# The Effect of Temperature Field on Low Amplitude Oscillatory Flow within a Parallel-Plate Heat Exchanger in a Standing Wave Thermoacoustic System

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## Abstract

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## 1. Introduction and Literature Review

## 2. Computational Model

^{−4}for the continuity and momentum equation, and 1 × 10

^{−7}for energy equation. The density is calculated using the second order upwind numerical scheme. Density is related to pressure and temperature by using the equation of state as shown in Equation (5). The transient solution of the flow problem is solved in a segregated way using the pressure-based solver [12,25]. The time step size was chosen so that solution converged within 15 to 18 iterations in every time step. If the time step size was too large, the solution was found not to converge in every time step. If the size was too small, convergence occurred too fast within a certain time step (sometimes only requiring one iteration in one time step). The best time step for convergence was determined and set at 1200 steps per acoustic cycle. The area-weighted-average of pressure at the end wall, known as the pressure antinode, was monitored until a steady state oscillatory flow was obtained. This is defined as a state when pressure, velocity, and temperature do not change much from cycle to cycle. By way of example, Figure 4 shows the history of oscillating pressure and velocity monitored for location “m” as defined in Figure 2. A steady oscillatory flow condition is achieved after six flow cycles. However, as will be discussed later, results presented in this paper are obtained after 70 flow cycles, when both the flow and thermal oscillatory flow conditions have reached a steady oscillatory state. The solutions are also monitored so that they converge in every time step at every flow cycle (through the selection of the time step size).

#### 2.1. Grid Independency

#### 2.2. CFD Model Validation

## 3. Results and Discussions

#### 3.1. Investigation of the Effect of the Initial Fluid Temperature on Flow and Heat Transfer

#### 3.2. The Effect of the Heat Exchanger Wall Temperature on the Flow Field

#### 3.2.1. Study of Flow Using the Adiabatic Model

#### 3.2.2. The Effect of Wall Temperature on the Flow and Heat Transfer

#### 3.3. The Effect of Gravity and Device Orientation on Flow and Heat Transfer

#### 3.4. Viscous Dissipation

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Dimensional Analysis

## References

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**Figure 1.**Schematic of a simple thermoacoustic cooler arrangement (

**top**). The acoustically induced compression and expansion of fluid elements causes heat pumping effects along the stack (

**bottom**).

**Figure 2.**(

**a**) A quarter-wavelength standing wave thermoacoustic rig; (

**b**) computational domain of the rig covering the 7.4 m length of the resonator; (

**c**) mesh generated in the area of heat exchangers and its neighbourhood; (

**d**) designations/locations within the individual channel used for analysis.

**Figure 4.**Oscillating pressure and velocity monitored for identification of the steady oscillatory condition.

**Figure 6.**Grid independency test for (

**a**) axial velocity amplitude at the centre of the channel for phases $\mathsf{\varphi}$6 and $\mathsf{\varphi}$ 16 and (

**b**) velocity profiles near the wall; both taken 15 mm from the joint above the cold plate.

**Figure 7.**Centreline velocity amplitude (1 mm from the joint, above the cold plate) obtained from the model with heat exchanger walls: (

**a**) adiabatic and (

**b**) with the temperature profile based on the experiment.

**Figure 8.**Temperature profiles for HHX and CHX—comparison between experimental (EXP) and computational (CFD) results at the location 15 mm from the joint (above the HHX and CHX plates).

**Figure 9.**Temperature contours at phase $\mathsf{\varphi}$1 from the experiment (

**a**) and three numerical models initialised with different temperature fields (

**b**–

**d**).

**Figure 10.**The effect of initial temperature on (

**a**) the oscillating velocity at location 1 mm from the joint above the cold plate and (

**b**) the velocity amplitude along the heat exchanger’s plate taken at the centre of the channel for $\mathsf{\varphi}$6 and $\mathsf{\varphi}$ 16.

**Figure 12.**Temperature profiles at the open area next to the (

**a**) cold and (

**b**) hot ends of the heat exchanger assembly—comparison between the experimental and numerical model with different approaches of temperature initialisation.

**Figure 13.**Illustration of the convective currents occurring at the top area of the heat exchanger assembly.

**Figure 14.**The effect of initial temperature on the temperature profiles between the heat exchanger plates at a location 10 mm from the joint above the cold (CHX), and hot (HHX) plates.

**Figure 15.**Temperature contours in the area bounded by the heat exchanger walls—comparison between (

**a**) experiment of Shi et al. [17] and (

**b**) simulation.

**Figure 16.**Velocity profiles from computational fluid dynamics (CFD) model (black and grey lines) and the experiment (red and black symbol) for all 20 phases of a flow cycle. The heat exchanger walls are adiabatic.

**Figure 17.**Vorticity contours within the channel with plates treated as adiabatic walls—comparison between the results from (

**a**) the experiment of Shi et al. [11] and (

**b**) CFD.

**Figure 18.**Velocity profiles from CFD (black and grey lines) and the experiment (red and black symbol) for 20 phases of a flow cycle with the influence from heat exchanger wall temperature.

**Figure 19.**The effect of the imposed temperature gradient (dT) on the velocity profile taken at 15 mm from the joint above the cold plate (CHX).

**Figure 20.**Velocity profile; (

**a**) for $\mathsf{\varphi}$9 at CHX and HHX, both at a location 15 mm away from the joint; (

**b**) at CHX only at the different flow direction.

**Figure 21.**Vorticity contour for the case of the heat exchanger set as the adiabatic wall at the (

**a**) hot and (

**b**) cold plates. (1 bar in nitrogen gas, 0.3% drive ratio, 13.1 Hz).

**Figure 22.**Vorticity contour for the case of the heat exchanger set at 200 °C at the (

**a**) hot and 30 °C at the (

**b**) cold plates. (1 bar in nitrogen gas, 0.3% drive ratio, 13.1 Hz, temperature profiles assumed “flat”).

**Figure 23.**Vorticity contour for the case with the heat exchanger set at 300 °C at the (

**a**) hot and 30 °C at the (

**b**) cold plates. (1 bar in nitrogen gas, 0.3% drive ratio, 13.1 Hz, temperature profiles assumed “flat”).

**Figure 24.**Axial velocity at the middle of the channel along the hot (HHX) and cold (CHX) heat exchangers; temperature profiles assumed “flat”.

**Figure 25.**Illustration of the thermoacoustic device orientation. Red and blue plates symbolise hot and cold heat exchangers, respectively.

**Figure 26.**The effect of gravity and device orientation on the velocity profiles at a location 15 mm from the joint above the cold (CHX) and hot (CHX) plates for phases (

**a**) $\mathsf{\varphi}$19 and (

**b**) $\mathsf{\varphi}$9.

**Figure 27.**The effect of gravity and device orientation on the wall heat flux of the hot (HHX) and cold (CHX) plates.

**Figure 28.**Temperature profiles for two selected phases, when there is no gravity (g = 0) and with gravity, g, at different tilt angles; 0°, 90°, 270°, at a location 15 mm from the joint above the cold (CHX) and hot (HHX) plates.

**Figure 30.**(

**a**) The effect of the heat exchanger wall temperature on viscous dissipation and (

**b**) the enlarged view for dissipations at the open area next to the heat exchangers.

**Figure 31.**(

**a**) The effect of the device’s orientation on viscous dissipation and (

**b**) the enlarged view for viscous dissipation at the open area next to the heat exchangers.

Drive Ratio (%) | Working Medium | Frequency (Hz) | Mean Pressure (bar) | Heat Exchanger Wall Temperature (°C) | |
---|---|---|---|---|---|

Hot, ${\mathit{T}}_{\mathit{H}}$ | Cold, ${\mathit{T}}_{\mathit{C}}$ | ||||

0.3 | Nitrogen | 13.1 | 1 | Adiabatic | Adiabatic |

200 | 30 | ||||

100 | 30 | ||||

300 | 30 | ||||

300 | 100 |

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**MDPI and ACS Style**

Mohd Saat, F.A.Z.; Jaworski, A.J.
The Effect of Temperature Field on Low Amplitude Oscillatory Flow within a Parallel-Plate Heat Exchanger in a Standing Wave Thermoacoustic System. *Appl. Sci.* **2017**, *7*, 417.
https://doi.org/10.3390/app7040417

**AMA Style**

Mohd Saat FAZ, Jaworski AJ.
The Effect of Temperature Field on Low Amplitude Oscillatory Flow within a Parallel-Plate Heat Exchanger in a Standing Wave Thermoacoustic System. *Applied Sciences*. 2017; 7(4):417.
https://doi.org/10.3390/app7040417

**Chicago/Turabian Style**

Mohd Saat, Fatimah A.Z., and Artur J. Jaworski.
2017. "The Effect of Temperature Field on Low Amplitude Oscillatory Flow within a Parallel-Plate Heat Exchanger in a Standing Wave Thermoacoustic System" *Applied Sciences* 7, no. 4: 417.
https://doi.org/10.3390/app7040417