Identiﬁcation of Long-term Behavior of Natural Circulation Loops: A Thresholdless Approach from an Initial Response

: Natural circulation loop (NCL) systems are buoyancy-driven heat exchangers that are 1 used in various industrial applications. The concept of passive heat exchange in NCL systems is 2 attractive, because there is no need for an externally driven equipment (e.g., a pump) to maintain 3 the ﬂuid circulation. However, relying on buoyancy as the sole driving force may lead to several 4 potential difﬁculties, one of which is generation of (possibly) time-varying nonlinearities in the 5 dynamical system, where a difference in the time scales of heat transfer and ﬂuid ﬂow causes 6 the ﬂow to change from a steady-state regime to either an oscillatory regime or a ﬂow-reversal 7 regime, both of which are undesirable. In this paper, tools of symbolic time series analysis (e.g., 8 probabilistic ﬁnite state automata (PFSA)) are proposed to identify selected regimes of operation 9 in NCL systems, where the underlying principle is built upon the concept of pattern classiﬁcation 10 from measurements of ﬂuid-ﬂow dynamics. The proposed method is shown to be capable of 11 identifying the current regime of operation from the initial time response under a given set of 12 operational parameters. The efﬁcacy of regime classiﬁcation is demonstrated by testing on two 13 data sets, generated from numerical simulation of a MATLAB SimuLink model that has previously 14 been validated with experimental data. The results of the proposed PFSA-based classiﬁcation are 15 compared with those of a hidden Markov model (HMM) that serves as the baseline.


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Natural circulation loops (NCL) are commonly used for passive heat exchange 20 and have found important applications in thermal systems, where there are a high-21 temperature heat source and a low-temperature heat sink that is located at a higher 22 elevation than the heat source. The fluid flow in the heat exchanger is gravity-driven 23 and is established as a balance of the buoyant force due to the temperature difference 24 (and hence the fluid-density difference) between the source and the sink, the inertial 25 force, and the frictional force. The NCL-based heat transfer requires no external power 26 or driving force and has no moving parts; thus NCL systems are relatively less difficult 27 to maintain and operate, and are also less expensive to manufacture. 28 Based on the concept of natural convection due to heating and cooling, the resulting to an unstable oscillatory condition. Increased heating may cause even larger density through the circulation loop have been used for STSA. Along this line, probabilistic finite 91 state automata (PFSA) models [18,19] are developed to identify the long term behavior of 92 the NCL system at hand. The rationale for choosing PFSA as the data-driven algorithm 93 is their inherent simplicity that ensures good classification accuracy, while still having 94 low computational complexity. In the past, PFSA-based methods have shown good 95 performance in various applications such as, analysis of combustion instabilities [20,21], 96 failure prognosis of structural materials [22] and rolling-element bearings [23], as well 97 as usage of sensor networks for detection of moving targets [24]. 98 An alternative data-driven method that is commonly used is hidden Markov mod-99 eling (HMM) which has shown good performance in several problems, such as speech in classification of chaotic system data [28] and pressure data in a multi-nozzle com-102 bustor [21]. It has been shown by Mondal et al. [29] that PFSA methods are able to 103 achieve classification accuracy nearly as close as HMM methods but at much lower 104 computational costs, being almost two orders of magnitude faster in both training and 105 testing than HMM. Thus, in this paper, the authors compare the results obtained using 106 PFSA to those of a baseline HMM. 107 The proposed algorithm is initially trained and tested on time-series data obtained 108 from a set of simulation runs conducted on the numerical model mentioned above 109 without radiation heat loss. In order to prove the efficacy of the model, it is later again 110 tested on more data generated from the same simulation program, but with different 111 process parameters and by incorporating the effects of radiation heat loss in the model. 112 These modifications cause significant changes in the system response, while leaving the 113 core underlying physics essentially the same. Testing the algorithm on this new set of 114 data proves that the algorithm is not dependent on the process parameters, but rather, 115 learns the underlying physics and is capable of effectively working across different NCL The long-term behavior of an NCL system is predicted from the initial transient 121 data.

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• Validation of the underlying algorithms on an experimentally validated NCL system sim-123 ulator: The validation process is based on testing with different sets of system 124 parameters and initial conditions. The test results demonstrate that the the per-125 formance is independent of the process parameters and that the predictions are 126 consistent with the physics of NCL systems.  The governing equations (e.g., conservation of mass, momentum, and energy) for  Conservation of mass (liquid phase): Conservation of momentum (liquid phase): The momentum equation is integrated over the entire loop to arrive at the current heat transfer. The first term on the right-hand side is the conduction heat transfer and 156 the second term on the right-hand side is the heat exchange due to convection.

Conservation of energy (liquid phase):
Conservation of energy (heat exchanger): Version January 14, 2021 submitted to Sci Conservation of energy (heat-exchanger tube wall): In Equation (5)  For evaluating the friction factor ( f ) the following correlations have been used: where the friction factors are computed as in Eqs.
where Nu l = 2.8 l n(1 + 2.8 C 1 Ra 1/4 (13) and The heat transfer from tube wall to ambient for the horizontal sections is estimated by the following correlation.

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In the first set of numerical simulation, radiation heat loss is ignored and accordingly,   shows steady-state flow; and an increased heater power first changes to the oscillatory 238 regime and then a further increase in heater power yields flow-reversal as seen in Figure   239 3. However, due to the radiation heat loss, the change-over points of flow dynamics (i.e., 240 the value of heater power at which flow regimes change, are different as seen in Table 1.

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A similarity between the trends in the cases with and without radiation is apparent.

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It is also noted that the duration of the initial transience is not constant. Another  A is a (nonempty) finite alphabet, i.e., its cardinality |A| is a positive integer.

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• Q is a (nonempty) finite set of states, i.e., its cardinality |Q| is a positive integer..

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• δ : Q × A → Q is a state transition map.

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Definition 3. A symbol block, also called a word, is a finite-length string of symbols belonging 265 to the alphabet A, where the length of a word w s 1 s 2 · · · s with every s i ∈ A is |w| = , and 266 the length of the empty word is | | = 0. The parameters of the FSA are extended as: The set of all words, constructed from symbols in A and including the empty word , is 268 denoted as A .

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• The set of all words, whose suffix (respectively, prefix) is the word w, is denoted as A w 270 (respectively, wA ).

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• The set of all words of (finite) length , where is a positive integer, is denoted as A . Equivalently, a D-Markov machine is a statistically stationary stochastic process S = · · · s −1 s 0 s 1 · · · , where the probability of occurrence of a new symbol depends only on the last consecutive (at most) D symbols, i.e., P[s n | · · · s n−D · · · s n−1 ] = P[s n | s n−D · · · s n−1 ] Consequently, for w ∈ A D (see Definition 3), the equivalence class A w of all (finite-length) 290 words, whose suffix is w, is qualified to be a D-Markov state that is denoted as w.

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For the PFSA method, there are primarily four choices as listed below:

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• Alphabet size (|A|): In order to separate out the regimes in the feature space, a 293 larger alphabet size is preferred but more data is required for training the model.

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For the purpose of this paper, an alphabet size |A| = 6 was sufficient.   HMMs are capable of learning and representing long-range dependencies between observations, with the underlying models being assumed to be probabilistic functions of the hidden states [37]. For a discrete-time representation of a data string Y = {y 1 , y 2 , . . . , y T } of T continuous (real-valued) observations, and assuming a firstorder Markov property [38] over the observations, the joint probability density function of Y is obtained as: where ∑ j a ij = 1 ∀i and a ij ≥ 0 ∀i, j.
is the probability density of the observation given the state: π is a 1 × |N| vector with ∑ i π |N| i=1 = 1 and π i ≥ 0 ∀i.

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Following a model λ, the corresponding joint probability distribution of states and observations has the form: p(Y, Z) = p(z 1:T )p(y 1:T |z 1:T ) = p(z 1 ) During the training phase, a commonly used expectation maximization (EM) proce- ∑ z 1 ,z 2 ,...,z T π z 1 b z 1 (y 1 )a z 1 z 2 b z 2 (y 2 ) . . . a z T−1 z T b z T (y T ) which is obtained by using the Forward Procedure [36] to compute the log likelihood (L k ) of the given time series data belonging to each of the K classes. The final decision, as to which class the unknown data belongs, is made by selecting the class with the largest log likelihood as follows: A continuous HMM formulation has been used in this paper, which uses a Gaussian

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Referring back to Figures 2 and 3, it is seen that the operational regimes of the NCL

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During the testing phase, the time-series (that belongs to an an unknown) is windowed in the same manner and a final morph matrix for the entire time-series, Π TS , is computed by averaging as described above. The final decision of which regime the test time-series belongs to, is made as follows: It is noted that the above approach to training a PFSA is different from those seen in  For the HMM based classification, the time series in its entirety is used for training of the three HMMs corresponding to each of the regimes. A windowed formulation is not needed here due to the fact that the HMM method inherently learns long-term dependencies. In the testing phase of HMM, the log-likelihood (L) of the time-series for each of the three regimes is computed, and the final classification decision is made by following  To test the efficacy of the regime classification algorithm, described in Section 5-A, 419 the PFSA models are trained exclusively using the no-radiation-loss data, where the data 420 set is split into 50% for training for both PFSA (i.e., morph matrices Π SS , Π OL and Π FR ) 421 and HMM, and the remaining 50% are used to test the performance of the respective 422 algorithms of PFSA and HMM. Each of the two confusion matrices in Table 2 shows the    Table 4 lists the confusion matrix reporting the classification accuracy of all (33) time 438 series incorporating radiation heat loss by using the PFSAs / HMMs that are optimally 439 trained with the no-radiation data.
440 Figure 4. Case-wise classification accuracy for both data-sets (without and with radiation) for PFSA Good classification accuracy is seen in Table 4 for the PFSA method, with a total 441 error of 9.09%, and an error of 15.15% for the HMM method. Thus, the PFSA method 442 apparently captures the system dynamics modestly better than the HMM method and 443 the results do not strongly depend on the actual system operating conditions. The 444 charts of Figure 4 show the classification accuracy of the optimal PFSAs for the data 445 sets, with and without radiation heat loss. It is concluded that the PFSA algorithm is    causality. The first identification occurs at ∼1,600 seconds, because the first ∼1,000 489 seconds are ignored (due to initial transience) and the window length needed to be 490 observed is 600 seconds.

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It is seen in Figures 6 and 7 that the initial transient region (i.e., first ∼ 3,000 seconds, 492 including the initial transience) does not allow for good accuracy of identification (i.e., 493 classification) using either PFSA or HMM. However, the accuracy almost monotonically longer to make the correction. In Figure 9a, it is seen that the HMM method completely 520 mis-classifies the steady time series to be oscillatory until 9,000 second.

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In view of the above observations, it is reasonable to conclude that the PFSA method 522 is capable of early identification of the final regime of operation of the NCL system, and 523 outperforms the (baseline) HMM method in many cases.  algorithm is significantly larger than that for the PFSA algorithm. It is concluded from 548 the observed results that PFSA is apparently more suitable than HMM for solving the 549 problem of regime identification/classification in NCL systems.

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While there are many areas of theoretical and experimental research, which must 551 be investigated before the proposed PFSA method can be implemented in real-life appli-552 cations, the following topics of future research are suggested to be pursued immediately.