1 / 30

Introduction to Dense Gas Flows: Exploitation of Non-Classical Properties for Energy Applications

Introduction to ORCDense GasesThermophysical properties and dynamic behaviorGoverning equations and thermodynamic modelNumerical MethodResultsStudy of turbulent transonic flows of a dense gas past a NACA0012 airfoilFlow through a turbine cascade . Outline. Organic Rankine Cycles .

andrew
Download Presentation

Introduction to Dense Gas Flows: Exploitation of Non-Classical Properties for Energy Applications

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

    1. Introduction to Dense Gas Flows: Exploitation of Non-Classical Properties for Energy Applications Pietro Marco CONGEDO, PhD Student Cha-Cha Days 2006 Charleston, October 1 2006

    2. Introduction to ORC Dense Gases Thermophysical properties and dynamic behavior Governing equations and thermodynamic model Numerical Method Results Study of turbulent transonic flows of a dense gas past a NACA0012 airfoil Flow through a turbine cascade Outline

    7. Dense Gas Dynamics Study of the dynamic behavior of gases at pressures/densities of the order of magnitude of the liquid/vapor critical point The perfect gas law is replaced by more complex EOS: van der Waals, Redlich-Kwong, Martin-Hou, … Dense gas dynamics is the study of …. Of course, in such thermodynamic conditions, the gas behavior can no longer be modeled through the perfect gas law, but it is necessary to consider more complex equations of state, in order to take into account attractive intermolecular forces and covolume effects.Dense gas dynamics is the study of …. Of course, in such thermodynamic conditions, the gas behavior can no longer be modeled through the perfect gas law, but it is necessary to consider more complex equations of state, in order to take into account attractive intermolecular forces and covolume effects.

    8. Dense Gases Inviscid properties The flow of dense gases is governed by the key parameter G (GAMMA, NON TAU, MI RACCOMANDO PIETRO!!), which For its importance, is usually referrd-to as the Fundamental Derivative of Gas Dynamics, and it is defined like this, with r being the fluid density, and a the sound speed. Gamma’s definition involves the first derivative of the sound speed taken at constant entropy. Using standard thermodynamic manipulations, and remembering the definition of the sound speed, it is also possible to write Gamma as a function of the second derivative of the pressure with respect to the specific volume, taken at constant entropy, that is, the concavity of the constant entropy lines in the Amagat diagram. As v and a are positive quantities, the sign of G is determined by the sign of this second derivative. From its definition, it appears that G represents a measure of the rate of change of the sound speed in isentropic perturbations. For perfect polytropic gases, G is precisely equal to g+1 divided by 2, with g the specific heat ratio. As g is always greter than 1, for themodynamic stability reasons, than G>1 as well. For dense gases, governed by more complex equations of state, G can become less than 1. This means that, for gas flows in such thermodynamic conditions, the sound speed decreases with increasing density, that is with increasing pressure, the opposite of what happens in “common” fluids.The flow of dense gases is governed by the key parameter G (GAMMA, NON TAU, MI RACCOMANDO PIETRO!!), which For its importance, is usually referrd-to as the Fundamental Derivative of Gas Dynamics, and it is defined like this, with r being the fluid density, and a the sound speed. Gamma’s definition involves the first derivative of the sound speed taken at constant entropy. Using standard thermodynamic manipulations, and remembering the definition of the sound speed, it is also possible to write Gamma as a function of the second derivative of the pressure with respect to the specific volume, taken at constant entropy, that is, the concavity of the constant entropy lines in the Amagat diagram. As v and a are positive quantities, the sign of G is determined by the sign of this second derivative. From its definition, it appears that G represents a measure of the rate of change of the sound speed in isentropic perturbations. For perfect polytropic gases, G is precisely equal to g+1 divided by 2, with g the specific heat ratio. As g is always greter than 1, for themodynamic stability reasons, than G>1 as well. For dense gases, governed by more complex equations of state, G can become less than 1. This means that, for gas flows in such thermodynamic conditions, the sound speed decreases with increasing density, that is with increasing pressure, the opposite of what happens in “common” fluids.

    10. Entropy change across a weak shock: If G<0 (Inversione zone): compression shocks forbidden, expansion shocks admissible, mixed shock/fan waves, sonic shocks ? Non classical hyperbolic behaviors | G |<<1 (G =O(?v))? Entropy change one order lower than normal: ?reduced losses (1) Dense Gases Inviscid properties In fact, shock theory and the second law of thermodynamics show that compression shocks are not allowed in flows with G<0 Another interesting effect, is that for shock waves with jump conditions in the vicinity of the transition line, it could be shock is of the order of Dv. Therefore, injecting this in the above formula for the entropy jump, we find that shock wave in such a region are one order less dissipative than normal.In fact, shock theory and the second law of thermodynamics show that compression shocks are not allowed in flows with G<0 Another interesting effect, is that for shock waves with jump conditions in the vicinity of the transition line, it could be shock is of the order of Dv. Therefore, injecting this in the above formula for the entropy jump, we find that shock wave in such a region are one order less dissipative than normal.

    11. Sutherland law for viscosity: Gas molecules act as noninteracting rigid spheres NO LONGER VALID!!! Viscosity of dense gases: intermediate between viscosity of liquids (decreases with T) and of gases (grows with T) Thermal conductivity behaves approximately as viscosity High Cp ? Small Eckert number in high-speed B-L: Ec=V2/(CpR ?TR) ?Reduced friction heating ?Reduced skin friction ?reduced losses (2) Dense Gases Viscous and heat transfer effects Concerning the viscous behavior of dense gases, a first remark is that the well known sutherland law, commonly used to represent viscosity variation with temperature in perfect gases, is no longer valid, as it is bsed upon the assuption that the gas molecules behave as noninteracting rigid spheres. So, more complicated laws are required. The complicate behavior of dense gases can be anticipated by recalling that the viscosity of liquids tends to decreas with T, wheras the viscosity of gases increases. Dense gases have an intermediate behavior. Thermal conductivity behaves approximately as the viscosity, so that the Prandtl number variation is governed by the Cp variation within the flow. This can significantly vary, especially for thermodynamic conditions close to the liquid vapor critical point. Dense gases close to saturation conditions, tipically posses large heat capacities. This implies a small Eckert number and, consequently, reduced friction heating and reduced BL thickness in high speed BLs. This implies in turb reduced viscous losses. (to give an idea, these fluids have a viscosity 3-to 10 times greater with respect to air, i.e. approximately the same order of magnitude). Concerning the viscous behavior of dense gases, a first remark is that the well known sutherland law, commonly used to represent viscosity variation with temperature in perfect gases, is no longer valid, as it is bsed upon the assuption that the gas molecules behave as noninteracting rigid spheres. So, more complicated laws are required. The complicate behavior of dense gases can be anticipated by recalling that the viscosity of liquids tends to decreas with T, wheras the viscosity of gases increases. Dense gases have an intermediate behavior. Thermal conductivity behaves approximately as the viscosity, so that the Prandtl number variation is governed by the Cp variation within the flow. This can significantly vary, especially for thermodynamic conditions close to the liquid vapor critical point. Dense gases close to saturation conditions, tipically posses large heat capacities. This implies a small Eckert number and, consequently, reduced friction heating and reduced BL thickness in high speed BLs. This implies in turb reduced viscous losses. (to give an idea, these fluids have a viscosity 3-to 10 times greater with respect to air, i.e. approximately the same order of magnitude).

    12. Governing equations Navier-Stokes equations for single-phase, non-reacting flows Thermal EOS Caloric EOS Compatibility relation: The governing equations are the Navier-Stokes equations for single phase, non reacting flows. In the equation, w is the conservative variable vector, and f is the inviscid flux and fv the viscous flux, including turbulent stresses. The above equations are completed by a termal eos, And a caloric eos, with T the absolute temperature. The caloric eos is determined from the thermal one from the following compatibility relations, once specified the variation with temperature of the specific heat in the low-pressure limit.The governing equations are the Navier-Stokes equations for single phase, non reacting flows. In the equation, w is the conservative variable vector, and f is the inviscid flux and fv the viscous flux, including turbulent stresses. The above equations are completed by a termal eos, And a caloric eos, with T the absolute temperature. The caloric eos is determined from the thermal one from the following compatibility relations, once specified the variation with temperature of the specific heat in the low-pressure limit.

    13. Equation of state Van der Waals equation of state for a polytropic gas p pressure, e internal energy, Zc compressibility factor Low complexity Explicit relationship between pressure and internal energy In the present work, we consider the realistic gas model of MH eos is also considered, which represent one of the best available models for saturated vapor behavior. The equation involves five virial terms and ten thermodynamic constraints. A, B, and C are gas dependent coefficients. Assuming a power law variation of the specific heat in the low-pressure limit, the compatible caloric equation of state is of the form you see here.In the present work, we consider the realistic gas model of MH eos is also considered, which represent one of the best available models for saturated vapor behavior. The equation involves five virial terms and ten thermodynamic constraints. A, B, and C are gas dependent coefficients. Assuming a power law variation of the specific heat in the low-pressure limit, the compatible caloric equation of state is of the form you see here.

    14. Equation of state Martin-Hou equation of state Thermal EOS (5 virial expansion terms): One of the best available models for Dense Gases (Emanuel, J. Fluids Eng., 1996) Caloric EOS: Power-law variation of the low-pressure heat coefficient In the present work, we consider the realistic gas model of MH eos is also considered, which represent one of the best available models for saturated vapor behavior. The equation involves five virial terms and ten thermodynamic constraints. A, B, and C are gas dependent coefficients. Assuming a power law variation of the specific heat in the low-pressure limit, the compatible caloric equation of state is of the form you see here.In the present work, we consider the realistic gas model of MH eos is also considered, which represent one of the best available models for saturated vapor behavior. The equation involves five virial terms and ten thermodynamic constraints. A, B, and C are gas dependent coefficients. Assuming a power law variation of the specific heat in the low-pressure limit, the compatible caloric equation of state is of the form you see here.

    15. Governing equations In the present work, we consider the realistic gas model of MH eos is also considered, which represent one of the best available models for saturated vapor behavior. The equation involves five virial terms and ten thermodynamic constraints. A, B, and C are gas dependent coefficients. Assuming a power law variation of the specific heat in the low-pressure limit, the compatible caloric equation of state is of the form you see here.In the present work, we consider the realistic gas model of MH eos is also considered, which represent one of the best available models for saturated vapor behavior. The equation involves five virial terms and ten thermodynamic constraints. A, B, and C are gas dependent coefficients. Assuming a power law variation of the specific heat in the low-pressure limit, the compatible caloric equation of state is of the form you see here.

    16. Governing equations Chung et al. models for viscosity and thermal conductivity Turbulence modelling: Working hypotheses Turbulence structure not modified by dense gas effects Equilibrium boundary layers ?Baldwin-Lomax algebraic model ?”Turbulent Fourier law” with Prt?1 The models of Chung et al. Are used to represent the variation of viscosity and thermal conductivity with temperature For turbulent flow, we introduce the working hypotheses that the turbulence structure is not affected by dense gas effects (which is quite true far enough from the critical point) and we use a classical eddy viscosity turbulence model, which gives sufficiently correct results for equilibrium boundary layers. Specifically we adopt the classical model of Baldwin and Lomax. The turbulent heat flux is modeled, as usual, by introducing a turbulent fourier law, with a Prantl number order 1. The models of Chung et al. Are used to represent the variation of viscosity and thermal conductivity with temperature For turbulent flow, we introduce the working hypotheses that the turbulence structure is not affected by dense gas effects (which is quite true far enough from the critical point) and we use a classical eddy viscosity turbulence model, which gives sufficiently correct results for equilibrium boundary layers. Specifically we adopt the classical model of Baldwin and Lomax. The turbulent heat flux is modeled, as usual, by introducing a turbulent fourier law, with a Prantl number order 1.

    17. Cell-centered FV spatial discretization Inviscid fluxes Third-order centered scheme: 4th-order centered approximation + 3rd-order scalar dissipation term Weighted discretization coefficients? high accuracy preserved on general grids Viscous fluxes Standard 2nd-order approximation Time integration Four-stage Runge-Kutta Local time stepping, Implicit residual smoothing, multigrid Numerical method The governing equations are solved using a celle centered finite volume scheme. The inviscid fluxes are approximated by a third-order-accurate centered scheme, which uses basically a 4th order centered approximation plus a 3rd order stablization term of Jameson’s type. The scheme uses weighted discrtization formulas, which take into account mesh deformations, in order to preserve the scheme’s high accuracy on general grids. Viscous fluxes are computed in a standard way. The equations are advanced in time using a four stage RK scheme with local time stepping, IRS and multigrid to speedup convergence to the steady state.The governing equations are solved using a celle centered finite volume scheme. The inviscid fluxes are approximated by a third-order-accurate centered scheme, which uses basically a 4th order centered approximation plus a 3rd order stablization term of Jameson’s type. The scheme uses weighted discrtization formulas, which take into account mesh deformations, in order to preserve the scheme’s high accuracy on general grids. Viscous fluxes are computed in a standard way. The equations are advanced in time using a four stage RK scheme with local time stepping, IRS and multigrid to speedup convergence to the steady state.

    18. Code validation Methodology validated for 1D and 2D dense gas problems P.Cinnella and P.M. Congedo, 2005, “A numerical solver for dense gas flows”, AIAA Journal, Vol.43, No.11, pp.2458-2461 P. Cinnella and P.M. Congedo, 2005, “Aerodynamic performance of transonic dense gas flows past an airfoil“ AIAA J., Vol.43, No.2, pp.370-378 Numerical method The methodology has been validated for a number of 1D and 2D dense gas problems. The methodology has been validated for a number of 1D and 2D dense gas problems.

    19. Results Flow past a NACA0012 Grids: 256x64 (y+?5), 512x128 (y+?2.5) Working fluid: PP10 Reference conditions (PFG): M=0.8, a=2.26°, Re=9x106 We now consider dense gas flows past a NACA0012 airfoil. Leggere i datiWe now consider dense gas flows past a NACA0012 airfoil. Leggere i dati

    20. Dense gas (G?=-0.017): shock suppressed – separation suppressed Results Flow past a NACA0012 The solution obtained for the PFG is characterized by a strong shock at the airfoil upper surface and by an extended post-shock separation bubble (mostrare il Cf negativo). The solution compares reasonably well with experimental data. The pressure distribution obtained for the dense gas (for upstream tehmodynamic conditions close to the transition line) is dramatically different and is much more similar to an incompressible distribution. The shock wave is suppressed and the flow is entirely subsonic. This is due to the reversed variation of the Mach number in flows with G<1. In the dense gas flow, in spite of the quite significant adverse pressure gradient at the upper surface, the flow remains attached. Thanks to these two effects, the lift grows from … to… Whereas the drag coefficient is reduced by abou 1/3.The solution obtained for the PFG is characterized by a strong shock at the airfoil upper surface and by an extended post-shock separation bubble (mostrare il Cf negativo). The solution compares reasonably well with experimental data. The pressure distribution obtained for the dense gas (for upstream tehmodynamic conditions close to the transition line) is dramatically different and is much more similar to an incompressible distribution. The shock wave is suppressed and the flow is entirely subsonic. This is due to the reversed variation of the Mach number in flows with G<1. In the dense gas flow, in spite of the quite significant adverse pressure gradient at the upper surface, the flow remains attached. Thanks to these two effects, the lift grows from … to… Whereas the drag coefficient is reduced by abou 1/3.

    21. Viscosity of PP10 quite close to viscosity of air (slightly higher) Inviscid effects (shock suppression) dominate Results Flow past a NACA0012 The significant gains observed for PP10 are mostly due to inviscid effects, and specifically to shock suppression. This contributes to drag reduction even if the viscosity of PP10 is slightly higher than that of air.The significant gains observed for PP10 are mostly due to inviscid effects, and specifically to shock suppression. This contributes to drag reduction even if the viscosity of PP10 is slightly higher than that of air.

    22. Parametric study Reference conditions: M=0.85, a=1°, Re=9x106 Freestream thermodynamic conditions vary at constant entropy Results Flow past a NACA0012 To investigate in more detail dense gas effets in airfoil flows we also performed a parametric study… leggereTo investigate in more detail dense gas effets in airfoil flows we also performed a parametric study… leggere

    23. Reference solution (perfect gas) Results Flow past a NACA0012 The reference solution is characterized by two strong shock waves that lead to boudary layer separation. The lift coefficient is negative as, because of the interaction of the upper shock with the strongly separated BL, the shock moves upstream, resulting in less suction on the airfoil. On the left we see the Mach number contours. On the left, the Cp distribution at the wall.The reference solution is characterized by two strong shock waves that lead to boudary layer separation. The lift coefficient is negative as, because of the interaction of the upper shock with the strongly separated BL, the shock moves upstream, resulting in less suction on the airfoil. On the left we see the Mach number contours. On the left, the Cp distribution at the wall.

    24. DG solution: G?<<1 Results Flow past a NACA0012 In DG results, for all the cases characterized by low freestream Gamma (say much less than 1) the flow is subsonic and attachedIn DG results, for all the cases characterized by low freestream Gamma (say much less than 1) the flow is subsonic and attached

    25. DG solution: G?=O(1) Results Flow past a NACA0012 When Ginf increases, shocks begin to appear. However, such shocks have jump conditions in the vicinity of the transition line, and are very weak. The flow remains attached. The lift is very high whereas the drag is still much lower than in the perfect gas caseWhen Ginf increases, shocks begin to appear. However, such shocks have jump conditions in the vicinity of the transition line, and are very weak. The flow remains attached. The lift is very high whereas the drag is still much lower than in the perfect gas case

    26. DG solution: G?>>1 Results Flow past a NACA0012 Finally, increasing further the pressure (and then the freestream G), strong shocks appear, the flow separates and the aerodynamic performance quicly deteriorates.Finally, increasing further the pressure (and then the freestream G), strong shocks appear, the flow separates and the aerodynamic performance quicly deteriorates.

    27. DG flow: aerodynamic performance Results Flow past a NACA0012 This behavior is clearly visible on these plots, showing the variation of the lift and drag coefficients with freestream gamma. (Dire che la pressione aumenta da una punta all’altra della curva) Commentare. Far notare i valori del PFGThis behavior is clearly visible on these plots, showing the variation of the lift and drag coefficients with freestream gamma. (Dire che la pressione aumenta da una punta all’altra della curva) Commentare. Far notare i valori del PFG

    28. Results Flow through turbine cascade VKI LS-59 Grids: 192x16 (y+?4), 384x32 (y+?2) Working fluid: PP10 Reference conditions (PFG): Misexit=1, ain=30°, Rein=7.44x105, pin/pexit: 1.8 The last series of results concerns transonic flow through the turbine cascade VKI LS-59. Dire le condizioni e le griglie Far notare che i calcoli DG sono fatti con lo stesso rapporto di pressioneThe last series of results concerns transonic flow through the turbine cascade VKI LS-59. Dire le condizioni e le griglie Far notare che i calcoli DG sono fatti con lo stesso rapporto di pressione

    29. Results Flow through turbine cascades Dense gas (p1/pc=1.17, r1/rc=0.892, G1=1.31): weaker shock waves Upstream Mach number about 3% lower Efficiency (real-to-ideal enthalpy drop) gains: from 91% to 93.5% Dire che sono le linee isoMach. Commentare i risultati facendo notare il Mach + basso e gli urti + deboli. Dire che sono le linee isoMach. Commentare i risultati facendo notare il Mach + basso e gli urti + deboli.

    30. Conclusions Analysis of viscous BZT flows Flow past NACA0012 Significant efficiency gains High lift Low drag Shocks completely suppressed or much weaker No BL separation Flow through turbine cascade DG effects reduced with respect to airfoil flow (if the turbine is operated with the same pressure ratio as for the PFG) Efficiency gains still significant (about 2.5%) We now draw our conclusions.We now draw our conclusions.

    31. Perspectives 3D computation of isolated airfoils and turbine cascades Shape Optimization for a turbine cascade by considering viscous effects We now draw our conclusions.We now draw our conclusions.

More Related