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Contents of /trunk/doc/cookbook/example02.tex

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Cookbook modifications Review -> Not quite finished yet.
New figure.
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8 % Primary Business: Queensland, Australia
9 % Licensed under the Open Software License version 3.0
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15 \section{Example 2: One Dimensional Heat Diffusion in an Iron Rod}
16 \sslist{example02.py}
18 \label{Sec:1DHDv0}
19 \begin{figure}[ht]
20 \centerline{\includegraphics[width=4.in]{figures/onedheatdiff002}}
21 \caption{Example 2: One dimensional model of an Iron bar.}
22 \label{fig:onedhdmodel}
23 \end{figure}
25 Our second example is of a cold iron bar at a constant temperature of $T\hackscore{ref}=20^{\circ} C$, see \reffig{fig:onedhdmodel}. The bar is perfectly insulated on all sides with a heating element at one end keeping the the temperature at a constant level $T\hackscore0=100^{\circ} C$. As heat is applied; energy will disperse along the bar via conduction. With time the bar will reach a constant temperature equivalent to that of the heat source.
27 This problem is very similar to the example of temperature diffusion in granite blocks presented in the previous section~\ref{Sec:1DHDv00}. Thus, it is possible to modify the script we have already developed for the granite blocks to suit the iron bar problem.
28 The obvious difference between the two problems are the dimensions of the domain and different materials involved. This will change the time scale of the model from years to hours.
29 The new settings are;
30 \begin{python}
31 #Domain related.
32 mx = 1*m #meters - model length
33 my = .1*m #meters - model width
34 ndx = 100 # mesh steps in x direction
35 ndy = 1 # mesh steps in y direction - one dimension means one element
36 #PDE related
37 rho = 7874. *kg/m**3 #kg/m^{3} density of iron
38 cp = 449.*J/(kg*K) # J/Kg.K thermal capacity
39 rhocp = rho*cp
40 kappa = 80.*W/m/K # watts/m.Kthermal conductivity
41 qH = 0 * J/(sec*m**3) # J/(sec.m^{3}) no heat source
42 Tref = 20 * Celsius # base temperature of the rod
43 T0 = 100 * Celsius # temperature at heating element
44 tend= 0.5 * day # - time to end simulation
45 \end{python}
46 We also need to alter the initial value for the temperature. Now we need to set the
47 temperature to $T\hackscore{0}$ at the left end of the rod where we have $x\hackscore{0}=0$ and
48 $T\hackscore{ref}$ elsewhere. Instead of \verb|whereNegative| function we use now the
49 \verb|whereZero| which returns the value one for those sample points where
50 the argument (almost) equals zero and the value zero elsewhere. The initial
51 temperature is set to;
52 \begin{python}
53 # ... set initial temperature ....
54 T= T0*whereZero(x[0])+Tref*(1-whereZero(x[0]))
55 \end{python}
57 \subsection{Dirichlet Boundary Conditions}
58 In the iron rod model we want to keep the initial temperature $T\hackscore0$ on the left side of the domain constant with time.
59 This implies that when we solve the PDE~\refEq{eqn:hddisc}, the solution must have the value $T\hackscore0$ on the left hand
60 side of the domain. As mentioned already in Section~\ref{SEC BOUNDARY COND} where we discussed
61 boundary conditions, this kind of scenario can be expressed using a \textbf{Dirichlet boundary condition}. Some people also
62 use the term \textbf{constraint} for the PDE.
64 To define a Dirichlet boundary condition we need to identify where to apply the condition and determine what value the
65 solution should have at these locations. In \esc we use $q$ and $r$ to define the Dirichlet boundary conditions
66 for a PDE. The solution $u$ of the PDE is set to $r$ for all sample points where $q$ has a positive value.
67 Mathematically this is expressed in the form;
68 \begin{equation}
69 u(x) = r(x) \mbox{ for any } x \mbox{ with } q(x) > 0
70 \end{equation}
71 In the case of the iron rod
72 we can set;
73 \begin{python}
74 q=whereZero(x[0])
75 r=T0
76 \end{python}
77 to prescribe the value $T\hackscore{0}$ for the temperature at the left end of the rod where $x\hackscore{0}=0$.
78 Here we use the \verb|whereZero| function again which we have already used to set the initial value.
79 Notice that $r$ is set to the constant value $T\hackscore{0}$ for all sample points. In fact,
80 values of $r$ are used only where $q$ is positive. Where $q$ is non-positive,
81 $r$ may have any value as these values are not used by the PDE solver.
83 To set the Dirichlet boundary conditions for the PDE to be solved in each time step we need
84 to add some statements;
85 \begin{python}
86 mypde=LinearPDE(rod)
87 A=zeros((2,2)))
88 A[0,0]=kappa
89 q=whereZero(x[0])
90 mypde.setValue(A=A, D=rhocp/h, q=q, r=T0)
91 \end{python}
92 It is important to remark here that the Dirichlet condition \textbf{overwrites} any Neuman boundary
93 condition \esc sets by default (or those defined by the user).
95 \begin{figure}
96 \begin{center}
97 \includegraphics[width=4in]{figures/ttrodpyplot150}
98 \caption{Example 2: Total Energy in the Iron Rod over Time (in seconds).}
99 \label{fig:onedheatout1 002}
100 \end{center}
101 \end{figure}
103 \begin{figure}
104 \begin{center}
105 \includegraphics[width=4in]{figures/rodpyplot001}
106 \includegraphics[width=4in]{figures/rodpyplot050}
107 \includegraphics[width=4in]{figures/rodpyplot200}
108 \caption{Example 2: Temperature ($T$) distribution in the iron rod at time steps $1$, $50$ and $200$.}
109 \label{fig:onedheatout 002}
110 \end{center}
111 \end{figure}
113 Besides some cosmetic modification this all we need to change. The total energy over time is shown in \reffig{fig:onedheatout1 002}. As heat
114 is transfered into the rod by the heater the total energy is growing over time but reaches a plateau
115 when the temperature is constant is the rod, see \reffig{fig:onedheatout 002}.
116 You will notice that the time scale of this model is several order of magnitudes faster than
117 for the granite rock problem due to the different length scale and material parameters.
118 In practice it can take a few models run before the right time scale has been chosen\footnote{An estimate of the
119 time scale for a diffusion problem is given by the formula $\frac{\rho c\hackscore{p} L\hackscore{0}^2}{4 \kappa}$, see
120 \url{http://en.wikipedia.org/wiki/Fick\%27s_laws_of_diffusion}}.
127 \section{For the Reader}
128 \begin{enumerate}
129 \item Move the boundary line between the two granite blocks to another part of the domain.
130 \item Split the domain into multiple granite blocks with varying temperatures.
131 \item Vary the mesh step size. Do you see a difference in the answers? What does happen with the compute time?
132 \item Insert an internal heat source (Hint: The internal heat source is given by $q\hackscore{H}$.)
133 \item Change the boundary condition for iron rod example such that the temperature
134 at the right end is kept at a constant level $T\hackscore{ref}$, which corresponds to the installation of a cooling element (Hint: Modify $q$ and $r$).
135 \end{enumerate}

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