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Multi-Physics Simulation of Ultra-Lightweight Carbon Nanotube Speakers.

1. INTRODUCTION

Carbon nanotube (CNT) speakers are a new type of non-moving, ultra-light-weight, flexible, and stretchable thin-film loudspeaker that produce sound with the thermoacoustic effect. Alternating current passes through the electrodes to the low heat capacity CNT thin film changing the surface temperature rapidly. Then, the air adjacent to the CNT surface expands and contracts and the pressure waves are produced. In comparison, conventional loudspeakers that we use mostly have a permanent magnet and a voice coil attached to the cone. By applying the current to the voice coil, the movement of the voice coil and cone will produce the pressure waves that are propagated into the medium. The simple structure of CNT speakers includes only two electrodes and CNT thin film with the mass per unit area of 1.5 [mu]g/[cm.sup.2] [1]. The CNT speakers are pure resistance up to 10 kHz and can produce the pressure with the frequencies from zero up to 100 KHz. These speakers are useful in many applications in the automotive and aerospace industry where the lightweight subsystems are needed. Active exhaust noise cancellation is one of the applications where the high temperature environment inside the exhaust will not damage the CNT speakers. The current exhaust noise cancellation systems use different external conventional speakers to cancel out the noise of the exhaust. Based on the space limitations inside the exhaust tube, a specific size and shape of the lightweight CNT speaker can be used instead of using many conventional speakers. Crandall [2] studied the thermoacoustic effect and the thermophone as a source of sound theoretically and experimentally in 1917. At that time, they could not find appropriate material for the thin film to make sound in the human audible range. In 2006, Yu et al. [3] found that a piece of CNT thin film can produce sound by applying a current across it at audio frequencies. Xiao et al [1,4] obtained the frequency dependent sound pressure level (SPL) and total harmonic distortion (THD) for different layers CNT loudspeaker. Also, the effect of different gases in the performance of the CNT speakers, was reported. Several studies have been reported experimentally the performance of the CNT speakers [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

To date, all modeling in the field of CNT speakers has focused on analytical 1-D lumped parameter estimations. Arnold and Crandall [2] obtained the temperature variation over the surface of the thermophone and pressure in the far field by applying the current through themoacoustic source. They showed that the sound pressure increases as frequency increases. Also, the sound pressure is inversely related to heat capacity of the thin film. The model was modified by Xiao [1] by including the rate of heat loss per unit area of the thin film and the instantaneous heat exchange per unit area between the thin film and its surrounding air due to thermal conduction. In addition, the pressure in the far field and near field was obtained in [5] by Lim by solving the coupled thermal-mechanical equations. However, all of these researchers reported the sound pressure level at a single location and assumed the source velocity was uniformly zero across the area of the thin-film speaker and the sound pressure level was uniformly non-zero across the same area. It is shown by Barnard et al [6] that the temperature distribution on the thin film is nonuniform due to the heat sink effect at the copper electrodes and free convection of a vertical CNT speaker. This indicates that the acoustic properties are also non-uniform in a realistic CNT thin film speaker.

It is also, shown using near field acoustic holography (NAH) measurements in [7] by Asgarisabet et al that the sound pressure level distribution is not uniform on the CNT thin film, which is contrary with the assumption of uniform acoustic properties in the lumped parameter models. Lumped parameter models are also effective for small and flat acoustic sources. However, the main benefit of CNT speakers is that they can easily be manufactured as large sources and can be conformal to any geometry, not just a flat plate. When evaluating conformal CNT speakers the simple 1-D lumped parameter estimations are no longer sufficient for predicting and designing the thermal or acoustic characteristics.

In this study, our goal is to develop a coupled electrical-mechanical-thermal-acoustic 3-D model of the CNT speaker in COMSOL Multi-Physics software. Simulations reduce the development cost and time and allow to study the effect of different parameters in the performance of the system. A planar sample will be modeled, built, and validated to ensure the material properties are correct and the modeling method is viable. First, the current lumped parameter models studied by Xiao [1] and Lim [5] will be discussed in the modeling section. Then, the results of the simulation will be compared with the experimental data and theoretical models. The effects of thin film surface area and material properties of the medium are also discussed with respect to the output sound power.

2. DESCRIPTION OF THE CNT SPEAKER

Figure 1 shows the CNT speaker, which was made of the CNT forests. The CNT forests were grown by NanoWorld Laboratories at the University of Cincinnati using a chemical vapor deposition (CVD) technique (14-15). To make the CNT speakers, the CNT forest was stretched and drawn over the copper electrodes in the dynamics systems laboratory at Michigan Tech. The speaker consists of seven ribbons of CNT, each overlaid with five layers of thin-film as shown in Fig 1.

3. MODELING

3.1. Analytical Lumped Parameter Model

By applying a sinusoidal alternating current (Isin([omega]t)) with the frequency of [omega] and the amplitude of I through the planar CNT thin film, the temperature on the thin film was obtained by Xiao et al [1],

[T.sub.f](f) = [T.sub.a] + |[[??].sub.f]|sin(2[omega]t + [phi]') (1)

where [T.sub.a] is the surface average temperature and |[[??].sub.f]| and <[phi]' are the amplitude and phase of the varying temperature on the surface of thin film:

[T.sub.a]=[[P.sub.input]/4a[[beta].sub.0] (2)

[mathematical expression not reproducible] (3)

[mathematical expression not reproducible] (4)

where a is the surface area of the this film and the input power of [P.sub.input] is the input power, which is defined as R[I.sup.2] where R is the resistance of the thin film. Also, [[beta].sub.0] is the rate of heat loss per unit area of the thin film due to conduction, convection and radiation. The parameters [[omega].sub.1] and [[omega].sub.2] can be expressed as:

[[omega].sub.1] =[2[alpha][[beta].sub.2.sub.0]/[K.sup.2]], [[omega].sub.2] = [2[[beta].sub.0]/[c.sub.s]](5)

where [alpha] and K are the thermal diffusivity and conductivity of the gas, respectively and [c.sub.s] is the heat capacity per unit area of the thin film.

In this model, the temperate on the thin film is only a function of time and uniform distribution of temperature over the thin film surface has been assumed. It is also shown that the frequency of the temperature is double the input frequency. Using the temperature and thermal expansion properties of the medium gas, the pressure in the far field at the distance of x was described as,

[mathematical expression not reproducible] (6)

where [[rho].sub.0] and [T.sub.0] are the density and the temperature of the gas, respectively. This shows that the acoustic pressure produced by the CNT speaker is proportional to the input power and increases as frequency increases. Also, it is shown that the heat capacity per unit area of the thin film and the pressure are inversely related. The heat capacity of the CNT thin film is orders of magnitude less than the platinum thin film used by Arnold and Crandall [2]. This makes these speakers practical to produce the high SPL in audio frequency range.

In a similar study, Lim [5] used another approach by using the coupled mechanical-thermal equations and obtained the pressure distribution in both near and far fields. Following equations show the pressure in the near (x [less than or equal to] [R.sub.0]) and far (x [greater than or equal to] [R.sub.0]) fields where [R.sub.0] is the Rayleigh distance and is defined as a/[lambda] where [lambda] is the acoustic wavelength:

[mathematical expression not reproducible] (7)

[mathematical expression not reproducible] (8)

where [gamma] is the heat ratio of the gas, [c.sub.0] is the isentropic sound velocity in the gas, which is 347 m/s for the air at 300 K. It was reported in [5] that the two models are in good correlation with the experimental results for small sources will small input power.

3.2. COMSOL Multi Physics Simulation

Our goal is to develop a coupled electrical-thermal-acoustic 3-D model of the CNT speaker in COMSOL. To simulate the CNT speaker, the AC/DC (Joule Heating), thermoviscous and pressure acoustic modules are used. In the AC/DC module, the Joule heating should be used to consider both Electric Current and Heat Transfer modules. The temperature variation on the CNT thin film is obtained by applying alternating electrical (AC) current to the CNT film in time domain. Then, the surface temperature is transformed to the frequency domain using the Time-to-Frequency study. Finally, the frequency dependent temperature variation on the surface is used to simulate the pressure distribution in the open medium using the thermoviscous and pressure acoustics modules. The results of simulation will be validated by analytical models and experimental results. All the material properties of the medium gas used in the simulation are shown in Table 1.

The material for the electrodes has been considered as copper and all material properties for the CNT thin film were obtained from the reported values in [1] and [5]. The most important material properties for the CNT thin film are the heat capacity [c.sub.s] and rate of heat loss [[beta].sub.0].

4. RESULTS AND DISCUSSION

The temperature distribution on the CNT thin film was obtained by applying the AC current between the electrodes at different frequencies. It is shown (Fig. 2) that the temperature distribution on the surface is not uniform and maximum in the centerline of the CNT thin film. This is due to the high thermal and electrical conductivity of the CNT thin film compared to the copper electrodes.

Also, the time variant temperature on the source surface for different input frequencies is shown in Fig 3. For each input frequency, the temperatures at different locations on the surface were obtained and then averaged over the surface. As shown, the output frequency of the temperature variation is double that of the input frequency, as discussed in the Eq. 1. In addition, the average temperature is not a function of frequency and is only related to the input power (Eq.2).

To consider the variation of temperate as a function of frequency, the normalized temperature variation [T.sub.fN] is defined as,

[mathematical expression not reproducible] (9)

The normalized temperature is a function of frequency, [[omega].sub.1] and [[omega].sub.2], which depends on [[beta].sub.0] and [c.sub.s]. Figure 4. Shows the effect of material properties for different frequencies on normalized temperature.

It is seen that for the frequencies less than 2000 Hz, the normalized temperature is very sensitive to [[beta].sub.0] for the constant [c.sub.s]. This shows that an accurate estimation of [[beta].sub.0] is critical to obtaining the correct results. In both studies by Xiao [1] and Lim [5], different values for the [[beta].sub.0] and [c.sub.s] have been used to obtain the better correlation between the models and experimental data.

In addition, it is shown in Eq. 9 that the normalized temperature decreases as frequency increases. Fig 5 shows the comparison between the simulation results and the analytical model (Eq. 9) for [[beta].sub.0] = 23 W/[m.sup.2].K.

The simulations result in a good correlation with the analytical model. Also, it is seen that for the all frequencies, the simulated normalized temperature is less the theoretical model. The reason is that the analytical model considers a uniform temperature distribution over the surface of the CNT speaker, while the temperature distribution as shown in Fig.2 is not uniform in the simulated model, resulting in a lower average temperature compared to the analytical model.

The SPL in the far field at 5 cm was compared to analytical models (Eq. 6 and Eq. 8) and experimental data [5] in Fig. 6. The CNT thin film is 3 x 3 [cm.sup.2] with the input power of 4.5 W. The frequency should be less than 19 kHz at 5 cm to be in the far field for this size of speaker. Also, [[beta].sub.0] = 23 W/[m.sup.2].K and [c.sub.s] = 7.7 x [10.sup.-3] J/[m.sup.2].K have been used as CNT material properties [18].

As shown in the Eqs. 6, 7, 8, SPL is a function of input power and by increasing the input power, SPL will increase. Figure 7 shows the SPL as a function of frequency for different input powers.

For all frequencies, increasing the input power, results in increasing of the SPL. Also, it is seen that for the all input power levels, SPL at at 10000 Hz is less than 8000 Hz. This is defined as a thermal saturation in the thin film by Brungart et al [17], where increasing the input power, does not result in the high sound pressure level at high frequencies. This can also be explained by the Eq. 6, where the average temperature of the thin film is in the denominator which is related to input power.

Figure 8. shows the SPL distribution of CNT speaker. It is seen that the directivity index increases as frequency increases. Also, the directivity is almost uniform for low frequencies and becomes more directional at higher frequencies.

One of the advantages of the simulations is to study the different shape and configuration of the system with low cost and time before manufacturing. Directivity pattern is one of the most important characteristics of the speaker in the design process which is defined based on the required SPL distribution from the customer. As shown in Fig. 8, the directivity pattern for the CNT speaker is not uniform at high frequencies. A specific configuration of the CNT speaker has been studied by considering four CNT thin films as shown in Fig. 9. It is shown that the directivity was improved at high frequencies; however, the SPL at low frequencies has been reduced considerably. Based on the required dimension of the speaker and the frequency content, an optimized geometry can be obtaining to improve the performance by simulations.

5. CONCLUSION

Carbon Nanotube (CNT) thin film speakers are a new type of speakers that produce sound via thermoacoustic effect. Because of the low heat capacity of the CNT thin film, the temperature of it can change very fast by applying the AC current through it. Then, the sound pressure waves are produced by heating and cooling the adjacent air to the film. These speakers are inexpensive, transparent, stretchable, flexible, magnet-free, and lightweight. CNT speakers are useful in the applications that require light weight subsystems such as automotive, aerospace and consumer electronics. To develop and improve the performance of these speakers, having a model to consider all of the parameters of the system will be necessary. To date, only analytical 1-D lumped parameter model of the CNT speakers have been studied. Also, the lumped parameter models have been only developed for the small Planar CNT speakers with low input power. In this study, a coupled electrical-mechanical-thermal-acoustic 3-D model of the Planar CNT speaker in COMSOL Multi-Physics software was developed. The temperature variation on the CNT thin film was obtained by applying alternating electrical current to the CNT film and compared to the analytical models. Then, surface temperature variation is used to simulate the pressure distribution in the open medium. The results showed that the simulation is in a good correlation between the theoretical models and experimental data. In addition, it has been obtained that for each frequency, increasing the input power will result in increasing the sound pressure level. Also, the effect of thermal saturation in the thin film has been studied for the frequencies higher than the 8000 Hz. Directivity patterns showed that for the simple planar speaker, directivity index increases as frequency increases. An experimentally validated model will help to study the different shapes and dimensions of the speaker in the short time and with low cost. The developed model will be improved to study other shapes and configurations of the CNT speaker in the future.

REFERENCES

(1.) Xiao L., Chen Z., Feng C., Liu L., Bai Z. Q., Wang Y., Lan L., Zhang Y., Li Q., Jiang K. and Fan S., Flexible, stretchable, transparent carbon nanotube thin film loudspeakers. Nano letters, 2008. 8(12): p. 4539-4545.

(2.) Arnold H. and Crandall I., The thermophone as a precision source of sound. Physical Review, 1917. 10(1): 22-38.

(3.) Yu 4X., Rajamani R., Stelson K., and Cui T., "Carbon nanotube-based transparent thin film acoustic actuators and sensors," Sensors Actuators A: Phys, 2006. 132(2), 626-631.

(4.) Xiao L., Liu P., Li Q., Feng Z., Fan S. and Jiang K., High frequency response of carbon nanotube thin film speaker in gases. Journal of Applied Physics, 2011. 110(8): p. 084311.

(5.) Lim C.W., Tong L., and Li Y., Theory of suspended carbon nanotube thin film as a thermal-acoustic source. Journal of Sound and Vibration, 2013. 332(21): 5451-5461.

(6.) Barnard A. R., Brungart T. A., McDevitt T. M., Aliev A. E., Jenkins D. M., Kline B. L. and Baughman R. H., Advancements toward a high-power, carbon nanotube, thin-film loudspeaker. Noise Control Engineering Journal, 2014. 62(5): 360-367.

(7.) Asgarisabet M., Barnard A. R., Bouman T. M., Near field acoustic holography measurements of canrob nanotube thin film speaker, Journal of Acoustical Society of America, 2016. 140 (6): 1-9.

(8.) Barnard A. R., Jenkins D. M., Brungart T. A., McDevitt T. M. and Kline B. L., Feasibility of a high-powered carbon nanotube thin-film loudspeaker. The Journal of the Acoustical Society of America, 2013. 134(3): EL276-EL281.

(9.) Barnard A. R., Brungart T. A., McDevitt T. M. and Jenkins D. M., Bachground and developement of a high-powered carbon nanotube thin-film loudspeaker. Proceedings of Inter Noise 2012, 1513-1524, Newyork, USA.

(10.) Barnard A. R., Brungart T. A., McDevitt T. M. and Jenkins D. M. and Kline B. L., Advancements toward a high-powered cabon nanotube thin-film loudspeaker. Proceedings of Noise Control. 2013, 796-802, Denver, USA.

(11.) Asadzadeh S. S., Moosavi A., Huynh C. and Saleki O., Thermo acoustic study of carbon nanotubes in near and far field: Theory, simulation, and experiment. Journal of Applied Physics, 2015. 117(9): p. 095101.

(12.) Yu X., Rajamani R., Stelson K. A. and Cui T., Carbon nanotube-based transparent thin film acoustic actuators and sensors. Sensors and Actuators A: Physical, 2006. 132(2): 626-631.

(13.) Bouman T. M., Barnard A. R., Asgarisabet M.. Experimental quantification of the true efficiency of carbon nanotube thin-film thermophones. Journal of the Acoustical Society of America. 2016; 139 :1353-1363.

(14.) Pohls J. H., Johnson M. B., White M. A., Malik R., Ruff B., Jayasinghe C., Schulz M. J. and Shanov V., Physical properties of carbon nanotube sheets drawn from nanotube arrays. Carbon, 2012. 50(11): 4175-4183.

(15.) Jakubinek M. B., White M. A., Li G., Jayasinghe C., Cho W., Schulz M. J. and Shanov V., Thermal and electrical conductivity of tall, vertically aligned carbon nanotube arrays. Carbon, 2010. 48(13): 3947-3952.

(16.) lackstock D.T., Fundamentals of physical acoustics. 2000, 440-464, John Wiley & Sons. New Jersey, US

(17.) Brungart, Timothy A., et al., Thermal saturation and its suppression in high-power, compact carbon nanotube thin-film thermophones, The Journal of the Acoustical Society of America. 2016. 140.4: 3034-3034.

(18.) Asgarisabet, Mahsa, and Barnard Andrew R., Multi-physics modeling of conformal, solid-state, and thin-film thermophones, The Journal of the Acoustical Society of America. 2016. 140.4: 3141-3141.

CONTACT INFORMATION

Mahsa Asgarisabet

Masgaris@mtu.edu

ACKNOWLEDGMENTS

The authors would like to acknowledge the contributions of many who made this work possible, including Dr. Vesselin Shanov and his employees at NanoWorld Laboratories at the University of Cincinnati for growing the CNT forests and PCB Piezotronics[R] for their donation of the measurement microphones used in this effort.

NOMENCLATURE

ABBREVIATIONS

CNT - Carbon Nanotube

NAH - Near-Field Acoustic Holography

SPL - Sound Pressure Level

THD - Total Harmonic Distortion

MWNT - Multi-Walled Nanotube

CVD - Chemical Vapor Deposition

AC - Alternating Current

dB - Decibels

SYMBOLS

I - Amplitude of the alternating current (A)

[omega] - Radial Frequency (rad/s)

t - Time (s)

[T.sub.f] - The temperature of the CNT thin film (K)

[T.sub.a] - Average temperature of the thin film (K)

|[[??].sub.f]|- Amplitude of the Varying temperature of the thin film (K)

[phi]'- Phase of the Varying temperature of the thin film (Rad)

R - The instantaneous resistance of the thin film ([OMEGA])

a - Area of the thin film ([m.sup.2])

[[beta].sub.0] - Rate of heat loss per unit area of the thin film (due to conduction, convection and radiation) per unit rise in temperature of the thin film (W/[m.sup.2].K)

[alpha] - Thermal diffusivity of the gas ([m.sup.2]/s)

K - Thermal conductivity of the ambient gas (W/m.K)

[c.sub.s] - Heat capacity per unit area of thin film (J/[m.sup.2].K)

[P.sub.rms] - Root mean square of the acoustic pressure (Pa)

[P.sub.input] - Input Power (W)

[[rho].sub.0] - Density of the gas (kg/[m.sup.3])

[T.sub.0] - Temperature of the gas (K)

x - Normal distance from the thin film (m)

[R.sub.0] - Rayleigh Distance (m)

[lambda] - Acoustic Wavelength (m)

[c.sub.0] - Isentropic Speed of Sound (m/s)

[gamma] - Heat ratio of the gas

Mahsa Asgarisabet and Andrew Barnard

Michigan Technological University

doi:10.4271/2017-01-1816
Table 1. Constant material properties of the gas (Air) at 300 K [5]

[[rho].sub.0] (kg/[m.sup.3])   1.16   K (W/m.K)
[gamma]                        1.14   [alpha] ([m.sup.2]/s)

[[rho].sub.0] (kg/[m.sup.3])   0.0262
[gamma]                        2.25 x [10.sup.-5]
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Author:Asgarisabet, Mahsa; Barnard, Andrew
Publication:SAE International Journal of Materials and Manufacturing
Article Type:Technical report
Date:Jul 1, 2017
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