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In this example, we simulate steady-state and transient heat flow in solids.  The simulation results are compared with analytic calculations.

## Steady-state heat flow in solids

### Single material

The simplest problem that can be solved by a thermal solver is to calculate the heat flow through a solid kept a two different temperatures at two ends.  This problem can be solved analytically by assuming the thermal conductivity of the solid, k to be a constant.  In such a case, the amount of heat flow through the solid can be calculated by, where Q is the amount of heat flow through the solid in Watts, A is the cross-section in m2, k is the thermal conductivity in W/m-K, T1 is the temperature at one end and T2 is the temperature at the other end both in Kelvin, and L is the length of the solid in meter.

To solve such a problem numerically, open and run the steady_state_heat_flow.ldev file.  The file contains a slab of silicon (solid_1) which is kept at two different temperatures at both ends.  The left edge is heated up to 350 K and the right edge is cooled to 280 K.  The temperatures at the two ends are defined by 'Fixed temperature' boundary conditions.  A 1D temperature monitor is placed along the length of the slab to capture the spatial temperature distribution.  Once the simulation is run, the (HEAT) solver region stores the simulation results.  The 'thermal' dataset provides the temperature profile and the 'boundaries' dataset provides the power flow at the end points.  To visualize the temperature profile, right-click on the HEAT solver and select visualize - 'thermal which will produce the plot on the left.  To compare the temperature profile with analytic values, use the script file plot_steady_state_temp.lsf to plot the temperature profile along x from the temperature monitor.  The script will generate the plot on the right.  It will also print the net heat flow in silicon from the simulation result and from analytic calculation. Fig: 2D temperature profile of the slab. Fig: 1D temperature profile of the slab compared with anaytic values.

Note that the temperature profile from the simulation is slightly nonlinear whereas the analytic result assumes linear variation of temperature (assuming constant value for k).  The HEAT solver uses a temperature dependent model for thermal conductivity which introduces the nonlinearity in the temperature profile.  To check this effect, open the material database and turn off the temperature dependency of thermal conductivity of silicon.  Re-run the simulation and the script.

### Two materials

Next, enable the object solid_2 (SiO2) and run the simulation again.  Keep the thermal conductivity for both Si and SiO2 constant (turn off the temperature dependency from the material properties).  The script file plot_steady_state_temp.lsf can be used again to plot the temperature profile along with analytic result.  The script will also print the net heat flow from the simulation result and from analytic calculation.  Note that the temperature variation in silicon is much smaller than in SiO2 owing to its larger thermal conductivity as compared to SiO2.

## Transient heat flow in solid

In this example, we will start with a slab of SiO2 kept at room temperature (300 K) at time t = 0 s.  At t = 1 us, the left side of the slab is heated up and suddenly brought to a fixed higher temperature of 400 K.  The temperature difference between both ends of the slab will result in a heat flow and the temperature profile of the entire slab will change with time until it reaches steady state.  The evolution of the temperature profile of the slab can be simulated by running the transient_heat_flow.ldev file.  The heat flow will be high initially as the entire slab starts to get heated up.  Once the slab reaches a steady state, the heat flow reduces to a constant value as seen in the plot below (right).  To get this plot, right click on the HEAT solver and choose visualize>boundaries.  Remove all other attributes except P_left.  Open the chart settings and choose 'log10y' and set the axis limits to be x min = -0.0001, x max = 0.005, y min = 0.01, and y max = 10. Fig: Schematic showing the oxide slab and the temperature variation with time. Fig: Transient heat flow through the oxide slab.

The same transient problem can also be solved analytically assuming that the thermal conductivity in the slab remains constant with variation in temperature (which is also the case in this simulation where the temperature dependence of the thermal conductivity of SiO2 was disabled intentionally.  The script file transient_temperature_comparison.lsf can be used to plot the temperature profile at different locations inside the slab at different times and compare with analytic results once the simulation is run. Fig: Transient behavior of the 1D temperature profile. Fig: Temperature at location x = 60 micron as a function of time.

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