## Heating Circuit

Thermal Analysis of Structural Mechanics Associated with an Electrical Circuit

Small heating circuits find use in many applications. For example, in manufacturing processes, they heat up reactive fluids. The device in this CMS example consists of an electrically resistive layer deposited on a glass plate. The layer results in Joule heating when a voltage is applied to the circuit, which results in a structural deformation. The layer’s properties determine the amount of heat produced.

This Multiphysics example simulates the electrical heat generation, heat transfer, and mechanical stresses and deformations of a heating circuit device. The model uses the Heat Transfer in Solids interface in combination with the Electric Currents in Layered Shells interface and the Solid Mechanics and Membrane interfaces. The Rigid Motion Suppression condition is automatically applied to a set of suitable constraints based on the geometry model and physics interfaces.

### Introduction

Small heating circuits find use in many applications. For example, in manufacturing processes they heat up reactive fluids. Figure 1 illustrates a typical heating device for this model. The device consists of an electrically resistive layer deposited on a glass plate. The layer causes Joule heating when a voltage is applied to the circuit. The layer’s properties determine the amount of heat produced.

In this particular model, you must observe three important design considerations:

• Noninvasive heating
• Minimal deflection of the heating device
• Avoidance of overheating the process fluid

The heater must also work without failure. You achieve the first and second requirements by inserting a glass plate between the heating circuit and the fluid; it acts as a conducting separator. Glass is an ideal material for both these purposes because it is nonreactive and has a low coefficient of thermal expansion. You must also avoid overheating due to the risk of self-ignition of the reactive fluid stream. Ignition is also the main reason for separating the electrical circuit from direct contact with the fluid. The heating device is tailored for each application, making virtual prototyping very important for manufacturers.

For heating circuits in general, detachment of the resistive layer often determines the failure rate. This is caused by excessive thermally induced interfacial stresses. Once the layer has detached, it gets locally overheated, which accelerates the detachment. Finally, in the worst case, the circuit might overheat and burn. From this perspective, it is also important to study the interfacial tension due to the different thermal-expansion coefficients of the resistive layer and the substrate as well as the differences in temperature. The geometric shape of the layer is a key parameter to design circuits for proper functioning. You can investigate all of the abovementioned aspects by modeling the circuit.

This Multiphysics example simulates the electrical heat generation, the heat transfer, and the mechanical stresses and deformations of a heating circuit device. The model uses the Heat Transfer in Solids interface of the Heat Transfer Module in combination with the Electric Currents, Layered Shell interface from the AC/DC Module and the Solid Mechanics and Membrane interfaces from the Structural Mechanics Module.

### Model Definition

Figure 2 shows a drawing of the modeled heating circuit.

The device consists of a serpentine-shaped Nichrome resistive layer, 10 μm thick and 5 mm wide, deposited on a glass plate. At each end, it has a silver contact pad measuring 10 mm-by-10 mm-by-10 μm. When the circuit is in use, the deposited side of the glass plate is in contact with surrounding air, and the back side is in contact with the heated fluid. Assume that the edges and sides of the glass plate are thermally insulated.

Table 1 gives the resistor’s dimensions.

During operation the resistive layer produces heat. Model the electrically generated heat using the Electric Currents, Layered Shell interface from the AC/DC Module. An electric potential of 12 V is applied to the pads. In the model, you achieve this effect by setting the potential at one edge of the first pad to 12 V and that of one edge of the other pad to 0 V.

To model the heat transfer in the thin conducting layer, use the Thin Layer feature from the Heat Transfer in Solids interface. The heat rate per unit area (measured in W/m2) produced inside the thin layer is given by

$\displaystyle {{q}{{prod}}}=d{{\vec{Q}}_{{DC}}}$

$\displaystyle {{\vec{Q}}_{{DC}}}=\vec{J}.\vec{E}=\sigma {{\left| {{{\nabla }_{t}}V} \right|}^{2}}(W/{{m}^{3}})$

Where QDC is the power density. The generated heat appears as an inward heat flux at the surface of the glass plate.

At steady state, the resistive layer dissipates the heat it generates in two ways: on its upside to the surrounding air (at 293 K), and on its down side to the glass plate. The glass plate is similarly cooled in two ways: on its circuit side by air, and on its back side by a process fluid (353 K). You model the heat fluxes to the surroundings using heat transfer coefficients, h. For the heat transfer to air, h = 5 W/(m2·K), representing natural convection. On the glass plate’s back side, h = 20 W/(m2·K), representing convective heat transfer to the fluid. The sides of the glass plate are insulated.

The model simulates thermal expansion using static structural-mechanics analyses. It uses the Solid Mechanics interface for the glass plate, and the Membrane interface for the circuit layer. The equations of these two interfaces are described in the Structural Mechanics Module User’s Guide in COMSOL Multiphysics. The stresses are set to zero at 293 K. You determine the boundary conditions for the Solid Mechanics interface by fixing one corner with respect to the x-, y-, and z-displacements and rotation.

Table 2 summarizes the material properties used in the model.

### Results and Discussion

Figure 3 shows the heat that the resistive layer generates. The highest heating power occurs at the inner corners of the curves due to the higher current density at these spots. The total generated heat, as calculated by integration, is approximately 13.8 W.

Figure 4 shows the temperature of the resistive layer and the glass plate at steady state.

The highest temperature is approximately 428 K, and it appears in the central section of the circuit layer. It is interesting to see that the differences in temperature between the fluid side and the circuit side of the glass plate are quite small because the plate is very thin.  Using boundary integration, the integral heat flux on the fluid side evaluates to approximately 8.5 W. This means that the device transfers the majority of the heat it generates — 8.5 W out of 13.8 W — to the fluid, which is good from a design perspective, although the thermal resistance of the glass plate results in some losses. The temperature rise also induces thermal stresses due the materials’ different coefficients of thermal expansion. As a result, mechanical stresses and deformations arise in the layer and in the glass plate. Figure 5 shows the effective stress distribution in the device and the resulting deformations. During operation, the glass plate bends towards the air side.

The highest effective stress, approximately 13 MPa, occurs at the inner corners of the curves of the Nichrome circuit. The yield stress for high quality glass is roughly 250 MPa, and for Nichrome it is 360 MPa. This means that the individual objects remain structurally intact for the simulated heating power loads.

You must also consider stresses in the interface between the resistive layer and the glass plate. Assume that the yield stress of the surface adhesion in the interface is in the region of 50 MPa — a value significantly lower than the yield stresses of the other materials in the device. If the effective stress increases above this value, the resistive layer locally detaches from the glass. Once it has detached, heat transfer is locally impeded, which can lead to overheating of the resistive layer and eventually cause the device to fail.

Figure 6 displays the effective forces acting on the adhesive layer during heater operation.  As the figure shows, the device experiences a maximum interfacial stress that is an order of magnitude smaller than the yield stress. This means that the device is OK in terms of adhesive stress.

Finally study the device’s deflections, shown in Figure 7.

The maximum deviation from being a planar surface, is approximately 50 μm. For high-precision applications, such as semiconductor processing, this might be a significant value that limits the device’s operating temperature.

Source: COMSOL Blogs, Models, Cyclopedia

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Computational Modeling and Simulation is the simulation center of Acamech that models Mechanical Engineering Problems, especially Multiphysics Simulation; including Fluid Flow (CFD), Structural Mechanics, Acoustics, Heat Transfer as a unique Multiphysics problem. We use COMSOL Multiphysics, MATLAB, and Autodesk Inventor.