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tutorial2
This tutorial introduces the following concepts:
- Arrays (discrete distribution domains)
- Distributed parameters
- Making equations more readable
- Degrees of freedom
- Setting an initial guess for variables (used by a DAE solver during an initial phase)
The model in this example is very similar to the model used in the tutorial 1.
The differences are:
- The heat capacity is temperature dependent
- Different boundary conditions are applied
The temperature plot (at t=100s, x=0.5, y=*):
.. image:: _static/tutorial2-results.png
:width: 500px
$\mathit{tutorial2}$BC_bottomNeumann boundary conditions at the bottom edge (constant flux)${\mathit{BC}}_{\mathit{bottom}}$eAlgebraic$${ \left( { \left( { \left( - { \lambda \left( { x, y } \right) } \right) } \right) \cdot \left( { { \partial { { \left( { T \left( { x, y } \right) } \right) } } } \over { \partial {y} } } \right) } \right) - { Q \left( { 0 } \right) } } = 0; {\forall { x } \in \left( { x } _{0}, { x } _{n} \right) }, {{ y } = { y } _{0}}$$x$\mathit{x}$$\mathit{x}$
xeOpenOpeny$\mathit{y}$$\mathit{y}$
yeLowerBoundBC_topNeumann boundary conditions at the top edge (constant flux)${\mathit{BC}}_{\mathit{top}}$eAlgebraic$${ \left( { \left( { \left( - { \lambda \left( { x, y } \right) } \right) } \right) \cdot \left( { { \partial { { \left( { T \left( { x, y } \right) } \right) } } } \over { \partial {y} } } \right) } \right) - { Q \left( { 1 } \right) } } = 0; {\forall { x } \in \left( { x } _{0}, { x } _{n} \right) }, {{ y } = { y } _{n}}$$x$\mathit{x}$$\mathit{x}$
xeOpenOpeny$\mathit{y}$$\mathit{y}$
yeUpperBoundBC_leftNeumann boundary conditions at the left edge (insulated)${\mathit{BC}}_{\mathit{left}}$eAlgebraic$${ { \partial { { \left( { T \left( { x, y } \right) } \right) } } } \over { \partial {x} } } = 0; {{ x } = { x } _{0}}, {\forall { y } \in \left[ { y } _{0}, { y } _{n} \right] }$$x$\mathit{x}$$\mathit{x}$
xeLowerBoundy$\mathit{y}$$\mathit{y}$
yeClosedClosedBC_right Neumann boundary conditions at the right edge (insulated)${\mathit{BC}}_{\mathit{right}}$eAlgebraic$${ { \partial { { \left( { T \left( { x, y } \right) } \right) } } } \over { \partial {x} } } = 0; {{ x } = { x } _{n}}, {\forall { y } \in \left[ { y } _{0}, { y } _{n} \right] }$$x$\mathit{x}$$\mathit{x}$
xeUpperBoundy$\mathit{y}$$\mathit{y}$
yeClosedClosedC_pEquation to calculate the specific heat capacity of the plate as a function of the temperature.${\mathit{C}}_{\mathit{p}}$eAlgebraic$${ \left( { { c_p \left( { x, y } \right) } - { a } } \right) - \left( { { b } \cdot { T \left( { x, y } \right) } } \right) } = 0; {\forall { x } \in \left[ { x } _{0}, { x } _{n} \right] }, {\forall { y } \in \left[ { y } _{0}, { y } _{n} \right] }$$x$\mathit{x}$$\mathit{x}$
xeClosedClosedy$\mathit{y}$$\mathit{y}$
yeClosedClosed