Lehrstuhl für THERMODYNAMIK Network Models in Thermoacoustics Ph.D. Camilo Silva Prof. Wolfgang Polifke 1
Lehrstuhl für THERMODYNAMIK Acoustics flame coupling Entropy-Acoustics coupling Acoustic BC downstream Acoustic BC upstream Fuel supply impedance Air supply impedance Ac. wave Ac. wave ¯ ˙ Q + u 0 + s 0 ˙ Q 0 � φ 0 Stable flame Unstable flame Unstable flame Why changing an injector position could make a flame unstable? Can entropy couple with the flame through acoustic waves? How to study combustion instabilities? 2
Lehrstuhl für THERMODYNAMIK How to study combustion instabilities? Experiments High fidelity CFD Network models All together may be the best solution !! For example … Helicopter Engine Network Models Flame dynamics from experiments or CFD 3D Acoustic Solvers 3
Lehrstuhl für THERMODYNAMIK How to study combustion instabilities? Experiments High fidelity CFD Network models All together may be the best solution !! For example … Helicopter Engine Network Models Flame dynamics from experiments or CFD or Network Models 4
Lehrstuhl für THERMODYNAMIK Full System Understand the System What we want to study? X Thermoacoustics Of longitudingal, transversal, azimuthal or radial acoustic waves? 5
Lehrstuhl für THERMODYNAMIK Full System Understand the System What we want to study? X Thermoacoustics X Of longitudinal plane acoustic waves Of short or long wavelengths? 6
Lehrstuhl für THERMODYNAMIK Full System Understand the System What we want to study? X Thermoacoustics X Of longitudinal plane acoustic waves X Of long wavelengths Main Assumption Acoustic compactness in most elements of the thermoacoustic system 7
Lehrstuhl für THERMODYNAMIK Full Thermoacoustic Understand the System System Thermoacoustic Network Decompose the models System 8
Lehrstuhl für THERMODYNAMIK Full Thermoacoustic Understand the System System Thermoacoustic Network Decompose the models System Acoustic Ducts White Box Compact Flames two-port Black Box White Box Nozzles Gray Box element Joints … 9
Lehrstuhl für THERMODYNAMIK Full Thermoacoustic Understand the System System Thermoacoustic Network Decompose the models System Acoustic Ducts White Box Compact Flames two-port Black Box White Box Nozzles Gray Box element Joints … ∂ t ( ρ A ) + ∂ ∂ WHITE BOX ∂ x ( ρ uA ) = 0 Quasi 1D ∂ t ( ρ uA ) + ∂ ∂ = − A ∂ p Q 0 ρ 0 u 0 p 0 s 0 ˙ � ρ u 2 A � Conservation ∂ x ∂ x Equations ∂ t ( ρ sA ) + ∂ ∂ ∂ x ( ρ usA ) = A T ˙ q Model Acoustic ∂ t ( ρ h t A ) + ∂ ∂ q − A ∂ p ∂ x ( ρ uh t A ) = A ˙ and entropy waves ∂ t 10
Lehrstuhl für THERMODYNAMIK Full Thermoacoustic Understand the System System Thermoacoustic Network Decompose the models System Acoustic Ducts White Box Compact Flames two-port Black Box White Box Nozzles Gray Box element Joints … ∂ t ( ρ A ) + ∂ ∂ WHITE BOX ∂ x ( ρ uA ) = 0 Quasi 1D ∂ t ( ρ uA ) + ∂ ∂ = − A ∂ p Q 0 ρ 0 u 0 p 0 s 0 ˙ � ρ u 2 A � Conservation ∂ x ∂ x Equations ∂ t ( ρ sA ) + ∂ ∂ ∂ x ( ρ usA ) = A T ˙ q Model Acoustic ∂ t ( ρ h t A ) + ∂ ∂ q − A ∂ p ∂ x ( ρ uh t A ) = A ˙ and entropy waves ∂ t 2 3 2 3 2 3 1 − R in 0 0 f 0 0 Gather all elements in a single − R out g 0 0 0 1 0 6 7 6 7 6 7 matrix and compute acoustic CONNEXIONS 5 = 6 7 6 7 6 7 T 11 T 12 f 3 − 1 0 0 4 5 4 4 5 response of the ensemble. T 21 T 22 0 − 1 g 3 0 | {z } M Note that under a suitable treatment, tens of elements can reduce to a 4 x 4 matrix ! 11
Lehrstuhl für THERMODYNAMIK Full Thermoacoustic Understand the System System Thermoacoustic Network Decompose the models System Acoustic Ducts White Box Compact Flames two-port Black Box White Box Nozzles Gray Box element Joints … ∂ t ( ρ A ) + ∂ ∂ WHITE BOX ∂ x ( ρ uA ) = 0 Quasi 1D ∂ t ( ρ uA ) + ∂ ∂ = − A ∂ p Q 0 ρ 0 u 0 p 0 s 0 ˙ � ρ u 2 A � Conservation ∂ x ∂ x Equations ∂ t ( ρ sA ) + ∂ ∂ ∂ x ( ρ usA ) = A T ˙ q Model Acoustic ∂ t ( ρ h t A ) + ∂ ∂ q − A ∂ p ∂ x ( ρ uh t A ) = A ˙ and entropy waves ∂ t 2 3 2 3 2 3 1 − R in 0 0 f 0 0 Gather all elements in a single − R out g 0 0 0 1 0 6 7 6 7 6 7 matrix and compute acoustic CONNEXIONS 5 = 6 7 6 7 6 7 T 11 T 12 f 3 − 1 0 0 4 5 4 4 5 response of the ensemble. T 21 T 22 0 − 1 g 3 0 | {z } M Freq. Unstable � Region Study stability of the system STABILITY ANALYSIS. Neg. Pos. 12 Growth rate
Lehrstuhl für THERMODYNAMIK OUTLINE ∂ t ( ρ A ) + ∂ ∂ Quasi 1D Conservation Equations WHITE BOX ∂ x ( ρ uA ) = 0 Spatial integration and the compact ∂ t ( ρ uA ) + ∂ ∂ = − A ∂ p Q 0 ρ 0 u 0 p 0 s 0 ρ u 2 A ˙ � � assumption ∂ x ∂ x ∂ t ( ρ sA ) + ∂ ∂ ∂ x ( ρ usA ) = A T ˙ q Linearization ∂ t ( ρ h t A ) + ∂ ∂ q − A ∂ p ∂ x ( ρ uh t A ) = A ˙ Model Acoustic Further Assumptions ∂ t and entropy waves ✓ ˆ p ◆ c (1+ M ) = 1 f = B + e − i ω x/ ¯ c + ˆ u Definition of waves 2 ρ ¯ ¯ ✓ ˆ p ◆ c (1 − M ) = 1 g = B − e i ω x/ ¯ c − ˆ u Isentropic ducts 2 ρ ¯ ¯ Boundary conditions ✓ ¯ ✓ ¯ ◆ ◆ T 2 γ ¯ p 1 T 2 ¯ ˙ u 1 A 1 c p ¯ Q = ¯ ρ 1 ¯ T 1 − 1 = ¯ u 1 A 1 − 1 ¯ ¯ ( γ − 1) T 1 T 1 Modeling of Flame dynamics 2 3 2 3 2 3 − R in f 0 1 0 0 0 Gather all elements in a single 0 0 − R out 1 g 0 0 6 7 6 7 6 7 CONNEXIONS matrix and compute acoustic 5 = 6 7 6 7 6 7 T 11 T 12 f 3 − 1 0 0 4 5 4 4 5 response of the ensemble. T 21 T 22 g 3 0 − 1 0 | {z } M Freq. Unstable � Region Study stability of the system STABILITY ANALYSIS. Neg. Pos. Growth rate 13
Lehrstuhl für THERMODYNAMIK OUTLINE ∂ t ( ρ A ) + ∂ ∂ Quasi 1D Conservation Equations WHITE BOX ∂ x ( ρ uA ) = 0 Spatial integration and the compact ∂ t ( ρ uA ) + ∂ ∂ = − A ∂ p Q 0 ρ 0 u 0 p 0 s 0 ρ u 2 A ˙ � � assumption ∂ x ∂ x ∂ t ( ρ sA ) + ∂ ∂ ∂ x ( ρ usA ) = A T ˙ q Linearization ∂ t ( ρ h t A ) + ∂ ∂ q − A ∂ p ∂ x ( ρ uh t A ) = A ˙ Model Acoustic Further Assumptions ∂ t and entropy waves ✓ ˆ p ◆ c (1+ M ) = 1 f = B + e − i ω x/ ¯ c + ˆ u Definition of waves 2 ρ ¯ ¯ ✓ ˆ p ◆ c (1 − M ) = 1 g = B − e i ω x/ ¯ c − ˆ u Isentropic ducts 2 ρ ¯ ¯ Boundary conditions ✓ ¯ ✓ ¯ ◆ ◆ T 2 γ ¯ p 1 T 2 ¯ ˙ u 1 A 1 c p ¯ Q = ¯ ρ 1 ¯ T 1 − 1 = ¯ u 1 A 1 − 1 ¯ ¯ ( γ − 1) T 1 T 1 Modeling of Flame dynamics 2 3 2 3 2 3 − R in f 0 1 0 0 0 Gather all elements in a single 0 0 − R out 1 g 0 0 6 7 6 7 6 7 CONNEXIONS matrix and compute acoustic 5 = 6 7 6 7 6 7 T 11 T 12 f 3 − 1 0 0 4 5 4 4 5 response of the ensemble. T 21 T 22 g 3 0 − 1 0 | {z } M Freq. Unstable � Region Study stability of the system STABILITY ANALYSIS. Neg. Pos. Growth rate 14
Lehrstuhl für THERMODYNAMIK Quasi 1D Conservation Equations From full Navier-Stokes equations Assumptions No viscous terms Quasi-1D ∂ t ( ρ A ) + ∂ ∂ Mass ∂ x ( ρ uA ) = 0 ∂ t ( ρ uA ) + ∂ ∂ = − A ∂ p ρ u 2 A � � ∂ x ∂ x ∂ t ( ρ sA ) + ∂ ∂ ∂ x ( ρ usA ) = A T ˙ q ∂ t ( ρ h t A ) + ∂ ∂ q − A ∂ p ∂ x ( ρ uh t A ) = A ˙ ∂ t 15
Lehrstuhl für THERMODYNAMIK Quasi 1D Conservation Equations From full Navier-Stokes equations Assumptions No viscous terms Quasi-1D ∂ t ( ρ A ) + ∂ ∂ Mass ∂ x ( ρ uA ) = 0 ∂ t ( ρ uA ) + ∂ ∂ = − A ∂ p ρ u 2 A � � Momentum ∂ x ∂ x ∂ t ( ρ sA ) + ∂ ∂ ∂ x ( ρ usA ) = A T ˙ q ∂ t ( ρ h t A ) + ∂ ∂ q − A ∂ p ∂ x ( ρ uh t A ) = A ˙ ∂ t 16
Lehrstuhl für THERMODYNAMIK Quasi 1D Conservation Equations From full Navier-Stokes equations Assumptions No viscous terms Quasi-1D ∂ t ( ρ A ) + ∂ ∂ Mass ∂ x ( ρ uA ) = 0 ∂ t ( ρ uA ) + ∂ ∂ = − A ∂ p ρ u 2 A � � Momentum ∂ x ∂ x ∂ t ( ρ sA ) + ∂ ∂ ∂ x ( ρ usA ) = A T ˙ q Entropy ∂ t ( ρ h t A ) + ∂ ∂ q − A ∂ p ∂ x ( ρ uh t A ) = A ˙ ∂ t 17
Lehrstuhl für THERMODYNAMIK Quasi 1D Conservation Equations From full Navier-Stokes equations Assumptions No viscous terms Quasi-1D ∂ t ( ρ A ) + ∂ ∂ Mass ∂ x ( ρ uA ) = 0 ∂ t ( ρ uA ) + ∂ ∂ = − A ∂ p ρ u 2 A � � Momentum ∂ x ∂ x ∂ t ( ρ sA ) + ∂ ∂ ∂ x ( ρ usA ) = A T ˙ q Entropy ∂ t ( ρ h t A ) + ∂ ∂ q − A ∂ p ∂ x ( ρ uh t A ) = A ˙ Total Enthalpy ∂ t 18
Lehrstuhl für THERMODYNAMIK Quasi 1D Conservation Equations From full Navier-Stokes equations Assumptions No viscous terms Quasi-1D ∂ t ( ρ A ) + ∂ ∂ ∂ x ( ρ uA ) = 0 ∂ t ( ρ uA ) + ∂ ∂ = − A ∂ p ρ u 2 A � � ∂ x ∂ x ∂ t ( ρ sA ) + ∂ ∂ ∂ x ( ρ usA ) = A T ˙ q ∂ t ( ρ h t A ) + ∂ ∂ q − A ∂ p ∂ x ( ρ uh t A ) = A ˙ Total Enthalpy ∂ t 19
Lehrstuhl für THERMODYNAMIK Quasi 1D Conservation Equations Assumptions No viscous terms ∂ t ( ρ h t A ) + ∂ ∂ q − A ∂ p ∂ x ( ρ uh t A ) = A ˙ Quasi-1D Total Enthalpy ∂ t 20
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