text:
Working America Photo credit: Julie Hunter
Overworked America
By Martin Hart-Landsberg (Posted Dec 05, 2019)
Originally published: Reports from the Economic Front (more by Reports from the Economic Front)
Culture, Imperialism, Inequality, LaborUnited StatesNewswire
Those living in the U.S. are encouraged to think that they live in the best country in the world with little to learn from the experiences of working people in other countries. This sense is reinforced by the fact that the mainstream media generally discusses U.S. problems without reference to developments or trends in other developed capitalist countries.
Here is one example: hours of work. It is a common complaint that Americans work too many hours. What is rarely noted, as Ryan Cooper points out in his study titled The Leisure Agenda, is that "Americans work far, far more than their counterparts in peer European nations."
The figure below, based on OECD reported data for the year 2018, shows just how much more. The average U.S. worker works roughly 110 hours a year more than the average Japanese worker, or some 2.6 weeks more; about 265 hours a year more than the average French worker, or some 6.6 weeks more; and 420 hours more than the average German worker, or some 10.5 weeks.
The OECD defines average annual hours worked per employed person as:
the total number of hours actually worked per year divided by the average number of people in employment per year. Actual hours worked include regular work hours of full-time, part-time and part-year workers, paid and unpaid overtime, hours worked in additional jobs, and exclude time not worked because of public holidays, annual paid leave, own illness, injury and temporary disability, maternity leave, parental leave, schooling or training, slack work for technical or economic reasons, strike or labor dispute, bad weather, compensation leave and other reasons. The data cover employees and self-employed workers.
The trend in average annual hours of work in the U.S. and other developed capitalist countries highlights just how far outside the mainstream the U.S. labor experience is. The following figure, again based on OECD data, is taken from Cooper's study. As he summarizes:
As most nations have gotten richer, their average worker has worked fewer hours. But this is not true of the United States. As shown [below], among wealthy OECD nations with data going back that far, the U.S. was in the middle of the pack among rich nations in 1970. Now, it works the most out of any in this cohort.
One reason for the higher average hours of work in the U.S. is that it is the only major OECD country that does not provide a federal statutory minimum annual leave entitlement to its workers, as illustrated in the following figure which is also from Cooper's study.
The annual U.S. work hours presented above is for the average worker, which means that it includes the work experience of people who are forced to work extremely long hours as well as those who cannot find full-time employment. In both cases, the U.S. employment situation is a major contributor to the stress, poor health, and weakening community ties experienced by growing numbers of workers.
The lower average annual hours of work in other OECD countries does not mean that workers in those countries don't have their own challenges, especially as many now confront governments that seek to undermine their past gains. At the same time, it does demonstrate that there is plenty of room for improvement in the United States.
In other countries a reduction in work hours and paid annual leave entitlements were won through aggressive workplace struggle and political pressure on governments. There is, of course, a long history of struggle for a shorter workday in the US, which came to be symbolized by May Day demonstrations and strike actions, and deserves renewed attention. It is also worth remembering that activists in that struggle were well aware that achieving a shorter work week was critical to securing for workers the time and energy needed to build a powerful working class movement for social transformation.
After his mysterious death, the media scrambles to get its story straight about White Helmets founder James Le Mesurier
Thinking and acting from Marxism today – Feminist proposals for a theoretical and strategic rearmament


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Issue No. 109 | Dec 2020 Issue No. 108 | Nov 2020 Issue No. 107 | Oct 2020 Issue No. 106 | Sep 2020 Issue No. 105 | Jul 2020 Issue No. 92 | Jul 2019 Past Issues
Scholar Updates
Reviving traditional craftsmanship through contemporary art
In a tech-savvy era overflowing with electronic products, how can historic, traditional craftsmanship carry on in society?
Funded by The Hong Kong Jockey Club Charities Trust, the Jockey Club ICH+ Innovative Heritage Education Programme is jointly launched by Lingnan University and Hong Kong Art School for a period of 4.5 years and has collaborated with schools in Hong Kong since 2018.
Through experiential learning that integrates contemporary art methods into traditional craftsmanship, the programme aims to revitalise the intangible cultural heritage (ICH) of Hong Kong and inherit the folk wisdom of ages.
The programme's inaugural exhibition was held from 27 June to 7 July at the Sam Tung Uk Museum in Tsuen Wan. Displayed under a large set of traditional bamboo scaffolding, an architectural technique longstanding in the local culture, were the three types of artistry featured in the ICH: paper crafting, paper cutting and cheongsam tailoring.
In the past year, a total of some 900 students from 19 schools participated in the programme. Among them, masterpieces by over a hundred students were presented at the exhibition, allowing the young generation to give the public a glimpse of the traditional culture and aesthetics of the city, turning a new page for traditional artistry using contemporary art.
The exhibition was kicked off with a cheongsam fashion show, with over 50 students modelling handmade cheongsams for a pioneering catwalk across the bamboo scaffolding stage.
Overseeing the project's overall operation, Prof Lau Chi-pang, Associate Vice-President (Academic Affairs and External Relations) and Professor of Department of History at Lingnan University, remarked at the opening ceremony that compared to today's material world where everything is easily within reach, people of the past had to rack their brains and use folk wisdom for entertainment given the simple lifestyle and limited resources at the time.
"With hands-on experience in ICH mastery such as bamboo scaffolding, paper cutting and cheongsam tailoring, we see the intelligence, wisdom and ingenuity of our past generations," said Lau.
In addition to the preservation of ICH, he hoped the young inherit and carry forward the traditional wisdom. For instance, when using mobile phones, he expressed that one should reflect on the purpose of its existence and live meaningfully instead of only looking at the surface.
The Jockey Club ICH+ Innovative Heritage Education Programme integrates education, revitalisation and investigation to preserve, promote, redefine and revitalise the ICH of Hong Kong. It actively engages traditional artisans, contemporary art practitioners and the general public to collaborate and revive the ICH, reclaim cultural esteem of artisanship and hence rediscover traditional values and local identity.
More courses on sugar-blowing, woodwork and flour doll craftsmanship are in the pipeline for the upcoming year.
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Karate athlete Eva Lam: 'Through trial and error, everyone can be a trailblazer!'
Collapse/Expand Scholar Updates
Student Achievements/Campus Life
Collapse/Expand Student Achievements/Campus Life
Four Lingnan students awarded the Hong Kong Jockey Club Scholarships
Collapse/Expand Event Highlights
Co-hosting the International Postgraduate Summer School with University of Oxford
Collapse/Expand Engagements
'Old Camp' at Lingnan promotes gerontechnology for elderly-youth harmony
Lingnan's global partnership sets foot on Peru
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Lingnan University ranks No.1 in teaching and learning experience

redpajama_set_name: RedPajamaCommonCrawl


Eventful is a useful event discovery app for your iPhone that informs you about concerts, movies, and family and sporting events near you.
One of my favorite apps used to be Songkick, because it sent me regular emails about bands performing in my area. But it has been replaced by Eventful, which is like Songkick, but with the addition of other non-music events.
Eventful 2.0 for iPhone (free, App Store) scans your music library and informs you about when your bands are in the area (like Songkick does), but that's where the similarities end. In addition to music, Eventful allows you to find out about numerous other events in your area, including movies, performing arts, family events, and sports.
It includes 26 additional categories, including sports, kids & family events, festivals, etc.
Eventful will alert you when your favorite bands are coming to town, and as an Eventful user, you will receive performer and recommended performer alerts (from its recommendation engine) via email. The addition of movies and family events are worth the price of admission. If you're looking for something to do this week (or weekend), I recommend that you give Eventful a spin.


text:
Untapped Territory: Social Recruiting with Instagram and Pinterest
Post author By Amanda Ashworth
Well there's no doubt about it, social recruiting has improved both the quantity and quality of candidates, and an online study by Jobvite suggests that one-third of human resource professionals tend to agree. This is because social profiles are giving recruiters more confidence in not only the professional fit of a candidate but the cultural fit too. Employers and candidates both actively seek this fit and a shifting trend in working life and employer brand positioning means that brands are opening their doors to the public and using the workplace as a communication tool to share their cultural values – and they're using social media to do it.
The survey study suggests that a staggering 93% of recruiters are likely to look at a candidate's social profile and 42% have reconsidered a candidate after viewing it. This lead to both positive and negative re-assessments, and funnily enough, it's not awkward photos that are deterring potential employers – it's often profanity, and grammar and punctuation errors that trigger negative reactions (over 60% of the time).
This upward trend means that the hiring environment is getting increasingly more competitive and defining your social media strategy is evermore important. Recruiters with clear social media strategy are reaching beyond the predicted platforms and into social environments like Pinterest and Instagram, which are being considered as "specialised", "localised" and "up-and-coming".
Pinterest and Instagram are not primarily designed to be recruiting sites, so unfortunately many recruiters fail to see the value in using them. But the fact is, plenty of desirable candidates and potential brands and clients use them, as much and sometimes more as other social sites. There is a strategic opportunity here, and it's based around the idea of this cultural fit – both these sites offer users the opportunity to express themselves, their interests, their creativity, their skills – even insight into their current, past and future work life. These platforms are being considered as an "innovators way of recruiting" – if you're using it already, you're one of the first to harness the opportunity.
Dan Finnigan, President and CEO of Jobvite, rightly says "It's no longer a question of 'are recruiters using social media?', it's a question of how."
So how are you using it? Have you considered using Pinterest or Instragram?
What is Pinterest?
Popular sharing site launched in 2011
Simple user interface makes it widely used
Sharing content is accomplished easily (photos can be pined to boards and then shared to Facebook and Twitter)
Highly visual images go viral
Videos can also be pinned to boards
Over 48 million registered members
80% of all pins are re-pins
What are the benefits of using Pinterest?
Driving Traffic with SEO: Like many social networks, it drives traffic to job openings and talent communities by tagging – in this case, with 'pins'. Recruiters are able to 'pin' pictures and information about a job or potential employment opportunities using photos and graphics. You can add links and other information either within the content of the image or in the description below it. This search engine optimization approach relies on the use of organic search on the Pinterest site to drive candidates to your posts.
Sourcing for Candidates: Pinterest is the perfect platform for our much-loved infographics. They are all over the social platform and are another reason to discuss the "death of the resume" and "rise of the infographic resume". It's a great idea and the trend is catching but at this stage, most applicant tracking systems are made to track down a graphic in lieu of a traditional resume format.
Pinterest Strategies:
Make the most important boards appear first.
Add relevant keywords to your boards and pin descriptions.
Reciprocal pinning: sharing only your own work will not encourage other pinners to share your content.
Pin when users are most active (2pm-4pm and 8pm-1am).
READ MORE: How to Recruit Using Pinterest [Top 3 Tips] / How To Use Pinterest for Social Recruiting [Cool Example]
What is Instagram?
Popular for its photo-editing features
Simplistic photo sharing ability allows users to upload images easily and quickly
Photos can be shared to other networks: Facebook, Twitter, Foursquare, Flickr & Tumblr
Creative images receives comments and likes
40 million photos shared every day
8,500 likes per second
1,000 comments per seconds
What are the benefits of using Instagram?
Employer brand image building: Being a photo-sharing app, Instagram certainly has the "cool factor" going for it. It draws trend leaders and allows brands an opportunity to outwardly portray its brand image to attract the right candidates. This reflects back on the idea of brands using their workplace as communication tool to express brand values. Instagram is one of the perfect places to do this.
Hashtag search for candidates: The platform uses the most simple keyword search process – the hashtag. It's as simple as searching "writer", "engineering", "PR", and you will not only receive an abundance of results to filter through, but recommendations from Instagram. The ability to follow and communicate directly with other users via the comment feature is also a great way to engage.
Instagram Strategies:
Follow others in your field/industry and network.
Post visually stimulating images that generate a significant amount of activity.
Stay active: Instagram feeds can become very cluttered.
Share images to other social networks such as Facebook and Twitter.
The rise in popularity of both these platforms gives more possibility in recruiting in terms of recruiting tools. There's lots to be learnt, lots to be explored and there is no doubt that when both platforms realise their potential as recruitment tools, more recruitment-focused features will be added to support us too.
These are both untapped potential candidate pipelines with little to no cost, of course with the exception of your time – definitely something to consider.
READ MORE: How to Recruit Using Instagram [5 Simple Steps]
Image: Shutterstock
Tags Instagram, Pinterest, Recruiting, social, Social Recruiting, Social Recruitment
By Amanda Ashworth
Amanda Ashworth is an agency recruiter with a passion for social recruiting, who now focuses on marketing. She blogs about the changes within the recruitment industry and social media. Follow Amanda on Twitter @RecSocially.
← Why the Cover Letter is Dead → 25 Odd Jobs that Make Good Money


In the Easter Sunday celebrations penitents wearing masks or antifaz prepare to begin the parade through the town from the church of El Salvador in Nerja on the Costa del Sol in Andalusia in Spain.
The small sandy beach at Playa El Salon below the cliffs and town at Nerja on the Costa del Sol in Andalusia in Spain.
Small fishermens cottages at the small sandy beach at Playa El Salon below the cliffs and town of Nerja on the Costa del Sol in Andalusia in Spain.


The essence of the brief behind this project was an office fitout on the top floor of 62 Victoria Street that potential future tenants would aspire to, whilst providing a new home for Redwood Group.
The space makes the most of glass, exposed concrete and a feature living wall while reinforcing an innovative and sustainable approach to construction and development. The combination of such organic, natural elements has resulted in a sophisticated workplace environment that achieves the high level of consistency and comfort desired by the client.


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\section{Bob's comments}

\section*{Introduction}
For transient modeling of gas flow through pipe networks, fluid dynamics and computational fluid dynamics, are well-established subjects focused on high-fidelity modeling given design and boundary conditions. They involve nonlinear partial differential equations (PDEs) and transport phenomena, which are not amenable to finite-dimensional control design but instead are targeted and tested for simulation.

Other more pragmatic modeling for gas pipeline distribution systems \cite{benner2019,kralik1988,Behbahaninejad2008AMS} is usually based on discretization and  yields a system of ordinary differential algebraic equations (DAEs), which again is not well-suited to control design. Although, it can be used directly for controller synthesis in some circumstances \cite{mpdDae} and, as noted in \cite{benner2019}, if the DAE is of index~1. Theorem~4.1 in Benner \text{et al.} \cite{benner2019} establishes that the DAEs describing the gas flow through interconnected pipes are indeed of index~1 and so it is possible to rewrite the DAE as an ordinary differential equation (ODE) without the algebraic constraints.

This fact is used in \cite{sven_bob_rob_gas} to rewrite the system of DAEs as a (state-space) system of linear ordinary differential equations (ODEs) \textit{subsuming} the algebraic constraints. Thus, to synthesize model-based controllers the rich literature on Linear Systems Theory can be exploited, such as Mason's Gain Formula for modeling interconnections. An equivalent approach for interconnections adapted to state space realizations is presented in \cite{sven_bob_rob_gas}.

This work builds on \cite{sven_bob_rob_gas} and provides linear control-oriented state space models of gas flow through standard equipment, such as valves, compressors, manifolds and non-trivial pipe geometries. After presenting the catalog of components, we show how to interconnect them: we briefly recall the matrix methodology derived in \cite{sven_bob_rob_gas}, accompanied by a Matlab example for a gas loop; secondly for the same example we provide an alternative approach using Matlab's \verb|connect| function.

All the aforementioned models are suited to model-based MIMO control design tools as they are in state-space form and they can be physically parametrized. {Conservation of mass is the source of the algebraic constraints in \cite{benner2019,kralik1988,Behbahaninejad2008AMS} and is a property subsumed in these new models. This is characterized in by model transfer function properties at frequency zero \cite{sven_bob_control}. Each of the component models is shown to posses this property}

{This document is meant to serve as an accompaniment to the material in \cite{sven_bob_rob_gas,sven_bob_control}. Details of assumptions are provided there, along with motivations and derivations from PDEs. Accordingly, it is organized by component type for the first eight sections with Section~9 devoted to the presentation of the approach to interconnection of the components into systems amenable to use with \mbox{\textsc{Matlab}}'s control system design tools.}

\newpage
\tableofcontents
\newpage
\section*{Notation}
We denote pressures by $\check p(x,t)$ and mass flows by $\check q(x,t)$, where $x$ is the spatial dimension and $t$ the time. For the nominal values we write $\bar p(x)$ and $\bar q(x)$, whereas deviations are described by $p(x,t)=\check p(x,t)-\bar p(x)$ and $q(x,t)=\check q(x,t)-\bar q(x)$. Assume $x\in[0,X]$ for some $X>0$. We define $p_\ell\doteq p(0,t)$ and $p_r \doteq p(X,t)$, and accordingly for the mass flow. {The same convention holds for the temperature $T(x,t)$ where applicable.} Signals related to component $i$ for some $i\in\mathbb{N}$ are denoted by the additional subscript $i$. By~$\mathbf{0}$ we denote the zero matrix of appropriate size; $\mathbf{0}_n$ is the related \emph{row vector}, and $\mathbf{0}_{n,m}$ is the related matrix of dimension $n\times m$. The same notation is used for the matrix of ones, $\mathbf{1}$. For matrix $A$, we write $[A]_{i,j}$ for the matrix element in row $i$ and column $j$, and similarly for vectors. The following table lists the meaning of common parameters.
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|l|l|}
\hline
Symbol&Meaning&SI-unit\\
\hline\hline
$A$&Cross-sectional area&$\scriptstyle[\text{m}^2]$\\
\hline
$c_v,c_p$&Specific heat capacities&$\scriptstyle[\text{J/(kg K)}]$\\
\hline
$D$&Pipe inside diameter&$\scriptstyle[\text{m}]$\\
\hline
$D_o$&Pipe outside diameter&$\scriptstyle[\text{m}]$\\
\hline
$E$&Total energy&$\scriptstyle[\text{J}]$\\
\hline
$g$&Gravity constant&$\scriptstyle[\frac{\text{m}}{\text{s}^2}]$\\
\hline
$h(x)$&Pipe elevation from $x=0$ to $x=X$&$\scriptstyle[\text{m}]$\\
\hline
$P_i$& Pipe $i$\\
\hline
$R_s$&Specific gas constant&$\scriptstyle[\frac{\text{m}^2}{\text{s}^2\text{K}}]$\\
\hline
$T_0$&Constant temperature&$\scriptstyle[\text{K}]$\\
\hline
$V$&Volume&$\scriptstyle[\text{m}^3]$\\
\hline
$X$&Pipe length&$\scriptstyle[\text{m}]$\\
\hline
$z_0$&Constant compressibility factor&$\scriptstyle[1]$\\
\hline
$\lambda$&Friction factor&$\scriptstyle[1]$\\
\hline
\end{tabular}
\end{center}
\caption{Definitions of variables and their SI-units.}
\label{tab:parameters}
\end{table}
\section{Single pipe}

\begin{tcolorbox}[title=State space model single pipe model]
Under the assumptions below the one-dimensional pipe flow can be described by the following state space model:
\begin{align*}
\dot x
&= \begin{bmatrix}
0 & -\DpCoeff{}\\
\DqCoeffpr{} & \DqCoeffql{}
\end{bmatrix}x+\begin{bmatrix}
0 & \DpCoeff{}\\
\DqCoeffpl{} & 0
\end{bmatrix}u,\\

&=x,
\end{align*}
with $x=\begin{bmatrix}
p_{r}& q_{\ell}
\end{bmatrix}^\top$ as the state and output vector, and $u=\begin{bmatrix}
  p_{\ell} & q_{r}
\end{bmatrix}^\top$ as the input vector.
The coefficients are
\begin{subequations}\label{eq:pipe_coeff}
\begin{align}
\DpCoeff{}&=-\frac{R_sT_0z_0}{AX},\quad \DqCoeffpr{}=-\frac{A}{X},\\
 \DqCoeffpl{}&=\frac{A}{X}+\frac{\lambda R_sT_0z_0}{2DA}\frac{ \bar q\vert  \bar q\vert}{ \bar p_{\ell}^2}-\frac{Agh}{R_sT_0z_0X},\\
\DqCoeffql{}&=-\frac{\lambda R_sT_0z_0}{DA} \frac{|\bar q|}{ \bar p_{\ell}}.
\end{align}
\end{subequations}
\end{tcolorbox}

\subsection{Assumptions}\label{subsec:pipe_ass}
\begin{ass}\label{ass:isoth}
The change in temperature along the pipe is negligible.
\end{ass}
\begin{ass}\label{ass:pipeFlow} For the one-dimensional pipe flow,
\begin{enumerate}[(i)]
\item the cross-sectional area of each pipe segment is constant;
\item at each point in $x$-dimension averaged velocities suffice;
\item friction along the pipe can be approximated by the Darcy-Weisbach equation, see e.g. \cite{rennels2012pipe};
\item the compressibility factor is constant along the pipe;
\item capillary, magnetic and electrical forces on the fluid are negligible;
\item the gas velocity is much smaller than the speed of sound.
\end{enumerate}
\end{ass}
\subsection{Derivation}
Under the assumptions above Benner \emph{et al.} \cite{benner2019} derive a nonlinear model for an isothermal one-dimensional pipe flow. Towards a discretization and linearization of the related nonlinear dynamics, let the boundary conditions $p_\ell$ and $q_r$ be given, whereas
$p_r$ and $q$ are to be determined through the model.
Spatial discretization and linearization of \cite[Eq. 3.2]{benner2019} yields
\begin{subequations}\label{eq:linODEs}
\begin{align}
\dot p_r&=\DpCoeff{}(q_r-q_l)\label{eq:dprdt_iso}\\
\dot q_\ell&=
\DqCoeffpr{} p_r+\DqCoeffpl{} p_\ell+\DqCoeffql{} q_\ell,\label{eq:dqldt_iso}
\end{align}
\end{subequations}
or equivalently,
\begin{subequations}
\begin{align*}
\dot x_{t} &= \begin{bmatrix}
0 & -\DpCoeff{}\\
\DqCoeffpr{} & \DqCoeffql{}
\end{bmatrix}x_{t}+\begin{bmatrix}
0 & \DpCoeff{}\\
\DqCoeffpl{} & 0
\end{bmatrix}u_{t},
\end{align*}
\end{subequations}
with terms as described above.
\subsection{Conservation of mass}
The DC gain from $u$ to $x$ can be readily extracted,
\begin{align*}
-A^{-1}B=-\frac{1}{\DpCoeff{}\DqCoeffpr{}}\begin{bmatrix}
\DpCoeff{}\DqCoeffpl{} & \DpCoeff{}\DqCoeffql{}\\
0 & -\DpCoeff{} \DqCoeffpr{}
\end{bmatrix}=\begin{bmatrix}
-\frac{\DqCoeffpl{}}{\DqCoeffpr{}} & -\frac{\DqCoeffql{}}{\DqCoeffpr{}}\\
0 & 1
\end{bmatrix},
\end{align*}
which shows that $T_{qp}(0)=0$ and $\mathbf{1} T_{qq}(0)=1$, where
\begin{align*}
X(s)&=T(s)U(s)=\begin{bmatrix}
T_{pp}(s)&T_{pq}(s)\\T_{qp}(s)&T_{qq}(s)
\end{bmatrix}U(s).
\end{align*}
By definition \cite{sven_bob_control} this is conservation of mass.

\section{Branch}\label{sec:branch}
Consider a branching pipe geometry as shown in Figure \ref{fig:branch_sketch}.
\begin{figure}[ht]
    \centering
   \resizebox{.4\columnwidth}{!}{

  \includegraphics{figures/branch.pdf}}
 \caption{Branch}
    \label{fig:branch_sketch}
    \end{figure}

\begin{tcolorbox}[title=State space model simple branch]
Under the assumptions from Section \ref{subsec:pipe_ass} and matching conditions at the junction
\begin{align*}
p_{0,r}&=p_{1,\ell}=p_{2,\ell},\\%\label{eq:c_branch_p}\\
q_{0,r}&=q_{1,\ell}+q_{2,\ell}
\end{align*}
the state space realization of the one-dimensional pipe flow through the branch depicted above is as follows:

\begin{subequations}\label{eq:ss_branch}
\begin{align*}
\dot x&=
\begin{multlined}[t]
\begin{bmatrix}0&0&0&
-\DpCoeff{0}& \DpCoeff{0} & \DpCoeff{0}\\
0&0&0&0 & -\DpCoeff{1} & 0\\
0&0&0&0 & 0 & -\DpCoeff{2}\\
\DqCoeffpr{0} & 0 & 0&\DqCoeffql{0} &0&0\\
\DqCoeffpl{1} & \DqCoeffpr{1} & 0&0&\DqCoeffql{1}&0 \\
\DqCoeffpl{2} & 0 & \DqCoeffpr{2}&0&0&\DqCoeffql{2}
\end{bmatrix}x
+
\begin{bmatrix}
0 & 0 & 0\\
0 & \DpCoeff{1} & 0\\
0 & 0 & \DpCoeff{2}\\
\DqCoeffpl{0} & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}u
\end{multlined}\\
y&=\begin{bmatrix}
0 & 1 & 0 & 0 & 0&0\\
0 & 0 & 1 & 0 & 0&0\\
0 & 0 & 0 & 1& 0&0
\end{bmatrix}x,
\end{align*}
\end{subequations}
with state $x=\begin{bmatrix}
p_{0,r} & p_{1,r} & p_{2,r} & q_{0.\ell}&q_{1,\ell}&q_{2,\ell}
\end{bmatrix}^\top$, input $u=\begin{bmatrix}
p_{0,\ell} & q_{1,r} & q_{2,r}
\end{bmatrix}^\top$ and output $y=\begin{bmatrix}
p_{1,r}&p_{2,r}&q_{0,\ell}
\end{bmatrix}^\top.$
\end{tcolorbox}
\subsection{Derivation}
 Building on \eqref{eq:linODEs}, consider Assumption \ref{ass:pipeFlow} and suppose that additional losses at the intersection are accommodated  by appropriately adjusting the friction factor for all pipes.
Continuity at the boundary of pressure and mass flow means
\begin{subequations}\label{eq:constraints_branch}
\begin{align*}
p_{0,r}&=p_{1,\ell}=p_{2,\ell},\\%\label{eq:c_branch_p}\\
q_{0,r}&=q_{1,\ell}+q_{2,\ell}
\end{align*}
\end{subequations}
which sets the input signal of the single pipe model of pipe $P_0$ as the sum of two other state variables. The related matrices for a branch model immediately follow. The extension to a split into more than two pipes is direct and hence for brevity not presented.
\subsection{Conservation of mass}
The state-space realization above satisfies conservation of mass, which follows from the first row of the steady state equation $\mathbf{0}=Ax+Bu$ and conservation of mass of the single pipe model.

\section{Joint}
We present models first for the case of two joining pipes and then consider the more complex case of multiple pipes joining.
\subsection{Two joining pipes}\label{subsec:joint}
\begin{figure}[ht!]
    \centering
    \resizebox{.4\columnwidth}{!}{

   \includegraphics{figures/joint.pdf}}
 \caption{Joint of 2 pipes merging into one}
    \label{fig:2joiningPipes_sketch}
\end{figure}
\begin{tcolorbox}[title=Simple joint model]
Under the assumptions in Section \ref{subsec:pipe_ass} and boundary conditions
\begin{align*}
p_{1,r}=p_{2,r}=p_{0,\ell},\\
q_{1,r}+q_{2,r}=q_{0,\ell},
\end{align*}
two pipes merging into a single pipe are described by the following state space model:
{\small
\begin{align*}
\dot{\begin{bmatrix}
p_{0,r}\\q_{0,\ell}\\p_{1,r}\\q_{1,\ell}\\q_{2,\ell}
\end{bmatrix}}&=
\begin{multlined}[t]
\begin{bmatrix}
0&-\DpCoeff{0}&0&0&0\\
\DqCoeffpr{0}&\DqCoeffql{0}& \DqCoeffpl{0}&0&0\\
0&\frac{\DpCoeff{1}\DpCoeff{2}}{\DpCoeff{1}+\DpCoeff{2}}&0 & \frac{\DpCoeff{1}^2}{\DpCoeff{1}+\DpCoeff{2}}-\DpCoeff{1}&-\frac{\DpCoeff{1}\DpCoeff{2}}{\DpCoeff{1}+\DpCoeff{2}}\\
0&0&\DqCoeffpr{1}&\DqCoeffql{1}&0\\
0 & 0 & \DqCoeffpr{2} & 0 & \DqCoeffql{2}
\end{bmatrix}
\begin{bmatrix}
p_{0,r}\\q_{0,\ell}\\p_{1,r}\\q_{1,\ell}\\q_{2,\ell}
\end{bmatrix}\\
+
\begin{bmatrix}
0 & 0 & \DpCoeff{0}\\
0 & 0 & 0\\
0 & 0 & 0\\
\DqCoeffpl{1} & 0 & 0\\
0 & \DqCoeffpl{2} & 0
\end{bmatrix}
\begin{bmatrix}
p_{1,\ell}\\p_{2,\ell}\\q_{0,r}
\end{bmatrix},
\end{multlined}\\
\begin{bmatrix}
p_{0,r}\\q_{1,\ell}\\q_{2,\ell}
\end{bmatrix}&=\begin{bmatrix}
1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 1\\
\end{bmatrix}
\begin{bmatrix}
p_{0,r}\\q_{0,\ell}\\p_{1,r}\\q_{1,\ell}\\q_{2,\ell}
\end{bmatrix}.
\end{align*}
}
\end{tcolorbox}
\subsubsection{Derivation}
Building on \eqref{eq:linODEs}, each joining pipe can be described by
\begin{subequations}\label{eq:njoint_ode}
\begin{align}
\dot p_{j, r}&=\DpCoeff{j} (q_{j,r}-q_{j,\ell})\label{eq:pdot},\\
\dot q_{j,\ell}&=\DqCoeffpr{j} p_{j,r}+\DqCoeffpl{j} p_{j,\ell}+\DqCoeffql{j} q_{j,\ell}\label{eq:qdot},
\end{align}
\end{subequations}
where the subscript $j$ corresponds to pipe $P_j$, with coefficients as defined above and $j\in\{0,1,\dots,n\}$.
Every $p_{j,r}$ and $q_{j,\ell}$ are state variables and every $p_{j,\ell}$ and $q_{j,r}$ are input variables related to pipe $P_j$.

Consider Assumption \ref{ass:pipeFlow} and algebraic constraints
\begin{subequations}
\begin{align}
p_{1,r}=p_{2,r}=p_{0,\ell},\label{eq:ACpn=2}\\
q_{1,r}+q_{2,r}=q_{0,\ell},\label{eq:ACqn=2}
\end{align}
\end{subequations}
which express continuity of the pressure and mass flow. Assume further that additional losses at the intersection are accommodated  by appropriately adjusting the friction factor for all pipes.

Note that \eqref{eq:ACqn=2} cannot directly be incorporated into the state space formulation since it is not immediate how the signal $q_{0,\ell}$ is split between the flow of pipes 1 and 2. To eliminate variables $q_{1,r},q_{r,2}$, consider \eqref{eq:ACpn=2}, take the derivative and use \eqref{eq:pdot} to obtain
\begin{subequations}\label{eq:derived_aeq_n2}
\begin{align}
(\DpCoeff{1}+\DpCoeff{2})q_{1,r}&=\DpCoeff{1}q_{1,\ell}+\DpCoeff{2}(q_{0,\ell}-q_{2,\ell}),\label{eq:q1r_n2}\\
(\DpCoeff{1}+\DpCoeff{2})q_{2,r}&=\DpCoeff{2}q_{2,\ell}+\DpCoeff{1}(q_{0,\ell}-q_{1, \ell})\label{eq:q2r_n2},
\end{align}
\end{subequations}
which yields the state space description above.
\subsubsection{Conservation of mass}
The state-space realization above satisfies conservation of mass, which follows from the third row of the steady state equation $\mathbf{0}=Ax+Bu$ and conservation of mass of the single pipe model.

\subsection{Multiple joining pipes}
\begin{figure}[ht!]
    \centering
    \resizebox{.4\columnwidth}{!}{

   \includegraphics{figures/njoint.pdf}}
 \caption{Joint of $n$ pipes merging into one}
    \label{fig:xjoiningPipes_sketch}
\end{figure}
\begin{tcolorbox}[title=Joint model]
Consider the assumptions in Section \ref{subsec:pipe_ass} and matching conditions at the junction
\begin{align*}
p_{1,r}=p_{2,r}=\dots=p_{n+1,r}=p_{0,\ell},\\
q_{1,r}+q_{2,3}+\dots+q_{r,n}+q_{r,n+1}=q_{0,\ell}.
\end{align*}
 A joint of $n$ pipes with dynamics \eqref{eq:njoint_ode} can be described by the state-space model,
\begin{align*}
\dot{x}&=\begin{bmatrix}
\mathbf{0}_{2,2} & A_{12}\\
A_{21} & A_{22}
\end{bmatrix} x+\begin{bmatrix}
B_1\\
B_2
\end{bmatrix} u,\\
y&=\begin{bmatrix}
\begin{matrix}
1 & 0 & 0 & 0\\
0 & 0 & 0 & 1
\end{matrix} & \mathbf{0}_{2,n-1}\\\mathbf{0}_{n-1,4}&I_{n-1}
\end{bmatrix} x,
\end{align*}
where
\begin{align*}
u&=
\begin{bmatrix}
p_{1,\ell}&p_{2,\ell}&\dots&p_{n,\ell}&q_{0,r}
\end{bmatrix}^\top\in\mathbb{R}^{n+1},\\
x&=
\begin{bmatrix}
p_{0,r}&p_{1,r}&q_{0,\ell}&q_{1,\ell}&q_{2,\ell}&\dots&q_{n,\ell}
\end{bmatrix}^\top\in\mathbb{R}^{n+3}, \\
y&=
\begin{bmatrix}
p_{0,r}&q_{1,\ell}&q_{2,\ell}&\dots&q_{n,\ell}
\end{bmatrix}^\top\in\mathbb{R}^{n+1},
\end{align*}
\begin{align*}
A_{12}&=
\begin{bmatrix}
-\DpCoeff{0} & \mathbf{0}_{n}\\
a & -a\mathbf{1}_{n}
\end{bmatrix},\quad
A_{21}=
\begin{bmatrix}
\DqCoeffpr{0} & 0 & 0 & \dots & 0\\
\DqCoeffpl{0} &  \DqCoeffpr{1} &  \DqCoeffpr{2}& \dots & \DqCoeffpr{n}
\end{bmatrix}^\top,\quad
A_{22}=\diag(\DqCoeffql{0} ,\dots,\DqCoeffql{n}) \\
B_1&=
\begin{bmatrix}
\mathbf{0}_n& \DpCoeff{0} \\
\mathbf{0}_n& 0
\end{bmatrix},\quad
B_2=\begin{bmatrix}
\mathbf{0}_{n+1}\\
\begin{matrix}
\begin{matrix}
\DqCoeffpl{1}&&\\
&\ddots&\\
&&\DqCoeffpl{n}
\end{matrix}&\mathbf{0}_n^\top
\end{matrix}
\end{bmatrix},
\end{align*}
with
\begin{align*}
a&=\DpCoeff{1}\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}\right)^{-1}\prod_{\substack{i=2}}^n \DpCoeff{i}.
\end{align*}
\end{tcolorbox}
\subsubsection{Derivation}
We wish to extend the model for $n=2$ joining pipes to any finite number of joining pipes. In particular, first we would like to obtain an expression for $q_{j,r}$ only depending on state variables and sources, as in \eqref{eq:derived_aeq_n2}.
\begin{defin}
 We say \textup{sources} for inputs into the joint model that are not related to states of any other pipes via algebraic constraints. We say \textup{internal variables} for input variables that are not sources.
\end{defin}
We derive the model by induction. Consider the algebraic constraints for $n=3$,
\begin{align*}
p_{1,r}=p_{2,r}=p_{3,r}=p_{0,\ell},\\
q_{1,r}+q_{2,r}+q_{3,r}=q_{0,\ell}.
\end{align*}
Then,
\begin{align*}
\dot p_{1,r}&=\dot p_{2,r},\\
(\DpCoeff{1}+\DpCoeff{2})q_{1,r}&=\DpCoeff{1}q_{1,\ell}+\DpCoeff{2}(q_{0,\ell}-q_{2,\ell}-q_{3,r}),\\[1em]
\dot p_{3,r}&=\dot p_{2,r},\\
(\DpCoeff{2}+\DpCoeff{3})q_{3,r}&=\DpCoeff{3}q_{3,\ell}+\DpCoeff{2}(q_{0,\ell}-q_{2,\ell}-q_{1,r})\\
&=\begin{multlined}[t]
\DpCoeff{3}q_{3,\ell}+\DpCoeff{2}\Big(q_{0,\ell}-q_{2,\ell}-(\DpCoeff{1}+\DpCoeff{2})^{-1
[\DpCoeff{1}q_{1,\ell}+\DpCoeff{2}(q_{0,\ell}-q_{2,\ell}-q_{3,r})]\Big).\end{multlined}
\end{align*}
so that
\begin{align}
\begin{multlined}[t]
(\DpCoeff{1}\DpCoeff{2}+\DpCoeff{1}\DpCoeff{3}+\DpCoeff{2}\DpCoeff{3})q_{3,r}=\DpCoeff{1}\DpCoeff{2}(q_{0,\ell}-q_{1,\ell}-q_{2,\ell}
+(\DpCoeff{1}\DpCoeff{3}+\DpCoeff{2}\DpCoeff{3})q_{3,\ell}.\end{multlined}\label{eq:q3r_n3}
\end{align}
Similarly,
\begin{align}
\begin{multlined}[t]
(\DpCoeff{1}\DpCoeff{2}+\DpCoeff{1}\DpCoeff{3}+\DpCoeff{2}\DpCoeff{3})q_{1,r}=\DpCoeff{2}\DpCoeff{3}(q_{0,\ell}-q_{2,\ell}-q_{3,\ell}
+(\DpCoeff{1}\DpCoeff{2}+\DpCoeff{1}\DpCoeff{3})q_{1,\ell},\end{multlined}\label{eq:q1r_n3}\\
\begin{multlined}[t]
(\DpCoeff{1}\DpCoeff{2}+\DpCoeff{1}\DpCoeff{3}+\DpCoeff{2}\DpCoeff{3})q_{2,r}=\DpCoeff{1}\DpCoeff{3}(q_{0,\ell}-q_{1,\ell}-q_{3,\ell}
+(\DpCoeff{1}\DpCoeff{2}+\DpCoeff{2}\DpCoeff{3})q_{2,\ell}.\end{multlined}\label{eq:q2r_n3}
\end{align}
Comparing equations \eqref{eq:q3r_n3}--\eqref{eq:q2r_n3} with \eqref{eq:derived_aeq_n2} we recognize a certain pattern, which we explore next. Facilitated by observations above we derive a general statement.
\begin{prop}\label{prop:qkr}
Consider $n$ joining pipes with individual dynamics \eqref{eq:pdot} - \eqref{eq:qdot}. Let $q_{j,\ell},\, j\in\{0,1,\dots,n\},$ be state variables. The internal variables $q_{k,r},\,k\in\{1,2,\dots,n\},$ can be described by state variables,
\begin{align}
\begin{multlined}[t]
\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}\right)q_{k,r}=\prod_{\substack{i=1\\i\neq k}}^n \DpCoeff{i}\left(q_{0,\ell}-\sum_{\substack{i=1\\i\neq k}}^n q_{i,\ell}\right
+\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}-\prod_{\substack{i=1\\i\neq k}}^n\DpCoeff{i}\right)q_{k,\ell}\end{multlined}\label{eq:qkr}
\end{align}
\end{prop}
\begin{proof}
We prove the theorem by induction. We observe that \eqref{eq:qkr} holds for $n\in\{2,3\}$. Hence, assume it holds for some $n$ and for $n+1$ note that interconnections dictate the algebraic constraints
\begin{subequations}
\begin{align}
p_{1,r}=p_{2,r}=\dots=&p_{n+1,r}=p_{0,\ell},\label{eq:ac_press_n}\\
q_{1,r}+q_{2,3}+\dots+q_{r,n}+q_{r,n+1}&\doteq q^n_{0,\ell}+q_{r,n+1}=q_{0,\ell}.
\end{align}
\end{subequations}
Hence, by the algebraic constraints on the pressure and the pipe dynamics,
\begin{align*}
q_{n+1,r}&=\frac{1}{\DpCoeff{n+1}}(\dot p_{k,r}-\dot p_{n+1,r})+q_{n+1,r}\\
&=\frac{1}{\DpCoeff{n+1}}\left(\DpCoeff{k}(q_{k,r}-q_{k,\ell})-\dot p_{n+1,r}\right)+q_{n+1,\ell}+\frac{\dot p_{n+1,r}}{\DpCoeff{n+1}}\\
&=\frac{1}{\DpCoeff{n+1}}\DpCoeff{k}(q_{k,r}-q_{k,\ell})+q_{n+1,\ell},
\end{align*}
where without loss of generality $k\in\{1,2\dots,n-1\}$.
Then \eqref{eq:qkr} yields
{\small
\begin{align*}
\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}\right)q_{k,r}&=\begin{multlined}[t]
\prod_{\substack{i=1\\i\neq k}}^n \DpCoeff{i}\left(q^n_{0,\ell}-\sum_{\substack{i=1\\i\neq k}}^n q_{i,\ell}\right
+\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}-\prod_{\substack{i=1\\i\neq k}}^n\DpCoeff{i}\right)q_{k,\ell}
\end{multlined}\\
&=\begin{multlined}[t]
\prod_{\substack{i=1\\i\neq k}}^n \DpCoeff{i}\left(q_{0,\ell}-q_{n+1,r}-\sum_{\substack{i=1\\i\neq k}}^n q_{i,\ell}\right
+\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}-\prod_{\substack{i=1\\i\neq k}}^n\DpCoeff{i}\right)q_{k,\ell}
\end{multlined}\\
&=\begin{multlined}[t]
-\prod_{\substack{i=1}}^n \DpCoeff{i}\frac{1}{\DpCoeff{n+1}}(q_{k,r}-q_{k,\ell}
+ \prod_{\substack{i=1\\i\neq k}}^n \DpCoeff{i}\left(q_{0,\ell}-q_{n+1,\ell}-\sum_{\substack{i=1\\i\neq k}}^n q_{i,\ell}\right
+\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}-\prod_{\substack{i=1\\i\neq k}}^n\DpCoeff{i}\right)q_{k,\ell}
\end{multlined}\\
&=\begin{multlined}[t]
-\frac{1}{\DpCoeff{n+1}}\prod_{\substack{i=1}}^n \DpCoeff{i}q_{k,r
+ \prod_{\substack{i=1\\i\neq k}}^n \DpCoeff{i}\left(q_{0,\ell}-\sum_{\substack{i=1\\i\neq k}}^{n+1} q_{i,\ell}\right
+\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}-\prod_{\substack{i=1\\i\neq k}}^n\DpCoeff{i}\right
\left.+\frac{1}{\DpCoeff{n+1}}\prod_{\substack{i=1}}^n \DpCoeff{i}\right)q_{k,\ell}\end{multlined}.
\end{align*}
}
Move the first term of the right-hand side to the left, multiply the equation by $\DpCoeff{n+1}$ and observe that the left-hand side is
\begin{align*}
\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}\DpCoeff{n+1}+\prod_{\substack{i=1}}^n \DpCoeff{i}=\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^{n+1}\DpCoeff{i}+\prod_{\substack{i=1}}^n \DpCoeff{i}=\sum_{j=1}^{n+1}\prod_{\substack{i=1\\i\neq j}}^{n+1}\DpCoeff{i}.
\end{align*}
This identity can also be used for the last term on the right-hand side. Then,
\begin{align*}
\left(\sum_{j=1}^{n+1}\prod_{\substack{i=1\\i\neq j}}^{n+1}\DpCoeff{i}\right)q_{k,r}&=
\begin{multlined}[t]
\prod_{\substack{i=1\\i\neq k}}^{n+1} \DpCoeff{i}\left(q_{0,\ell}-\sum_{\substack{i=1\\i\neq k}}^{n+1} q_{i,\ell}\right
+ \left(\sum_{j=1}^{n+1}\prod_{\substack{i=1\\i\neq j}}^{n+1}\DpCoeff{i}-\prod_{\substack{i=1\\i\neq k}}^{n+1}\DpCoeff{i}\right)q_{k,\ell},
\end{multlined}
\end{align*}
which is equivalent to our induction hypothesis and thus concludes the proof.
\end{proof}
The state space realization above is a direct consequence of Proposition \ref{prop:qkr}, the pipe dynamics in \eqref{eq:njoint_ode}, the algebraic constraint of equal pressure at the intersection in \eqref{eq:ac_press_n}, and the observation that for the ODE of $p_{1,r}$,
\begin{align*}
\begin{multlined}[t]
\DpCoeff{1}\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}\right)^{-1}\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}-\prod_{\substack{i=2}}^n\DpCoeff{i}\right)-\DpCoeff{1
=-\DpCoeff{1}\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}\right)^{-1}\prod_{\substack{i=2}}^n \DpCoeff{i}.\end{multlined}
\end{align*}

\subsubsection{Conservation of mass}
As for $n=2$ joining pipes, the state-space realization above satisfies conservation of mass, which follows from the third row of the steady state equation $\mathbf{0}=Ax+Bu$ and conservation of mass of the single pipe model.
\section{Star junction}
Consider the pipe geometry illustrated in Figure \ref{fig:star_junction_sketch}, combining the branch and joint model to so called star junction.
\begin{figure}[ht!]
    \centering
    \resizebox{.4\columnwidth}{!}{

   \includegraphics{figures/star.pdf}}
 \caption{Star junction of $n+m$ pipes}
    \label{fig:star_junction_sketch}
\end{figure}
\begin{tcolorbox}[title=Star junction model]
Consider the assumptions in Section \ref{subsec:pipe_ass} and matching conditions at the junction
\begin{align*}
p_{1,r}=\dots=p_{n,r}=p_{n+1,\ell}=\dots=p_{n+m,\ell},\\
q_{1,r}+\dots+q_{r,n}=q_{n+1,\ell}+\dots+q_{n+m,\ell}.
\end{align*}
A star junction of $n$ pipes connecting to $m$ pipes with individual dynamics \eqref{eq:pdot} - \eqref{eq:qdot} can be represented by the state-space model,
\begin{align*}
\dot x_1&=A_1x_1+B_1u,\\
\dot x_2&=A_1x_2+B_2u,\\
\dot x_3&=A_1x_3+B_3u,\\
\dot x_4&=A_1x_4+B_4u,\\
y&=C(x_1+x_2+x_3+x_4),
\end{align*}
where
{\small
\begin{align*}
x_1&=x_2=x_3=x_4\\
&=\begin{multlined}[t]
[\begin{matrix}
p_{1,r}&p_{n+1,r}&p_{n+2,r}&\dots&p_{n+m,r}&q_{1,\ell}\end{matrix
\begin{matrix}
q_{2,\ell}&\dots&q_{n+m,\ell}
\end{matrix}]^\top\in\mathbb{R}^{n+2m+1},
\end{multlined}\\
u&=\begin{multlined}[t]
[\begin{matrix}
p_{1,\ell}&p_{2,\ell}&\dots&p_{n,\ell}&q_{n+1,r}&q_{n+2,r}\end{matrix
\begin{matrix}
\dots&q_{n+m,r}
\end{matrix}]^\top\in\mathbb{R}^{n+m},
\end{multlined} \\
y&=
\begin{multlined}[t]
[\begin{matrix}
p_{n+1,r}&p_{n+2,r}&\dots&p_{n+m,r}&q_{1,\ell}&q_{2,\ell}\end{matrix
\begin{matrix}
\dots&q_{n,\ell}
\end{matrix}]^\top\in\mathbb{R}^{n+m},\end{multlined}
\end{align*}
\begin{align*}
A_1&=
\begin{bmatrix}
\begin{array}{c}
\begin{matrix}
\mathbf{0}_{m+1} & -a & -a\mathbf{1}_{n-1} & a\mathbf{1}_{m}
\end{matrix}\\
\hline
\mathbf{0}_{n+2m,n+2m+1}
\end{array}
\end{bmatrix},
\quad
B_1=\mathbf{0}_{n+2m+1,n+m},
\end{align*}
\begin{align*}
A_2 &= \begin{bmatrix}
\begin{array}{c|c}
\mathbf{0}_{n+2m+1,n+m+1} &
\begin{array}{c}
\begin{matrix}
\mathbf{0}_{m}\\
\hline
\begin{matrix}
-\DpCoeff{n+1}&&\mathbf{0}\\
&\ddots&\\
\mathbf{0}&&-\DpCoeff{n+m}
\end{matrix}
\end{matrix}\\
\hline
\mathbf{0}_{n+m}
\end{array}
\end{array}
\end{bmatrix},\quad
B_2=
\begin{bmatrix}
\begin{array}{c}
\mathbf{0}_{n+m}\\
\hline
\begin{array}{c|c}
\mathbf{0}_{m,n}
&
\begin{matrix}
\DpCoeff{n+1}&&\mathbf{0}\\
&\ddots&\\
\mathbf{0}&&\DpCoeff{n+m}
\end{matrix}
\end{array}\\
\hline
\mathbf{0}_{n+m,n+m}
\end{array}
\end{bmatrix},
\end{align*}
\begin{align*}
A_3&=
\begin{bmatrix}
\begin{array}{c}
\mathbf{0}_{m+1,n+2m+1}\\
\hline
\begin{array}{c|c|c|c}
\begin{matrix}
\DqCoeffpr{1}\\
\vdots\\
\DqCoeffpr{n}
\end{matrix}
&
\mathbf{0}_{n,m}
&
\begin{matrix}
\DqCoeffql{1}&&\mathbf{0}\\
&\ddots&\\
\mathbf{0} &&\DqCoeffql{n}
\end{matrix}
&
\mathbf{0}_{n,m}
\end{array}\\
\hline
\mathbf{0}_{m,n+2m+1}
\end{array}
\end{bmatrix}, \quad
B_3=
\begin{bmatrix}
\begin{array}{c}
\mathbf{0}_{m+1,n+m}\\
\hline
\begin{array}{c|c}
\begin{matrix}
\DqCoeffpl{1}&&\mathbf{0}\\
&\ddots&\\
\mathbf{0}&&\DqCoeffpl{n}
\end{matrix}
&
\mathbf{0}_{n,m}
\end{array}\\
\hline
\mathbf{0}_{m,n+m}
\end{array}
\end{bmatrix},
\end{align*}
\begin{align*}
A_4&=\begin{multlined}[t]
\left[
\begin{matrix}
\begin{array}{c}
\mathbf{0}_{n+m+1,n+2m+1}\\
\hline
\begin{array}{c|c|c|}
\begin{matrix}
\DqCoeffpl{n+1}\\
\vdots\\
\DqCoeffpl{n+m}
\end{matrix}
&
\begin{matrix}
\DqCoeffpr{n+1}&&\mathbf{0}\\
&\ddots&\\
\mathbf{0} &&\DqCoeffpr{n+m}
\end{matrix}
&
\mathbf{0}_{m,n}
\end{array}
\end{array}
\end{matrix}\right
\left.
\begin{matrix}
\begin{array}{c}\\
\hline
\begin{matrix}
\DqCoeffql{n+1}&&\mathbf{0}\\
&\ddots&\\
\mathbf{0} &&\DqCoeffql{n+m}
\end{matrix}
\end{array}
\end{matrix}\right]
\end{multlined},\quad
B_4=\mathbf{0}_{n+2m+1,n+m},
\end{align*}
\begin{align*}
C&=\begin{bmatrix}
\mathbf{0}_{n+m} & I_{n+m} & \mathbf{0}_{n+m,m}
\end{bmatrix},
\end{align*}
}
\end{tcolorbox}

\subsubsection{Derivation}
Proposition \ref{prop:qkr} also facilitates a statement about a general star junction, which consists of $n$ pipes connecting to $m$ pipes, as illustrated in Figure \ref{fig:star_junction_sketch}. Towards a state-space realization and akin to the case of a joint, we wish to express internal variables in terms of state variables.
\begin{corollary}\label{col:pkr_qkr}
Consider a star junction with $n$ joining pipes connecting to $m$ branching pipes with individual dynamics \eqref{eq:pdot} - \eqref{eq:qdot}. Let $q_{i,\ell},\, i\in\{1,2,\dots,n+m\}$, and $p_{1,r}$ be state variables. The internal variables $q_{k,r}, \,k\in\{1,2,\dots,n\}$ and $p_{j,\ell},\,j\in\{n+1,n+2,\dots,n+m\},$ can be described by state variables through
\begin{align}
\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}\right)q_{k,r}&=
\begin{multlined}[t]
\prod_{\substack{i=1\\i\neq k}}^n \DpCoeff{i}\left(\sum_{i=n+1}^{n+m}q_{i,\ell}-\sum_{\substack{i=1\\i\neq k}}^n q_{i,\ell}\right
+\left(\sum_{j=1}^n\prod_{\substack{i=1\\i\neq j}}^n\DpCoeff{i}-\prod_{\substack{i=1\\i\neq k}}^n\DpCoeff{i}\right)q_{k,\ell},\end{multlined}\label{eq:qkr_star_junction}\\
p_{j,\ell}&=p_{1,r}\label{eq:pjl_star}.
\end{align}
\end{corollary}
\begin{proof}
Consider \eqref{eq:qkr} from Proposition \ref{prop:qkr} and note that therein that $q_{0,\ell}=\sum_{j=n+1}^{n+m}q_{j,\ell}$. The result for the first equation follows immediately. The second equation is a direct consequence of the algebraic constraint of all pressures being equal at the intersection.
\end{proof}
The state space realization above follows directly: the system related to $x_1$ describes the solution to $\dot p_{1,r}$ and directly follows from \eqref{eq:qkr_star_junction}. The system corresponding to $x_2$ describes $p_{j,r},\,j\in\{n+1,n+2,\dots,n+m\}$ and is immediately obtained by the dynamics \eqref{eq:pdot}. The systems for $x_3$ and $x_4$ relate to $q_{j,\ell},\,j\in\{1,2,\dots,n+m\}$ and are a consequence of the dynamics \eqref{eq:qdot} and the algebraic constraint \eqref{eq:pjl_star}.
\subsubsection{Conservation of mass}
Conservation of mass can be established from the branch and joint above and is omitted for brevity.

\section{Control valve}\label{sec:valve}
We present two models for control valves: firstly, we use a static gain (less or equal than 1) on the pressure and unity gain on mass flow; secondly, we provide a more complex model with the mass flow varying with in- and outlet pressure and cross-sectional area.
\subsection{Static model}
\begin{tcolorbox}[title=Static valve model]
For the static gain model let the states and inputs be:
\begin{align}
x=\begin{bmatrix}
p_r&q_\ell
\end{bmatrix}^\top,\quad u=\begin{bmatrix}
p_\ell & q_r
\end{bmatrix}^\top,
\end{align}
and define the static relationship
\begin{align*}
x=\begin{bmatrix}
k_v&0\\0 & 1
\end{bmatrix}u,
\end{align*}
where $k_v\in(0,1)$.
\end{tcolorbox}
Conservation of mass is immediate.
\subsection{Dynamic model}
Here we develop a model based on a static relationship commonly used for orifices combined with a first order low pass filter approximating the dynamics between command and actuation.

\begin{tcolorbox}[title=Dynamic valve model
\footnotemark] Assume an isentropic expansion and approximate the command-to-actuation relationship by a first order model. Define the state, about and input signals as
\begin{align*}
x=\begin{bmatrix}
A_o
\end{bmatrix},\quad y=\begin{bmatrix}
q_v
\end{bmatrix},\quad u=\begin{bmatrix}
u_v&p_{\ell}&p_{r}
\end{bmatrix}^\top.
\end{align*}
The dynamics are
\begin{subequations}\label{eq:valv_linmod}
\begin{align}
\dot x&=\begin{bmatrix}
-1/\tau
\end{bmatrix}x+\begin{bmatrix}
K/\tau & 0& 0
\end{bmatrix}u,\\
 y&=\begin{multlined}[t]\begin{bmatrix}
g_o(A_o,\bar p_{\ell},\bar p_{r})/A_o
\end{bmatrix}
+\begin{bmatrix}
0&\zeta_o(A_o,\bar p_{\ell},\bar p_{r})&\xi_o(A_o,\bar p_{\ell},\bar p_{r})
\end{bmatrix}u,\end{multlined}
\end{align}
\end{subequations}
where $g_0$ is the orifice equation from \eqref{eq:q-orifice}, $\tau$ and $K$ the time constant and gain from transfer function \eqref{eq:lpf_actuator} between the control command and actual actuation, $q_v$ the mass flow through the valve, $u_v\in(0,1)$ the control input, $A_o$ the cross-sectional area of the valve, and $\xi_o$ and $\zeta_o$ the linearization terms from \eqref{eq:orifice_lin_pr} and \eqref{eq:orifice_lin_pl}.
\end{tcolorbox}
\footnotetext{The dynamic valve model has not been fully tested in operational models.}
We note that conservation of mass does not apply to this model, as there are no mass flows as model inputs.

\subsubsection{Derivation}

The static mapping is justified by pipe time constants and the usual sampling time, which both vastly exceed the low time constants of the valve \cite{funk72_valve}.

In \cite[Equ. (3-34)]{mattingly}, assuming an isentropic expansion the mass flow through an orifice is approximated by
{\small
\begin{align}
q_o&=Cp_{o,\ell} A_o\sqrt{\frac{2}{R_sT_0z_0}\frac{\mu}{\mu-1}\left[\left(\frac{p_{o,r}}{p_{o,\ell}}\right)^{2/\mu}-\left(\frac{p_{o,r}}{p_{o,\ell}}\right)^{(\mu+1)/\mu}\right]},\label{eq:q-orifice}\\
&\doteq g_o(A_o,p_r,p_\ell)
\end{align}
}
where $C\doteq C_d/\sqrt{1-(D_0/D_1)^4}$ corrects for the head loss, with $C_d$ being the discharge coefficient, $\mu$ the ratio of specific heat, $\mu=c_p/c_v$ and $D_0(D_1)$ the orifice (pipe) diameter.

For the control valve let the cross-sectional area, $A_o$, from \eqref{eq:q-orifice} represent the manipulated variable. To approximate the dynamics of the actuator itself, i.e., the relationship between the control command and actual actuation, consider a first-order low pas filter with transfer function,
\begin{align}
K\frac{1}{\tau s+1},\label{eq:lpf_actuator}
\end{align}
where $\tau>0$ is the time constant, $K=A_{o,max}$ is the gain. The control input is constrained, i.e., $u_v\in(0,1)$, where $0$ is closed and $1$ is open.
Towards a linear model:
\begin{align}
\frac{\partial g_o(A_o,p_\ell,p_r)}{\partial p_r}&\doteq \xi_o(A_o,p_\ell,p_r),\label{eq:orifice_lin_pr}\\
\frac{\partial g_o(A_o,p_\ell,p_r)}{\partial p_\ell}&\doteq \zeta_o(A_o,p_\ell,p_r).\label{eq:orifice_lin_pl}
\end{align}
Combining these equations with \eqref{eq:lpf_actuator} yields the state space model above.

\section{Tank}
We next introduce two tank models: the first model approximates pressure changes while assuming a constant temperature; the second model additionally admits temperature changes.
\subsection{Isothermal tank}
\begin{tcolorbox}[title=Isothermal tank model]
Assume there are multiple in- and outlets at the tank and the temperature is constant. Under the assumptions below, for the pressure as the single state, $x=p$, and input vector
\begin{align*}
u=\begin{multlined}[t]\left[\begin{matrix}
q_{1,\ell}\dots&q_{n_\ell,\ell}&q_{1,r}&\dots&q_{n_r,r}
\end{matrix}\right],\end{multlined}
\end{align*}
with $n_\ell\in\mathbb{N}$ as the number of inlets and $n_r\in\mathbb{N}$ as the number of outlets, the linear state space model is
\begin{align*}
\dot x=\frac{R_s z_0T_0}{V}\begin{bmatrix}
\mathbf{1}_{n_\ell} & -\mathbf{1}_{n_r}
\end{bmatrix} u,
\end{align*}
with $V$ as the constant volume.
\end{tcolorbox}
\subsubsection{Assumptions}
\begin{ass}\label{ass:tank} Suppose:
\begin{enumerate}[label=\ref{ass:tank}(\roman*)]
\item there is perfect mixing inside the volume;\label{ass:mix}
\item the compressibility factor, $z$, changes negligibly or if necessary can be represented by a time-varying parameter;\label{ass:z}
\item the gas is described by the ideal gas equation; \label{ass:rge}
\end{enumerate}
\end{ass}
\subsubsection{Derivation}
For the pressure inside the constant volume, we have
\begin{align}
p&=\rho R_s T_0z_0,\nonumber\\
\dot p&=  R_s z_0T_0\dot \rh
 = R_s z_0T_0\frac{q}{V}
 = \frac{R_s z_0T_0}{V} \left(\sum_{j=1}^{n_\ell}q_{j,\ell} - \sum_{k=1}^{n_r}q_{k,r}\right)\label{eq:p_tank_iso}
\end{align}
where the first equation is the ideal gas equation. The derivative uses Assumption \ref{ass:z} and the last line conservation of mass: change in mass equals mass flow in minus mass flow out. The variables are: $\rho$ density; $\Sigma_jq_{j,\ell}$ ($\Sigma_kq_{k,r}$) sum of mass flowing in (out). The state space model above and conservation of mass follow directly.
\subsection{Non-isothermal tank}
We now drop the isothermal assumption and incorporate temperature dynamics. This also has an effect on the pressure dynamics themselves.

\begin{tcolorbox}[title=Non-isothermal tank model]
Consider Assumptions \ref{ass:tank} and \ref{ass:noniso_tank}. We obtain a nonlinear representation relating the entering mass flows and temperatures, $q_{i,\ell}$ and $\;T_{i,\ell},i=\{1,\dots,n_\ell\}$, and exiting mass flows, $q_{i,r},i=\{1,\dots,n_r\}$, to the pressure and temperature inside the tank, $p$ and $T$:
\begin{align*}
\dot p &= \frac{R_s z_0\mu}{V} \left(\sum_{j=1}^{n_\ell}q_{j,\ell}T_{j,\ell} - \sum_{k=1}^{n_r}q_{k,r} T\right),\\
\dot T &=\begin{multlined}[t]\frac{R_sTz_0}{pVc_v}\left(\sum_{j=1}^{n_\ell}q_{i,j}\left(c_{p,j}T_j -c_vT\right)- R_s T\sum_{k=1}^{n_r}q_{o,k}\right).\end{multlined}
\end{align*}
Therefore, with $x=\begin{bmatrix}
p & T
\end{bmatrix}^\top$ and
\begin{align*}
u=\begin{multlined}[t]\left[\begin{matrix}
q_{1,\ell}&T_{1,\ell}&\dots&q_{n_\ell,\ell}&T_{n_i,\ell}\end{matrix}\qquad \right
\left.\begin{matrix}&q_{1,r}&\dots&q_{n_r,r}
\end{matrix}\right],\end{multlined}
\end{align*}
we can use the linearized dynamics for pressure and temperature to generate a linear state space model.
\end{tcolorbox}

\subsubsection{Assumptions}
\begin{ass}\label{ass:noniso_tank} Suppose:
\begin{enumerate}[label=\ref{ass:noniso_tank}(\roman*)]
\item potential and kinetic energy are negligible;\label{ass:energy}
\item the reference states for internal energy and enthalpy  are set to zero at zero degree Kelvin; \label{ass:t0u0}
\item specific heats are constant;\label{ass:cv}
\item the heat flux to the environment is negligible. \label{ass:Qdot_zero}
\end{enumerate}
\end{ass}

\subsubsection{Derivation}

The pressure dynamics are akin to the isothermal case but differ as the temperature is not a constant anymore:
\begin{align}
\dot p &=  R_s z_0\left(\frac{\sum_{j=1}^{n_\ell}q_{j,\ell} - \sum_{k=1}^{n_r}q_{k,r}}{V} T+\rho \dot T\right)\nonumber\\
 &= R_s z_0\left(\frac{\sum_{j=1}^{n_\ell}q_{j,\ell} - \sum_{k=1}^{n_r}q_{k,r}}{V} T+\frac{p}{R_sTz_0} \dot T\right)\label{eq:pdot_w_Tdot}
\end{align}
We next derive an expression for $\dot T$, which we will plug back in.

\paragraph*{Temperature dynamics}
As mentioned in Assumptions~\ref{ass:rge} and \ref{ass:t0u0} we assume ideal gas and select the reference states for the internal energy and enthalpy to be zero at zero degree Kelvin: $T_{\rm ref}=0\implies U_{\rm ref},H_{\rm ref}=0$ (see for example \cite[Ch. 3.6.3]{moranfundamentals}). Then, the derivative of the total energy of a system, $E$, is \cite[Equ. (2.27)]{moranfundamentals}:
\begin{align}
\frac{d}{dt}E&\stackrel{Ass. \ref{ass:energy}}{=}\frac{d}{dt}U\\%=\frac{d}{dt}(mu)\nonumber\\
&\stackrel{Ass. \,\ref{ass:mix}, \ref{ass:t0u0}}{=}\frac{d}{dt}(m c_vT)\nonumber\\
&\stackrel{Ass. \,\ref{ass:cv}}{=}\left(\sum_{j=1}^{n_\ell}q_{j,\ell} - \sum_{k=1}^{n_r}q_{k,r}\right)c_vT+mc_v\dot T\label{eq:Udot_tot_energy}
\end{align}
with $U$ being the internal energy. Moreover, conservation of energy implies \cite[Equ. (4.9)-(4.12)]{moranfundamentals}
\begin{align}
\frac{d}{dt}U&\stackrel{Ass. \ref{ass:energy}}{=} \dot Q+ \sum_{j=1}^{n_\ell}q_{j,\ell}h_j - \sum_{k=1}^{n_r}q_{k,r}h_k\nonumber\\
&\stackrel{Ass. \ref{ass:mix}}{=}\dot Q +c_{p}\sum_{j=1}^{n_\ell}q_{j,\ell}T_j - c_pT\sum_{k=1}^{n_r}q_{k,r},\label{eq:Udot_conserve}
\end{align}
where $h$ is the specific enthalpy, and $\dot Q$ is the rate of heat transfer to the environment, which is a function of the temperature (deviations) inside and outside the volume. Usually, empirical formulations are used for conduction, convection and radiation (see \cite[Ch. 2]{moranfundamentals}). Here for brevity we assume it is negligible (cf. Assumption \ref{ass:Qdot_zero}).
Combining \eqref{eq:Udot_tot_energy} and \eqref{eq:Udot_conserve} yields
\begin{align}
\dot T&=\begin{multlined}[t]\frac{1}{mc_v}\left[c_vT \left(\sum_{k=1}^{n_r}q_{k,r}-\sum_{j=1}^{n_\ell}q_{j,\ell}\right)+\sum_{j=1}^{n_\ell}q_{j,\ell}c_{p}T_j \right
\left.- c_pT\sum_{k=1}^{n_r}q_{k,r}\right]\end{multlined}\nonumber\\
&=\begin{multlined}[t]\frac{1}{mc_v}\left[\sum_{j=1}^{n_\ell}q_{j,\ell}\left(c_{p}T_j -c_vT\right)\ - R_s T\sum_{k=1}^{n_r}q_{k,r}\right]\end{multlined}\nonumber\\
&=\begin{multlined}[t]\frac{R_sTz_0}{pVc_v}\left[\sum_{j=1}^{n_\ell}q_{j,\ell}\left(c_{p}T_j -c_vT\right)- R_s T\sum_{k=1}^{n_r}q_{k,r}\right],\end{multlined}\label{eq:Tdot}
\end{align}
which represents the nonlinear temperature dynamics from the model. Note that \eqref{eq:Tdot} also coincides with that in \cite[Ch. 13.4.5]{egeland2002}. Using this equation in \eqref{eq:pdot_w_Tdot} yields the stated nonlinear dynamics for the pressure.

\section{Valve manifold}\label{sec:composite_models}
Consider a valve manifold as illustrated in Figure~\ref{fig:manifold}.
\begin{figure}[ht!]
	\centering
   \resizebox{.7\columnwidth}{!}

   {\includegraphics{figures/vmani.pdf}}
 \caption{Manifold}
    \label{fig:manifold}
\end{figure}
The following model is derived by using the tank models from above for Tanks 1 and 2, the single pipe model for each pipe-valve-pipe connection with adjusted friction factor, and the orifice equation for the entrances of the tanks. We provide more details in the derivation subsequently.
\begin{tcolorbox}[title=Valve manifold model]
Consider Assumptions \ref{ass:pipeFlow}--\ref{ass:tank} (for isothermal pipes and tanks) and approximate the entering mass flow into Tanks~1 and 2 as an isentropic expansion described by orifice equation \eqref{eq:q-orifice}. Further, let each pipe-valve-pipe connection be approximated by single pipe models with adjusted friction factors. Let the state, output and input vectors be defined as
\vspace{-5pt}
\begin{align*}
x &= \begin{bmatrix}
p_{T_1}&p_{1,r}& q_{1,\ell}&p_{2,r}&q_{2,\ell}&p_{T_2}
\end{bmatrix}^\top,\\
y &= \begin{bmatrix}
p_{T_2}&q_{T_1,\ell}
\end{bmatrix}^\top,\\
u&=\begin{bmatrix}
p_{T_1,\ell}&q_{T_2,r}
\end{bmatrix}^\top,
\end{align*}
where subscript $i=\{1,2\}$ refers to middle section $P_{ia}-V_i-P_{ib}$ and other terms as shown in Figure \ref{fig:manifold}.
 Then the overall dynamics are described by
{
\begin{subequations
\begin{align*}
\dot x&=
\begin{multlined}[t]
\begin{bmatrix}
\alpha_{T_1}\xi_{T_1}&0&-\alpha_{T_1}&0&-\alpha_{T_1}&0\\
0 &\alpha_1\zeta_{T_2}&-\alpha_1&0&0&\alpha_1\xi_{T_2}\\
\kappa_1&\beta_1&\gamma_1&0&0&0\\
0&0&0&\alpha_2\zeta_{T_2}&-\alpha_2&\alpha_2\xi_{T_2}\\
\kappa_2&0&0&\beta_2&\gamma_2&0\\
0&\alpha_{T_2}\zeta_{T_2}&0&\alpha_{T_2}\zeta_{T_2}&0&2\alpha_{T_2}\xi_{T_2}
\end{bmatrix}
+\begin{bmatrix}
\alpha_{T_1}\zeta_{T_1}&0&0&0&0&0\\
0&0&0&0&0&-\alpha_{T_2}
\end{bmatrix}^\top u,\end{multlined}\\
y& = \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 1\\
\xi_{T_1} & 0 & 0 & 0 & 0 & 0
\end{bmatrix}x+\begin{bmatrix}
0&0\\ \zeta_{T_1}&0
\end{bmatrix}u,
\end{align*}
\end{subequations}
}
where $\xi_{T_1}=\xi_{T_1}(\bar p_{T_1,\ell},\bar p_{T_1})$ from \eqref{eq:orifice_lin_pr} and $\zeta_{T_1}=\zeta_{T_1}(\bar p_{T_1,\ell},\bar p_{T_1})$ from \eqref{eq:orifice_lin_pl} are related to the linearized orifice equation for the entrance into Tank 1, and accordingly for Tank~2. The parameters $\DpCoeff{i},\DqCoeffpr{i},\DqCoeffpl{i}$ and $\DqCoeffql{i}, i=\{1,2,3\}$ are as defined in \eqref{eq:pipe_coeff} for pipes, whereas $\alpha_{T_i}=\frac{R_sz_0T_0}{V_{T_i}},i=\{1,2\}$.
\end{tcolorbox}

\subsection{Derivation}
The model is based on the assumptions of each component, stated in the corresponding sections above. The primary motivation for including the orifice model for the entrance but not the exit is to obtain compatible boundary conditions. Our focus here is application and control orientation; an experimental validation is out of the scope of this work. {Without loss of generality}, we assume a positive flow, two middle sections, and that all outlets of Tank~1 and inlets of Tank~2 are of identical geometry, resp.
\paragraph*{Tanks}
For Tank~1, \eqref{eq:p_tank_iso} from the tank model above yields
\begin{align}
\dot p_{T_1}&=\alpha_{T_1}(q_{T_1,\ell}-q_{T_1,1,r}-q_{T_1,2,r}),\label{eq:T1-com}
\end{align}
where $q_{T_1,\ell}$ is the mass flow at the inlet of Tank~1 and $q_{T_1,i,r}$ is the mass flow at its $i^{\text{th}}$ outlet. We interpret the tank entrance as an orifice, and as in \eqref{eq:orifice_lin_pr} and \eqref{eq:orifice_lin_pl} define
{\small
\begin{align}
\frac{\partial g_{o}(p_{T_1},p_{T_1,\ell})}{\partial p_{T_1}}\doteq \xi_{T_1}, \quad \frac{\partial g_{o}(p_{T_1},p_{T_1,\ell})}{\partial p_{T_1,\ell}}\doteq \zeta_{T_1}.\label{eq:coeff_tank}
\end{align}}

Therefore, with \eqref{eq:T1-com} -- \eqref{eq:coeff_tank}, we obtain a model for Tank~1:
\begin{align*}
x&=\begin{bmatrix}
p_{T_1}
\end{bmatrix},\\
y&=\begin{bmatrix}
p_{T_1}&q_{T_1,\ell}
\end{bmatrix}^\top,\\
u&=\begin{bmatrix}
p_{T_1,\ell}&q_{T_1,1,r}&q_{T_1,2,r},
\end{bmatrix}^\top
\end{align*} with dynamics
{\small
\begin{subequations}\label{eq:tank1_model}
\begin{align}
\dot x&=\begin{bmatrix}
\alpha_{T_1}\xi_{T_1}
\end{bmatrix}x+\begin{bmatrix}
\alpha_{T_1}\zeta_{T_1} & -\alpha_{T_1} & -\alpha_{T_1}
\end{bmatrix}u,\\
 y&=\begin{bmatrix}
1 \\ \xi_{T_1}
\end{bmatrix}x+\begin{bmatrix}
0&0&0\\ \zeta_{T_1}&0 &0
\end{bmatrix}u.
\end{align}
\end{subequations}
}
Similarly, for Tank~2 we have:
\begin{align*}
x&=\begin{bmatrix}
p_{T_2}
\end{bmatrix},\\
y&=\begin{bmatrix}
p_{T_2}&q_{T_2,1,\ell}&q_{T_2,2,\ell}
\end{bmatrix}^\top\\
u&=\begin{bmatrix}
p_{T_2,1,\ell}&p_{T_2,2,\ell}&q_{T_2,r}
\end{bmatrix}^\top,
\end{align*}
with dynamics
{\small
\begin{subequations}\label{eq:tank2_model}
\begin{align}
\dot x&=\alpha_{T_2}(\xi_{T_2}+\xi_{T_2})
x+\begin{bmatrix}
\alpha_{T_2}\zeta_{T_2} & \alpha_{T_2}\zeta_{T_2} & -\alpha_{T_2}
\end{bmatrix}u,\\
 y&=\begin{bmatrix}
1 \\ \xi_{T_2}\\\xi_{T_2}
\end{bmatrix}x+\begin{bmatrix}
0&0&0\\ \zeta_{T_2}&0 &0\\ 0&\zeta_{T_2}&0
\end{bmatrix}u.
\end{align}
\end{subequations}
}
\paragraph*{Composite valve manifold model}
The state-space models for Tanks~1 and 2 in \eqref{eq:tank1_model} and \eqref{eq:tank2_model} as well as the single pipe model from \eqref{eq:linODEs} for each connection, $P_{1a}-V_1-P_{1b}$ and $P_{2a}-V_2-P_{2b}$, resp., share the following boundary conditions,
\vspace{-10pt}
\begin{align*}
p_{T_1,i,r}=p_{ia,\ell},\quad p_{T_2,i,\ell}=p_{ib,\ell}\\
q_{T_1,i,r}=q_{ia,\ell},\quad q_{T_2,i,\ell}=q_{ib,\ell},
\end{align*}
for $i=\{1,2\}$. Connecting tank and pipe models accordingly generates the  model stated above.
{
\section{Compressor}\label{sec:compressor}
Similar to the valve modeling, we present a static gain model as well as a more complex dynamic model for the compressor.
\subsection{Static model}
\begin{tcolorbox}[title=Static compressor model]
For the static gain model let the states and inputs be :
\begin{align}
x=\begin{bmatrix}
p_r&q_\ell
\end{bmatrix}^\top,\quad u=\begin{bmatrix}
p_\ell & q_r
\end{bmatrix}^\top,
\end{align}
where the suction side corresponds to subscript $\ell$ and discharge side to subscript $r$.
We define the static relationship
\begin{align*}
x=\begin{bmatrix}
k_c&0\\0 & 1
\end{bmatrix}u,
\end{align*}
where $k_c>1$.
\end{tcolorbox}
Conservation of mass is immediate.
\subsection{Dynamic model}
\begin{figure}[ht]
    \centering

   {\includegraphics{figures/compr.pdf}}
 \caption{Compressor system consisting of compressor, duct and plenum. The compression is isentropic, whereas duct and plenum are isothermal. We consider boundary conditions $p_1,q_{3,r}$ and $T_1$ as known. Variables at internal boundaries are continuous.}
    \label{fig:compressor}
\end{figure}
Consider a compressor system as shown in Figure \ref{fig:compressor}, which consists of a compressor, a duct and a plenum. Similar to \cite{greitzer1976} and \cite{egeland2002}, we model each of these elements separately and then connect them via boundary conditions, akin to the valve manifold model above.
\begin{tcolorbox}[title=Dynamic compressor model]
Consider Assumptions~\ref{ass:pipeFlow} (for duct), \ref{ass:tank} (for isothermal plenum) and \ref{ass:compr} (for compression) below.
Define state, input and output vectors as
\begin{align}
x&= \begin{bmatrix}
 p_3&
 q_2
\end{bmatrix}^\top,\\
u &= \begin{bmatrix}
p_1&q_{3,r}&T_1& \omega
\end{bmatrix} ^\top,\\
y &= \begin{bmatrix}
p_3&q_{2} & T_3
\end{bmatrix} ^\top,
\end{align}
with terms as shown in Figure \ref{fig:compressor}, and $\omega$ the compressor speed. The the nonlinear dynamics are
\begin{align}
\dot x& = \begin{bmatrix}
\frac{R_sT_{2}z_0}{V_p}(q_{2}-q_{3,r})\\
\frac{A_2}{L_2}(\Phi(q_2,\omega)-p_3)
\end{bmatrix}\\
y&=\begin{bmatrix}
x\\
T_1\left(\frac{\Phi(q_2,\omega)}{p_1}\right)^{(\eta-1)/\eta}
\end{bmatrix},
\end{align}
where $\Phi(q_2,\omega)=p_{2,\ell}$ is the suction pressure after the isentropic compression modeled as a static map. A linear state space model can be derived by linearization.
\end{tcolorbox}
\subsubsection{Assumptions}
\begin{ass}\label{ass:compr} Suppose:
\begin{enumerate}[label=\ref{ass:compr}(\roman*)]
\item The compression is isentropic.
\item The discharge pressure of the compressor, $p_{2,\ell}$, is defined by a static relation between speed, $\omega$, suction pressure, $p_1$, and mass flow, $q_2$: $\Phi(q_2,\omega): \;\mathbb{R}_{>0}\times \mathbb{R}_{>0}\to\mathbb{R}_{>0}$.
\item The friction in the duct is negligible.
\item The temperature change in the duct is negligible.
\item The gas in the duct and plenum is perfectly mixed.
\end{enumerate}

\end{ass}

\subsubsection{Derivation}
For the plenum we obtain from the isothermal tank model in \eqref{eq:p_tank_iso},
\begin{align}
\dot p_3 = \frac{R_sT_{2}z_0}{V_p}(q_{2}-q_{3,r}).\label{eq:p3dot}
\end{align}
For the duct, by \eqref{eq:dqldt_iso} and boundary condition $p_{2,r}=p_3$,
\begin{align}
\dot q_2 = \frac{A_2}{L_2}(p_{2,\ell}-p_3).
\end{align}
The suction pressure after the isentropic compression is modeled as a static map, $p_{2.\ell}=\Phi(q_2,\omega)$, with compressor speed $\omega$. The mapping is usually based on empirical formulas \cite{rennels2012pipe}, or compressor geometry \cite[Ch. 13]{egeland2002}. It follows that
\begin{align}
\dot q_2 = \frac{A_2}{L_2}(\Phi(q_2,\omega)-p_3).\label{eq:q2dot}
\end{align}
By the hypothesis of isentropic compression and isothermal duct and plenum,
\begin{align}
T_3=T_1\left(\frac{p_{2,\ell}}{p_1}\right)^{(\eta-1)/\eta}=T_1\left(\frac{\Phi(q_2,\omega)}{p_1}\right)^{(\eta-1)/\eta},\label{eq:T3}
\end{align}
with isentropic coefficient $\eta$. The model above immediately follows. Note that conservation of mass is immediate: by hypothesis and \eqref{eq:p3dot}, $\bar q_{1}=\bar q_{2}=\bar q_{3,r}$.
}
{
\section{Heat exchanger}
We provide a heat exhanger model based on the non-isothermal one-dimensional pipe model presented in \cite{sven_bob_rob_gas}. The assumptions are therefore similar to those for the pipe in Section \ref{subsec:pipe_ass}. Given the fact that under usual operating conditions no phase {change} occurs {within the gas} our model is similar to that in \cite{Rasmussen2004} in the context of air conditioning systems. While we provide a nonlinear formulation, linearization around the nominal operating point directly leads to a linear state-space realization.

{Our rudimentary model replaces the heat exchanger with a single pipe, with the geometric and conductivity properties} accommodated by adjusting the heat transfer coefficient, $k_{\text{rad}}$, and friction factor, $\lambda${, to parametrize the exchanger design and nominal operating point}. The temperature output of the model can be used to set the temperature parameter for {subsequent downstream} isothermal {(or other)} pipe models.
\begin{tcolorbox}[title=Heat exchanger]
Considering Assumption \ref{ass:pipeFlow}, if the heat flux {is} dominated by radial conduction we define the state and input vectors {to be}
\begin{align*}
x&=\begin{bmatrix}
p_r & q_\ell & T_r
\end{bmatrix},\\
u &= \begin{bmatrix}
p_\ell & q_r & T_\ell
\end{bmatrix}.
\end{align*}
Then, the nonlinear dynamics of the heat exchanger are governed by
\begin{subequations}\label{eq:3d_model_eq}
\begin{align}
\dot{p}_r&=\begin{multlined}[t] \frac{R_sz_0}{Ac_v}\left[k_\text{rad}  \pi D_o (T_\text{amb}-T_r)
-\frac{q_r-q_\ell}{L}T_r\left(c_v+R_sz_0\right)+\frac{p_r-p_\ell}{L} \frac{R_sz_0T_rq_r}{p_r}\right.\\
\left.-\frac{T_r-T_\ell}{L}q_r\left(c_v+R_sz_0\right)+\frac{\lambda R_s^2z_0^2T_r^2q_r^2|q_r|}{2DA^2p_r^2}\right]
\end{multlined}\label{eq:ode_p_noniso}
\\
\dot{q}_\ell&=-A\frac{p_r-p_\ell}{L}-\frac{\lambda R_sT_\ell z_0}{2DA}\frac{q_\ell\vert q_\ell\vert}{p_\ell}-\frac{Ag}{R_sT_\ell z_0} \frac{d h}{dx} p_\ell\label{eq:ode_q_noniso}\\
\dot{T}_r&=\begin{multlined}[t] \frac{R_sz_0T_r}{Ac_vp_r}\left[k_\text{rad}  \pi D_o (T_\text{amb}-T_r)-\frac{q_r-q_\ell}{L}T_rR_sz_0
+\frac{p_r-p_\ell}{L} \frac{R_sz_0T_rq_r}{p_r}-\frac{T_r-T_\ell}{L}q_r\left(c_v+R_sz_0\right)\right.\\
\left.+\frac{\lambda R_s^2z_0^2 T_r^2q_r^2|q_r|}{2DA^2p_r^2}\right]\end{multlined}\label{eq:ode_T_noniso
\end{align}
\end{subequations}
\end{tcolorbox}

\subsection{Derivation}
{This is the spatially discretized model of non-isothermal pipe flow developed} in \cite{sven_bob_rob_gas}. {As remarked above, the heat transfer coefficient, $k_\text{rad}$, is deliberately non-negligible and the friction factor, $\lambda$, can be significant depending on exchanger geometry.}
}
\section{Interconnections}
Suppose each component is described by a state space model with signals describing pressures, $p$, and mass flows, $q$. In this section we show how these component models can be interconnected to build an entire network model. Towards this goal, we recall interconnection rules from \cite{sven_bob_control} which translate allowable boundary conditions to signal flows.

\subsection{Directed pipe connections and `ports'}
The signal flow graph models derived later have directions associated with each signal. Thus, $p_\ell$ and $q_r$ are input signals, indicating that they are specified from outside the component, and $p_r$ and $q_\ell$ are output signals, meaning that they are determined by the component itself and the input signals. The spatially localized connection sites, however, possess one input signal and one output signal, namely $(p_\ell,q_\ell)$ at the left end and $(p_r,q_r)$ at the right. We further appropriate circuit terminology and identify two distinct location types, which we term \textit{ports}. As in \cite{benner2019}, every element in our interconnected system presents signal interfaces to other elements and to the outside world.

\begin{defin}[Ports]
\begin{description}
\item
\item[$\boldsymbol p$-port]  of a component possesses two signals: an input pressure signal $p_\ell$ and an output flow signal $q_\ell$.
\item[$\boldsymbol q$-port] of a component possesses two signals: an input flow signal $q_r$ and an output pressure signal $p_r$.
\end{description}
\end{defin}

Internal series connection of two components, 1 and 2, will involve the cascading of signals $p_{2,\ell}=p_{1,r}$ and $q_{1,r}=q_{2,\ell}$ at the junction point. This describes a $p$-port to $q$-port connection. Likewise, connection to the outside of the network must respect the type and causality of the signals. These rules are specified below.

\subsection{Interconnection rules}
\begin{tcolorbox}[title=Interconnection rules]
\begin{enumerate}[label=\Roman*.]
\item \vskip -2mm
Connections are permitted only between:
\begin{enumerate}[label=\roman*.]
\item a $p$-port and a $q$-port, or
\item a $p$-port and an external pressure source/input signal plus an external flow sink/output signal, or
\item a $q$-port and an external flow source/input signal plus an external pressure sink/output signal.
\end{enumerate}
\item Pressure input signals must connect to pressure output signals, and flow input signals must connect to flow output signals.
\item Connection of one variable of a port requires connection of the other.
\item All ports must be connected and algebraic loops avoided.
\end{enumerate}
\end{tcolorbox}

These rules conform to the connections examined in \cite{sven_bob_control} to formulate the systematic interconnection of state-space models. Each component model possesses input and output signals and, in Proposition~3 \cite{sven_bob_rob_gas}, it is shown how a (possibly non-minimal) state-space realization of the interconnection of gas system elements can be directly constructed with the above rules. This construction replaces and extends the graph-theoretic DAE methods of \cite{benner2019} and yields a new input-output transfer function satisfying Mason's Gain Formula, \cite[Proposition~4]{sven_bob_rob_gas}.

\subsection{Matrix methodology}
Per \cite{sven_bob_rob_gas}, we begin by stacking the state-space models of the individual network components:
\begin{align}\label{eq:totalSystem}
\dot{ x}=A x+B w,\quad  y=C x+D w
\end{align}
where
\begin{subequations}\label{eq:matrices_stacked_up}
\begin{align}
A&=\blkdiag(A^{(1)}, A^{(2)}, \dots, A^{(N)}),\\
B&=\blkdiag(B^{(1)}, B^{(2)}, \dots, B^{(N)}),\\
C&=\blkdiag(C^{(1)}, C^{(2)}, \dots, C^{(N)}),\\
D&=\blkdiag(D^{(1)}, D^{(2)}, \dots, D^{(N)}),
\end{align}
\end{subequations}
with superscript $(i),i=\{1,\dots,N\},$ denoting the respective matrix of component $i$, and $x\in\mathbb{R}^{n_x},w\in\mathbb{R}^{n_w}$ and $y\in\mathbb{R}^{n_y}$.
Interconnections and external sources $u\in\mathbb{R}^{n_u}$ {and sinks $z\in\mathbb{R}^{n_z}$} are described by
\begin{align}\label{eq:FG_network}
 w = F y+G u,\;\;\; z = Hx+J u,
\end{align}
with {structured matrices} $[F,G,H,J]$ {with 0-1 elements}:
\begin{align}\label{eq:connection_matrices}
[F]_{i,j}&=\begin{cases}
1, & \text{if } [ y]_j=[ w]_i,\\
0, & \text{otherwise}.
\end{cases}
\end{align}
\begin{tcolorbox}[title=Matrix interconnection methodology]
A (perhaps non-minimal) state-space realization of \eqref{eq:totalSystem} and \eqref{eq:FG_network} is given by
\begin{align}\label{eq:matrix_framework}
x=\bar A  x+\bar B  u, \quad  y=\bar C  x+\bar D  u,
\end{align}
where
\begin{align*}
\bar A&=A+BF(I-DF)^{-1}C, \quad \bar C=(I-DF)^{-1}C,\\
\bar B&=B\left[I+F(I-DF)^{-1}D\right]G,\quad \bar D=(I-DF)^{-1}DG.
\end{align*}
\end{tcolorbox}
We next show an example applying the methodology.
\subsubsection{Example: two pipes in series}
Consider two pipes in series, 1 and 2, with aggregated input and state/output vectors,
\begin{align}
x =y=\begin{bmatrix}
p_{1,r}\\q_{1,\ell}\\p_{2,r}\\q_{2,\ell}
\end{bmatrix}, \quad w=\begin{bmatrix}
p_{1,\ell}\\q_{1,r}\\p_{2,\ell}\\q_{2,r}
\end{bmatrix},
\end{align}
We further define the sources and sinks:
\begin{align}
u = \begin{bmatrix}
p_{1,\ell}\\q_{2,r}
\end{bmatrix}, \quad z=\begin{bmatrix}
p_{2,r}\\
q_{1,\ell}
\end{bmatrix}.
\end{align}
As $p_{1,r}=p_{2,\ell}$ and $q_{1,r}=q_{2,\ell}$, this leads to the following matrices:
\begin{align}
F&=\begin{bmatrix}
0&0&0&0\\
0 &0& 0& 1 \\
1 & 0 & 0 & 0\\
0&0&0&0
\end{bmatrix},\quad
G=\begin{bmatrix}
1&0\\
0 &0 \\
0 & 0 \\
0&1
\end{bmatrix}, \\
H&=\begin{bmatrix}
0&0&1&0\\
0 &1&0&0 \\
\end{bmatrix},\quad  J=\mathbf{0}.
\end{align}
The aggregate system can directly be implemented in Matlab using \eqref{eq:matrix_framework}.
\subsubsection{Matlab example: {vented} gas loop}
We recall the example from \cite{sven_bob_rob_gas} in Figure \ref{fig:control_loop}.

\begin{figure}[ht!]
    \centering
   \includegraphics{figures/09distrubance.pdf}

  \caption{Pipe network with compressor and valve $\bowtie$. In process control parlance, the fill pressure and vent flow are manipulated variables, the suction and distal pressures are controlled variables, and the flow from $P_6$ is a disturbance signal.}
    \label{fig:control_loop}
\end{figure}
The gas flows clockwise, entering through pipe $P_1$ and exiting through pipes $P_6$  and $P_9$. The compressor and valve, whose corresponding variables are respectively labeled by subscripts $c$ and $v$, are modeled as static gains as in Sections~\ref{sec:valve} and \ref{sec:compressor},
\begin{align*}
D_c=\begin{bmatrix}
k_c&0\\0 & 1
\end{bmatrix},\quad D_v=\begin{bmatrix}
k_v&0\\0 & 1
\end{bmatrix},
\end{align*}
where $k_c=4$ and $k_v=0.8$. Further, pipes $(P_1,P_2,P_3)$ are modeled as a joint, as in Subsection~\ref{subsec:joint}, and $(P_5,P_6,P_7)$ and $(P_8,P_9,P_{10})$ as branches, as in Section~\ref{sec:branch}.
Composing the system according to \eqref{eq:matrices_stacked_up}, results in the component input vector,
\begin{align*}
 w&=\left[\begin{matrix}
p_{1,\ell}& p_{2,\ell} & q_{3,r} & p_{c,\ell} & q_{c,r} & p_{4,\ell} & q_{4,r} \end{matrix}\right
\left.\begin{matrix} p_{v,\ell} & q_{v,r} & p_{5,\ell} & q_{6,r} & q_{7,r} & p_{8,\ell}q_{9,r} & q_{10,r}
\end{matrix}\right]^\top,
\end{align*}
and the total output vector,
\begin{align*}
 y&=\left[\begin{matrix}
p_{3,r}& q_{1,\ell} & q_{2,\ell} & p_{c,r} & q_{c,\ell} & p_{4,r} & q_{4,\ell} \end{matrix}\right
 \left.\begin{matrix} p_{v,r} & q_{v,\ell} & p_{6,r} & p_{7,r} & q_{5,\ell} & p_{9,r}&p_{10,r} & q_{8,\ell}
\end{matrix}\right]^\top.
\end{align*}
The external sources are
\begin{align*}
\bar u_t&=\begin{bmatrix}
p_{1,\ell} & q_{6,r} & q_{9,r}
\end{bmatrix}^\top.
\end{align*}
\paragraph*{Matlab code}\mbox{}\\
\lstinputlisting[style=Matlab-editor]{matlab/sloop_matrix.m}

\subsection{Matlab's connect function}
As an alternative to the matrix methodology we can use the \verb|connect| function in Matlab. This is best explained by the example below. For each component we name the input and output signals, respecting the boundary conditions. We then define the input and output signal names of the aggregate system and call the \verb|connect| function.
\subsubsection{Matlab example: two pipes in series

\lstinputlisting[style=Matlab-editor]{matlab/connect_sample.m}
\subsubsection{Matlab example: {vented} gas loop
\lstinputlisting[style=Matlab-editor]{matlab/sloop_connect.m}

\section{Conclusion}
We derived composite state-space models for the transient dynamics of gas flow through intersecting pipe geometries, valves, compressors, and valve manifolds that are well-suited candidates for model-based control design. They also capture conservation of mass at steady state by subsuming algebraic constraints that would otherwise appear as part of a system of DAEs. Additionally, we provide examples of how to use the matrix methodology introduced in \cite{sven_bob_rob_gas} in Matlab and alternatively show how to use Matlab's \verb|connect| function.

\section*{Acknowledgement}
This research was supported by funding from Solar Turbines {Incorporated}.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Holbrook, Henry Harrison
Age: 21, credited to Hyde Park, VT
Unit(s): 11th VT INF
Service: enl 5/30/63, m/i 6/10/63, Pvt, Co. L, 11th VT INF, d/dis 1/28/64 (typhoid fever/measles)
Birth: 02/27/1842, Eden, VT
Burial: Holbrook Cemetery, Hyde Park, VT
Gravestone researcher/photographer: Deanna French
Pension?: Yes
Portrait?: Unknown
Webmaster's Note: The 11th Vermont Infantry was also known as the 1st Vermont Heavy Artillery; the names were used interchangably for most of its career
Holbrook Cemetery, Hyde Park, VT
FROM THE 11TH REGIMENT
FORT SLOCUM.
Died in Regimental Hospital at this post, since Jan. 28th, of Typhoid Pneumonia --
Jan. 28, Private Henry H. Holbrook, Hyde Park, aged 21 years and 6 months: Chester King, Coventry; Syndall Burns, Troy; Russell D. Warren, Elmore, Feb 24, Solon W. Hutchins, Elmore; Robert H. Hutchins, Lunenburg; Feb. 26, Hollis Bagley, Newport; all members of Battery L, 1st Artillery, 11th Regt. Vt. Vols. This Battery has now thirteen in hospital 52 on sick list.
Young Holbrook was the first to go; was one of the first to come at the formation of the battery, in May 1863. He was the first ill of fever, and before he had recovered from that he was taken with measles; and again, when about to return to his battery, the fatal disease, typhoid pneumonia, which is now sweeping away so many of the noblest boys of our regiment, set in, and in spite of the efforts of our skillful surgeon Parks, the tender care of the nurses, and the kind attention of his fellow soldiers, he too has fallen victim to its unyielding grasp. Being the first death that occurred since our organization, it cast a heavy gloom over our minds which remains with us daily another goes, and sometimes two. Henry, although a stranger to the nurses when he entered the hospital, soon won their affection by the quiet and uncomplaining spirit which he manifested. It was my privilege to be often with him during his illness, to notice the fortitude with which he bore his sufferings, and his grateful expressions in his rational moments for the kind attention given him.
In young Holbrook, Warren, Burns, King, Bagley, and the two Hutchins, their comrades have lost kind friends, cheerful companions, willing workmen, and faithful soldiers, and while deeply mourning their loss, we as a battery, tender our heartfelt sympathies to the circle of mourning friends, wives, and children at home.
Cushing Nichols, Bat.,L.,1st Art.
Source: Lamoille Newsdealer, 2 Mar 1864.
Courtesy of Deanna French
Mortality of the 1st Vt. Artillery, from the 1st of September, 1862 to the 13th of March, 1864.
March, 18th, 1864
Mr. Editor:-- Below is a record of all the deaths, that have occurred since the time above stated, with the exception of a few cases of varioloid, which have not been sent to us from General Hospital as yet. I will give their names and age, with their diseases and the date of their death:--
Henry H. Holbrook, L, aged 21, of typhoid fever -- sent home Jan. 28th, 1864.
The above is a true record of all that have died in our hospital, as before state, making in all, ninety-three cases in one year and a half. Perhaps some may think this a pretty large amount of sickness and death in so short a time; but it seems that thus it is. I have been connected with the hospital department all the time, with the exception of a few weeks. We have things very convenient at present, so that the sick are, or can be, well taken care of; and I think that they do have all done for them that can be, for the place.
Yours &c., W. J. Cheney.
Source: Lamoille Newsdealer, 30 Mar 1864.


text:
Book Chapter 3
Stewart Clem 2
Margaret Nicola Abruzzo 1
William R Smith 1
Moral Philosophy 7[remove]
Thomas Aquinas 2
University of Notre Dame Press 3
Cambridge University Press 1
Graduate 1
Subject: Moral Philosophy remove ×
Truth as a Virtue: A Thomistic Framework for the Ethics of Lying and Truthtelling
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Celia Deane-Drummond, John O'Callaghan, Gerald McKenny, Jean Porter
Dropping the Debt: A New Conundrum in Kant's Rational Religion
In this article, I argue that Immanuel Kant fails to provide a satisfactory account of 'moral debt' in Religion within the Boundaries of Mere Reason. More precisely, he fails to answer the question of why we should assume that a debt exists in the first place. In light of recent scholarship on this area of his thought, I sketch some possible readings of Kant on the nature of moral transformation that suggest how he might account for this debt. I then argue that these accounts fail to justify …
Legitimizing Medical Practice
William R Smith
Robert Audi, Fritz Warfield, Paul Weithman
Complicity and Moral Accountability
Gregory Mellema
Responsibility, Accomplices
In Complicity and Moral Accountability, Gregory Mellema presents a philosophical approach to the moral issues involved in complicity. Starting with a taxonomy of Thomas Aquinas, according to whom there are nine ways for one to become complicit in the wrongdoing of another, Mellema analyzes each kind of complicity and examines the moral status of someone complicit in each of these ways. Mellema's central argument is that one must perform a contributing action to qualify as an accomplice, a…
Christopher Kutz, Complicity, H.D. Lewis, Christian Theology, Karl Jaspers, Law, Thomas Aquinas, Harm Responsibility, Ethics, Philosophy, Moral Philosophy, Politics
I Want You to Be: On the God of Love
Tomáš Halík
Love--Religious aspects--Christianity, God (Christianity)--Love, Christianity and culture
In his two previous books translated into English, Patience with God and Night of the Confessor, best-selling Czech author and theologian Tom\00E1\0161 Halík focused on the relationship between faith and hope. Now, in I Want You to Be, Halík examines the connection between faith and love, meditating on a statement attributed to St. Augustine-amo, volo ut sis, "I love you: I want you to be"-and its importance for contemporary Christian practice. Halík suggests that because Go…
Moral Philosophy, Religion, Fortitude, Justice, Prudence, Philosophy, Temperance
Gerald Turner
Realizing the Distinctive University: Vision and Values, Strategy and Culture
Mark William Roche
Education, Humanistic--United States, Education, Higher--United States, Universities and colleges--United States--Administration
In Realizing the Distinctive University: Vision and Values, Strategy and Culture, Mark William Roche changes the terms of the debate about American higher education. A former dean of the College of Arts and Letters at the University of Notre Dame, Roche argues for the importance of an institutional vision, not simply a brand, and while he extols the value of entrepreneurship, he defines it in contrast to the corporate drive toward commercialization and demands for business management models…
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Gail Bederman, James Turner, John T. McGreevy, Thomas P. Slaughter, Robert E. Sullivan


text:
Home/News/Facebook Disruption: I'm not worried about money lost- Zuckerberg
MetroNewsSociety
Facebook Disruption: I'm not worried about money lost- Zuckerberg
goodnewsOctober 6, 2021Last Updated: October 6, 2021
Mark Zuckerberg, Silicon Valley Techpreneur has said that he was not worried about the money lost or the number of users who left to other social media platforms during a seven-hour disruption of his services on Monday.
He made this known via a Facebook page on Tuesday, adding that he was rather concerned about those who rely on his products to connect with family and loved ones.
According to The PUNCH, it was reported that about three billion social media users regained access to their online community late Monday as Facebook-owned services, WhatsApp and Instagram, came back online after the record seven-hour outage.
With Monday's disruption of his apps, Zuckerberg's personal wealth has fallen by nearly $7 billion, knocking him down a notch on the list of the world's richest people. Information Guide Nigeria
The stock slide on Monday sent Zuckerberg's worth down to $120.9 billion, dropping him below Bill Gates to No. 5 on the Bloomberg Billionaires Index. He's lost about $19 billion of wealth since September 13, when he was worth nearly $140 billion, according to the index.
See Other Top News:
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As reported by The Punch, Zukerberg wrote: "The SEV that took down all our services yesterday (Monday) was the worst outage we've had in years.
"We've spent the past 24 hours debriefing how we can strengthen our systems against this kind of failure. This was also a reminder of how much our work matters to people.
"The deeper concern with an outage like this isn't how many people switch to competitive services or how much money we lose, but what it means for the people who rely on our services to communicate with loved ones, run their businesses, or support their communities." JAMB Result
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Essien, Goodnews is a gruaduate of Communication Arts, University of Uyo, Uyo; Currently working at Infoguide Media as a news writer.
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BREAKING: Buhari fires agriculture, power ministers


It's, once again, been too long since I've blogged. I've been so busy, with all of the iPhone users downloading iTransaX - our recently released credit card app for iPhone.
One of my colleagues recently sent me an interesting article on the iPhone phenomenon. Of course, I wholeheartedly disagree with the article - I think the iPhone rocks! And not just because we have an iPhone application, with more to come, in the iTunes store!


text:
Chelsea Skin Center Blog, New York dermatology clinic, 212-675-0549
Dr. Michael Eidelman
Dr. Brian Ginsberg
Dominic P. Moccia, RPA-C
How to educate men on aesthetic treatment
by John Jesitus
http://dermatologytimes.modernmedicine.com/dermatology-times/news/how-educate-men-aesthetic-treatment?page=0%2C0
Men are flocking to dermatologists for aesthetic procedures, and they usually know what they want aesthetic procedures to accomplish, but they may not know how to express what they want or what options are available, says an expert at the American Society for Dermatologic Surgery annual meeting in Chicago.
Michael Eidelman, M.D., says that with a patient population that's 40% men, increasing numbers of male patients ask him to help refresh their appearance. "Men don't want to look tired. They know they can do something; they just don't know what it is." He is medical director of Chelsea Skin & Laser in New York and assistant clinical professor of dermatology, Icahn School of Medicine at Mount Sinai.
To bridge the gap between a man's current appearance and a younger photograph, "I'm looking at the lines on his face—the depth of the wrinkles, what muscles are pulling and causing some of the changes; and the bone structure—the brow shape and position, eyelid position, shadows and generalized volume loss. I put that together with what they're telling me their concerns are to try to help them understand how we can make some improvements."
Regarding patient expectations, he says, male patients generally want results that "won't get called out in the locker room or at work." Along with the patient's budget, he considers what's realistic and how to achieve the biggest impact with the least downtime. He builds trust by crafting the treatment plan around the patient's lifestyle. "Many men don't have as much information available to them about how things work, as well as how quickly they work and what side effects may occur," says Dr. Eidelman, he carefully reviews with them each treatment's onset time and potential adverse effects. Dr. Eidelman also queries men more aggressively than women about their work and social calendar. "Many men are in the office and ready to go—they don't think 'I have a board meeting tomorrow night.'"
Regarding neuromodulators, Dr. Eidelman says that because men have larger facial muscle mass, they have more receptors that require saturation. To avoid under-dosing men, he generally increases labeled doses by 50%, for starters. "Doing a conservative treatment and then having the patient return for a two week follow-up to see how much improvement has occurred is often helpful in getting a final dosage that works best for the patient," he says.
Dr. Eidelman adds that, because male forehead muscles may be longer, "We may need to use additional injection points (up to nine, versus the usual five for females) to achieve optimal results."
Additionally, he says, "the frontalis muscle is often a bit more complicated to treat in men." Because men have very long, broad muscles, frontalis fibers often intermix with those of the upper face and periocular area, he says: "As a result, you must be very mindful of where your toxin will diffuse. You might get some effects in both muscle areas because they're all in very close proximity." When injecting the frontalis, Dr. Eidelman says, "I start in the upper third of the muscle body to avoid hitting any of the lower muscle fibers, as well as to create a more natural look. When the patient returns for follow-up, I decide if I want to treat any of the lower fibers."
As with neuromodulators, Dr. Eidelman says that, "because men have larger faces and more volume loss, we may need to use more product." Soft-tissue imaging studies also have shown that men tend to age less rapidly around the mouth than women, perhaps because beard hair and follicles provide extra structural support, he says.
For men in their 30s, says Dr. Eidelman, common complaints include periocular changes—which are often related to subtle volume loss in the cheek. "The two are best treated simultaneously," using products such as hyaluronic acid/HA (Juvéderm and Juvéderm Voluma, Allergan; Restylane Lyft, Galderma), or poly-L-lactic acid (Sculptra, Galderma) in the cheek. In the thin skin around the eye, hyaluronic acid (Belotero, Merz) can be used to soften periocular shadowing and fine etched lines.
In one particularly challenging case, Dr. Eidelman recounts a 60-year-old male who presented with irregular contours, significant sun damage and redundant skin: "He had diffuse changes over his upper face, midface and lower face. Someone had filled his midface and nasolabial folds with a permanent filler that caused irreversible lumpiness. I had to creatively find a way to avoid that area, but also to give him some lift and restore the contour irregularities." Dr. Eidelman used the following treatments:
Calcium hydroxylapatite (Radiesse, Merz) to provide structure in his cheeks posteriorly and superior to the lumpy areas
Poly-L-lactic acid (Sculptra, Galderma) in his posterior fat pads to replace lost volume and restore his jawline
A neuromodulator to reduce dynamic lines in his upper face and crow's feet, also applied to the depressor anguli oris to elevate the corners of his mouth.
"Using all those different products in different ways, I was able to create a softer, more balanced version of his face," Dr. Eidelman says.
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