Simulation of a traveling-wave thermoacoustic engine using computational fluid dynamics

Transcription

1 ECN-RX Simulation of a traveling-wave thermoacoustic engine using computational fluid dynamics J.A. Lycklama à Nijeholt a) M.E.H. Tijani b)c) S. Spoelstra b) ~) Nuc]ear Research & Consultancy Group, P.O. Box 1, 1755 ZG Petten, The Nethedands b) Energy Research Centre of the Netherlands, P.O. Box 1, 1755 ZG Petten, The Netherlands ~) Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Published in Joumal of Aooustical Society of America, Volume 11û, Issue 4, pp Revisions A ~18 January 2005, draft version B 12 Ma 2~005, final version Made by:/ií////~ I Approved by:,/ S. Spoels~ a "~ P.W. Bach Checked by: j Issued by: ECN Energy Efficiency in Industry Waste heat technology W P~ffí. Alderliesten OCTOBER 2005

2 Abstract A two-dimensional computational fluid dynamics (CFD) simulation study of a traveling-wave thermoacoustic engine is presented. The computations show an increase of the dynamic pressure when a linear temperature difference is applied across the regenerator. An amplification of the acoustic power through the engine is also illustrated. A satisfactory agreement between the calculated and expected gains of the traveling-wave thermoacoustic engine is obtained. The expected gain is defined as the ratio of the absolute temperatures at the ends of the regenerator. Nonlinear phenomena that cannot be captured by existing linear theory, like streaming mass flows and vortices formation are also visualized. It is concluded that CFD codes could be used in the future to predict and optimize thermoacoustic systems. This is an important step towards the development of nonlinear simulation tools for the high-amplitude thermoacoustic systems that are needed for practical use. PACS numbers: Ud. 2 ECN-RX

3 Contents List of tables 4 List of figures 4 1. Introduction 5 2. CFD model Numerical model for the traveling-wave thermoacoustic engine Resonator Engine Regenerator and heat exchangers CFD model parameters Initial and boundary conditions Postprocessing Results and discussion Acoustic results Multidimensional and nonlinear results Conclusions 15 References 16 ECN-RX

4 List of tables Table 2.1 Geometrical parameters of the heat exchangers and the regenerator 7 Table 3.1 Time-averaged acoustic power in the axial direction at the ambient and hot side of the regenerator and their ratio 10 kpa pressure amplitude 12 Table 3.2 Phase difference between velocity and pressure at the positions defined in Figure List of figures Figure 2.1 Schematic view of the traveling-wave thermoacoustic engine 6 Figure 2.2 Geometrical dimensions and computation mesh of the traveling-wave thermoacoustic engine. The dimensions are in millimeters 7 Figure 2.3 Prescribed temperature profile in the regenerator and the heat exchangers 9 Figure 3.1 Onset of oscillations for the traveling-wave thermoacoustic engine 11 Figure 3.2 Time-averaged acoustic power density (W/m 2 ) in the axial 10kPa pressure amplitude 12 Figure 3.3 Phase difference monitoring positions around the engine 12 Figure 3.4 Velocity fields (m/s) in the 10 kpa pressure amplitude 13 Figure 3.5 Time-averaged temperature 10 kpa pressure amplitude 14 Figure 3.6 Time-averaged air mass flow rate (kg/m 2 s) in the axial 10kPa pressure amplitude 14 4 ECN-RX

5 1. Introduction During the past two decades significant progress has been made in the development of thermoacoustic systems. A number of thermoacoustic standing-wave engines and coolers have been built 1-6. The thermal efficiency of these stack-based systems is limited to about 20%. This is due in part to the intrinsic irreversibility of the heat transfer in the stack. In contrast, a reversible Stirling cycle is executed in a traveling-wave system. However, traveling-wave systems have long suffered from high viscous losses in the regenerator 7. The combination of a standing- and traveling wave by making use of a feedback inertance parallel to the regenerator opened the way to practical traveling-wave systems 8,9. Researchers at Los Alamos National Laboratory have constructed a thermoacoustic traveling-wave engine with high efficiency 9. This regeneratorbased engine has shown a 50% efficiency improvement over the best standing-wave counterpart. This efficiency improvement has been obtained by suppressing a number of undesirable nonlinear effects. Further improvements would be possible if nonlinear effects could be better understood. Numerical simulations and visualizations of the unsteady flow field can help in understanding these phenomena; this could lead to a more rigorous design and optimization of thermoacoustic systems. The aim of this paper is to present a computational fluid dynamics (CFD) study of a travelingwave thermoacoustic engine. This study forms a first step towards the application of CFD as a full numerical nonlinear three-dimensional tool for the design and prediction of the behavior of thermoacoustic engines and heat pumps. Most of the numerical calculations done to date concern standing-wave systems and are based on the linear theory of thermoacoustics developed by Rott 10,11 and implemented into design tools such as DELTAE 12. At low amplitude, the linear theory gives a good understanding of the physical phenomena involved and can be used to predict the behavior of thermoacoustic systems. However, thermoacoustic systems developed for practical use operate at high amplitude. At such conditions nonlinear effects, like turbulence, vortices, and streaming, will be present and the actual behavior will deviate from the linear theory. In this case the full nonlinear Navier-Stokes equations have to be considered. In the past some nonlinear models have been developed to simulate flows in the resonator, and around the stack, and to simulate nonlinear phenomena like streaming in stack-based systems To our knowledge, these are the first CFD results to be presented for a complete system containing an active, acoustic-power producing, thermoacoustic element. In the present work a commercially available CFD code 16 has been used. In this so-called structured CFD code, the engine is spatially discretized into a large number of hexahedral computational cells, and the governing unsteady Navier-Stokes equations and the equation for the total enthalpy are solved by the CFD code. The results of the computations show the onset of selfoscillations accompanied by an increase of the dynamic pressure. Amplification of the acoustic power through the regenerator is also shown. Nonlinear phenomena like streaming mass flows and vortices formation are also visualized. In Sec. II, the CFD model for the thermoacoustic traveling-wave engine is described. Special attention is given to the CFD sub-model for the regenerator and heat exchangers. Section III presents the computational results. Finally, in Sec. IV some conclusions are drawn. ECN-RX

6 2. CFD model In this section the construction of the CFD model for the thermoacoustic engine is described. The geometry and boundary conditions used for the engine are also presented. ANSYS's computational fluid dynamics code CFX 4.4 is used to run all simulations. All computations are performed using a single CPU on a Linux cluster. Roughly 1 day on a single Pentium 4 processor is needed to run the simulation for the traveling-wave thermoacoustic engine. Fixed time stepping with the Cranck-Nicolson central differencing scheme has been used for the time integration. An upwind differencing scheme has been used for the spatial discretization. Upwind differencing was needed to numerically stabilize the flow near the porous region of the regenerator. Test calculations have shown that the artificial acoustic losses associated with the application of the upwind scheme are rather small. Numerical tests revealed that in particular first-order time differencing caused large artificial acoustic losses. Using the second-order Cranck-Nicolson scheme diminishes these losses. The numerical tests showed that application of the first-order upwind differencing scheme resulted in a small increase in the acoustic losses compared to a second-order spatial differencing scheme. This increase was small compared to the difference between first- and second-order time differencing. The SIMPLEC algorithm has been used for updating pressure and correcting the velocity components for continuity. 2.1 Numerical model for the traveling-wave thermoacoustic engine This subsection describes the modeled geometries for the three components, namely the resonator, the engine, and the regenerator and heat exchangers Resonator Figure 2.1 shows a schematic illustration of the thermoacoustic system that has been modeled in the CFD computations. The system is composed of a traveling-wave thermoacoustic engine placed in a double-helmholtz resonator. In order to simplify the modeling and to save computational time, a system without acoustic load has been considered. This results in very low damping of the acoustic wave. An axisymmetric grid has been applied for the modeling of the system. The resonator is filled with air at an average pressure of 1 bar and at 300 K initially. The 2D mesh is relatively small and consists of only 3462 computational cells for the total system in order to achieve a reasonable computational time. Figure 2.1 Schematic view of the traveling-wave thermoacoustic engine 6 ECN-RX

7 2.1.2 Engine Figure 2.2 shows the geometrical dimensions and the mesh of the engine part of the model. The engine consists of an ambient (300 K) and hot (500 K) heat exchanger (HX) with the regenerator placed in between. The engine is placed in a tube with a length of 204 mm and a diameter of 104 mm. This tube is placed concentrically in the resonator at the right side of the system (cf. Figure 2.1). The annular gap between the tube and the resonator serves as a feedback inertance connected to the compliance at the right side of the engine. Porous metallic structures are used for both the heat exchangers and the regenerator. Figure 2.2 Geometrical dimensions and computation mesh of the traveling-wave thermoacoustic engine. The dimensions are in millimeters Regenerator and heat exchangers A CFD submodel for the regenerator and the heat exchangers has been developed. This submodel accounts for the flow resistance of the air in the porous metal structure and the thermal interaction between this structure and the air. The very detailed structure in the regenerator and in the heat exchangers is not geometrically modeled. Instead, a porous region is defined in which body forces account for the flow resistance of air in the porous metallic structure. Volume porosity is prescribed in this region to account for the reduction of the flow area in the heat exchangers and the regenerator, see Table 2.1. Table 2.1 Geometrical parameters of the heat exchangers and the regenerator Porosity [%] Hydraulic radius [µm] Thickness [mm] Heat exchangers Regenerator Flow resistance The flow resistance is implemented as a function of the local air velocity in the heat exchangers and regenerator. This implementation is realized by means of body forces at the location of the regenerator and heat exchangers. The following one-dimensional equation is considered for the pressure gradient in the porous structure: 17 dp dx 2 Csf ρu = C fd + Re (1) 4d w ECN-RX

8 where C fd =0.572, and C sf = The Reynolds number Re is given by ρd ε Re = w u ( 1 ε )µ (2) where ρ is the gas density, u is the gas velocity in the x-direction, ε and d w are the porosity and the wire diameter of the porous structure respectively. The pressure drop from expression (1) is implemented in two dimensions by using the so-called body forces given by the following expressions: 16 B and dp = = ( RC + RF v u (3) dx x ) dp B y = = ( RC + RF v )v (4) dy where u and v are the gas velocities in the x- and y-directions respectively. v is the magnitude of the velocity. The parameters R c and R F are given by the expressions 17 R c ( 1 ε ) Csf µ = (5) 2 4d ε w and R F ρc fd = (6) 4d w Expressions (5) and (6) apply in a Cartesian as well as in a cylindrical coordinate system. Thermal model The CFD model for the thermal interaction between the porous structure and the air is based on the following idealized assumptions: Perfect thermal contact between the porous metallic structure and the gas. The thermal penetration depth at the prevailing conditions at the ambient side of the regenerator and at the prevailing frequency amounts to about 350 µm. This distance is a factor of 8 more than the hydraulic radius of the regenerator (cf. Table 2.1). The perfect thermal contact assumption (infinite heat transfer) therefore seems justified. Very large heat capacity of the metallic structure. The heat capacity of any real regenerator will be much larger than air at 1 bar and the prevailing temperature conditions, so this assumption also seems justified. A linear temperature profile is assumed in the regenerator by means of two heat exchangers as shown in Figure 2.3. No heat conduction through the metal structure of the regenerator. In reality there will be a heat leakage through the regenerator from the hot to the ambient side. This heat does not participate in the thermoacoustic process. However, in a real regenerator this heat loss should be much less than 10% of the total heat. For the purpose of this study, it seems justified to neglect this heat loss. 8 ECN-RX

9 Figure 2.3 Prescribed temperature profile in the regenerator and the heat exchangers A temperature profile T s as shown in Figure 2.3 is assumed for the heat exchangers, regenerator material, and the air inside these structures. The air inside the porous metallic structures is forced to adapt to this prescribed temperate profile. This is achieved by means of timedependent heat sources in the enthalpy equation for air. The source is active in each grid cell located in the heat exchangers and regenerator regions. The heat source has the following form S H ( T T )) = α (7) g s where α is the heat transfer coefficient, T g and T s are the gas and solid matrix temperatures, respectively. 2.2 CFD model parameters The following CFD modeling conditions are used for the computations: Time-dependent flow Axisymmetric flow Laminar compressible air flow Ideal-gas approximation Constant material properties Nonbuoyant flow Air heat conduction and convection Porous region for heat exchanger and regenerator ECN-RX

10 The unsteady Navier-Stokes equations and the equation for total enthalpy are solved by the CFD code. The flow oscillations at a frequency of 56 Hz are solved with fixed time steps of s, resulting in 107 time integration points per oscillation period. Initially, at low-pressure amplitudes, the flow is laminar in the system. However, at high-pressure amplitudes the flow velocities at the pressure node in the resonator and in the feedback inertance will become turbulent, with Reynolds numbers exceeding In this first study, no attempt has been made to predict the very complicated transition from laminar to turbulent flow. Therefore, the flow has been approximated as being laminar during the whole transient. 2.3 Initial and boundary conditions At t=0, a small pressure and temperature perturbation is present in the system. The initial perturbation is prescribed according to a standing wave with temporary zero velocity in the whole system and ½-wavelength sine shaped axial profiles for temperature and pressure. The tube containing the regenerator is modeled as a cylindrical surface (tube with zero thickness) with no heat conduction. All other walls are also assumed to be adiabatic. Acoustic losses are minimized in the resonator by application of a slip boundary condition for velocity. All other walls including the regenerator inner tube are treated as no-slip walls for velocity. The calculation is stopped at the moment that the pressure amplitude exceeds 10 kpa. No attempt has been made to reach stationary conditions, since this would require too much computational time for the present model. 2.4 Postprocessing The local acoustic power density P ac is calculated from the CFD results for each computational cell according to the equation P ac piui = i N (8) In this equation the product of dynamic pressure p and axial velocity u is averaged over all time steps i during one cycle. Positive values of the acoustic power correspond with transport of acoustic energy to the right side in Figure 2.1. The local time-averaged acoustic power P ac according to equation (8) has been integrated at two cross sections to obtain the net acoustic power at these cross sections (P cross ): P cross = P da (9) A cross ac The same procedure as for the local acoustic power is applied to obtain time-averaged mass flux and time-averaged temperature distributions. 10 ECN-RX

11 3. Results and discussion The results of the calculations are discussed in two parts. The first part deals with the acoustic results like pressure amplitude, local acoustic power, and phase differences between gas velocity and dynamic pressure. The second part concerns nonlinear and multidimensional effects like flow fields, resulting vortices, and streaing. 3.1 Acoustic results Figure 3.1 shows the development with time of the pressure at the antinode position as predicted by the CFD modeling. A strong increase of the pressure amplitude is predicted. The pressure increases up to 10 kpa (10% drive ratio) in only 1.5 s. Eventually, this increase in pressure amplitude will continue until the acoustic power generated by the engine is balanced by the dissipation within the resonator. Since there is very little damping in the system, this process will lead to very high pressure amplitudes. Figure 3.1 Onset of oscillations for the traveling-wave thermoacoustic engine The acoustic power is calculated at two cross sections and integrated with time over one period. One cross section is located at the right side of the ambient heat exchanger and the other cross section at the left side of the hot heat exchanger. The resulting acoustic power and gain can be found in Table 3.1. The calculated gain of 1.5 is somewhat below the expected amplification factor of 1.67 due to acoustic losses in the regenerator. The expected amplification factor is defined as the ratio of the absolute temperatures at the ends of the regenerator 7,9. ECN-RX

12 Table 3.1 Time-averaged acoustic power in the axial direction at the ambient and hot side of the regenerator and their ratio 10 kpa pressure amplitude Cross section regenerator Acoustic power (watt) Gain Ambient side 38.5 Hot side A detailed plot of the space distribution of acoustic power (time averaged for one period) in the engine is shown in Figure 3.2. Positive values correspond with transport of acoustic energy to the right side. This figure shows that acoustic power is transported to the right side via the feedback inertance, and back to the left side through the heat exchangers and regenerator. Some of the phenomena observed in this figure can be understood by looking at the velocity fields, which are described in the next section. Figure 3.2 Time-averaged acoustic power density (W/m 2 ) in the axial 10kPa pressure amplitude Figure 3.3 Phase difference monitoring positions around the engine Table 3.2 Phase difference between velocity and pressure at the positions defined in Figure 3.3 Position Phase difference ( o ) A-End of Thermal Buffer Tube 121 B-Hot heat exchanger 146 C-Ambient heat exchanger 160 D-Compliance 118 E-Feedback inertance left side 94 F-Feedback inertance right side ECN-RX

13 Figure 3.3 defines six positions at which the phase difference between velocity and pressure is calculated from the CFD results. The calculated phase differences are given in Table 3.2. In an optimized traveling-wave thermoacoustic engine, conditions are established in the regenerator corresponding to a local phase shift of 0 or 180 deg (depending on the direction of the acoustic power). In the present system, a 180-deg phase difference between velocity and pressure would be optimal. A value between 146 and 160 deg across the regenerator is found from the present CFD results. This phase difference can be optimized by tuning the parameters of the acoustic network consisting of the feedback inertance, the compliance, and the flow resistance of the regenerator. 3.2 Multidimensional and nonlinear results The calculated flow profile is shown in Figure 3.4. The time period between each figure is ¼ of a full cycle and the figure at the top is taken at a maximum positive pressure at the engine side of the system. Note that not all grid points are used for the vector plots. The spatial resolution of the grid exceeds the spatial resolution of the vectors. Figure 3.4 Velocity fields (m/s) in the 10 kpa pressure amplitude As can be seen from Figure 3.4, the flow is strongly nonlinear. During the whole period a vortex can be observed in the compliance at the right side. This vortex shifts somewhat in position; and also, the vortex s magnitude fluctuates during the period. In the first and third vector plots a vortex has also temporarily developed at the left side of the feedback inertance. These vortices are believed to be driven by the jet coming out of the feedback inertance, where velocities are rather high (on the order of 15 m/s). The vortex that is created in the thermal buffer tube leads to a strong two-dimensional temperature distribution of the air. The thermal buffer tube is the part of the tube containing the regenerator to the left of the hot heat exchanger in Figure 2.1. Figure 3.5 shows the time-averaged gas temperature around the engine. Clearly visible is the convection of heat driven by the vortex. ECN-RX

14 Figure 3.5 Time-averaged temperature 10 kpa pressure amplitude One known nonlinear phenomenon in traveling-wave thermoacoustic systems is Gedeon or dc streaming 18. This effect is due to a time-averaged pressure difference across the regenerator resulting in a mass flow circulating from the hot side of the regenerator through the feedback inertance to the ambient side of the regenerator. In order to identify this phenomenon, the timeaveraged air mass flow rate in the axial direction is calculated at 10 kpa pressure amplitude and presented in Figure 3.6. This figure is somewhat hard to interpret, partly due to the fact that the system is not stationary. Also the fact that the largest part of the area is located on the outer part of the cylinder makes it hard to get an impression of the total mass flow from the local mass fluxes. When the mass flow rate is integrated over the cross section, it turns out that there is a net negative flow (to the left side) within the thermal buffer tube and a net positive flow (to the right side) within the feedback inertance. Thus, dc-streaming is indeed occurring. Figure 3.6 Time-averaged air mass flow rate (kg/m 2 s) in the axial 10kPa pressure amplitude 14 ECN-RX

15 4. Conclusions The main purpose of the study presented is to find out whether a commercially available CFD code is able to model a thermoacoustic system. Both the oscillatory flow behavior and the interaction between the sound wave and the porous structure are taken into account. This paper clearly shows that this can indeed be done by the code used. Although the model used is an idealization of the real situation, the phenomena observed correspond to a great extent to the expectations. The agreement between the calculated and expected gains of the traveling-wave thermoacoustic engine strengthens confidence in the computational results. The CFD results clearly show strong nonlinear effects at high pressure amplitudes. Both dc streaming and jet-driven streaming are visualized. The jet-driven streaming results in vortices that lead to unwanted heat convection. We think that CFD may prove to be a useful tool in the study of thermoacoustic phenomena that cannot be captured by today s one-dimensional linear codes. In the near future, the present CFD model will be extended to include temperature-dependent material properties, finite heat transfer in the regenerator and the heat exchangers, losses in the resonator and the effect of gravity. In addition, higher average pressures and different working media will be used. And, last but not least, the CFD model should be validated against experimental data to obtain a design tool for a real thermoacoustic system. ECN-RX

16 References [1] Hofler, T.J. (1986): Thermoacoustic refrigerator design and performance. PhD. Thesis, Physics department, University of California, San Diego. [2] Garrett, S.L, J.A Adeff and T.J. Hofler (1993): Thermoacoustic refrigerator for space applications. J. of Thermophysics and Heat Transfer 7, [3] Tijani, M.E.H. (2001): Loudspeaker-driven thermo-acoustic refrigeration. PhD. Thesis, Department of Applied Physics, Eindhoven University of Technology. [4] Tijani, M.E.H., J.C.H. Zeegers and A.T.A.M. de Waele (2002): Design of thermoacoustic refrigerators. Cryogenics 42, [5] Tijani, M.E.H., J.C.H. Zeegers and A.T.A.M. de Waele (2002): Construction and performance of a thermoacoustic refrigerator. Cryogenics 42, [6] Swift, W. (1992): Analysis and performance of a large thermoacoustic engine. J. Acoust. Soc. Am. 92, [7] Ceperley, P.H. (1979): A pistonless Stirling engine-the traveling wave heat engine. J. Acoust. Soc. Am. 66, [8] Yazaki, T., A. Iwata, T. Maekawa and A. Tominaga (1998): Traveling wave thermoacoustic engine in a looped tube. Phys. Rev. Lett. 81, [9] Backhaus, S. and G.W. Swift (2000): A thermoacoustic-stirling heat engine: Detailed study. J. Acoust. Soc. Am. 107, [10] Rott, N. (1969): Damped and thermally driver acoustic oscillations in wide and narrow tubes. Z. Angew. Math. Phys. 20, [11] Rott, N. (1975): Thermally driven acoustic oscillations, Part II: Stability limit for helium. Z. Angew. Math. Phy. 24, (1973); "Thermally driven acoustic oscillations, part III: Second-order heat flux", Z. Angw. Math. Phy. 26, [12] Ward, W.C. and G.W. Swift (1994): Design environment for low amplitude thermoacoustic engines. J. Acoust. Soc. Am. 96, [13] Worlikar, A.S. and O.M. Knio (1999): Numerical study of oscillatory flow and heat transfer in a loaded thermoacoustic stack. Numerical Heat Transfer Part A 35, [14] Hamilton, M.F., Y.A. Ilinskii and E.A. Zabolotskaya (2002): Nonlinear two-dimensional model for thermoacoustic engines. J. Acoust. Soc. Am. 111, [15] Blanc-Benon, P., E. Besnoin and O.M. Knio (2003): Experimental and computational visualization of the flow field in a thermoacoustic stack. C.R. Mecanique 331, [16] CFX4.4 User Guide, Computational Fluid Dynamics Services, AEA Technology, Harwell Laboratory, Oxfordshire OX11 0RA, United Kindom, (2001). [17] Thomas, B. and D. Pittman (2000): Update on the evaluation of different correlations for the flow friction factor and heat transfer of Stirling engine regenerators. AIAA [18] Gedeon, D., (1997): DC Gas Flows in Stirling and Pulse Tube. Cryocoolers 9, (R. G. Ross Plenum Press, New York, 1997), ECN-RX