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Investigation on thermo-acoustic instability dynamic characteristics of hydrocarbon fuel flowing in scramjet cooling channel based on wavelet entropy method
Accepted Manuscript Investigation on thermo-acoustic instability dynamic characteristics of hydrocarbon fuel flowing in scramjet cooling channel based on wavelet entropy method Hao Zan, Haowei Li, Yuguang Jiang, Meng Wu, Weixing Zhou, Wen Bao PII:
S0094-5765(17)31864-7
DOI:
10.1016/j.actaastro.2018.03.015
Reference:
AA 6757
To appear in:
Acta Astronautica
Received Date: 19 December 2017 Revised Date:
2 February 2018
Accepted Date: 9 March 2018
Please cite this article as: H. Zan, H. Li, Y. Jiang, M. Wu, W. Zhou, W. Bao, Investigation on thermoacoustic instability dynamic characteristics of hydrocarbon fuel flowing in scramjet cooling channel based on wavelet entropy method, Acta Astronautica (2018), doi: 10.1016/j.actaastro.2018.03.015. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT The Title: Investigation on thermo-acoustic instability dynamic characteristics of hydrocarbon fuel
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flowing in scramjet cooling channel based on wavelet entropy method
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The Authors: Hao Zan a, Haowei Li a, Yuguang Jiang a, Meng Wu b, Weixing Zhou b,*, Wen Bao a
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Affiliation： ：a College of Energy Science and Engineering, Harbin Institute of Technology
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b
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College of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology
Corresponding Author: Weixing Zhou
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Address: No.92, West Da-Zhi Street, Harbin, Heilongjiang, 150001, P.R.China.
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E-mail: [email protected]
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Investigation on thermo-acoustic instability dynamic characteristics
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of hydrocarbon fuel flowing in scramjet cooling channel based on
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wavelet entropy method
b
College of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, P. R. China
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Hao Zan a, Haowei Li a, Yuguang Jiang a, Meng Wu b, Weixing Zhou b,*, Wen Bao a a
College of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, Harbin 150001, P. R. China
Abstract
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As part of our efforts to find ways and means to further improve the regenerative cooling technology in
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scramjet, the experiments of thermo-acoustic instability dynamic characteristics of hydrocarbon fuel flowing
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have been conducted in horizontal circular tubes at different conditions. The experimental results indicate that
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there is a developing process from thermo-acoustic stability to instability. In order to have a deep
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understanding on the developing process of thermo-acoustic instability, the method of Multi-scale Shannon
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Wavelet Entropy (MSWE) based on Wavelet Transform Correlation Filter (WTCF) and Multi-Scale Shannon
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Entropy (MSE) is adopted in this paper. The results demonstrate that the developing process of
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thermo-acoustic instability from noise and weak signals is well detected by MSWE method and the
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differences among the stability, the developing process and the instability can be identified. These properties
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render the method particularly powerful for warning thermo-acoustic instability of hydrocarbon fuel flowing
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in scramjet cooling channels. The mass flow rate and the inlet pressure will make an influence on the
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developing process of the thermo-acoustic instability. The investigation on thermo-acoustic instability
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dynamic characteristics at supercritical pressure based on wavelet entropy method offers guidance on the
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control of scramjet fuel supply, which can secure stable fuel flowing in regenerative cooling system.
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Keywords: Thermo-acoustic instability, Developing process, Hydrocarbon fuel, Scramjet engine
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cooling channel, Wavelet entropy method
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Research highlights
1 The developing process from thermo-acoustic stability to instability is detected.
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2 A wavelet entropy method is used to analyze the developing process.
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3 The mass flow rate and inlet pressure have an influence on developing process.
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Nomenclature
=
index scale number of wavelet decomposition
Corr
=
correlation coefficient
D
=
scale wavelet coefficient
e
=
scale factor of coarse-grained signals
H
=
multi-scale Shannon value of signal
k
=
index block number
l
=
the number of scales involved in the direct multiplication
L
=
Test section heated length (m)
m
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a
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=
mass flow rate (g/s)
=
total scale number of wavelet decomposition
=
point number of discrete signal
n
=
translation index
p
=
Probability
P
=
inlet pressure (MPa)
M N
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3
s
=
block number of data divided
th
=
threshold ratio
T
=
Temperature ( )
x
=
Signals
y
=
coarse-grained signals
Y
=
wavelet transform data
λ
block parameter
µ
bifurcation parameter
f
Filtered
h
stable stage
i
i-th
in
inlet
j
j-th
max min out
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Subscripts
index block number the maximum value the minimum value outlet
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Greek symbols
k
3
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power of correlation coefficient
EP
2
=
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1
PCorr
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1.
Introduction
Scramjet is a promising hypersonic air-breathing propulsion system for hypersonic missiles, aircrafts and
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even reusable space transport vehicles because of its good performance at hypersonic region [1, 2]. Yet, when
4
flight velocities increase beyond supersonic speed, the thermal protection becomes a critical problem because
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combustion chamber is exposed to very high temperature. In this sense, regenerative cooling is one of the
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most effective cooling techniques, using propellant as a coolant. As a high energy density fuel and potential
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coolant, hydrocarbon can also be used to lower the structure temperature of combustor wall [3-5]. Before
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injected into the combustor, hydrocarbon fuel is forced into the cooling channels distributed in the hot wall of
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combustor for absorbing heat [6]. Indeed, the strong coupling between fuel-coolant flowing (in the cooling
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channels) and fuel-coolant combustion (in the combustor) makes system control a very hard task. For
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example, fuel mixing has important influence on combustion efficiency and heat release distribution, which
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further affects fuel flowing in cooling channels [7, 8]. Studies on hydrocarbon fuel flowing are of great
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importance, especially for acquiring a better understanding on the fuel flowing behavior in scramjet cooling
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channels. Different kinds of hydrocarbon fuel heating experiments are conducted, which show that flowing
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instability occurs during heating of supercritical hydrocarbon fuel. It leads to severe problems such as
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structural vibrations, difficulty in controlling fuel flow and generation of acoustic noise [9-12].
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The exploration of hydrocarbon fuel flowing instability has attracted considerable attentions from
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engineering and physical research fields because of its significant importance [13]. The high temperature fuel
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flowing instability mechanism has been revealed gradually. Hitch et al. [14] experimentally investigated heat
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transfer in supercritical hydrocarbon fuel and identified two previously observed oscillation modes, which are
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Helmholtz oscillation mode and the acoustic oscillation mode. Although Helmholtz mode oscillation could be
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eliminated relatively easily [15], fuel temperature and pressure could still be unstable near the critical point.
ACCEPTED MANUSCRIPT The transient pressure behavior quickly changes between large and slow oscillations to nearly pure tones
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associated with the acoustic vibration mode. Zhou et al. [16] observed pressure-drop type instability in the
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hydrocarbon fuel cooling system. By numerical simulation and analytical solutions, the author found that
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instability of the cooling system was caused by the interaction of positive and negative feedback. Liu et al.
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[17] found flow instability, pressure drop deduction, and heat transfer deterioration happened simultaneously
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accompanied with acoustic flow instability and peculiarly diabatic pressure drop deduction due to the steep
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thermodynamic properties. Wang et al. [18, 19] investigated thermo-acoustic instability of hydrocarbon fuel
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(RP-3) in a vertical circular tube at supercritical pressure. The influences of mass flow rate and operating
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pressure on thermo-acoustic instability were found. Moreover, the pressure drops fluctuations can be used to
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represent the characteristics of thermo-acoustic instability. Some researchers claimed that the thermo-acoustic
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instability was related to the formation and collapse of pseudo bubbles, propagated in the fluid with the speed
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of sound [20-22]. Others believed that large variations in thermo-physical properties were significant reasons
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[23]. By conducting a series of heat transfer experiments, Linne et al. [24] found flow instability could not be
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attributed to one single reason. He suggested a statistical model to predict the flow instability in supercritical
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hydrocarbon fuel flowing.
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Despite the existing results, there are still significant challenges in the study of hydrocarbon fuel flowing
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instability. Previous work was mainly focused on the instability phenomenon and mechanism with
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experimental method, but fewer researchers paid attention to the developing process from thermos-acoustic
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stability to instability. In literature [18], thermo-acoustic instability is not a sudden change, but a developing
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process. The developing process of the thermo-acoustic instability can be divided into four stages, namely,
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thermo-stable stage, thermo-unstable stage, preliminary thermo-acoustic stage and developed thermo-acoustic
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stage. However, there are few investigations on characteristics in different stages, and it is difficult to do the
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ACCEPTED MANUSCRIPT classification in experimental conditions. As thermo-acoustic instability pressure fluctuation signals contain a
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lot of flowing information of hydrocarbon fuel, the information of pressure loss, flow rate and fuel
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temperature can be acquired from them [25]. Real time measurement of pressure fluctuation signals can be
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implemented easily, based on which, the developing process of thermo-acoustic instability can be detected
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[26]. With the development of wavelet theory and entropy statistical theory, the wavelet entropy method is
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widely applied to the analysis and processing of transient signals in many fields [27-29]. Wavelet Transform
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Correlation Filter (WTCF) has superiority of extracting weak fault feature of pressure signals and high SNR
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(Signal-Noise Ratio) scales wavelet coefficients are gained [30]. Multi-scale Shannon entropy (MSE) has
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been applied to the two-phase flow field to characterize flow behavior [31]. The method of Multi-scale
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Shannon Wavelet Entropy (MSWE) based on WTCF and MSE is put forward to analyze the thermo-acoustic
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instability dynamic characteristics of hydrocarbon fuel flowing.
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Based on previous studies, experiments of thermo-acoustic instability of hydrocarbon fuel under
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different conditions are conducted firstly. Then, the developing process from thermo-acoustic stability to
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instability is analyzed with MSWE method, and characteristics of different stages are obtained in this paper.
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At last, the influences of mass flow rate and inlet pressure on the developing process are studied. This work is
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to get a better understanding of the developing process and offers guidance on adjusting time for stable fuel
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supply to control thermo-acoustic instability. For example, we will change the equivalent ratio as soon as the
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developing process is detected, which can avoid the occurrence of developed thermos-acoustic instability.
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2.
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2.1 Wavelet Transform Correlation Filter (WTCF)
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MSWE method
The method of Wavelet Transform Correlation Filter has been described previously [30, 32, 33]. Based
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scale, WTCF is the direct spatial correlation of wavelet coefficients at several adjacent scales. Edges and
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significant features of the signal are enhanced while noise and small sharp features, edges and features gained
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from the noise are suppressed. By threshold inspection, SNR of wavelet coefficients of WTCF is greatly
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higher than wavelet transform. In this paper, locations of edges and significant features of the signal are
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detected by spatial correlation Corrl (a, n) of wavelet transform contents at several adjacent scales. n = 1, 2 L , N
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l −1
Corrl ( a ,n ) = ∏ Y ( a + i , n )
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i =0
(1)
where Y denotes the wavelet transform data, a is the scale index, N is the point of discrete signal, n is the
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translation index, l is the number of scales involved in the direct multiplication, M is the total number of
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scales and a
Step 1: Compute all Corr2 (a, n) for all every wavelet scale a, enhanced significant features and
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suppressed noise are gained.
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Step 2: Rescale the power of Corr2 (a, n) to that of Y (a, n) and get NewCorr2 (a, n). NewCorr2 ( a,n ) = Corr2 ( a , n ) PY ( a ) / PCorr ( a )
(2)
PY ( a ) = ∑ Y ( a , n )
(3)
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2
n
PCorr2 ( a ) = ∑ Corr2 ( a , n )
2
(4)
n
where PY(a) is ath scale wavelet coefficients and PCorr2(a) is the ath power of Corr2(a, n).
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Step 3: If NewCorr2 ( a,n ) = Y ( a, n ) , the point is accepted as edge. Pass Y(a, n) to Yf, then reset Y(a, n)
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and Corr2(a, n) to 0. Otherwise, we suppose Y(a, n) is produced by noise and then retain Y(a, n) and Corr2(a,
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n).
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Step 4: Return to step 1, repeat steps 2 and 3 until value of
PY ( a ) reaches threshold ratio th(a)
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which is correlated with some reference noise power at ath wavelet scale. Energy normalization, comparison of data and extracting information edge will be iterated multiple
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times until the power of Y(a, n) reaches some reference noise power at the ath wavelet scale. As introduced in
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literature [30], the reference noise power should be estimated. After pressure signal is processed by WTCF,
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the scale wavelet coefficient D1,D2,…,Dj, which contain information from high frequency to low frequency
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and are the estimation of local energy in different scale can be gained.
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2.2 Multi-Scale Shannon Entropy (MSE)
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The basic procedure for calculating multi-scale Shannon entropy from signals is as follows: for a signal
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x(i), i=1, 2,…, N, we first perform a coarse-grained process to define temporal scales and further obtain the
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coarse-grained signals as follows:
je 1 xi ∑ e i = ( j −1) e +1
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y ej =
λ=
max
j =1,K, N / e
{ y ( j )} −
min
k =1,K, N / e
{ y ( j )}
(5)
(6)
s
where e is the scale factor and 1≤j≤N/e. The signals are equally divided into s blocks, 1≤s≤N/e. The range
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of each block is between ymin+(k-1)λ and ymin+kλ (k=1, 2, …, s). The data number of signals in each block is
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counted, and the probability (pk) of signal data in every block is calculated.
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pk =
eN k N
(7)
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where Nk denotes the number of signal in each part. Multi-scale Shannon entropy of signal x(i) can be showed
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as follows: s
H = − ∑ Pk ln ( Pk ) k =1
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H will reach the maximum when Pk is 1/s. H is normalization processed by dividing lns.
(8)
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H ≤1 ln s
(9)
where H denotes the degree of randomness of signal x(i). The smaller H is, the more regular the signal is.
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Otherwise, the signal is more random. The change of H shows subtle change of signal.
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2.3 Computation process
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The WTCF has been applied in the field of signal feature extraction, because the method has a strong
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ability of analysis in the time-frequency domain. Combined with the property of the MSE which is useful for the
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analysis of signal changing, a method of MSWE can be designed shown in Fig. 1.
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Data Acquistion
Wavelet Transform Correlation Filter
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high SNR scales wavelet coefficients D
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Multi-scale Shannon Wavelet Entropy E
△E≥△Ethreshold?
No
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Yes Warning
Fig. 1. Computation process
Step 1: The pressure fluctuation signal is sampled by pressure sensors. Step 2: The time series data, corresponding to pressure fluctuation signal, then processed by WTCF with a five-level decomposition using Eqs. (1) - (4). Step 3: Following the computation of wavelet transform, the scale wavelet coefficients D1, D2, …, Dj, which contain information from high frequency to low frequency in different scale, can be gained.
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the MSE values of the wavelet coefficients D1, D2, …, Dj can be computed by Eqs. (5) - (9).
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Step 5: The absolute value of the relative MSWE △E is given in Eq. (10) aiming to take the difference into consideration.
∆E = Eh − Ew
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(10)
Eh and Ew denote the MSWE values of the stable state and the developing process of thermo-acoustic instability, respectively.
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Step 4: To grasp the signal entropy difference of the stability, the developing process and the instability,
Step 6: As shown in Fig. 1, the focus on this computation process becomes the determination of threshold.
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Once the relative MSWE value of the hydrocarbon fuel pressure is larger than this threshold, a forewarning
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signal will be immediately delivered to the fuel supply system so that the appropriate actions can be performed
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at an early stage.
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3.
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Experiments and data acquisition
The tests were conducted in the HIT. The experiments of dynamic characteristics of thermo-acoustic
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instability of hydrocarbon fuel flowing have been conducted in horizontal circular tubes at different
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conditions (as shown in Fig. 2). The hydrocarbon fuel (RP-3) is used as the working substance for the
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experiment. The critical pressure and critical temperature of hydrocarbon fuel are about 2.4 MPa and 372
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respectively, which are consistent with the critical properties of general kerosene hydrocarbons in the
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reference [34]. Fill the fuel sump with nitrogen. Start the pump and set the mass flow rate and let the fuel
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flow through the whole system. Turn on the DC power and heat the test section. When the outlet temperature
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becomes steady, the heating power is raised to a higher level. Record the experimental data when
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thermo-acoustic instability occurs.
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Test Section
P
1
2
3
4
T
Coller
T
P
Insulator
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Turbine Flowmeter
Outlet temperature
Inlet pressure
Different Pressure Control Valve
Cutoff Valve
Filter
P
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Check Valve
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Data Acquisition System
Fuel Sump
Cutoff Valve
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Fig. 2. Schematic diagram of the experimental set-up
The geometric configuration of cooling channels has important influence on heat exchange between
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combustion and hydrocarbon fuel. Different physical configuration of cooling channels should be considered
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to investigate flowing instability. The detailed physical parameters are referred to the studies of Huang
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[35-38]. The test section is a horizontal straight superalloy (GH3128) circular tube having a length of 1~3 m,
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an internal diameter of 12 mm and a thickness of 2 mm and is heated by DC power during experiment. Some
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parameters are measured per second using Nl c-rio data acquisition system. The sampling frequency range is
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1-100 Hz. Four K-type thermocouples (Accuracy: ±5 ) are spot welded directly to the outer surface of the
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test section to detect wall outside temperature. The outlet temperature refers to outlet fluid temperature
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measured by K-type armored thermocouples (Accuracy: ±5 ). Pressure sensors (Accuracy: ±0.5% of FS: 5
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MPa) are installed in the inlet/outlet test section and the fuel sump. The inlet pressure refers to the inlet fluid
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pressure measured by the pressure sensor. Turbine flow meters are installed at the inlet of the tube to measure
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ACCEPTED MANUSCRIPT the mass flow rate. The accuracy of the flow meter is 1.0% of the full scale (500 g/s). The response time of
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the pressure sensor (MPM489) and armored thermocouples is 2 ms and 0.5s, respectively. The response time
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of the pressure sensor is enough to catch the dynamic process of thermo-acoustic instability. The internal
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diameter of the test section is larger than traditional geometric configuration of the cooling channels in
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scramjet. The experiments are our preliminary work, and we will conduct the experiment with a dimension of
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millimeter in further.
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A total of 15 tests were conducted to investigate the thermo-acoustic instability dynamic characteristics.
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The operating conditions and results are shown in Table 1 with mass flow rate (min), inlet pressure (P) and
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tube length (L) varying from 50 g/s to 200 g/s, 2.6 MPa to 4.5 Mpa, and 1 m to 3 m respectively. The inlet fuel temperature (Tin) is 28 .
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Table 1 Operating conditions and results Test
1
2
3
4
5
min(g/s)
50
50
50
10
100
P(MPa)
2.6
3
L(m)
3
3
0.6
0.5
oscillation amplitude(MPa)
9
10
11
12
13
14
15
10
20
20
200
20
20
200
20
10
50
0
0
0
0
0
0
0
2.6
3
4.5
2.6
3
4.5
2.6
3
4.5
3
3
3
3
3
3
3
3
3
3
2
2
2
1
1
1
1.2
1.6
1
2
1.5
2
2.5
0.9
1.2
0.6
0.3
1
32
12
16
26
21
15
11
8~1
3~
4~
Frequency (Hz)
5
8
7
22
7~2 3
21~3 7
13~2 4
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4.5
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4.
Results and discussion
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4.1 The developing process of thermo-acoustic instability
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In these 15 tests, tube oscillation with the abnormal sound similar to sharp metal hammering was
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observed. The abnormal sound is strongly interrelated with the thermo-dynamic instability, and the sharing
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features of thermo-acoustic instability are also suitable for the abnormal sound in this paper [18]. The large
0.0 5
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pressure fluctuations appear synchronously with the abnormal sounds. The time series of temperature and
2
pressure of thermo-acoustic instabilities under four conditions are shown in Fig. 3. As can be seen in enlarged
3
images A, B, C and D, inlet pressure begins to oscillate, accompanied by the increase of outlet temperature.
4
The outlet temperature is about 200
5
while in Fig. 3b the outlet temperature is about 120 . The reason may be that the critical pressure is
6
approached. In our opinion, the increase of the heating power will affect the dynamic process. In this paper,
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the rate of increasing heating power is about 2 KW/s. The curves of heating power have been given together
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with the pressure and outlet temperature in below.
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when inlet pressure begins to oscillate in Fig. 3a, Fig. 3c, and Fig. 3d,
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Fig. 3. The phenomenon of thermo-acoustic instability, (a) min=200 g/s, P=3.0 MPa, L=3 m, (b) min=100 g/s, P=3 MPa,
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L=1 m, (c) min=50 g/s, P=2.6 MPa, L=3 m, (d) min=200 g/s, P=4.5 MPa, L=2 m
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As shown in Fig. 4, the signals of inlet pressure and outlet temperature present the developing process of
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the thermo-acoustic instability. As can be seen in Fig. 4a, we define region
15
temperature is stable, region
as the stable state in which
as the developing process in which the temperature changed but pressure did
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not oscillate significantly, region
2
significantly.
3
as unstable state in which temperature and pressure oscillated
The pressure change can be detected by pressure fluctuation amplitude and the complexity of pressure
4
time series. In region
5
time series shows random behavior. It shows that pressure time series in the stable state is disturbed by noise.
6
In region
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pressure time series changes from random to periodic behavior. This means that abnormal sound occurs and
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thermo-acoustic instability starts. In region
9
complexity of pressure time series shows periodic behavior. This clearly demonstrates that thermo-acoustic
10
instability is fully developed. It has been shown that complexity change of pressure time series is prior to the
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pressure fluctuation amplitude. Thus, measure of complexity (MSWE method) is developed to investigate
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dynamic characteristics of thermos-acoustic instability.
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(Fig. 4b), pressure fluctuation amplitude is very small, and complexity of pressure
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(Fig. 4c), there is no obvious change in pressure fluctuation amplitude, but complexity of
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(Fig. 4d), pressure fluctuation amplitude is very large, and
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Fig. 4. The developing process of thermo-acoustic instability (min=200 g/s, P=3.0 MPa, L=3 m)
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4.2 Pressure signal analysis based on MSWE method
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4.2.1
Validation of multi-scale Shannon entropy method
In order to obtain the thermo-acoustic instability dynamic characteristics based on MSWE method, the
4
validation of MSE method is carried out by comparing several entropy methods in detecting change of the
5
logistic model. As shown in Fig. 5a, the logistic map is a polynomial mapping (equivalently, recurrence
6
relation) of degree 2, often cited as an archetypal example of how complex, chaotic behavior can arise from
7
very simple non-linear dynamical equations [39]. Mathematically, the logistic map is written:
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(11)
where xn is a number between zero and one that represents the ratio of existing population to the maximum
9
possible population. The values of interest for the parameter µ are those in the interval [3, 4]. The µ values of
10
Bifurcation point A, Bifurcation point B and Onset of Chaos are 3.449, 3.544 and 3.57 respectively. With µ
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value between 3 and 3.449, x will approach permanent oscillations between two values. With µ value between
12
3.449 and 3.544, x will approach permanent oscillations between four values. When the value of µ reaches
13
3.57, we no longer see oscillations of finite period.
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Fig. 5. Validation of multi-scale Shannon entropy method
Several entropy methods detecting change of the logistic map are compared in Fig. 5b. This figure
4
indicates that Shannon entropy method is an effective method to show change of the logistic map with µ
5
value between 3 and 4. Phenomenon of period-doubling bifurcation (critical point A, critical point B) and the
6
onset of chaos are accurately captured. At the point of chaos onset, the Shannon entropy value is about 0.7.
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When µ>3.57, the Shannon entropy value is stable at 0.9. However, permutation entropy is irregular,
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approximate entropy and sample entropy do not change between µ=3 and µ=3.57. Therefore, Shannon
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entropy method is a reliable method to detect the signal change.
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As can be seen in Fig. 5c, the odd scale factor and even scale factor have the different values between
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µ=3 and µ=3.57. Entropy values of odd scale factors are about the same, while values of even scale change
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before bifurcation point and chaos onset point, which should be paid more attention.
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4.2.2
Effect of parameters on MSWE
As shown in Figs. 6 - 9, effect of several parameters of wavelet coefficients (D), data number (N), scale
4
factor (e) and block number (s) on MSWE values are analyzed using pressure signals under the condition of
5
min=200 g/s, P=3.0 MPa and L=3 m. Fig. 6 indicates that entropy values for signal and wavelet coefficient D=1
6
change at about 20.8s, which is slower than 19.5s and 19.7s for wavelet coefficients D=2 and D=3, respectively.
7
The reason is that noise still has an important effect on signal and wavelet coefficient D=1. However, MSWE
8
values for wavelet coefficients D=4 and D=5 don’t change during the developing process of thermo-acoustic
9
instability, which indicates that WTCF method has filtered the characteristics of the signal. Thus, wavelet
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coefficients D=2 and D=3 are an effective choice for analysis developing process of thermo-acoustic instability.
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Fig. 6. Plot of entropy values vs. wavelet coefficient D (min=200 g/s, P=3.0 MPa, L=3 m)
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In order to study the effect of data number variations on MSWE values, scale factor e is considered. Fig. 7
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indicates that a raise of data number N leads to a little increase of MSWE values under stable condition, a steady
15
of MSWE values under critical condition, while a decrease of MSWE values under unstable condition. This
16
figure also shows that the MSWE values (0.65~0.85) under critical condition (when data number N increases
17
from 500 to 1000) are lower than MSWE values (0.85~0.95) under stable condition, while are much higher than
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the stable, the critical and the unstable conditions have different MSWE values when data number N is more
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than 500.
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Fig. 7. MSWE values of different scale factor for different data number (min=200 g/s, P=3.0 MPa, L=3 m)
As shown in Fig. 8, for a given data number, the MSWE value of stable, developing process and unstable
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states are related to the scale factor. When the scale factor is between 1 and 4, MSWE values of the three stages
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are obvious different. The difference of the values under even scale factor condition is a little larger than odd
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scale factor. When scale factor increases from 5 to 10, MSWE values cannot be distinguished well between the
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stable state and the developing process, because the coarse-graining procedure reduces the sample size of
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templates [40].
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Fig. 8. MSWE values of different data number for scale factor (min=200 g/s, P=3.0 MPa, L=3 m)
As shown in Fig. 9, the block number is an important parameter in MSWE method. For a given data
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0.06 and 0.13.
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Fig. 9. MSWE values of different state for different block number (min=200 g/s, P=3.0 MPa, L=3 m)
Analysis results
In order to obtain thermo-acoustic instability dynamic characteristics, pressure signals of all experiments
7
are analyzed by MSWE method, in which wavelet coefficient is D3, scale factor e is 2, data number N is 800 and
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block number s is 80. MSWE value shows complex behavior of fuel flowing. The larger the MSWE value, the
9
more irregular behavior of the fuel flowing. Regions
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,
and
are showed in Fig.4a. The pressure measured
during this experiment is given, as a function of time, in Figs. 10a -d, representing the fuel pressure at the outlet
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of the tube. This figure indicates that MSWE values reach steady state in region
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This means the complexity of pressure time series shows random behavior in thermo-acoustic stability.
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Thermo-acoustic stability is stage without abnormal sound. For these experiments, thermo-acoustic stability is
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the stage besides the developing process and developed thermo-acoustic instability. At this stage, the system is
15
stable, and the pressure fluctuation amplitude and the complexity of pressure time series don’t change. There is
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background noise in this stage, which is almost inaudible and caused by the experiment system (pump, tube,
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valve, fuel and so on).
, which is between 0.8 and 0.9.
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Fig. 10. The thermo-acoustic instability dynamic characteristics based on MSWE (a) min=100 g/s, P=3 MPa, L=1 m,
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At the beginning of region
(b) min=50 g/s, P=2.6 MPa, L=3 m, (c) min=200 g/s, P=4.5 MPa, L=2 m, (d) min=200 g/s, P=3 MPa, L=3 m
, MSWE values are still stable at 0.8~0.9 although fuel temperature
increases. Thus, there is no thermo-acoustic instability at the beginning of region
6
to change obviously at change point, which indicates the behavior of fuel flowing become change. However,
7
we cannot observe fuel flow instability directly from pressure signals and temperature signals in region
8
because pressure fluctuation amplitude is very small. In region
9
decreases slowly and then decreases steeply for inlet pressure is 3MPa and 4.5MPa. However, MSWE value
10
decreases slowly and then decreases steeply for inlet pressure is 2.6MPa. In general, the value of
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△Ethreshold=0.1 was chosen as the threshold, which means the developing process occurs. We define the time
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between △E=△Ethreshold and onset of region
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the thermo-acoustic stability to instability. MSWE value has an evident decreasing during region
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the fuel pressure signals change from randomly to periodically. Adjusting time for controlling instability
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should be within warn time. MSWE values are about 0.3~0.6 in region
. The MSWE values begin
,
, the MSWE value decreases steeply, then
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as warn time, which indicates the developing process time from , because
. This means pressure signals
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oscillate periodically and pressure fluctuation amplitude is large.
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4.3 Effects of flow parameters on developing process
For thermo-acoustic instability in a test section, all the parameters (mass flow rate, pressure, heat flux, inlet
4
fluid temperature, the increase rate of the heating power, and the ratio of fluid mass over metal mass) will affect
5
the dynamic process. The interaction of each parameter on dynamic process is a kind of nonlinear relationship
6
[10]. In our opinion, this relationship is prior to the pressure fluctuations change. There is a time delay involved
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between the formation of nonlinear relationship and pressure oscillation. The warning time that obtained with
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wavelet entropy method from pressure time series shows the time delay.
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The effect of mass flow rate and inlet pressure on the developing process is shown in Fig. 11. For a given
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inlet pressure, the warn time decreases with the increasing of the mass flow, which indicates that developing
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process time from thermo-acoustic stability to instability is getting shorter as mass flow increases. This means
12
that the time of developing process (warn time) is negatively correlated with the mass flow rate. This figure also
13
shows that warn time is strongly affected by inlet pressure. For example, the warn time for 2.6 MPa is much
14
smaller than 3.0 MPa and 4.5 MPa. It can be expected that the developing process at 2.6 MPa is very fast. This
15
is a consequence of the large variations of thermal physical properties near the critical pressure (2.4 MPa [34]).
16
It is important to observe that warn time for P=3.0 MPa decreases steeply from 50 g/s to 100 g/s and then
17
changes little from 100 g/s to 200 g/s. Meanwhile, the warn time for 3.0 MPa and 4.5 MPa is about the same at
18
50 g/s and 200 g/s. This indicates that a raise of inlet pressure leads to a raise of warn time. Indeed, when inlet
19
pressure is above critical pressure, a raise of pressure leads to a decrease of variations of thermal physical
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property. The system tends to a steady state under the high pressure condition, and the developing process of
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thermo-acoustic instability cost more time.
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Fig. 11. Warn time vs. mass flow rate for different inlet pressure (L=3 m)
In general, the tube length associated with fuel flow time is always considered and analyzed for dynamic
4
characteristics of fuel flow instability. As shown in Fig. 12, the warn time for L=3 m decreases from 50 g/s to
5
100 g/s and then change little from 100 g/s to 200 g/s. The warn time for 1 m increases from 50 g/s to 100 g/s
6
and then decreases from 100 g/s to 200 g/s. It seems that an increase of tube length causes the warn time to
7
increase.
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Fig. 12. Warn time vs. mass flow rate for different tube lengths (P=3 MPa)
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In order to investigate the developing process of thermo-acoustic instability for different mass flow rates,
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the distribution of MSWE values of different mass flow rates for P=3.0 MPa and L=3 m are shown in Fig. 13 as
12
boxplot. The lower and upper lines of the “box” are the 25th and 75th percentiles of the sample. The distance
13
between the top and bottom of the box is the interquartile range. The line and square in the middle of the box are
14
the sample median and mean, respectively. Outliers are cases with values that are more than 1.5 times the
ACCEPTED MANUSCRIPT interquartile range. As shown in the figure, the distribution of stable MSWE values for different mass flow rates
2
is close to 0.9, which indicates that pressure fluctuation in the stable state is caused by noise. The distribution
3
distance between developing process and unstable stage increases with increasing of the mass flow rate. By
4
taking into consideration complex behavior distribution in Fig. 4, the complex behavior changes of developing
5
process from thermo-acoustic stability to instability increase with increasing of mass flow rate. However, it can
6
be seen from Fig. 9 and Fig. 10 that the developing process time decreases with increasing of mass flow rate.
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Thus, the developing process gets more intense with increasing of mass flow rate. Meanwhile, the value of
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△Ethreshold should be determined smaller with increasing of mass flow rate because the distribution range of the
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developing process and the unstable stage increases.
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Fig. 13. Distribution of MSWE values for different mass flow rates (P=3.0 MPa, L=3 m)
It is demonstrated in Fig. 14 that the distribution of stable MSWE values for different inlet pressure is close
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to 0.9, which indicates that pressure fluctuation in the stable state is caused by noise. The ranges of MSWE
14
values distribution for stable and unstable stage at different pressures are similar. However, the range of MSWE
15
values distribution for developing process at different pressures are significant different. Specially, the range of
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MSWE values distribution for developing process at 2.6 MPa is very small. The reason may be that large
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variations of thermal physical properties near critical pressure will decrease the complexity change from
18
stability to instability. Based on the analysis above, the warn time at the 2.6 MPa is also small. These indicate
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ACCEPTED MANUSCRIPT that the system tends to unstable at 2.6 MPa. The range of MSWE values distribution for developing process at
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3 MPa is about 0.3 (0.5~0.8), which is much larger than 4.5 MPa. However, Fig. 11 indicates that the warn time
3
of 3 MPa is similar with 4.5 MPa. These mean that large pressure may inhibit the complexity change.
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Fig. 14. Distribution of MSWE values for different inlet pressure (min=200 g/s, L=3 m)
5.
Conclusions
The experiments of thermo-acoustic instability dynamic characteristics of hydrocarbon fuel flowing have
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been conducted in horizontal circular tubes at different conditions. In order to further analyze the developing
9
process of the thermo-acoustic instability, the method of MSWE based on WTCF and MSE is developed. Then,
10
the developing process of thermo-acoustic instability and the influence of flow parameters on the developing
11
process are studied. The investigation leads to the following conclusions:
12
1) Experimental results indicate that thermo-acoustic instability can be divided into three states, namely
13
stability, developing process and instability. The inlet pressure signal of stability is random, developing
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process is random and period, and instability is period.
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2) The developing process of thermo-acoustic instability from noise and weak signals is well detected by
16
MSWE method. The method can be used to warn thermo-acoustic instability before the instability is
17
developed. In order to get a valid warning effect, the wavelet coefficient is selected from 2 to 3, data
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number larger than 400, scale factor from 1 to 4, block number about 0.1 times of the data number. The
2
MSWE values of the stable stage, the developing process and the unstable stage are different. 3) The mass flow rate and inlet pressure have an important influence on the developing process. The warn
4
time of thermo-acoustic instability is negatively correlated with the mass flow rate, while is positively
5
correlated with inlet pressure. Large variations of thermal physical properties near the critical pressure
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make developing process of thermo-acoustic very fast. The complex behavior change increases with
7
increasing of mass flow rate, while decreases by improving inlet pressure.
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Acknowledgments
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This research work is supported by National Natural Science Foundation of China (Grants No. 91741204), and the authors thank the reviewers for their valuable advice on this paper.
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DOI:
10.1016/j.actaastro.2018.03.015
Reference:
AA 6757
To appear in:
Acta Astronautica
Received Date: 19 December 2017 Revised Date:
2 February 2018
Accepted Date: 9 March 2018
Please cite this article as: H. Zan, H. Li, Y. Jiang, M. Wu, W. Zhou, W. Bao, Investigation on thermoacoustic instability dynamic characteristics of hydrocarbon fuel flowing in scramjet cooling channel based on wavelet entropy method, Acta Astronautica (2018), doi: 10.1016/j.actaastro.2018.03.015. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT The Title: Investigation on thermo-acoustic instability dynamic characteristics of hydrocarbon fuel
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flowing in scramjet cooling channel based on wavelet entropy method
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The Authors: Hao Zan a, Haowei Li a, Yuguang Jiang a, Meng Wu b, Weixing Zhou b,*, Wen Bao a
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Affiliation： ：a College of Energy Science and Engineering, Harbin Institute of Technology
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b
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College of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology
Corresponding Author: Weixing Zhou
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Address: No.92, West Da-Zhi Street, Harbin, Heilongjiang, 150001, P.R.China.
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E-mail: [email protected]
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Investigation on thermo-acoustic instability dynamic characteristics
1 2
of hydrocarbon fuel flowing in scramjet cooling channel based on
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wavelet entropy method
b
College of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, P. R. China
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Hao Zan a, Haowei Li a, Yuguang Jiang a, Meng Wu b, Weixing Zhou b,*, Wen Bao a a
College of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, Harbin 150001, P. R. China
Abstract
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As part of our efforts to find ways and means to further improve the regenerative cooling technology in
10
scramjet, the experiments of thermo-acoustic instability dynamic characteristics of hydrocarbon fuel flowing
11
have been conducted in horizontal circular tubes at different conditions. The experimental results indicate that
12
there is a developing process from thermo-acoustic stability to instability. In order to have a deep
13
understanding on the developing process of thermo-acoustic instability, the method of Multi-scale Shannon
14
Wavelet Entropy (MSWE) based on Wavelet Transform Correlation Filter (WTCF) and Multi-Scale Shannon
15
Entropy (MSE) is adopted in this paper. The results demonstrate that the developing process of
16
thermo-acoustic instability from noise and weak signals is well detected by MSWE method and the
17
differences among the stability, the developing process and the instability can be identified. These properties
18
render the method particularly powerful for warning thermo-acoustic instability of hydrocarbon fuel flowing
19
in scramjet cooling channels. The mass flow rate and the inlet pressure will make an influence on the
20
developing process of the thermo-acoustic instability. The investigation on thermo-acoustic instability
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dynamic characteristics at supercritical pressure based on wavelet entropy method offers guidance on the
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control of scramjet fuel supply, which can secure stable fuel flowing in regenerative cooling system.
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Keywords: Thermo-acoustic instability, Developing process, Hydrocarbon fuel, Scramjet engine
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cooling channel, Wavelet entropy method
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Research highlights
1 The developing process from thermo-acoustic stability to instability is detected.
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2 A wavelet entropy method is used to analyze the developing process.
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3 The mass flow rate and inlet pressure have an influence on developing process.
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Nomenclature
=
index scale number of wavelet decomposition
Corr
=
correlation coefficient
D
=
scale wavelet coefficient
e
=
scale factor of coarse-grained signals
H
=
multi-scale Shannon value of signal
k
=
index block number
l
=
the number of scales involved in the direct multiplication
L
=
Test section heated length (m)
m
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a
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=
mass flow rate (g/s)
=
total scale number of wavelet decomposition
=
point number of discrete signal
n
=
translation index
p
=
Probability
P
=
inlet pressure (MPa)
M N
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s
=
block number of data divided
th
=
threshold ratio
T
=
Temperature ( )
x
=
Signals
y
=
coarse-grained signals
Y
=
wavelet transform data
λ
block parameter
µ
bifurcation parameter
f
Filtered
h
stable stage
i
i-th
in
inlet
j
j-th
max min out
TE D
Subscripts
index block number the maximum value the minimum value outlet
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Greek symbols
k
3
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power of correlation coefficient
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2
=
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PCorr
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1.
Introduction
Scramjet is a promising hypersonic air-breathing propulsion system for hypersonic missiles, aircrafts and
3
even reusable space transport vehicles because of its good performance at hypersonic region [1, 2]. Yet, when
4
flight velocities increase beyond supersonic speed, the thermal protection becomes a critical problem because
5
combustion chamber is exposed to very high temperature. In this sense, regenerative cooling is one of the
6
most effective cooling techniques, using propellant as a coolant. As a high energy density fuel and potential
7
coolant, hydrocarbon can also be used to lower the structure temperature of combustor wall [3-5]. Before
8
injected into the combustor, hydrocarbon fuel is forced into the cooling channels distributed in the hot wall of
9
combustor for absorbing heat [6]. Indeed, the strong coupling between fuel-coolant flowing (in the cooling
10
channels) and fuel-coolant combustion (in the combustor) makes system control a very hard task. For
11
example, fuel mixing has important influence on combustion efficiency and heat release distribution, which
12
further affects fuel flowing in cooling channels [7, 8]. Studies on hydrocarbon fuel flowing are of great
13
importance, especially for acquiring a better understanding on the fuel flowing behavior in scramjet cooling
14
channels. Different kinds of hydrocarbon fuel heating experiments are conducted, which show that flowing
15
instability occurs during heating of supercritical hydrocarbon fuel. It leads to severe problems such as
16
structural vibrations, difficulty in controlling fuel flow and generation of acoustic noise [9-12].
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The exploration of hydrocarbon fuel flowing instability has attracted considerable attentions from
18
engineering and physical research fields because of its significant importance [13]. The high temperature fuel
19
flowing instability mechanism has been revealed gradually. Hitch et al. [14] experimentally investigated heat
20
transfer in supercritical hydrocarbon fuel and identified two previously observed oscillation modes, which are
21
Helmholtz oscillation mode and the acoustic oscillation mode. Although Helmholtz mode oscillation could be
22
eliminated relatively easily [15], fuel temperature and pressure could still be unstable near the critical point.
ACCEPTED MANUSCRIPT The transient pressure behavior quickly changes between large and slow oscillations to nearly pure tones
2
associated with the acoustic vibration mode. Zhou et al. [16] observed pressure-drop type instability in the
3
hydrocarbon fuel cooling system. By numerical simulation and analytical solutions, the author found that
4
instability of the cooling system was caused by the interaction of positive and negative feedback. Liu et al.
5
[17] found flow instability, pressure drop deduction, and heat transfer deterioration happened simultaneously
6
accompanied with acoustic flow instability and peculiarly diabatic pressure drop deduction due to the steep
7
thermodynamic properties. Wang et al. [18, 19] investigated thermo-acoustic instability of hydrocarbon fuel
8
(RP-3) in a vertical circular tube at supercritical pressure. The influences of mass flow rate and operating
9
pressure on thermo-acoustic instability were found. Moreover, the pressure drops fluctuations can be used to
10
represent the characteristics of thermo-acoustic instability. Some researchers claimed that the thermo-acoustic
11
instability was related to the formation and collapse of pseudo bubbles, propagated in the fluid with the speed
12
of sound [20-22]. Others believed that large variations in thermo-physical properties were significant reasons
13
[23]. By conducting a series of heat transfer experiments, Linne et al. [24] found flow instability could not be
14
attributed to one single reason. He suggested a statistical model to predict the flow instability in supercritical
15
hydrocarbon fuel flowing.
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Despite the existing results, there are still significant challenges in the study of hydrocarbon fuel flowing
17
instability. Previous work was mainly focused on the instability phenomenon and mechanism with
18
experimental method, but fewer researchers paid attention to the developing process from thermos-acoustic
19
stability to instability. In literature [18], thermo-acoustic instability is not a sudden change, but a developing
20
process. The developing process of the thermo-acoustic instability can be divided into four stages, namely,
21
thermo-stable stage, thermo-unstable stage, preliminary thermo-acoustic stage and developed thermo-acoustic
22
stage. However, there are few investigations on characteristics in different stages, and it is difficult to do the
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ACCEPTED MANUSCRIPT classification in experimental conditions. As thermo-acoustic instability pressure fluctuation signals contain a
2
lot of flowing information of hydrocarbon fuel, the information of pressure loss, flow rate and fuel
3
temperature can be acquired from them [25]. Real time measurement of pressure fluctuation signals can be
4
implemented easily, based on which, the developing process of thermo-acoustic instability can be detected
5
[26]. With the development of wavelet theory and entropy statistical theory, the wavelet entropy method is
6
widely applied to the analysis and processing of transient signals in many fields [27-29]. Wavelet Transform
7
Correlation Filter (WTCF) has superiority of extracting weak fault feature of pressure signals and high SNR
8
(Signal-Noise Ratio) scales wavelet coefficients are gained [30]. Multi-scale Shannon entropy (MSE) has
9
been applied to the two-phase flow field to characterize flow behavior [31]. The method of Multi-scale
10
Shannon Wavelet Entropy (MSWE) based on WTCF and MSE is put forward to analyze the thermo-acoustic
11
instability dynamic characteristics of hydrocarbon fuel flowing.
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Based on previous studies, experiments of thermo-acoustic instability of hydrocarbon fuel under
13
different conditions are conducted firstly. Then, the developing process from thermo-acoustic stability to
14
instability is analyzed with MSWE method, and characteristics of different stages are obtained in this paper.
15
At last, the influences of mass flow rate and inlet pressure on the developing process are studied. This work is
16
to get a better understanding of the developing process and offers guidance on adjusting time for stable fuel
17
supply to control thermo-acoustic instability. For example, we will change the equivalent ratio as soon as the
18
developing process is detected, which can avoid the occurrence of developed thermos-acoustic instability.
19
2.
20
2.1 Wavelet Transform Correlation Filter (WTCF)
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MSWE method
The method of Wavelet Transform Correlation Filter has been described previously [30, 32, 33]. Based
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2
scale, WTCF is the direct spatial correlation of wavelet coefficients at several adjacent scales. Edges and
3
significant features of the signal are enhanced while noise and small sharp features, edges and features gained
4
from the noise are suppressed. By threshold inspection, SNR of wavelet coefficients of WTCF is greatly
5
higher than wavelet transform. In this paper, locations of edges and significant features of the signal are
6
detected by spatial correlation Corrl (a, n) of wavelet transform contents at several adjacent scales. n = 1, 2 L , N
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l −1
Corrl ( a ,n ) = ∏ Y ( a + i , n )
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i =0
(1)
where Y denotes the wavelet transform data, a is the scale index, N is the point of discrete signal, n is the
8
translation index, l is the number of scales involved in the direct multiplication, M is the total number of
9
scales and a
Step 1: Compute all Corr2 (a, n) for all every wavelet scale a, enhanced significant features and
11 12
suppressed noise are gained.
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Step 2: Rescale the power of Corr2 (a, n) to that of Y (a, n) and get NewCorr2 (a, n). NewCorr2 ( a,n ) = Corr2 ( a , n ) PY ( a ) / PCorr ( a )
(2)
PY ( a ) = ∑ Y ( a , n )
(3)
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2
n
PCorr2 ( a ) = ∑ Corr2 ( a , n )
2
(4)
n
where PY(a) is ath scale wavelet coefficients and PCorr2(a) is the ath power of Corr2(a, n).
15
Step 3: If NewCorr2 ( a,n ) = Y ( a, n ) , the point is accepted as edge. Pass Y(a, n) to Yf, then reset Y(a, n)
16
and Corr2(a, n) to 0. Otherwise, we suppose Y(a, n) is produced by noise and then retain Y(a, n) and Corr2(a,
17
n).
18
Step 4: Return to step 1, repeat steps 2 and 3 until value of
PY ( a ) reaches threshold ratio th(a)
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which is correlated with some reference noise power at ath wavelet scale. Energy normalization, comparison of data and extracting information edge will be iterated multiple
3
times until the power of Y(a, n) reaches some reference noise power at the ath wavelet scale. As introduced in
4
literature [30], the reference noise power should be estimated. After pressure signal is processed by WTCF,
5
the scale wavelet coefficient D1,D2,…,Dj, which contain information from high frequency to low frequency
6
and are the estimation of local energy in different scale can be gained.
7
2.2 Multi-Scale Shannon Entropy (MSE)
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The basic procedure for calculating multi-scale Shannon entropy from signals is as follows: for a signal
9
x(i), i=1, 2,…, N, we first perform a coarse-grained process to define temporal scales and further obtain the
10
coarse-grained signals as follows:
je 1 xi ∑ e i = ( j −1) e +1
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λ=
max
j =1,K, N / e
{ y ( j )} −
min
k =1,K, N / e
{ y ( j )}
(5)
(6)
s
where e is the scale factor and 1≤j≤N/e. The signals are equally divided into s blocks, 1≤s≤N/e. The range
12
of each block is between ymin+(k-1)λ and ymin+kλ (k=1, 2, …, s). The data number of signals in each block is
13
counted, and the probability (pk) of signal data in every block is calculated.
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pk =
eN k N
(7)
14
where Nk denotes the number of signal in each part. Multi-scale Shannon entropy of signal x(i) can be showed
15
as follows: s
H = − ∑ Pk ln ( Pk ) k =1
16
H will reach the maximum when Pk is 1/s. H is normalization processed by dividing lns.
(8)
ACCEPTED MANUSCRIPT 0≤ H =
H ≤1 ln s
(9)
where H denotes the degree of randomness of signal x(i). The smaller H is, the more regular the signal is.
2
Otherwise, the signal is more random. The change of H shows subtle change of signal.
3
2.3 Computation process
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The WTCF has been applied in the field of signal feature extraction, because the method has a strong
5
ability of analysis in the time-frequency domain. Combined with the property of the MSE which is useful for the
6
analysis of signal changing, a method of MSWE can be designed shown in Fig. 1.
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Data Acquistion
Wavelet Transform Correlation Filter
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△E≥△Ethreshold?
No
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Yes Warning
Fig. 1. Computation process
Step 1: The pressure fluctuation signal is sampled by pressure sensors. Step 2: The time series data, corresponding to pressure fluctuation signal, then processed by WTCF with a five-level decomposition using Eqs. (1) - (4). Step 3: Following the computation of wavelet transform, the scale wavelet coefficients D1, D2, …, Dj, which contain information from high frequency to low frequency in different scale, can be gained.
ACCEPTED MANUSCRIPT 1
the MSE values of the wavelet coefficients D1, D2, …, Dj can be computed by Eqs. (5) - (9).
3 4
Step 5: The absolute value of the relative MSWE △E is given in Eq. (10) aiming to take the difference into consideration.
∆E = Eh − Ew
5
(10)
Eh and Ew denote the MSWE values of the stable state and the developing process of thermo-acoustic instability, respectively.
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Step 4: To grasp the signal entropy difference of the stability, the developing process and the instability,
Step 6: As shown in Fig. 1, the focus on this computation process becomes the determination of threshold.
8
Once the relative MSWE value of the hydrocarbon fuel pressure is larger than this threshold, a forewarning
9
signal will be immediately delivered to the fuel supply system so that the appropriate actions can be performed
10
at an early stage.
11
3.
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Experiments and data acquisition
The tests were conducted in the HIT. The experiments of dynamic characteristics of thermo-acoustic
13
instability of hydrocarbon fuel flowing have been conducted in horizontal circular tubes at different
14
conditions (as shown in Fig. 2). The hydrocarbon fuel (RP-3) is used as the working substance for the
15
experiment. The critical pressure and critical temperature of hydrocarbon fuel are about 2.4 MPa and 372
16
respectively, which are consistent with the critical properties of general kerosene hydrocarbons in the
17
reference [34]. Fill the fuel sump with nitrogen. Start the pump and set the mass flow rate and let the fuel
18
flow through the whole system. Turn on the DC power and heat the test section. When the outlet temperature
19
becomes steady, the heating power is raised to a higher level. Record the experimental data when
20
thermo-acoustic instability occurs.
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ACCEPTED MANUSCRIPT Power Source
Test Section
P
1
2
3
4
T
Coller
T
P
Insulator
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Turbine Flowmeter
Outlet temperature
Inlet pressure
Different Pressure Control Valve
Cutoff Valve
Filter
P
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Check Valve
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Data Acquisition System
Fuel Sump
Cutoff Valve
1 2
Fig. 2. Schematic diagram of the experimental set-up
The geometric configuration of cooling channels has important influence on heat exchange between
4
combustion and hydrocarbon fuel. Different physical configuration of cooling channels should be considered
5
to investigate flowing instability. The detailed physical parameters are referred to the studies of Huang
6
[35-38]. The test section is a horizontal straight superalloy (GH3128) circular tube having a length of 1~3 m,
7
an internal diameter of 12 mm and a thickness of 2 mm and is heated by DC power during experiment. Some
8
parameters are measured per second using Nl c-rio data acquisition system. The sampling frequency range is
9
1-100 Hz. Four K-type thermocouples (Accuracy: ±5 ) are spot welded directly to the outer surface of the
10
test section to detect wall outside temperature. The outlet temperature refers to outlet fluid temperature
11
measured by K-type armored thermocouples (Accuracy: ±5 ). Pressure sensors (Accuracy: ±0.5% of FS: 5
12
MPa) are installed in the inlet/outlet test section and the fuel sump. The inlet pressure refers to the inlet fluid
13
pressure measured by the pressure sensor. Turbine flow meters are installed at the inlet of the tube to measure
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ACCEPTED MANUSCRIPT the mass flow rate. The accuracy of the flow meter is 1.0% of the full scale (500 g/s). The response time of
2
the pressure sensor (MPM489) and armored thermocouples is 2 ms and 0.5s, respectively. The response time
3
of the pressure sensor is enough to catch the dynamic process of thermo-acoustic instability. The internal
4
diameter of the test section is larger than traditional geometric configuration of the cooling channels in
5
scramjet. The experiments are our preliminary work, and we will conduct the experiment with a dimension of
6
millimeter in further.
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A total of 15 tests were conducted to investigate the thermo-acoustic instability dynamic characteristics.
8
The operating conditions and results are shown in Table 1 with mass flow rate (min), inlet pressure (P) and
9
tube length (L) varying from 50 g/s to 200 g/s, 2.6 MPa to 4.5 Mpa, and 1 m to 3 m respectively. The inlet fuel temperature (Tin) is 28 .
11
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Table 1 Operating conditions and results Test
1
2
3
4
5
min(g/s)
50
50
50
10
100
P(MPa)
2.6
3
L(m)
3
3
0.6
0.5
oscillation amplitude(MPa)
9
10
11
12
13
14
15
10
20
20
200
20
20
200
20
10
50
0
0
0
0
0
0
0
2.6
3
4.5
2.6
3
4.5
2.6
3
4.5
3
3
3
3
3
3
3
3
3
3
2
2
2
1
1
1
1.2
1.6
1
2
1.5
2
2.5
0.9
1.2
0.6
0.3
1
32
12
16
26
21
15
11
8~1
3~
4~
Frequency (Hz)
5
8
7
22
7~2 3
21~3 7
13~2 4
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4.
Results and discussion
14
4.1 The developing process of thermo-acoustic instability
15
In these 15 tests, tube oscillation with the abnormal sound similar to sharp metal hammering was
16
observed. The abnormal sound is strongly interrelated with the thermo-dynamic instability, and the sharing
17
features of thermo-acoustic instability are also suitable for the abnormal sound in this paper [18]. The large
0.0 5
14
ACCEPTED MANUSCRIPT 1
pressure fluctuations appear synchronously with the abnormal sounds. The time series of temperature and
2
pressure of thermo-acoustic instabilities under four conditions are shown in Fig. 3. As can be seen in enlarged
3
images A, B, C and D, inlet pressure begins to oscillate, accompanied by the increase of outlet temperature.
4
The outlet temperature is about 200
5
while in Fig. 3b the outlet temperature is about 120 . The reason may be that the critical pressure is
6
approached. In our opinion, the increase of the heating power will affect the dynamic process. In this paper,
7
the rate of increasing heating power is about 2 KW/s. The curves of heating power have been given together
8
with the pressure and outlet temperature in below.
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when inlet pressure begins to oscillate in Fig. 3a, Fig. 3c, and Fig. 3d,
10
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Fig. 3. The phenomenon of thermo-acoustic instability, (a) min=200 g/s, P=3.0 MPa, L=3 m, (b) min=100 g/s, P=3 MPa,
12
L=1 m, (c) min=50 g/s, P=2.6 MPa, L=3 m, (d) min=200 g/s, P=4.5 MPa, L=2 m
13
As shown in Fig. 4, the signals of inlet pressure and outlet temperature present the developing process of
14
the thermo-acoustic instability. As can be seen in Fig. 4a, we define region
15
temperature is stable, region
as the stable state in which
as the developing process in which the temperature changed but pressure did
ACCEPTED MANUSCRIPT 1
not oscillate significantly, region
2
significantly.
3
as unstable state in which temperature and pressure oscillated
The pressure change can be detected by pressure fluctuation amplitude and the complexity of pressure
4
time series. In region
5
time series shows random behavior. It shows that pressure time series in the stable state is disturbed by noise.
6
In region
7
pressure time series changes from random to periodic behavior. This means that abnormal sound occurs and
8
thermo-acoustic instability starts. In region
9
complexity of pressure time series shows periodic behavior. This clearly demonstrates that thermo-acoustic
10
instability is fully developed. It has been shown that complexity change of pressure time series is prior to the
11
pressure fluctuation amplitude. Thus, measure of complexity (MSWE method) is developed to investigate
12
dynamic characteristics of thermos-acoustic instability.
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(Fig. 4b), pressure fluctuation amplitude is very small, and complexity of pressure
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(Fig. 4c), there is no obvious change in pressure fluctuation amplitude, but complexity of
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(Fig. 4d), pressure fluctuation amplitude is very large, and
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Fig. 4. The developing process of thermo-acoustic instability (min=200 g/s, P=3.0 MPa, L=3 m)
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4.2 Pressure signal analysis based on MSWE method
2
4.2.1
Validation of multi-scale Shannon entropy method
In order to obtain the thermo-acoustic instability dynamic characteristics based on MSWE method, the
4
validation of MSE method is carried out by comparing several entropy methods in detecting change of the
5
logistic model. As shown in Fig. 5a, the logistic map is a polynomial mapping (equivalently, recurrence
6
relation) of degree 2, often cited as an archetypal example of how complex, chaotic behavior can arise from
7
very simple non-linear dynamical equations [39]. Mathematically, the logistic map is written:
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(11)
where xn is a number between zero and one that represents the ratio of existing population to the maximum
9
possible population. The values of interest for the parameter µ are those in the interval [3, 4]. The µ values of
10
Bifurcation point A, Bifurcation point B and Onset of Chaos are 3.449, 3.544 and 3.57 respectively. With µ
11
value between 3 and 3.449, x will approach permanent oscillations between two values. With µ value between
12
3.449 and 3.544, x will approach permanent oscillations between four values. When the value of µ reaches
13
3.57, we no longer see oscillations of finite period.
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Fig. 5. Validation of multi-scale Shannon entropy method
Several entropy methods detecting change of the logistic map are compared in Fig. 5b. This figure
4
indicates that Shannon entropy method is an effective method to show change of the logistic map with µ
5
value between 3 and 4. Phenomenon of period-doubling bifurcation (critical point A, critical point B) and the
6
onset of chaos are accurately captured. At the point of chaos onset, the Shannon entropy value is about 0.7.
7
When µ>3.57, the Shannon entropy value is stable at 0.9. However, permutation entropy is irregular,
8
approximate entropy and sample entropy do not change between µ=3 and µ=3.57. Therefore, Shannon
9
entropy method is a reliable method to detect the signal change.
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As can be seen in Fig. 5c, the odd scale factor and even scale factor have the different values between
11
µ=3 and µ=3.57. Entropy values of odd scale factors are about the same, while values of even scale change
ACCEPTED MANUSCRIPT 1
before bifurcation point and chaos onset point, which should be paid more attention.
2
4.2.2
Effect of parameters on MSWE
As shown in Figs. 6 - 9, effect of several parameters of wavelet coefficients (D), data number (N), scale
4
factor (e) and block number (s) on MSWE values are analyzed using pressure signals under the condition of
5
min=200 g/s, P=3.0 MPa and L=3 m. Fig. 6 indicates that entropy values for signal and wavelet coefficient D=1
6
change at about 20.8s, which is slower than 19.5s and 19.7s for wavelet coefficients D=2 and D=3, respectively.
7
The reason is that noise still has an important effect on signal and wavelet coefficient D=1. However, MSWE
8
values for wavelet coefficients D=4 and D=5 don’t change during the developing process of thermo-acoustic
9
instability, which indicates that WTCF method has filtered the characteristics of the signal. Thus, wavelet
10
coefficients D=2 and D=3 are an effective choice for analysis developing process of thermo-acoustic instability.
12
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Fig. 6. Plot of entropy values vs. wavelet coefficient D (min=200 g/s, P=3.0 MPa, L=3 m)
13
In order to study the effect of data number variations on MSWE values, scale factor e is considered. Fig. 7
14
indicates that a raise of data number N leads to a little increase of MSWE values under stable condition, a steady
15
of MSWE values under critical condition, while a decrease of MSWE values under unstable condition. This
16
figure also shows that the MSWE values (0.65~0.85) under critical condition (when data number N increases
17
from 500 to 1000) are lower than MSWE values (0.85~0.95) under stable condition, while are much higher than
ACCEPTED MANUSCRIPT MSWE values (0.48~0.65) under unstable condition. It is important to observe that pressure time series under
2
the stable, the critical and the unstable conditions have different MSWE values when data number N is more
3
than 500.
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Fig. 7. MSWE values of different scale factor for different data number (min=200 g/s, P=3.0 MPa, L=3 m)
As shown in Fig. 8, for a given data number, the MSWE value of stable, developing process and unstable
7
states are related to the scale factor. When the scale factor is between 1 and 4, MSWE values of the three stages
8
are obvious different. The difference of the values under even scale factor condition is a little larger than odd
9
scale factor. When scale factor increases from 5 to 10, MSWE values cannot be distinguished well between the
10
stable state and the developing process, because the coarse-graining procedure reduces the sample size of
11
templates [40].
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Fig. 8. MSWE values of different data number for scale factor (min=200 g/s, P=3.0 MPa, L=3 m)
As shown in Fig. 9, the block number is an important parameter in MSWE method. For a given data
ACCEPTED MANUSCRIPT number and scale factor, MSWE values of different states are stable and distinguished well when s/N between
2
0.06 and 0.13.
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4.2.3
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Fig. 9. MSWE values of different state for different block number (min=200 g/s, P=3.0 MPa, L=3 m)
Analysis results
In order to obtain thermo-acoustic instability dynamic characteristics, pressure signals of all experiments
7
are analyzed by MSWE method, in which wavelet coefficient is D3, scale factor e is 2, data number N is 800 and
8
block number s is 80. MSWE value shows complex behavior of fuel flowing. The larger the MSWE value, the
9
more irregular behavior of the fuel flowing. Regions
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,
and
are showed in Fig.4a. The pressure measured
during this experiment is given, as a function of time, in Figs. 10a -d, representing the fuel pressure at the outlet
11
of the tube. This figure indicates that MSWE values reach steady state in region
12
This means the complexity of pressure time series shows random behavior in thermo-acoustic stability.
13
Thermo-acoustic stability is stage without abnormal sound. For these experiments, thermo-acoustic stability is
14
the stage besides the developing process and developed thermo-acoustic instability. At this stage, the system is
15
stable, and the pressure fluctuation amplitude and the complexity of pressure time series don’t change. There is
16
background noise in this stage, which is almost inaudible and caused by the experiment system (pump, tube,
17
valve, fuel and so on).
, which is between 0.8 and 0.9.
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1 2 3
Fig. 10. The thermo-acoustic instability dynamic characteristics based on MSWE (a) min=100 g/s, P=3 MPa, L=1 m,
4
At the beginning of region
(b) min=50 g/s, P=2.6 MPa, L=3 m, (c) min=200 g/s, P=4.5 MPa, L=2 m, (d) min=200 g/s, P=3 MPa, L=3 m
, MSWE values are still stable at 0.8~0.9 although fuel temperature
increases. Thus, there is no thermo-acoustic instability at the beginning of region
6
to change obviously at change point, which indicates the behavior of fuel flowing become change. However,
7
we cannot observe fuel flow instability directly from pressure signals and temperature signals in region
8
because pressure fluctuation amplitude is very small. In region
9
decreases slowly and then decreases steeply for inlet pressure is 3MPa and 4.5MPa. However, MSWE value
10
decreases slowly and then decreases steeply for inlet pressure is 2.6MPa. In general, the value of
11
△Ethreshold=0.1 was chosen as the threshold, which means the developing process occurs. We define the time
12
between △E=△Ethreshold and onset of region
13
the thermo-acoustic stability to instability. MSWE value has an evident decreasing during region
14
the fuel pressure signals change from randomly to periodically. Adjusting time for controlling instability
15
should be within warn time. MSWE values are about 0.3~0.6 in region
. The MSWE values begin
,
, the MSWE value decreases steeply, then
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as warn time, which indicates the developing process time from , because
. This means pressure signals
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oscillate periodically and pressure fluctuation amplitude is large.
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4.3 Effects of flow parameters on developing process
For thermo-acoustic instability in a test section, all the parameters (mass flow rate, pressure, heat flux, inlet
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fluid temperature, the increase rate of the heating power, and the ratio of fluid mass over metal mass) will affect
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the dynamic process. The interaction of each parameter on dynamic process is a kind of nonlinear relationship
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[10]. In our opinion, this relationship is prior to the pressure fluctuations change. There is a time delay involved
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between the formation of nonlinear relationship and pressure oscillation. The warning time that obtained with
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wavelet entropy method from pressure time series shows the time delay.
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The effect of mass flow rate and inlet pressure on the developing process is shown in Fig. 11. For a given
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inlet pressure, the warn time decreases with the increasing of the mass flow, which indicates that developing
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process time from thermo-acoustic stability to instability is getting shorter as mass flow increases. This means
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that the time of developing process (warn time) is negatively correlated with the mass flow rate. This figure also
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shows that warn time is strongly affected by inlet pressure. For example, the warn time for 2.6 MPa is much
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smaller than 3.0 MPa and 4.5 MPa. It can be expected that the developing process at 2.6 MPa is very fast. This
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is a consequence of the large variations of thermal physical properties near the critical pressure (2.4 MPa [34]).
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It is important to observe that warn time for P=3.0 MPa decreases steeply from 50 g/s to 100 g/s and then
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changes little from 100 g/s to 200 g/s. Meanwhile, the warn time for 3.0 MPa and 4.5 MPa is about the same at
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50 g/s and 200 g/s. This indicates that a raise of inlet pressure leads to a raise of warn time. Indeed, when inlet
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pressure is above critical pressure, a raise of pressure leads to a decrease of variations of thermal physical
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property. The system tends to a steady state under the high pressure condition, and the developing process of
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thermo-acoustic instability cost more time.
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Fig. 11. Warn time vs. mass flow rate for different inlet pressure (L=3 m)
In general, the tube length associated with fuel flow time is always considered and analyzed for dynamic
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characteristics of fuel flow instability. As shown in Fig. 12, the warn time for L=3 m decreases from 50 g/s to
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100 g/s and then change little from 100 g/s to 200 g/s. The warn time for 1 m increases from 50 g/s to 100 g/s
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and then decreases from 100 g/s to 200 g/s. It seems that an increase of tube length causes the warn time to
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increase.
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Fig. 12. Warn time vs. mass flow rate for different tube lengths (P=3 MPa)
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In order to investigate the developing process of thermo-acoustic instability for different mass flow rates,
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the distribution of MSWE values of different mass flow rates for P=3.0 MPa and L=3 m are shown in Fig. 13 as
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boxplot. The lower and upper lines of the “box” are the 25th and 75th percentiles of the sample. The distance
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between the top and bottom of the box is the interquartile range. The line and square in the middle of the box are
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the sample median and mean, respectively. Outliers are cases with values that are more than 1.5 times the
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is close to 0.9, which indicates that pressure fluctuation in the stable state is caused by noise. The distribution
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distance between developing process and unstable stage increases with increasing of the mass flow rate. By
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taking into consideration complex behavior distribution in Fig. 4, the complex behavior changes of developing
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process from thermo-acoustic stability to instability increase with increasing of mass flow rate. However, it can
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be seen from Fig. 9 and Fig. 10 that the developing process time decreases with increasing of mass flow rate.
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Thus, the developing process gets more intense with increasing of mass flow rate. Meanwhile, the value of
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△Ethreshold should be determined smaller with increasing of mass flow rate because the distribution range of the
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developing process and the unstable stage increases.
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Fig. 13. Distribution of MSWE values for different mass flow rates (P=3.0 MPa, L=3 m)
It is demonstrated in Fig. 14 that the distribution of stable MSWE values for different inlet pressure is close
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to 0.9, which indicates that pressure fluctuation in the stable state is caused by noise. The ranges of MSWE
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values distribution for stable and unstable stage at different pressures are similar. However, the range of MSWE
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values distribution for developing process at different pressures are significant different. Specially, the range of
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MSWE values distribution for developing process at 2.6 MPa is very small. The reason may be that large
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variations of thermal physical properties near critical pressure will decrease the complexity change from
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stability to instability. Based on the analysis above, the warn time at the 2.6 MPa is also small. These indicate
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3 MPa is about 0.3 (0.5~0.8), which is much larger than 4.5 MPa. However, Fig. 11 indicates that the warn time
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of 3 MPa is similar with 4.5 MPa. These mean that large pressure may inhibit the complexity change.
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Fig. 14. Distribution of MSWE values for different inlet pressure (min=200 g/s, L=3 m)
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Conclusions
The experiments of thermo-acoustic instability dynamic characteristics of hydrocarbon fuel flowing have
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been conducted in horizontal circular tubes at different conditions. In order to further analyze the developing
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process of the thermo-acoustic instability, the method of MSWE based on WTCF and MSE is developed. Then,
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the developing process of thermo-acoustic instability and the influence of flow parameters on the developing
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process are studied. The investigation leads to the following conclusions:
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1) Experimental results indicate that thermo-acoustic instability can be divided into three states, namely
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stability, developing process and instability. The inlet pressure signal of stability is random, developing
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process is random and period, and instability is period.
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2) The developing process of thermo-acoustic instability from noise and weak signals is well detected by
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MSWE method. The method can be used to warn thermo-acoustic instability before the instability is
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developed. In order to get a valid warning effect, the wavelet coefficient is selected from 2 to 3, data
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number larger than 400, scale factor from 1 to 4, block number about 0.1 times of the data number. The
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MSWE values of the stable stage, the developing process and the unstable stage are different. 3) The mass flow rate and inlet pressure have an important influence on the developing process. The warn
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time of thermo-acoustic instability is negatively correlated with the mass flow rate, while is positively
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correlated with inlet pressure. Large variations of thermal physical properties near the critical pressure
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make developing process of thermo-acoustic very fast. The complex behavior change increases with
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increasing of mass flow rate, while decreases by improving inlet pressure.
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Acknowledgments
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This research work is supported by National Natural Science Foundation of China (Grants No. 91741204), and the authors thank the reviewers for their valuable advice on this paper.
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