Thermal Controller, Reduced Order Model

This example demonstrates controlling the temperature in a heated metal block using a thermal controller and how to use reduced order modeling to shorten computing time for additional simulations.

The model studies the controlled temperature response to a sinusoidal variation of the external temperature for two candidate thermostat positions and two temperature setpoints. One candidate location for the thermostat is between the surface with the external temperature variation and the heater, and in the other location both the heater and the surface with the external temperature are located on the same side.

Model Definition

The dynamic system consists of a metal block that exchanges heat with the environment. A heater and a thermostat switch are situated inside the glass-enclosed system. The system works as follows: The thermostat turns the heater on or off when the temperature becomes too low or too high.

The finite-element model of the metal block requires two inputs:

•

The state of the heater, which can be On (1) or Off (0)

•

The exterior temperature, Tout

As its output, the model supplies the temperature at the thermostat’s location.

Figure 1: Block diagram for the thermal controller system.

The PDE describes the overall system’s temperature distribution given the temperature of the heater and the exterior environment. If the heat transfer is so fast that the heat distribution is more or less constant (in space, not in time), a single state is sufficient. Otherwise, controlling the temperature requires modeling a PDE in COMSOL Multiphysics.

Domain Equations

The heat equation is

Boundary conditions

The boundary conditions come from the level of insulation around the system. On well-insulated sides the temperature flux is zero, which gives the Neumann boundary condition

. The poorly insulated sides involve the Neumann condition

, where kG and lG are the thermal conductivity and the thickness of the glass sheet that separates the metal block and the exterior.

Figure 2: Geometry of the thermal controller system. The figure shows one of the two candidate thermostat positions.

Because only the temperature distribution in the xy-plane is of interest, you can use a 2D model. For the units to make sense, think of the domain as having a depth (z direction) of 1 m.

Controlling Temperature

The temperature is controlled by switching the heater on and off depending on a temperature measurement (T) relative to a temperature setpoint (Tset). In order to avoid switching on an off as soon as there is a small deviation, a deadband (+/- dT) is often used to define an acceptable deviation from the setpoint. Introducing indicator functions (lowtemp and higtemp) switching signs when the temperature changes to a value outside the acceptable range, events can be used to switch the heater on and off.

lowtemp = (Tset – dT) – T

hightemp = T – (Tset + dT)

The event triggered when detecting lowtemp > 0 switches the heater on and the event triggered when detecting hightemp > 0 switches the heater off.

Reduced Order Modeling (Model reduction)

Large finite-element simulations can be costly, and if repeated simulations are needed it can be beneficial to use reduced-order models (ROMs). ROMs are typically valid only in the vicinity of their design conditions and have lower accuracy, but the simulation time is significantly shorter. The objective for model reduction is to provide a sufficiently accurate representation of the input-output dynamics of the unreduced model in a given parameter range with a minimal total computational cost, including the cost of creating the reduced model. The characteristics of the unreduced model as well as the value of the reduced model guide the choice of model reduction method. Nonlinearities require special treatment, and if the model is to be valid in a large parameter range it can be costly to produce basis functions or input-output samples. Model reduction can, for example, be performed by linearization of the finite-element model and projection of the resulting system matrices onto a limited set of base functions representing the dynamics of significance for the application and defining the relevant inputs and outputs of interest.

In the present example the unreduced model as well as the outputs are linear and the dependence on the input parameters has an affine representation. A small number of eigenmodes are used as the basis functions and the inputs represent the exterior temperature (Tout) and the fraction of maximum power of the heater (HeatState). The outputs are defined as the temperatures in two candidate points for thermostat placement (T1, T2). The control strategy described in the previous section is however clearly nonlinear and not a candidate for model reduction since it only has two states.

Results and Discussion

The tutorial shows how to use a reduced model rather than a full finite-element simulation to evaluate a control strategy. In this case the controlled system is linear but the controller has nonlinear dynamics. It is also illustrated how increasing the number of basis functions can improve the transient response of the reduced model when compared to a finite-element simulation. The external temperature variation is slow and smooth but the dynamics of the controller can introduce high frequency transients. The dynamic response of the reduced model is determined by the eigenmodes included in the basis, and increasing the number of modes extends the dynamic range of the reduced model and improves the accuracy of the transient response. From the comparison with the response of the unreduced model it is clear that an increase from 6 to 30 eigenmodes brings the reduced model response closer to that of the unreduced model. The final comparison is shown in Figure 3.

Figure 3: Comparing the reduced model outputs with the temperatures from the FEM model with the thermostat shows good agreement when using 30 eigenmodes, a setpoint of 300 K, and placing the thermostat at position 2.

COMSOL_Multiphysics/Multiphysics/thermal_controller_rom

Modeling Instructions

From the menu, choose .

New

In the window, click

Root

Add a 2D component to set up the geometry and use the Heat Transfer in Solids physics interface in a time-dependent study.

Add Component

In the toolbar, click

and choose .

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.3.

In the text field, type 0.2.

Square 1 (sq1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.04.

Locate the section. In the text field, type 0.1.

In the text field, type 0.1.

From the list, choose .

Point 1 (pt1)

Add 2 points to represent the candidate thermostat positions 1 and 2.

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.05,0.2.

In the text field, type 0.1,0.1.

In the toolbar, click

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Heat Equation (hteq)

In the window for , click to expand the section.

In the text field, type T.

Global Definitions

Add model parameters for the temperature setpoint, thermal conductivity, density, and heat capacity.

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Tset	293.15[K]	293.15 K	Setpoint temperature
kiso	4.92e3[W/(m*K)]	4920 W/(m·K)	Thermal conductivity
rho	7.82e3[kg/m^3]	7820 kg/m³	Density
Cp	449[J/(kg*K)]	449 J/(kg·K)	Heat capacity