Amplification and saturation of the thermoacoustic instability in a - PowerPoint PPT Presentation

Experiments Theory Simulations of transient regimes Conclusion Amplification and saturation of the thermoacoustic instability in a standing-wave prime mover. edra ( a ) , Thibaut Devaux, Guillaume Penelet, Pierrick Lotton Matthieu Gu´ Laboratoire d’Acoustique de l’Universit´ e du Maine, UMR CNRS 6613 Avenue Olivier Messiaen 72085 Le Mans Cedex 9, FRANCE ( a ) matthieu.guedra.etu@univ-lemans.fr 1 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

Experiments Theory Simulations of transient regimes Conclusion Framework Experiments 1 description of the thermoacoustic device transient regimes measurements Theory 2 description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime Simulations of transient regimes 3 Conclusion 4 2 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

Experiments Theory description of the thermoacoustic device Simulations of transient regimes transient regimes measurements Conclusion Experiments 1 description of the thermoacoustic device transient regimes measurements Theory 2 description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime Simulations of transient regimes 3 Conclusion 4 3 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

Experiments Theory description of the thermoacoustic device Simulations of transient regimes transient regimes measurements Conclusion Experiments description of the thermoacoustic device 600 CPSI glass tube ceramic stack L 59 cm l s 4 . 8 cm R 2 . 6 cm r s 0 . 45 mm Figure: (a) Photograph of the experimental Microphone apparatus. (b) Photograph of the hot end of the Bruel & Kjaer – 1 / 4 inch stack. Acquisition PC SoundCard Q microphone ��� ��� ��� ��� ��� ��� Lx 0 x s x h Figure: Schematic drawing of the standing-wave prime mover. 4 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

Experiments Theory description of the thermoacoustic device Simulations of transient regimes transient regimes measurements Conclusion Experiments description of the thermoacoustic device 50 Q onset ( W ) 40 30 20 Figure: (a) Photograph of the experimental apparatus. (b) Photograph of the hot end of the 10 stack. 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 x s ( m ) Q microphone Figure: Stability curve as function of the location x s of the stack. ��� ��� ��� ��� ��� ��� Lx 0 x s x h Figure: Schematic drawing of the standing-wave prime mover. 4 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

Experiments Theory description of the thermoacoustic device Simulations of transient regimes transient regimes measurements Conclusion Experiments transient regimes measurements Figure: Q ( t = 0) = 18 W (slightly below Figure: Q ( t = 0) = 16 W (slightly below Q onset = 19 . 6 W ). (a) ∆ Q/Q = 16% , (b) Q onset = 16 . 9 W ). (a) ∆ Q/Q = 16% , (b) ∆ Q/Q = 24% ,(c) ∆ Q/Q = 30% . ∆ Q/Q = 34% ,(c) ∆ Q/Q = 53% . Q microphone Q microphone ��� ��� �� �� ��� ��� �� �� ��� ��� �� �� x x x h 0 x s L x s x h 21.6cm 0 L 31.6cm 5 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

Experiments description of acoustic propagation using transfer matrices Theory amplification/attenuation of the acoustic wave Simulations of transient regimes determination of the onset threshold Conclusion transient regime Experiments 1 description of the thermoacoustic device transient regimes measurements Theory 2 description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime Simulations of transient regimes 3 Conclusion 4 6 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

Experiments description of acoustic propagation using transfer matrices Theory amplification/attenuation of the acoustic wave Simulations of transient regimes determination of the onset threshold Conclusion transient regime Theory description of acoustic propagation using transfer matrices Harmonic plane wave assumption � p 1 ( x ) e − iωt � � ξ 1 ( x, y ) e − iωt � ˜ p 1 ( x, t ) = ℜ ˜ and ξ 1 ( x, y, t ) = ℜ , (1) where ξ 1 = v 1 ,x , ρ 1 , τ 1 , s 1 . T h T c ���� ���� ���� ���� ���� ���� L x 0 x s x h � � � � p 1 ( L ) ˜ ˜ p 1 (0) = M w × M s × M 1 × ˜ u 1 ,x ( L ) u 1 ,x (0) ˜ � � cos( kxs ) iZc sin( kxs ) M1 = iZ − 1 sin( kxs ) cos( kxs ) c Ms and Mw derived from the linear thermoacoustic propagation equation transformed into a Volterra integral equation of the second kind [Penelet et al. , Acust. Acta Acust. (2005)]. 7 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

Experiments description of acoustic propagation using transfer matrices Theory amplification/attenuation of the acoustic wave Simulations of transient regimes determination of the onset threshold Conclusion transient regime Theory description of acoustic propagation using transfer matrices Harmonic plane wave assumption � p 1 ( x ) e − iωt � � ξ 1 ( x, y ) e − iωt � ˜ p 1 ( x, t ) = ℜ ˜ and ξ 1 ( x, y, t ) = ℜ , (1) where ξ 1 = v 1 ,x , ρ 1 , τ 1 , s 1 . T h T c ���� ���� ���� ���� ���� ���� L x 0 x s x h � � � � � � p 1 ( L ) ˜ M pp ( ω, T ( x )) M pu ( ω, T ( x )) ˜ p 1 (0) = × u 1 ,x ( L ) ˜ M up ( ω, T ( x )) M uu ( ω, T ( x )) u 1 ,x (0) ˜ Appropriate boundary conditions rigid wall : ˜ u 1 ,x ( L ) = 0 = ⇒ M uu ( ω, T ( x )) = 0 . no radiation : ˜ p 1 (0) = 0 7 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

Experiments description of acoustic propagation using transfer matrices Theory amplification/attenuation of the acoustic wave Simulations of transient regimes determination of the onset threshold Conclusion transient regime Theory amplification/attenuation of the acoustic wave M uu ( ω, T ( x )) = 0 . (2) A solution ( ω, T ) of Eq. (2) represents an operating point of the system. In the Fourier domain ( ω ∈ R ), it describes an equilibrium point : either unstable (onset threshold), or stable (steady state), corresponding to an acoustic wave which is neither amplified, nor attenuated in both cases. 8 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

Experiments description of acoustic propagation using transfer matrices Theory amplification/attenuation of the acoustic wave Simulations of transient regimes determination of the onset threshold Conclusion transient regime Theory amplification/attenuation of the acoustic wave M uu ( ω, T ( x )) = 0 . (2) A solution ( ω, T ) of Eq. (2) represents an operating point of the system. In the Fourier domain ( ω ∈ R ), it describes an equilibrium point : either unstable (onset threshold), or stable (steady state), corresponding to an acoustic wave which is neither amplified, nor attenuated in both cases. “quasi-steady” state assumption p 1 ( x ) e − i Ω t � , p 1 ( x, t ) = e ǫ g t ℜ � ˜ ω = Ω + iǫ g ⇒ ǫ g << Ω on the time scale of few acoustic periods. For a fixed temperature distribution T ( x ) , the solution of M uu (Ω , ǫ g ) = 0 gives the angular frequency of the oscillations and the amplification rate. 8 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

Experiments description of acoustic propagation using transfer matrices Theory amplification/attenuation of the acoustic wave Simulations of transient regimes determination of the onset threshold Conclusion transient regime Theory determination of the onset threshold ǫ g ( T ( x )) = 0 . (3) 400 T ( ◦ C ) 200 50 0 x h 0 x s L 200 T ( ◦ C ) 100 40 0 x s x h 0 L T ( ◦ C ) 120 80 40 Q onset ( W ) 0 30 x s x h 0 L 20 10 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 x s ( m ) Figure: Stability curve as function of the location x s of the stack. 9 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

Experiments description of acoustic propagation using transfer matrices Theory amplification/attenuation of the acoustic wave Simulations of transient regimes determination of the onset threshold Conclusion transient regime Theory transient regime Ordinary differential equation for the acoustic pressure amplitude dP 1 − ǫ g ( T ( x, t )) P 1 ( t ) = 0 , with P 1 ( t ) = | p 1 ( L, t ) | . (4) dt 10 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine