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− | + | {{Stub}} | |
− | { | + | #LyX 2.1 created this file. For more info see http://www.lyx.org/ |
− | + | \lyxformat 474 | |
− | + | \begin_document | |
− | + | \begin_header | |
− | + | \textclass article | |
− | + | \use_default_options true | |
− | + | \maintain_unincluded_children false | |
− | + | \language english | |
− | + | \language_package default | |
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− | + | \font_roman default | |
− | + | \font_sans default | |
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+ | \font_default_family default | ||
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+ | \font_tt_scale 100 | ||
+ | \graphics default | ||
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+ | \use_package mathtools 1 | ||
+ | \use_package mhchem 1 | ||
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+ | \use_package stmaryrd 1 | ||
+ | \use_package undertilde 1 | ||
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+ | \cite_engine_type default | ||
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+ | \paperorientation portrait | ||
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+ | \justification true | ||
+ | \use_refstyle 1 | ||
+ | \index Index | ||
+ | \shortcut idx | ||
+ | \color #008000 | ||
+ | \end_index | ||
+ | \secnumdepth 3 | ||
+ | \tocdepth 3 | ||
+ | \paragraph_separation indent | ||
+ | \paragraph_indentation default | ||
+ | \quotes_language english | ||
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+ | \end_header | ||
+ | |||
− | + | \begin_body | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | \begin_layout Standard | |
+ | The state determinted by two temperatures (refering | ||
+ | \begin_inset Formula $T_{0}$ | ||
+ | \end_inset | ||
+ | |||
− | + | as constant heat sink) : | |
+ | \begin_inset Formula | ||
+ | $$ | ||
+ | \begin{cases} | ||
+ | C\frac{dT}{dt}+K\left(T-T_{0}\right)=K_{h}\left(T_{h}-T\right) & \, | ||
− | + | C_{h}\frac{dT_{h}}{dt}=P_{h}-K_{h}\left(T_{h}-T\right) & \, | |
+ | \end{cases} | ||
+ | $$ | ||
+ | |||
− | + | \end_inset | |
− | + | ||
− | + | Matrix representation: | |
+ | \begin_inset Formula | ||
+ | $$ | ||
+ | $$ | ||
+ | |||
− | + | \end_inset | |
− | + | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | \end_layout | |
+ | |||
− | + | \begin_layout Standard | |
+ | Eigenvalues of | ||
+ | \begin_inset Formula $M$ | ||
+ | \end_inset | ||
+ | |||
− | + | : | |
+ | \begin_inset Formula | ||
+ | $$\begin{array}{rcl} | ||
+ | \lambda & = & -\frac{CK_{h}+C_{h}K+C_{h}K_{h}\pm\sqrt{C^{2}K_{h}^{2}-2CC_{h}KK_{h}+2CC_{h}K_{h}^{2}+C_{h}^{2}K^{2}+2C_{h}^{2}KK_{h}+C_{h}^{2}K_{h}^{2}}}{2CC_{h}} | ||
− | = | + | \lambda & = & -\frac{1}{2}\left[\frac{K_{h}}{C_{h}}+\frac{K}{C}+\frac{K_{h}}{C}\pm\sqrt{\frac{K_{h}^{2}}{C_{h}^{2}}-2\frac{KK_{h}}{CC_{h}}+2\frac{K_{h}^{2}}{CC_{h}}+\frac{K^{2}}{C^{2}}+2\frac{KK_{h}}{C^{2}}+\frac{K_{h}^{2}}{C^{2}}}\right] |
− | + | ||
− | * | + | \, & \, & \mbox{denoting}\;\begin{cases} |
+ | \gamma & =\frac{K}{C} | ||
+ | \gamma_{m} & =\frac{K_{h}}{C} | ||
+ | \gamma_{h} & =\frac{K_{h}}{C_{h}} | ||
+ | \end{cases} | ||
+ | \lambda & = & -\frac{1}{2}\left[\gamma_{h}+\gamma+\gamma_{m}\pm\sqrt{\gamma_{h}^{2}-2\gamma\gamma_{h}+2\gamma_{m}\gamma_{h}+\gamma^{2}+2\gamma\gamma_{m}+\gamma_{m}^{2}}\right] | ||
+ | |||
+ | \lambda & = & -\frac{1}{2}\left[\gamma+\gamma_{m}+\gamma_{h}\pm\sqrt{\left(\gamma+\gamma_{m}+\gamma_{h}\right)^{2}-4\gamma\gamma_{h}}\right] | ||
+ | |||
+ | \, & \, & \mbox{denotinng}\begin{cases} | ||
+ | \gamma+\gamma_{m}+\gamma_{h}=\gamma_{+} & \, | ||
+ | |||
+ | \gamma\gamma_{h}=\gamma_{*}^{2} & \, | ||
+ | \end{cases} | ||
+ | \, & \, & \gamma_{+}\gg\gamma_{*} | ||
+ | \lambda & = & -\frac{1}{2}\left[\gamma_{+}\pm\sqrt{\gamma_{+}^{2}-4\gamma_{*}^{2}}\right] | ||
+ | |||
+ | \lambda & \approx & -\frac{1}{2}\left[\gamma_{+}\pm\left(\gamma_{+}-2\frac{\gamma_{*}^{2}}{\gamma_{+}}\right)\right] | ||
+ | |||
+ | \, & \, & \begin{cases} | ||
+ | \lambda_{+} & =-\gamma_{+} | ||
+ | \lambda_{-} & =-\frac{\gamma_{*}^{2}}{\gamma_{+}}=-\frac{\gamma\gamma_{h}}{\gamma+\gamma_{m}+\gamma_{h}}=-\frac{\gamma\gamma_{h}}{\gamma+\gamma_{m}+\gamma_{h}} | ||
+ | \end{cases} | ||
+ | \end{array}$$ | ||
+ | |||
+ | |||
+ | \end_inset | ||
+ | |||
+ | |||
+ | |||
+ | \end_layout | ||
+ | |||
+ | |||
+ | \end_body | ||
+ | \end_document | ||
+ | |||
+ | </wikitex> | ||
+ | == References == | ||
+ | {{#RefList:}} | ||
+ | [[Category:]] |
Revision as of 07:59, 7 June 2015
This page has not been refereed. The information given here might be incomplete or provisional. |
- LyX 2.1 created this file. For more info see http://www.lyx.org/
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\begin_body
\begin_layout Standard
The state determinted by two temperatures (refering \begin_inset Formula $T_{0}$ \end_inset
as constant heat sink) :
\begin_inset Formula $$ \begin{cases} C\frac{dT}{dt}+K\left(T-T_{0}\right)=K_{h}\left(T_{h}-T\right) & \,
C_{h}\frac{dT_{h}}{dt}=P_{h}-K_{h}\left(T_{h}-T\right) & \,
\end{cases} $$
\end_inset
Matrix representation:
\begin_inset Formula $$ $$
\end_inset
\end_layout
\begin_layout Standard
Eigenvalues of \begin_inset Formula $M$ \end_inset
\begin_inset Formula $$\begin{array}{rcl} \lambda & = & -\frac{CK_{h}+C_{h}K+C_{h}K_{h}\pm\sqrt{C^{2}K_{h}^{2}-2CC_{h}KK_{h}+2CC_{h}K_{h}^{2}+C_{h}^{2}K^{2}+2C_{h}^{2}KK_{h}+C_{h}^{2}K_{h}^{2}}}{2CC_{h}}
\lambda & = & -\frac{1}{2}\left[\frac{K_{h}}{C_{h}}+\frac{K}{C}+\frac{K_{h}}{C}\pm\sqrt{\frac{K_{h}^{2}}{C_{h}^{2}}-2\frac{KK_{h}}{CC_{h}}+2\frac{K_{h}^{2}}{CC_{h}}+\frac{K^{2}}{C^{2}}+2\frac{KK_{h}}{C^{2}}+\frac{K_{h}^{2}}{C^{2}}}\right]
\, & \, & \mbox{denoting}\;\begin{cases}
\gamma & =\frac{K}{C} \gamma_{m} & =\frac{K_{h}}{C} \gamma_{h} & =\frac{K_{h}}{C_{h}} \end{cases} \lambda & = & -\frac{1}{2}\left[\gamma_{h}+\gamma+\gamma_{m}\pm\sqrt{\gamma_{h}^{2}-2\gamma\gamma_{h}+2\gamma_{m}\gamma_{h}+\gamma^{2}+2\gamma\gamma_{m}+\gamma_{m}^{2}}\right]
\lambda & = & -\frac{1}{2}\left[\gamma+\gamma_{m}+\gamma_{h}\pm\sqrt{\left(\gamma+\gamma_{m}+\gamma_{h}\right)^{2}-4\gamma\gamma_{h}}\right]
\, & \, & \mbox{denotinng}\begin{cases}
\gamma+\gamma_{m}+\gamma_{h}=\gamma_{+} & \,
\gamma\gamma_{h}=\gamma_{*}^{2} & \,
\end{cases} \, & \, & \gamma_{+}\gg\gamma_{*} \lambda & = & -\frac{1}{2}\left[\gamma_{+}\pm\sqrt{\gamma_{+}^{2}-4\gamma_{*}^{2}}\right]
\lambda & \approx & -\frac{1}{2}\left[\gamma_{+}\pm\left(\gamma_{+}-2\frac{\gamma_{*}^{2}}{\gamma_{+}}\right)\right]
\, & \, & \begin{cases}
\lambda_{+} & =-\gamma_{+} \lambda_{-} & =-\frac{\gamma_{*}^{2}}{\gamma_{+}}=-\frac{\gamma\gamma_{h}}{\gamma+\gamma_{m}+\gamma_{h}}=-\frac{\gamma\gamma_{h}}{\gamma+\gamma_{m}+\gamma_{h}} \end{cases} \end{array}$$
\end_inset
\end_layout
\end_body
\end_document
</wikitex>
References
[[Category:]]