Abstract
Based on the heattosound and virtualsource principles developed in previous works, this paper designs a thermoacoustic traducer for measurement of fasttime heatflux from/to heating/cooling sources. The inverse thermoacoustic algorithm in this instrumentation is fulfilled by a PIDadaptive Luenberger observer, which is newly developed here. It is able to realtime measure the heatflux with oscillating frequency larger than 10 Hz, beyond the capability of thermoelectric sensors. Such an elegant performance is then utilized to clarify the following two doubts: one is about the ad hoc energytransfer process in selfexcited thermoacoustics, and the other is for the existence of thermalinductance materials in nature
1. Introduction
The heattosound principle explains how a heatflux into an air tube can drive acoustic waves [1, 2]. That is, on the interface of heating source and the air tube, the heatflux gives rise to entropyrate that changes the fluid densityrate there, which initiates acoustic velocity due to mass continuity. With such an acoustic source on the boundary, acousticpressure is then developed inside the whole tube. Therein the boundary inhomogeneity specified by the heatflux can be virtually realized as a delta source of heat generation on the interface. A detailed exploration on this realization can refer to [1]. With this realization, the dynamics from the heatflux to the acoustic pressure can be modelled as an inputoutput relationship. This inputoutput model can easily be identified in the Laplace domain, based on which an inputstate observer can be designed to realtime measure the heatflux with an acousticpressure sensor. This inputstate observer can be programmed into a dsPIC controller with Euler discretization [3], and then works as an inverse thermoacoustic algorithm to online deduce the heatflux. Besides the air tube and a circuit board that implements the inverse thermoacoustic algorithm, the thermoacoustic transducer comprises a porous media with ultralow thermalresistance to allow for measurement of the heatflux into a cooling source.
It is popular in industry and academy to measure heatflux through thermoelectric traducers, such as thinfilm sensors, Gardon sensors, and coaxial thermocouples. They measure temperature gradients from electricresistance variations. To measure powerful heatflux, Gardon heatflux sensor usually includes a costly watercooling heatsink. For faster measurement, the coaxial thermocouple was developed as a new material to replace thin films. For realtime measurements, these sensors are embedded with the inverse heattransfer programs that calculate the heatflux source out of temperature signals [46]. Even so, it was found that thermoelectric transducers show incapability of measuring highfrequency (say 10 Hz) or impulsive fluctuations of heatflux, since temperaturetoelectric materials are ultralowpass filters in nature. For modern practices of thermo science and engineering, reliable measurements of fasttime heatflux are needed, for example, the access to the transient behavior of thermally inductive materials that store entropyflow inside [7, 8]. Therefore, in this paper is the thermoacoustic transducer developed to solve the situation upon fasttime measurement of heatflux.
Of the thermoacoustic transducer, the inverse thermoacoustic algorithm (ITAA) is analogous to the inverse heatconduction problems (IHCP) of thermoelectric transducers. There are three types of IHCP methodologies in this decade: inverse filter solutions in Laplace domain [913], Luenberger inputstate observers based on statespace models [14], and neural networks backpropagated trained to do the inverse [15]. Among these, the neural networks are intelligent but with the tradeoff of reliability. The inverse filters are sensitive, but suffer noises and uncertainties from improper transferfunctions and initial conditions, respectively. Luenberger observers, such as Kalman filters, become rather conservative for nonlinear dynamics, although they are excellent in rejection of noises and uncertainties. Since the thermoacoustic dynamics can be considered as a linear process, Luenberger stateinput observer is chosen here as the methodology for the ITAA design as to accommodate reliability, sensitivity, and rejection of noises and uncertainties.
Luenberger observer is originally developed to estimate the state as the input can be online measured and sent into the observer [16]. To extend the Luenberger observer to estimate unknown inputs, the state equation can be expanded to contain the dynamics of the tobeestimated input [17]. For the reduction of steadystate error of state estimation, the observer gain can be extended to be a proportionintegral (PI) type from the proportional (P) type [18]. To shorten the transience in the input estimation, the dynamics of tobeestimated input can be replaced by adaptive approaches such as the gradient rule [19]. Therefore, for simultaneous reduction of steadystate error and transient time of state estimation, unknown input PI observers combined with the adaptive approaches were developed in, say, [20]. In fact, the gradient rule can be extended to be a proportionintegral (PI) type from the proportional (P) type, to further suppress the steadystate error of input estimation. In this work, the inverse thermoacoustic algorithm is virtually a Luenberger stateinput observer combined with proportionintegralderivative (PID) gradient rule that further improves both of the steadystate and transient performance of input estimation. It is found from the experiments that such a PIDadaptive Luenberger works well for ITAA.
Including this section, this paper is organized into five sections. In Section 2 is the design of thermoacoustic transducer, based on the heattosound and virtualsource principles. Section 3 synthesizes the inverse thermoacoustic algorithm structured as a PIDadaptive Luenberger observer. Section 4 applies the inverse thermoacoustic algorithm to measure the heatflux oscillations in a selfexcited thermoacoustic engine, and to verify the existence of thermal inductance.
2. Thermoacoustic transducers
The thermoacoustic transducer designed for measuring fasttime heatflux is plotted in Fig. 1. It is assembled by a porous media, an air tube, an acoustic damper, an acousticpressure sensor, two thermocouples, and a circuit board for signal processing.
In the interface between the air tube and the porous media, $x=\text{0}$, the tobemeasured heatflux $u$ flows into the air tube. Based on energy conservation, the heatflux on the boundary $x=0$ can be realized into a delta heatgeneration $u\delta \left(x\right)$ in conjunction with a Robin homogeneous boundary, which keeps the acoustic dynamics inside the tube almost the same. Based furthermore on the heattosound principle in [1], on the boundary $x\in [{0}^{},{0}^{+}]$ this virtual heatgeneration gives rise to entropyrate that changes densityrate thereon, which initiates acoustic velocity at $x={0}^{+}$. Due to the inertia and stiffness of fluid fluctuation, traveling wave or standing wave of acoustic pressure is developed inside the air tube. That is, in the virtualsource realization, the heatflux $u$ becomes an input rather than a part of boundary condition. Let the acousticpressure at some location is online measured, and denoted by $y$. The dynamics $G$ from the input $u$ to the output $y$ will be identified in Laplace domain. This inputoutput model serves for the synthesis of stateinput observer that plays as the inverse thermoacoustic algorithm (ITAA).
The function of porous media is to allow for reverse acousticvelocity across the boundary, while the heat flows into a cooling source from the air tube. To decrease the difference between the heatflux out of heating source and that into the airtube, the porous media is of ultralow thermalresistance and porosity. The massspringdamper attached to the tail of air tube has the purpose of model regulation. The damper is for decreasing the number of dominated acousticmodes, the mass is for amplifying the frequency responses in lowfrequency region, and the spring is to suppress the meanflow effect. As for the circuit board, it comprises analog circuits for signal conditioning and a MicrochipdsPIC chip that implements the ITAA.
Fig. 1Real time measurement of surface heatflux oscillations with thermoacoustic transducer embedded with inverse thermoacoustics
As for the identification of the dynamics $G$ from the input $u$ to the output $y$, it is a routine work in control engineering, referring to [2], which is proceeded as follows:
Step 1: Give an input $u$ being able to be measured by a thinfilm heatflux sensor, and collect the responses of $y$.
Step 2: Calculate the frequency spectrums of $u\left(t\right)$ and $y\left(t\right)$, denote them by $u\left(j\omega \right)$ and $y\left(j\omega \right)$. The frequency response of the dynamics $G$ is $G\left(j\omega \right)=y\left(j\omega \right)/u\left(j\omega \right)$.
Step 3: The transferfunction of the dynamics $G$, $G\left(s\right)$, can be obtained through curvefitting of its frequency response $G(j\omega $) in the Bode plot.
Step 4: Derive an observable statespace realization of $G\left(s\right)$, such as the observability canonical form. Choose a proper samplingtime ${T}_{s}$, and discretize this statespace model to be:
where $x$ represents the state vector, the cardinality of which is identical to the order of the identified dynamics $G$.
3. Inverse thermoacoustic algorithm
The next task is to develop the dynamics that is capable of deducing the boundary heatflux $u$ from the pointed sensing of acoustic pressure $y$ based on the identified dynamics $G$ in Eq. (1). These dynamics plays as an online inverse of the forward dynamics $G$, named here by Inverse Thermoacoustic Algorithm (ITAA). As mentioned in the introduction section, several onlineinverse processes were ever derived for inverse heatconduction problems and others, among which many researchers like the method of inverse transferfunction. Here the inverse transferfunction ${G}^{1}\left(s\right)$ must be improper, i.e. the order of denominator is larger than that of the nominator, so that significant noises will arise from online differentiation. Moreover, the method of inverse transferfunction treats the initial state as modelling uncertainty, which also severely affects measurement accuracy. On the other hand, this work improves the Luenberger stateinput observer with PID adaptive to obtain an onlineinverse attributed to reliability, sensitivity, and rejection of noises and uncertainties.
In the sequel is the ITAA programmed into a dsPICcontroller. Denote the estimated state and the estimate input by $\widehat{x}$ and $\widehat{u}$, respectively, Make a guess of initial state and input, ${\widehat{x}}_{0}$ and ${\widehat{u}}_{0}$. At the present time $k$, within a sampling period ${T}_{s}$, two consecutive stages are coded as follows:
Stage 1: Luenberger observer for state estimation:
where the ${\widehat{x}}_{k+1}$ represents the update of estimated state for the next instant. The observergain $L$ is chosen such that $ALC$ is Hurwitz, i.e. all of its eigenvalues are inside the unit circle in the complex plane. Specially, Kalmanfiltering gains are all candidates.
Stage 2: Gradient rule for input estimation:
where the ${\widehat{u}}_{k+1}$ represents the update of estimated input for the next instant $k+1$. Therein $e$ stands for the error away from dynamic equality, $S$ for the summation of errors in history, and $d$ for the difference of errors at two consecutive instants. The parameters $({k}_{P},{k}_{I},{k}_{D})$ represent the tobeadjusted PIDgains that are positive definite.
Subtraction Eq. (2) from Eq. (1) yields:
where $(ALC)$ is Hurwitz. Suppose the estimated input $\widehat{u}$ coincides with the real input $u$ at some time, then the estimated state $\widehat{x}$ will converge to the real state $x$ with the transience determined by the eigenvalues of $(ALC)$. This explains the rationale of Eq. (2). During the choice of an observergain $L$, note that there is always a tradeoff between fast convergence and rejection of sensor noises.
The information of the identified plant $G$ in Eq. (1) can be utilized again to update the estimated input $\widehat{u}$ by adaptive approaches. At every instant the update of the estimated input is to minimize the 2norms of the error $e$ defined in Eq. (3a). That is:
where $\eta $ is known as the dynamic parameter in the gradient rule. Eq. (5) can be rephrased as:
where ${k}_{P}$ is the proportional gain of the gradient rule. To improve the transient performance and reduce the steadystate error, we can add a derivative gain ${k}_{D}$ and an integral gain ${k}_{I}$, respectively, to the gradient rule, such that:
which is just the Eq. (3d). As for the choice of PID gains in this PIDgradient rule, we can let the Pgain coarsely adjust the response, and then finely tune the Igain for zero steadystate error and the Dgain to shorten the converging time. This PIDgradient rule is newly developed in this paper, which works especially well for measurement of fasttime heatflux, as shown in the sequel.
The thermoacoustic transducer depicted in Fig. 1 and the ITAA presented in Eqs. (2) and (3) are then put into a pilotrun. Therein an electrically heating source is onoff switched by a buck chopper to excite heatflux into the airtube. A purchased thinfilm thermoelectric sensor and the selfdesigned thermoacoustic transducer are taken to realtime measure the heatflux on the boundary. As the switching frequency of the heating source is as low as 1 Hz, the thermoacoustic transducer with the samplingtime ${T}_{s}=\mathrm{}$0.001 sec and the PIDgains of gradient rule $\left({k}_{P},{k}_{I},{k}_{D}\right)=\mathrm{}$(8000, 0.3, 8000) performs almost as well as the thinfilm heatflux sensor, as shown in Fig. 2. However, as the switching frequency is increased to 5 Hz, it becomes obvious that the thermoelectric sensor is unable to catch such a fasttime heatflux. This fact is shown in Fig. 3, which verifies the value of this new sensor applied to modern thermoindustry.
Fig. 2Measurement of slowtime heatflux with thermoelectric transducer for system identification and for verification of inverse thermoacoustic algorithm
Fig. 3Measurement of fasttime heatflux with thermoacoustic and thermoelectric transducers
Besides the Luenberger observer teamed with the PIDgradient rule, a whitenoise filter is always programmed into the embedded controller. Therein, the acousticpressure signal sent to the Luenberger observer is taken as an average of, say, 100 measured samples to reject whitenoise contamination. Thanks to the high samplingrate in the peripheral, the averaging routine can be performed in a realtime fashion. Without whitenoises, large observergains thus become feasible to remove the conventional tradeoff between fast convergence and rejection of sensor noises regarding to the choice of observergains. Moreover, the Luenberger observer is simultaneously a lowpass filter, besides the state estimator, whereby it rejects the highfrequency noises in the aftermath of whitenoise filtering by averaging. It also becomes insensitive to the orderreduction at the stage of system identification in frequency domain, when highfrequency components were cut off from the nominated model. Furthermore, the convergence time of stateinput estimation is almost independent of the initialstate guessing, since Luenberger observer functions as a linearly dynamical inverse. This allows for resetting the clock to zero at any time, which is necessary for realtime operation. For the methods of inverse filters on the other hand, the program of inverse transferfunction will severely amplify white or highfrequency noises due to the improperness of the transferfunction. Meanwhile, the initial state is an unavoidable uncertainty that destroys the measurement in practice.
4. Applications
The ITAA is now put into applications. This section presents two applications: one is to verify an adhoc energytransfer mechanism in selfexcited thermoacoustic processes, and the other is to prove the existence of thermal inductance.
A selfexcited thermoacoustic engine to harvest solar power is designed as in Fig. 4. It is assembled by the porous media, the acoustic resonator, the sunlight collector, and the mechatronic load. The thermoacoustic dynamics inside such a thermoacoustic engine can be represented into a feedbackloop interconnected by the heattransfer dynamics of the porous media and the acoustic dynamics of working gas. On the interface with the acoustic resonator, the porous media has the heattransfer affected by the temperature fluctuation that is algebraically dependent on the acoustic pressure as well as the entropy fluctuation thereon. Meanwhile, the heatflux fluctuation from the porous media initiates acoustic velocity that excites the acoustic motions distributed in the chamber [1], and thus forms the feedback loop. Imposed on those meanflow conditions which characterize large loopgains, the thermoacoustic dynamics can become linearly unstable up to limitcycling (nonlinear vibration), or even up to meanflow buckling (turbulence, streaming, vortex, etc.). Such thermoacoustic instability can be utilized for fasttime propulsion in thermoacoustic engines, but should be suppressed in combustion chambers to avert a variety of combustion instabilities.
Fig. 4Measurement of the heatflux out of porous media in a selfexcited thermoacoustic engine with inverse thermoacoustic algorithm
The ITAA developed in Section 3 is then employed to measure the heatflux out of the porous media, which is recorded in Fig. 5. It is found that PDgradient rule catches the real heatflux with a phase lag, which is known from the measurement via PIDgradient rule. Fig. 6 is for the study on the sensitivity of converging times to Pgains in PIDadaptive observer. It shows that the converging time can be effectively shortened by enlargement of the Pgain. However, the converging time is unable to be arbitrarily small by adjusting the Pgain or Dgain, beyond some values of which the system exhibits in instability. Furthermore, the converging time is observed independent of the initialstate preset inside the PIDadaptive observer, as shown in the Fig. 7 wherein canonical responses of two initialstate guesses are plotted. This means that the sensor of entropyflow can be turned on at any time, a necessary property for realtime operation.
Fig. 5Measurement of the porousmedia heatflux in Fig. 4 with PD and PID adaptive observers
Fig. 6Converging times versus the Pgains of the PIDadaptive observer in heatflux measurement of Fig. 4 (kD= 1000; kI= 100)
Fig. 5 demonstrates that the heatflux is of a sinusoidal shape, which verifies the limitcycle oscillation. Furthermore, it is found by measurement that the entropyflow across the interface between porous media and acoustic resonator is zero in average. That is, when the thermoacoustic process is fasttime drawing the internal energy in the thermal storage into vibration energy of fluid, the thermodynamics process is virtually isentropic. Therefore, the selfexcited thermoacoustic engine is the most energyefficient mechanism for converting heat to work, without considering the issue of power ratings. In the field of fluid fluctuation, it seems reasonable to assume that there is always some mechanism that can convert the internal energy into mechanical energy virtually without irreversibility, provided that the timescale of fluid fluctuation is short enough.
Fig. 7Responses with different guesses of initial state in the heatflux measurement of Fig. 4 (kP= 10000; kI= 100; kD= 1000)
The second application is for measurement of impulsive heatflux. With the advent of thermally inductive materials [7], it becomes possible to design a thermal boost chopper as an electrical analogy to a stepup power converter. In the design, as shown in Fig. 8, Table 1, an electrically controlled microblind is chosen as the switching thermalrelay, and the thermally inductive material is to store and then to deliver heatflux along with the onoff of thermal relay in a cycle. According to the principle of boost choppers, the temperature drop from the thermal storage to the thermoacoustic transducer at their interface is impulsively large, by which a vigorous stream of heatflux will then be pumped into the thermoacoustic transducer at the frequency of thermalrelay switching. Fig. 9 shows the measured heatflux from the thermal storage to the thermoacoustic transducer. Even though the measured data are not clear enough to quantify the value of the thermal inductance, they are obviously impulsive. With such an impulsive heatflux, there is no doubt that thermalinductance materials exist in nature.
Fig. 8Thermal boost chopper designed to study thermal inductance
Table 1Thermal parameters measured at 25 degrees centigrade
Material  Thermal resistance (m∙K/W)  Thermal capacitance (J/m^{3}∙k)  Thermal inductance (s∙m∙K/W) 
Agar  1.646  4170451  6.01 
Processed meat  1.179  4069800  3.83 
Sand  0.8  1488960  1.24 
NaHCO3  1.42  2246400  2.37 
Fig. 9Measurement of thermal booster response with thermoacoustic transducer
5. Conclusion
Four tools are applied to design the thermoacoustic transducer for measurement of highfrequency or impulsive heatflux: (1) the heattosound principle; (2) the virtualsource principle; (3) the structure of Luenberger observer; and (4) the PIDgradient rule (newly developed). This thermoacoustic transducer is then applied to verify the flowing two facts: (1) the isentropic process in the selfexcited thermoacoustic instability; and (2) the existence of thermal inductance in nature.
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