diff --git a/lrftubes.bib b/lrftubes.bib index bd4c64b..9669a45 100644 --- a/lrftubes.bib +++ b/lrftubes.bib @@ -31,7 +31,7 @@ journaltitle = {Journal of Sound and Vibration}, author = {Beltman, W. M.}, urldate = {2011-07-20}, - date = {1999-10-28}, + date = {1999}, keywords = {Printed}, file = {BELTMAN - 1999 - VISCOTHERMAL WAVE PROPAGATION INCLUDING ACOUSTO-EL.pdf:/home/anne/.literature/storage/VUEH48TH/BELTMAN - 1999 - VISCOTHERMAL WAVE PROPAGATION INCLUDING ACOUSTO-EL.pdf:application/pdf;ScienceDirect Snapshot:/home/anne/.literature/storage/7RZVXTAA/S0022460X99923556.html:text/html} } @@ -112,6 +112,17 @@ The full solution of the problem has been obtained by Kirchhoff (1868) in the fo file = {ingentaconnect An Efficient Finite Element Model for Viscothermal Acoustics:/home/anne/.literature/storage/68KJ2IVT/art00009.html:text/html;Kampinga et al. - 2011 - An Efficient Finite Element Model for Viscothermal.pdf:/home/anne/.literature/storage/QRGF9MR6/Kampinga et al. - 2011 - An Efficient Finite Element Model for Viscothermal.pdf:application/pdf} } +@book{bird_transport_2007, + location = {New York, {NY}, {USA}}, + edition = {2nd}, + title = {Transport phenomena}, + isbn = {978-0-470-11539-8}, + series = {Wiley International edition}, + publisher = {J. Wiley}, + author = {Bird, R.B. and Stewart, W.E. and Lightfoot, E.N.}, + date = {2007} +} + @article{bossart_hybrid_2003, title = {Hybrid numerical and analytical solutions for acoustic boundary problems in thermo-viscous fluids}, volume = {263}, @@ -179,6 +190,21 @@ The full solution of the problem has been obtained by Kirchhoff (1868) in the fo file = {Ward e.a. - 2017 - DeltaEC Users Guide version 6.4b2.7.pdf:/home/anne/.literature/storage/MQKGHJ9I/Ward e.a. - 2017 - DeltaEC Users Guide version 6.4b2.7.pdf:application/pdf} } +@article{licht_variation_1944, + title = {The Variation of the Viscosity of Gases and Vapors with Temperature.}, + volume = {48}, + issn = {0092-7325}, + doi = {10.1021/j150433a004}, + pages = {23--47}, + number = {1}, + journaltitle = {The Journal of Physical Chemistry}, + shortjournal = {J. Phys. Chem.}, + author = {Licht, William and Stechert, Dietrich G.}, + urldate = {2015-01-08}, + date = {1944}, + file = {ACS Full Text Snapshot:/home/anne/.literature/storage/4JAIZ6VC/j150433a004.html:text/html;Licht and Stechert - 1944 - The Variation of the Viscosity of Gases and Vapors.pdf:/home/anne/.literature/storage/MKPFZ6SN/Licht and Stechert - 1944 - The Variation of the Viscosity of Gases and Vapors.pdf:application/pdf} +} + @thesis{van_der_eerden_noise_2000, location = {Enschede, The Netherlands}, title = {Noise reduction with coupled prismatic tubes}, @@ -205,7 +231,6 @@ The full solution of the problem has been obtained by Kirchhoff (1868) in the fo title = {Numerical modeling of thermoacoustic systems}, rights = {All rights reserved}, url = {http://doc.utwente.nl/96275/}, - abstract = {The subject of this thesis is a relatively new class of heat engines and refrigerators, called thermoacoustic ({TA}) systems. {TA} systems have gained commercial interest due to their low number of moving parts and potentially high efficiency. In the case of a {TA} engine, heat is converted to acoustic power. This power can subsequently be converted to electricity using a ?reversed? loudspeaker, called a linear alternator. In a {TA} refrigerator, a speaker or linear alternator is used to generate a strong acoustic wave, which is used to pump heat. To achieve competitive power densities, thermoacoustic systems are generally run at such high amplitudes, that performance deteriorating nonlinear effects can no longer be neglected. To accu- rately predict performance in the nonlinear regime, nonlinear models are required. This thesis describes two contributions to the field of thermoacoustic system modeling. Firstly, a one-dimensional heat transfer model has been developed. This model can be used to estimate the performance of often used parallel-plate heat exchangers for thermoacoustic systems. These heat exchangers are located close to the stack or regenerator of a {TA} system and are responsible for the heat in/output required to let the system execute its thermodynamic cycle. The results of the model show a good match with a different heat transfer model from the literature, and the model provides guidelines for future heat exchanger design. Secondly, a nonlinear frequency domain method is developed with which the initial transient start-up process can be skipped in the simulations. The method can be used to directly simulate a {TA} system in its periodic steady-state. This significantly reduces computational cost, since the initial transient regime often involves several hundred oscillation cycles. The method is applied to a one-dimensional nonlinear model of {TA} systems. The model is used to simulate an experimental standing wave thermoacoustic engine from the literature. The obtained results are in agreement with literature results.}, institution = {Universiteit Twente}, type = {phdthesis}, author = {De Jong, J.A.}, @@ -338,4 +363,22 @@ The full solution of the problem has been obtained by Kirchhoff (1868) in the fo journaltitle = {{IEEE} Transactions on audio and electroacoustics}, author = {Welch, Peter}, date = {1967} +} + +@article{kino_investigation_2009, + title = {Investigation of non-acoustical parameters of compressed melamine foam materials}, + volume = {70}, + issn = {0003682X}, + url = {https://linkinghub.elsevier.com/retrieve/pii/S0003682X08001497}, + doi = {10.1016/j.apacoust.2008.07.002}, + abstract = {A series of careful non-acoustical parameters measurements using 5 ‘Illtec’ melamine foam and 10 ‘Basotect {TG}’ melamine foam samples have been made. Flow resistivity, tortuosity, porosity, viscous characteristic length and thermal characteristic length of two types of compressed melamine foam materials with different foam magnifications have been investigated. It has been found that a relationship between the flow resistivity, fibre equivalent diameter and bulk density exists for each of the compressed melamine foam materials.}, + pages = {595--604}, + number = {4}, + journaltitle = {Applied Acoustics}, + shortjournal = {Applied Acoustics}, + author = {Kino, Naoki and Ueno, Takayasu and Suzuki, Yasuhiro and Makino, Hiroshi}, + urldate = {2019-11-28}, + date = {2009-04}, + langid = {english}, + file = {Kino et al. - 2009 - Investigation of non-acoustical parameters of comp.pdf:/home/anne/.literature/storage/I9P5SZAE/Kino et al. - 2009 - Investigation of non-acoustical parameters of comp.pdf:application/pdf} } \ No newline at end of file diff --git a/lrftubes.lyx b/lrftubes.lyx index 537bae4..7397351 100644 --- a/lrftubes.lyx +++ b/lrftubes.lyx @@ -7,6 +7,7 @@ \textclass memoir \begin_preamble \input{tex/preamble.tex} +\usepackage{ar} \end_preamble \options a4paper \use_default_options true @@ -58,9 +59,11 @@ \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 -\cite_engine basic -\cite_engine_type default +\cite_engine biblatex +\cite_engine_type authoryear \biblio_style plain +\biblatex_bibstyle numeric +\biblatex_citestyle numeric \use_bibtopic false \use_indices false \paperorientation portrait @@ -79,8 +82,8 @@ \headsep 1cm \secnumdepth 3 \tocdepth 3 -\paragraph_separation indent -\paragraph_indentation default +\paragraph_separation skip +\defskip smallskip \is_math_indent 0 \math_numbering_side default \quotes_style english @@ -114,7 +117,7 @@ thispagestyle{empty} \end_layout \begin_layout Title -LRFTubes documentation - v1.0 +LRFTubes documentation - v1.1 \end_layout \begin_layout Author @@ -126,7 +129,7 @@ Dr.ir. \begin_layout Standard \align center \begin_inset Graphics - filename img_default/ascee_beeldmerk.pdf + filename img/LRFTubes.pdf width 65text% \end_inset @@ -172,11 +175,11 @@ ASCEE \begin_inset Newline newline \end_inset -Vildersveenweg 19 +Maximastraat 1 \begin_inset Newline newline \end_inset -7443 RZ Nijverdal +7442 NW Nijverdal \begin_inset Newline newline \end_inset @@ -1312,7 +1315,7 @@ Introduction \begin_layout Standard Welcome to the documentation of \begin_inset ERT -status collapsed +status open \begin_layout Plain Layout @@ -1382,7 +1385,7 @@ This documentation serves as a reference for the implemented models. \begin_inset CommandInset href LatexCommand href name "README" -target "https://github.com/asceenl/lrftubes" +target "https://code.ascee.nl/ASCEE/lrftubes/raw/branch/master/LICENSE" literal "false" \end_inset @@ -1599,6 +1602,1496 @@ reference "chap:Provided-acoustic-models" For each of the segments, the resulting transfer matrix model is derived. \end_layout +\begin_layout Chapter +Material properties +\end_layout + +\begin_layout Section +Air +\end_layout + +\begin_layout Standard +Nonlinearity parameter: +\end_layout + +\begin_layout Section +Exhaust gas +\end_layout + +\begin_layout Subsection +Composition +\end_layout + +\begin_layout Standard +Definitions: +\end_layout + +\begin_layout Itemize +\begin_inset Formula $\omega_{i}$ +\end_inset + + mass fraction of species +\begin_inset Formula $i$ +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $x_{i}$ +\end_inset + + molar / volume fraction of species +\begin_inset Formula $i$ +\end_inset + + (assuming ideal gas behavior) +\end_layout + +\begin_layout Itemize +\begin_inset Formula $\overline{M}$ +\end_inset + + average molar mass of (exhaust gas) mixture +\end_layout + +\begin_layout Itemize +\begin_inset Formula $M_{i}$ +\end_inset + + molar mass of species +\begin_inset Formula $i$ +\end_inset + + +\end_layout + +\begin_layout Standard +The following equations hold in a mixture: +\begin_inset Formula +\begin{align} +\sum_{i}\omega_{i} & =1\\ +\sum_{i}x_{i} & =1\\ +\overline{M} & =\sum\nolimits _{i}x_{i}M_{i}\label{eq:molar_mass_comp} +\end{align} + +\end_inset + +We can convert mass fractions to mole fractions with the following rule: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +The total mass is ( +\begin_inset Formula $N$ +\end_inset + +) is the total number of moles +\begin_inset Formula +\[ +m=x_{i}M_{i}N +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +The total number of moles is: +\begin_inset Formula +\[ +N=\frac{m}{\overline{M}} +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +The average molar mass is: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +\overline{M}=\frac{m}{N}=\sum_{i}x_{i}M_{i} +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +The mass fraction to mole fraction is: +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\omega_{i}=x_{i}\frac{M_{i}}{\overline{M}}\qquad\Longleftrightarrow\qquad x_{i}=\omega_{i}\frac{\overline{M}}{M_{i}}\label{eq:massfr_to_molarfr_viceversa} +\end{equation} + +\end_inset + +Henceforth, what is often used, is to compute the average molar mass given + only the mass fractions: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $\omega_{i}m=N_{i}M_{i}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\omega_{i}}{M_{i}}=\frac{N_{i}}{m}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\sum_{i}\frac{\omega_{i}}{M_{i}}=\frac{N}{m}=\frac{1}{\overline{M}}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\overline{M}=\frac{1}{\sum\nolimits _{i}\frac{\omega_{i}}{M_{i}}}\label{eq:molar_mass_vs_massfr} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Ideal gas mixtures +\end_layout + +\begin_layout Standard +For an ideal gas, the components of a gas mixture can be represented by + their +\begin_inset Quotes eld +\end_inset + +partial pressure +\begin_inset Quotes erd +\end_inset + +, which is the total pressure times the volume fraction of the component + in the mixture. + For an ideal gas, the volume fraction equals to mole fraction. + Hence: +\begin_inset Formula +\begin{equation} +\frac{V_{i}}{V}\overset{\mathrm{ideal\,gas}}{=}x_{i}=\frac{p_{i}}{R_{u}T} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The mass fraction can be computed from the mole fraction. +\end_layout + +\begin_layout Subsection +Transport properties +\end_layout + +\begin_layout Standard +\begin_inset Float table +wide false +sideways false +status open + +\begin_layout Plain Layout +\noindent +\align center +\begin_inset Tabular + + + + + + + + + +\begin_inset Text + +\begin_layout Plain Layout +Substance +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $M$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $T_{c}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $G$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $C_{r}$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Carbon dioxide +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +SI{44.01e-3}{kg +\backslash +per +\backslash +mole} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +SI{304}{ +\backslash +K} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +44.6 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0.766 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Oxygen +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +SI{32.00e-3}{kg +\backslash +per +\backslash +mole} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +SI{154}{ +\backslash +K} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +32.8 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0.712 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Nitrogen +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +SI{28.02e-3}{kg +\backslash +per +\backslash +mole} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +SI{126}{ +\backslash +K} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +24.6 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0.881 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Water vapor +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +SI{18.02e-3}{kg +\backslash +per +\backslash +mole} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +SI{647}{ +\backslash +K} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +52.2 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +1.018 +\end_layout + +\end_inset + + + + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Critical values and constants of common diatomic gases +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "tab:crit_values_diatom_gas" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +Dynamic viscosity of pure gases +\end_layout + +\begin_layout Standard +Here we assume the dynamic viscosity of a pure substance can be modeled + using Sutherland's equation: +\begin_inset Formula +\begin{equation} +\mu=\mu_{c}\left(\frac{T_{0}+C}{T+C}\right)\left(\frac{T}{T_{0}}\right)^{3/2}, +\end{equation} + +\end_inset + +where the subscript +\begin_inset Formula $c$ +\end_inset + + denotes the value at its +\begin_inset Quotes eld +\end_inset + +critical point +\begin_inset Quotes erd +\end_inset + +. + In convenient form we solve: +\begin_inset Formula +\begin{equation} +\mu=\mu_{c}\mu_{r}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\mu_{c}$ +\end_inset + + is the critical viscosity and +\begin_inset Formula $\mu_{r}$ +\end_inset + + is the +\begin_inset Quotes eld +\end_inset + +reduced viscosity defined as +\begin_inset Formula $\mu/\mu_{c}$ +\end_inset + +. + For +\begin_inset Formula $\mu_{c}$ +\end_inset + + we have the reduced form of Sutherland's equation: +\begin_inset Formula +\begin{equation} +\mu_{c}=\frac{1+C_{r}}{T_{r}+C_{r}}T_{r}^{3/2} +\end{equation} + +\end_inset + + The value for +\begin_inset Formula $\mu_{c}$ +\end_inset + + can be calculated as: +\begin_inset Formula +\begin{equation} +\mu_{c}=\num{3.5e-6}G +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Values for +\begin_inset Formula $T_{r}$ +\end_inset + +, +\begin_inset Formula $C_{r}$ +\end_inset + + and +\begin_inset Formula $G$ +\end_inset + + are listed in Table +\begin_inset CommandInset ref +LatexCommand ref +reference "tab:crit_values_diatom_gas" + +\end_inset + + +\begin_inset CommandInset citation +LatexCommand cite +key "licht_variation_1944" +literal "false" + +\end_inset + +. +\end_layout + +\begin_layout Subsubsection +Dynamic viscosity of a gas mixture +\end_layout + +\begin_layout Standard +The dynamic viscosity of a gas mixture can be derived from the dynamic viscosity + of pure gases as +\begin_inset CommandInset citation +LatexCommand cite +after "p. 27" +key "bird_transport_2007" +literal "false" + +\end_inset + +: +\begin_inset Formula +\begin{equation} +\mu_{\mathrm{mix}}=\sum_{n=0}^{N-1}\frac{x_{n}\mu_{n}}{\sum_{m=0}^{N-1}\Phi_{nm}x_{m}},\label{eq:mumix} +\end{equation} + +\end_inset + +where +\begin_inset Formula $\mu_{n}$ +\end_inset + +denotes the pure substance dynamic viscosity of species +\begin_inset Formula $n$ +\end_inset + +, and +\begin_inset Formula $x_{n}$ +\end_inset + +denotes its mole fraction in the mixture. + +\begin_inset Formula $\Phi_{mn}$ +\end_inset + + is defined as: +\begin_inset Formula +\begin{equation} +\Phi_{mn}=\frac{1}{\sqrt{8}}\left(1+\frac{M_{n}}{M_{m}}\right)^{-1/2}\left[1+\left(\frac{\mu_{n}}{\mu_{m}}\right)^{1/2}\left(\frac{M_{m}}{M_{n}}\right)^{1/4}\right]^{2}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $M_{i}$ +\end_inset + + is the molar mass of species +\begin_inset Formula $i$ +\end_inset + +. + The denominator of Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:mumix" + +\end_inset + + can efficiently be solved by noting that +\begin_inset Formula $d_{n}=\sum_{m=0}^{N-1}\Phi_{nm}x_{m}$ +\end_inset + + is a matrix-vector product, which can be written as +\begin_inset Formula $\boldsymbol{d}=\boldsymbol{\Phi}\cdot\boldsymbol{x}$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Combustion +\end_layout + +\begin_layout Standard +To compute the gas constant, first the mixture components of the exhaust + gas need to be computed. + We assume that the oxidizer is air with 79% vol of nitrogen (molecules) + and 21% oxygen molecules. + The tiny part of argon and other components is ignored. + Then, the gross formula for combustion is: +\begin_inset Formula +\begin{equation} +\underbrace{x_{f,C}C+x_{f,O}O+x_{H,f}H+x_{f,N}N}_{\mathrm{fuel}}+\underbrace{y_{\mathrm{ox}}\left(0.79N_{2}+0.21O_{2}\right)}_{\mathrm{oxidizer}}\rightarrow\underbrace{y_{g,\mathrm{water}}H_{2}O+y_{g,CO_{2}}CO_{2}+y_{g,N_{2}}N_{2}}_{\mathrm{exhaust\,gas}}.\label{eq:combustion} +\end{equation} + +\end_inset + +Above reaction formula can be read as: +\begin_inset Quotes eld +\end_inset + +take +\begin_inset Formula $x_{f,C}$ +\end_inset + + moles of carbon in the fuel, add +\begin_inset Formula $y_{\mathrm{ox}}$ +\end_inset + + moles of air, and it should result in +\begin_inset Formula $y_{g,CO_{2}}$ +\end_inset + + moles of +\begin_inset Formula $CO_{2}$ +\end_inset + + +\begin_inset Quotes erd +\end_inset + + And so on for the other elements. + The mole fractions in the fuel composition can be derived from its mass + fractions, upon utilizing Eqs. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:massfr_to_molarfr_viceversa" + +\end_inset + + and +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:molar_mass_vs_massfr" + +\end_inset + +. + From Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:combustion" + +\end_inset + +, the following system of equations can be created: +\begin_inset Formula +\begin{equation} +\left\{ \begin{array}{c} +x_{f,C}\\ +x_{f,O}\\ +x_{f,H}\\ +x_{f,N} +\end{array}\right\} +\left[\begin{array}{cccc} +0 & 0 & -1 & 0\\ +2\times0.21 & -1 & -2 & 0\\ +0 & -2 & 0 & 0\\ +2\times0.79 & 0 & 0 & -2 +\end{array}\right]\left\{ \begin{array}{c} +y_{\mathrm{ox}}\\ +y_{g,\mathrm{water}}\\ +y_{g,CO_{2}}\\ +y_{g,N_{2}} +\end{array}\right\} =\left\{ \begin{array}{c} +0\\ +0\\ +0\\ +0 +\end{array}\right\} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Solving this results in: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $x_{f,O}+2\times0.21y_{\mathrm{ox}}-y_{g,\mathrm{water}}-2y_{g,CO_{2}}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $2\times0.21y_{\mathrm{ox}}=\frac{1}{2}x_{f,H}+2x_{f,C}+x_{f,O}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +– +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $x_{f,N}+2\times0.79y_{\mathrm{ox}}-2y_{g,N_{2}}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $y_{g,N_{2}}=0.79y_{\mathrm{ox}}+\frac{1}{2}x_{f,N}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align} +y_{g,CO_{2}} & =x_{f,C}\\ +y_{g,\mathrm{water}} & =\frac{1}{2}x_{f,H}\\ +y_{\mathrm{ox}}= & \frac{\frac{1}{2}x_{f,H}+2x_{f,C}-x_{f,O}}{2\times0.21}\\ +y_{g,N_{2}}= & 0.79y_{\mathrm{ox}}+\frac{1}{2}x_{f,N} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +Note that the mole fractions are +\emph on +unnormalized +\emph default + (that is why we use symbol +\begin_inset Formula $y$ +\end_inset + +, not +\begin_inset Formula $x$ +\end_inset + +): they denote the number of moles required to burn 1 mole of fuel. + To compute the mole fractions in the exhaust gas, +\begin_inset Formula +\begin{equation} +x_{g,\mathrm{water}}=\frac{y_{1}}{y_{1}+y_{2}+y_{3}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Table +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "tab:fuel_components" + +\end_inset + + gives an overview of the composition of typical combustion fuels. + Once the molar fractions of the exhaust gas are known, the average molar + mass can be computed using Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:molar_mass_comp" + +\end_inset + +. + Then, the specific gas constant can be computed according to: +\begin_inset Formula +\begin{equation} +R_{s}=\frac{R_{u}}{\overline{M}}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $R_{u}$ +\end_inset + + is the universal gas constant. +\end_layout + +\begin_layout Standard +\begin_inset Float table +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Tabular + + + + + + + +\begin_inset Text + +\begin_layout Plain Layout +Mass fraction +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Wood +\begin_inset Foot +status collapsed + +\begin_layout Plain Layout +https://www.engineeringtoolbox.com/co2-emission-fuels-d_1085.html +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Dutch Natural gas +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Carbon +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +50 % +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Oxygen +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +42 % +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0 % +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Hydrogen +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +6 % +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Nitrogen +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0 % +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + + + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Mixture mass composition of fuels +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "tab:fuel_components" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +Specific heat ratio +\end_layout + +\begin_layout Standard +The specific heat is build-up according to mass percentages of the flue + gas. + Carbon dioxide has a +\begin_inset Formula $c_{p}$ +\end_inset + + of 840 J/kg/K, water vapor of 1930: +\begin_inset Formula +\begin{equation} +\overline{c}_{p}=\sum\nolimits _{i}\omega_{i}c_{p,i}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Sound absorbing solid materials +\end_layout + +\begin_layout Standard +High porosity soft materials can be modeled adequately with the Delaney-Bazley-M +iki model. + The model has a single input, namely the static flow resistivity. + Table +\end_layout + +\begin_layout Standard +\begin_inset Float table +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Tabular + + + + + + +\begin_inset Text + +\begin_layout Plain Layout +Name +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Basotect TG +\begin_inset Foot +status collapsed + +\begin_layout Plain Layout +A.k.a.Flamex Basic (akoestiekwinkel.nl) +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Description +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Melamine resin foam (fire retardant) +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Density [ +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +si{ +\backslash +kg +\backslash +per +\backslash +cubic +\backslash +m} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Foot +status collapsed + +\begin_layout Plain Layout +https://www.forman.co.nz/media/emizen_banner/b/a/basf_basotect_datasheet.pdf +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Flow resistivity [ +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +si{ +\backslash +pascal +\backslash +s +\backslash +per +\backslash +meter} +\end_layout + +\end_inset + +] +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +num{8.5e3} +\end_layout + +\end_inset + +, source: +\begin_inset CommandInset citation +LatexCommand cite +key "kino_investigation_2009" +literal "false" + +\end_inset + +, Table 2 average value. +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + + + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Resistivity values are given for room temperature +\end_layout + +\end_inset + + +\end_layout + +\end_inset + +Conversion +\end_layout + \begin_layout Chapter The transfer matrix method \end_layout @@ -1666,13 +3159,14 @@ where is a source term. In the code and in this documentation -\begin_inset Formula $e^{+i\omega t}$ +\begin_inset Formula $e^{{\color{red}+}i\omega t}$ \end_inset convention is used. A common choice of state variables is such that their product has the unit of power. - For the acoustic systems in this work the state variables are acoustic + For all systems in this code, the state variables satisfy this property. + For example in an acoustic segment, the power is the product of acoustic pressure \begin_inset Formula $p\left(\omega\right)$ \end_inset @@ -1682,7 +3176,7 @@ where \end_inset . - The acoustic power flow can then be computed as: + For complex phasors and, the acoustic power flow can then be computed as: \begin_inset Formula \begin{equation} E=\frac{1}{2}\Re\left[pU^{*}\right], @@ -2127,6 +3621,231 @@ lrftubess field inside a non-lumped segment, such as an acoustic duct. \end_layout +\begin_layout Section +Input impedance, output impedance +\end_layout + +\begin_layout Standard +The acoustic input impedance +\begin_inset Formula $Z_{\mathrm{in}}\equiv p_{L}/U_{L}$ +\end_inset + + on the left side of a segment is defined as the impedance a connecting + segment +\begin_inset Quotes eld +\end_inset + +feels +\begin_inset Quotes erd +\end_inset + + for a certain boundary condition on the right side. + +\begin_inset Foot +status open + +\begin_layout Plain Layout +Note that the definitions of open and closed below are relating to electrical + circuits, not open or closed in the acoustical sense. + I.e. + an open impedance corresponds to a hard acoustic wall (which is acoustically + closed). +\end_layout + +\end_inset + + There are two special load cases for the segment, either on the right side, + the circuit is open, resulting in +\begin_inset Formula $U_{R}=0$ +\end_inset + +, or the circuit is shorted, which results in +\begin_inset Formula $p_{R}=0$ +\end_inset + +. + For the open circuit, the input impedance can be computed from the transfer + matrix as: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Open case: ( +\begin_inset Formula $U_{R}=0$ +\end_inset + +_ +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $p_{R}=T_{11}p_{L}+T_{12}U_{L}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $U_{R}=0=T_{21}p_{L}+T_{22}U_{L}\Rightarrow\frac{p_{L}}{U_{L}}=-\frac{T_{22}}{T_{21}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Shorted case ( +\begin_inset Formula $p_{R}=0$ +\end_inset + + ): +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $0=T_{11}p_{L}+T_{12}U_{L}\Rightarrow\frac{p_{L}}{U_{L}}=-\frac{T_{12}}{T_{11}}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align} +Z_{\mathrm{in},\mathrm{open}} & =-\frac{T_{22}}{T_{21}}\\ +Z_{\mathrm{in},\mathrm{short}} & =-\frac{T_{12}}{T_{11}} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +For a passive component (and passive load on the right side), the real part + of the input impedance should be positive: +\begin_inset Formula +\begin{equation} +\Re\left[Z_{\mathrm{in}}\right]\geq0. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The acoustic output impedance +\begin_inset Formula $Z_{\mathrm{out}}\equiv p_{R}/U_{R}$ +\end_inset + + on the right side of a segment is defined as the impedance a connecting + segment +\begin_inset Quotes eld +\end_inset + +feels +\begin_inset Quotes erd +\end_inset + + for a certain boundary condition on the left side. + +\begin_inset Formula +\begin{align} +Z_{\mathrm{out},\mathrm{open}} & =\frac{T_{11}}{T_{21}}\\ +Z_{\mathrm{out},\mathrm{short}} & =\frac{T_{\mathrm{12}}}{T_{22}} +\end{align} + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Open case left side, means +\begin_inset Formula $U_{L}=0$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $p_{R}=T_{11}p_{L}+T_{12}U_{L}$ +\end_inset + + –> +\begin_inset Formula $p_{R}=T_{11}p_{L}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $U_{R}=T_{21}p_{L}+T_{22}U_{L}$ +\end_inset + + –> +\begin_inset Formula $U_{R}=T_{21}p_{L}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +================================== +\end_layout + +\begin_layout Plain Layout +Shorted case, means +\begin_inset Formula $p_{L}=0$ +\end_inset + +, +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $p_{R}=T_{11}p_{L}+T_{12}U_{L}$ +\end_inset + + –> +\begin_inset Formula $p_{R}=T_{12}U_{L}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $U_{R}=T_{21}p_{L}+T_{22}U_{L}$ +\end_inset + + –> +\begin_inset Formula $U_{R}=T_{22}U_{L}$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +For passive segments, the real part of the output impedance should be +\emph on +negative: +\emph default + +\begin_inset Formula +\begin{equation} +\Re\left[Z_{\mathrm{out}}\right]\leq0. +\end{equation} + +\end_inset + + +\end_layout + \begin_layout Chapter Provided acoustic models \begin_inset CommandInset label @@ -2477,6 +4196,261 @@ reference "subsec:Thermal-relaxation-effect" can be set to 0. \end_layout +\begin_layout Subsection +Other cross-sectional geometries +\end_layout + +\begin_layout Subsubsection +Rectangular duct +\end_layout + +\begin_layout Standard +Analytical functions exist for prismatic geometries, such as parallel plates, + rectangular holes, and even triangular holes. + For parallel plates with sides +\begin_inset Formula $2y_{0}\times2z_{0}$ +\end_inset + +, the Rott function reads: +\begin_inset Formula +\begin{equation} +f=1-\frac{64}{\pi^{4}}\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{1}{\left(2m-1\right)^{2}}\frac{1}{\left(2n-1\right)^{2}C_{mn}}, +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +C_{mn}=1-\frac{i\pi^{2}\delta^{2}}{8y_{0}^{2}z_{0}^{2}}\left(\left(2m-1\right)^{2}z_{0}^{2}+\left(2n-1\right)^{2}y_{0}^{2}\right). +\end{equation} + +\end_inset + +The hydraulic radius is related to +\begin_inset Formula $y_{0}$ +\end_inset + + and +\begin_inset Formula $z_{0}$ +\end_inset + + as: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $r_{h}=\frac{S}{\Pi}=\frac{4y_{0}z_{0}}{4y_{0}+4z_{0}}=$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +r_{h}=\frac{y_{0}z_{0}}{y_{0}+z_{0}} +\end{equation} + +\end_inset + +Defining the aspect ratio as +\begin_inset Formula $\AR=z_{0}/y_{0}$ +\end_inset + +, a useful equation is to derive +\begin_inset Formula $y_{0}$ +\end_inset + + and +\begin_inset Formula $z_{0}$ +\end_inset + + from +\begin_inset Formula $r_{h}$ +\end_inset + + and +\begin_inset Formula $\AR$ +\end_inset + +: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $r_{h}=\frac{y_{0}A}{\left(1+A\right)}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $y_{0}=r_{h}\frac{\left(1+A\right)}{A}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $z_{0}=r_{h}\left(1+A\right)$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align} +y_{0} & =r_{h}\frac{\left(1+\AR\right)}{\AR}\\ +z_{0} & =r_{h}\left(1+\AR\right) +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +Annular ring +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $i\omega\rho_{0}u=-\frac{\mathrm{d}p}{\mathrm{d}x}+\mu_{0}\nabla_{\perp}^{2}u$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Fill in: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +u=\frac{i}{\omega\rho_{0}}\left(1-h_{\nu}\right)\frac{\mathrm{d}p}{\mathrm{d}x} +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Note that +\begin_inset Formula $h_{\nu}|_{\mathrm{wall}}\equiv1$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $0=h_{\nu}+\frac{i\delta_{\nu}^{2}}{2}\nabla_{\perp}^{2}h_{\nu}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $h_{\nu}+\frac{i\mu_{0}}{\omega\rho_{0}}\nabla_{\perp}^{2}h_{\nu}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +- +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{i\mu_{0}}{\omega\rho_{0}}\nabla_{\perp}^{2}h_{\nu}+h_{\nu}=0$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The differential equation that is required to be solved +\begin_inset Formula +\begin{equation} +\frac{i\mu_{0}}{\omega\rho_{0}}\nabla_{\perp}^{2}h_{\nu}+h_{\nu}=0,\qquad h_{\nu|\mathrm{wall}}=0 +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +For an annular duct the Rott function reads: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +h_{\nu}=\frac{\left(J_{0}\left(\frac{r_{0}\left(1-i\right)}{\delta_{\nu}}\right)-J_{0}\left(\frac{r_{1}\left(1-i\right)}{\delta_{\nu}}\right)\right)Y_{0}\left(\frac{r\left(1-i\right)}{\delta_{\nu}}\right)+\left(Y_{0}\left(\frac{r_{1}\left(1-i\right)}{\delta_{\nu}}\right)-Y_{0}\left(\frac{r_{0}\left(1-i\right)}{\delta_{\nu}}\right)\right)J_{0}\left(\frac{r\left(1-i\right)}{\delta_{\nu}}\right)}{J_{0}\left(\frac{r_{0}\left(1-i\right)}{\delta_{\nu}}\right)Y_{0}\left(\frac{r_{1}\left(1-i\right)}{\delta_{\nu}}\right)-J_{0}\left(\frac{r_{1}\left(1-i\right)}{\delta_{\nu}}\right)Y_{0}\left(\frac{r_{0}\left(1-i\right)}{\delta_{\nu}}\right)} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Where +\begin_inset Formula +\begin{align*} +\alpha_{0} & =\frac{r_{0}\left(1-i\right)}{\delta_{i}}\\ +\alpha_{1} & =\frac{r_{1}\left(1-i\right)}{\delta_{i}} +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Standard +And: +\begin_inset Formula +\begin{align} +C_{1} & =\frac{Y_{0}\left(\alpha_{1}\right)-Y_{0}\left(\alpha_{0}\right)}{J_{0}\left(\alpha_{0}\right)Y_{0}\left(\alpha_{1}\right)-J_{0}\left(\alpha_{1}\right)Y_{0}\left(\alpha_{0}\right)}\\ +C_{2} & =\frac{J_{0}\left(\alpha_{0}\right)-J_{0}\left(\alpha_{1}\right)}{J_{0}\left(\alpha_{0}\right)Y_{0}\left(\alpha_{1}\right)-J_{0}\left(\alpha_{1}\right)Y_{0}\left(\alpha_{0}\right)} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +f_{i}=\delta_{i}\left(1+i\right)\frac{\left\{ H_{0}^{(1)}\left(\alpha_{0}\right)-H_{0}^{(1)}\left(\alpha_{1}\right)\right\} \left[r_{0}H_{-1}^{(2)}\left(\alpha_{0}\right)-r_{1}H_{-1}^{(2)}\left(\alpha_{1}\right)\right]+\left\{ H_{0}^{(2)}\left(\alpha_{0}\right)-H_{0}^{(2)}\left(\alpha_{1}\right)\right\} \left[r_{1}H_{-1}^{(1)}\left(\alpha_{1}\right)-r_{0}H_{-1}^{(1)}\left(\alpha_{0}\right)\right]}{\left(r_{1}^{2}-r_{0}^{2}\right)\left[H_{0}^{(1)}\left(\alpha_{0}\right)H_{0}^{(2)}\left(\alpha_{1}\right)-H_{0}^{(1)}\left(\alpha_{1}\right)H_{0}^{(2)}\left(\alpha_{0}\right)\right]} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Transfer matrix +\end_layout + \begin_layout Standard Upon solving for Eqs. @@ -2733,7 +4707,7 @@ reference "eq:Z_c_prismduct" the wave number corrected for viscothermal losses: \begin_inset Formula \begin{equation} -\Gamma=k\sqrt{\frac{1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\epsilon_{s}}}{1-f_{\nu}}}.\label{eq:Gamma} +\Gamma=\frac{\omega}{c_{0}}\sqrt{\frac{1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\epsilon_{s}}}{1-f_{\nu}}}.\label{eq:Gamma} \end{equation} \end_inset @@ -3035,7 +5009,7 @@ reference "fig:im_gamma" \begin_inset Float figure wide false sideways false -status open +status collapsed \begin_layout Plain Layout \align center @@ -3110,20 +5084,14 @@ name "fig:im_gamma" \end_layout -\begin_layout Standard -\begin_inset Note Note -status open - -\begin_layout Plain Layout +\begin_layout Section \series bold -Duct with conical cross-sectional area +Duct with varying cross-sectional area \end_layout -\begin_layout Plain Layout -For conical ducts, i.e. - ducts with quadratic variation in the cross-sectional area (linear variation - in the diameter, or cross-sectional length scale), an approximately valid +\begin_layout Standard +For ducts with variation in the cross-sectional area, an approximately valid ordinary differential equation can be derived, which is a viscothermal correction to Webster's horn equation \begin_inset CommandInset citation @@ -3135,9 +5103,90 @@ literal "true" \end_inset : +\begin_inset Formula +\begin{equation} +\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}+\frac{1}{S_{f}}\frac{\mathrm{d}S_{f}}{\mathrm{d}x}\frac{\mathrm{d}p}{\mathrm{d}x}+\Gamma^{2}p=0 +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Exponential duct (horn) +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +S_{f}=\exp\left(\alpha x\right) +\end{equation} + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}+\alpha\frac{\mathrm{d}p}{\mathrm{d}x}+\Gamma^{2}p=0$ +\end_inset + + \end_layout \begin_layout Plain Layout +Filling in: +\begin_inset Formula $p=a\exp\left(\beta x\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +\beta^{2}+\alpha\beta+\Gamma^{2}=0 +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Solving for +\begin_inset Formula $\beta$ +\end_inset + +: +\begin_inset Formula +\[ +\beta=\frac{1}{2}\left(-\alpha\pm\sqrt{\alpha^{2}-4\Gamma^{2}}\right) +\] + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +Conical ducts +\end_layout + +\begin_layout Standard +For conical ducts, i.e. + ducts with quadratic variation in the cross-sectional area (linear variation + in the diameter, or cross-sectional length scale), +\end_layout + +\begin_layout Standard \begin_inset Note Note status collapsed @@ -3393,14 +5442,55 @@ And we find volume flow from \end_layout -\begin_layout Plain Layout +\begin_layout Standard +such that for a conical tube the radius +\begin_inset Formula $r(x)$ +\end_inset + + varies as: \begin_inset Formula \begin{equation} -\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}+\frac{1}{S_{f}}\frac{\mathrm{d}S_{f}}{\mathrm{d}x}\frac{\mathrm{d}p}{\mathrm{d}x}+\Gamma^{2}p=0 +r(x)=r_{0}+\eta x, \end{equation} \end_inset +where +\begin_inset Formula +\begin{equation} +\eta=\frac{x}{L}\left(r_{1}-r_{0}\right) +\end{equation} + +\end_inset + +Filling in for +\begin_inset Formula $S_{f}=\pi\left(r_{0}+\eta x\right)^{2}$ +\end_inset + + yields +\begin_inset Formula +\begin{equation} +\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}+\frac{2\eta}{r_{0}+\eta x}\frac{\mathrm{d}p}{\mathrm{d}x}+\Gamma^{2}p=0, +\end{equation} + +\end_inset + +for which the solution is: +\begin_inset Formula +\begin{equation} +p=\frac{C_{1}\exp\left(-i\Gamma x\right)+C_{1}\exp\left(-i\Gamma x\right)}{r_{0}+\eta x} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout If we assume \begin_inset Formula $S_{f}=\pi\left(r_{0}+\eta x\right)^{2}$ \end_inset @@ -3569,7 +5659,7 @@ Derivation transfer matrix: \begin_layout Plain Layout \lang english -\begin_inset Formula $U_{1}=\frac{i\left(1-f_{\nu}\right)\pi S_{f}}{\omega\rho_{m}}\frac{dp_{1}}{dx}$ +\begin_inset Formula $U_{1}=\frac{i\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{m}}\frac{dp_{1}}{dx}=\frac{i\left(1-f_{\nu}\right)}{kZ_{0}}\frac{dp_{1}}{dx}$ \end_inset @@ -4039,33 +6129,68 @@ U_{1} \end_inset +\end_layout + +\begin_layout Plain Layout +According to sympy, for pR: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\Gamma p_{L}r_{0}\cos\left(\Gamma L\right)+\eta p_{L}\sin\left(\Gamma L\right)}{\Gamma r_{1}}-iZ_{c,0}\frac{r_{0}\sin\left(\Gamma L\right)}{r_{1}}U_{L}$ +\end_inset + + +\end_layout + +\end_inset + +, klopt! +\end_layout + +\begin_layout Plain Layout +According to sympy, for UR: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula +\begin{align*} +U_{R} & =-i\frac{\Gamma\left(L\eta+r_{0}\right)\left(\Gamma p_{L}r_{0}\sin{\left(\Gamma L\right)}+\left(i\Gamma U_{L}Z_{c0}r_{0}-\eta p_{L}\right)\cos{\left(\Gamma L\right)}\right)+\eta\left(\Gamma p_{L}r_{0}\cos{\left(\Gamma L\right)}-\left(i\Gamma U_{L}Z_{c0}r_{0}-\eta p_{L}\right)\sin{\left(\Gamma L\right)}\right)}{\Gamma^{2}Z_{c0}r_{0}^{2}}\\ +\end{align*} + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + \begin_inset Formula \begin{equation} \mathbf{T}_{\mbox{cone}}=\left[\begin{array}{cc} -\frac{r_{0}\cos\left(\Gamma_{0}L\right)}{r_{0}+\eta L}+\frac{\alpha}{\Gamma}\frac{\sin\left(\Gamma_{0}L\right)}{r_{0}+\eta L} & -iZ_{c,0}\frac{r_{0}\sin\left(\Gamma_{0}L\right)}{r_{0}+\eta L}\\ --iZ_{c,0}^{-1}\left(1+\frac{\eta L}{r_{0}}+\frac{\eta^{2}}{\Gamma_{0}^{2}r_{0}^{2}}\right)\sin\left(\Gamma_{0}L\right)+i\frac{\eta^{2}L\cos\left(\Gamma_{0}L\right)}{r_{0}^{2}\Gamma_{0}Z_{c,0}}\,\,\,\,\,\, & \left(1+\frac{\eta L}{r_{0}}\right)\cos\left(\Gamma_{0}L\right)-\frac{\eta}{\Gamma r_{0}}\sin\left(\Gamma_{0}L\right) -\end{array}\right] +\frac{\Gamma r_{0}\cos\left(\Gamma L\right)+\eta\sin\left(\Gamma L\right)}{\Gamma r_{1}} & -iZ_{c,0}\frac{kr_{0}\sin\left(\Gamma L\right)}{\Gamma r_{1}}\\ +\frac{iL\eta^{2}\cos\left(\Gamma L\right)}{\Gamma Z_{c0}r_{0}^{2}}-\frac{i}{Z_{c0}}\left(\frac{r_{1}}{r_{0}}+\frac{\eta^{2}}{\Gamma^{2}r_{0}^{2}}\right)\sin\left(\Gamma L\right)\,\,\,\, & \frac{r_{1}}{r_{0}}\cos\left(\Gamma L\right)-\frac{\eta\sin\left(\Gamma L\right)}{\Gamma r_{0}} +\end{array}\right], \end{equation} \end_inset where -\end_layout - -\begin_layout Plain Layout \begin_inset Formula \begin{equation} -Z_{c,0}=\frac{kz_{0}}{\left(1-f_{\nu}\right)\pi r_{0}^{2}\Gamma_{0}} +Z_{c,0}=\frac{\omega\rho_{0}}{\left(1-f_{\nu}\right)S_{f,0}\Gamma_{0}} \end{equation} \end_inset -\end_layout - -\end_inset - - \end_layout \begin_layout Section @@ -4241,7 +6366,7 @@ where \end_inset . - This is the characteristic eqation for + This is the characteristic equation for \begin_inset Formula $\epsilon R$ \end_inset @@ -4314,7 +6439,242 @@ where 630. \end_layout -\begin_layout Subsection +\begin_layout Section +Prismatic duct with flow +\end_layout + +\begin_layout Itemize +Assuming fully developed plug flow in a duct the linearized governing equations + in frequency domain read: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +i\omega\rho+\rho_{0}\frac{\mathrm{d}u}{\mathrm{d}x}+u_{0}\frac{\mathrm{d}\rho}{\mathrm{d}x} & =0\\ +i\rho_{0}\omega u+\rho_{0}u_{0}\frac{\mathrm{d}u}{\mathrm{d}x}+\frac{\mathrm{d}p}{\mathrm{d}x} & =0\\ +p & =c_{0}^{2}\rho +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Itemize +With subscript 0 are the mean flow variables. + Eliminating +\begin_inset Formula $\rho$ +\end_inset + +: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +\frac{1}{c_{0}^{2}}\left(i\omega p+u_{0}\frac{\mathrm{d}p}{\mathrm{d}x}\right)+\rho_{0}\frac{\mathrm{d}u}{\mathrm{d}x} & =0\\ +\rho_{0}\left(i\omega u+u_{0}\frac{\mathrm{d}u}{\mathrm{d}x}\right)+\frac{\mathrm{d}p}{\mathrm{d}x} & =0 +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Itemize +Taking spatial derivative of momentum and subtracting the convective derivative + of the continuity equation from it yields the convective wave equation: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $i\omega\rho_{0}\frac{\mathrm{d}u}{\mathrm{d}x}+u_{0}\rho_{0}\frac{\mathrm{d}^{2}u}{\mathrm{d}^{2}x}+\frac{\mathrm{d}^{2}p}{\mathrm{d}^{2}x}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Take the convected time derivative of the continuity: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(i\omega+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\right)^{2}\frac{1}{c_{0}^{2}}p+\rho_{0}\left(i\omega+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\right)\frac{\mathrm{d}u}{\mathrm{d}x}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +– +\end_layout + +\begin_layout Plain Layout +Subtract: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(i\omega+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\right)^{2}\frac{1}{c_{0}^{2}}p-\frac{\mathrm{d}^{2}p}{\mathrm{d}^{2}x}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Try +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\left(i\omega+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\right)^{2}\frac{1}{c_{0}^{2}}p-\frac{\mathrm{d}^{2}p}{\mathrm{d}^{2}x}=0 +\end{equation} + +\end_inset + +For constant +\begin_inset Formula $u_{0}$ +\end_inset + +, we try solutions of the form: +\begin_inset Formula +\begin{equation} +p=A\exp\left(\alpha x\right), +\end{equation} + +\end_inset + +which yields the characteristic equation for +\begin_inset Formula $\alpha$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +\underbrace{\left(M^{2}-1\right)}_{a}\alpha^{2}+\underbrace{2Mki}_{b}\alpha\underbrace{-k^{2}}_{c}=0, +\end{equation} + +\end_inset + +where +\begin_inset Formula $M$ +\end_inset + + denotes the Mach number +\begin_inset Formula $u_{0}/c_{0}$ +\end_inset + +. + The solutions for +\begin_inset Formula $\alpha$ +\end_inset + + are: +\begin_inset Formula +\begin{equation} +\alpha=i\frac{Mk\pm k}{1-M^{2}}=\pm ik\frac{1}{1\mp M} +\end{equation} + +\end_inset + +Written out: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $\left(i\omega+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\right)^{2}\frac{1}{c_{0}^{2}}A\exp\left(\alpha x\right)-\frac{\mathrm{d}^{2}A\exp\left(\alpha x\right)}{\mathrm{d}^{2}x}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(i\omega+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\right)\left(i\omega\frac{1}{c_{0}^{2}}A\exp\left(\alpha x\right)+\frac{u_{0}}{c_{0}^{2}}\alpha A\exp\left(\alpha x\right)\right)-\frac{\mathrm{d}^{2}A\exp\left(\alpha x\right)}{\mathrm{d}^{2}x}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(i\omega i\omega\frac{1}{c_{0}^{2}}\exp\left(\alpha x\right)+u_{0}\alpha i\omega\frac{1}{c_{0}^{2}}\exp\left(\alpha x\right)\right)+\left(i\omega\frac{u_{0}}{c_{0}^{2}}\alpha\exp\left(\alpha x\right)+u_{0}\frac{u_{0}}{c_{0}^{2}}\alpha^{2}\exp\left(\alpha x\right)\right)-\frac{\mathrm{d}^{2}\exp\left(\alpha x\right)}{\mathrm{d}^{2}x}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(M^{2}-1\right)\alpha^{2}+2Mk\alpha i-k^{2}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Regular form: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\underbrace{\left(M^{2}-1\right)}_{a}\alpha^{2}+\underbrace{2Mki}_{b}\alpha\underbrace{-k^{2}}_{c}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Solutions are: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\alpha=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{-2Mki\pm\sqrt{\left(2Mki\right)^{2}+4\left(M^{2}-1\right)k^{2}}}{2\left(M^{2}-1\right)}=\frac{\pm ik-Mki}{\left(M^{2}-1\right)}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align} +p=A\exp\left(-\frac{ik}{1+M}x\right)+B\exp\left(\frac{ik}{1-M}x\right), +\end{align} + +\end_inset + +and the volume flow: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\mathrm{d}u}{\mathrm{d}x}=-\frac{1}{\rho_{0}c_{0}^{2}}\left(i\omega\left(A\exp\left(-\frac{ik}{1+M}x\right)+B\exp\left(\frac{ik}{1-M}x\right)\right)+u_{0}\frac{\mathrm{d}}{\mathrm{d}x}\left(A\exp\left(-\frac{ik}{1+M}x\right)+B\exp\left(\frac{ik}{1-M}x\right)\right)\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\mathrm{d}u}{\mathrm{d}x}=-\frac{1}{\rho_{0}c_{0}^{2}}i\omega\left(A\exp\left(-\frac{ik}{1+M}x\right)+B\exp\left(\frac{ik}{1-M}x\right)\right)+-\frac{1}{\rho_{0}c_{0}^{2}}u_{0}\left(-\frac{ik}{1+M}A\exp\left(-\frac{ik}{1+M}x\right)+B\frac{ik}{1-M}\exp\left(\frac{ik}{1-M}x\right)\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $u=\frac{A}{z_{0}}\exp\left(-\frac{ik}{1+M}x\right)-\frac{B}{z_{0}}\exp\left(\frac{ik}{1-M}x\right)+C$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section Cremers impedance \end_layout @@ -4399,6 +6759,14 @@ Such that the total impedance is \end_layout +\begin_layout Section +Cavity silencer +\end_layout + +\begin_layout Standard +- +\end_layout + \begin_layout Section Compliance volume \begin_inset CommandInset label @@ -4639,6 +7007,14 @@ It should be noticed that in practice, a compliance volume often functions is 0. \end_layout +\begin_layout Section +Membrane +\end_layout + +\begin_layout Standard +A membrane is a mechanical +\end_layout + \begin_layout Section End corrections and discontinuities \begin_inset CommandInset label @@ -5216,19 +7592,285 @@ name "fig:hardwall" \end_layout +\begin_layout Section +Spherical wave propagation models +\end_layout + \begin_layout Standard +For spherical waves, the Helmholtz equation reads +\begin_inset Formula +\begin{equation} +\left(\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}+\frac{2}{r}\frac{\mathrm{d}}{\mathrm{d}r}+\Gamma^{2}\right)p=0.\label{eq:hh_spher} +\end{equation} + +\end_inset + +The solution of Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:hh_spher" + +\end_inset + + reads: +\begin_inset Formula +\begin{equation} +p=\frac{C_{1}\exp\left(-i\Gamma r\right)+C_{2}\exp\left(-i\Gamma r\right)}{r}. +\end{equation} + +\end_inset + +The acoustic volume flow can be computed as \begin_inset Note Note status open \begin_layout Plain Layout -\begin_inset CommandInset bibtex -LatexCommand bibtex -bibfiles "lrftubes" -options "plain" +\begin_inset Formula $u_{r}=\frac{i}{\omega\rho_{0}}\frac{\mathrm{d}p}{\mathrm{d}r}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $u_{r}=\frac{i}{kz_{0}}\frac{\mathrm{d}p}{\mathrm{d}r}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $U_{r}=4\pi r^{2}\alpha\frac{i}{kz_{0}}\frac{\mathrm{d}p}{\mathrm{d}r}$ +\end_inset + + +\end_layout \end_inset +\begin_inset Formula +\begin{equation} +U=i\frac{\alpha4\pi r^{2}}{\Gamma z_{c}}\frac{\mathrm{d}p}{\mathrm{d}r}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\alpha=1$ +\end_inset + + for a full sphere and +\begin_inset Formula $\alpha=\frac{1}{2}$ +\end_inset + + for a hemisphere. + We can derive the following transfer matrix for +\begin_inset Formula $p$ +\end_inset + + and +\begin_inset Formula $U$ +\end_inset + +: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\frac{C_{1}\exp\left(-i\Gamma r_{L}\right)+C_{2}\exp\left(-i\Gamma r_{L}\right)}{\Gamma r_{L}}=p_{L}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{C_{1}\exp\left(-i\Gamma r_{R}\right)+C_{2}\exp\left(-i\Gamma r_{R}\right)}{\Gamma r_{R}}=p_{R}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(U_{L}\frac{e^{-i\Gamma\left(r_{L}+r_{R}\right)}}{8\pi\Gamma\alpha r_{L}r_{R}}\left(i\Gamma z_{c}e^{2i\Gamma r_{L}}-i\Gamma z_{c}e^{2i\Gamma r_{R}}\right)+p_{L}\left(\frac{r_{L}e^{i\Gamma\left(r_{L}+r_{R}\right)}}{2r_{R}}+\frac{i}{2\Gamma r_{R}}\left(e^{i\Gamma\left(r_{L}-r_{R}\right)}-e^{i\Gamma\left(r_{R}+r_{L}\right)}\right)\right)\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $p_{R}=\frac{iU_{L}z_{c}}{4\pi\alpha r_{L}r_{R}}\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right)+p_{L}\left[\frac{r_{L}}{r_{R}}\cos\left(\Gamma\left(r_{L}-r_{R}\right)\right)-\frac{1}{\Gamma r_{R}}\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right)\right]$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +and: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $U_{R}=U_{L}\left(\frac{r_{R}}{r_{L}}\cos\left(\Gamma\left(r_{L}-r_{R}\right)\right)+\frac{1}{\Gamma r_{L}}\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right)\right)+\frac{4i\pi\alpha}{z_{c}}p_{L}\left[\left(r_{L}r_{R}+\frac{1}{\Gamma^{2}}\right)\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right)+\frac{\left(r_{R}-r_{L}\right)}{\Gamma}\cos\left(\Gamma\left(r_{L}-r_{R}\right)\right)\right]$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\left\{ \begin{array}{c} +p\\ +U +\end{array}\right\} _{R}=\left[\begin{array}{cc} +M_{11} & M_{12}\\ +M_{21} & M_{22} +\end{array}\right]\left\{ \begin{array}{c} +p\\ +U +\end{array}\right\} _{L}, +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{align} +M_{11} & =\frac{r_{L}}{r_{R}}\cos\left(\Gamma\left(r_{L}-r_{R}\right)\right)-\frac{1}{\Gamma r_{R}}\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right),\\ +M_{12} & =\frac{iz_{c}\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right)}{4\pi\alpha r_{L}r_{R}},\\ +M_{21} & =\frac{4\pi i\alpha}{z_{c}}\left[\left(r_{L}r_{R}+\frac{1}{\Gamma^{2}}\right)\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right)+\frac{r_{R}-r_{L}}{\Gamma}\cos\left(\Gamma\left(r_{L}-r_{R}\right)\right)\right]\\ +M_{22} & =\frac{r_{R}}{r_{L}}\cos\left(\Gamma\left(r_{L}-r_{R}\right)\right)+\frac{1}{\Gamma r_{L}}\sin\left(\Gamma\left(r_{L}-r_{R}\right)\right), +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Section +Boundary conditions +\end_layout + +\begin_layout Subsection +Radiation impedance of a baffled piston +\end_layout + +\begin_layout Itemize +\begin_inset Formula $a$ +\end_inset + +: radius of the exit [m] +\end_layout + +\begin_layout Itemize +\begin_inset Formula $S$ +\end_inset + +: +\begin_inset Formula $\pi a^{2}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +p=Z_{\mathrm{rad}}U, +\end{equation} + +\end_inset + + +\begin_inset Formula +\begin{equation} +Z_{\mathrm{rad}}=\frac{z_{0}}{S}\left[1-\frac{2J_{1}\left(2ka\right)}{2ka}+i\frac{2H_{1}(2ka)}{2ka}\right] +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +In the low frequency range, a power series expansion of +\begin_inset Formula $H_{1}$ +\end_inset + + yields [Aarts]: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +H_{1}(x)=\frac{2}{\pi}\left[\frac{x^{2}}{3}-\frac{x^{4}}{45}+\frac{x^{6}}{1575}-\dots\right] +\end{equation} + +\end_inset + +Filling this in, we obtain the following low-frequency approximation to + +\begin_inset Formula $Z_{\mathrm{rad}}$ +\end_inset + +: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +Z_{\mathrm{rad}}=\frac{z_{0}}{S}\left[i\frac{8ka}{3\pi}+\frac{1}{2}\left(ka\right)^{2}+\mathcal{O}\left(\left(ka\right)^{3}\right)\right] +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Chapter +Speaker +\end_layout + +\begin_layout Section +As an active element, with voltage control +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename img/spk.pdf + width 100text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Electrical and mechanical model of the speaker +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + \end_layout \end_inset @@ -5237,6 +7879,114 @@ options "plain" \end_layout \begin_layout Standard +The speaker generates electromotive force +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +F_{\mathrm{emf}} & =B\ell I, +\end{align} + +\end_inset + +where +\begin_inset Formula $B\ell$ +\end_inset + + is the +\begin_inset Quotes eld +\end_inset + +motor constant +\begin_inset Quotes erd +\end_inset + +, or force factor, in units +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +si{ +\backslash +newton +\backslash +per +\backslash +ampere} +\end_layout + +\end_inset + +, or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +si{ +\backslash +volt +\backslash +second +\backslash +per +\backslash +meter} +\end_layout + +\end_inset + +. + The back-emf +\begin_inset Quotes eld +\end_inset + +force +\begin_inset Quotes erd +\end_inset + +: +\begin_inset Formula +\begin{equation} +V_{\mathrm{bemf}}=B\ell u +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The +\begin_inset Quotes eld +\end_inset + +circuit equation +\begin_inset Quotes erd +\end_inset + +: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +V_{\mathrm{in}}-V_{\mathrm{bemf}}=Z_{\mathrm{el}}I, +\end{equation} + +\end_inset + +where +\begin_inset Formula $Z_{\mathrm{el}}$ +\end_inset + + is the equivalent impedance of the electrical circuit in \begin_inset ERT status open @@ -5244,11 +7994,3248 @@ status open \backslash -printbibliography +si{ +\backslash +ohm} \end_layout \end_inset +. + The mechanical impedance comprises a stiffness part, a damping part and + a mass part. + The equation of motion is: +\begin_inset Formula +\begin{equation} +z_{m}u=F_{\mathrm{emf}}+p_{l}S-p_{r}S, +\end{equation} + +\end_inset + +where +\begin_inset Formula $u$ +\end_inset + + denotes the velocity phasor of the membrane. + The mechanical impedance +\begin_inset Formula $z_{m}$ +\end_inset + + is defined as: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +z_{m}=i\omega m_{m}+r_{m}+\frac{k_{m}}{i\omega}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $m_{m}$ +\end_inset + + is the moving mass, +\begin_inset Formula $r_{m}$ +\end_inset + + the damping force and +\begin_inset Formula $k_{m}$ +\end_inset + + the spring constant. + +\begin_inset Formula $z_{m}$ +\end_inset + + can equivalently be written as: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $z_{m}=i\omega m+R_{m}+\frac{K_{m}}{i\omega}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $z_{m}=m\left(i\omega+\frac{R_{m}}{m}+\frac{\omega_{r}^{2}}{i\omega}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $z_{m}=m\left(i\omega+\frac{R_{m}}{m}+\frac{\omega_{r}^{2}}{i\omega}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +using: +\begin_inset Formula $\omega_{r}^{2}=\frac{K_{m}}{m}\Rightarrow m=\frac{K_{m}}{\omega_{r}^{2}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $z_{m}=\frac{m}{i\omega}\left(-\omega^{2}+i\omega\frac{R_{m}}{m}+\omega_{r}^{2}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $z_{m}=\frac{m}{i\omega}\left(-\omega^{2}+i\omega\frac{R_{m}\omega_{r}^{2}}{K_{m}}+\omega_{r}^{2}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Now, writing +\begin_inset Formula $R_{m}$ +\end_inset + + as: +\begin_inset Formula $R_{m}=2\zeta\sqrt{K_{m}m}$ +\end_inset + +: +\begin_inset Formula $\zeta=\frac{1}{2}\frac{r_{m}}{\sqrt{k_{m}m_{m}}}\Rightarrow\zeta=\frac{1}{2}\frac{r_{m}}{\sqrt{k_{m}m_{m}}}=\frac{1}{2}\frac{r_{m}}{\omega_{r}m_{m}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $z_{m}=\frac{m}{i\omega}\left(\omega_{r}^{2}-\omega^{2}+2i\omega\zeta\omega_{r}\right)$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +z_{m}=m\left(i\omega+2\zeta\omega_{r}+\frac{\omega_{r}^{2}}{i\omega}\right), +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +\omega_{r}^{2}=\frac{k_{m}}{m_{m}}\qquad;\qquad\zeta=\frac{r_{m}}{2\sqrt{k_{m}m_{m}}}=\frac{r_{m}}{2\omega_{r}m_{m}}=\frac{\omega_{r}r_{m}}{2k_{m}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +After some algebraic manipulations we find: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $z_{m}u=\left(p_{l}-p_{r}\right)S+B\ell I$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{1}{Z_{\mathrm{el}}}\left(V_{\mathrm{in}}-V_{\mathrm{bemf}}\right)=I$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +where +\begin_inset Formula $V_{\mathrm{bemf}}=B\ell u$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Results in: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}\right)u=\left(p_{l}-p_{r}\right)S+\frac{B\ell}{Z_{\mathrm{el}}}V_{\mathrm{in}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +To acoustic variables +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +\frac{1}{S}\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}\right)U=\left(p_{l}-p_{r}\right)S+\frac{B\ell}{Z_{\mathrm{el}}}V_{\mathrm{in}} +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +To transfer matrix notation: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $p_{r}=p_{l}-\frac{1}{S^{2}}\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}\right)U+\frac{B\ell}{Z_{\mathrm{el}}S}V_{\mathrm{in}}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align} +\frac{1}{S_{l}}\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}\right)U_{l} & =p_{l}S_{l}-p_{r}S_{r}+\frac{B\ell}{Z_{\mathrm{el}}}V_{\mathrm{in}},\\ +U_{r}-U_{l} & =0, +\end{align} + +\end_inset + +which is in transfer matrix notation: +\begin_inset Formula +\begin{equation} +\left\{ \begin{array}{c} +p_{r}\\ +U_{r} +\end{array}\right\} =\boldsymbol{T}\left\{ \begin{array}{c} +p_{l}\\ +U_{l} +\end{array}\right\} +\boldsymbol{s}, +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +\boldsymbol{T}=\left[\begin{array}{cc} +1 & -\frac{1}{S^{2}}\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}\right)\\ +0 & 1 +\end{array}\right]\qquad;\qquad\boldsymbol{s}=\left\{ \begin{array}{c} +\frac{B\ell}{Z_{\mathrm{el}}S}V_{\mathrm{in}}\\ +0 +\end{array}\right\} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +As antireciprocal segment +\end_layout + +\begin_layout Standard +As antireciprocal segment, a voltage controlled speaker has electrical connectio +ns on the left side, and acoustical connections on the right side: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\left\{ \begin{array}{c} +p\\ +U +\end{array}\right\} _{R}=\boldsymbol{T}_{\mathrm{spk}}\left\{ \begin{array}{c} +V\\ +I +\end{array}\right\} _{L}. +\end{equation} + +\end_inset + +A model us used for the back cavity pressure build-up which can be added + as an extra impedance, placed in series with the effective acoustic impedance + of the front side, hence the force balance reads: +\begin_inset Formula +\begin{equation} +F_{\mathrm{emf}}=Z_{\mathrm{back}}U+Z_{\mathrm{front}}U +\end{equation} + +\end_inset + +The transfer matrix reads: +\begin_inset Formula +\begin{equation} +\boldsymbol{T}_{\mathrm{spk}}=\left[\begin{array}{cc} +-\frac{S^{2}Z_{\mathrm{back}}+z_{m}}{SB\ell} & \frac{\left(B\ell\right)^{2}+Z_{\mathrm{el}}\left(z_{m}+S^{2}Z_{\mathrm{back}}\right)}{B\ell S}\\ +\frac{S}{B\ell} & -\frac{SZ_{\mathrm{el}}}{B\ell} +\end{array}\right] +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +Determinant: +\begin_inset Formula +\[ +\frac{Z_{\mathrm{el}}\left(S^{2}Z_{\mathrm{back}}+z_{m}\right)}{B\ell^{2}}-\left(1+\frac{Z_{\mathrm{el}}\left(S^{2}Z_{\mathrm{back}}+z_{m}\right)}{B\ell^{2}}\right)=-1 +\] + +\end_inset + + +\end_layout + +\end_inset + +For a closed back-cavity volume, the back-cavity is: +\end_layout + +\begin_layout Standard +Then again: +\begin_inset Note Note +status open + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +Compute determinant: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\mathrm{det}=-S$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Chapter +Optimized reactive silencers +\end_layout + +\begin_layout Section +Parallel Helmholtz resonator transfer function and transmission loss +\end_layout + +\begin_layout Standard +Equations for a side branch Helmholtz resonator: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +p_{R} & =p_{L},\\ +U_{R} & =U_{L}-p_{L}/Z_{h}, +\end{align} + +\end_inset + +where +\begin_inset Formula $Z_{h}$ +\end_inset + + is the side branch impedance of the Helmholtz resonator, defined as +\begin_inset Formula +\begin{equation} +Z_{h}=\left(\frac{\rho_{0}z_{0}}{i\omega V}+R_{v}+i\omega m_{\mathrm{neck}}\right), +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +m_{\mathrm{neck}}=\frac{\rho_{0}\ell_{\mathrm{eff},\mathrm{neck}}}{S_{\mathrm{neck}}}, +\end{equation} + +\end_inset + +and for relatively large holes, air at STP, the resistance term can be estimated + as: +\begin_inset Formula +\begin{equation} +R_{v}\approx7.2\times10^{-3}z_{0}/S_{h}, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Now, the following substitutions are made: +\begin_inset Formula +\begin{align} +C & =\frac{V}{\rho_{0}z_{0}},\\ +m_{\mathrm{neck}} & =\frac{1}{\omega_{r}^{2}C}\\ +\zeta & =\frac{1}{2}\omega_{r}CR_{v}. +\end{align} + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\frac{2\zeta}{\omega_{r}C}=R_{v}.$ +\end_inset + + +\end_layout + +\end_inset + +such that we can write: +\begin_inset Formula +\begin{equation} +Z_{h}=\frac{1}{\omega_{r}C}\left(\frac{\omega_{r}}{i\omega}+2\zeta+\frac{i\omega}{\omega_{r}}\right) +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The quality factor of the resonator is the ratio of the resonance frequency + to its bandwidth measure. + If we take +\begin_inset Formula +\begin{equation} +Q\overset{\mathrm{def}}{=}\frac{f_{r}}{\Delta f}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\Delta f$ +\end_inset + + is the full width at half the maximum value, i.e. + the frequency distance between two points lying at +\begin_inset Formula $-3$ +\end_inset + + +\begin_inset space ~ +\end_inset + +dB w.r.t. + the maximum value. + The damping ratio +\begin_inset Formula $\zeta$ +\end_inset + + is related to +\begin_inset Formula $Q$ +\end_inset + + as: +\begin_inset Formula +\begin{equation} +\zeta=\frac{1}{2Q}=\frac{1}{2}\frac{\Delta f}{f_{r}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Assembling the transfer matrix +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\left\{ \begin{array}{c} +p\\ +U +\end{array}\right\} _{R}=\left[\begin{array}{cc} +T_{11} & T_{12}\\ +T_{21} & T_{22} +\end{array}\right]\left\{ \begin{array}{c} +p\\ +U +\end{array}\right\} _{L}, +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{align} +T_{11} & =1\\ +T_{12} & =0\\ +T_{21} & =-Z_{h}^{-1}\\ +T_{22} & =1 +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Transmission loss +\end_layout + +\begin_layout Standard +The transmission coefficient can be computed as: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\tau=\frac{C}{A}=\frac{Z_{0}\left(T_{21}p_{L}+T_{22}U_{L}\right)}{\frac{1}{2}\left(p_{L}+Z_{0}U_{L}\right)}, +\end{equation} + +\end_inset + +using +\begin_inset Formula +\begin{equation} +T_{11}p_{L}+T_{12}U_{L}=p_{R}=Z_{0}U_{R}=Z_{0}\left(T_{21}p_{L}+T_{22}U_{L}\right), +\end{equation} + +\end_inset + +we get +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $Z_{0}\left(T_{21}p_{L}+T_{22}U_{L}\right)=T_{11}p_{L}+T_{12}U_{L}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +– +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $U_{L}=\frac{\left(T_{11}-Z_{0}T_{21}\right)}{\left(Z_{0}T_{22}-T_{12}\right)}p_{L}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +U_{L}=\frac{\left(T_{11}-Z_{0}T_{21}\right)}{\left(Z_{0}T_{22}-T_{12}\right)}p_{L}, +\end{equation} + +\end_inset + +filling in: +\begin_inset Formula +\begin{equation} +\tau=\frac{2}{Z_{0}}\frac{T_{11}T_{22}-T_{12}T_{21}}{T_{11}-T_{12}/Z_{0}-T_{21}Z_{0}+T_{22}}, +\end{equation} + +\end_inset + +assuming that the determinant of the transfer matrix be unity +\begin_inset Formula $(T_{11}T_{22}-T_{12}T_{21}\equiv1$ +\end_inset + +), this can be further simplified: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\tau=\frac{2}{T_{11}-T_{12}/Z_{0}-T_{21}Z_{0}+T_{22}}, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +For a Helmholtz resonator, this results in: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula +\[ +\tau=\frac{2}{T_{11}-T_{12}/Z_{0}-T_{21}Z_{0}+T_{22}}, +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Filling in: +\begin_inset Formula $T_{11}=1$ +\end_inset + +, +\begin_inset Formula $T_{12}=0$ +\end_inset + +, +\begin_inset Formula $T_{21}=-1/Z_{h}$ +\end_inset + + +\begin_inset Formula $T_{22}=1$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +\tau=\frac{2Z_{h}}{2Z_{h}+Z_{0}}, +\] + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\tau(\omega)=\frac{2Z_{h}(\omega)}{Z_{0}+2Z_{h}(\omega)}, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Filling in the Helmholtz resonator equation: +\begin_inset Formula +\begin{equation} +\tau(\omega)=\frac{2\left(1+2\frac{\omega}{\omega_{r}}\zeta-\left(\frac{\omega}{\omega_{r}}\right)^{2}\right)}{2\left(1+2\frac{\omega}{\omega_{r}}\zeta-\left(\frac{\omega}{\omega_{r}}\right)^{2}\right)+i\frac{\omega}{\omega_{r}}\left(\frac{Cz_{0}\omega_{r}}{S}\right)} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The peak height, filling in for +\begin_inset Formula $\omega/\omega_{r}=1$ +\end_inset + +: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\tau=\frac{4\zeta}{4\zeta+\beta}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\beta$ +\end_inset + + is defined as the resonator strength: +\begin_inset Formula +\begin{equation} +\beta=\frac{V\omega_{r}}{Sc_{0}} +\end{equation} + +\end_inset + +In terms of transmission loss: +\begin_inset Formula +\begin{equation} +\mathrm{TL}_{\omega=\omega_{r}}=20\log\left(\frac{\beta+4\zeta}{4\zeta}\right) +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The peak half width is the distance over which the transmission loss has + dropped 3 +\begin_inset space ~ +\end_inset + +dB w.r.t. + the transmission loss at the resonance frequency. + This is an important design parameter. + We can compute it by setting +\begin_inset Formula +\begin{equation} +|\frac{\tau|_{\omega_{r}+\Delta\omega}}{\tau|_{\omega_{r}}}|=\sqrt{2}, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +For light relative damping, and +\begin_inset Formula $\Delta\omega/\omega_{r}\approx1$ +\end_inset + +, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +|\frac{\tau|_{\omega_{r}+\Delta\omega}}{\tau|_{\omega_{r}}}|\approx\frac{\alpha-1}{\zeta}, +\end{equation} + +\end_inset + +So given the -3 +\begin_inset space ~ +\end_inset + +dB point, and the maximum required transmission loss, we can compute +\begin_inset Formula $\zeta$ +\end_inset + + and +\begin_inset Formula $\beta$ +\end_inset + +: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +Eq 1: +\end_layout + +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $\frac{\alpha_{-3\mathrm{dB}}-1}{\zeta}=\sqrt{2}\Rightarrow\zeta=\frac{\alpha_{-3\mathrm{dB}}-1}{\sqrt{2}}$ +\end_inset + + +\begin_inset Newline newline +\end_inset + +Eq 2: +\end_layout + +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $\mathrm{TL}_{\mathrm{max}}=20\log\left(\frac{\beta+4\zeta}{4\zeta}\right)\Rightarrow\frac{\beta+4\zeta}{4\zeta}=10^{\frac{\mathrm{TL}_{\mathrm{max}}}{20}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\beta=4\zeta\left(10^{\frac{\mathrm{TL}_{\mathrm{max}}}{20}}-1\right)$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $\zeta=\frac{\alpha_{-3\mathrm{dB}}-1}{\sqrt{2}}$ +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $\beta=4\zeta\left(10^{\frac{\mathrm{TL}_{\mathrm{max}}}{20}}-1\right)$ +\end_inset + + +\end_layout + +\begin_layout Standard +Required volume in terms of resonator strength: +\begin_inset Formula +\begin{equation} +V=\frac{Sc_{0}\beta}{\omega_{r}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Insertion loss +\end_layout + +\begin_layout Standard +For computation of the insertion loss, we require two more parameters: +\end_layout + +\begin_layout Itemize +The load impedance at the downstream end of the silencer +\end_layout + +\begin_layout Itemize +The output impedance of the source ( +\begin_inset Formula $Z_{\mathrm{rad}}$ +\end_inset + +) +\end_layout + +\begin_layout Standard +Suppose the source strength is defined by +\begin_inset Formula $\mathcal{S}$ +\end_inset + +. + Situation without silencer: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +U_{L} & =\mathcal{S}/\left(Z_{s}+Z_{l}\right),\\ +U_{R} & =U_{L},\\ +p_{R} & =Z_{\mathrm{rad}}U_{R}, +\end{align} + +\end_inset + +where +\begin_inset Formula $Z_{s}$ +\end_inset + + denotes the source output impedance, and +\begin_inset Formula $Z_{l}$ +\end_inset + + denotes the load impedance as felt by the source. +\end_layout + +\begin_layout Standard +For the reference case, the load impedance equals the radiation impedance, + and the radiated acoustic power is: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[p_{R}U_{R}^{*}\right]$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[Z_{\mathrm{rad}}\left(\mathcal{S}/Z_{s}\right)\left(\mathcal{S}/Z_{s}\right)^{*}\right]$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}|\mathcal{S}/Z_{s}|^{2}\Re\left[Z_{\mathrm{rad}}\right]$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +P_{\mathrm{ref}}=\frac{1}{2}\frac{|\mathcal{S}|^{2}}{|Z_{\mathrm{rad}}+Z_{s}|^{2}}\Re\left[Z_{\mathrm{rad}}\right] +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Now, situation including silencer, with in general, transfer matrix +\begin_inset Formula $\boldsymbol{T}$ +\end_inset + +. +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[p_{R}U_{R}^{*}\right]$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[Z_{\mathrm{rad}}U_{R}U_{R}^{*}\right]$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Using: +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +P_{\mathrm{with\,silencer}}=\frac{1}{4}|\mathcal{S}|^{2}\frac{\Re\left[Z_{\mathrm{rad}}\right]}{|T_{22}Z_{\mathrm{rad}}-T_{12}+Z_{s}\left(T_{11}-T_{21}Z_{\mathrm{rad}}\right)|^{2}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +From that, computing the power ratio, that +\begin_inset Formula $\det\boldsymbol{T}\equiv1$ +\end_inset + + for a reciprocal system: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +R_{P}=\frac{P_{\mathrm{with\,silencer}}}{P_{\mathrm{ref}}}=\frac{|Z_{\mathrm{rad}}+Z_{s}|^{2}}{|T_{22}Z_{\mathrm{rad}}-T_{12}+Z_{s}\left(T_{11}-T_{21}Z_{\mathrm{rad}}\right)|^{2}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Insertion loss for a Helmholtz side branch resonator +\end_layout + +\begin_layout Standard +Filling in for a simple Helmholtz side branch resonator: +\begin_inset Formula +\begin{equation} +R_{P,\mathrm{Helmholtz}}=\frac{|Z_{\mathrm{rad}}+Z_{s}|^{2}}{|Z_{\mathrm{rad}}+Z_{s}\left(1+\frac{Z_{\mathrm{rad}}}{Z_{h}}\right)|^{2}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Comparing this to the transmission loss curve: +\begin_inset Formula +\begin{equation} +|\tau|_{\mathrm{Helmholtz}}^{2}=\frac{4|Z_{h}|^{2}}{|2Z_{h}+Z_{0}|^{2}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +High output impedance limit +\begin_inset Formula $(Z_{s}\gg Z_{\mathrm{rad}})$ +\end_inset + +, volume flow source +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +R_{P,\mathrm{Helmholtz}}=\frac{|Z_{h}|^{2}}{|Z_{h}+Z_{\mathrm{rad}}|^{2}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +Low output impedance limit +\begin_inset Formula $(Z_{s}\ll Z_{\mathrm{rad}})$ +\end_inset + +, pressure source +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +R_{P,\mathrm{Helmholtz}}=\frac{|Z_{h}|^{2}}{|Z_{h}+Z_{s}|^{2}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +Special case: barrier in an infinite space +\begin_inset Formula $(Z_{s}=Z_{\mathrm{rad}})$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +R_{P,\mathrm{Helmholtz}}=\frac{|Z_{h}|^{2}}{|Z_{h}+\frac{1}{2}Z_{\mathrm{rad}}|^{2}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Comparing limits to power transmission ratio +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +|\tau|^{2}=\frac{|Z_{h}|^{2}}{|Z_{h}+\frac{1}{2}Z_{0}|^{2}}, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +So the transmission loss is the reduction in transmitted sound power for + the situation where the source output impedance equals the radiation impedance + on the other side of the silencer. +\end_layout + +\begin_layout Subsection +Multiple Helmholtz resonators at a single inlet +\end_layout + +\begin_layout Standard +In case multiple resonators are connected to the same inlet, the parallel + impedance can be computed by computing the equivalent parallel impedance: +\begin_inset Formula +\begin{equation} +\frac{1}{Z_{h,\mathrm{tot}}}=\frac{1}{Z_{h,1}}+\frac{1}{Z_{h,2}}+\dots +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Transmission of the duct +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\left\{ \begin{array}{c} +p_{R}\\ +U_{R} +\end{array}\right\} =\left[\begin{array}{cc} +\cos\left(kL\right) & -iZ_{0}\sin\left(kL\right)\\ +-iZ_{0}^{-1}\sin\left(kL\right) & \cos\left(kL\right) +\end{array}\right]\left\{ \begin{array}{c} +p_{L}\\ +U_{L} +\end{array}\right\} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Chapter +(Micro)-perforated plate design +\end_layout + +\begin_layout Standard +Given +\begin_inset Formula $\beta$ +\end_inset + +, +\begin_inset Formula $\zeta$ +\end_inset + + and +\begin_inset Formula $\omega_{r}$ +\end_inset + +, a proper acoustic mass has to be chosen. + Given the resonator equations +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula +\begin{align*} +C & =\frac{V}{\rho_{0}z_{0}},\\ +m_{\mathrm{neck}} & =\frac{1}{\omega_{r}^{2}C}\\ +\zeta & =\frac{1}{2}\omega_{r}CR_{v}. +\end{align*} + +\end_inset + +and +\begin_inset Formula +\[ +Z_{h}=\frac{1}{\omega_{r}C}\left(\frac{\omega_{r}}{i\omega}+2\zeta+\frac{i\omega}{\omega_{r}}\right) +\] + +\end_inset + + +\end_layout + +\end_inset + +, the viscous resistance and required acoustic mass can be determined. + This results in requirements for the (effective) acoustic mass and resistance + of the perforate. + For arbitrary hole sizes, the definition of the acoustic impedance of a + perforate is: +\begin_inset Formula +\begin{equation} +z=\frac{\Delta p}{\overline{u}}.\label{eq:perforate_impedance_definition} +\end{equation} + +\end_inset + +where +\begin_inset Formula $\overline{u}$ +\end_inset + + denotes the acoustic volume flow per unit of area through the perforate + (uncorrected yet for porosity), such that the area-averaged velocity +\emph on +in a hole +\emph default + is +\begin_inset Formula $u_{h}=\overline{u}/\phi$ +\end_inset + +, where +\begin_inset Formula $\phi$ +\end_inset + + denotes the porosity. + In Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:perforate_impedance_definition" + +\end_inset + +, it is assumed that the acoustic wavelength is typically much larger than + the length scale(s) of the perforate. + The model for the impedance of a perforate, in the linear range is +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +The COMSOL language, partially translated: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $z=-\rho_{0}c_{0}\frac{2i\sin\left(\frac{k_{c}t_{p}}{2}\right)}{\sqrt{\left(\gamma-\left(\gamma-1\right)\Psi_{h}\right)\Psi_{v}}}-\rho_{0}c_{0}\frac{i\omega}{c_{0}C_{D}\phi}\frac{2\delta}{\Psi_{v}}f_{\mathrm{int}},$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Using the fact that: +\begin_inset Formula $\Psi_{v}\equiv f_{\nu}-1$ +\end_inset + + and equivalently: +\begin_inset Formula $\Psi_{h}\equiv f_{\kappa}-1$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $z=\rho_{0}c_{0}\frac{2i\sin\left(\Gamma\frac{t_{w}}{2}\right)}{\sqrt{\left(\gamma-\left(\gamma-1\right)\left(f_{\kappa}-1\right)\right)\left(f_{\nu}-1\right)}}+\rho_{0}c_{0}\frac{i\omega}{c_{0}C_{D}\phi}\frac{2\delta}{1-f_{\nu}}f_{\mathrm{int}},$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +where +\begin_inset Formula $k_{c}$ +\end_inset + + is our +\begin_inset Formula $\Gamma$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $z=\rho_{0}c_{0}\frac{2i\sin\left(\Gamma\frac{t_{w}}{2}\right)}{\sqrt{\left(1+\left(\gamma-1\right)f_{\kappa}\right)\left(1-f_{\nu}\right)}}+\frac{i\omega}{c_{0}C_{D}\phi}\frac{2\delta}{1-f_{\nu}}f_{\mathrm{int}},$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\delta$ +\end_inset + + is the end correction length for one side: +\begin_inset Formula $\delta=4D/(3\pi)$ +\end_inset + +. + For small plate thicknesses: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $z=\rho_{0}c_{0}\frac{2i\sin\left(\Gamma\frac{t_{w}}{2}\right)}{\sqrt{\left(1+\left(\gamma-1\right)f_{\kappa}\right)\left(1-f_{\nu}\right)}}+\rho_{0}c_{0}\frac{i\omega}{c_{0}C_{D}\phi}\frac{2\delta}{1-f_{\nu}}f_{\mathrm{int}},$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +Z_{c}=\frac{kZ_{0}}{\left(1-f_{\nu}\right)\Gamma}. +\] + +\end_inset + +viscothermal wave number, i.e. + the wave number corrected for viscothermal losses: +\begin_inset Formula +\[ +\Gamma=\frac{\omega}{c_{0}}\sqrt{\frac{1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\epsilon_{s}}}{1-f_{\nu}}}. +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +For small plate thicknesses: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $z=i\omega\rho_{0}\frac{t_{w}+\frac{2\delta f_{\mathrm{int}}}{C_{D}}}{\left(1-f_{\nu}\right)},$ +\end_inset + + +\end_layout + +\end_inset + +: +\begin_inset Formula +\begin{equation} +z=\frac{i\omega\rho_{0}}{\phi}\left[\frac{t_{w}}{\left(1-f_{\nu}\right)}+2\delta f_{\mathrm{int}}\right]+\alpha\frac{\rho_{0}\omega\delta_{\nu}}{\phi}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $f_{\mathrm{int}}$ +\end_inset + + is the hole-hole interaction function which +\begin_inset Formula $\to1$ +\end_inset + + for +\begin_inset Formula $\phi\to0$ +\end_inset + +, and +\begin_inset Formula $\delta$ +\end_inset + + is the single-sided hole (therefore, the factor 2 in front) end correction + due to the added mass effect, for the situation of negligible hole-hole + interaction. + [Paper: Tayong, 2013]. + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +f_{\mathrm{int}}(\phi) & =1-1.4092\sqrt{\phi}+0.33818\sqrt{\phi}^{3}+0.06793\sqrt{\phi}^{5}.\\ + & -0.02287\sqrt{\phi}^{6}+0.063015\sqrt{\phi}^{7}-0.01614\sqrt{\phi}^{8} +\end{align} + +\end_inset + +For square holes: +\end_layout + +\begin_layout Standard +where +\begin_inset Formula +\begin{equation} +\xi^{2}=\frac{\pi D^{2}}{4P^{2}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\frac{D}{P}=\sqrt{\frac{4\phi}{\pi}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +For circular large holes with diameter +\begin_inset Formula $D$ +\end_inset + +, the end correction for both sides is +\begin_inset Formula +\begin{equation} +2\delta=\frac{8}{3\pi}D\approx0.85D. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +For circular large holes with diameter +\begin_inset Formula $D$ +\end_inset + +, the end correction for both sides is +\begin_inset Formula +\begin{equation} +2\delta=\frac{8}{3\pi}D\approx0.85D. +\end{equation} + +\end_inset + +Here we use a more advanced model, which includes the shear wave number. + For unrounded edges and a perforate thickness of +\begin_inset Formula $t_{p}$ +\end_inset + +, the added mass end correction can be computed as: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +Equation according to Temiz for added mass effect: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $2\delta=\frac{\delta_{\mathrm{temiz}}}{2}D$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Where: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\delta_{\mathrm{temiz}}=0.97\exp\left(-0.2S_{h}\right)+1.54-0.003\frac{D}{t_{p}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $S_{h}=\frac{D}{2}\sqrt{\frac{\rho_{0}\omega}{\mu_{0}}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +——– +\end_layout + +\begin_layout Plain Layout +Ours: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $2\delta=\frac{8}{3\pi}D$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +2\delta=\frac{1}{2}\left[0.97\exp\left(-0.14\frac{D}{\delta_{\nu}}\right)+1.54-0.003\frac{D}{t_{p}}\right]D +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The factor +\begin_inset Formula $\alpha$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\alpha=5.08\left(\frac{D}{\sqrt{2}\delta_{\nu}}\right)^{-1.45}+1.70-0.002\frac{D}{t_{p}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Tuning the hole diameter for large holes and the negligible hole-hole interactio +n +\end_layout + +\begin_layout Standard +The coarse 0impedance of a Helmholtz resonator repeated here: +\begin_inset Formula +\begin{equation} +Z(\omega)=\underbrace{i\omega m_{A}+R_{v}}_{Z_{h}}+\frac{\rho_{0}c_{0}^{2}}{i\omega V}, +\end{equation} + +\end_inset + +The resistive and reacting part +\begin_inset Formula $i\omega m_{A}+R_{v}$ +\end_inset + + is due to the resonator holes, +\begin_inset Formula +\begin{equation} +Z_{h}=i\omega m_{A}+R_{v}\approx\frac{1}{S}\left[\frac{i\omega\rho_{0}}{\phi}\left[\frac{t_{w}}{\left(1-f_{\nu}\right)}+2\delta f_{\mathrm{int}}\right]+\frac{\alpha\rho_{0}\omega\delta_{\nu}}{\phi}\right].\label{eq:Zhole} +\end{equation} + +\end_inset + +In the large hole limit, or high shear wave number: +\begin_inset Formula +\[ +\Re\left[i\omega m_{A}+R_{v}\right]\approx\frac{\rho_{0}\delta_{\nu}\omega}{\phi S}\left[\alpha+\frac{2t_{w}}{\left(D-4\delta_{\nu}\right)}\right]\underbrace{\propto}_{\mathrm{approx}.}\sqrt{\omega}. +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +In the large hole limit, without hole-hole interaction and +\begin_inset Formula $\delta_{\nu}\to0$ +\end_inset + +, we the resonance frequency of the system is: +\begin_inset Formula +\begin{equation} +\omega_{r,\mathrm{lh}}^{2}=\frac{\phi Sc_{0}^{2}}{V\left(1.54D+t_{w}\right)}\label{eq:omgr_largeholes} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +– +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $\frac{c_{0}^{2}\rho_{0}}{V\omega_{\mathrm{r,lh}}^{2}}\left[\frac{\omega_{\mathrm{r,lh}}^{2}}{i\omega}+\omega\left\{ \frac{\alpha\delta_{\nu}}{2\delta f_{\mathrm{int}}+t_{w}}+i\frac{Dt_{w}+2\delta f_{\mathrm{int}}\left(D-2\delta_{\nu}\left(1-i\right)\right)}{\left(D-2\delta_{\nu}\left(1-i\right)\right)\left(2\delta f_{\mathrm{int}}+t_{w}\right)}\right\} \right]$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{c_{0}^{2}\rho_{0}}{V\omega_{\mathrm{r,lh}}^{2}}\left[\frac{\omega_{\mathrm{r,lh}}^{2}}{i\omega}+\frac{\omega\alpha\delta_{\nu}}{2\delta f_{\mathrm{int}}+t_{w}}+i\omega\left(1+t_{w}\frac{2\delta_{\nu}\left(1-i\right)}{\left(D-2\delta_{\nu}\left(1-i\right)\right)\left(2\delta f_{\mathrm{int}}+t_{w}\right)}\right)\right]$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +Z_{\mathrm{large\,holes},\mathrm{res}}(\omega)=\frac{c_{0}^{2}\rho_{0}}{V\omega_{r,\mathrm{lh}}^{2}}\left[\frac{\omega_{r,\mathrm{lh}}^{2}}{i\omega}+\frac{i\omega t_{w}}{\left\{ 1+2\frac{\delta_{\nu}\left(i-1\right)}{D}\right\} \left(2\delta f_{\mathrm{int}}+t_{w}\right)}+\frac{i\omega\left[2\delta f_{\mathrm{int}}-i\delta_{\nu}\alpha\right]}{2\delta f_{\mathrm{int}}+t_{w}}\right]\label{eq:Zlargeholes_forres} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +COMSOL boundary condition to useful +\end_layout + +\begin_layout Standard +When using COMSOL to compute Helmholtz resonances, the added mass effect + is included just by solving the Helmholtz equation. + Therefore, to model the holes, only the final wall thickness part of the + added mass (and hole-hole interaction), and the resistive part of the impedance + should be added to the simulation. + If we look at Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:Zhole" + +\end_inset + +, it means only the following part: +\begin_inset Formula +\begin{equation} +z_{\mathrm{bc,\,COMSOL}}=i\omega\rho_{0}\frac{t_{w}}{1-f_{\nu}}+\alpha\rho_{0}\omega\delta_{\nu}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Porosity estimator constraint +\end_layout + +\begin_layout Standard +An estimation for the porosity is a good requirement, as a too large porosity + leads to too much hole-hole interaction and shift away from proper Helmholtz + resonators. + First of all, we set the surface area at the inner duct, which is available + for holes as +\begin_inset Formula +\begin{equation} +S=\Pi L_{h}, +\end{equation} + +\end_inset + +and we fix +\begin_inset Formula $L_{h}$ +\end_inset + + to +\begin_inset Formula +\begin{equation} +L_{h}=\lambda_{r}/20=\frac{2\pi c_{0}}{20\omega_{r,\mathrm{lh}}}=\frac{\pi c_{0}}{10\omega_{r,\mathrm{lh}}}. +\end{equation} + +\end_inset + +Rewriting Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:omgr_largeholes" + +\end_inset + + to +\begin_inset Formula $\phi$ +\end_inset + + yields +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\phi\approx\frac{V\left(1.54D+t_{w}\right)\omega_{r,\mathrm{lh}}^{2}}{Sc_{0}^{2}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Fill in for +\begin_inset Formula $S=\Pi L_{h}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\phi\approx\frac{V\left(1.54D+t_{w}\right)\omega_{r,\mathrm{lh}}^{2}}{\Pi L_{h}c_{0}^{2}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +And for +\begin_inset Formula $L_{h}$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $L_{h}=\frac{\pi c_{0}}{10\omega_{r,\mathrm{lh}}}.$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\phi\approx\frac{10}{\pi}\frac{V\left(1.54D+t_{w}\right)\omega_{r,\mathrm{lh}}^{3}}{\Pi c_{0}^{3}}$ +\end_inset + + +\end_layout + +\end_inset + +: +\begin_inset Formula +\begin{equation} +\phi_{\mathrm{estimation}}\approx\frac{10}{\pi}\frac{V\left(1.54D+t_{w}\right)\omega_{r,\mathrm{lh}}^{3}}{\Pi c_{0}^{3}}\leq0.1 +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +See what this constraint does...* +\end_layout + +\begin_layout Section +Large hole (boundary layer) limit +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\phi=\frac{S_{\mathrm{hole}}}{S_{\mathrm{tot}}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\delta_{\nu}\ll D$ +\end_inset + +. + Given +\begin_inset Formula $\zeta$ +\end_inset + + and +\begin_inset Formula $\omega_{r}$ +\end_inset + +. +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +Now, the following substitutions are made: +\begin_inset Formula +\begin{align*} +C & =\frac{V}{\rho_{0}z_{0}},\\ +m_{\mathrm{neck}} & =\frac{1}{\omega_{r}^{2}C}\\ +\zeta & =\frac{1}{2}\omega_{r}CR_{v}. +\end{align*} + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $ $ +\end_inset + + +\end_layout + +\end_inset + +such that we can write: +\begin_inset Formula +\[ +Z_{h}=\frac{1}{\omega_{r}C}\left(\frac{\omega_{r}}{i\omega}+2\zeta+\frac{i\omega}{\omega_{r}}\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +Z_{h}=\frac{1}{\omega_{r}C}\left(\frac{\omega_{r}}{i\omega}+2\zeta+\frac{i\omega}{\omega_{r}}\right)=\frac{1}{i\omega C}+\frac{2\zeta\omega_{r}}{C} +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{2\zeta}{\omega_{r}C}=R_{v}.$ +\end_inset + + Or: +\begin_inset Formula +\[ +\zeta=\frac{1}{2}\omega_{r}R_{v}C +\] + +\end_inset + +But: +\begin_inset Formula $\frac{1}{Cm_{A}}=\omega_{r}^{2}$ +\end_inset + +Such that: +\begin_inset Formula $\frac{1}{C}=\omega_{r}^{2}m_{A}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +\zeta=\frac{1}{2}\frac{R_{v}}{\omega_{r}m_{A}} +\] + +\end_inset + + +\end_layout + +\end_inset + + Note that: +\begin_inset Formula +\begin{equation} +\zeta=\frac{1}{2}\frac{R}{m_{A}\omega_{r}}\approx\frac{1}{2}\frac{\Re\left[z\right]}{\Im\left[z\right]} +\end{equation} + +\end_inset + +Procedure: +\end_layout + +\begin_layout Standard +In the boundary layer limit: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +f_{\nu}=\frac{\left(1-i\right)\delta_{\nu}}{2r_{h}}, +\end{equation} + +\end_inset + +such that: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +z_{\mathrm{perforate}}=\frac{i\omega\rho_{0}}{\phi}\frac{t_{w}+2\delta f_{\mathrm{int}}}{\left(1-\frac{\delta_{\nu}}{2r_{h}}+\frac{i\delta_{\nu}}{2r_{h}}\right)} +\end{equation} + +\end_inset + + +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $z=\frac{i\omega\rho_{0}}{\phi}\frac{t_{w}+\frac{2\delta f_{\mathrm{int}}}{C_{D}}\left(1-\frac{\delta_{\nu}}{2r_{h}}+\frac{i\delta_{\nu}}{2r_{h}}\right)}{\left(1-\frac{\delta_{\nu}}{2r_{h}}+\frac{i\delta_{\nu}}{2r_{h}}\right)}$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Typical resistance: fill in +\begin_inset Formula $\omega=\omega_{r}$ +\end_inset + +. + Filling in: +\begin_inset Formula +\begin{equation} +\zeta\approx\frac{\delta_{\nu}}{D}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The real part of the perforate impedance is the resistive part. + In a 3D simulation, this impedance can be added to a surface of the hole, + to model the hole +\emph on +resistance +\emph default + in an otherwise inviscid simulation. + The real part is: +\begin_inset Formula +\begin{equation} +\frac{}{} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Lots of holes +\end_layout + +\begin_layout Standard +Hereby, once we know the hole diameter, the required acoustic mass can be + tuned using the porosity: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +m_{A}\approx\frac{\Im\left[z(\omega=\omega_{r}\right]}{\omega S_{\mathrm{t}}}\approx\frac{1}{S_{\mathrm{tot}}\phi}\left(\frac{\rho_{0}8Df_{\mathrm{int}}(\phi)}{3\pi}+\rho_{0}t_{w}\right) +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +So that the porosity can be computed as: +\begin_inset Formula +\begin{equation} +\phi\approx F(\phi)=\frac{D\rho_{0}\left(D-2\delta_{\nu}\right)\left(8Df_{\mathrm{int}}+3\pi t_{w}\right)}{3\pi S_{\mathrm{tot}}m_{A}\left(D^{2}-4D\delta_{\nu}+8\delta_{\nu}^{2}\right)}\approx\frac{\rho_{0}\left(8Df_{\mathrm{int}}(\phi)+3\pi t_{w}\right)}{3\pi S_{\mathrm{tot}}m_{A}}. +\end{equation} + +\end_inset + +Note that this is a trancendental equation in +\begin_inset Formula $\phi$ +\end_inset + +, which can easily be solved by iterating +\begin_inset Formula $\phi$ +\end_inset + +: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +\phi_{1} & =F(1)\\ +\phi_{2} & =F(\phi_{1})\\ +\phi_{3} & =F(\phi_{2})\\ +\vdots & =\vdots +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Some holes +\end_layout + +\begin_layout Standard +For only +\begin_inset Quotes eld +\end_inset + +some holes +\begin_inset Quotes erd +\end_inset + +, far away from each other, we fill in for +\begin_inset Formula $\phi=\frac{1}{4}N_{\mathrm{hole}}\pi D^{2}/S_{\mathrm{tot}}$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +m_{A}\approx\frac{\rho_{0}}{3\pi N_{\mathrm{hole}}D}\left(\frac{32}{\pi}+\frac{12t_{w}}{D}\right) +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +So the number of holes can be chosen as: +\begin_inset Formula +\begin{equation} +N_{\mathrm{holes}}\approx\frac{4\rho_{0}\left(8Df_{\mathrm{int}}+3\pi t_{w}\right)}{3\pi^{2}D^{2}m_{A}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Small hole limit +\end_layout + +\begin_layout Standard +In the small hole limit, +\begin_inset space ~ +\end_inset + + +\begin_inset Formula +\begin{equation} +f_{\nu}\approx1-\frac{iD^{2}}{16\delta_{\nu}^{2}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Such that: +\begin_inset Formula +\begin{equation} +\zeta=\frac{1}{2}\frac{R}{m_{A}\omega_{r}}\approx\frac{1}{2}\frac{\Re\left[z(\omega=\omega_{r}\right]}{\Im\left[z(\omega=\omega_{r}\right]}\approx\frac{3\pi\delta_{\nu}^{2}t_{w}}{D^{3}f_{\mathrm{int}}} +\end{equation} + +\end_inset + +Such that: +\begin_inset Formula +\begin{equation} +D=\sqrt[3]{\frac{6\pi\delta_{\nu}^{2}t_{w}}{6\zeta}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +And: +\begin_inset Formula +\begin{equation} +m_{A}=\rho_{0}\frac{8Df_{\mathrm{int}}}{3\pi S_{\mathrm{tot}}\phi} +\end{equation} + +\end_inset + +Such that: +\begin_inset Formula +\begin{equation} +\phi\approx\rho_{0}\frac{8Df_{\mathrm{int}}}{3\pi S_{\mathrm{tot}}m_{A}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Geometry of hole patterns +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +ss +\begin_inset Graphics + filename img/hexagonal_pattern.pdf + width 50text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Geometry details of a hexagonal hole pattern +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "fig:hexagonal_pitch" + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +For a square hole pattern, with hole-hole pitch +\begin_inset Formula $P$ +\end_inset + +, the overall surface of a unit cell +\begin_inset Formula $S_{\mathrm{unit}}=P^{2}$ +\end_inset + +. + For a certain porosity, the pitch can then be computed as: +\begin_inset Formula +\begin{equation} +P=\sqrt{\frac{\pi}{4\phi}}D. +\end{equation} + +\end_inset + +For a hexagonal hole pattern (Fig. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:hexagonal_pitch" + +\end_inset + +) with hole-hole pitch +\begin_inset Formula $P$ +\end_inset + +, the overall surface of a unit cell +\begin_inset Formula $S_{\mathrm{unit}}=\frac{\sqrt{3}}{2}P^{2}$ +\end_inset + +. + Henceforth, the pitch can be computed from the porosity and the hole diameter + as: +\begin_inset Formula +\begin{equation} +P=\sqrt{\frac{\sqrt{3}\pi}{6\phi}}D. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The most important design parameters of a perforate are the porosity and + the hole diameter. +\end_layout + +\begin_layout Section +Addition of acoustic hole resistance in an otherwise inviscid simulation +\end_layout + +\begin_layout Standard +We assume that in a 3D FEM simulation, the imaginary acoustic impedance + of a single hole +\begin_inset Formula +\begin{equation} +Z_{\mathrm{hole}}=i\omega\rho_{0}\frac{4}{\pi D^{2}}\left[\frac{t_{w}}{\left(1-f_{\nu}\right)}+\frac{8Df_{\mathrm{int}}}{3\pi C_{D}}\right], +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\Re[z_{\mathrm{hole}}]=\frac{2D\delta_{\nu}\omega\rho_{0}t_{w}}{\left(4\delta_{\nu}^{2}+\left(D-2\delta_{\nu}\right)^{2}\right)}, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Over-all transmission matrix +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} + & & & & \left\{ \begin{array}{c} +p_{R}\\ +U_{R} +\end{array}\right\} _{1} & = & \boldsymbol{T}_{1}\left\{ \begin{array}{c} +p_{L}\\ +U_{L} +\end{array}\right\} _{1}\\ + & & \left\{ \begin{array}{c} +p_{R}\\ +U_{R} +\end{array}\right\} _{2} & & =\boldsymbol{T}_{2}\left\{ \begin{array}{c} +p_{R}\\ +U_{R} +\end{array}\right\} _{1}\\ +\left\{ \begin{array}{c} +p_{R}\\ +U_{R} +\end{array}\right\} _{3} & =\boldsymbol{T}_{3} & \left\{ \begin{array}{c} +p_{R}\\ +U_{R} +\end{array}\right\} _{2}\\ +\end{align} + +\end_inset + +, hence +\begin_inset Formula +\begin{equation} +\left\{ \begin{array}{c} +p_{R}\\ +U_{R} +\end{array}\right\} _{3}=\underbrace{\boldsymbol{T}_{3}\cdot\boldsymbol{T}_{2}\cdot\boldsymbol{T}_{1}}_{\boldsymbol{T}}\left\{ \begin{array}{c} +p_{L}\\ +U_{L} +\end{array}\right\} _{1} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Chapter +Miscellaneous models for acoustic components +\end_layout + +\begin_layout Section +Acoustic impedance of small orifices +\end_layout + +\begin_layout Subsection +Rectangular orifice +\end_layout + +\begin_layout Subsection +Slit orifice +\end_layout + +\begin_layout Chapter +Lookup model +\end_layout + +\begin_layout Standard +\align left +LRFTubes allows importing transfer matrix data from externally computed + sources (i.e. + finite element model results). + We focus on the use of COMSOL Multiphysics here. + The output data from COMSOL should be created using the +\begin_inset Quotes eld +\end_inset + +Port Sweep +\begin_inset Quotes erd +\end_inset + + functionality. + Implementation is only for 2 ports, as this is the only case for which + COMSOL is able to export data. + In COMSOL, the transfer matrix is defined as: +\end_layout + +\begin_layout Standard +\align center +\begin_inset Graphics + filename img/comsol_transfermatrix.png + +\end_inset + + +\end_layout + +\begin_layout Standard +\align left +\begin_inset Formula +\begin{equation} +\left\{ \begin{array}{c} +p_{i}\\ +Q_{i} +\end{array}\right\} =\left[\begin{array}{cc} +T_{11} & T_{12}\\ +T_{21} & T_{22} +\end{array}\right]\left\{ \begin{array}{c} +p_{o}\\ +Q_{o} +\end{array}\right\} , +\end{equation} + +\end_inset + +hence the transfer matrix definition of +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +lrftubes +\end_layout + +\end_inset + + is the +\emph on +inverse +\emph default + of the definition of COMSOL Multiphysics: +\begin_inset Formula +\begin{equation} +\boldsymbol{T}_{\mathrm{\lrftubes}}=\boldsymbol{T}_{\mathrm{COMSOL}}^{-1} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +To properly use the Lookup model, in COMSOL port 1 should be corresponding + to the LEFT side of a segment, and port 2 should be corresponding to the + RIGHT side of a segment. + Then, the data should be exported to a +\emph on +txt +\emph default + file with the columns in the following order: frequency, T11, T12, T21, + T22. + A file of this format, as exported by COMSOL can be passed to the constructor + of +\family typewriter +\emph on +LookupModel +\family default +. +\end_layout + +\begin_layout Chapter +IEC Coupler impedances +\end_layout + +\begin_layout Standard +The Comsol model with which this data is gathered exports the input impedance + correctly, but the transfer impedance is actually the +\emph on +negative +\emph default +of the actual transfer impedance. + This is due to Comsol, which was only interested in the magnitude of the + impedance values, and due to us (sloppy work). + The input impedance is defined as: +\begin_inset Formula +\begin{equation} +Z_{\mathrm{in}}=\frac{p_{\mathrm{coupler,entrance}}}{U_{\mathrm{coupler,entrance}}} +\end{equation} + +\end_inset + +and the transfer impedance as: +\begin_inset Formula +\begin{equation} +Z_{\mathrm{tr}}=\frac{p_{\mathrm{DRP}}}{U_{\mathrm{coupler,entrance}}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Chapter +Kampinga's SLNS model in our notation +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +Apply equation of state: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\begin{align*} +i\omega\rho+\rho_{0}\nabla\cdot\boldsymbol{u} & =0\\ +i\omega\rho_{0}\boldsymbol{u} & =-\nabla p+\mu_{0}\nabla^{2}\boldsymbol{u}+\left(\frac{1}{3}\mu+\zeta\right)\nabla\left(\nabla\cdot\boldsymbol{u}\right)\\ +i\omega\rho_{0}c_{p}T & =i\omega p+\kappa\nabla^{2}T\\ +\frac{\rho}{\rho_{0}} & =\frac{p}{p_{0}}-\frac{T}{T_{0}} +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Solving for +\begin_inset Formula $i\omega\rho_{0}c_{p}T=i\omega p+\kappa\nabla^{2}T$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +: +\begin_inset Formula $T=\frac{1}{\rho_{0}c_{p}}\left(1-h_{\kappa}\right)p$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Where +\begin_inset Formula $\frac{i\delta_{\kappa}^{2}}{2}\nabla^{2}h_{\kappa}+h_{\kappa}=0$ +\end_inset + + and +\end_layout + +\begin_layout Plain Layout +Same for velocity, negliging +\begin_inset Quotes eld +\end_inset + +bulk +\begin_inset Quotes erd +\end_inset + + viscosity terms: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $i\omega\rho_{0}\boldsymbol{u}=-\nabla p+\mu_{0}\nabla^{2}\boldsymbol{u}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +More or less solution: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\boldsymbol{u}=\frac{i}{\rho_{0}\omega}\left(1-h_{\nu}\right)\nabla p$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Where +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{2i}{\delta_{\nu}^{2}}\nabla^{2}h_{\nu}+h_{\nu}=0$ +\end_inset + + and +\begin_inset Formula $h_{\nu}|_{\mathrm{wall}}=1$ +\end_inset + + for a no-slip b.c. + and 0 for a slip b.c. +\end_layout + +\begin_layout Plain Layout +Filling in the expression for eq of state, +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\rho=\frac{1}{c_{0}^{2}}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Substituting that one, for +\begin_inset Formula $\rho$ +\end_inset + + in continuity eq: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $i\omega\frac{1}{c_{0}^{2}}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p+\rho_{0}\nabla\cdot\boldsymbol{u}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\rho_{0}\nabla\cdot\boldsymbol{u}+i\frac{k}{c_{0}}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Fill in for momentum: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\rho_{0}\nabla\cdot\left(\frac{i}{\rho_{0}\omega}\left(1-h_{\nu}\right)\nabla p\right)+i\frac{k}{c_{0}}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\nabla\cdot\left(\left(1-h_{\nu}\right)\nabla p\right)+k^{2}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +Multiplying with weight factor, applying greens theorem: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{V}p_{w}k\left(1+\left(\gamma-1\right)h_{\kappa}\right)p-iz_{0}\nabla\cdot\boldsymbol{u}p_{w}\mathrm{d}V=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{V}p_{w}k\left(1+\left(\gamma-1\right)h_{\kappa}\right)p+iz_{0}\nabla p_{w}\cdot\boldsymbol{u}\mathrm{d}V=iz_{0}\oint_{S}p_{w}\boldsymbol{u}\cdot\boldsymbol{n}\mathrm{d}S$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{V}p_{w}k\left(1+\left(\gamma-1\right)h_{\kappa}\right)p+iz_{0}\nabla p_{w}\cdot\boldsymbol{u}\mathrm{d}V=iz_{0}\oint_{S}p_{w}\boldsymbol{u}\cdot\boldsymbol{n}\mathrm{d}S$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Filling in +\begin_inset Formula $\boldsymbol{u}$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{i}{\rho_{0}\omega}\nabla p\left(1-\psi_{v}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{V}p_{w}k^{2}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p-\left(1-\psi_{v}\right)\nabla p_{w}\cdot\nabla p\mathrm{d}V=ikz_{0}\oint_{S}p_{w}\boldsymbol{u}\cdot\boldsymbol{n}\mathrm{d}S$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Axially symmetric: +\begin_inset Formula $\int_{z}\int_{r=0}^{a}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{z}\int_{r=0}^{a}\left(p_{w}k^{2}\left(1+\left(\gamma-1\right)\psi_{T}\right)p-\left(1-\psi_{v}\right)\nabla p_{w}\cdot\nabla p\right)2\pi r\mathrm{d}r\mathrm{d}z=ikz_{0}\oint_{S}p_{w}\boldsymbol{u}\cdot\boldsymbol{n}\mathrm{d}S$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +—– Which +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $i\omega\frac{1}{c_{0}^{2}}p\left(1+\left(\gamma-1\right)h_{\kappa}\right)+\rho_{0}\nabla\cdot\left(\frac{i}{\rho_{0}\omega}\nabla p\left(1-\psi_{v}\right)\right)=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $k^{2}p\left(1+\left(\gamma-1\right)h_{\kappa}\right)+\rho_{0}\nabla\cdot\left(\nabla p\left(1-\psi_{v}\right)\right)=0$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +From +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Model +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +\nabla^{2}h_{v}+\frac{2}{i\delta_{\nu}^{2}}h_{v} & =0,\\ +\nabla^{2}h_{\kappa}+\frac{2}{i\delta_{\kappa}^{2}}h_{\kappa} & =0,\\ +\frac{1}{k}\nabla\cdot\left(\left(1-h_{\nu}\right)\nabla p\right)+k\left(1+\left(\gamma-1\right)h_{\kappa}\right)p & =0\label{eq:slns} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +The velocity is: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\boldsymbol{u}=\frac{i}{\rho_{0}\omega}\left(1-h_{\nu}\right)\nabla p +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +With boundary conditions: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +h_{\nu} & =1\qquad\mathrm{at\,the\,wall}\\ +h_{\kappa} & =1\qquad\mathrm{at\,the\,wall} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +For pressure / velocity b.c.'s +\begin_inset Formula +\begin{equation} +\boldsymbol{u}=\frac{i}{\rho_{0}\omega}\left(1-h_{\nu}\right)\nabla p +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Combine with pressure acoustics: +\end_layout + +\begin_layout Plain Layout +Weak form: +\end_layout + +\begin_layout Plain Layout +(-acpr.gradpx*acpr.gradtestpx-acpr.gradpy*acpr.gradtestpy-acpr.gradpz*acpr.gradtestpz- +acpr.p_t*test(pac)*acpr.ik^2)*acpr.delta/acpr.rho_c +\end_layout + +\begin_layout Plain Layout +(-acpr.gradpx*acpr.gradtestpx-acpr.gradpy*acpr.gradtestpy-acpr.gradpz*acpr.gradtestpz- +acpr.p_t*test(pac)*acpr.ik^2)*acpr.delta/acpr.rho_c +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{V}\left[-\nabla p_{t}\cdot\nabla p-p_{t}p\left(ik\right)\right]\frac{\delta}{\rho_{0}c_{0}}\mathrm{dV}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Weak form of SLNS: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{V}p_{t}\left[\nabla\cdot\left(\left(1-h_{\nu}\right)\nabla p\right)+k^{2}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p\right]\frac{\delta}{\rho_{0}c_{0}}\mathrm{d}V$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{V}\left[-\nabla p_{t}\cdot\left(\left(1-h_{\nu}\right)\nabla p\right)+p_{t}k^{2}\left(1+\left(\gamma-1\right)h_{\kappa}\right)p\right]\frac{\delta}{\rho_{0}c_{0}}\mathrm{d}V$ +\end_inset + ++Boundary term. +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{V}\left[\underbrace{-\nabla p_{t}\cdot\nabla p-p_{t}\left(ik\right)^{2}p}_{\mathrm{already\,there}}+\nabla p_{t}\cdot\left(h_{\nu}\nabla p\right)-p_{t}\left(ik\right)^{2}p\left(\left(\gamma-1\right)h_{\kappa}\right)\right]\frac{\delta}{\rho_{0}c_{0}}\mathrm{d}V$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Makes the weak contribution equal to: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{V}\left[\nabla p_{t}\cdot\left(h_{\nu}\nabla p\right)+p_{t}\left(ik\right)^{2}p\left(\left(1-\gamma\right)h_{\kappa}\right)\right]\frac{\delta}{\rho_{0}c_{0}}\mathrm{d}V$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Written out: +\end_layout + +\begin_layout Plain Layout +(hnu*(test(px)*px+test(py)*py+pz*test(pz))+test(p)*p*acpr.ik^2*(1-gamma)*hkappa)* +acpr.delta/acpr.rho_c +\end_layout + +\end_inset + + +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +DEPRECATED, we doen het met de pressure acoustics interface en een enkele + weak contribution! +\end_layout + +\begin_layout Section +Comsol implementation - General Form PDE +\end_layout + +\begin_layout Plain Layout +Model in Comsol: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\begin{equation} +e_{a}\frac{\partial^{2}p}{\partial t^{2}}+d_{a}\frac{\partial p}{\partial t}+\nabla\cdot\boldsymbol{\Gamma}=f +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Comparing with Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:slns" + +\end_inset + + results in: +\begin_inset Formula +\begin{align} +\boldsymbol{\Gamma} & =\frac{1}{k}\left(1-h_{\nu}\right)\nabla p\\ +f & =-k\left(1+\left(\gamma-1\right)h_{\kappa}\right)p +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Section +Comsol implementation - prescribed velocity +\end_layout + +\begin_layout Plain Layout +Flux / source term form in Comsol: +\begin_inset Formula +\begin{equation} +-\boldsymbol{n}\cdot\boldsymbol{\Gamma}=g-qp +\end{equation} + +\end_inset + +From the mathematics, we find: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $k\boldsymbol{\Gamma}=\left(1-h_{\nu}\right)\nabla p$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\boldsymbol{u}=\frac{i}{\rho_{0}\omega}\left(1-h_{\nu}\right)\nabla p$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +– Combine: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\rho_{0}\omega}{i}\boldsymbol{u}=\left(1-h_{\nu}\right)\nabla p$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +– +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\boldsymbol{\Gamma}=-iz_{0}\boldsymbol{u}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +-\boldsymbol{n}\cdot\boldsymbol{\Gamma}=iz_{0}\boldsymbol{u}\cdot\boldsymbol{n}\label{eq:Gam_vs_un} +\end{equation} + +\end_inset + +Such that: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\boldsymbol{u}\cdot\boldsymbol{n}=\frac{i}{\rho_{0}\omega}\left(1-h_{\nu}\right)\nabla p\cdot\boldsymbol{n}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Note that: +\begin_inset Formula +\[ +k\boldsymbol{\Gamma}=\left(1-h_{\nu}\right)\nabla p +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Fill in: +\begin_inset Formula $iz_{0}\boldsymbol{u}\cdot\boldsymbol{n}=-\boldsymbol{\Gamma}\cdot\boldsymbol{n}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align} +q & =0\\ +g & =iz_{0}\boldsymbol{u}\cdot\boldsymbol{n} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Moreover, at such a boundary, we need to set +\begin_inset Formula $h_{\nu}$ +\end_inset + + and +\begin_inset Formula $h_{\kappa}$ +\end_inset + + to 0. +\end_layout + +\begin_layout Section +Normal impedance b.c. +\end_layout + +\begin_layout Plain Layout +We set +\begin_inset Formula +\begin{equation} +z\boldsymbol{u}\cdot\boldsymbol{n}=p +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Upon using Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:Gam_vs_un" + +\end_inset + +, we find: +\end_layout + +\begin_layout Plain Layout +Yields: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\frac{i}{z_{0}}\boldsymbol{n}\cdot\boldsymbol{\Gamma}=\boldsymbol{u}\cdot\boldsymbol{n}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +-\boldsymbol{n}\cdot\boldsymbol{\Gamma}=-i\frac{z_{0}}{z}p +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Such that: +\begin_inset Formula +\begin{align} +q & =i\frac{z_{0}}{z}\\ +g & =0 +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Section +Interior impedance jump +\end_layout + +\begin_layout Plain Layout +Equation: +\begin_inset Formula +\begin{equation} +p_{\mathrm{up}}-p_{\mathrm{down}}=z\boldsymbol{u}\cdot\boldsymbol{n}_{\mathrm{up}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +It should be implemented as a +\begin_inset Quotes eld +\end_inset + +weak contribution +\begin_inset Quotes erd +\end_inset + +. + For that we refer the the weak form equation: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Reverse engineering comsols weak contribution of such a split: +\end_layout + +\begin_layout Plain Layout +-acpr.delta*acpr.iomega*(down(acpr.p_t)-up(acpr.p_t))*(down(test(acp))-up(test(acp)) +)/acpr.Zi +\end_layout + +\begin_layout Plain Layout +waar: delta = 1/omega^2 +\end_layout + +\begin_layout Plain Layout +Leest: +\end_layout + +\begin_layout Plain Layout +-i/omega*(down(p)-up(p))*(down(test(p))-up(test(p))) /z +\end_layout + +\begin_layout Plain Layout +We hebben altijd op een rand: +\begin_inset Formula +\[ +\] + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Weak contribution in pressure acoustics interface: +\end_layout + +\begin_layout Standard + +\family typewriter +(hnu*(test(px)*px+test(py)*py+pz*test(pz))+test(p)*p*acpr.ik^2*(1-gamma)*hkappa)* +acpr.delta/acpr.rho_c +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset CommandInset bibtex +LatexCommand bibtex +btprint "btPrintCited" +bibfiles "lrftubes" +options "plain" + +\end_inset + \end_layout diff --git a/tex/preamble.tex b/tex/preamble.tex index 63528e0..87575e1 100644 --- a/tex/preamble.tex +++ b/tex/preamble.tex @@ -65,40 +65,33 @@ \usepackage{xcolor} -% If we use non-tex fonts we need this 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