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If it's 1, each line will be numbered stringstyle=\color{mymauve}, % string literal style tabsize=2, % sets default tabsize to 2 spaces title=\lstname % show the filename of files included with \lstinputlisting; also try caption instead of title } \end_preamble \options a4paper \use_default_options true \maintain_unincluded_children false \language english \language_package default \inputencoding auto \fontencoding global \font_roman "default" "default" \font_sans "default" "default" \font_typewriter "libertine-mono" "default" \font_math "auto" "auto" \font_default_family default \use_non_tex_fonts false \font_sc false \font_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 \graphics default \default_output_format default \output_sync 1 \bibtex_command bibtex \index_command default \paperfontsize default \spacing single \use_hyperref true \pdf_title "TaSMET User's guide" \pdf_author "J.A. de Jong" \pdf_bookmarks true \pdf_bookmarksnumbered false \pdf_bookmarksopen false \pdf_bookmarksopenlevel 1 \pdf_breaklinks false \pdf_pdfborder false \pdf_colorlinks false \pdf_backref false \pdf_pdfusetitle true \papersize default \use_geometry true \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine basic \cite_engine_type default \biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date true \justification true \use_refstyle 1 \index Index \shortcut idx \color #008000 \end_index \leftmargin 2cm \rightmargin 2cm \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \listings_params "breaklines=true,captionpos=b,frame=tb,language=Python,keywordstyle={\color{blue}},morekeywords={ta, TaSystem, Globalconf, Solver, SolverConfiguration, setSc, Solve}" \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \begin_body \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash thispagestyle{empty} \end_layout \begin_layout Plain Layout \backslash begin{titlingpage} \end_layout \begin_layout Plain Layout \backslash titleM \end_layout \end_inset \begin_inset Separator latexpar \end_inset \end_layout \begin_layout Standard \align center \begin_inset Graphics filename fig/tasys_artist.eps width 50text% \end_inset \family typewriter \size huge \begin_inset Newline newline \end_inset Version 0.1 \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash end{titlingpage} \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset toc LatexCommand tableofcontents \end_inset \end_layout \begin_layout Part Tutorial \end_layout \begin_layout Chapter Introduction \end_layout \begin_layout Standard Welcome to the user's guide of the Thermoacoustic System Modeling Environment Twente, or \noun on TaSMET \noun default . \noun on TaSMET \noun default is a computer code to model thermoacoustic (TA) engines, refrigerators and combined systems by providing nonlinear models for laminar/turbulent oscillating flow in ducts, heat exchangers and stacks/regenerators. A coupling to the mechanical and electrical domain is provided with a piston model. \end_layout \begin_layout Standard The nonlinear sub-models can be connected to form a model of a complete TA or Stirling system. The main ideas of this code are developed as part of my PhD work \begin_inset Quotes eld \end_inset \noun on Numerical Modeling of Thermoacoustic Systems \begin_inset Quotes erd \end_inset \noun default \begin_inset CommandInset citation LatexCommand cite key "jong_numerical_2015" \end_inset . \end_layout \begin_layout Standard The code has been developed with a strong focus on computational cost. Hence, it uses an efficient modeling technique, called the Nonlinear Frequency Domain (NLFD) method, to directly simulate the periodic steady state of a TA system. \end_layout \begin_layout Standard The modular design makes it easy to create any system for which a similar model can be made using the DELTAEC computer code. The main differences between \noun on TaSMET \noun default and other TA modeling computer codes are \end_layout \begin_layout Itemize Open source and free to use. As the author is aware of the possibilities created by using open source codes, the choice of publishing this computer code as open source was not a hard one. \end_layout \begin_layout Itemize Fast modeling in the nonlinear regime. \end_layout \begin_layout Itemize No graphical user interface (GUI), but the Python programming language as modeling glue. GUI's require a considerable time to implement, but they do not add any fundamental features. As we provide all modeling classes in Python, the user can therefore decide whether he still wants a GUI. Using Matplotlib, we can plot everything we want and by building post-processin g scripts we can derive any results from the solved model. \end_layout \begin_layout Section Purpose of this code \end_layout \begin_layout Itemize Expand modeling capabilities of DeltaEC to the nonlinear regime -> More detailed modeling of nonlinear effects \end_layout \begin_layout Itemize More in-depth insight in behavior of TA systems \end_layout \begin_layout Standard In this user's guide, we assume that the reader has already gained some knowledge and experience with (modeling of) TA systems. Moreover, we expect that the user of \noun on TaSMET \noun default has already modeled TA systems with a linear TA code, such as the well-known \noun on DELTAEC \noun default code. A great deal of inspiration of \noun on TaSMET \noun default has been obtained by the way \noun on DELTAEC \noun default is designed. \end_layout \begin_layout Section Bug reporting and contributing \end_layout \begin_layout Standard If you are interested in working and contributing to \noun on TaSMET \noun default , please contact me by email. My email address is anne(at)amdj(dot)nl. I am looking forward to cooperate! \end_layout \begin_layout Chapter Basic ideas \end_layout \begin_layout Standard With \noun on TaSMET \noun default , the periodic steady-state of a TA system is simulated using a numerical model. The convergence speed is much accelerated by assuming a periodic steady state of the system. When a system is in periodic steady state, every physical quantity \begin_inset Formula $\xi$ \end_inset can be described by a Fourier series: \begin_inset Formula \begin{equation} \xi(t)=\sum_{n=0}^{\infty}\Re\left[\hat{\xi}_{n}e^{in\omega t}\right], \end{equation} \end_inset where \begin_inset Formula $\hat{\xi}_{n}$ \end_inset denote the Fourier coefficients of the quantity \begin_inset Formula $\xi$ \end_inset , which can be computed as \begin_inset Note Note status collapsed \begin_layout Plain Layout \begin_inset Formula $\int\limits _{0}^{T}\xi(t)e^{-im\omega t}\mathrm{d}t=\int\limits _{0}^{T}\sum\limits _{n=0}^{\infty}\Re\left[\hat{\xi}_{n}e^{in\omega t}\right]e^{-im\omega t}\mathrm{d}t$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\int\limits _{0}^{T}\xi(t)e^{-im\omega t}\mathrm{d}t=\int\limits _{0}^{T}\sum\limits _{n=0}^{\infty}\frac{1}{2}\left(\hat{\xi}_{n}e^{in\omega t}+\hat{\xi}_{n}^{*}e^{-in\omega t}\right)e^{-im\omega t}\mathrm{d}t$ \end_inset \end_layout \begin_layout Plain Layout For \begin_inset Formula $m=0$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\int\limits _{0}^{T}\xi(t)\mathrm{d}t=T\hat{\xi}_{0}$ \end_inset \end_layout \begin_layout Plain Layout And for \begin_inset Formula $m\neq0$ \end_inset : \end_layout \begin_layout Plain Layout \begin_inset Formula $\hat{\xi}_{n}=\frac{2}{T}\int\limits _{0}^{T}\xi(t)e^{-im\omega t}\mathrm{d}t$ \end_inset \end_layout \end_inset \begin_inset Formula \begin{equation} \hat{\xi}_{n}=\frac{2}{T}\int\limits _{0}^{T}\xi(t)e^{-im\omega t}\mathrm{d}t \end{equation} \end_inset Now, to solve the Fourier coefficients, only a finite number of terms ( \begin_inset Formula $N$ \end_inset ) can be taken into account. This also results in a finite time resolution. Writing both the time and frequency domain in a discrete form: \begin_inset Note Note status collapsed \begin_layout Plain Layout \begin_inset Formula $t=m\Delta t$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $T=M\Delta t$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $M=\Delta t/T=\Delta t\omega$ \end_inset \end_layout \end_inset \begin_inset Formula \begin{equation} \xi_{m}\equiv\xi(t_{m})=\sum_{n=0}^{N-1}\Re\left[\hat{\xi}_{n}e^{inm/M}\right]. \end{equation} \end_inset This can be written in matrix-vector form as \begin_inset Formula \begin{equation} \xi_{m}=\boldsymbol{f}_{m}^{-1}\cdot\hat{\boldsymbol{\xi}}, \end{equation} \end_inset or, for all time-instances \begin_inset Formula \begin{equation} \boldsymbol{\xi}=\boldsymbol{\mathcal{F}}^{-1}\cdot\sum_{n=0}^{N-1}\Re\left[in\omega\hat{\xi}_{n}e^{in\omega t}\right] \end{equation} \end_inset The time-derivative can be easily obtained \begin_inset Formula \begin{equation} \frac{\partial\boldsymbol{\xi}}{\partial t}=\boldsymbol{\mathcal{F}}^{-1}\cdot\boldsymbol{\text{\omega}}\cdot\boldsymbol{\mathcal{F}}\cdot\boldsymbol{\xi}, \end{equation} \end_inset where \begin_inset Formula \begin{equation} \boldsymbol{\omega}=\left[\begin{array}{ccccc} 0 & 0 & 0 & 0 & \dots\\ 0 & i\omega & 0 & 0 & \dots\\ 0 & 0 & 2i\omega & 0 & \dots\\ 0 & 0 & 0 & 3i\omega & \dots\\ \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right] \end{equation} \end_inset \end_layout \begin_layout Chapter Tutorial \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard In this chapter, the usage of \noun on TaSMET \noun default is described by example. Two TA systems will be described to show the main capabilities of \noun on TaSMET \noun default . Focus will also be given on the post-processing capabilities. As all \noun on TaSMET \noun default objects are exposed to Python, the user of \noun on TaSMET \noun default can easily create script to customize post-processing. \end_layout \begin_layout Standard Each TA system in \noun on TaSMET \noun default is an object of type \noun on TaSystem \noun default , a \noun on TaSystem \noun default contains data about the global configuration of the system, such as the working gas in the system and the fundamental oscillation frequency ( \begin_inset Formula $\omega$ \end_inset ). The physical configuration of a typical TA system is defined by the segments in a \noun on TaSystem \noun default . Each segment can represent a part of the TA system, such as resonator tubes, heat exchangers, stacks/regenerators and pistons. The combination of these segments, including the connecting and boundary conditions and the working gas results in a (nonlinear) system of equations. This system of equations can be solved by a \noun on Solver \noun default object. Often, the user creates a Python scripts which has approximately the following shape: \end_layout \begin_layout Standard The main difference between connectors and segments is that only the latter has provides both \emph on equations \emph default as well as \emph on degrees of freedom \emph default (DOFs), while connectors often only provide equations. Hence, a complete system comprises at least one segment. So far, \noun on TaSMET \noun default contains three main type of segments. A short overview of these segments is given below. \end_layout \begin_layout Subsection \noun on Duct \end_layout \begin_layout Standard A Duct is probably the most important type of segment. In a \noun on Duct \noun default , 1D dynamic gas flow can be modeled including its interaction with a solid. Depending on the geometry, a \noun on Duct \noun default can model resonators with variable cross-sectional area, stacks, and heat exchangers. The interaction model of the flow with its surrounding solid is provided in an object-oriented way with a derived class. Examples of derived classes are \noun on LaminarDuct \noun default , \noun on IsentropicDuct \noun default , and \noun on TurbulentDuct. \end_layout \begin_layout Subsection \noun on Piston \end_layout \begin_layout Standard This \noun on Piston \noun default provides a means to exchange mechanical energy between the gas domain and the mechanical domain. The implementation of the \noun on Piston \noun default segment is done in such a way that both the front and back volume can be used. The front and back volume are assumed to be small compared to the wavelength, because no momentum equation is solved for the gas volumes. A reference of this class is given in Sec. \end_layout \begin_layout Subsection \noun on ConnectorVolume \end_layout \begin_layout Standard A \noun on ConnectorVolume \noun default is used to connect multiple \noun on Ducts \noun default together. This allows for 'branching' of multiple \noun on Ducts \noun default and it is an essential feature of traveling wave thermoacoustic engines. A \noun on ConnectorVolume \noun default with only one \noun on Duct \noun default connected to it serves as a \begin_inset Quotes eld \end_inset compliance \begin_inset Quotes erd \end_inset , i.e. an expansion volume in which the fluid motion is brought to rest. \end_layout \begin_layout Section Resonance tube \end_layout \begin_layout Section Hofler's refrigerator (Hofler1) \end_layout \begin_layout Standard In this section, \end_layout \begin_layout Standard \begin_inset CommandInset bibtex LatexCommand bibtex bibfiles "bib/TaSMET" options "plain" \end_inset \end_layout \begin_layout Part The API \end_layout \begin_layout Chapter \noun on Duct \end_layout \begin_layout Section Geometry \end_layout \begin_layout Chapter \noun on Piston \end_layout \begin_layout Section Introduction \noun on \begin_inset CommandInset label LatexCommand label name "sec:Piston_ref" \end_inset \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 fig/piston.eps width 80text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Piston \end_layout \end_inset \begin_inset CommandInset label LatexCommand label name "fig:piston_overview" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard An overview of the model of a \noun on Piston \noun default is schematically shown in Figure ( \begin_inset CommandInset ref LatexCommand ref reference "fig:piston_overview" \end_inset ). A \noun on Duct \noun default can be connected on both the front, as well as on the back side of the \noun on Piston \noun default . This way, the segment has been made flexible. The following geometrical, and mechanical parameters are required for the model: \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 Symbol \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Token \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Meaning \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Unit \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $S_{l}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \family typewriter Sl \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Piston surface area on the left side \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout m \begin_inset Formula $^{2}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $S_{r}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \family typewriter Sr \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Piston surface area on the right side \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout m \begin_inset Formula $^{2}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $V_{o,l}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \family typewriter V0l \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Left volume. Note: the left volume should be defined with a size such that at the minimum \begin_inset Formula $x$ \end_inset , the volume does not become negative! \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout m \begin_inset Formula $^{3}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $V_{0,r}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \family typewriter V0r \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Right volume. Note: the right volume should be defined such that for the maximum piston excursion \begin_inset Formula $x$ \end_inset , the volume does not become negative! \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout m \begin_inset Formula $^{3}$ \end_inset \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 \family typewriter M \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Total moving mass of the piston \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout kg \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $K_{m}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \family typewriter Km \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Piston spring constant \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout N/m \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $C_{m}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \family typewriter Cm \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Piston damping \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout N \begin_inset Formula $\cdot$ \end_inset s/m \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $S_{t,l}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \family typewriter Stl \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Total contact area of the fluid with the solid in the left volume. This variable is used to compute the thermal relaxation dissipation in the piston volume. \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout m \begin_inset Formula $^{2}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $S_{t,r}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \family typewriter Str \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Total contact area of the fluid with the solid in the right volume. This variable is used to compute the thermal relaxation dissipation in the piston volume. \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout m \begin_inset Formula $^{2}$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Overview of all the parameters required for the Piston model. \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard To initialize a \noun on Piston, \noun default first a helper \noun on struct \noun default needs to be defined, called the \noun on PistonConfiguration. \noun default The initialization of a \noun on PistonConfiguration \noun default is \end_layout \begin_layout Standard \begin_inset listings inline false status open \begin_layout Plain Layout pc=PistonConfiguration(Sl,Sr,V0l,V0r,M,Km,Cm,Stl,Str) \end_layout \end_inset \end_layout \begin_layout Standard . Then we can build a new \noun on Piston \noun default segment with \begin_inset listings inline false status open \begin_layout Plain Layout p=Piston(pc) \end_layout \end_inset . \end_layout \begin_layout Section DuctPistonConnector \end_layout \begin_layout Standard The \noun on DuctistonConnector \noun default connects a Piston segment to a Tube. The following syntax is used to initialize a \noun on DuctPistonConnector: \end_layout \begin_layout Standard \begin_inset listings inline false status open \begin_layout Plain Layout tpc=DuctPistonConnector(id_duct,duct_pos,id_piston,piston_pos,KDuctPiston,KPisto nDuct) \end_layout \end_inset , where the id's are strings by which a \noun on Duct \noun default or a \noun on Piston \noun default is identified. \end_layout \begin_layout Subsection Piston energy balance \end_layout \begin_layout Standard \begin_inset Formula \[ mH_{p}+Q_{p\rightarrow t}+s\cdot mH_{t}=0 \] \end_inset \end_layout \begin_layout Section Mechanical boundary conditions \end_layout \begin_layout Subsection \noun on MechBc \end_layout \begin_layout Standard Using a \noun on MechBc, \noun default a simple boundary condition can be set on the mechanical domain of the \noun on Piston \noun default . It can be used to set either an external force, the piston displacement, or a mechanical impedance. The syntax is \begin_inset listings inline false status open \begin_layout Plain Layout m=MechBc(piston_id,contraint_var,boundary_condition) \end_layout \end_inset , where \family typewriter boundary_condition \family default is the b.c. \family typewriter var \family default object, and \family typewriter constraint_var \family default is either \family typewriter Varnr_x, Varnr_F, \family default or \family typewriter Varnr_Z. \family default The latter induces the boundary condition to solve \begin_inset Formula \begin{equation} F-Zx=0, \end{equation} \end_inset and can be used to model a passive electrical domain. \end_layout \begin_layout Subsection VCMNetwork \end_layout \begin_layout Standard The current version does not yet provide this model. In a future version we will provide a voice coil motor (VCM) network to the list of segments. \end_layout \begin_layout Standard \begin_inset Note Note status open \begin_layout Plain Layout To include the model of the electrical domain, it is \end_layout \begin_layout Plain Layout \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \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 \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $L$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout L \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Voice coil inductance \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout H \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $R_{e}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Re \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Electrical current resistance \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\Omega$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $Bl$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Bl \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Motor constant (Newton per Amp or Volt per meter). \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $N/A$ \end_inset \end_layout \end_inset \end_inset \end_layout \end_inset \end_layout \begin_layout Chapter Connecting it all together \end_layout \begin_layout Chapter A system \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard - A system comprises all segments in a physical sense \end_layout \begin_layout Standard - A system : amount of mass \end_layout \begin_layout Standard - Number of harmonics to solve for \end_layout \begin_layout Section \noun on TaSystem \end_layout \begin_layout Standard A \noun on TaSystem \noun default is a class which initializes and contains all segments and connectors. The TaSystem provides basic information about the nonlinear system of equations which has to be solved to obtain the value of all dependent variables in the individual segments. A \noun on TaSystem \noun default object is created with \begin_inset listings inline false status open \begin_layout Plain Layout import TaSMET \end_layout \begin_layout Plain Layout gc=TaSMET.Globalconf(...) \end_layout \begin_layout Plain Layout sys=TaSMET.TaSystem(gc) \end_layout \end_inset Then individual segments can be added to a \noun on TaSystem \noun default by \begin_inset listings inline false status open \begin_layout Plain Layout sys+=seg1 \end_layout \begin_layout Plain Layout sys+=seg2 \end_layout \begin_layout Plain Layout sys+=... \end_layout \begin_layout Plain Layout sys+=con1 \end_layout \begin_layout Plain Layout ... \end_layout \end_inset \end_layout \begin_layout Subsection Mass conservation \end_layout \begin_layout Section EngineSystem \end_layout \begin_layout Standard The EngineSystem class solves for the unknown frequency as well. \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \uuline off \uwave off \noun off \color none m \end_layout \begin_layout Standard We define a new system of equations in which the fundamental frequency is added as unknown and the timing constraint as an equation. So the augmented solution vector is \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \mathbf{y}=\left(\mathbf{x},\omega\right), \end{equation} \end_inset and the augmented residual is \begin_inset Formula \begin{equation} \mathbf{M}=\left(\mathbf{L}(\mathbf{x}),p_{\mathrm{cd}}\right), \end{equation} \end_inset where \begin_inset Formula $p_{\mathrm{cd}}$ \end_inset denotes the phase constraint degree of freedom. \end_layout \begin_layout Standard To search for the solution, Newton iterations are done using the Jacobian of the augmented residual operator \begin_inset Formula $\mathbf{M}$ \end_inset : \begin_inset Formula \begin{equation} \delta\mathbf{y}=-\frac{\mathrm{d}\mathbf{M}}{\mathrm{d}\mathbf{y}}^{-1}\cdot\mathbf{M}, \end{equation} \end_inset where \begin_inset Formula \begin{equation} \frac{\mathbf{\mathrm{d}}\mathbf{M}}{\mathrm{d}\mathbf{y}}=\left[\begin{array}{cc} \frac{\partial\mathbf{L}}{\partial\mathbf{x}} & \frac{\partial\mathbf{L}}{\partial\omega}\\ \frac{\partial p_{\mathrm{cd}}}{\partial\mathbf{x}} & 0 \end{array}\right].\label{eq:newjac} \end{equation} \end_inset In Eq. \begin_inset space ~ \end_inset ( \begin_inset CommandInset ref LatexCommand ref reference "eq:newjac" \end_inset ), \begin_inset Formula $\frac{\partial p_{\mathrm{cd}}}{\partial\mathbf{x}}$ \end_inset is a single row which is zero everywhere, but is one at the global degree of freedom number corresponding to \begin_inset Formula $\Im\left(\tilde{p}_{1}(x=0)\right)$ \end_inset . The column \begin_inset Formula $\frac{\partial\mathbf{L}}{\partial\omega}$ \end_inset , is the sensitivity of the residual to a change in frequency. For brevity we only show the semi-discrete form of these sensitivities. \begin_inset Note Note status open \begin_layout Plain Layout Due to the normalization of the time-derivative matrix t \end_layout \end_inset These sensitivities can \begin_inset Note Note status open \begin_layout Plain Layout easily \end_layout \end_inset be derived from the governing equations. \end_layout \begin_layout Standard For the continuity equation this sensitivity is \begin_inset Formula \begin{equation} \frac{\partial\mathbf{L}_{c,i}}{\partial\omega}=V_{f}\check{\mathbf{D}}\cdot\tilde{\boldsymbol{\rho}}_{i}. \end{equation} \end_inset For the momentum and energy equation, however, we neglect the sensitivity to the operators \begin_inset Formula $\boldsymbol{\tilde{\mathcal{D}}}$ \end_inset , \begin_inset Formula $\boldsymbol{\tilde{\mathcal{H}}}$ \end_inset and \begin_inset Formula $\boldsymbol{\tilde{\mathcal{Q}}}$ \end_inset , so for the momentum equation we use \begin_inset Formula \begin{equation} \frac{\partial\mathbf{L}_{m,L}}{\partial\omega}\simeq\left(x_{i}-x_{i-1}\right)\check{\mathbf{D}}\cdot\boldsymbol{\mathcal{F}}\cdot\tilde{\mathbf{m}}, \end{equation} \end_inset and for the energy equation \begin_inset Formula \begin{equation} \frac{\partial\mathbf{L}_{e,i}}{\partial\omega}\simeq\check{\mathbf{D}}\cdot\left(\frac{S_{f}}{\gamma-1}\tilde{\mathbf{p}}_{i}+\left(x_{R}-x_{L}\right)\left(\mathbf{mu}\right)_{i}\right), \end{equation} \end_inset and finally for the equation of state \begin_inset Formula \begin{equation} \frac{\partial\mathbf{L}_{s,i}}{\partial\omega}=\mathbf{0}. \end{equation} \end_inset \end_layout \begin_layout Subsection Setting a phase constraint \end_layout \begin_layout Itemize Create a PhaseConstraint object: \end_layout \begin_layout Verbatim pc=PhaseConstraint(Varnr, freqnr, left) \end_layout \begin_layout Itemize Apply this contraint to a segment which can accept them, for example a Tube: \end_layout \begin_layout Verbatim t1.setPhaseConstraint(pc) \end_layout \begin_layout Itemize And you're done. Note: only one phase constraint can be used in an EngineSystem. For a TaSystem, the phase constraint is ignored. \end_layout \begin_layout Standard – \end_layout \begin_layout Chapter The \noun on Solver \noun default class \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard .. \end_layout \begin_layout Standard \begin_inset listings inline false status open \begin_layout Plain Layout sc=TaSMET.SolverConfiguration() \end_layout \begin_layout Plain Layout sc.setFunTol(...) \end_layout \begin_layout Plain Layout sc.setRelTol(...) \end_layout \begin_layout Plain Layout sol=TaSMET.Solver(sc) \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout sol.Solve(sys) \end_layout \end_inset \end_layout \begin_layout Section Solver statistics \end_layout \begin_layout Part Model reference \end_layout \begin_layout Chapter Ducts \end_layout \begin_layout Section General 3D conservation equations \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\int_{V(t)}\rho\mathrm{d}V & +\int_{S}\rho\boldsymbol{u}\cdot\boldsymbol{n}\mathrm{d}S=0\\ \frac{\partial\rho\boldsymbol{u}}{\partial t}+\nabla\cdot\rho\boldsymbol{u}\otimes\boldsymbol{u}+\nabla p & =\nabla\cdot\underline{\boldsymbol{\tau}}\\ \frac{\partial\rho E}{\partial t}+\nabla\cdot\left(\rho\boldsymbol{u}E+p\boldsymbol{u}\right)+\nabla\cdot\boldsymbol{q}= & \nabla\cdot\left(\underline{\boldsymbol{\tau}}\cdot\boldsymbol{u}\right) \end{align} \end_inset \end_layout \begin_layout Standard Integrating over a piece of tube length: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} S_{f}\frac{\partial\overline{\rho}}{\partial t}+\frac{\partial}{\partial x}\left(S_{f}\overline{\rho u}\right)=0 \end{equation} \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} S_{f}\frac{\partial\overline{\rho u}}{\partial t}+ \end{equation} \end_inset \end_layout \begin_layout Section Some definitions \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} s=r_{h}\sqrt{\frac{\rho_{0}\omega}{\mu}} \end{equation} \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash begin{subequations} \end_layout \end_inset \begin_inset Formula \begin{eqnarray} s & = & \sqrt{2}\frac{r_{h}}{\delta_{\nu}}\\ s_{t} & = & s\sqrt{\Pr}=\sqrt{2}\frac{r_{h}}{\delta_{\kappa}}\\ s_{s} & = & \sqrt{2}\frac{r_{h,s}}{\delta_{s}} \end{eqnarray} \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash end{subequations} \end_layout \end_inset \end_layout \begin_layout Section Duct geometry \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 fig/duct_grid.eps width 75text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Geometry and discretization of a duct \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Itemize Cell vertices halfway between the cell walls \end_layout \begin_layout Itemize Cross-sectional area jumps at the cell walls \end_layout \begin_layout Standard What variables do live where? \end_layout \begin_layout Standard \noun on Duct \end_layout \begin_layout Subsection Continuity equation \end_layout \begin_layout Standard Continuity equation lives at the vertex \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} S_{f}\frac{\partial\rho}{\partial t}+\frac{\partial S_{f}\rho u}{\partial x}=0, \end{equation} \end_inset where \begin_inset Formula $x$ \end_inset is the axial position, \begin_inset Formula $S_{f}$ \end_inset the cross-sectional area occupied by fluid, \begin_inset Formula $\rho$ \end_inset is the density and \begin_inset Formula $m$ \end_inset the mass flow. Quasi-discrete form: \begin_inset Formula \begin{equation} \Delta x_{i}S_{f,i}\frac{\partial\rho_{i}}{\partial t}+\left(S_{f}\rho u\right){}_{i+1}-\left(S_{f}\rho u\right){}_{i}=0, \end{equation} \end_inset \end_layout \begin_layout Subsection Momentum equation \end_layout \begin_layout Standard Momentum equation lives at the cell wall. It conserves the momentum in a \begin_inset Quotes eld \end_inset cell \begin_inset Quotes erd \end_inset with left and right walls which are at the corresponding vertices. \begin_inset Note Note status collapsed \begin_layout Plain Layout Check units: \end_layout \begin_layout Plain Layout \begin_inset Formula $\left[\frac{\partial m}{\partial t}\right]=\frac{kg}{s^{2}}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\left[S_{f}\frac{\partial p}{\partial x}\right]=m^{2}\frac{N}{m^{2}m}=\frac{N}{m}=\frac{kgm}{s^{2}m}=\frac{kg}{s^{2}}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{\partial\rho u}{\partial t}+\frac{\partial\rho u^{2}}{\partial x}+\frac{\partial p}{\partial x}+\frac{1}{S_{f}}\mathcal{R}=0 \end{equation} \end_inset where, \begin_inset Formula $p$ \end_inset is the pressure and \begin_inset Formula $\mathcal{R}$ \end_inset is the viscous resistance coefficient. \end_layout \begin_layout Standard Quasi-discrete form \begin_inset Formula \begin{equation} \left(x_{i+1}-x_{i}\right)\frac{\partial\rho u}{\partial t}+\left(\rho u^{2}\right)_{i+1}-\left(\rho u^{2}\right)_{i}+\left(p_{i+1}-p_{i}\right)+\frac{\left(x_{i+1}-x_{i}\right)}{S_{f,i}}\mathcal{R}_{i}=0. \end{equation} \end_inset \end_layout \begin_layout Subsection Energy equation \end_layout \begin_layout Standard Since \begin_inset Formula \begin{equation} \frac{1}{2}\rho u^{2}S_{f}\equiv mu, \end{equation} \end_inset The energy equation can be written as \begin_inset Note Note status collapsed \begin_layout Plain Layout \begin_inset Formula $\frac{\partial}{\partial t}\left(S_{f}\rho E\right)+\frac{\partial}{\partial x}\left(mH\right)-\frac{\partial}{\partial x}\left(\kappa S_{f}\frac{\partial T}{\partial x}\right)=Q_{in}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $S_{f}\rho E=S_{f}\rho\left(c_{v}T+\frac{1}{2}u^{2}\right)=S_{f}\left(\rho c_{v}T+\frac{1}{2}\rho u^{2}\right)=\left(S_{f}\frac{p}{\gamma-1}+\frac{1}{2}S_{f}\rho u^{2}\right)$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $S_{f}\rho E=S_{f}\rho\left(c_{v}T+\frac{1}{2}u^{2}\right)=S_{f}\left(\rho c_{v}T+\frac{1}{2}\rho u^{2}\right)=\left(S_{f}\frac{p}{\gamma-1}+\frac{1}{2}S_{f}\rho u^{2}\right)=S_{f}\frac{p}{\gamma-1}+\frac{1}{2}mu$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\frac{\partial}{\partial t}\left(S_{f}\rho E\right)+\frac{\partial}{\partial x}\left(mH\right)-\frac{\partial}{\partial x}\left(\kappa\frac{\partial T}{\partial x}\right)=Q_{in}$ \end_inset \end_layout \end_inset \begin_inset Formula \begin{equation} \frac{S_{f}}{\gamma-1}\frac{\partial p}{\partial t}+\frac{1}{2}\frac{\partial\rho S_{f}u^{2}}{\partial t}+\frac{\partial}{\partial x}\left(\rho uS_{f}c_{p}T+\frac{1}{2}\rho u^{3}S_{f}+Q_{\mathrm{ax}}\right)-Q_{s\to f}=0, \end{equation} \end_inset , and \begin_inset Formula $Q_{\mathrm{ax}}$ \end_inset is the axial conduction: \begin_inset Formula \begin{equation} Q_{\mathrm{ax}}=-\kappa S_{f}\frac{\partial T}{\partial x}, \end{equation} \end_inset and, \begin_inset Formula $Q_{s\to f}$ \end_inset is the heat flow from the solid to the fluid. \end_layout \begin_layout Standard The quasi-discretized form of this equation is \begin_inset Formula \begin{multline} \frac{V_{f}}{\gamma-1}\frac{\partial p_{i}}{\partial t}+\frac{1}{2}\left(x_{r}-x_{l}\right)\frac{\partial\left(mu\right)_{i}}{\partial t}+\\ c_{p}m_{R}\left(W_{r,L}T_{i}+W_{r,R}T_{i+1}\right)-c_{p}m_{l}\left(W_{l,R}T_{i}+W_{l,L}T_{i-1}\right)+\\ \kappa_{R}\left(W_{c,Rl}T_{i}+W_{c,Rr}T_{i+1}\right)-\kappa_{L}\left(W_{c,Lr}T_{i}+W_{c,Ll}T_{i-1}\right)+\\ m_{r}E_{\mathrm{kin},r}-mE_{\mathrm{kin},l}-\left(x_{R}-x_{L}\right)Q_{s\to f}=0, \end{multline} \end_inset , where \begin_inset Formula \begin{equation} W_{R,l}=\frac{x_{i+1}-x_{R}}{x_{i+1}-x_{i}}+\quad;\quad W_{R,r}=1-W_{R,l} \end{equation} \end_inset , and \begin_inset Formula \begin{equation} \kappa_{R}=W_{R,l}\kappa_{i}+W_{R,r}\kappa_{i+1} \end{equation} \end_inset , and \begin_inset Formula \begin{equation} W_{c,Rl}=\frac{S_{f,R}}{x_{i+1}-x_{i}}\quad;\quad W_{c,Rr}=-W_{c,Rl} \end{equation} \end_inset , and \begin_inset Formula \begin{equation} W_{c,Ll}=\frac{S_{f,L}}{x_{i}-x_{i-1}}\quad;\quad W_{c,Lr}=-W_{c,Ll} \end{equation} \end_inset The flow of kinetic energy is computed as \begin_inset Formula \begin{equation} E_{\mathrm{kin},R}=\frac{1}{2}m_{R}\left(\frac{m_{R}}{S_{f,R}\left(W_{R,l}\rho_{i}+W_{R,r}\rho_{i+1}\right)}\right)^{2}=\frac{1}{2}m_{R}^{3}S_{f,R}\left(W_{R,l}\rho_{i}+W_{R,r}\rho_{i+1}\right)^{-2}, \end{equation} \end_inset So \begin_inset Formula \begin{eqnarray} \frac{\partial E_{\mathrm{kin},R}}{\partial m} & = & \frac{3}{2}S_{f,R}\left(W_{R,l}\rho_{i}+W_{R,r}\rho_{i+1}\right)m_{R}^{2}\\ \frac{\partial E_{\mathrm{kin},R}}{\partial\rho_{i}} & = & -m_{R}^{3}S_{f,R}\left(W_{R,l}\rho_{i}+W_{R,r}\rho_{i+1}\right)^{-3}W_{R,l}\\ \frac{\partial E_{\mathrm{kin},R}}{\partial\rho_{i+1}} & = & -m_{R}^{3}S_{f,R}\left(W_{R,l}\rho_{i}+W_{R,r}\rho_{i+1}\right)^{-3}W_{R,r} \end{eqnarray} \end_inset similarly, the flux through the left wall is computed as \begin_inset Formula \begin{equation} E_{\mathrm{kin},L}=\frac{1}{2}m_{L}\left(\frac{m_{L}}{S_{f,L}\left(W_{L,l}\rho_{i-1}+W_{L,l}\rho_{i}\right)}\right)^{2}, \end{equation} \end_inset \end_layout \begin_layout Standard The last term, \begin_inset Formula $Q_{s\to f}$ \end_inset is dependent on the specific model implemented in a derived class of \noun on Duct \noun default . See for example the HopkinsLaminarDuct.Laminar flow \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \tilde{\boldsymbol{\mathcal{R}}}=\tilde{\mathbf{R}}\diamond\tilde{\mathbf{m}} \end{equation} \end_inset \begin_inset Note Note status collapsed \begin_layout Plain Layout \begin_inset Formula $\hat{\mathcal{R}}_{n}=\frac{\mu}{r_{h}^{2}}\frac{is_{n}^{2}f_{\nu,n}}{1-f_{\nu,n}}\hat{U}_{n}$ \end_inset where \begin_inset Formula $s_{n}^{2}=r_{h}^{2}\frac{\rho_{0}\omega}{\mu}$ \end_inset \end_layout \begin_layout Plain Layout Substituting that: \end_layout \begin_layout Plain Layout \begin_inset Formula $\hat{\mathcal{R}}_{n}=\rho_{0}\frac{i\omega nf_{\nu,n}}{1-f_{\nu,n}}\hat{U}_{n}$ \end_inset \end_layout \end_inset where \begin_inset Formula \begin{equation} \tilde{R}_{n}=\frac{i\omega nf_{\nu,n}}{1-f_{\nu,n}} \end{equation} \end_inset \end_layout \begin_layout Subsection Isentropic state equation \end_layout \begin_layout Standard \begin_inset Formula \[ \hat{\mathbf{1}}+\frac{\hat{\mathbf{p}}}{p_{0}}-\mathcal{\boldsymbol{F}}\cdot\left(\frac{\boldsymbol{\rho}}{\rho_{0}}\right)^{\gamma}=\mathbf{0} \] \end_inset \end_layout \begin_layout Subsection Discretization \end_layout \begin_layout Standard Since pressure and density live on the vertices, to compute \begin_inset Formula $f_{\nu,n}$ \end_inset at each cell wall, we take for \begin_inset Formula $\delta_{\nu,\kappa}$ \end_inset the weighted average of the neighboring vertices. \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray} \hat{T}_{0,L} & \approx & w_{i-1}\hat{T}_{0,i-1}+w_{i}T_{0,i},\\ \hat{p}_{0,L} & \approx & w_{i-1}\hat{T}_{0,i-1}+w_{i}T_{0,i}, \end{eqnarray} \end_inset where \begin_inset Note Note status open \begin_layout Plain Layout \begin_inset Formula $w_{i}=1-w_{i-1}=\frac{x_{i+1}-x_{L}}{x_{i+1}-x_{i-1}}$ \end_inset \end_layout \end_inset \begin_inset Formula \begin{equation} w_{i-1}=\frac{x_{i+1}-x_{L}}{x_{i+1}-x_{i-1}}\quad w_{i}=1-w_{i-1} \end{equation} \end_inset \begin_inset Note Note status collapsed \begin_layout Section Turbulent flow \end_layout \begin_layout Plain Layout From paper Characteristic-based non-linear simulation of large-scale standing-wa ve thermoacoustic engine \end_layout \begin_layout Plain Layout \begin_inset Formula \begin{equation} \mathcal{D}=-S_{f}\rho f \end{equation} \end_inset , with \begin_inset Note Note status collapsed \begin_layout Plain Layout \begin_inset Formula \[ C_{f}=\frac{1}{4}\frac{fd}{u|u|}\Rightarrow f=\frac{4C_{f}}{d}u|u| \] \end_inset \end_layout \end_inset \begin_inset Formula \[ f=\frac{4C_{f}}{d}u|u \] \end_inset , this becomes \begin_inset Formula \begin{equation} \mathcal{D}=-m\frac{4C_{f}}{d}|u| \end{equation} \end_inset \end_layout \end_inset \end_layout \begin_layout Section LaminarDuct \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Insulated \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Not insulated \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout With solid \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Impossible combination \end_layout \end_inset \begin_inset Text \begin_layout Itemize Wall temperature is determined by balance in heat flow from fluid domain to solid domain \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Without solid \end_layout \end_inset \begin_inset Text \begin_layout Itemize Time-averaged heat transfer from fluid to solid is zero by setting wall temperature equal to fluid temperature. Hence \begin_inset Formula $\frac{d\hat{T}_{w,0}}{dx}=\frac{d\hat{T}_{0}}{dx}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Itemize Wall temperature is prescribed, heat flow through solid material is zero \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Overview \end_layout \end_inset \end_layout \begin_layout Plain Layout \end_layout \end_inset \end_layout \begin_layout Subsection Viscous resistance \end_layout \begin_layout Subsection Wall temperature prescribed \end_layout \begin_layout Standard In the HopkinsLaminarDuct, \begin_inset Formula \begin{equation} \hat{Q}_{s\to f}=\hat{\mathcal{H}}-\hat{Q}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}\hat{m}, \end{equation} \end_inset \begin_inset Note Note status collapsed \begin_layout Plain Layout \series bold \begin_inset Formula $\hat{\mathbf{Q}}_{s\to f}=\mathcal{H}-\hat{Q}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}\hat{m}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\frac{\partial\hat{\mathbf{Q}}_{s\to f}}{\partial\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}}=-\hat{Q}\hat{m}$ \end_inset \end_layout \begin_layout Plain Layout – \begin_inset Formula $T_{w,0}=\left[\begin{array}{cccc} 1 & 0 & 0 & \dots\end{array}\right]\hat{\mathbf{T}}_{w}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\frac{\partial\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}}{\partial\frac{\mathrm{d}\hat{\mathbf{T}}}{\mathrm{d}x}}=$ \end_inset \end_layout \end_inset where \begin_inset Formula \begin{align} \hat{\boldsymbol{\mathcal{H}}} & =-S_{f}\hat{H}\hat{T} & ; & n>0\\ \hat{\mathcal{\boldsymbol{\mathcal{H}}}}_{0} & =S_{f}\hat{H}_{0}\left(\hat{T}_{w,0}-\hat{T}_{0}\right) & ; & n=0 \end{align} \end_inset where \begin_inset Formula \begin{equation} \hat{H}=\frac{i\hat{\rho}_{0}c_{p}\frac{f_{\kappa}}{1+\epsilon_{s}}}{1-\frac{f_{\kappa}}{1+\epsilon_{s}}} \end{equation} \end_inset where \begin_inset Formula \[ \tilde{Q}_{n}=\frac{c_{p}}{1-\Pr}\left(\frac{f_{\nu,n}}{1-f_{\nu,n}}-\Pr\nolimits _{0}\frac{f_{\kappa,n}}{1-f_{\kappa,n}}\right) \] \end_inset \end_layout \begin_layout Section TurbulentDuct \end_layout \begin_layout Section \noun on Duct \noun default s with solids \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \rho_{s}c_{s}S_{s}\frac{\partial T_{s}}{\partial t}-\frac{\partial}{\partial x}\left(\kappa_{s}S_{s}\frac{\partial T_{s}}{\partial x}\right)=-Q_{s\to f} \end{equation} \end_inset \begin_inset Note Note status open \begin_layout Plain Layout Discretized: \end_layout \begin_layout Plain Layout \begin_inset Formula \[ \rho_{s}c_{s}S_{s}\frac{\partial T_{s}}{\partial t}-\frac{\partial}{\partial x}\left(\kappa_{s}S_{s}\frac{\partial T_{s}}{\partial x}\right)=-Q_{s\to f} \] \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula \[ V_{vs}\rho_{s}c_{s}\frac{\partial T_{s,i}}{\partial t}+Q_{r}-Q_{l}+Q_{s\to f}=0 \] \end_inset \end_layout \end_inset where \begin_inset Formula $T_{s}$ \end_inset is the area-averaged temperature of the solid and \begin_inset Formula \begin{equation} Q_{s\to f}=Q_{s\to f}\left(T_{w},\dots\right)=S_{s}\hat{H}_{s}\left(\hat{T}_{s}-\hat{T}_{w}\right) \end{equation} \end_inset is the transverse heat transfer, which is a function of the \emph on wall \emph default temperature. The wall temperature, on its turn is a function of the time-averaged temperatur e of the solid and the transverse heat transfer: \begin_inset Formula \begin{equation} T_{w}=T_{w}\left(T_{s},Q_{s\to f}\right) \end{equation} \end_inset \end_layout \begin_layout Standard \begin_inset Note Note status collapsed \begin_layout Plain Layout Local \begin_inset Formula $T_{s}$ \end_inset is \begin_inset Formula $T_{s,l}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $i\omega T_{s,l}-\alpha_{s}\frac{\partial^{2}T_{s,l}}{\partial y^{2}}=0$ \end_inset with symmetric boundary conditions: \begin_inset Formula $T_{s,l}\left(\pm y_{0}\right)=T_{w}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\Rightarrow T_{s,l}=A+B\cosh\left(\left(1+i\right)\frac{y}{\delta_{\kappa,s}}\right)$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\left\langle T_{s,l}\right\rangle =T_{s}$ \end_inset \end_layout \begin_layout Plain Layout —– \end_layout \begin_layout Plain Layout \begin_inset Formula $T_{s,l}=A+B\cosh\left(\left(1+i\right)\frac{y}{\delta_{\kappa,s}}\right)$ \end_inset \end_layout \begin_layout Plain Layout – Using boundary conditions: \end_layout \begin_layout Plain Layout \begin_inset Formula $T_{s,l}(y_{0})=T_{w}=A+B\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $A=T_{w}-B\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)$ \end_inset \end_layout \begin_layout Plain Layout such that: \end_layout \begin_layout Plain Layout \begin_inset Formula $T_{s,l}=T_{w}+B\left(\cosh\left(\left(1+i\right)\frac{y}{\delta_{\kappa,s}}\right)-\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)\right)$ \end_inset \end_layout \begin_layout Plain Layout and: \end_layout \begin_layout Plain Layout \begin_inset Formula $\left\langle T_{s,l}\right\rangle =T_{s}=T_{w}+B\left[\frac{\sinh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)}{\left(1+i\right)\frac{y}{\delta_{\kappa,s}}}-B\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)\right]\Rightarrow\frac{T_{s}-T_{w}}{\frac{\sinh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)}{\left(1+i\right)\frac{y}{\delta_{\kappa,s}}}-\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)}=B$ \end_inset \end_layout \begin_layout Plain Layout Sucht that \end_layout \begin_layout Plain Layout \begin_inset Formula $T_{s,l}=T_{w}+\left(T_{s}-T_{w}\right)\frac{1-\frac{\cosh\left(\left(1+i\right)\frac{y}{\delta_{\kappa,s}}\right)}{\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)}}{1-\frac{\sinh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)}{\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)}}=T_{w}+\left(T_{s}-T_{w}\right)\frac{1-h_{\kappa,s}}{1-f_{\kappa,s}}$ \end_inset \end_layout \end_inset The temperature distribution in the solid obeys \begin_inset Formula \begin{equation} T_{s,l}=T_{w}+\left(T_{s}-T_{w}\right)\frac{1-h_{s}}{1-f_{s}}, \end{equation} \end_inset \end_layout \begin_layout Standard And the heat input equals \begin_inset Note Note status collapsed \begin_layout Plain Layout \begin_inset Formula $Q_{s\to f}=\kappa_{s}\Pi\frac{\partial T_{s,l}}{\partial n}=-\kappa_{s}\Pi\frac{\partial T_{s}}{\partial y}|_{y_{0}}\overset{\mathrm{par.\,plates}}{=}=$ \end_inset \end_layout \begin_layout Plain Layout with \begin_inset Formula $\mathbf{n}$ \end_inset pointing from the fluid into the solid (do not know if this is according to my definition) \end_layout \begin_layout Plain Layout for parallel plates: \end_layout \begin_layout Plain Layout \begin_inset Formula $T_{s,l}=T_{w}+\left(T_{s}-T_{w}\right)\frac{1-h_{s}}{1-f_{s}}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $s_{t}=r_{h}\sqrt{\frac{\rho_{0}c_{p}\omega}{\kappa}}=\sqrt{\Pr}s$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $s_{s}=r_{h,s}\sqrt{\frac{\rho_{s}c_{s}\omega}{\kappa_{s}}}$ \end_inset \end_layout \begin_layout Plain Layout For parallel plates: \end_layout \begin_layout Plain Layout \begin_inset Formula $Q_{s\to f}=-\kappa_{s}\Pi\frac{\partial T_{s,l}}{\partial y}|_{y=r_{h,s}}=-\kappa_{s}\frac{S_{s}}{r_{h,s}}\left(T_{s}-T_{w}\right)\frac{1}{1-f_{s}}\frac{\partial}{\partial y}\left(\frac{-\cosh\left(\sqrt{i}s_{s}\frac{y}{r_{h,s}}\right)}{\cosh\left(\sqrt{i}s_{s}\right)}\right)|_{y=r_{h,s}}$ \end_inset \end_layout \begin_layout Plain Layout — For parallel-plates \end_layout \begin_layout Plain Layout \begin_inset Formula $Q_{s\to f}=\kappa_{s}\frac{S_{s}}{r_{h,s}}\left(T_{s}-T_{w}\right)\frac{1}{1-f_{s}}\frac{\partial}{\partial y}\left(\frac{\cosh\left(\sqrt{i}s_{s}\frac{y}{r_{h,s}}\right)}{\cosh\left(\sqrt{i}s_{s}\right)}\right)|_{y=r_{h,s}}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $Q_{s\to f}=\frac{\kappa_{s}S_{s}}{r_{h,s}^{2}}\left(T_{s}-T_{w}\right)\frac{\sqrt{i}s_{s}}{1-f_{s}}\tanh\left(\sqrt{i}s_{s}\right)$ \end_inset \end_layout \begin_layout Plain Layout — \end_layout \begin_layout Plain Layout Using: \begin_inset Formula $f_{s}=\frac{\tanh\left(\sqrt{i}s_{s}\right)}{\sqrt{i}s_{s}}\Rightarrow\tanh\left(\sqrt{i}s_{s}\right)=f_{s}\sqrt{i}s_{s}$ \end_inset \end_layout \begin_layout Plain Layout Fill in: \begin_inset Formula $Q_{s\to f}=\frac{\kappa_{s}S_{s}}{r_{h,s}^{2}}\left(T_{s}-T_{w}\right)\frac{is_{s}^{2}f_{s}}{1-f_{s}}$ \end_inset \end_layout \end_inset \begin_inset Formula \begin{equation} Q_{s\to f}\overset{\mathrm{par.\,plates}}{=}=-\kappa_{s}\Pi\frac{\partial T_{s}}{\partial y}|_{y_{0}}=\frac{\kappa_{s}S_{s}}{r_{h,s}^{2}}\left(T_{s}-T_{w}\right)\frac{is_{s}^{2}f_{s}}{1-f_{s}} \end{equation} \end_inset Hence \begin_inset Formula \begin{equation} Q_{s\to f}=S_{s}H_{s}\left(T_{s}-T_{w}\right) \end{equation} \end_inset Such that \begin_inset Formula \begin{equation} H_{s}=\frac{\kappa_{s}}{r_{h,s}^{2}}\frac{is_{s}^{2}f_{s}}{1-f_{s}} \end{equation} \end_inset \end_layout \begin_layout Subsection Wall temperature not prescribed \end_layout \begin_layout Standard \begin_inset Note Note status collapsed \begin_layout Plain Layout Check if this is in agreement with other one \end_layout \begin_layout Plain Layout \begin_inset Formula \[ Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega\frac{f_{\kappa}}{1+\epsilon_{s}}}{1-\frac{f_{\kappa}}{1+\epsilon_{s}}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\dfrac{f_{\kappa}}{1-\frac{f_{\kappa}}{1+\epsilon_{s}}}\left(\frac{1-\lambda f_{\kappa}-\Pr\left(1-f_{\nu}\right)}{1-f_{\nu}}\right)-\frac{f_{\kappa}-f_{\nu}}{1-f_{\nu}}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x} \] \end_inset \end_layout \begin_layout Plain Layout Using: \begin_inset Formula $\lambda=\frac{1+\frac{\epsilon_{s}f_{\nu}}{f_{\kappa}}}{1+\epsilon_{s}}$ \end_inset , \begin_inset Formula $\lambda\to1$ \end_inset , \begin_inset Formula $\epsilon_{s}=0$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula \[ Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega f_{\kappa}}{1-f_{\kappa}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\dfrac{f_{\kappa}}{1-f_{\kappa}}\left(\frac{1-f_{\kappa}-\Pr\left(1-f_{\nu}\right)}{1-f_{\nu}}\right)-\frac{f_{\kappa}-f_{\nu}}{1-f_{\nu}}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x} \] \end_inset \end_layout \begin_layout Plain Layout Work under same denominator: \end_layout \begin_layout Plain Layout \begin_inset Formula \[ Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}\omega c_{p}f_{\kappa,n}}{1-f_{\kappa,n}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\frac{f_{\kappa}\left(1-f_{\kappa}-\Pr\left(1-f_{\nu}\right)\right)-\left(1-f_{\kappa}\right)\left(f_{\kappa}-f_{\nu}\right)}{\left(1-f_{\kappa}\right)\left(1-f_{\nu}\right)}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x} \] \end_inset \end_layout \begin_layout Plain Layout Split off Prandtl term: \end_layout \begin_layout Plain Layout \begin_inset Formula \[ Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega f_{\kappa}}{1-f_{\kappa}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\frac{f_{\kappa}\left(1-f_{\kappa}\right)-\left(1-f_{\kappa}\right)\left(f_{\kappa}-f_{\nu}\right)}{\left(1-f_{\kappa}\right)\left(1-f_{\nu}\right)}-\frac{\Pr f_{\kappa}}{\left(1-f_{\kappa}\right)}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x} \] \end_inset \end_layout \begin_layout Plain Layout – Work out rest \end_layout \begin_layout Plain Layout \begin_inset Formula \[ Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega f_{\kappa}}{1-f_{\kappa}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\frac{\left(1-f_{\kappa}\right)f_{\nu}}{\left(1-f_{\kappa}\right)\left(1-f_{\nu}\right)}-\frac{\Pr f_{\kappa}}{\left(1-f_{\kappa}\right)}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x} \] \end_inset \end_layout \begin_layout Plain Layout Finally: \end_layout \begin_layout Plain Layout \begin_inset Formula \[ Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega f_{\kappa}}{1-f_{\kappa}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\frac{f_{\nu}}{\left(1-f_{\nu}\right)}-\frac{\Pr f_{\kappa}}{\left(1-f_{\kappa}\right)}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x} \] \end_inset \end_layout \end_inset \begin_inset Note Note status collapsed \begin_layout Plain Layout What happens for \begin_inset Formula $\omega\to0$ \end_inset ? \end_layout \begin_layout Plain Layout \begin_inset Formula \[ \lim_{\omega\to0}\epsilon_{s}=\sqrt{\frac{\kappa_{0}\rho_{0}c_{p}}{\kappa_{s}\rho_{s}c_{s}}}\frac{\tanh\left(\sqrt{i\frac{\rho_{0}c_{p}\omega}{\kappa}}r_{h,s}\right)}{\tanh\left(\sqrt{i\frac{\rho_{s}c_{s}\omega}{\kappa_{s}}}r_{h,s}\right)}=\sqrt{\frac{\kappa_{0}\rho_{0}c_{p}}{\kappa_{s}\rho_{s}c_{s}}}\frac{\tanh\left(\sqrt{i\frac{\rho_{0}c_{p}\omega}{\kappa}}r_{h,s}\right)}{\tanh\left(\sqrt{i\frac{\rho_{s}c_{s}\omega}{\kappa_{s}}}r_{h,s}\right)}=\sqrt{\frac{\kappa_{0}\rho_{0}c_{p}}{\kappa_{s}\rho_{s}c_{s}}}\frac{\sqrt{i\frac{\rho_{0}c_{p}\omega}{\kappa_{0}}}r_{h}}{\sqrt{i\frac{\rho_{s}c_{s}\omega}{\kappa_{s}}}r_{h,s}}=\frac{\rho_{0}c_{p}r_{h}}{\rho_{s}c_{s}r_{h,s}} \] \end_inset \end_layout \begin_layout Plain Layout For \begin_inset Formula $\omega\to0$ \end_inset , \begin_inset Formula $\lambda\to1$ \end_inset , \begin_inset Formula $\epsilon_{s}\to g_{s}\frac{\rho_{0}c_{p}r_{h}}{\rho_{s}c_{s}r_{h,s}}$ \end_inset , \begin_inset Formula $f_{\nu,\kappa}\to1$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\lim\limits _{\omega\to0}Q_{s\to f}==S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega f_{\kappa}}{1+\epsilon_{s}-f_{\kappa}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\dfrac{f_{\kappa}}{1-\frac{f_{\kappa}}{1+\epsilon_{s}}}\left(\frac{1-\lambda f_{\kappa}-\Pr\left(1-f_{\nu}\right)}{1-f_{\nu}}\right)-\frac{f_{\kappa}-f_{\nu}}{1-f_{\nu}}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\lambda=\frac{1+\frac{\epsilon_{s}f_{\nu}}{f_{\kappa}}}{1+\epsilon_{s}}$ \end_inset , \end_layout \begin_layout Plain Layout For parallel plates \end_layout \begin_layout Plain Layout \begin_inset Formula \[ \epsilon_{s}=\sqrt{\frac{\kappa_{0}\rho_{0}c_{p}}{\kappa_{s}\rho_{s}c_{s}}}\frac{\tanh\left(\sqrt{i\Pr}s\right)}{\tanh\left(\sqrt{i}s_{t,s}\right)} \] \end_inset \end_layout \begin_layout Plain Layout Limit for \begin_inset Formula $s$ \end_inset and \begin_inset Formula $s_{t,s}$ \end_inset to zero gives: \end_layout \begin_layout Plain Layout Filling this in in \end_layout \begin_layout Plain Layout \begin_inset Formula \[ Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega f_{\kappa}}{1+\epsilon_{s}-f_{\kappa}}\hat{T} \] \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Note Note status collapsed \begin_layout Plain Layout which is \begin_inset Formula $Q_{s\to f}=\frac{i\hat{\rho}_{0}\omega f_{\kappa}}{1-f_{\kappa}}\left(\hat{T}-\hat{T}_{w}\right)-\frac{c_{p}}{1-\Pr}\left(\frac{f_{\nu,n}}{1-f_{\nu,n}}-\Pr\nolimits _{0}\frac{f_{\kappa,n}}{1-f_{\kappa,n}}\right)m\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}$ \end_inset \end_layout \begin_layout Plain Layout Filling in for \begin_inset Formula $T_{w}$ \end_inset : \end_layout \begin_layout Plain Layout \begin_inset Formula $Q_{s\to f}=\frac{i\hat{\rho}_{0}\omega f_{\kappa}}{1-f_{\kappa}}\left(\hat{T}-\left(T_{s}-\frac{Q_{s\to f}}{J_{s}S_{f}}\right)\right)-\frac{c_{p}}{1-\Pr}\left(\frac{f_{\nu,n}}{1-f_{\nu,n}}-\Pr\nolimits _{0}\frac{f_{\kappa,n}}{1-f_{\kappa,n}}\right)m\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Boundary conditions \end_layout \begin_layout Subsubsection \noun on Duct \noun default with solid to \noun on Duct \noun default without solid \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} Q_{s}=hS_{s}\left(T_{s}-T\right) \end{equation} \end_inset \end_layout \begin_layout Subsubsection \noun on Duct \noun default with solid to \noun on Duct \noun default with solid \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray} Q_{s,1} & = & Q_{s,2}\\ T_{s,1} & = & T_{s,2} \end{eqnarray} \end_inset \end_layout \begin_layout Section \noun on Regenerator \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} k_{\mathrm{tort}}\left(\frac{\partial m}{\partial t}+\frac{\partial mu}{\partial x}\right)+\frac{\partial p}{\partial x}+\mathcal{R}=0 \end{equation} \end_inset \end_layout \begin_layout Standard If \begin_inset Formula $k_{\mathrm{tort}}$ \end_inset not given, it is computed according to Eq. (14) of Swift and Ward: \begin_inset Formula \begin{equation} k_{\mathrm{tort}}=1+\frac{\left(1-\phi\right)^{2}}{2\left(2\phi-1\right)} \end{equation} \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \mathcal{R}=-\frac{1}{2}\frac{f}{r_{h}}m|u| \end{equation} \end_inset \end_layout \begin_layout Standard With \begin_inset Formula \begin{equation} Re=\frac{4\rho|u|r_{h}}{\mu} \end{equation} \end_inset \end_layout \begin_layout Standard Using \begin_inset Formula \begin{equation} f=\frac{c_{1}}{Re}+c_{2} \end{equation} \end_inset \end_layout \begin_layout Standard Filling in: \begin_inset Formula \begin{equation} \mathcal{R}=S_{f}\left(\frac{c_{1}\mu}{8r_{h}^{2}}\frac{m}{\rho}+\frac{c_{2}}{2r_{h}}\mathrm{sgn}\left(m\right)mu\right) \end{equation} \end_inset \end_layout \begin_layout Section \noun on Duct \noun default boundary conditions \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 fig/duct_vars.eps width 80text% \end_inset \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \begin_inset Quotes eld \end_inset node \begin_inset Quotes erd \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Variables \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Equations \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset ERT status open \begin_layout Plain Layout \backslash huge{ \end_layout \end_inset \begin_inset Formula $\bullet$ \end_inset \begin_inset ERT status open \begin_layout Plain Layout } \end_layout \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\rho,mu,T,p,Ts$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Continuity, Energy, \begin_inset Formula $mu=\frac{m^{2}}{\rho S_{f}}$ \end_inset , Solid energy \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\blacksquare$ \end_inset \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 Momentum \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\Box$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $m,mH,T$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Dependent on b.c. \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Staggered grid equations and variables \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection PressureBc \end_layout \begin_layout Standard Momentum equation (prescribes pressure \begin_inset Formula $p_{p}$ \end_inset ) \begin_inset Formula \begin{equation} \Delta x\frac{\partial\mathbf{m}_{L}}{\partial t}+\left(\widehat{\mathbf{mu}}_{0}-\widehat{\mathbf{mu}}_{L}\right)+S_{f,L}\left(\hat{\mathbf{p}}_{0}-\hat{\mathbf{p}}_{p}\right)+\Delta x\mbox{\textbf{\mathcal{R}}}_{L}=\mathbf{0} \end{equation} \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $mH$ \end_inset extrapolated from inside \end_layout \begin_layout Itemize \begin_inset Formula $T=T_{p}$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $T_{s}=T_{s,p}$ \end_inset \end_layout \begin_layout Standard And the boundary condition for the temperature is computed assuming adiabatic compression-expansion. Currently, this is implemented for thermally perfect gases only: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} c_{p}\left(T\right)\mathrm{d}T=\frac{\mathrm{d}p}{\rho} \end{equation} \end_inset FIlling in the perfect gas law and a bit of bookkeeping results in \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{1}{R_{s}}\int\limits _{T_{0}}^{T_{p}}\frac{c_{p}\left(T\right)}{T}\mathrm{d}T-\ln\left(\frac{p_{p}-p_{0}}{p_{0}}\right)=0 \end{equation} \end_inset Note that in general, to solve this equation for the temperature requires a numerical integration, however for the currently implemented gases, \begin_inset Formula $c_{p}$ \end_inset is a polynomial function of \begin_inset Formula $T$ \end_inset : \begin_inset Formula \begin{equation} c_{p}(T)=\sum_{i=0}^{N_{c_{p}}}c_{p,i}T^{i} \end{equation} \end_inset In that case \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{1}{R_{s}}\left(c_{p,0}\ln\left(\frac{T_{p}}{T_{0}}\right)+\sum_{i=1}^{N_{c_{p}}}\frac{c_{p,i}\left(T_{p}^{i}-T_{0}^{i}\right)}{i}\right)-\ln\left(\frac{p_{p}-p_{0}}{p_{0}}\right)=0.\label{eq:T-p-adiabatic} \end{equation} \end_inset \end_layout \begin_layout Standard Equation \begin_inset CommandInset ref LatexCommand ref reference "eq:T-p-adiabatic" \end_inset is solved using a one-dimensional root finding algorithm (see Section \begin_inset CommandInset ref LatexCommand ref reference "sec:One-dimensional-function-solvers" \end_inset ). Note that for an ideal gas an explicit formula is available: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} T_{p,\mathrm{ideal}}=T_{0}\left(\frac{p_{0}+p_{p}}{p_{0}}\right)^{\frac{\gamma_{0}-1}{\gamma_{0}}}.\label{eq:ideal-gas-isentropic-p-T} \end{equation} \end_inset Looking closely at Equation \begin_inset CommandInset ref LatexCommand ref reference "eq:ideal-gas-isentropic-p-T" \end_inset , we find that \begin_inset Formula $T_{p,\mathrm{ideal}}$ \end_inset provides a good guess for the final solution. \end_layout \begin_layout Subsection AdiabaticWall \end_layout \begin_layout Itemize \begin_inset Formula $m=0$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $mH=0$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $Q=0$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $Q_{s}=0$ \end_inset \end_layout \begin_layout Subsection IsoTWall \end_layout \begin_layout Itemize \begin_inset Formula $m=0$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $mH=0$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $T=T_{p}$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $T_{s}=T_{s,p}$ \end_inset \end_layout \begin_layout Subsection ImpedanceBc \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \hat{\mathbf{p}}=\mathbf{Z}\cdot\hat{\mathbf{u}} \end{equation} \end_inset \begin_inset Formula \begin{equation} \mathbf{R}=\hat{\mathbf{p}}-\mathbf{Z}\cdot\mathcal{\boldsymbol{F}}\cdot\left(\frac{\mathbf{m}_{bc}}{S_{f}\left(w_{1}\mathbf{\rho}_{1}+w_{2}\mathbf{\rho}_{2}\right)}\right)=\mathbf{0} \end{equation} \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{\partial\mathbf{R}}{\partial\hat{\mathbf{p}}_{bc}}=\mathbf{I} \end{equation} \end_inset \begin_inset Formula \begin{equation} \frac{\partial\mathbf{R}}{\partial\hat{\mathbf{m}}_{bc}}=-\mathbf{Z}\cdot\boldsymbol{\mathcal{F}}\cdot\mbox{diag}\left(\frac{1}{S_{f}\boldsymbol{\rho}_{bc}}\right)\cdot\boldsymbol{\mathcal{F}}^{-1} \end{equation} \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{\partial\mathbf{R}}{\partial\boldsymbol{\rho}_{1,2}}=w_{1,2}\mathbf{Z}\cdot\boldsymbol{\mathcal{F}}\cdot\mbox{diag}\left(\frac{\mathbf{m}_{bc}}{S_{f}\boldsymbol{\rho}_{bc}^{2}}\right)\cdot\boldsymbol{\mathcal{F}}^{-1} \end{equation} \end_inset \end_layout \begin_layout Standard — \end_layout \begin_layout Standard \begin_inset Note Note status open \begin_layout Plain Layout \begin_inset Formula $\mathbf{Z}^{-1}\cdot\hat{\mathbf{p}}=\boldsymbol{\mathcal{F}}^{-1}\cdot\hat{\mathbf{u}}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\boldsymbol{\mathcal{F}}\cdot\left[\left(\boldsymbol{\mathcal{F}}^{-1}\left(\mathbf{Z}^{-1}\cdot\hat{\mathbf{p}}\right)\right)\circ\mathbf{m}\right]=\boldsymbol{\mathcal{F}}^{-1}\cdot\mathbf{u}\circ\mathbf{m}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\boldsymbol{\mathcal{F}}\cdot\left[\left(\boldsymbol{\mathcal{F}}^{-1}\left(\mathbf{Z}^{-1}\cdot\hat{\mathbf{p}}\right)\right)\circ\mathbf{m}\right]=\widehat{\mathbf{m}\mathbf{u}}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard Temperature at the b.c. computed from pressure: \end_layout \begin_layout Standard \begin_inset Note Note status open \begin_layout Plain Layout \begin_inset Formula $T_{bc}=T_{0}\left(\frac{p_{bc}+p_{p}}{p_{0}}\right)^{\frac{\gamma_{0}-1}{\gamma_{0}}}$ \end_inset , dus \end_layout \end_inset \begin_inset Formula \begin{equation} \mathbf{R}=\hat{\mathbf{p}}_{bc}-\boldsymbol{\mathcal{F}}\cdot\left[p_{0}\left(\left(\frac{\mathbf{T}_{bc}}{T_{0}}\right)^{\frac{\gamma_{0}}{\gamma_{0}-1}}-\mathbf{1}\right)\right]=\mathbf{0} \end{equation} \end_inset and \begin_inset Note Note status open \begin_layout Plain Layout \begin_inset Formula $\mathbf{R}=\hat{\mathbf{p}}_{bc}-\boldsymbol{\mathcal{F}}\cdot\left[p_{0}\left(\left(\frac{\mathbf{T}_{bc}}{T_{0}}\right)^{\frac{\gamma_{0}}{\gamma_{0}-1}}-\mathbf{1}\right)\right]$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\frac{\partial\mathbf{R}}{\partial\mathbf{T}_{bc}}=\boldsymbol{\mathcal{F}}\cdot\mbox{diag}\left[\frac{\gamma_{0}}{\gamma_{0}-1}\frac{p_{0}}{T_{0}}\left(\left(\frac{\mathbf{T}_{bc}}{T_{0}}\right)^{\frac{1}{\gamma_{0}-1}}\right)\right]\cdot\boldsymbol{\mathcal{F}}^{-1}$ \end_inset \end_layout \end_inset \begin_inset Formula \begin{equation} \frac{\partial\mathbf{R}}{\partial\mathbf{T}_{bc}}=\boldsymbol{\mathcal{F}}\cdot\mbox{diag}\left[\frac{\gamma_{0}}{\gamma_{0}-1}\frac{p_{0}}{T_{0}}\left(\left(\frac{\mathbf{T}_{bc}}{T_{0}}\right)^{\frac{1}{\gamma_{0}-1}}\right)\right]\cdot\boldsymbol{\mathcal{F}}^{-1} \end{equation} \end_inset \end_layout \begin_layout Standard Finally: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \mathbf{R}=\boldsymbol{\mathcal{F}}\cdot\left[\left(\boldsymbol{\mathcal{F}}^{-1}\left(\mathbf{Z}^{-1}\cdot\hat{\mathbf{p}}\right)\right)\circ\left(\boldsymbol{\mathcal{F}}^{-1}\cdot\mathbf{m}\right)\right]-\widehat{\mathbf{m}\mathbf{u}}=\mathbf{0} \end{equation} \end_inset And \begin_inset Formula \begin{eqnarray} \frac{\partial\mathbf{R}}{\partial\hat{\mathbf{p}}} & = & \boldsymbol{\mathcal{F}}\cdot\mathrm{diag}\left(\mathbf{m}\right)\cdot\boldsymbol{\mathcal{F}}^{-1}\cdot\mathbf{Z}^{-1}\\ \frac{\partial\mathbf{R}}{\partial\hat{\mathbf{m}}} & = & \boldsymbol{\mathcal{F}}\cdot\left[\mbox{diag}\left(\boldsymbol{\mathcal{F}}^{-1}\cdot\mathbf{Z}^{-1}\cdot\hat{\mathbf{p}}\right)\cdot\boldsymbol{\mathcal{F}}^{-1}\right]\\ \frac{\partial\mathbf{R}}{\partial\widehat{\mathbf{m}\mathbf{u}}} & = & -\mathbf{I} \end{eqnarray} \end_inset \end_layout \begin_layout Subsection VelocityBc \end_layout \begin_layout Standard Prescribed velocity, adiabatic compression/expansion. \begin_inset Formula \begin{equation} \mathbf{R}=\boldsymbol{\mathcal{F}}\cdot\left(\frac{\mathbf{m}_{bc}}{S_{f}\mathbf{\rho}_{bc}}\right)-\hat{\mathbf{u}}_{bc}=0 \end{equation} \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{\partial\mathbf{R}}{\partial\hat{\mathbf{m}}_{bc}}=\boldsymbol{\mathcal{F}}\cdot\mbox{diag}\left(\frac{1}{S_{f}\boldsymbol{\rho}_{bc}}\right)\cdot\boldsymbol{\mathcal{F}}^{-1} \end{equation} \end_inset \end_layout \begin_layout Chapter Piston \end_layout \begin_layout Standard For the piston, the equation of motion \begin_inset Formula \begin{equation} M\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+C_{m}\frac{\mathrm{d}x}{\mathrm{d}t}+K_{m}x=S_{l}p_{l}-S_{r}p_{r}+F \end{equation} \end_inset is solved, where \begin_inset Formula $F$ \end_inset is the external force applied to the Piston. This external force can be either boundary condition, or applied by electromagn etic interaction. For both the right as well as the left volume, conservation of mass, energy and the thermal equation of state is solved. The continuity equations are \begin_inset Formula \begin{eqnarray} \frac{\partial\rho_{r}\left(V_{0r}-xS_{r}\right)}{\partial t}+m_{f} & = & 0,\\ \frac{\partial\rho_{l}\left(V_{0l}+xS_{l}\right)}{\partial t}+m_{l} & = & 0, \end{eqnarray} \end_inset . \end_layout \begin_layout Subsection Piston volume mass conservation \end_layout \begin_layout Standard Variables \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} x,\\ p_{r},T_{r},\rho_{r} & & m_{r},mH_{r}\\ p_{l},T_{l},\rho_{l} & & m_{l},mH_{l} \end{eqnarray*} \end_inset \end_layout \begin_layout Standard and \begin_inset Formula \begin{eqnarray} p_{r}-\rho_{r}R_{s}T_{r} & = & 0\\ p_{l}-\rho_{l}R_{s}T_{l} & = & 0 \end{eqnarray} \end_inset and \end_layout \begin_layout Subsection Piston volume energy conservation \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{p_{r}}{\gamma-1}V_{r}\right)+p_{r}\frac{\mathrm{d}V_{r}}{\mathrm{d}t}+mH_{r}=0 \end{equation} \end_inset or \begin_inset Formula \[ \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{p_{r}}{\gamma-1}V_{r}\right)-\frac{\mathrm{d}x_{p}}{\mathrm{d}t}S_{r}p_{r}+mH_{r}=0 \] \end_inset \end_layout \begin_layout Standard \begin_inset Note Note status collapsed \begin_layout Plain Layout To isentropic (and no mass flow): \end_layout \begin_layout Plain Layout \begin_inset Formula $\mathrm{d}\left(\frac{p_{r}}{\gamma-1}V_{r}\right)+p_{r}\mathrm{d}V_{r}=0=\frac{\mathrm{d}p_{r}}{p_{r}\left(\gamma-1\right)}+\frac{1}{\gamma-1}\frac{\mathrm{d}V_{r}}{V_{r}}+\frac{\mathrm{d}V_{r}}{V_{r}}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\frac{\mathrm{d}p_{r}}{p_{r}}+\gamma\frac{\mathrm{d}V_{r}}{V_{r}}=0$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\frac{p}{p_{0}}=\left(\frac{V_{0}}{V}\right)^{\gamma}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula \[ \frac{dp}{p}=-\gamma\frac{dV}{V} \] \end_inset \end_layout \end_inset \end_layout \begin_layout Standard left side: \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray} \frac{1}{\gamma-1}\mathbf{D}\cdot\boldsymbol{\mathcal{F}}\cdot\left[\mathbf{p}_{l}\circ\left(V_{0r}-\mathbf{x}S_{l}\right)\right]+\boldsymbol{\mathcal{F}}\cdot\left[\mathbf{p}_{l}\circ\left(\boldsymbol{\mathcal{F}}^{-1}\cdot\mathbf{D}\cdot\left(V_{0l}-\hat{\mathbf{x}}S_{l}\right)\right)\right]+\boldsymbol{\mathcal{F}}\cdot\mathbf{mH}_{l} & = & 0 \end{eqnarray} \end_inset right side: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{\partial\frac{p_{l}}{\gamma-1}\left(V_{0l}+xS_{l}\right)}{\partial t}+mH_{l}=0 \end{equation} \end_inset \end_layout \begin_layout Subsubsection Mass in piston volumes \end_layout \begin_layout Standard The time-averaged amount of mass in a piston volume is \begin_inset Formula \[ m=\mathcal{F}_{0}\cdot\left[(V_{0}\pm S\mathbf{x})\circ\mathbf{\rho}\right] \] \end_inset where \begin_inset Formula $\boldsymbol{\mathcal{F}}_{0}$ \end_inset is the first row in the Fourier transform matrix. Therefore, \begin_inset Formula \begin{equation} \frac{\partial m}{\partial\hat{\boldsymbol{\rho}}}=\mathcal{F}_{0}\cdot\left[\mathrm{diag}(V_{0}\pm S\mathbf{x})\cdot\boldsymbol{\mathcal{F}}^{-1}\right] \end{equation} \end_inset \end_layout \begin_layout Subsection Unconnected side \end_layout \begin_layout Standard For example, left side is not connected: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \rho_{l}\left(V_{0l}+xS_{l}\right)=m_{l} \end{equation} \end_inset \end_layout \begin_layout Standard Isentropic: \end_layout \begin_layout Standard \begin_inset Formula \[ p-p_{0}\left(\frac{\rho}{\rho_{0}}\right)^{\gamma}=0 \] \end_inset \end_layout \begin_layout Standard Later: \end_layout \begin_layout Chapter Connectors \end_layout \begin_layout Section Connector theory \end_layout \begin_layout Subsection Conservation of mass \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} sm_{1}=sm_{2} \end{equation} \end_inset \end_layout \begin_layout Subsection Conservation of energy \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} smH_{1}=smH_{2} \end{equation} \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} sQ_{1}=sQ_{2} \end{equation} \end_inset \end_layout \begin_layout Standard mH at the boundary is half the \begin_inset Formula \begin{equation} mH_{boundary}=\frac{1}{2}\left(mH_{1}+mH_{2}\right) \end{equation} \end_inset \end_layout \begin_layout Standard Heat flow between \noun on Duct \noun default s \end_layout \begin_layout Subsection Minor loss - conversion of kinetic energy to thermal energy \end_layout \begin_layout Standard Minor loss generates a decrease in the total (stagnation) pressure: \begin_inset Formula \begin{equation} \Delta p_{\mathrm{tot}}=-\frac{1}{2}K\rho_{u}u_{u}^{2}. \end{equation} \end_inset For incompressible flow, the stagnation pressure over density is \begin_inset Formula \begin{equation} p_{\mathrm{tot}}=p+\frac{1}{2}\rho u^{2}. \end{equation} \end_inset For compressible flow, the stagnation pressure over density is \begin_inset Note Note status collapsed \begin_layout Plain Layout \begin_inset Formula $p_{tot}=p\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{\gamma-1}}\Rightarrow\frac{p_{tot}}{p}=\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{\gamma-1}}\Rightarrow\frac{p}{p_{tot}}=\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{1-\gamma}}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\rho=\rho_{tot}\left(\frac{p}{p_{tot}}\right)^{\frac{1}{\gamma}}\Rightarrow\rho_{tot}=\rho\left(\frac{p_{tot}}{p}\right)^{\frac{1}{\gamma}}\Rightarrow\rho_{tot}=\rho\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{\gamma-1}\frac{1}{\gamma}}=\rho\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{1}{\gamma-1}}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\frac{p_{tot}}{\rho_{tot}}=\frac{p\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{\gamma-1}}}{\rho\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{1}{\gamma-1}}}=\frac{p}{\rho}\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{\gamma-1}-\frac{1}{\gamma-1}}=\frac{p}{\rho}\left(1+\frac{\gamma-1}{2}M^{2}\right)$ \end_inset \end_layout \end_inset \begin_inset Formula \begin{equation} p_{\mathrm{tot}}=p\left(1+\frac{1}{2}\frac{u^{2}}{c_{p}T}\right)^{\frac{\gamma}{\gamma-1}} \end{equation} \end_inset \end_layout \begin_layout Standard A change in total pressure over density results in an increase in entropy. For a callorically perfect gas, the entropy is \begin_inset Formula \begin{equation} s=s_{0}+c_{v}\ln\left[\left(\frac{p}{p_{0}}\right)\left(\frac{\rho_{0}}{\rho}\right)^{\gamma}\right], \end{equation} \end_inset so the change in entropy from state 1 to 2 is \begin_inset Formula \begin{equation} s_{2}-s_{1}=c_{v}\ln\left[\left(\frac{p_{2}}{p_{1}}\right)\left(\frac{\rho_{1}}{\rho_{2}}\right)^{\gamma}\right]=-R_{s}\ln\left(\frac{p_{\mathrm{tot},2}}{p_{\mathrm{tot},1}}\right) \end{equation} \end_inset \begin_inset Note Note status collapsed \begin_layout Plain Layout Using Hoeijmakers eq 6.15e: \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Note Note status collapsed \begin_layout Plain Layout For incompressible flow the total enthalpy is \begin_inset Formula \begin{equation} dH=T\mathrm{d}s+\rho^{-1}\mathrm{d}p+u\mathrm{d}u \end{equation} \end_inset hence from the internal energy we can derive \begin_inset Formula \begin{equation} s=s_{0}+c\ln\left(T/T_{0}\right), \end{equation} \end_inset which is, since \begin_inset Formula $H_{d}=H_{u}$ \end_inset \begin_inset Formula \begin{equation} s_{d}-s_{u}=c\ln\left(\frac{T_{d}}{T_{u}}\right)=-c\ln\left(\frac{\frac{p_{\mathrm{tot}}}{\rho}_{d}}{\frac{p_{\mathrm{tot}}}{\rho}_{u}}\right)=-c\ln\left(1+\frac{\Delta p_{\mathrm{tot}}}{p_{\mathrm{tot},u}}\right) \end{equation} \end_inset such that the change in exergy flow is \begin_inset Formula \begin{equation} m\Delta E_{x}=-T_{0}m\left(s_{d}-s_{u}\right), \end{equation} \end_inset so \begin_inset Formula \begin{equation} \Delta E_{x}=T_{0}c\ln\left(1-\frac{1}{2}\frac{Ku_{u}^{2}}{\left(\frac{p_{\mathrm{tot},u}}{\rho}\right)}\right) \end{equation} \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Note Note status collapsed \begin_layout Subsubsection Generalization for compressible flow \end_layout \begin_layout Plain Layout At a sharp interface, generally turbulence is created, which converts kinetic energy to thermal energy. To model this effect, minor loss coefficients are introduced. If the minor loss coefficient is zero, no entropy is generated at an interface. \end_layout \begin_layout Plain Layout \begin_inset Formula \begin{equation} \mathrm{d}h=T\mathrm{d}s+\rho^{-1}\mathrm{d}p \end{equation} \end_inset \end_layout \begin_layout Plain Layout Isentropically: \end_layout \begin_layout Plain Layout \begin_inset Formula \begin{equation} \frac{T}{T_{0}}=\left(\frac{p}{p_{0}}\right)^{\frac{\gamma-1}{\gamma}} \end{equation} \end_inset \end_layout \begin_layout Plain Layout For a calorically perfect gas, the entropy is \begin_inset Note Note status collapsed \begin_layout Plain Layout From Hoeijmakers: \end_layout \begin_layout Plain Layout \begin_inset Formula $s=s_{0}+c_{v}\ln\left[\left(\frac{p}{p_{0}}\right)\left(\frac{\rho_{0}}{\rho}\right)^{\gamma}\right]$ \end_inset \end_layout \begin_layout Plain Layout Rework to: \end_layout \begin_layout Plain Layout \begin_inset Formula $s=s_{0}+c_{v}\ln\left[\left(\frac{p}{p_{0}}\right)\left(\frac{Tp_{0}}{T_{0}p}\right)^{\gamma}\right]$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $s=s_{0}+c_{v}\ln\left[\left(\frac{p}{p_{0}}\right)\left(\frac{p_{0}}{p}\right)^{\gamma}\left(\frac{T}{T_{0}}\right)^{\gamma}\right]$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $s=s_{0}+c_{v}\ln\left[\left(\frac{p}{p_{0}}\right)^{1-\gamma}\left(\frac{T}{T_{0}}\right)^{\gamma}\right]$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $s=s_{0}+-c_{v}\ln\left[\left(\frac{p_{0}}{p}\right)^{1-\gamma}\left(\frac{T}{T_{0}}\right)^{-\gamma}\right]$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $s=s_{0}+\left(\gamma-1\right)c_{v}\ln\left[\left(\frac{p_{0}}{p}\right)\left(\frac{T}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $s=s_{0}+R_{s}\ln\left[\left(\frac{p_{0}}{p}\right)\left(\frac{T}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]$ \end_inset \end_layout \end_inset \begin_inset Formula \begin{equation} s=s_{0}+R_{s}\ln\left[\left(\frac{p_{0}}{p}\right)\left(\frac{T}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]\label{eq:entropy_idealgas} \end{equation} \end_inset \begin_inset Note Note status collapsed \begin_layout Plain Layout Isentropic change in temperature: \end_layout \begin_layout Plain Layout \begin_inset Formula $T=T_{0}\left(\frac{p}{p_{0}}\right)^{\frac{\gamma-1}{\gamma}}$ \end_inset \end_layout \begin_layout Plain Layout Such that: \end_layout \begin_layout Plain Layout \begin_inset Formula $\Delta h_{irr}=c_{p}\Delta T=c_{p}T_{0}\left(\left(\frac{p}{p_{0}}\right)^{\frac{\gamma-1}{\gamma}}-1\right)$ \end_inset \end_layout \begin_layout Plain Layout Fully non-isentropic change in temperature: \end_layout \begin_layout Plain Layout \begin_inset Formula $dh=Tds+0$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $c_{p}\frac{dT}{T}=dS$ \end_inset \end_layout \begin_layout Plain Layout with: \begin_inset Formula $ds=\frac{}{}$ \end_inset \end_layout \begin_layout Plain Layout and: \end_layout \begin_layout Plain Layout \begin_inset Formula $\Delta H=\Delta\left(c_{p}T+\frac{1}{2}u^{2}\right)=0$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $c_{p}\left(T-T_{0}\right)+\frac{1}{2}\left(u^{2}-u_{0}^{2}\right)=0$ \end_inset \end_layout \begin_layout Plain Layout – Fill in above: \end_layout \begin_layout Plain Layout \begin_inset Formula $\underbrace{c_{p}T_{0}\left(\left(\frac{p}{p_{0}}\right)^{\frac{\gamma-1}{\gamma}}-1\right)}_{\Delta h_{irr}}+\frac{1}{2}\left(u^{2}-u_{0}^{2}\right)=0$ \end_inset \end_layout \begin_layout Plain Layout Hence: \end_layout \begin_layout Plain Layout \begin_inset Formula $c_{p}T_{0}\left(\left(\frac{p}{p_{0}}\right)^{\frac{\gamma-1}{\gamma}}-1\right)=-\frac{1}{2}\left(u^{2}-u_{0}^{2}\right)$ \end_inset \end_layout \begin_layout Plain Layout For very small \begin_inset Formula $\Delta p$ \end_inset , this becomes: \end_layout \begin_layout Plain Layout \begin_inset Formula $c_{p}T_{0}\frac{\gamma-1}{\gamma}\frac{\Delta p}{p_{0}}=-\frac{1}{2}\left(u^{2}-u_{0}^{2}\right)$ \end_inset \end_layout \begin_layout Plain Layout is \end_layout \begin_layout Plain Layout \begin_inset Formula $p=p_{0}-\frac{1}{2}\rho\left(u^{2}-u_{0}^{2}\right)\Rightarrow p+\frac{1}{2}\rho u^{2}=p_{0}+\frac{1}{2}\rho u_{0}^{2}$ \end_inset \end_layout \begin_layout Plain Layout which is Bernouillis law \end_layout \end_inset The result of minor losses is a reduction in Exergy. To generalize minor loss to the full compressible flow, minor loss is modeled as \begin_inset Formula \begin{equation} \Delta E_{x}=-K\left(\frac{T_{0}}{T_{u}}\right)\frac{1}{2}u_{u}^{2}, \end{equation} \end_inset where subscript \begin_inset Formula $u$ \end_inset denotes the upstream condition and \begin_inset Formula $\Delta E_{x}$ \end_inset is the change in exergy per unit mass: \begin_inset Formula \begin{equation} E_{x}=h+\frac{1}{2}u^{2}-T_{0}s, \end{equation} \end_inset which is also called the available energy. \begin_inset Formula $K$ \end_inset is the traditional minor loss coefficient. Since \begin_inset Formula $H=h+\frac{1}{2}u^{2}$ \end_inset is constant, the change in exergy is \begin_inset Formula \begin{equation} \Delta E_{x}=-T_{0}\left(s_{d}-s_{u}\right), \end{equation} \end_inset where subscript \begin_inset Formula $d$ \end_inset denotes the downstream condition. Filling in Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:entropy_idealgas" \end_inset ) for the entropy of an ideal gas results in \begin_inset Note Note status open \begin_layout Plain Layout \begin_inset Formula $-T_{0}(s_{d}-s_{u})=-T_{0}R_{s}\left(\ln\left[\left(\frac{p_{0}}{p_{d}}\right)\left(\frac{T_{d}}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]-\ln\left[\left(\frac{p_{0}}{p_{u}}\right)\left(\frac{T_{u}}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]\right)$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $-T_{0}(s_{d}-s_{u})=T_{0}R_{s}\left(\ln\left[\left(\frac{p_{0}}{p_{u}}\right)\left(\frac{T_{u}}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]-\ln\left[\left(\frac{p_{0}}{p_{d}}\right)\left(\frac{T_{d}}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]\right)$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $-T_{0}(s_{d}-s_{u})=T_{0}R_{s}\ln\left[\left(\frac{p_{0}}{p_{u}}\right)\left(\frac{T_{u}}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\left(\frac{p_{d}}{p_{o}}\right)\left(\frac{T_{0}}{T_{d}}\right)^{\frac{\gamma}{\gamma-1}}\right]$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $-T_{0}(s_{d}-s_{u})=T_{0}R_{s}\ln\left[\left(\frac{p_{d}}{p_{u}}\right)\left(\frac{T_{u}}{T_{d}}\right)^{\frac{\gamma}{\gamma-1}}\right]$ \end_inset \end_layout \end_inset \begin_inset Formula \begin{equation} \Delta E_{x}=E_{x,d}-E_{x,u}=T_{0}R_{s}\ln\left[\left(\frac{p_{d}}{p_{u}}\right)\left(\frac{T_{u}}{T_{d}}\right)^{\frac{\gamma}{\gamma-1}}\right]=-K\left(\frac{T_{0}}{T_{u}}\right)\frac{1}{2}u_{u}^{2}.\label{eq:DeltaE_steady} \end{equation} \end_inset If we assume incompressible flow and if we take \begin_inset Formula $p_{d}=p_{u}+\Delta p$ \end_inset , with \begin_inset Formula $\Delta p\ll1$ \end_inset , this equation can be linearized to obtain \begin_inset Note Note status collapsed \begin_layout Plain Layout \begin_inset Formula $\Delta E_{x}=T_{0}c_{p}\ln\left[\left(\frac{p_{d}}{p_{u}}\right)^{\gamma-1}\left(\frac{T_{u}}{T_{d}}\right)\right]$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\Delta E_{x}=T_{0}c_{p}\left(\left(\gamma-1\right)\frac{\Delta p}{p_{u}}-\frac{\Delta T}{T_{u}}\right)$ \end_inset \end_layout \begin_layout Plain Layout In the limit of incompressible flow, \begin_inset Formula $\gamma\to1$ \end_inset , hence \end_layout \begin_layout Plain Layout \begin_inset Formula $\Delta E_{x}=-\frac{T_{0}}{T_{u}}c_{p}\Delta T$ \end_inset \end_layout \end_inset \begin_inset Formula \begin{equation} \Delta E_{x}=-\frac{T_{0}}{T_{u}}c\Delta T=-K\left(\frac{T_{0}}{T_{u}}\right)\frac{1}{2}u_{u}^{2}. \end{equation} \end_inset Moreover, for incompressible flow \begin_inset Formula \begin{equation} H=cT+\frac{p}{\rho}+\frac{1}{2}u^{2}=\mathrm{const}, \end{equation} \end_inset the minor loss directly converts kinetic energy to thermal energy for an incompressible flow. \end_layout \end_inset \end_layout \begin_layout Subsection Generalization for oscillating flow \end_layout \begin_layout Standard For oscillating flow, there is no real \begin_inset Quotes eld \end_inset upstream \begin_inset Quotes erd \end_inset and \begin_inset Quotes eld \end_inset downstream \begin_inset Quotes erd \end_inset position. To generalize Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:DeltaE_steady" \end_inset ): \begin_inset Note Note status collapsed \begin_layout Plain Layout Flow from 1 to 2 \end_layout \begin_layout Plain Layout \begin_inset Formula $p_{\mathrm{tot},2}-p_{\mathrm{tot},1}=-K_{1\to2}\frac{1}{8}\rho_{1}\left(s|u_{1}|+u_{1}\right)^{2}$ \end_inset \end_layout \begin_layout Plain Layout Flow from 2 to 1 \end_layout \begin_layout Plain Layout \begin_inset Formula $p_{\mathrm{tot},1}-p_{\mathrm{tot},2}=-K_{2\to1}\rho_{2}\frac{1}{8}\left(s|u_{2}|+u_{2}\right)^{2}$ \end_inset \end_layout \begin_layout Plain Layout Mutual exclusive sum: \end_layout \begin_layout Plain Layout \begin_inset Formula $p_{\mathrm{tot},2}-p_{\mathrm{tot},1}=K_{2\to1}\rho_{2}\frac{1}{8}\left(s|u_{2}|+u_{2}\right)^{2}-K_{1\to2}\rho_{1}\frac{1}{8}\left(s|u_{1}|+u_{1}\right)^{2}$ \end_inset \end_layout \end_inset \begin_inset Formula \begin{equation} p_{\mathrm{tot},2}-p_{\mathrm{tot},1}=K_{2\to1}\frac{\rho_{2}}{8}\left(s|u_{2}|+u_{2}\right)^{2}-K_{1\to2}\frac{\rho_{1}}{8}\left(s|u_{1}|+u_{1}\right)^{2}. \end{equation} \end_inset \end_layout \begin_layout Section TubeConnector \end_layout \begin_layout Standard The SimpleTubeConnector can connect two Tube segments together. At the interface, continuity of mass and energy flow is enforced. \end_layout \begin_layout Section Transition of kinetic energy to heat \end_layout \begin_layout Section TubePistonConnector \end_layout \begin_layout Section ConnectorVolume \end_layout \begin_layout Standard A \noun on ConnectorVolume \noun default is a special kind of gas volume on which \noun on Tube \noun default s and \noun on Piston \noun default s can be connected. This way, branches can be created \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} V_{c}\frac{\partial\rho}{\partial t}+\sum\nolimits _{i}m_{i}=0 \end{equation} \end_inset energy: \begin_inset Formula \begin{equation} \frac{V_{c}}{\gamma-1}\frac{\partial p}{\partial t}+\sum\nolimits _{i}mH_{i}+Q_{i}=0 \end{equation} \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ mH_{i}=\mathrm{extrapolated\,from\,tube} \] \end_inset \end_layout \begin_layout Standard Minor loss \end_layout \begin_layout Chapter Systems \end_layout \begin_layout Chapter Solvers \end_layout \begin_layout Section One-dimensional function solvers \begin_inset CommandInset label LatexCommand label name "sec:One-dimensional-function-solvers" \end_inset \end_layout \begin_layout Subsection Gradient-based \end_layout \begin_layout Subsection Gradient free \begin_inset CommandInset label LatexCommand label name "subsec:Gradient-free" \end_inset \end_layout \begin_layout Standard As a gradient free one-dimensional function solver, we use Brent's mehhod. Brent's method combines root bracketing, bisection and inverse quadratic interpolation to find the root of the function without using the gradient. See Wikipedia for more information. \end_layout \begin_layout Section Minimizers \end_layout \end_body \end_document