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The Rijke Tube Investigating ThermoAcoustic Dynamics and Control Bassam Bamieh Mechanical Engineering & Center for Control, Dynamical Systems & Computation UNIVERSITY OF CALIFORNIA AT SANTA BARBARA SpongFest 2012, UTD () SpongFest, Nov ’12 1 / 24

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Page 1: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

The Rijke TubeInvestigating ThermoAcoustic Dynamics and Control

Bassam Bamieh

Mechanical Engineering &Center for Control, Dynamical Systems & Computation

UNIVERSITY OF CALIFORNIA AT SANTA BARBARA

SpongFest 2012, UTD

() SpongFest, Nov ’12 1 / 24

Page 2: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Outline

1 The Rijke Tube Experiment, Empirically

2 ThermoAcoustic Energy Conversion

3 ThermoAcoustic Dynamics & Control

4 Optimal Periodic Control

() SpongFest, Nov ’12 2 / 24

Page 3: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

THE RIJKE TUBE EXPERIMENT, EMPIRICALLY

() SpongFest, Nov ’12 3 / 24

Page 4: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

The Rijke Tube Experiment

Observations:Heated coil induces convective flow upwardsWhen coil is sufficient hot, tube begins to hum loudly

() SpongFest, Nov ’12 4 / 24

Page 5: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Observing the ThermoAcoustic Instability

vp

v

qqv =

Acoustic mode: standing wave1/2 wavelength = L

q: Heat release from coildepends on velocity fluctuations v

(convective heat transfer)

Acoustic Dynamics Coil Heat Releasev

q

TheromAcoustic Instabilities are caused by an unstable feedback betweenAcoustic Dynamics and a Flow-Dependent Heat Release Process

() SpongFest, Nov ’12 5 / 24

Page 6: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Observing the ThermoAcoustic Instability

vp

v

qqv =

Acoustic mode: standing wave1/2 wavelength = L

q: Heat release from coildepends on velocity fluctuations v

(convective heat transfer)

Acoustic Dynamics Coil Heat Releasev

q

TheromAcoustic Instabilities are caused by an unstable feedback betweenAcoustic Dynamics and a Flow-Dependent Heat Release Process

() SpongFest, Nov ’12 5 / 24

Page 7: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Observing the ThermoAcoustic Instability

vp

v

qqv =

Acoustic mode: standing wave1/2 wavelength = L

q: Heat release from coildepends on velocity fluctuations v

(convective heat transfer)

Acoustic Dynamics Coil Heat Releasev

q

TheromAcoustic Instabilities are caused by an unstable feedback betweenAcoustic Dynamics and a Flow-Dependent Heat Release Process

() SpongFest, Nov ’12 5 / 24

Page 8: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Empirical (ID-based) Approach to the Rijke Tube

Amp

mic

speaker

heatingcoil

w

Rijke Tube

-k1k2

speaker micpre-amp/

variable amppower amp

w

With simple proportional acoustic feedback K, can stabilize this system!

0 Kfundamental mode unstable stable higher harmonic is unstable

Kmin Kmax

Observation: Which higher harmonic becomes unstable at high gain (e.g. 3’rd or 5’th, ..)depends on mic placement!

() SpongFest, Nov ’12 6 / 24

Page 9: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Empirical (ID-based) Approach to the Rijke Tube

Amp

mic

speaker

heatingcoil

w

Rijke Tube

-k1k2

speaker micpre-amp/

variable amppower amp

w

With simple proportional acoustic feedback K, can stabilize this system!

0 Kfundamental mode unstable stable higher harmonic is unstable

Kmin Kmax

Observation: Which higher harmonic becomes unstable at high gain (e.g. 3’rd or 5’th, ..)depends on mic placement!

() SpongFest, Nov ’12 6 / 24

Page 10: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Identification of the Rijke Tube

Rijke Tube

-k1k2

speaker micpre-amp/

variable amppower amp

w PK+w y

Identify Stabilized Closed Loop T :=PK

1− PK, K ∈ [Kmin,Kmax]

calculate P = KT

1 + TK is uncalibrated → can determined poles and zeros of P, but not OL gain

() SpongFest, Nov ’12 7 / 24

Page 11: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Identification of the Rijke Tube (cont.)

Validating a frequency-fitted, finite-dimensional, open-loop Rijke Tube model

Unstable fundamental, stable higher harmonic modes

Parabolic root pattern characteristic of wave equation with Kelvin-Voigt damping

OL zero locations depend on mic location, OL poles do not

() SpongFest, Nov ’12 8 / 24

Page 12: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Identification of the Rijke Tube (cont.)

Further validation: which harmonic is predicted to go unstable at high gain?

Identified model must predict particular higher harmonic instability

Changing mic location allows for repeat experimentsmicrophone position

position 4: +1 inposition 3: +1/2 inposition 2: 0 inposition 1: -1/2 in

top end of Rijke tube

() SpongFest, Nov ’12 9 / 24

Page 13: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Identification of the Rijke Tube (cont.)

Further validation: which harmonic is predicted to go unstable at high gain?

Identified model must predict particular higher harmonic instability

Changing mic location allows for repeat experimentsmicrophone position

position 4: +1 inposition 3: +1/2 inposition 2: 0 inposition 1: -1/2 in

top end of Rijke tube

() SpongFest, Nov ’12 9 / 24

Page 14: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

The Rijke Tube as a Controls Laboratory Experiment

A versatile experiment to illustrate many important controls conceptsEasy/cheap to buildVery rich and deep (see analysis section)

cf. upcoming Control Systems Magazine article (with K. Åstörm)

() SpongFest, Nov ’12 10 / 24

Page 15: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

THERMOACOUSTIC ENERGY CONVERSION

() SpongFest, Nov ’12 11 / 24

Page 16: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Basic Physics of the ThermoAcoustic Instability

vp

v

qqv =

Acoustic mode: standing wave1/2 wavelength = L

q: Heat release from coildepends on velocity fluctuations v

(convective heat transfer)

Acoustic Dynamics Coil Heat Releasev

q

TheromAcoustic Instabilities are caused by an unstable feedback betweenAcoustic Dynamics and a Flow-Dependent Heat Release Process

() SpongFest, Nov ’12 12 / 24

Page 17: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

ThermoAcoustic Engines and Heat Pumps

Stack-based systems:Temperature gradients inside acoustic cavities acoustic wave

() SpongFest, Nov ’12 13 / 24

Page 18: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

ThermoAcoustic Engines and Heat Pumps

Similar principle as “pulse-tube refrigerators”

() SpongFest, Nov ’12 13 / 24

Page 19: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

ThermoAcoustic Engines and Heat Pumps

A heat-powered refrigerator with no moving parts!

+

acoustic wave

engine heat pump

+

() SpongFest, Nov ’12 13 / 24

Page 20: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

ThermoAcoustic Engines and Heat Pumps

Vision:Replace traditional piston/crank/turbines with pressure/displacement waves

“The power of sound”, Garrett & Backhaus, American Scientist, 2000

Why is this not more widely known? Why hasn’t there been more progress?

() SpongFest, Nov ’12 13 / 24

Page 21: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

ThermoAcoustic Engines and Heat Pumps

Vision:Replace traditional piston/crank/turbines with pressure/displacement waves

“The power of sound”, Garrett & Backhaus, American Scientist, 2000

Why is this not more widely known? Why hasn’t there been more progress?

The Los Alamos Prototype:

Acoustic impedances and couplingsvery carefully designed!

Mechanical design is forone operating point!

() SpongFest, Nov ’12 13 / 24

Page 22: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

ThermoAcoustic Engines and Heat Pumps

Vision:Replace traditional piston/crank/turbines with pressure/displacement waves

“The power of sound”, Garrett & Backhaus, American Scientist, 2000

Why is this not more widely known? Why hasn’t there been more progress?

Acoustics need to adapt to changes in operating conditions

A delicate dance between pressure, displacement and solid/gas heat transfer with nokinematics to enforce timing

Answer: Active Control is needed!

Very little work has been done in this area“[tunable thermoacoustic cooler]", Li, Rotea, Chiu, Mongeau, Paek, ’05

() SpongFest, Nov ’12 13 / 24

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THERMOACOUSTIC DYNAMICS & CONTROL

() SpongFest, Nov ’12 14 / 24

Page 24: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Acoustic & Thermal Dynamics

vp

v

qqv =

Acoustic Dynamics Velocity-dependentheat release

Acoustic Dynamics Coil Heat Releasev

q

() SpongFest, Nov ’12 15 / 24

Page 25: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

1d Gas Dynamics

x1 x2

Ap(x1) p(x2)

v(x2)v(x1)

x

q

Pressure form: (there’s an equivalent Temperature form)

∂t

ρvp

= −

v ρ 00 v 1

ρ

0 γp v

∂xρ∂xv∂xp

+

0β v

ρ

γq

:

mass conservationmomentum balance

energy balance

A PDE in the Hamilton-Jacobi form with q as source term

Linearizations behave like Wave Equation

Point-to-point Transfer Function (non-dimensionalized)

1cosh(s/2)

q v

() SpongFest, Nov ’12 16 / 24

Page 26: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

1d Gas Dynamics

x1 x2

Ap(x1) p(x2)

v(x2)v(x1)

x

q

Pressure form: (there’s an equivalent Temperature form)

∂t

ρvp

= −

v ρ 00 v 1

ρ

0 γp v

∂xρ∂xv∂xp

+

0β v

ρ

γq

:

mass conservationmomentum balance

energy balance

A PDE in the Hamilton-Jacobi form with q as source term

Linearizations behave like Wave Equation

Point-to-point Transfer Function (non-dimensionalized)

1cosh(s/2)

q v

() SpongFest, Nov ’12 16 / 24

Page 27: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Convective Heat Transfer

Hot-wire in cross flow

qe = (Ts−T)(κd + κv

√|v|)

=: (Ts−T) f (v),

v

v

f(v)

“King’s Law” (for steady flow)

() SpongFest, Nov ’12 17 / 24

Page 28: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Convective Heat Transfer (unsteady)

Hot-wire in fluctuating cross flow

Frequency response

vthermal boundary layer

Lighthill ’53

The Royal Society is collaborating with JSTOR to digitize, preserve, and extend access toProceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences.

www.jstor.org®

on August 28, 2012rspa.royalsocietypublishing.orgDownloaded from

() SpongFest, Nov ’12 18 / 24

Page 29: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Convective Heat Transfer (unsteady)

Hot-wire in fluctuating cross flow

Frequency response

vthermal boundary layer

Lighthill ’53Calculation of the frequency response (matched asymptotic expansions)

on August 28, 2012rspa.royalsocietypublishing.orgDownloaded from

() SpongFest, Nov ’12 18 / 24

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Convective Heat Transfer (unsteady)

Hot-wire in fluctuating cross flow

Frequency response

vthermal boundary layer

Lighthill ’53Approximate first-order lag with time delay

() SpongFest, Nov ’12 18 / 24

Page 31: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Convective Heat Transfer (a simple model)

Use King’s law for heat transfer into a thermalboundary layer

Conductive heat transfer from thermal boundarylayer to bulk flow

This introduces a “thermal inertia” around wire Ts

Tb

T

Qs→bQb→f

v

Tube Wall

Thus:

1s/↵+1

f(.)qv

v

f(v)

v

() SpongFest, Nov ’12 19 / 24

Page 32: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Predicting the ThermoAcoustic Instability

1cosh(s/2)

1s/↵+1

f(.)

Correct prediction of the unstable fundamental mode

() SpongFest, Nov ’12 20 / 24

Page 33: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

OPTIMAL PERIODIC CONTROL

() SpongFest, Nov ’12 21 / 24

Page 34: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Control to Enhance Heat-to-Acoustic Power Conversion

Design control to enhance heat-to-acoustic powerconversion

Trade off control effort

Design of an engine’s “optimal limit cycle”

Details of heat transfer are probably important

Similar to problems in (periodic) motion planningWALKING ROBOTS ↔ HUMMING ENGINES

Controller

() SpongFest, Nov ’12 22 / 24

Page 35: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Optimal Periodic Control

More popular in days past (’70s)(is cycling (periodic) operation better than steady (constant) operation? )

ApproachesI Periodic Boundary Value ProblemsI Flatness-based

Our current approach:Frequency domain and flatness-based method for PDE systems

Example:non-sinusoidal operation may be optimal

(“A frequency domain method for optimal periodic control”, Epperline, BB, ACC ’12)

() SpongFest, Nov ’12 23 / 24

Page 36: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Optimal Periodic Control

More popular in days past (’70s)(is cycling (periodic) operation better than steady (constant) operation? )

ApproachesI Periodic Boundary Value ProblemsI Flatness-based

Our current approach:Frequency domain and flatness-based method for PDE systems

Example:non-sinusoidal operation may be optimal

(“A frequency domain method for optimal periodic control”, Epperline, BB, ACC ’12)

() SpongFest, Nov ’12 23 / 24

Page 37: The Rijke Tube - UCSB College of Engineeringbamieh/talks/1211...Amp mic speaker heating coil w Rijke Tube k 2-k 1 speaker mic pre-amp/ variable amp power amp w With simple proportional

Concluding Thoughts

# of times the word “thermoacoustic” appears in the ACC or CDCproceedings (all PDF files) in the past 2 years?

≈ 2-3This area deserves more attention

() SpongFest, Nov ’12 24 / 24