The system being modeled in this Simulink example is a variable-speed fan cooling a CPU which, depending on the model and the operating point, can generate variable amount of heat. Each component of the system is modeled using techniques that easily incorporate empirical data and technical specifications, and all assumptions are stated in the following discussion regarding levels of modeling fidelity.
The plant in this system is the thermal dynamics of the heat-generating CPU and the heat-dissipating heat sink. These dynamics are modeled by a finite element formulation of the heat equation on the 3D geometry. The geometry, meshing, and discretization in this example are done using MATLAB’s PDE Toolbox. The finite element matrices generated can then be solved using Simulink’s Descriptor State Space block. For more information on time-dependent finite-element formulations, please see the MathWorks documentation. The dominant mode of heat transfer in this scenario is due to convection. A convective boundary condition is constructed in Simulink for the ducted airflow where forced convection coefficients would be applied on the internal fin surfaces, and free convection coefficients are applied on the external facing surfaces of the system. The heat generation of the CPU can be controlled by the user in this formulation as an input Wattage. This demonstration provides examples of different loadings of the processor that lead to different heat generation, thus lending itself useful for design decisions in determining if a certain heat sink is suitable for a specific CPU.
This example is organized inside a Simulink Project. Once the project is opened, running “top_script” will load all the variables needed for the simulation and then execute a parameter sweep. Six scenarios are included in this example. The six simulations run varied levels of heat generation from the CPU as input to the plant subsystem. Upon completion of the parameter sweep, the Simulation Manager will launch. The socket temperature signal is logged using Simulink Data Inspector and can be visualized by selecting the simulations of interest and clicking the “Show Results” option. The Signal Editor block, data logging, and adding additional systems to the model can all provide opportunities to expand on physical system modeling and the engineering design decision process that goes along with it. Note, there is an included “debug” plant model that uses the same parameters, but a simpler geometry with fewer mesh nodes to reduce the number of degrees of freedom. This is useful for debugging the model and iterating on other parameters in the system outside of the plant. This feature can be utilized by commenting out “heat_sink_defs” in the “top_script” and replacing it with “heat_sink_defs_DEBUG”.
This example using a finite element formulation of the heat equation on the heat sink is quite computationally intensive, requiring matrix equations of size 18,559 x 18,559. This method therefore naturally presents an opportunity to apply a model order reduction technique in order to reduce the resources needed to simulate. An option to choose between the full-order finite element model or a reduced-order version of the FE model was added in the "Choose fidelity" block in the CPU + Heat Sink subsystem. The user now has an option to choose the size of the reduced model such that the the execution time of running "top_script" goes from the order of one day down to the order of one minute. The directory "MOR_tests" contains a test script that examines the relative error of reduced order models as a function of reduced order as well as other model order reduction parameters. The model order reduction parameters appear when "Reduced Order Model" is chosen from the drop down menu in the "Choose fidelity" block in the plant subsytem. The first parameter chosen by the user is a frequency range range of interest for the modes to be retained after modal truncation. Choosing this range requires prior knowledge about the system of interest from the user. The remaining two parameters are the number of modes retained, corresponding to the size of the reduced system.
Contents:
Chip_temperature_control.prj: the Simulink project file
CFMvsRPM.csv, RPMvsV.csv: data tables used to construct 1D lookup tables in Simulink model, can be digitized data from plots
fan.slx, heat_sink.slx, sensor.slx: subsystem files referenced by the top level model
fan_defs.m, heat_sink_defs.m, heat_sink_defs_DEBUG.m, sensor_defs.m: MATLAB scripts to initialize subsystem parameters
top_system.slx: top level model of complete control systems
HeatInput.mat: MAT file that defines Signal Editor scenarios
top_script.m: MATLAB script that calls subsytem scripts, runs simulations with different input scenarios, and opens Simulation Manager
Demo_description.pdf: Detailed write-up of model development and design decisions made
MOR_tests: directory that contains script to determine acceptable reduced order model