COMSOL 6.4 - Thermoacoustic Engine and Heat Pump

Thermoacoustic devices are interesting systems which can generate sound waves from temperature gradients or use sound waves to create temperature gradients. The devices have no moving parts; this feature not only makes the devices look mysterious but also allow the systems to keep working without frequent maintenance. Although a thermoacoustic engine has a long history from pioneering work in the 1970s (Ref. 1), the research field is still active and developing; see, for example, Ref. 2 and Ref. 3.

There are two types of thermoacoustic systems: those that use standing waves and those that use traveling waves. In this model, a standing wave-type thermoacoustic system is simulated. The thermoacoustic system in the model consists of both a thermoacoustic engine and a thermoacoustic heat pump, and the heat pump is driven by the power generated by the engine. The interface is used to simulate the steady temperature field around the engine, and then the interface is coupled with the interface to reproduce the pressure amplification by the engine and the cooling effect by the heat pump. In the interface, nonlinear terms are taken into consideration by using the feature, which is necessary for simulating the heat pump effect.

Model Definition

The model is a 2D axisymmetric thermoacoustic device, as shown in Figure 1. The geometry dimensions are listed in Table 1. The device consists of an 8 m-long tube and two sets of thermal stacks made of acrylic. In the figure, it is also noted that one of the stacks is omitted from the simulation. This is because the temperature distribution in the stack which functions as an engine is determined by an external input and can be modeled simply by giving boundary conditions on the surface. The model uses the same geometry as the pipe and stack of the Simple Thermoacoustic Engine tutorial model available in the on-line COMSOL Application Gallery, which refers to the geometry in Ref. 4.

Figure 1: Sketch of the analysis domain (left) and the thermal stack (right). For sake of visibility the figures are not to scale, and some of the plates in the thermal stacks are omitted from the left figure.

The spacing between the plates of the stacks is 1.5 mm, which is roughly two and a half time the thickness of the thermal boundary layer

(1)

where κ, ρ, and Cp are the thermal conductivity, density, and isobaric heat capacity of the air, respectively, and ω is the angular frequency of the standing wave.

Table 1: Geometry Dimensions of the thermoacoustic system.
Parameter	Value	Description
Ltube	8 m	Tube length
Lstack	140 mm	Stack length
Rtube	20.25 mm	Tube radius
tstack	1.5 mm	Plate thickness
wstack	1.5 mm	Plate spacing
zstack	1.4 m	Stack placement

The relevant physical quantities are listed in Table 2. The boundary temperature of the engine stack varies by 270 K in the axial direction, which generates a temperature gradient large enough to amplify the oscillation. In the other stack, the temperature is fixed at a portion of the surface at 293.15 K, while the plates and the air are thermally coupled elsewhere. Around the heat pump stack, the node is enabled. It adds the nonlinear advection term to the governing equations, this term is important for modeling the thermoacoustic heat pump.

Table 2: Physical quantities of the thermoacoustic system.
Parameter	VAlue	Description
p0	1 kPa	Initial pressure amplitude
T0	293.15 K	Ambient Temperature
Tengine	270 K	Temperature difference

In actual applications, the oscillation would be initiated by an external device, such as a speaker. In this model, however, the initial pressure is given a sinusoidal distribution whose wave length is half the pipe length. The upper end of the pipe is set to the maximum pressure, 1 kPa, while the minimum pressure, −1 kPa, is set at the lower end. This condition enables the model to reach a steady acoustic pressure amplitude in a few oscillations.

Results and Discussion

Figure 2 shows the instantaneous distribution of the magnitude of the acoustic velocity around the heat pump stack. Because the Thermoviscous Acoustics interface solves the transportation equation of the velocity, the flow field induced by the oscillation is resolved. It is confirmed that the air flows through the stack due to the oscillation, and the magnitude of the velocity becomes large inside the stack passages. Capturing the behavior of the acoustic velocity is important for thermoacoustic simulations, because the advection due to the oscillatory flow plays a key role in transporting heat.

Figure 2: Instantaneous acoustic velocity around the heat pump stack which is induced by the standing wave.

In Figure 3, the pressure history at a point on the lower face of the heat pump stacks (z = 6.46 m) is plotted. The figure compares two different conditions: the blue line corresponds to the original setting, while the green line shows the result without any temperature gradient in the engine stack. It is confirmed that the temperature gradient in the engine stack is maintaining the pressure amplitude. Without the temperature gradient in the engine stack, the pressure oscillation is damped by viscous losses.

Figure 3: The pressure as a function of time at the heat pump stack for a temperature gradient in the engine stack (blue) and without the temperature gradient (green).

Figure 4 shows the temperature at the lower end of the heat pump stack (where the temperature becomes lowest). The figure also shows how the temperature is affected by the nonlinear terms. It is shown that the nonlinear terms must be considered when simulating the cooling effect by the heat pump. The nonlinear terms include a convection term responsible for the heat transport due to the coupling between the acoustic velocity field and the acoustic temperature field. Therefore, the temperature will not be decreased without the nonlinear terms.

Figure 4: The temperature as a function of time at the heat pump stack with the nonlinear contributions included (blue) and without the nonlinear contribution (green).

In a thermoacoustic heat pump the material of the stack is important, especially the thermal conductivity. In Figure 5 the temperature at the cooling end of the heat stack is shown for two different stack materials, acrylic plastic and copper. The temperature decrease is largest for the acrylic plastic. If the model with acrylic plastic is run for 60 s the temperature at the heat pump is cooled down with approximately 2 K.