x axis
0 Lx
Points 'B' at the bottom edge of the plate (for y = 0), and the points 'T' at the
top edge of the plate (for y = Ly) represent the points where the heat is applied.
The plate is considered insulated at the left (x = 0) and the right edges (x = Lx)
of the plate (points 'L' and 'R'). To model this type of problem, we have to write
a heat balance equation for all interior points except the left, right, top and
bottom edges, where we need to specify boundary conditions.
In this problem we have to define two distribution domains:
- x (x axis, length Lx = 0.1 m)
- y (y axis, length Ly = 0.1 m)
the following parameters:
- rho: copper density, 8960 kg/m3
- cp: copper specific heat capacity, 385 J/(kgK)
- k: copper heat conductivity, 401 W/(mK)
- Qb: heat flux at the bottom edge, 1E6 W/m2 (or 100 W/cm2)
- Tt: temperature at the top edge, 300 K
and a single variable:
- T: the temperature of the plate (distributed on x and y domains)
The model consists of 5 equations (1 distributed equation + 4 boundary conditions):
1) Heat balance::
rho * cp * dT(x,y)/dt = k * [d2T(x,y)/dx2 + d2T(x,y)/dy2]; x in (0, Lx), y in (0, Ly)
2) Neumann boundary conditions at the bottom edge::
-k * dT(x,y)/dy = Qb; x in (0, Lx), y = 0
3) Dirichlet boundary conditions at the top edge::
T(x,y) = Tt; x in (0, Lx), y = Ly
4) Neumann boundary conditions at the left edge (insulated)::
dT(x,y)/dx = 0; y in [0, Ly], x = 0
5) Neumann boundary conditions at the right edge (insulated)::
dT(x,y)/dx = 0; y in [0, Ly], x = Lx
The temperature plot (at t=100s, x=0.5, y=*):
.. image:: _static/tutorial1-results.png
:width: 500px
]]>eAlgebraic$${ \left( { \left( { \left( - { \lambda_p } \right) } \right) \cdot \left( { { \partial { { \left( { T \left( { x, y } \right) } \right) } } } \over { \partial {y} } } \right) } \right) - { Q_b } } = 0; {\forall { x } \in \left( { x } _{0}, { x } _{n} \right) }, {{ y } = { y } _{0}}$$eOpenOpeneLowerBoundeAlgebraic$${ { T \left( { x, y } \right) } - { T_t } } = 0; {\forall { x } \in \left( { x } _{0}, { x } _{n} \right) }, {{ y } = { y } _{n}}$$eOpenOpeneUpperBoundeAlgebraic$${ { \partial { { \left( { T \left( { x, y } \right) } \right) } } } \over { \partial {x} } } = 0; {{ x } = { x } _{0}}, {\forall { y } \in \left[ { y } _{0}, { y } _{n} \right] }$$eLowerBoundeClosedClosedeAlgebraic$${ { \partial { { \left( { T \left( { x, y } \right) } \right) } } } \over { \partial {x} } } = 0; {{ x } = { x } _{n}}, {\forall { y } \in \left[ { y } _{0}, { y } _{n} \right] }$$eUpperBoundeClosedClosed