5162 lines
102 KiB
Plaintext
5162 lines
102 KiB
Plaintext
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Tutorial
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\end_layout
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\begin_layout Chapter
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Introduction
|
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\end_layout
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||
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\begin_layout Standard
|
||
Welcome to the user's guide of the Thermoacoustic System Modeling Environment
|
||
Twente, or
|
||
\noun on
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||
TaSMET
|
||
\noun default
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||
.
|
||
|
||
\noun on
|
||
TaSMET
|
||
\noun default
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||
is a computer code to model thermoacoustic (TA) engines, refrigerators
|
||
and combined systems by providing nonlinear models for laminar/turbulent
|
||
oscillating flow in ducts, heat exchangers and stacks/regenerators.
|
||
A coupling to the mechanical and electrical domain is provided with a piston
|
||
model.
|
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\end_layout
|
||
|
||
\begin_layout Standard
|
||
The nonlinear sub-models can be connected to form a model of a complete
|
||
TA or Stirling system.
|
||
The main ideas of this code are developed as part of my PhD work
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
|
||
\noun on
|
||
Numerical Modeling of Thermoacoustic Systems
|
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\begin_inset Quotes erd
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key "jong_numerical_2015"
|
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.
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\end_layout
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|
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The code has been developed with a strong focus on computational cost.
|
||
Hence, it uses an efficient modeling technique, called the Nonlinear Frequency
|
||
Domain (NLFD) method, to directly simulate the periodic steady state of
|
||
a TA system.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The modular design makes it easy to create any system for which a similar
|
||
model can be made using the DELTAEC computer code.
|
||
The main differences between
|
||
\noun on
|
||
TaSMET
|
||
\noun default
|
||
and other TA modeling computer codes are
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Open source and free to use.
|
||
As the author is aware of the possibilities created by using open source
|
||
codes, the choice of publishing this computer code as open source was not
|
||
a hard choice.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Fast modeling in the nonlinear regime.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
No graphical user interface (GUI), but the Python programming language as
|
||
modeling glue.
|
||
GUI's require a considerable time to implement, but they do not add any
|
||
fundamental features.
|
||
As we provide all modeling classes in Python, the user can therefore decide
|
||
whether he still wants a GUI.
|
||
Using Matplotlib, we can plot everything we want and by building post-processin
|
||
g scripts we can derive any results from the solved model.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Purpose of this code
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Expand modeling capabilities of DeltaEC to the nonlinear regime -> More
|
||
detailed modeling of nonlinear effects
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
More in-depth insight in behavior of TA systems
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In this user's guide, we assume that the reader has already gained some
|
||
knowledge and experience with (modeling of) TA systems.
|
||
Moreover, we expect that the user of
|
||
\noun on
|
||
TaSMET
|
||
\noun default
|
||
has already modeled TA systems with a linear TA code, such as the well-known
|
||
|
||
\noun on
|
||
DELTAEC
|
||
\noun default
|
||
code.
|
||
A great deal of inspiration of
|
||
\noun on
|
||
TaSMET
|
||
\noun default
|
||
has been obtained by the way
|
||
\noun on
|
||
DELTAEC
|
||
\noun default
|
||
is designed.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Bug reporting and contributing
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
If you are interested in working and contributing to
|
||
\noun on
|
||
TaSMET
|
||
\noun default
|
||
, please contact me by email.
|
||
My email address is anne(at)amdj(dot)nl.
|
||
I am looking forward to cooperate!
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Basic ideas
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
With
|
||
\noun on
|
||
TaSMET
|
||
\noun default
|
||
, the periodic steady-state of a TA system is simulated using a numerical
|
||
model.
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Tutorial
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Introduction
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In this chapter, the usage of
|
||
\noun on
|
||
TaSMET
|
||
\noun default
|
||
is described by example.
|
||
Two TA systems will be described to show the main capabilities of
|
||
\noun on
|
||
TaSMET
|
||
\noun default
|
||
.
|
||
Focus will also be given on the post-processing capabilities.
|
||
As all
|
||
\noun on
|
||
TaSMET
|
||
\noun default
|
||
objects are exposed to Python, the user of
|
||
\noun on
|
||
TaSMET
|
||
\noun default
|
||
can easily create script to customize post-processing.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Each TA system in
|
||
\noun on
|
||
TaSMET
|
||
\noun default
|
||
is an object of type
|
||
\noun on
|
||
TaSystem
|
||
\noun default
|
||
, a
|
||
\noun on
|
||
TaSystem
|
||
\noun default
|
||
contains data about the global configuration of the system, such as the
|
||
working gas in the system and the fundamental oscillation frequency (
|
||
\begin_inset Formula $\omega$
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||
\end_inset
|
||
|
||
).
|
||
The physical configuration of a typical TA system is defined by the segments
|
||
in a
|
||
\noun on
|
||
TaSystem
|
||
\noun default
|
||
.
|
||
Each segment can represent a part of the TA system, such as resonator tubes,
|
||
heat exchangers, stacks/regenerators and pistons.
|
||
The combination of these segments, including the connecting and boundary
|
||
conditions and the working gas results in a (nonlinear) system of equations.
|
||
This system of equations can be solved by a
|
||
\noun on
|
||
Solver
|
||
\noun default
|
||
object.
|
||
Often, the user creates a Python scripts which has approximately the following
|
||
shape:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The main difference between connectors and segments is that only the latter
|
||
has provides both
|
||
\emph on
|
||
equations
|
||
\emph default
|
||
as well as
|
||
\emph on
|
||
degrees of freedom
|
||
\emph default
|
||
(DOFs), while connectors often only provide equations.
|
||
Hence, a complete system comprises at least one segment.
|
||
So far,
|
||
\noun on
|
||
TaSMET
|
||
\noun default
|
||
contains three main type of segments.
|
||
A short overview of these segments is given below.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
|
||
\noun on
|
||
Duct
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A Duct is probably the most important type of segment.
|
||
In a
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
, 1D dynamic gas flow can be modeled including its interaction with a solid.
|
||
Depending on the geometry, a
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
can model resonators with variable cross-sectional area, stacks, and heat
|
||
exchangers.
|
||
The interaction model of the flow with its surrounding solid is provided
|
||
in an object-oriented way with a derived class.
|
||
Examples of derived classes are
|
||
\noun on
|
||
LaminarDuct
|
||
\noun default
|
||
,
|
||
\noun on
|
||
IsentropicDuct
|
||
\noun default
|
||
, and
|
||
\noun on
|
||
TurbulentDuct.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
|
||
\noun on
|
||
Piston
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
This
|
||
\noun on
|
||
Piston
|
||
\noun default
|
||
provides a means to exchange mechanical energy between the gas domain and
|
||
the mechanical domain.
|
||
The implementation of the
|
||
\noun on
|
||
Piston
|
||
\noun default
|
||
segment is done in such a way that both the front and back volume can be
|
||
used.
|
||
The front and back volume are assumed to be small compared to the wavelength,
|
||
because no momentum equation is solved for the gas volumes.
|
||
A reference of this class is given in Sec.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
|
||
\noun on
|
||
ConnectorVolume
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A
|
||
\noun on
|
||
ConnectorVolume
|
||
\noun default
|
||
is used to connect multiple
|
||
\noun on
|
||
Ducts
|
||
\noun default
|
||
together.
|
||
This allows for 'branching' of multiple
|
||
\noun on
|
||
Ducts
|
||
\noun default
|
||
and it is an essential feature of traveling wave thermoacoustic engines.
|
||
A
|
||
\noun on
|
||
ConnectorVolume
|
||
\noun default
|
||
with only one
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
connected to it serves as a
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
compliance
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
, i.e.
|
||
an expansion volume in which the fluid motion is brought to rest.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Resonance tube
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Hofler's refrigerator (Hofler1)
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In this section,
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset bibtex
|
||
LatexCommand bibtex
|
||
bibfiles "bib/TaSMET"
|
||
options "plain"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Part
|
||
The API
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
|
||
\noun on
|
||
Duct
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Geometry
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
|
||
\noun on
|
||
Piston
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Introduction
|
||
\noun on
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "sec:Piston_ref"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename fig/piston.eps
|
||
width 80text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Piston
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:piston_overview"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
An overview of the model of a
|
||
\noun on
|
||
Piston
|
||
\noun default
|
||
is schematically shown in Figure (
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:piston_overview"
|
||
|
||
\end_inset
|
||
|
||
).
|
||
A
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
can be connected on both the front, as well as on the back side of the
|
||
\noun on
|
||
Piston
|
||
\noun default
|
||
.
|
||
This way, the segment has been made flexible.
|
||
The following geometrical, and mechanical parameters are required for the
|
||
model:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float table
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Tabular
|
||
<lyxtabular version="3" rows="10" columns="4">
|
||
<features tabularvalignment="middle">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="left" valignment="top" width="60text%">
|
||
<column alignment="center" valignment="top">
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Symbol
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Token
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Meaning
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Unit
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $S_{l}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family typewriter
|
||
Sl
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Piston surface area on the left side
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
m
|
||
\begin_inset Formula $^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $S_{r}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family typewriter
|
||
Sr
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Piston surface area on the right side
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
m
|
||
\begin_inset Formula $^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $V_{o,l}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family typewriter
|
||
V0l
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Left volume.
|
||
Note: the left volume should be defined with a size such that at the minimum
|
||
|
||
\begin_inset Formula $x$
|
||
\end_inset
|
||
|
||
, the volume does not become negative!
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
m
|
||
\begin_inset Formula $^{3}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $V_{0,r}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family typewriter
|
||
V0r
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Right volume.
|
||
Note: the right volume should be defined such that for the maximum piston
|
||
excursion
|
||
\begin_inset Formula $x$
|
||
\end_inset
|
||
|
||
, the volume does not become negative!
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
m
|
||
\begin_inset Formula $^{3}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $M$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family typewriter
|
||
M
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Total moving mass of the piston
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
kg
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $K_{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family typewriter
|
||
Km
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Piston spring constant
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
N/m
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $C_{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family typewriter
|
||
Cm
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Piston damping
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
N
|
||
\begin_inset Formula $\cdot$
|
||
\end_inset
|
||
|
||
s/m
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $S_{t,l}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family typewriter
|
||
Stl
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Total contact area of the fluid with the solid in the left volume.
|
||
This variable is used to compute the thermal relaxation dissipation in
|
||
the piston volume.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
m
|
||
\begin_inset Formula $^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $S_{t,r}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family typewriter
|
||
Str
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Total contact area of the fluid with the solid in the right volume.
|
||
This variable is used to compute the thermal relaxation dissipation in
|
||
the piston volume.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
m
|
||
\begin_inset Formula $^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Overview of all the parameters required for the Piston model.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
To initialize a
|
||
\noun on
|
||
Piston,
|
||
\noun default
|
||
first a helper
|
||
\noun on
|
||
struct
|
||
\noun default
|
||
needs to be defined, called the
|
||
\noun on
|
||
PistonConfiguration.
|
||
|
||
\noun default
|
||
The initialization of a
|
||
\noun on
|
||
PistonConfiguration
|
||
\noun default
|
||
is
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset listings
|
||
inline false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
pc=PistonConfiguration(Sl,Sr,V0l,V0r,M,Km,Cm,Stl,Str)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
.
|
||
Then we can build a new
|
||
\noun on
|
||
Piston
|
||
\noun default
|
||
segment with
|
||
\begin_inset listings
|
||
inline false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
p=Piston(pc)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
DuctPistonConnector
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The
|
||
\noun on
|
||
DuctistonConnector
|
||
\noun default
|
||
connects a Piston segment to a Tube.
|
||
The following syntax is used to initialize a
|
||
\noun on
|
||
DuctPistonConnector:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset listings
|
||
inline false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
tpc=DuctPistonConnector(id_duct,duct_pos,id_piston,piston_pos,KDuctPiston,KPisto
|
||
nDuct)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
, where the id's are strings by which a
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
or a
|
||
\noun on
|
||
Piston
|
||
\noun default
|
||
is identified.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Piston energy balance
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
mH_{p}+Q_{p\rightarrow t}+s\cdot mH_{t}=0
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Mechanical boundary conditions
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
|
||
\noun on
|
||
MechBc
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Using a
|
||
\noun on
|
||
MechBc,
|
||
\noun default
|
||
a simple boundary condition can be set on the mechanical domain of the
|
||
|
||
\noun on
|
||
Piston
|
||
\noun default
|
||
.
|
||
It can be used to set either an external force, the piston displacement,
|
||
or a mechanical impedance.
|
||
The syntax is
|
||
\begin_inset listings
|
||
inline false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
m=MechBc(piston_id,contraint_var,boundary_condition)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
, where
|
||
\family typewriter
|
||
boundary_condition
|
||
\family default
|
||
is the b.c.
|
||
|
||
\family typewriter
|
||
var
|
||
\family default
|
||
object, and
|
||
\family typewriter
|
||
constraint_var
|
||
\family default
|
||
is either
|
||
\family typewriter
|
||
Varnr_x, Varnr_F,
|
||
\family default
|
||
or
|
||
\family typewriter
|
||
Varnr_Z.
|
||
|
||
\family default
|
||
The latter induces the boundary condition to solve
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
F-Zx=0,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and can be used to model a passive electrical domain.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
VCMNetwork
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The current version does not yet provide this model.
|
||
In a future version we will provide a voice coil motor (VCM) network to
|
||
the list of segments.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
To include the model of the electrical domain, it is
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Tabular
|
||
<lyxtabular version="3" rows="4" columns="4">
|
||
<features tabularvalignment="middle">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top" width="60text%">
|
||
<column alignment="center" valignment="top">
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $L$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
L
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Voice coil inductance
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
H
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $R_{e}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Re
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Electrical current resistance
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Omega$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $Bl$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Bl
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Motor constant (Newton per Amp or Volt per meter).
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $N/A$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Connecting it all together
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
A system
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Introduction
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
- A system comprises all segments in a physical sense
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
- A system : amount of mass
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
- Number of harmonics to solve for
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
|
||
\noun on
|
||
TaSystem
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A
|
||
\noun on
|
||
TaSystem
|
||
\noun default
|
||
is a class which initializes and contains all segments and connectors.
|
||
The TaSystem provides basic information about the nonlinear system of equations
|
||
which has to be solved to obtain the value of all dependent variables in
|
||
the individual segments.
|
||
A
|
||
\noun on
|
||
TaSystem
|
||
\noun default
|
||
object is created with
|
||
\begin_inset listings
|
||
inline false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
import TaSMET
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
gc=TaSMET.Globalconf(...)
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
sys=TaSMET.TaSystem(gc)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
Then individual segments can be added to a
|
||
\noun on
|
||
TaSystem
|
||
\noun default
|
||
by
|
||
\begin_inset listings
|
||
inline false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
sys+=seg1
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
sys+=seg2
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
sys+=...
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
sys+=con1
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
...
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Mass conservation
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
EngineSystem
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The EngineSystem class solves for the unknown frequency as well.
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
m
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
We define a new system of equations in which the fundamental frequency is
|
||
added as unknown and the timing constraint as an equation.
|
||
So the augmented solution vector is
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathbf{y}=\left(\mathbf{x},\omega\right),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and the augmented residual is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathbf{M}=\left(\mathbf{L}(\mathbf{x}),p_{\mathrm{cd}}\right),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $p_{\mathrm{cd}}$
|
||
\end_inset
|
||
|
||
denotes the phase constraint degree of freedom.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
To search for the solution, Newton iterations are done using the Jacobian
|
||
of the augmented residual operator
|
||
\begin_inset Formula $\mathbf{M}$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\delta\mathbf{y}=-\frac{\mathrm{d}\mathbf{M}}{\mathrm{d}\mathbf{y}}^{-1}\cdot\mathbf{M},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\mathbf{\mathrm{d}}\mathbf{M}}{\mathrm{d}\mathbf{y}}=\left[\begin{array}{cc}
|
||
\frac{\partial\mathbf{L}}{\partial\mathbf{x}} & \frac{\partial\mathbf{L}}{\partial\omega}\\
|
||
\frac{\partial p_{\mathrm{cd}}}{\partial\mathbf{x}} & 0
|
||
\end{array}\right].\label{eq:newjac}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
In Eq.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
(
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:newjac"
|
||
|
||
\end_inset
|
||
|
||
),
|
||
\begin_inset Formula $\frac{\partial p_{\mathrm{cd}}}{\partial\mathbf{x}}$
|
||
\end_inset
|
||
|
||
is a single row which is zero everywhere, but is one at the global degree
|
||
of freedom number corresponding to
|
||
\begin_inset Formula $\Im\left(\tilde{p}_{1}(x=0)\right)$
|
||
\end_inset
|
||
|
||
.
|
||
The column
|
||
\begin_inset Formula $\frac{\partial\mathbf{L}}{\partial\omega}$
|
||
\end_inset
|
||
|
||
, is the sensitivity of the residual to a change in frequency.
|
||
For brevity we only show the semi-discrete form of these sensitivities.
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Due to the normalization of the time-derivative matrix t
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
These sensitivities can
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
easily
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
be derived from the governing equations.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For the continuity equation this sensitivity is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial\mathbf{L}_{c,i}}{\partial\omega}=V_{f}\check{\mathbf{D}}\cdot\tilde{\boldsymbol{\rho}}_{i}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
For the momentum and energy equation, however, we neglect the sensitivity
|
||
to the operators
|
||
\begin_inset Formula $\boldsymbol{\tilde{\mathcal{D}}}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\boldsymbol{\tilde{\mathcal{H}}}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\boldsymbol{\tilde{\mathcal{Q}}}$
|
||
\end_inset
|
||
|
||
, so for the momentum equation we use
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial\mathbf{L}_{m,L}}{\partial\omega}\simeq\left(x_{i}-x_{i-1}\right)\check{\mathbf{D}}\cdot\boldsymbol{\mathcal{F}}\cdot\tilde{\mathbf{m}},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and for the energy equation
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial\mathbf{L}_{e,i}}{\partial\omega}\simeq\check{\mathbf{D}}\cdot\left(\frac{S_{f}}{\gamma-1}\tilde{\mathbf{p}}_{i}+\left(x_{R}-x_{L}\right)\left(\mathbf{mu}\right)_{i}\right),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and finally for the equation of state
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial\mathbf{L}_{s,i}}{\partial\omega}=\mathbf{0}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Setting a phase constraint
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Create a PhaseConstraint object:
|
||
\end_layout
|
||
|
||
\begin_layout Verbatim
|
||
pc=PhaseConstraint(Varnr, freqnr, left)
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Apply this contraint to a segment which can accept them, for example a Tube:
|
||
\end_layout
|
||
|
||
\begin_layout Verbatim
|
||
t1.setPhaseConstraint(pc)
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
And you're done.
|
||
Note: only one phase constraint can be used in an EngineSystem.
|
||
For a TaSystem, the phase constraint is ignored.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
–
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
The
|
||
\noun on
|
||
Solver
|
||
\noun default
|
||
class
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Introduction
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
..
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset listings
|
||
inline false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
sc=TaSMET.SolverConfiguration()
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
sc.setFunTol(...)
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
sc.setRelTol(...)
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
sol=TaSMET.Solver(sc)
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
sol.Solve(sys)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Solver statistics
|
||
\end_layout
|
||
|
||
\begin_layout Part
|
||
Model reference
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Ducts
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
General 3D conservation equations
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\frac{\mathrm{d}}{\mathrm{d}t}\int_{V(t)}\rho\mathrm{d}V & +\int_{S}\rho\boldsymbol{u}\cdot\boldsymbol{n}\mathrm{d}S=0\\
|
||
\frac{\partial\rho\boldsymbol{u}}{\partial t}+\nabla\cdot\rho\boldsymbol{u}\otimes\boldsymbol{u}+\nabla p & =\nabla\cdot\underline{\boldsymbol{\tau}}\\
|
||
\frac{\partial\rho E}{\partial t}+\nabla\cdot\left(\rho\boldsymbol{u}E+p\boldsymbol{u}\right)+\nabla\cdot\boldsymbol{q}= & \nabla\cdot\left(\underline{\boldsymbol{\tau}}\cdot\boldsymbol{u}\right)
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Integrating over a piece of tube length:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
S_{f}\frac{\partial\overline{\rho}}{\partial t}+\frac{\partial}{\partial x}\left(S_{f}\overline{\rho u}\right)=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
S_{f}\frac{\partial\overline{\rho u}}{\partial t}+
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Some definitions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
s=r_{h}\sqrt{\frac{\rho_{0}\omega}{\mu}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
begin{subequations}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
s & = & \sqrt{2}\frac{r_{h}}{\delta_{\nu}}\\
|
||
s_{t} & = & s\sqrt{\Pr}=\sqrt{2}\frac{r_{h}}{\delta_{\kappa}}\\
|
||
s_{s} & = & \sqrt{2}\frac{r_{h,s}}{\delta_{s}}
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
end{subequations}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Duct geometry
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename fig/duct_grid.eps
|
||
width 75text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Geometry and discretization of a duct
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Cell vertices halfway between the cell walls
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Cross-sectional area jumps at the cell walls
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
What variables do live where?
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
|
||
\noun on
|
||
Duct
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Continuity equation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Continuity equation lives at the vertex
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
S_{f}\frac{\partial\rho}{\partial t}+\frac{\partial S_{f}\rho u}{\partial x}=0,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $x$
|
||
\end_inset
|
||
|
||
is the axial position,
|
||
\begin_inset Formula $S_{f}$
|
||
\end_inset
|
||
|
||
the cross-sectional area occupied by fluid,
|
||
\begin_inset Formula $\rho$
|
||
\end_inset
|
||
|
||
is the density and
|
||
\begin_inset Formula $m$
|
||
\end_inset
|
||
|
||
the mass flow.
|
||
Quasi-discrete form:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Delta x_{i}S_{f,i}\frac{\partial\rho_{i}}{\partial t}+\left(S_{f}\rho u\right){}_{i+1}-\left(S_{f}\rho u\right){}_{i}=0,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Momentum equation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Momentum equation lives at the cell wall.
|
||
It conserves the momentum in a
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
cell
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
with left and right walls which are at the corresponding vertices.
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Check units:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left[\frac{\partial m}{\partial t}\right]=\frac{kg}{s^{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left[S_{f}\frac{\partial p}{\partial x}\right]=m^{2}\frac{N}{m^{2}m}=\frac{N}{m}=\frac{kgm}{s^{2}m}=\frac{kg}{s^{2}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial\rho u}{\partial t}+\frac{\partial\rho u^{2}}{\partial x}+\frac{\partial p}{\partial x}+\frac{1}{S_{f}}\mathcal{R}=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where,
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
is the pressure and
|
||
\begin_inset Formula $\mathcal{R}$
|
||
\end_inset
|
||
|
||
is the viscous resistance coefficient.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Quasi-discrete form
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\left(x_{i+1}-x_{i}\right)\frac{\partial\rho u}{\partial t}+\left(\rho u^{2}\right)_{i+1}-\left(\rho u^{2}\right)_{i}+\left(p_{i+1}-p_{i}\right)+\frac{\left(x_{i+1}-x_{i}\right)}{S_{f,i}}\mathcal{R}_{i}=0.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Energy equation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Since
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{1}{2}\rho u^{2}S_{f}\equiv mu,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
The energy equation can be written as
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\partial}{\partial t}\left(S_{f}\rho E\right)+\frac{\partial}{\partial x}\left(mH\right)-\frac{\partial}{\partial x}\left(\kappa S_{f}\frac{\partial T}{\partial x}\right)=Q_{in}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $S_{f}\rho E=S_{f}\rho\left(c_{v}T+\frac{1}{2}u^{2}\right)=S_{f}\left(\rho c_{v}T+\frac{1}{2}\rho u^{2}\right)=\left(S_{f}\frac{p}{\gamma-1}+\frac{1}{2}S_{f}\rho u^{2}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $S_{f}\rho E=S_{f}\rho\left(c_{v}T+\frac{1}{2}u^{2}\right)=S_{f}\left(\rho c_{v}T+\frac{1}{2}\rho u^{2}\right)=\left(S_{f}\frac{p}{\gamma-1}+\frac{1}{2}S_{f}\rho u^{2}\right)=S_{f}\frac{p}{\gamma-1}+\frac{1}{2}mu$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\partial}{\partial t}\left(S_{f}\rho E\right)+\frac{\partial}{\partial x}\left(mH\right)-\frac{\partial}{\partial x}\left(\kappa\frac{\partial T}{\partial x}\right)=Q_{in}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{S_{f}}{\gamma-1}\frac{\partial p}{\partial t}+\frac{1}{2}\frac{\partial\rho S_{f}u^{2}}{\partial t}+\frac{\partial}{\partial x}\left(\rho uS_{f}c_{p}T+\frac{1}{2}\rho u^{3}S_{f}+Q_{\mathrm{ax}}\right)-Q_{s\to f}=0,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
, and
|
||
\begin_inset Formula $Q_{\mathrm{ax}}$
|
||
\end_inset
|
||
|
||
is the axial conduction:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Q_{\mathrm{ax}}=-\kappa S_{f}\frac{\partial T}{\partial x},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and,
|
||
\begin_inset Formula $Q_{s\to f}$
|
||
\end_inset
|
||
|
||
is the heat flow from the solid to the fluid.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The quasi-discretized form of this equation is
|
||
\begin_inset Formula
|
||
\begin{multline}
|
||
\frac{V_{f}}{\gamma-1}\frac{\partial p_{i}}{\partial t}+\frac{1}{2}\left(x_{r}-x_{l}\right)\frac{\partial\left(mu\right)_{i}}{\partial t}+\\
|
||
c_{p}m_{R}\left(W_{r,L}T_{i}+W_{r,R}T_{i+1}\right)-c_{p}m_{l}\left(W_{l,R}T_{i}+W_{l,L}T_{i-1}\right)+\\
|
||
\kappa_{R}\left(W_{c,Rl}T_{i}+W_{c,Rr}T_{i+1}\right)-\kappa_{L}\left(W_{c,Lr}T_{i}+W_{c,Ll}T_{i-1}\right)+\\
|
||
m_{r}E_{\mathrm{kin},r}-mE_{\mathrm{kin},l}-\left(x_{R}-x_{L}\right)Q_{s\to f}=0,
|
||
\end{multline}
|
||
|
||
\end_inset
|
||
|
||
, where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
W_{R,l}=\frac{x_{i+1}-x_{R}}{x_{i+1}-x_{i}}+\quad;\quad W_{R,r}=1-W_{R,l}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
, and
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\kappa_{R}=W_{R,l}\kappa_{i}+W_{R,r}\kappa_{i+1}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
, and
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
W_{c,Rl}=\frac{S_{f,R}}{x_{i+1}-x_{i}}\quad;\quad W_{c,Rr}=-W_{c,Rl}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
, and
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
W_{c,Ll}=\frac{S_{f,L}}{x_{i}-x_{i-1}}\quad;\quad W_{c,Lr}=-W_{c,Ll}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
The flow of kinetic energy is computed as
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
E_{\mathrm{kin},R}=\frac{1}{2}m_{R}\left(\frac{m_{R}}{S_{f,R}\left(W_{R,l}\rho_{i}+W_{R,r}\rho_{i+1}\right)}\right)^{2}=\frac{1}{2}m_{R}^{3}S_{f,R}\left(W_{R,l}\rho_{i}+W_{R,r}\rho_{i+1}\right)^{-2},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
So
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
\frac{\partial E_{\mathrm{kin},R}}{\partial m} & = & \frac{3}{2}S_{f,R}\left(W_{R,l}\rho_{i}+W_{R,r}\rho_{i+1}\right)m_{R}^{2}\\
|
||
\frac{\partial E_{\mathrm{kin},R}}{\partial\rho_{i}} & = & -m_{R}^{3}S_{f,R}\left(W_{R,l}\rho_{i}+W_{R,r}\rho_{i+1}\right)^{-3}W_{R,l}\\
|
||
\frac{\partial E_{\mathrm{kin},R}}{\partial\rho_{i+1}} & = & -m_{R}^{3}S_{f,R}\left(W_{R,l}\rho_{i}+W_{R,r}\rho_{i+1}\right)^{-3}W_{R,r}
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
similarly, the flux through the left wall is computed as
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
E_{\mathrm{kin},L}=\frac{1}{2}m_{L}\left(\frac{m_{L}}{S_{f,L}\left(W_{L,l}\rho_{i-1}+W_{L,l}\rho_{i}\right)}\right)^{2},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The last term,
|
||
\begin_inset Formula $Q_{s\to f}$
|
||
\end_inset
|
||
|
||
is dependent on the specific model implemented in a derived class of
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
.
|
||
See for example the HopkinsLaminarDuct.Laminar flow
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\tilde{\boldsymbol{\mathcal{R}}}=\tilde{\mathbf{R}}\diamond\tilde{\mathbf{m}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\hat{\mathcal{R}}_{n}=\frac{\mu}{r_{h}^{2}}\frac{is_{n}^{2}f_{\nu,n}}{1-f_{\nu,n}}\hat{U}_{n}$
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $s_{n}^{2}=r_{h}^{2}\frac{\rho_{0}\omega}{\mu}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Substituting that:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\hat{\mathcal{R}}_{n}=\rho_{0}\frac{i\omega nf_{\nu,n}}{1-f_{\nu,n}}\hat{U}_{n}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\tilde{R}_{n}=\frac{i\omega nf_{\nu,n}}{1-f_{\nu,n}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Isentropic state equation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\hat{\mathbf{1}}+\frac{\hat{\mathbf{p}}}{p_{0}}-\mathcal{\boldsymbol{F}}\cdot\left(\frac{\boldsymbol{\rho}}{\rho_{0}}\right)^{\gamma}=\mathbf{0}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Discretization
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Since pressure and density live on the vertices, to compute
|
||
\begin_inset Formula $f_{\nu,n}$
|
||
\end_inset
|
||
|
||
at each cell wall, we take for
|
||
\begin_inset Formula $\delta_{\nu,\kappa}$
|
||
\end_inset
|
||
|
||
the weighted average of the neighboring vertices.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
\hat{T}_{0,L} & \approx & w_{i-1}\hat{T}_{0,i-1}+w_{i}T_{0,i},\\
|
||
\hat{p}_{0,L} & \approx & w_{i-1}\hat{T}_{0,i-1}+w_{i}T_{0,i},
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $w_{i}=1-w_{i-1}=\frac{x_{i+1}-x_{L}}{x_{i+1}-x_{i-1}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
w_{i-1}=\frac{x_{i+1}-x_{L}}{x_{i+1}-x_{i-1}}\quad w_{i}=1-w_{i-1}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Section
|
||
Turbulent flow
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
From paper Characteristic-based non-linear simulation of large-scale standing-wa
|
||
ve thermoacoustic engine
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathcal{D}=-S_{f}\rho f
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
, with
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
C_{f}=\frac{1}{4}\frac{fd}{u|u|}\Rightarrow f=\frac{4C_{f}}{d}u|u|
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\[
|
||
f=\frac{4C_{f}}{d}u|u
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
, this becomes
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathcal{D}=-m\frac{4C_{f}}{d}|u|
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
LaminarDuct
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float table
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Tabular
|
||
<lyxtabular version="3" rows="3" columns="3">
|
||
<features tabularvalignment="middle" tabularwidth="100text%">
|
||
<column alignment="left" valignment="top" width="0pt">
|
||
<column alignment="left" valignment="top" width="40text%">
|
||
<column alignment="left" valignment="top" width="40text%">
|
||
<row>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Insulated
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Not insulated
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
With solid
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Impossible combination
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Itemize
|
||
Wall temperature is determined by balance in heat flow from fluid domain
|
||
to solid domain
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Without solid
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Itemize
|
||
Time-averaged heat transfer from fluid to solid is zero by setting wall
|
||
temperature equal to fluid temperature.
|
||
Hence
|
||
\begin_inset Formula $\frac{d\hat{T}_{w,0}}{dx}=\frac{d\hat{T}_{0}}{dx}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Itemize
|
||
Wall temperature is prescribed, heat flow through solid material is zero
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Overview
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Viscous resistance
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Wall temperature prescribed
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In the HopkinsLaminarDuct,
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{Q}_{s\to f}=\hat{\mathcal{H}}-\hat{Q}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}\hat{m},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\series bold
|
||
\begin_inset Formula $\hat{\mathbf{Q}}_{s\to f}=\mathcal{H}-\hat{Q}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}\hat{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\partial\hat{\mathbf{Q}}_{s\to f}}{\partial\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}}=-\hat{Q}\hat{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
–
|
||
\begin_inset Formula $T_{w,0}=\left[\begin{array}{cccc}
|
||
1 & 0 & 0 & \dots\end{array}\right]\hat{\mathbf{T}}_{w}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\partial\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}}{\partial\frac{\mathrm{d}\hat{\mathbf{T}}}{\mathrm{d}x}}=$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{align}
|
||
\hat{\boldsymbol{\mathcal{H}}} & =-S_{f}\hat{H}\hat{T} & ; & n>0\\
|
||
\hat{\mathcal{\boldsymbol{\mathcal{H}}}}_{0} & =S_{f}\hat{H}_{0}\left(\hat{T}_{w,0}-\hat{T}_{0}\right) & ; & n=0
|
||
\end{align}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{H}=\frac{i\hat{\rho}_{0}c_{p}\frac{f_{\kappa}}{1+\epsilon_{s}}}{1-\frac{f_{\kappa}}{1+\epsilon_{s}}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula
|
||
\[
|
||
\tilde{Q}_{n}=\frac{c_{p}}{1-\Pr}\left(\frac{f_{\nu,n}}{1-f_{\nu,n}}-\Pr\nolimits _{0}\frac{f_{\kappa,n}}{1-f_{\kappa,n}}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
TurbulentDuct
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
s with solids
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\rho_{s}c_{s}S_{s}\frac{\partial T_{s}}{\partial t}-\frac{\partial}{\partial x}\left(\kappa_{s}S_{s}\frac{\partial T_{s}}{\partial x}\right)=-Q_{s\to f}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Discretized:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\rho_{s}c_{s}S_{s}\frac{\partial T_{s}}{\partial t}-\frac{\partial}{\partial x}\left(\kappa_{s}S_{s}\frac{\partial T_{s}}{\partial x}\right)=-Q_{s\to f}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
V_{vs}\rho_{s}c_{s}\frac{\partial T_{s,i}}{\partial t}+Q_{r}-Q_{l}+Q_{s\to f}=0
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $T_{s}$
|
||
\end_inset
|
||
|
||
is the area-averaged temperature of the solid and
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Q_{s\to f}=Q_{s\to f}\left(T_{w},\dots\right)=S_{s}\hat{H}_{s}\left(\hat{T}_{s}-\hat{T}_{w}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
is the transverse heat transfer, which is a function of the
|
||
\emph on
|
||
wall
|
||
\emph default
|
||
temperature.
|
||
The wall temperature, on its turn is a function of the time-averaged temperatur
|
||
e of the solid and the transverse heat transfer:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
T_{w}=T_{w}\left(T_{s},Q_{s\to f}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Local
|
||
\begin_inset Formula $T_{s}$
|
||
\end_inset
|
||
|
||
is
|
||
\begin_inset Formula $T_{s,l}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $i\omega T_{s,l}-\alpha_{s}\frac{\partial^{2}T_{s,l}}{\partial y^{2}}=0$
|
||
\end_inset
|
||
|
||
with symmetric boundary conditions:
|
||
\begin_inset Formula $T_{s,l}\left(\pm y_{0}\right)=T_{w}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Rightarrow T_{s,l}=A+B\cosh\left(\left(1+i\right)\frac{y}{\delta_{\kappa,s}}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left\langle T_{s,l}\right\rangle =T_{s}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
—–
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $T_{s,l}=A+B\cosh\left(\left(1+i\right)\frac{y}{\delta_{\kappa,s}}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
– Using boundary conditions:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $T_{s,l}(y_{0})=T_{w}=A+B\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $A=T_{w}-B\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
such that:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $T_{s,l}=T_{w}+B\left(\cosh\left(\left(1+i\right)\frac{y}{\delta_{\kappa,s}}\right)-\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
and:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\left\langle T_{s,l}\right\rangle =T_{s}=T_{w}+B\left[\frac{\sinh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)}{\left(1+i\right)\frac{y}{\delta_{\kappa,s}}}-B\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)\right]\Rightarrow\frac{T_{s}-T_{w}}{\frac{\sinh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)}{\left(1+i\right)\frac{y}{\delta_{\kappa,s}}}-\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)}=B$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Sucht that
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $T_{s,l}=T_{w}+\left(T_{s}-T_{w}\right)\frac{1-\frac{\cosh\left(\left(1+i\right)\frac{y}{\delta_{\kappa,s}}\right)}{\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)}}{1-\frac{\sinh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)}{\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\cosh\left(\left(1+i\right)\frac{y_{0}}{\delta_{\kappa,s}}\right)}}=T_{w}+\left(T_{s}-T_{w}\right)\frac{1-h_{\kappa,s}}{1-f_{\kappa,s}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
The temperature distribution in the solid obeys
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
T_{s,l}=T_{w}+\left(T_{s}-T_{w}\right)\frac{1-h_{s}}{1-f_{s}},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
And the heat input equals
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $Q_{s\to f}=\kappa_{s}\Pi\frac{\partial T_{s,l}}{\partial n}=-\kappa_{s}\Pi\frac{\partial T_{s}}{\partial y}|_{y_{0}}\overset{\mathrm{par.\,plates}}{=}=$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
with
|
||
\begin_inset Formula $\mathbf{n}$
|
||
\end_inset
|
||
|
||
pointing from the fluid into the solid (do not know if this is according
|
||
to my definition)
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
for parallel plates:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $T_{s,l}=T_{w}+\left(T_{s}-T_{w}\right)\frac{1-h_{s}}{1-f_{s}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $s_{t}=r_{h}\sqrt{\frac{\rho_{0}c_{p}\omega}{\kappa}}=\sqrt{\Pr}s$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $s_{s}=r_{h,s}\sqrt{\frac{\rho_{s}c_{s}\omega}{\kappa_{s}}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
For parallel plates:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $Q_{s\to f}=-\kappa_{s}\Pi\frac{\partial T_{s,l}}{\partial y}|_{y=r_{h,s}}=-\kappa_{s}\frac{S_{s}}{r_{h,s}}\left(T_{s}-T_{w}\right)\frac{1}{1-f_{s}}\frac{\partial}{\partial y}\left(\frac{-\cosh\left(\sqrt{i}s_{s}\frac{y}{r_{h,s}}\right)}{\cosh\left(\sqrt{i}s_{s}\right)}\right)|_{y=r_{h,s}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
— For parallel-plates
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $Q_{s\to f}=\kappa_{s}\frac{S_{s}}{r_{h,s}}\left(T_{s}-T_{w}\right)\frac{1}{1-f_{s}}\frac{\partial}{\partial y}\left(\frac{\cosh\left(\sqrt{i}s_{s}\frac{y}{r_{h,s}}\right)}{\cosh\left(\sqrt{i}s_{s}\right)}\right)|_{y=r_{h,s}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $Q_{s\to f}=\frac{\kappa_{s}S_{s}}{r_{h,s}^{2}}\left(T_{s}-T_{w}\right)\frac{\sqrt{i}s_{s}}{1-f_{s}}\tanh\left(\sqrt{i}s_{s}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
—
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Using:
|
||
\begin_inset Formula $f_{s}=\frac{\tanh\left(\sqrt{i}s_{s}\right)}{\sqrt{i}s_{s}}\Rightarrow\tanh\left(\sqrt{i}s_{s}\right)=f_{s}\sqrt{i}s_{s}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Fill in:
|
||
\begin_inset Formula $Q_{s\to f}=\frac{\kappa_{s}S_{s}}{r_{h,s}^{2}}\left(T_{s}-T_{w}\right)\frac{is_{s}^{2}f_{s}}{1-f_{s}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Q_{s\to f}\overset{\mathrm{par.\,plates}}{=}=-\kappa_{s}\Pi\frac{\partial T_{s}}{\partial y}|_{y_{0}}=\frac{\kappa_{s}S_{s}}{r_{h,s}^{2}}\left(T_{s}-T_{w}\right)\frac{is_{s}^{2}f_{s}}{1-f_{s}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Hence
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Q_{s\to f}=S_{s}H_{s}\left(T_{s}-T_{w}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Such that
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
H_{s}=\frac{\kappa_{s}}{r_{h,s}^{2}}\frac{is_{s}^{2}f_{s}}{1-f_{s}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Wall temperature not prescribed
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Check if this is in agreement with other one
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega\frac{f_{\kappa}}{1+\epsilon_{s}}}{1-\frac{f_{\kappa}}{1+\epsilon_{s}}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\dfrac{f_{\kappa}}{1-\frac{f_{\kappa}}{1+\epsilon_{s}}}\left(\frac{1-\lambda f_{\kappa}-\Pr\left(1-f_{\nu}\right)}{1-f_{\nu}}\right)-\frac{f_{\kappa}-f_{\nu}}{1-f_{\nu}}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Using:
|
||
\begin_inset Formula $\lambda=\frac{1+\frac{\epsilon_{s}f_{\nu}}{f_{\kappa}}}{1+\epsilon_{s}}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\lambda\to1$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\epsilon_{s}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega f_{\kappa}}{1-f_{\kappa}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\dfrac{f_{\kappa}}{1-f_{\kappa}}\left(\frac{1-f_{\kappa}-\Pr\left(1-f_{\nu}\right)}{1-f_{\nu}}\right)-\frac{f_{\kappa}-f_{\nu}}{1-f_{\nu}}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Work under same denominator:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}\omega c_{p}f_{\kappa,n}}{1-f_{\kappa,n}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\frac{f_{\kappa}\left(1-f_{\kappa}-\Pr\left(1-f_{\nu}\right)\right)-\left(1-f_{\kappa}\right)\left(f_{\kappa}-f_{\nu}\right)}{\left(1-f_{\kappa}\right)\left(1-f_{\nu}\right)}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Split off Prandtl term:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega f_{\kappa}}{1-f_{\kappa}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\frac{f_{\kappa}\left(1-f_{\kappa}\right)-\left(1-f_{\kappa}\right)\left(f_{\kappa}-f_{\nu}\right)}{\left(1-f_{\kappa}\right)\left(1-f_{\nu}\right)}-\frac{\Pr f_{\kappa}}{\left(1-f_{\kappa}\right)}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
– Work out rest
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega f_{\kappa}}{1-f_{\kappa}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\frac{\left(1-f_{\kappa}\right)f_{\nu}}{\left(1-f_{\kappa}\right)\left(1-f_{\nu}\right)}-\frac{\Pr f_{\kappa}}{\left(1-f_{\kappa}\right)}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Finally:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega f_{\kappa}}{1-f_{\kappa}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\frac{f_{\nu}}{\left(1-f_{\nu}\right)}-\frac{\Pr f_{\kappa}}{\left(1-f_{\kappa}\right)}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
What happens for
|
||
\begin_inset Formula $\omega\to0$
|
||
\end_inset
|
||
|
||
?
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\lim_{\omega\to0}\epsilon_{s}=\sqrt{\frac{\kappa_{0}\rho_{0}c_{p}}{\kappa_{s}\rho_{s}c_{s}}}\frac{\tanh\left(\sqrt{i\frac{\rho_{0}c_{p}\omega}{\kappa}}r_{h,s}\right)}{\tanh\left(\sqrt{i\frac{\rho_{s}c_{s}\omega}{\kappa_{s}}}r_{h,s}\right)}=\sqrt{\frac{\kappa_{0}\rho_{0}c_{p}}{\kappa_{s}\rho_{s}c_{s}}}\frac{\tanh\left(\sqrt{i\frac{\rho_{0}c_{p}\omega}{\kappa}}r_{h,s}\right)}{\tanh\left(\sqrt{i\frac{\rho_{s}c_{s}\omega}{\kappa_{s}}}r_{h,s}\right)}=\sqrt{\frac{\kappa_{0}\rho_{0}c_{p}}{\kappa_{s}\rho_{s}c_{s}}}\frac{\sqrt{i\frac{\rho_{0}c_{p}\omega}{\kappa_{0}}}r_{h}}{\sqrt{i\frac{\rho_{s}c_{s}\omega}{\kappa_{s}}}r_{h,s}}=\frac{\rho_{0}c_{p}r_{h}}{\rho_{s}c_{s}r_{h,s}}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
For
|
||
\begin_inset Formula $\omega\to0$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\lambda\to1$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\epsilon_{s}\to g_{s}\frac{\rho_{0}c_{p}r_{h}}{\rho_{s}c_{s}r_{h,s}}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $f_{\nu,\kappa}\to1$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\lim\limits _{\omega\to0}Q_{s\to f}==S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega f_{\kappa}}{1+\epsilon_{s}-f_{\kappa}}\hat{T}-\frac{c_{p}}{\left(1-\Pr\right)\left(1+\epsilon_{s}\right)}\left[\dfrac{f_{\kappa}}{1-\frac{f_{\kappa}}{1+\epsilon_{s}}}\left(\frac{1-\lambda f_{\kappa}-\Pr\left(1-f_{\nu}\right)}{1-f_{\nu}}\right)-\frac{f_{\kappa}-f_{\nu}}{1-f_{\nu}}\right]\hat{m}\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\lambda=\frac{1+\frac{\epsilon_{s}f_{\nu}}{f_{\kappa}}}{1+\epsilon_{s}}$
|
||
\end_inset
|
||
|
||
,
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
For parallel plates
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\epsilon_{s}=\sqrt{\frac{\kappa_{0}\rho_{0}c_{p}}{\kappa_{s}\rho_{s}c_{s}}}\frac{\tanh\left(\sqrt{i\Pr}s\right)}{\tanh\left(\sqrt{i}s_{t,s}\right)}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Limit for
|
||
\begin_inset Formula $s$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $s_{t,s}$
|
||
\end_inset
|
||
|
||
to zero gives:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Filling this in in
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
Q_{s\to f}=S_{f}\frac{i\hat{\rho}_{0}c_{p}\omega f_{\kappa}}{1+\epsilon_{s}-f_{\kappa}}\hat{T}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
which is
|
||
\begin_inset Formula $Q_{s\to f}=\frac{i\hat{\rho}_{0}\omega f_{\kappa}}{1-f_{\kappa}}\left(\hat{T}-\hat{T}_{w}\right)-\frac{c_{p}}{1-\Pr}\left(\frac{f_{\nu,n}}{1-f_{\nu,n}}-\Pr\nolimits _{0}\frac{f_{\kappa,n}}{1-f_{\kappa,n}}\right)m\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Filling in for
|
||
\begin_inset Formula $T_{w}$
|
||
\end_inset
|
||
|
||
:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $Q_{s\to f}=\frac{i\hat{\rho}_{0}\omega f_{\kappa}}{1-f_{\kappa}}\left(\hat{T}-\left(T_{s}-\frac{Q_{s\to f}}{J_{s}S_{f}}\right)\right)-\frac{c_{p}}{1-\Pr}\left(\frac{f_{\nu,n}}{1-f_{\nu,n}}-\Pr\nolimits _{0}\frac{f_{\kappa,n}}{1-f_{\kappa,n}}\right)m\frac{\mathrm{d}\hat{T}_{w,0}}{\mathrm{d}x}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Boundary conditions
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
with solid to
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
without solid
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Q_{s}=hS_{s}\left(T_{s}-T\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
with solid to
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
with solid
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
Q_{s,1} & = & Q_{s,2}\\
|
||
T_{s,1} & = & T_{s,2}
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
|
||
\noun on
|
||
Regenerator
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
k_{\mathrm{tort}}\left(\frac{\partial m}{\partial t}+\frac{\partial mu}{\partial x}\right)+\frac{\partial p}{\partial x}+\mathcal{R}=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
If
|
||
\begin_inset Formula $k_{\mathrm{tort}}$
|
||
\end_inset
|
||
|
||
not given, it is computed according to Eq.
|
||
(14) of Swift and Ward:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
k_{\mathrm{tort}}=1+\frac{\left(1-\phi\right)^{2}}{2\left(2\phi-1\right)}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathcal{R}=-\frac{1}{2}\frac{f}{r_{h}}m|u|
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
With
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Re=\frac{4\rho|u|r_{h}}{\mu}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Using
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
f=\frac{c_{1}}{Re}+c_{2}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Filling in:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathcal{R}=S_{f}\left(\frac{c_{1}\mu}{8r_{h}^{2}}\frac{m}{\rho}+\frac{c_{2}}{2r_{h}}\mathrm{sgn}\left(m\right)mu\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
boundary conditions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename fig/duct_vars.eps
|
||
width 80text%
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset VSpace medskip
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Tabular
|
||
<lyxtabular version="3" rows="4" columns="3">
|
||
<features tabularvalignment="middle">
|
||
<column alignment="left" valignment="top">
|
||
<column alignment="left" valignment="top">
|
||
<column alignment="left" valignment="top" width="0pt">
|
||
<row>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
node
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Variables
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Equations
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="left" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
huge{
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $\bullet$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\rho,mu,T,p,Ts$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Continuity, Energy,
|
||
\begin_inset Formula $mu=\frac{m^{2}}{\rho S_{f}}$
|
||
\end_inset
|
||
|
||
, Solid energy
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="left" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\blacksquare$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $m$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Momentum
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Box$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $m,mH,T$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Dependent on b.c.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Staggered grid equations and variables
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
PressureBc
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Momentum equation (prescribes pressure
|
||
\begin_inset Formula $p_{p}$
|
||
\end_inset
|
||
|
||
)
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Delta x\frac{\partial\mathbf{m}_{L}}{\partial t}+\left(\widehat{\mathbf{mu}}_{0}-\widehat{\mathbf{mu}}_{L}\right)+S_{f,L}\left(\hat{\mathbf{p}}_{0}-\hat{\mathbf{p}}_{p}\right)+\Delta x\mbox{\textbf{\mathcal{R}}}_{L}=\mathbf{0}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $mH$
|
||
\end_inset
|
||
|
||
extrapolated from inside
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $T=T_{p}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $T_{s}=T_{s,p}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
And the boundary condition for the temperature is computed assuming adiabatic
|
||
compression-expansion.
|
||
Currently, this is implemented for thermally perfect gases only:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
c_{p}\left(T\right)\mathrm{d}T=\frac{\mathrm{d}p}{\rho}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
FIlling in the perfect gas law and a bit of bookkeeping results in
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{1}{R_{s}}\int\limits _{T_{0}}^{T_{p}}\frac{c_{p}\left(T\right)}{T}\mathrm{d}T-\ln\left(\frac{p_{p}-p_{0}}{p_{0}}\right)=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Note that in general, to solve this equation for the temperature requires
|
||
a numerical integration, however for the currently implemented gases,
|
||
\begin_inset Formula $c_{p}$
|
||
\end_inset
|
||
|
||
is a polynomial function of
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
c_{p}(T)=\sum_{i=0}^{N_{c_{p}}}c_{p,i}T^{i}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
In that case
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{1}{R_{s}}\left(c_{p,0}\ln\left(\frac{T_{p}}{T_{0}}\right)+\sum_{i=1}^{N_{c_{p}}}\frac{c_{p,i}\left(T_{p}^{i}-T_{0}^{i}\right)}{i}\right)-\ln\left(\frac{p_{p}-p_{0}}{p_{0}}\right)=0.\label{eq:T-p-adiabatic}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Equation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:T-p-adiabatic"
|
||
|
||
\end_inset
|
||
|
||
is solved using a one-dimensional root finding algorithm (see Section
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "sec:One-dimensional-function-solvers"
|
||
|
||
\end_inset
|
||
|
||
).
|
||
Note that for an ideal gas an explicit formula is available:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
T_{p,\mathrm{ideal}}=T_{0}\left(\frac{p_{0}+p_{p}}{p_{0}}\right)^{\frac{\gamma_{0}-1}{\gamma_{0}}}.\label{eq:ideal-gas-isentropic-p-T}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Looking closely at Equation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:ideal-gas-isentropic-p-T"
|
||
|
||
\end_inset
|
||
|
||
, we find that
|
||
\begin_inset Formula $T_{p,\mathrm{ideal}}$
|
||
\end_inset
|
||
|
||
provides a good guess for the final solution.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
AdiabaticWall
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $m=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $mH=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $Q=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $Q_{s}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
IsoTWall
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $m=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $mH=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $T=T_{p}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
\begin_inset Formula $T_{s}=T_{s,p}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
ImpedanceBc
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\hat{\mathbf{p}}=\mathbf{Z}\cdot\hat{\mathbf{u}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathbf{R}=\hat{\mathbf{p}}-\mathbf{Z}\cdot\mathcal{\boldsymbol{F}}\cdot\left(\frac{\mathbf{m}_{bc}}{S_{f}\left(w_{1}\mathbf{\rho}_{1}+w_{2}\mathbf{\rho}_{2}\right)}\right)=\mathbf{0}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial\mathbf{R}}{\partial\hat{\mathbf{p}}_{bc}}=\mathbf{I}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial\mathbf{R}}{\partial\hat{\mathbf{m}}_{bc}}=-\mathbf{Z}\cdot\boldsymbol{\mathcal{F}}\cdot\mbox{diag}\left(\frac{1}{S_{f}\boldsymbol{\rho}_{bc}}\right)\cdot\boldsymbol{\mathcal{F}}^{-1}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial\mathbf{R}}{\partial\boldsymbol{\rho}_{1,2}}=w_{1,2}\mathbf{Z}\cdot\boldsymbol{\mathcal{F}}\cdot\mbox{diag}\left(\frac{\mathbf{m}_{bc}}{S_{f}\boldsymbol{\rho}_{bc}^{2}}\right)\cdot\boldsymbol{\mathcal{F}}^{-1}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
—
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\mathbf{Z}^{-1}\cdot\hat{\mathbf{p}}=\boldsymbol{\mathcal{F}}^{-1}\cdot\hat{\mathbf{u}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\boldsymbol{\mathcal{F}}\cdot\left[\left(\boldsymbol{\mathcal{F}}^{-1}\left(\mathbf{Z}^{-1}\cdot\hat{\mathbf{p}}\right)\right)\circ\mathbf{m}\right]=\boldsymbol{\mathcal{F}}^{-1}\cdot\mathbf{u}\circ\mathbf{m}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\boldsymbol{\mathcal{F}}\cdot\left[\left(\boldsymbol{\mathcal{F}}^{-1}\left(\mathbf{Z}^{-1}\cdot\hat{\mathbf{p}}\right)\right)\circ\mathbf{m}\right]=\widehat{\mathbf{m}\mathbf{u}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Temperature at the b.c.
|
||
computed from pressure:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $T_{bc}=T_{0}\left(\frac{p_{bc}+p_{p}}{p_{0}}\right)^{\frac{\gamma_{0}-1}{\gamma_{0}}}$
|
||
\end_inset
|
||
|
||
, dus
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathbf{R}=\hat{\mathbf{p}}_{bc}-\boldsymbol{\mathcal{F}}\cdot\left[p_{0}\left(\left(\frac{\mathbf{T}_{bc}}{T_{0}}\right)^{\frac{\gamma_{0}}{\gamma_{0}-1}}-\mathbf{1}\right)\right]=\mathbf{0}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\mathbf{R}=\hat{\mathbf{p}}_{bc}-\boldsymbol{\mathcal{F}}\cdot\left[p_{0}\left(\left(\frac{\mathbf{T}_{bc}}{T_{0}}\right)^{\frac{\gamma_{0}}{\gamma_{0}-1}}-\mathbf{1}\right)\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\partial\mathbf{R}}{\partial\mathbf{T}_{bc}}=\boldsymbol{\mathcal{F}}\cdot\mbox{diag}\left[\frac{\gamma_{0}}{\gamma_{0}-1}\frac{p_{0}}{T_{0}}\left(\left(\frac{\mathbf{T}_{bc}}{T_{0}}\right)^{\frac{1}{\gamma_{0}-1}}\right)\right]\cdot\boldsymbol{\mathcal{F}}^{-1}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial\mathbf{R}}{\partial\mathbf{T}_{bc}}=\boldsymbol{\mathcal{F}}\cdot\mbox{diag}\left[\frac{\gamma_{0}}{\gamma_{0}-1}\frac{p_{0}}{T_{0}}\left(\left(\frac{\mathbf{T}_{bc}}{T_{0}}\right)^{\frac{1}{\gamma_{0}-1}}\right)\right]\cdot\boldsymbol{\mathcal{F}}^{-1}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Finally:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathbf{R}=\boldsymbol{\mathcal{F}}\cdot\left[\left(\boldsymbol{\mathcal{F}}^{-1}\left(\mathbf{Z}^{-1}\cdot\hat{\mathbf{p}}\right)\right)\circ\left(\boldsymbol{\mathcal{F}}^{-1}\cdot\mathbf{m}\right)\right]-\widehat{\mathbf{m}\mathbf{u}}=\mathbf{0}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
And
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
\frac{\partial\mathbf{R}}{\partial\hat{\mathbf{p}}} & = & \boldsymbol{\mathcal{F}}\cdot\mathrm{diag}\left(\mathbf{m}\right)\cdot\boldsymbol{\mathcal{F}}^{-1}\cdot\mathbf{Z}^{-1}\\
|
||
\frac{\partial\mathbf{R}}{\partial\hat{\mathbf{m}}} & = & \boldsymbol{\mathcal{F}}\cdot\left[\mbox{diag}\left(\boldsymbol{\mathcal{F}}^{-1}\cdot\mathbf{Z}^{-1}\cdot\hat{\mathbf{p}}\right)\cdot\boldsymbol{\mathcal{F}}^{-1}\right]\\
|
||
\frac{\partial\mathbf{R}}{\partial\widehat{\mathbf{m}\mathbf{u}}} & = & -\mathbf{I}
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
VelocityBc
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Prescribed velocity, adiabatic compression/expansion.
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathbf{R}=\boldsymbol{\mathcal{F}}\cdot\left(\frac{\mathbf{m}_{bc}}{S_{f}\mathbf{\rho}_{bc}}\right)-\hat{\mathbf{u}}_{bc}=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial\mathbf{R}}{\partial\hat{\mathbf{m}}_{bc}}=\boldsymbol{\mathcal{F}}\cdot\mbox{diag}\left(\frac{1}{S_{f}\boldsymbol{\rho}_{bc}}\right)\cdot\boldsymbol{\mathcal{F}}^{-1}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Piston
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For the piston, the equation of motion
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
M\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+C_{m}\frac{\mathrm{d}x}{\mathrm{d}t}+K_{m}x=S_{l}p_{l}-S_{r}p_{r}+F
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
is solved, where
|
||
\begin_inset Formula $F$
|
||
\end_inset
|
||
|
||
is the external force applied to the Piston.
|
||
This external force can be either boundary condition, or applied by electromagn
|
||
etic interaction.
|
||
For both the right as well as the left volume, conservation of mass, energy
|
||
and the thermal equation of state is solved.
|
||
The continuity equations are
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
\frac{\partial\rho_{r}\left(V_{0r}-xS_{r}\right)}{\partial t}+m_{f} & = & 0,\\
|
||
\frac{\partial\rho_{l}\left(V_{0l}+xS_{l}\right)}{\partial t}+m_{l} & = & 0,
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Piston volume mass conservation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Variables
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
x,\\
|
||
p_{r},T_{r},\rho_{r} & & m_{r},mH_{r}\\
|
||
p_{l},T_{l},\rho_{l} & & m_{l},mH_{l}
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
and
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
p_{r}-\rho_{r}R_{s}T_{r} & = & 0\\
|
||
p_{l}-\rho_{l}R_{s}T_{l} & = & 0
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Piston volume energy conservation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{p_{r}}{\gamma-1}V_{r}\right)+p_{r}\frac{\mathrm{d}V_{r}}{\mathrm{d}t}+mH_{r}=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
or
|
||
\begin_inset Formula
|
||
\[
|
||
\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{p_{r}}{\gamma-1}V_{r}\right)-\frac{\mathrm{d}x_{p}}{\mathrm{d}t}S_{r}p_{r}+mH_{r}=0
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
To isentropic (and no mass flow):
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\mathrm{d}\left(\frac{p_{r}}{\gamma-1}V_{r}\right)+p_{r}\mathrm{d}V_{r}=0=\frac{\mathrm{d}p_{r}}{p_{r}\left(\gamma-1\right)}+\frac{1}{\gamma-1}\frac{\mathrm{d}V_{r}}{V_{r}}+\frac{\mathrm{d}V_{r}}{V_{r}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{\mathrm{d}p_{r}}{p_{r}}+\gamma\frac{\mathrm{d}V_{r}}{V_{r}}=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{p}{p_{0}}=\left(\frac{V_{0}}{V}\right)^{\gamma}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\frac{dp}{p}=-\gamma\frac{dV}{V}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
left side:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
\frac{1}{\gamma-1}\mathbf{D}\cdot\boldsymbol{\mathcal{F}}\cdot\left[\mathbf{p}_{l}\circ\left(V_{0r}-\mathbf{x}S_{l}\right)\right]+\boldsymbol{\mathcal{F}}\cdot\left[\mathbf{p}_{l}\circ\left(\boldsymbol{\mathcal{F}}^{-1}\cdot\mathbf{D}\cdot\left(V_{0l}-\hat{\mathbf{x}}S_{l}\right)\right)\right]+\boldsymbol{\mathcal{F}}\cdot\mathbf{mH}_{l} & = & 0
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
right side:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial\frac{p_{l}}{\gamma-1}\left(V_{0l}+xS_{l}\right)}{\partial t}+mH_{l}=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Mass in piston volumes
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The time-averaged amount of mass in a piston volume is
|
||
\begin_inset Formula
|
||
\[
|
||
m=\mathcal{F}_{0}\cdot\left[(V_{0}\pm S\mathbf{x})\circ\mathbf{\rho}\right]
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\boldsymbol{\mathcal{F}}_{0}$
|
||
\end_inset
|
||
|
||
is the first row in the Fourier transform matrix.
|
||
Therefore,
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{\partial m}{\partial\hat{\boldsymbol{\rho}}}=\mathcal{F}_{0}\cdot\left[\mathrm{diag}(V_{0}\pm S\mathbf{x})\cdot\boldsymbol{\mathcal{F}}^{-1}\right]
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Unconnected side
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For example, left side is not connected:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\rho_{l}\left(V_{0l}+xS_{l}\right)=m_{l}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Isentropic:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
p-p_{0}\left(\frac{\rho}{\rho_{0}}\right)^{\gamma}=0
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Later:
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Connectors
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Connector theory
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Conservation of mass
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
sm_{1}=sm_{2}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Conservation of energy
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
smH_{1}=smH_{2}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
sQ_{1}=sQ_{2}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
mH at the boundary is half the
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
mH_{boundary}=\frac{1}{2}\left(mH_{1}+mH_{2}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Heat flow between
|
||
\noun on
|
||
Duct
|
||
\noun default
|
||
s
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Minor loss - conversion of kinetic energy to thermal energy
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Minor loss generates a decrease in the total (stagnation) pressure:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Delta p_{\mathrm{tot}}=-\frac{1}{2}K\rho_{u}u_{u}^{2}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
For incompressible flow, the stagnation pressure over density is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{\mathrm{tot}}=p+\frac{1}{2}\rho u^{2}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
For compressible flow, the stagnation pressure over density is
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p_{tot}=p\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{\gamma-1}}\Rightarrow\frac{p_{tot}}{p}=\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{\gamma-1}}\Rightarrow\frac{p}{p_{tot}}=\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{1-\gamma}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\rho=\rho_{tot}\left(\frac{p}{p_{tot}}\right)^{\frac{1}{\gamma}}\Rightarrow\rho_{tot}=\rho\left(\frac{p_{tot}}{p}\right)^{\frac{1}{\gamma}}\Rightarrow\rho_{tot}=\rho\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{\gamma-1}\frac{1}{\gamma}}=\rho\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{1}{\gamma-1}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\frac{p_{tot}}{\rho_{tot}}=\frac{p\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{\gamma-1}}}{\rho\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{1}{\gamma-1}}}=\frac{p}{\rho}\left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{\gamma-1}-\frac{1}{\gamma-1}}=\frac{p}{\rho}\left(1+\frac{\gamma-1}{2}M^{2}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{\mathrm{tot}}=p\left(1+\frac{1}{2}\frac{u^{2}}{c_{p}T}\right)^{\frac{\gamma}{\gamma-1}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A change in total pressure over density results in an increase in entropy.
|
||
For a callorically perfect gas, the entropy is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
s=s_{0}+c_{v}\ln\left[\left(\frac{p}{p_{0}}\right)\left(\frac{\rho_{0}}{\rho}\right)^{\gamma}\right],
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
so the change in entropy from state 1 to 2 is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
s_{2}-s_{1}=c_{v}\ln\left[\left(\frac{p_{2}}{p_{1}}\right)\left(\frac{\rho_{1}}{\rho_{2}}\right)^{\gamma}\right]=-R_{s}\ln\left(\frac{p_{\mathrm{tot},2}}{p_{\mathrm{tot},1}}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Using Hoeijmakers eq 6.15e:
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
For incompressible flow the total enthalpy is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
dH=T\mathrm{d}s+\rho^{-1}\mathrm{d}p+u\mathrm{d}u
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
hence from the internal energy we can derive
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
s=s_{0}+c\ln\left(T/T_{0}\right),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
which is, since
|
||
\begin_inset Formula $H_{d}=H_{u}$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
s_{d}-s_{u}=c\ln\left(\frac{T_{d}}{T_{u}}\right)=-c\ln\left(\frac{\frac{p_{\mathrm{tot}}}{\rho}_{d}}{\frac{p_{\mathrm{tot}}}{\rho}_{u}}\right)=-c\ln\left(1+\frac{\Delta p_{\mathrm{tot}}}{p_{\mathrm{tot},u}}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
such that the change in exergy flow is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
m\Delta E_{x}=-T_{0}m\left(s_{d}-s_{u}\right),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
so
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Delta E_{x}=T_{0}c\ln\left(1-\frac{1}{2}\frac{Ku_{u}^{2}}{\left(\frac{p_{\mathrm{tot},u}}{\rho}\right)}\right)
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Subsubsection
|
||
Generalization for compressible flow
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
At a sharp interface, generally turbulence is created, which converts kinetic
|
||
energy to thermal energy.
|
||
To model this effect, minor loss coefficients are introduced.
|
||
If the minor loss coefficient is zero, no entropy is generated at an interface.
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\mathrm{d}h=T\mathrm{d}s+\rho^{-1}\mathrm{d}p
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Isentropically:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{T}{T_{0}}=\left(\frac{p}{p_{0}}\right)^{\frac{\gamma-1}{\gamma}}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
For a calorically perfect gas, the entropy is
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
From Hoeijmakers:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $s=s_{0}+c_{v}\ln\left[\left(\frac{p}{p_{0}}\right)\left(\frac{\rho_{0}}{\rho}\right)^{\gamma}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Rework to:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $s=s_{0}+c_{v}\ln\left[\left(\frac{p}{p_{0}}\right)\left(\frac{Tp_{0}}{T_{0}p}\right)^{\gamma}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $s=s_{0}+c_{v}\ln\left[\left(\frac{p}{p_{0}}\right)\left(\frac{p_{0}}{p}\right)^{\gamma}\left(\frac{T}{T_{0}}\right)^{\gamma}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $s=s_{0}+c_{v}\ln\left[\left(\frac{p}{p_{0}}\right)^{1-\gamma}\left(\frac{T}{T_{0}}\right)^{\gamma}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $s=s_{0}+-c_{v}\ln\left[\left(\frac{p_{0}}{p}\right)^{1-\gamma}\left(\frac{T}{T_{0}}\right)^{-\gamma}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $s=s_{0}+\left(\gamma-1\right)c_{v}\ln\left[\left(\frac{p_{0}}{p}\right)\left(\frac{T}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $s=s_{0}+R_{s}\ln\left[\left(\frac{p_{0}}{p}\right)\left(\frac{T}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
s=s_{0}+R_{s}\ln\left[\left(\frac{p_{0}}{p}\right)\left(\frac{T}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]\label{eq:entropy_idealgas}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Isentropic change in temperature:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $T=T_{0}\left(\frac{p}{p_{0}}\right)^{\frac{\gamma-1}{\gamma}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Such that:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Delta h_{irr}=c_{p}\Delta T=c_{p}T_{0}\left(\left(\frac{p}{p_{0}}\right)^{\frac{\gamma-1}{\gamma}}-1\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Fully non-isentropic change in temperature:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $dh=Tds+0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $c_{p}\frac{dT}{T}=dS$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
with:
|
||
\begin_inset Formula $ds=\frac{}{}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
and:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Delta H=\Delta\left(c_{p}T+\frac{1}{2}u^{2}\right)=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $c_{p}\left(T-T_{0}\right)+\frac{1}{2}\left(u^{2}-u_{0}^{2}\right)=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
– Fill in above:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\underbrace{c_{p}T_{0}\left(\left(\frac{p}{p_{0}}\right)^{\frac{\gamma-1}{\gamma}}-1\right)}_{\Delta h_{irr}}+\frac{1}{2}\left(u^{2}-u_{0}^{2}\right)=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Hence:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $c_{p}T_{0}\left(\left(\frac{p}{p_{0}}\right)^{\frac{\gamma-1}{\gamma}}-1\right)=-\frac{1}{2}\left(u^{2}-u_{0}^{2}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
For very small
|
||
\begin_inset Formula $\Delta p$
|
||
\end_inset
|
||
|
||
, this becomes:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $c_{p}T_{0}\frac{\gamma-1}{\gamma}\frac{\Delta p}{p_{0}}=-\frac{1}{2}\left(u^{2}-u_{0}^{2}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
is
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p=p_{0}-\frac{1}{2}\rho\left(u^{2}-u_{0}^{2}\right)\Rightarrow p+\frac{1}{2}\rho u^{2}=p_{0}+\frac{1}{2}\rho u_{0}^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
which is Bernouillis law
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
The result of minor losses is a reduction in Exergy.
|
||
To generalize minor loss to the full compressible flow, minor loss is modeled
|
||
as
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Delta E_{x}=-K\left(\frac{T_{0}}{T_{u}}\right)\frac{1}{2}u_{u}^{2},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where subscript
|
||
\begin_inset Formula $u$
|
||
\end_inset
|
||
|
||
denotes the upstream condition and
|
||
\begin_inset Formula $\Delta E_{x}$
|
||
\end_inset
|
||
|
||
is the change in exergy per unit mass:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
E_{x}=h+\frac{1}{2}u^{2}-T_{0}s,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
which is also called the available energy.
|
||
|
||
\begin_inset Formula $K$
|
||
\end_inset
|
||
|
||
is the traditional minor loss coefficient.
|
||
Since
|
||
\begin_inset Formula $H=h+\frac{1}{2}u^{2}$
|
||
\end_inset
|
||
|
||
is constant, the change in exergy is
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Delta E_{x}=-T_{0}\left(s_{d}-s_{u}\right),
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where subscript
|
||
\begin_inset Formula $d$
|
||
\end_inset
|
||
|
||
denotes the downstream condition.
|
||
Filling in Eq.
|
||
(
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:entropy_idealgas"
|
||
|
||
\end_inset
|
||
|
||
) for the entropy of an ideal gas results in
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $-T_{0}(s_{d}-s_{u})=-T_{0}R_{s}\left(\ln\left[\left(\frac{p_{0}}{p_{d}}\right)\left(\frac{T_{d}}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]-\ln\left[\left(\frac{p_{0}}{p_{u}}\right)\left(\frac{T_{u}}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $-T_{0}(s_{d}-s_{u})=T_{0}R_{s}\left(\ln\left[\left(\frac{p_{0}}{p_{u}}\right)\left(\frac{T_{u}}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]-\ln\left[\left(\frac{p_{0}}{p_{d}}\right)\left(\frac{T_{d}}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\right]\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $-T_{0}(s_{d}-s_{u})=T_{0}R_{s}\ln\left[\left(\frac{p_{0}}{p_{u}}\right)\left(\frac{T_{u}}{T_{0}}\right)^{\frac{\gamma}{\gamma-1}}\left(\frac{p_{d}}{p_{o}}\right)\left(\frac{T_{0}}{T_{d}}\right)^{\frac{\gamma}{\gamma-1}}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $-T_{0}(s_{d}-s_{u})=T_{0}R_{s}\ln\left[\left(\frac{p_{d}}{p_{u}}\right)\left(\frac{T_{u}}{T_{d}}\right)^{\frac{\gamma}{\gamma-1}}\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Delta E_{x}=E_{x,d}-E_{x,u}=T_{0}R_{s}\ln\left[\left(\frac{p_{d}}{p_{u}}\right)\left(\frac{T_{u}}{T_{d}}\right)^{\frac{\gamma}{\gamma-1}}\right]=-K\left(\frac{T_{0}}{T_{u}}\right)\frac{1}{2}u_{u}^{2}.\label{eq:DeltaE_steady}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
If we assume incompressible flow and if we take
|
||
\begin_inset Formula $p_{d}=p_{u}+\Delta p$
|
||
\end_inset
|
||
|
||
, with
|
||
\begin_inset Formula $\Delta p\ll1$
|
||
\end_inset
|
||
|
||
, this equation can be linearized to obtain
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Delta E_{x}=T_{0}c_{p}\ln\left[\left(\frac{p_{d}}{p_{u}}\right)^{\gamma-1}\left(\frac{T_{u}}{T_{d}}\right)\right]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Delta E_{x}=T_{0}c_{p}\left(\left(\gamma-1\right)\frac{\Delta p}{p_{u}}-\frac{\Delta T}{T_{u}}\right)$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
In the limit of incompressible flow,
|
||
\begin_inset Formula $\gamma\to1$
|
||
\end_inset
|
||
|
||
, hence
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\Delta E_{x}=-\frac{T_{0}}{T_{u}}c_{p}\Delta T$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\Delta E_{x}=-\frac{T_{0}}{T_{u}}c\Delta T=-K\left(\frac{T_{0}}{T_{u}}\right)\frac{1}{2}u_{u}^{2}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Moreover, for incompressible flow
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
H=cT+\frac{p}{\rho}+\frac{1}{2}u^{2}=\mathrm{const},
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
the minor loss directly converts kinetic energy to thermal energy for an
|
||
incompressible flow.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Generalization for oscillating flow
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For oscillating flow, there is no real
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
upstream
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
downstream
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
position.
|
||
To generalize Eq.
|
||
(
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:DeltaE_steady"
|
||
|
||
\end_inset
|
||
|
||
):
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Flow from 1 to 2
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p_{\mathrm{tot},2}-p_{\mathrm{tot},1}=-K_{1\to2}\frac{1}{8}\rho_{1}\left(s|u_{1}|+u_{1}\right)^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Flow from 2 to 1
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p_{\mathrm{tot},1}-p_{\mathrm{tot},2}=-K_{2\to1}\rho_{2}\frac{1}{8}\left(s|u_{2}|+u_{2}\right)^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Mutual exclusive sum:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $p_{\mathrm{tot},2}-p_{\mathrm{tot},1}=K_{2\to1}\rho_{2}\frac{1}{8}\left(s|u_{2}|+u_{2}\right)^{2}-K_{1\to2}\rho_{1}\frac{1}{8}\left(s|u_{1}|+u_{1}\right)^{2}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
p_{\mathrm{tot},2}-p_{\mathrm{tot},1}=K_{2\to1}\frac{\rho_{2}}{8}\left(s|u_{2}|+u_{2}\right)^{2}-K_{1\to2}\frac{\rho_{1}}{8}\left(s|u_{1}|+u_{1}\right)^{2}.
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
TubeConnector
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The SimpleTubeConnector can connect two Tube segments together.
|
||
At the interface, continuity of mass and energy flow is enforced.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Transition of kinetic energy to heat
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
TubePistonConnector
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
ConnectorVolume
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A
|
||
\noun on
|
||
ConnectorVolume
|
||
\noun default
|
||
is a special kind of gas volume on which
|
||
\noun on
|
||
Tube
|
||
\noun default
|
||
s and
|
||
\noun on
|
||
Piston
|
||
\noun default
|
||
s can be connected.
|
||
This way, branches can be created
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
V_{c}\frac{\partial\rho}{\partial t}+\sum\nolimits _{i}m_{i}=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
energy:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\frac{V_{c}}{\gamma-1}\frac{\partial p}{\partial t}+\sum\nolimits _{i}mH_{i}+Q_{i}=0
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
mH_{i}=\mathrm{extrapolated\,from\,tube}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Minor loss
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Systems
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Solvers
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
One-dimensional function solvers
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "sec:One-dimensional-function-solvers"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Gradient-based
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Gradient free
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "subsec:Gradient-free"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
As a gradient free one-dimensional function solver, we use Brent's mehhod.
|
||
Brent's method combines root bracketing, bisection and inverse quadratic
|
||
interpolation to find the root of the function without using the gradient.
|
||
See Wikipedia for more information.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Minimizers
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|