Symposium on the Acousticsof Poro-Elastic Materials2017/12/6-8, Le Mans

Characterisation of inhomogeneous ducts and porous mediaand extrapolation to unavailable thermal conditions

Jacques Cuenca, Laurent De Ryck, Arnaud Perdigon, Thibaud Le Scolan

Siemens Industry Software, Interleuvenlaan 68, B-3001 Leuven, Belgium

jacques.cuenca@siemens.com, laurent.deryck@siemens.com

ObjectiveAim: Predict the acoustic behaviour of two-port inhomogeneous one-dimensionalmedia in experimentally unavailable thermal conditions.

Applications: Experiment-based simulation of the acoustic behaviour of mufflercomponents and inhomogeneous media, including porous materials.

Method:1 — Standard sound transmission loss (STL) experiment at room temperature2 — Inverse estimation of the parameters of a multi-layer model3 — Simulation in arbitrary thermal conditions

STL

frequency

STL

frequency

STL

frequency

Unknown object Fitted model Extrapolated model

ModelInhomogeneous 1D components and porous materials are here modelled as piece-wise homogeneous equivalent fluids with constant temperature. Thermal diffusivityis assumed to vary slowly with space and the overall acoustic behaviour is computedby means of a transfer matrix model.

Equivalent fluid model of rigid-frame porous material1

(Defines the properties of each layer of the global model)

Z (ω, φ, σ,K ) =ρc

φ

F

Gk(ω, φ, σ,K ) =

ω

cFG

Specific impedance Wavenumber

F =

√K(

1− iωµ

ω

)G =

√γ

(1− 1− 1/γ

1− iωθ/ω

) ωµ =σφ

ρK, ωθ =

σ

ρPrviscous and thermal

characteristic frequencies

φ = porosityσ = flow resistivity (N·s·m−4)K = pore shape factor

Thermal conduction2 (equivalent fluid multi-layered medium)(Defines the temperature profile in the multi-layer medium)∂τ

∂t(x , t) = δ(x)

∂2τ

∂x2(x , t) Heat equation, with thermal diffusivity δ(x)

τ (x = 0, t) = τin Imposed inlet temperature

τ (x , t = 0) = τ (∞, t) = τatm Room temperature at rest and at infinity

Hypothesis: thermal diffusivity δ(x) varies slowly in x , e.g. constant in each layer

τ (x) = τatm + (τin − τatm) erfc

(1

2√t

∫ x

0

dξ√δ(ξ)

)Temperature profile of the multi-layer medium

Solution via theLaplace domain

Transfer matrix method3

(Describes the acoustic response of the two-port multi-layer component)[pSv

]in

= T

[pSv

]out

Transfer matrix formalism(pressure+flow convention)

T(ω) =N∏

n=1

Tn(ω, xn)

Transfer matrix of a multi-layertwo-port component

Tn(ω, xn) =

cos(kLn)−ZiSn

sin(kLn)

iSnZ

sin(kLn) cos(kLn)

Transfer matrix of layer n

Properties of the layer:xn = {Ln, Sn, φn,Kn, σn}

STL = 20 log10

(1

2

∣∣∣∣T11 + T12Sout

Zout+ T21

Zin

Sin+ T22

Zin

Zout

Sout

Sin

∣∣∣∣)Sound transmission loss

Properties of airρ = 1.2 kg·m−3 (density)γ = 1.4 (ratio of specific heats)R = 287 J·K−1mol−1 (gas constant)c =√γRτ (m·s−1) (speed of sound)

Pr= 0.71 (Prandtl number)κ = 0.023 W·m−1·K−1 (thermal conductivity)cP = 1.005 kJ·kg−1·K−1 (isobaric specific heat capacity)δ = κ/cPρ (m2·s−1) (thermal diffusivity)

Inverse estimation methodThe parameter search is formulated as a minimisation problem4

min fobj(x) =∑m

|STL(x, ωm)− STL0(x0, ωm)|2

• x0 = true parameters, accessible by direct measurement only

• x = model parameters, including design variables of all layers

• STL0(x0, ωm): measured sound transmission loss (STL) at frequency ωm

• STL(x, ωm): STL predicted by the model at frequency ωm

The problem is solved for x using an optimisation algorithm5, leading to anapproximation of x0, limited by the model accuracy.

Examples

Double expansion chamber

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9280

300

320

340

360

380

y(m

)

x (m)

air air air air air

τ(K

)

τatm

• atmospheric temperatureIdentified geometry (inverse estimation)

• Temperature profile (extrapolation)Adaptive spatial discretisation

• Experiment at room temperature (τatm = 20◦C)

• Design variables: length and cross-section area ofexpansion chambers and intermediate duct

L1 (m) S1 (m2) L2 (m) S2 (m

2) L3 (m) S3 (m2)

x0 0.230 0.017 0.220 0.958·10−3 0.400 0.0907

x 0.239 0.012 0.250 1.2·10−3 0.395 0.0925

• Simulation with τin = 100◦C after 1 s

0

10

20

30

40

50

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

STL(dB)

f (kHz)ExperimentModel (inverse estimation)Model (extrapolation)

Expansion chamber + porous material

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3 0.4 0.50 0.1 0.2 0.3 0.4 0.5280

300

320

340

360

380

y(m

)

x (m)

air air air hb air

τ(K

)

τatm

• atmospheric temperatureIdentified geometry (inverse estimation)

• Temperature profile (extrapolation)Adaptive spatial discretisation

• Numerical experiment at room temperature

• Design variables: expansion chamber length andcross-section area, porous material properties

L (m) S (m) φ σ (N·s·m−4) K

x0 0.23 0.0248846 0.98 60000 3

x 0.229 0.0249 0.979 59933 2.97

• Simulation with τin = 100◦C after 10 s

0

10

20

30

40

50

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

STL(dB)

f (kHz)SimulationModel (inverse estimation)Model (extrapolation)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3 0.4 0.50 0.1 0.2 0.3 0.4 0.5280

300

320

340

360

380

y(m

)

x (m)

air air air hb air

τ(K

)

• Thermal diffusivity: δfoam = δair/10

• Simulation with 100◦C inlet temperature at a rangeof time intervals, from 1 ms to 1 h

1 ms

10 ms

100 ms

1 s

10 s

100 s

1000 s

0

10

20

30

40

50

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

STL(dB)

f (kHz)

Temperature profilesTemperature profilesTemperature profilesTemperature profilesTemperature profilesTemperature profilesTemperature profiles Model (extrapolation for different time intervals)Model (extrapolation for different time intervals)Model (extrapolation for different time intervals)Model (extrapolation for different time intervals)Model (extrapolation for different time intervals)Model (extrapolation for different time intervals)Model (extrapolation for different time intervals)

Discussion• STL is sensitive to thermal conditions; therefore standard room-temperature measurements

are often not representative of operating conditions. Model extrapolation allows to simulate alaboratory object in operating thermal conditions, or to retrieve ISO-comparable tests fromthermally-inhomogeneous measurements.

• Possible applications include experiment-based design of multi-layer media, for instance forperformance optimisation in realistic conditions.

• Only thermal conduction is considered here. Further work would require a refined model ofthermal processes in porous media.

• Experimental and 3D numerical validations are required in order to determine the range ofvalidity of the extrapolation procedure.

1. J.F. Hamet and M. Berengier. Acoustical characteristics of porous pavements: A new phenomenological model.In Internoise 93, pages 641–646, Leuven, 1993.

2. L. Theodore. Heat Transfer Applications for the Practicing Engineer. John Wiley & Sons, 2011.

3. M.L. Munjal. Acoustics of Ducts and Mufflers With Application to Exhaust and Ventilation System Design. AWiley-Interscience publication. Wiley, 1987.

4. J. Cuenca, C. Van der Kelen, and P. Goransson. A general methodology for inverse estimation of the elasticand anelastic properties of anisotropic open-cell porous materials — with application to a melamine foam. J.Appl. Phys., 115(8):084904, 2014.

5. K. Svanberg. A class of globally convergent optimization methods based on conservative convex separableapproximations. SIAM Journal on Optimization, 12(2):555, 2002.