 6 ) (Eqn. 15) κ dimensionless number ( = 1/3 for upwind biased scheme) λ eigenvalue of the Jacobian flux matrix ρ mixture density (vapor + liquid) (kg/m 3 ) ρ g gas (vapor) density (kg/m 3 ) φ the flux limiter function (Eqn. 15) χ the quality (mass fraction of vapor to that of the mixture) q increment in primitive variables (Eqns. 8, 14 and 15) t integration time step (s) (Eqns. 6 and 7) x spatial step size (m) (Eqns. 6 and 7) δw wave amplitude vector (Eqn. 9) Subscripts dry non-condensable case, Eqn. 23 E East face of control volume W West face of control volume EE East of east face of control volume f fluid (liquid) fg interval of latent heat g gas (vapor) in inflow j nodal index (Eqns. 8, 14 and 15) m mixture out outflow sat saturation 0 stagnation (or total) condition Superscripts ˆ l L Roe s averaged state liquid Left side of east cell face 12

13 n time level representing iteration number R Right side of east cell face s isentropic process, Eqn. 18 Appendix A Saturated Pressure Value. The saturation pressure for steam is determined by a fifth order polynomial least square curve fit to the steam data taken from [24], given by: p sat = A 5 (T t 0 ) 5 + A 4 (T t 0 ) 4 + A 3 (T t 0 ) 3 + A 2 (T t 0 ) 2 + A 1 (T t 0 ) + A 0, (24) where p and T are in terms of Pa and K, t 0 = K, accurately predicting saturated pressure in the range of T = [293, 423] K. The coefficients are A 5 = E-6 A 4 = E-4 A 3 = E-3 A 2 = E+0 A 1 = E+0 A 0 = E+2 (25) Internal Energy. [24], as Internal energy is obtained by a parabolic curve fit to the steam data taken from e fg = E 2 (T t 0 ) 2 + E 1 (T t 0 ) + E 0, (26) where T is measured in K, e fg is in J/kg, E 2 = E+0 E 1 = E+3 E 0 = E+6 (27) Therefore, h fg = e fg + RT, where R = J/kg.K for vapor. Entropy. The entropy of the mixture is determined from s = s f +χs fg, where s fg = h fg /T and s g is obtained from s g = C p ln T R ln p, and s f = s g s fg. 13

14 P 0 in T T 0 in Inlet P in T in S P e Throat T e Exit s Figure 1: Schematic of an isentropic expansion of two-phase pure steam under equilibrium thermodynamic model, in which the liquid phase is generated as soon as the saturation line is crossed (Point S). The flow contains a finite content of moisture at the throat. Speed of Sound for Steam Flow along an Isobar line P= 14 kpa (dp/drho) s Guha (1995) Speed of Sound (m/s) Dry Region Frozen Value Wet Region Mixture Density (kg/m 3 ) Figure 2: The speed of sound along an isobar line, determined in three different ways. 14

15 0.14 Point X (m) S (m 2 ) S (m 2 ) Nozzle A Nozzle D (inlet) S, Cross Sectional Area (m 2 ) (throat) (exit) Nozzle D Nozzle A X (m) Figure 3: Geometry of the nozzles A and D used in the present computations, taken from Moore et al. (1973). 15

16 Pressure distribution along the Moore A nozzle P 0 in = 25 kpa, T 0 in = K Present computation (Equilibrium Model) Experiment, Moore et. al. (1973) p (Pa) X (m) Pressure distribution along the Moore D nozzle P 0 in = 25 kpa, T 0 in = K Present computation (Equilibrium Model) Experiment, Moore et. al. (1973) p (Pa) X (m) Figure 4: Comparison of pressure distribution along the nozzle centerline. (Top)- Nozzle Moore A, and (Bottom)- nozzle Moore D. 16

17 p (Pa) Corrected pressure, based on enlarged throat area to meet mass flow of: kg / s Pressure, based on original throat area giving the mass flow of: kg / s X (m) Figure 5: Corrected pressure distribution for Moore A nozzle, compared with those profiles given in Fig. 4-(Top) Moisture Prediction Wetness Fraction Throat Nozzle (A) Nozzle (D) 0 S A S D X (m) Figure 6: Wetness fraction (1 χ) profile along the nozzle axes. Points S A and S D correspond to the moisture onset points. The flow is wet at the nozzle throat (X = 0) according to the present equilibrium model. 17

18 h h g, h m, h 0 (kj/kg) S A u 2 / 2 h g h m s g, s m kj/(kg.k) S A s g s m x (m) x (m) Figure 7: (Left)- Enthalpy profile along the nozzle A axis (it is noted that h 0 = h m + u 2 /2; where h m = h f + χh fg ). (Right)- Entropy profile along the nozzle A axis (it is noted that S m = S f + χs fg ). Points S A and X = 0 denote condensation onset point and throat, respectively. Nozzle (A) 2 ( P 0 ) in = 25 kpa, ( T 0 ) in = K Mach S A x (m) Figure 8: The Mach number profile along the nozzle A axis obtained based on a frozen speed of sound. The value of the Mach number at the throat is not unity because the speed of sound at the wet-throat differs from its frozen value. Points S A and X = 0 denote the condensation onset point and throat location, respectively. 18