monitoring temperature

computation
Article
Monitoring of Temperature Measurements for Different Flow
Regimes in Water and Galinstan with Long Short-Term Memory
Networks and Transfer Learning of Sensors
Stella Pantopoulou 1,2 , Victoria Ankel 1,3 , Matthew T. Weathered 1 , Darius D. Lisowski 1 , Anthonie Cilliers 4 ,
Lefteri H. Tsoukalas 2 and Alexander Heifetz 1, *
1
2
3
4
*
Citation: Pantopoulou, S.; Ankel, V.;
Weathered, M.T.; Lisowski, D.D.;
Cilliers, A.; Tsoukalas, L.H.; Heifetz,
A. Monitoring of Temperature
Measurements for Different Flow
Regimes in Water and Galinstan with
Long Short-Term Memory Networks
and Transfer Learning of Sensors.
Computation 2022, 10, 108.
https://doi.org/10.3390/
computation10070108
Academic Editors: George Tsakalidis
and Kostas Vergidis
Received: 7 May 2022
Nuclear Science and Engineering Division, Argonne National Laboratory, Lemont, IL 60439, USA;
[email protected] (S.P.); [email protected] (V.A.); [email protected] (M.T.W.);
[email protected] (D.D.L.)
School of Nuclear Engineering, Purdue University, West Lafayette, IN 47906, USA; [email protected]
Department of Physics, University of Chicago, Chicago, IL 60637, USA
Kairos Power, Alameda, CA 94501, USA; [email protected]
Correspondence: [email protected]
Abstract: Temperature sensing is one of the most common measurements of a nuclear reactor
monitoring system. The coolant fluid flow in a reactor core depends on the reactor power state. We
investigated the monitoring and estimation of the thermocouple time series using machine learning
for a range of flow regimes. Measurement data were obtained, in two separate experiments, in a
flow loop filled with water and with liquid metal Galinstan. We developed long short-term memory
(LSTM) recurrent neural networks (RNNs) for sensor predictions by training on the sensor’s own
prior history, and transfer learning LSTM (TL-LSTM) by training on a correlated sensor’s prior history.
Sensor cross-correlations were identified by calculating the Pearson correlation coefficient of the time
series. The accuracy of LSTM and TL-LSTM predictions of temperature was studied as a function
of Reynolds number (Re). The root-mean-square error (RMSE) for the test segment of time series of
each sensor was shown to linearly increase with Re for both water and Galinstan fluids. Using linear
correlations, we estimated the range of values of Re for which RMSE is smaller than the thermocouple
measurement uncertainty. For both water and Galinstan fluids, we showed that both LSTM and
TL-LSTM provide reliable estimations of temperature for typical flow regimes in a nuclear reactor.
The LSTM runtime was shown to be substantially smaller than the data acquisition rate, which allows
for performing estimation and validation of sensor measurements in real time.
Keywords: fluid flow measurements; predictive models; recurrent neural networks; long short-term
memory networks; transfer learning
Accepted: 23 June 2022
Published: 29 June 2022
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1. Introduction
The performance of nuclear reactors could be enhanced through automation of monitoring tasks for the early detection of incipient signs of failure in the reactor monitoring
system data streams [1–3]. Temperature sensing is one of the most common types of measurements in a reactor monitoring system [4]. A nuclear system typically has hundreds
of temperature sensors, the most common of which are thermocouples. When repeatedly
exposed to high temperatures and ionizing radiation, thermocouples are prone to drift and
de-calibration. The challenges in the detection of sensor fault include uncertainties and
noise inherent in the physics of measurements, as well as fluctuations in sensor readings
due to normal transients in the fluid. One approach to system monitoring is through development of a model, which typically involves the numerical solution of a nonlinear system
of differential equations [5,6]. An example is the development of virtual sensors through
a computational fluid dynamics solution of Navier–Stokes equations [7]. However, the
Computation 2022, 10, 108. https://doi.org/10.3390/computation10070108
https://www.mdpi.com/journal/computation
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model-based detection of sensor faults is difficult to accomplish because exact knowledge
of a complex system, such as a nuclear power plant, is required [1,2]. On the other hand,
data-driven methods learn directly from experimental observations without any prior
knowledge of the system.
A number of supervised and unsupervised machine learning (ML) strategies for sensor
fault detection in nuclear power plants (NPPs) have been proposed recently. A review of
the literature can be found in [1–3]. Some specific methods include anomaly detection and
discrimination with Hotelling’s T2 test and interquartile range (IQR) method [8], principal
component analysis (PCA) [9], enhanced singular value decomposition (ESVD) [10], and
correlation analysis and deep belief networks [11]. In addition, there have been studies
specifically focused on NPP thermocouples using deep learning [12] and SVD [13]. In
particular, ML-based strategies have been proposed for the detection of thermocouple
sensor failures from time series analysis [14,15], which include strategies based on recurrent
neural networks (RNN) [16], and a special case of RNN, long short-term memory (LSTM)
networks [17,18]. In general, LSTM networks are highly efficient in predicting the evolution
of dynamic systems because they contain “feedback loops” within their structure [19].
LSTM’s analysis of time series has been benchmarked in the literature [20,21]. Recent
studies have investigated LSTM’s performance in various applications, including nuclear
reactor control [22], fiber-optic sensor monitoring [18,23,24], failure detection and remaining
useful lifetime estimation [25,26], detection of wind turbine blade icing [27], short-term
forecasting of solar energy systems [28,29], and water distillation [30,31].
In this paper, we develop thermocouple monitoring LSTM models trained on the same
sensor, and through transfer learning (TL-LSTM) by training on a correlated sensor [11].
In transfer learning (TL), a pre-trained model is utilized for different applications [32].
This is advantageous for NPP sensor monitoring when limited data for model training
are available, such as for the monitoring of newly installed sensors. Recent work has
explored TL in conjunction with convolutional neural networks (CNNs) [33], and in conjunction with LSTM networks for time series forecasting [34], pandemic modeling [35], and
hydrology [36].
We study the accuracy of LSTM and TL-LSTM prediction of a nuclear-grade type
K thermocouple sensor in water and Galinstan fluids for different flow regimes. In the
envisioned nuclear energy application, ML methods developed in this work are used for
monitoring the same type K thermocouple sensors installed in high-temperature sodium,
molten salt, or water fluids. Water and Galinstan fluids, which are used for LSTM model
validation in this paper, have been shown to be good surrogates for sodium when performing thermal hydraulic experiments, possessing very similar density and viscosity to within
an order of magnitude [37,38]. We investigate the sensor monitoring accuracy dependence
on the flow rate because, in nuclear applications, depending on the reactor power state, the
same thermocouple sensors will experience different types of coolant fluid flow regimes.
During start-up and shut-down, when the reactor is subcritical, the coolant fluid velocity
in reactor core channels is relatively low. During full-power operation when the reactor is
critical, the fluid velocity in core channels is higher.
In this study, the accuracy of time series prediction is evaluated with the root-meansquare error (RMSE), which is compared to the uncertainty of thermocouple sensor fluid
temperature measurements. For the development of TL-LSTM for water and Galinstan
loops, we identify correlated sets of sensors by calculating the Pearson correlation coefficient
for the sensor time series [11]. We show that LSTM and TL-LSTM perform similarly in
one-step-ahead prediction of thermocouple time series. Flow regimes can be characterized
by the nondimensional Reynolds number (Re) [39]. We show that the RMSE of LSTM
prediction increases linearly with the value of Re for both water and Galinstan fluids.
However, the values of RMSE as well as all errors in the time series in each test segment are
smaller than the sensor measurement uncertainty. Deriving the correlation between RMSE
and Re allows the estimation to be made of the range of flow regimes for which LSTM
provides a reliable estimation of temperature. Benchmarking of the LSTM performance
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shows that one-step-ahead predictions can be performed in real time, i.e., faster than sensor
data acquisition, which is limited by a sensor’s time constant.
This paper is organized as follows. Section 2 describes temperature and flow rate
measurements in water and Galinstan flow loops. Section 3 discusses the development
of LSTM and TL-LSTM networks using training data from the same sensor and training
data from a correlated sensor. Sections 4 and 5 discuss the results of LSTM and TL-LSTM
monitoring of thermocouple time series in water and Galinstan loops, respectively. Section 6
contains a discussion of the main results. Section 7 concludes the paper and discusses
future work.
2. Data Collection in Thermal Hydraulic Flow Loop
A thermal hydraulic test flow loop was assembled and instrumented with temperature
and flow sensors [18,40]. The loop allows temperature sensor time series to be monitored
for different flow regimes, with data from all sensors recorded with a data acquisition
system with the LabVIEWTM interface. A schematic diagram of the loop is shown in
Figure 1. The loop was constructed with polycarbonate pipe sections, which have an
operating temperature range between 0 ◦ C and 121 ◦ C. In one set of experiments, the loop
was filled with water, and in another separate set of tests, the loop was filled with liquid
metal Galinstan. Water and Galinstan are frequently used for proof-of-principle studies of
fluid sensing at room temperature [18,40]. Galinstan (Ga 68.5%, In 21.5%, and Sn 10%) is a
liquid metal at room temperature, which has a melting point of −19 ◦ C. Galinstan has a
higher density and thermal conductivity than water, but lower heat capacity.
Figure 1. Schematics of the flow loop. Locations of temperature sensors TC-1 through TC-7 and flow
rate sensors FT-1 and FT-2 are indicated in the diagram. Correlated sensors are highlighted with the
same color.
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Fluid in the loop was circulated clockwise against gravity with a variable-speed 1.5HP
stainless circulation pump. A split in the main loop created “hot leg” and “cold leg” flows.
Fluid pumped through the “hot leg“ was heated with a variable power Watlow FLC-16
heater, with a maximum power rating of 4 kWe. Fluid in the “hot leg” was eventually
mixed with the “cold leg” fluid in a thermal Tee. The inner diameters of the “hot leg” and
“cold leg” pipes were 0.75 in and 1.5 in, respectively. After mixing, fluid in the main loop
was brought back to room temperature with a 130,000 BTU Shell & Tube standard heat
exchanger, which was coupled to a 11,300 BTU/hr chiller.
Temperature measurements in the flow loop were performed using type K thermocouples, which are commonly found in nuclear applications because of their relative radiation
hardness. This thermocouple, which is rated for operation at temperatures up to 1100 ◦ C,
consists of two dissimilar metallic contacts enclosed in a protective metallic sheath. The
positive contact is ChromelTM (nickel and chromium alloy) and the negative contact is
AlumelTM (nickel, manganese, silicon, and aluminum alloy) (Concept Alloys Inc., Whitmore Lake, MI, USA). The protective sheath, which is in contact with high-temperature
fluid, is made of corrosion-resistant stainless steel or nickel alloy. The accuracy of measurements with the type K thermocouple is given as max (±1.1 ◦ C, 0.4%). For fluid temperature
ranges in this study, the thermocouple measurement uncertainty was ±1.1 ◦ C. The flow
was measured using Turbine Blancett B110-500-1/2 flowmeters, which operate at temperatures up to 177 ◦ C. The accuracy of flow measurements is ±1.0% of the reading and the
repeatability is ±0.1%.
Flow rate and temperature data were sampled using sensors at different points
throughout the loop. Locations of the sensors are indicated in the diagram in Figure 1.
Seven thermocouples are labeled as TC-1 through TC-7 and two flowmeters are labeled
as FT-1 and FT-2. Correlated sensors (discussed in Section 3.1) are highlighted with the
same color in Figure 1 (color online). Temperatures of the “cold leg” were measured with
thermocouples TC-1 and TC-2, in the “hot leg” with thermocouples TC-6 and TC-7, and in
the mixing zone and downstream with TC-3, TC-4, and TC-5. Thermocouples TC-6, TC-2,
and TC-3 were located 14.75 in, 5.5 in, and 17.75 in, respectively, from the thermal mixing
tee. The mass flow rate in the main loop and “cold leg” was measured with two volumetric
flow meters FT-1 and FT-2, respectively. In this study, the ranges of operating temperatures
in the “hot leg” (TC-6) and “cold leg” (TC-2) were approximately 30 ◦ C and 50 ◦ C, and
20 ◦ C to 30 ◦ C, respectively. The fluids were maintained at ambient pressure.
Temperature and flow rate transients were generated by varying the heater power and
pump speed, respectively, from within the LabVIEWTM interface. Transient temperature
and flow rate data were collected in separate experiments with water and liquid metal
Galinstan fluids, consisting of time series of 2100 and 1470 points, respectively, which were
sampled every second. Measurements from all sensors were collected concurrently.
3. Development of LSTM Networks for Temperature Sensors Validation
3.1. Identification of Correlated Sensors
To identify redundancies in sensor measurements, we calculated the Pearson correlation coefficient to determine the pairwise similarity between different sensor time series.
For two signals x, y, the Pearson correlation coefficient r is defined as [41]
r ( x, y) = q
∑iN=1 ( xi − x )(yi − y)
q
2
2
N
∑i=1 ( xi − x ) ∑iN=1 (yi − y)
(1)
where xi , yi are the measurements of signals x and y at time i, respectively; N is the total
number of samples; x, y are the mean values for each signal. The coefficient r takes on
a value in the range of −1 ≤ r ≤ 1. The similarity between two signals increases as the
value of the coefficient approaches 1. Identical signals produce a Pearson correlation
coefficient r = 1. Correlations were calculated between respective temperature sensors and
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flow sensors. Values of Pearson correlation coefficients calculated for sensors in water and
Galinstan loops are shown in Figure 2a,b, respectively.
Figure 2. Pearson correlation coefficient for sensors in (a) water loop and (b) Galinstan loop. Correlated sensors are highlighted with the same color (color online).
Our analysis showed that for both water and Galinstan loops, correlated sensor sets
were {TC-1, TC-2}, {TC-3, TC-4, TC-5}, {TC-6, TC-H}, and {FT-1, FT-2}. Correlated sensors
are indicated with the same colors in Figure 1. Thus, the basis set (linearly independent
sensors) for this loop consisted of [TC-2, TC-3, TC-6, FT-1].
3.2. Development of LSTM Networks
In this work, we focused on the validation of temperature sensor TC-3, which was located downstream of the mixing zone. LSTM networks were developed with the MATLAB
Deep Learning Toolbox. Sensor time series were subdivided into training (80% of data) and
validation (10% of data) segments for the development of the LSTM, and test (10% of data)
segments for which the LSTM was used to perform monitoring/validation function. This
is a common data partitioning ratio for machine learning of time series [42]. The structure
of the LSTM neural network used for this study is shown in Figure 3.
Figure 3. Cont.
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Figure 3. (a) Structure of LSTM neural network for prediction of time series of temperature sensor;
(b) structure of LSTM unit cell.
The network is comprised of an LSTM layer, a fully connected layer, and a regression
layer. The one-step LSTM layer is able to learn the dependencies between the time steps
in the sequence data. For both water and Galinstan fluids, we determined that the best
performance of the LSTM network for all sensors was achieved for a number of hidden
nodes in that layer equal to 15. The state activation function for the LSTM cell was specified
as tanh and the gate activation function was specified as sigmoid. The solver selected for
this problem was Adam and the number of epochs was 250. When training the LSTM, we
used the loss function of the root-mean-square error (RMSE), which is defined as
s
RMSE =
∑iN=1 ( xn − x̂n )
N
2
(2)
where xn is the actual data, x̂n is the forecasted data, and N is the total number of points.
An LSTM unit cell is comprised of the forget, input, and output gates [19,20]. The
structure of an LSTM cell is shown in Figure 3b. The forget gate (f ) makes a decision
whether to retain previous information or not. Equation (3a) gives the expression for the
forget gate:
f t = sigmoid( x t ·U f + H t−1 ·W f
(3a)
where xt is the current input, U f is the weight corresponding to that input, Ht−1 is the
hidden state of the previous step, and W f is the weight matrix for that state.
The input gate (i) is described by Equation (3b):
it = sigmoid( x t ·Ui + H t−1 ·Wi )
(3b)
The updated state of the LSTM cell is described by Equation (3c):
Ct = f t ·Ct−1 +it ·tanh( xt ·Uc + Ht−1 ·Wc )
(3c)
The activation function used to express the information that passes through the cell
state is tanh. The subscripts i and c indicate relevance to the input gate and to the updated
state, respectively.
The output gate (o) is described by Equation (3d):
ot = σ ( xt ·Uo + Ht−1 ·Wo )
(3d)
The updated hidden state of the LSTM cell is shown in Equation (3e):
Ht = ot ·tanh(Ct )
(3e)
The accuracy of sensor prediction with LSTM was evaluated with RMSE calculated
over the test segment. In one approach, we studied the prediction of TC-3 using the LSTM
developed with training and validation on TC-3 data in water and Galinstan. Because
TC-3 and TC-4 were highly correlated in both water and Galinstan (r = 0.99 according
to Figure 2), we investigated the transfer learning prediction of TC-3 with the TL-LSTM
network trained and validated on TC-4 data.
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3.3. Selection of Time Series Test Segments
Flow regimes in the pipes can be characterized by the nondimensional Reynolds
number (Re), which can be calculated as
Re =
ρQD
µA
(4)
where ρ is the fluid density, Q is the volumetric flow rate, D is the hydraulic diameter (inner
diameter of the pipe), A = πD2 /4 is the pipe cross-sectional area, and µ is the dynamic
viscosity of the fluid. Typical criteria are that the flow in a pipe is laminar when Re < 2300,
transitional when 2300 < Re < 4000, and turbulent when Re > 4000 [39]. Table 1 shows
the values of ρ and µ, as well as heat capacity c and thermal conductivity k, for water
and Galinstan at room temperature and ambient pressure, and the diameter of the pipe
containing the TC-3 thermocouple.
Table 1. Fluid thermophysical properties and pipe diameter.
Fluid
Water
Galinstan
ρ (kg/m3 )
1000
6440
µ (Pa·s)
10−4
8.9 ×
2.4 × 10−3
c (J/kg·K)
k (W/m·K)
D (in)
4183
296
0.6
16.5
1.61
1.61
Depending on the reactor power state, the coolant flow velocity in coolant channels
and, hence, Re numbers take on different values. For pressurized water reactors (PWRs),
the range of values is approximately 103 < Re < 106 . For liquid metal fast breeder reactors
(LMFBRs), which use liquid sodium as a coolant, the range of values is approximately
103 < Re < 105 [39].
The graph of time-dependent values of Re for water flow in the pipe with TC-3 is
shown in Figure 4. Values of Re for water flow were calculated using Equation (4) and
parameters in Table 1, and values for Q were taken from FT-1 time series of water loop
measurements. Five time segments with nearly constant flows are labeled 1 through 5, in
order of increasing values of Re.
Figure 4. Reynolds number Re for water flow in pipe with TC-3. The intervals of constant Re are
labeled as segments 1 through 5, in order of increasing Re.
Note that in segment 1, the flow was laminar. For segment 2, the flow was in the
transitional regime, while for segments 2 through 5, the flow was turbulent. Five time
intervals, each 210 s long or 10% of the total times series of 2100 s, were selected as subsets
of constant-flow time intervals. These 210 s long time intervals are used in Section 4 as
TC-3 test segments for LSTM development. The numbers of segments, constant-flow time
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intervals, 210 s long time series test segments, and corresponding values of Re are listed in
Table 2.
Table 2. Time segments and corresponding Re values for water flow near TC-3.
Segment
Constant-Flow Time Interval (s)
Time Series Test Segments (s)
Re
1
2
3
4
5
1836–2100
1444–1661
1124–1393
811–1027
1–739
1870–2079
1444–1653
1130–1339
811–1020
200–409
1680
2600
5600
11,900
18,300
The graph of time-dependent values of Re for Galinstan flow in a pipe with TC-3 is
shown in Figure 5. Values of Re for Galinstan flow were calculated using Equation (4) and
parameters in Table 1. Values of Q were taken from FT-1 time series of measurements in
the Galinstan flow loop. Four intervals of nearly constant flow are labeled in Figure 5 as
segments 1 through 4, in order of increasing values of Re.
Figure 5. Reynolds number Re for Galinstan flow in pipe with TC-3. The intervals of constant Re are
labeled as segments 1 through 4, in order of increasing Re.
All four constant-flow segments in Figure 5 corresponded to the turbulent flow regime.
Four time intervals for the temperature test, each 147 s long or 10% of the total times series
of 1470 s, were selected as subsets of the constant-flow segments. These 147 s long time
intervals are used in Section 5 as TC-3 test segments for LSTM development. Segment
numbers, constant-flow time intervals, time series test segments, and the corresponding
values of Re are listed in Table 3.
Table 3. Time segments and corresponding Re values for Galinstan flow near TC-3.
Segment
Constant-Flow Time Interval (s)
Time Series Test Segments (s)
Re
1
2
3
4
276–791
1170–1471
1–216
918–1100
300–446
1200–1346
50–196
950–1096
5800
9100
14,500
18,200
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4. Monitoring of Thermocouple Time Series in Water Flow Loop
4.1. LSTM Prediction of Thermocouple Time Series
Time series of TC-3 and TC-4 measurements in the water loop are shown in Figure 6a,b,
respectively. Locations of each of the five 210 s long test segments of temperature measurements, numbered 1 through 5, are indicated in red (color online). The numbers of test
segments in Figure 6a,b correspond to the numbers of time segments in Table 2.
Figure 6. Time series of (a) TC-3 and (b) TC-4 measurements in water loop indicating locations of five
210 s long tests segments. Test segments numbered 1 through 5, which correspond to the numbering
of time segments in Table 2, are indicated in red (color online).
Results of TC-3 time series prediction for each of the five test segments with LSTM
developed using TC-3 data and TL-LSTM developed with transfer learning using TC-4
data are shown in Figure 7. Note that a separate LSTM network was developed for the
prediction of each of the five test segments. In each case, the training and validation data
sets excluded the corresponding test segment. The performances of LSTM and TL-LSTM
predictions are quantified by listing RMSE values in Table 4. RMSE is a measure of the
distance between LSTM-predicted and observed values. In addition, statistics of the LSTM
performance are investigated by listing the mean values µ and standard deviations σ for
each time segment in Table 4. Error mean value µ indicates if LSTM, on average, underpredicts or over-predicts the observed values. Standard deviation σ is an indicator of the
spread of the instantaneous errors around the mean value µ.
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Figure 7. Prediction of five test segments (1 through 5) of TC-3 in water loop with LSTM trained on TC-3
data and TL-LSTM trained on TC-4 data. Time series of observed (black), LSTM (blue), and TL-LSTM
(orange) predictions are shown in the left column. Error time series (error = predicted − observed) for LSTM
(blue) and TL-LSTM (orange) are in the middle column. Error histograms for LSTM (blue) and TL-LSTM
(orange) along with Gaussian fits are in the right column. RMSE, mean, and standard deviation values
are listed in Table 4.
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Table 4. Statistics of thermocouple prediction errors with LSTM and TL-LSTM in water.
Segment
1
2
3
4
5
RMSE (10−2 ) (◦ C)
LSTM
TL-LSTM
1.96
2.01
4.18
6.72
9.07
2.08
3.64
4.60
7.85
9.34
µ(·10−2 ) (◦ C)
LSTM
TL-LSTM
−1.49
0.25
−1.32
5.25
3.14
0.84
−0.01
−0.09
2.82
5.17
σ(·10−2 ) (◦ C)
LSTM
TL-LSTM
1.28
2.00
3.98
4.20
8.53
1.91
3.65
4.61
7.34
7.80
Measurement Uncertainty (◦ C)
±1.1
±1.1
±1.1
±1.1
±1.1
The left column of Figure 7 labeled “Test Segment” shows the graphs of the observed
(black), predicted-with-LSTM (blue), and predicted-with-TL-LSTM (orange) time series
for each of the test segments 1 through 5 of TC-3. The middle column labeled “Error Time
Series” shows the graphs of real-time errors for LSTM (blue) and TL-LSTM (orange). The
errors at each time step are calculated as error = predicted – observed. The right column
labeled “Error Histogram” shows the histogram of prediction errors of LSTM (blue) and
TL-LSTM (orange). A Gaussian fit curve is added to each histogram.
The measurement uncertainty of TC-3 for all five test segments was max (±1.1 ◦ C,
0.4%) = ±1.1 ◦ C. For reference, this number is listed in Table 4. The trend for the LSTM
prediction errors listed in Table 4 showed that RMSE and σ increased with the increase in
Re value (the time segments are numbered in the order of increasing values of Re number).
However, the RMSE values as well instantaneous errors (middle column in Figure 7) for
each test segment were smaller than the TC-3 measurement uncertainty error by at least
an order of magnitude. The RMSE, µ, and σ in predictions with LSTM and TL-LSTM had
comparable values for all five time segments. The values of RMSE and σ for LSTM were
smaller than the corresponding values for TL-LSTM for all test segments. The exception
was segment 5, for which TL-LSTM had a smaller σ value. From the error histogram
displayed in the last column in Figure 7, prediction errors for both LSTM and TL-LSTM
for time segment 5 had more outliers compared to other time segments. These outliers
occurred mostly in the beginning of test segment 5. Error mean values µ indicated that both
LSMT and TL-LSTM over-predicted the actual time series values for time segments 4 and 5.
4.2. Relationship between RMSE, σ, and Re
To investigate the performance of the LSTM for different flow regimes in the water
loop, we plot Table 4 values of RMSE as a function of Re in Figure 8. We obtained linear fits
for the LSTM (dashed blue line) and the TL-LSTM (dashed red line), with R2 = 0.99 and
R2 = 0.96, respectively. High values of R2 indicate strong linearity in the data.
Linear correlations for RMSE as a function of Re are listed in Table 5. Note that the
slopes of the two linear correlations were almost the same, with the difference appearing in
the second significant digit. Using linear correlations, we estimated the range of applicability of LSTM-based monitoring by setting the condition that RMSE < 1.1 ◦ C (thermocouple
measurement uncertainty). Solving for Re numbers using linear correlations, we obtained
the upper bound values, which are listed in the last column of Table 5. The upper bounds
for Re numbers were almost the same for both LSTMs, with a difference appearing in the
second significant digit.
As an additional consideration, for the data in Table 4, we investigated the correlation
between σ and Re. The graphs of σ vs. Re for LSTM (blue color) and TL-LSTM (red color)
LSTMs are shown in Figure 9. Linear fits were obtained with R2 = 0.91 for LSTM and
R2 = 0.89 for TL-LSTM. This indicated a relatively weak linearity of the data, compared to
the data in Figure 8.
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Figure 8. Correlation between RMSE and Re for estimations with LSTM (blue) and TL-LSTM (red) in
water (color online).
Table 5. Correlation between RMSE and Re in water.
Model
RMSE (◦ C)
RMSE < 1.1 ◦ C
LSTM
TL-LSTM
4.38 × 10−6 Re + 0.013
4.21 × 10−6 Re + 0.022
Re < 2.48 × 105
Re < 2.56 × 105
Figure 9. Correlations between σ and Re for estimations with LSTM (blue) and TL-LSTM (red) in
water (color online).
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4.3. Benchmarking LSTM Runtime
The runtime of the LSTM network was benchmarked on an Intel© Xeon© CPU E52687W v3 @ 3.10 GHz with 64 GB RAM. A representative benchmark LSTM runtime in
the water loop is shown in Figure 10. The graph displays the runtime of LSTM prediction
of each sample in the 210 s long test segment 3 of the TC-3 time series. For the data in
Figure 10, we obtained the LSTM per sample runtime mean value µ = 3.4 ms with standard
deviation σ = 0.85 ms. Note that the LSTM runtime for the prediction of each sample was
at least an order of magnitude smaller than the acquisition time of 1 s. Similar runtime
characteristics were observed in benchmarking of the LSTM prediction of other TC-3 test
segments. Therefore, in the online monitoring scenario, validation of the measurements
with LSTM can be, in principle, performed in real time.
Figure 10. Benchmark of LSTM runtime for test segment 3 of TC-3 prediction in water with LSTM.
Runtimes for the samples in test interval have µ = 3.4 ms and σ = 0.85 ms.
5. Monitoring of Thermocouple Time Series in Galinstan Flow Loop
5.1. LSTM Prediction of Thermocouple Time Series
Time series of TC-3 and TC-4 in Galinstan loop are shown in Figure 11a,b, respectively.
The location of each of the five 147 s long test segments, numbered 1 through 4, is indicated
in red (color online). The numbers of test segments in Figure 11a,b correspond to the
numbers of time segments in Table 3.
The results of TC-3 time series prediction for each of the four test segments with the
LSTM developed using TC-3 data and TL-LSTM developed using TC-4 data are shown in
Figure 12. Note that a separate LSTM network was developed for the prediction of each
of the four test segments. In each case, the training and validation datasets excluded the
corresponding test segment. Similar to the case of the water loop considered in Section 4,
we considered the RMSE over each test segment, as well as mean value of the error µ and
standard deviation σ of errors from the mean. Statistics of LSTM and TL-LSTM predictions
were quantified by listing RMSE, µ, and σ values for each time segment in Table 6.
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Figure 11. Time series of (a) TC-3 and (b) TC-4 in Galinstan loop indicating temporal locations of
147 s long test segments. Test segments numbered 1 through 4, which correspond to the numbering
of time segments in Table 3, are indicated in red (color online).
Table 6. Statistics of thermocouple prediction errors with LSTM and TL-LSTM in Galinstan.
Segment
1
2
3
4
RMSE (10−2 ) (◦ C)
LSTM
TL-LSTM
3.47
6.01
10.55
15.39
4.51
8.44
11.97
16.18
µ(·10−2 ) (◦ C)
LSTM
TL-LSTM
1.64
3.81
3.58
0.96
−2.93
−0.89
−1.03
−5.51
σ(·10−2 ) (◦ C)
LSTM
TL-LSTM
3.07
4.66
9.95
15.42
3.44
8.42
11.97
15.27
Measurement Uncertainty (◦ C)
±1.1
±1.1
±1.1
±1.1
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Figure 12. Prediction of four test segments (1 through 4) of TC-3 in Galinstan loop with LSTM
trained on TC-3 data, and TL-LSTM trained on TC-4 data. Time series of observed (black),
LSTM (blue), and TL-LSTM (orange) predictions are shown in the left column. Error time series
(error = predicted-observed) for LSTM (blue) and TL-LSTM (orange) are in the middle column. Error
histograms for LSTM (blue) and TL-LSTM (orange) along with Gaussian fits are in the right column.
RMSE, mean, and standard deviation values are listed in Table 6.
The left column of Figure 12 labeled “Test Segment” shows the graphs of the observed
(black), predicted-with-LSTM (blue), and predicted-with-TL-LSTM (orange) time series
for each of the test segments 1 through 4 of TC-3. The middle column labeled “Error Time
Series” shows the graphs of real-time errors for the LSTM (blue) and TL-LSTM (orange).
The errors at each time step were calculated as error = predicted − observed. The right column
labeled “Error Histogram” shows the histogram of prediction errors of the LSTM (blue)
and TL-LSTM (orange). A Gaussian fit curve is added to each histogram.
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The measurement uncertainty of TC-3 for all four test segments was max (±1.1 ◦ C,
0.4%) = ±1.1 ◦ C. For reference, this number is listed in Table 6. The trend for the LSTM
prediction errors listed in Table 6 showed that RMSE and σ increased with the increase in
Re value (the time segments are numbered in the order of increasing values of Re number).
However, the RMSE values as well instantaneous errors (middle column in Figure 12) for
each test segment were smaller than the TC-3 measurement uncertainty error by at least
an order of magnitude. The RMSE, µ, and σ in predictions with the LSTM and TL-LSTM
had comparable values for all four time segments. The values of RMSE and σ for LSTM
were smaller than the corresponding values for TL-LSTM for all test segments. From the
error histogram displayed in the last column in Figure 12, prediction errors for both LSTM
and TL-LSTM for all time segments had a Gaussian-like shape for all time segments. Error
mean values µ indicated that the LSTM over-predicted while the TL-LSTM under-predicted
the actual time series values for all time segments.
5.2. Relationship between RMSE, σ, and Re
To investigate the performance of the LSTM for different flow regimes in the water
loop, we plot Table 6 values of RMSE as a function of Re in Figure 13. We obtained linear
fits for the LSTM (dashed blue line) and TL-LSTM (dashed red line), for both of which
R2 = 0.99. High values of R2 indicate strong linearity in the data. Linear correlations
for RMSE as a function of Re are listed in Table 7. Note that the slopes of the two linear
correlations were close in value.
Figure 13. Correlation between RMSE and Re for estimations with LSTM (blue) and TL-LSTM (red)
in Galinstan (color online).
Table 7. Correlation between RMSE and Re in Galinstan.
Model
RMSE (◦ C)
RMSE < 1.1 ◦ C
LSTM
TL-LSTM
9.44 × 10−6 Re − 0.024
8.95 × 10−6 Re − 0.004
Re < 1.19 × 105
Re < 1.23 × 105
Solving for Re numbers using linear correlations, we obtained the upper bound values,
which are listed in the last column of Table 7. The upper bounds for Re numbers were almost
the same for both LSTMs, with the difference appearing in the second significant digit.
As an additional consideration, for the data in Table 6, we investigated the correlation
between σ and Re. The graphs of σ vs. Re for LSTM (blue color) and TL-LSTM (red color)
are shown in Figure 14. Linear fits were obtained with R2 = 0.97 for both the LSTM and TL-
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Please check if this should be capital. It capital in other places (including figures and tables). It for
17 of 21
whole text.
We removed the background color of table, please confirm.
LSTM. Unlike the case of the water loop, we observed a strong linearity in the dependence
between σ and Re in the Galinstan loop.
Figure 14. Correlations between σ and Re for estimations with LSTM (blue) and TS-LSTM (red) in
Galinstan (color online).
5.3. Benchmarking LSTM Runtime
A representative benchmark LSTM runtime in the Galinstan loop is shown in Figure 15.
The graph displays the runtime of the TC-3/TC-3 LSTM prediction of each sample in the
147 s long test segment 1 of the TC-3 time series. For the data in Figure 15, we obtained
the LSTM per sample runtime mean value µ = 3.5 ms with standard deviation σ = 1.1 ms.
These values were comparable to those of the water loop benchmark in Figure 10. Note that
the LSTM runtime for prediction of each sample was at least an order of magnitude smaller
than the acquisition time of one second. Similar runtime characteristics were observed in
the benchmarking of LSTM prediction of other TC-3 test segments.
3
Figure 15. Benchmark of LSTM runtime for test segment 1 of TC-3 prediction in Galinstan. Runtimes
for the samples in test interval have µ = 3.5 ms and σ = 1.1 ms.
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6. Discussion
In Sections 4 and 5, we investigated monitoring of the thermocouple times series
with the LSTM in water and Galinstan fluids. In both fluids, RMSE increased linearly
with Re. Comparing the LSTM performance in different fluids (Figures 7 and 12) over the
common range of Re values (approximately 6000 < Re< 18,000), RMSE of the same sensor’s
prediction in the Galinstan was consistently larger than RMSE in the water loop. This can
be potentially attributed to higher noise in the temperature measurements in Galinstan.
Using the MATLAB function, we calculated the 99% occupancy bandwidth (BW) for each
test segment of TC-3 in water to be BW = 2.4 mHz. On the other hand, for each test segment
of TC-3 in Galinstan, we obtained BW = 3.4 mHz.
For developing a qualitative explanation for the observed increase in RMSE with
increasing Re, we considered the dependence of the thermocouple time constant τ on Re.
We showed that τ decreases with increasing Re (sensor responds faster). At the same time,
as Re increases, the fluid flow becomes more turbulent [39]. Thus, our hypothesis is that the
LSTM prediction accuracy decreases because the thermocouple becomes more sensitive to
temperature fluctuations in an increasingly turbulent flow. In prior work, we showed that
τ of the cylindrical resistance temperature detector (RTD) decreases with the increase in the
fluid-to-RTD heat transfer coefficient [4]. For the cylindrical-shaped type K thermocouple
probe, the heat transfer coefficient h in forced convective heat transfer for a fluid flow
normal to a circular cylinder is [4]
k·Nu
h=
(5a)
d
where k is the fluid thermal conductivity, d is the cylinder diameter, and Nu is the
Nusselt number:
Nu = 0.43 + C · Rem · Pr0.31
(5b)
Here, Pr is the Prandtl number given as
Pr =
c·µ
k
(5c)
where c is the specific heat capacity, µ is the dynamic viscosity, and C and m are fitting
constants. Values of C and m for different ranges of Re number are given in Table 8.
Table 8. Values of constants for heat transfer coefficient correlation.
Re
C
m
35–5 × 103
5 × 103 –5 × 104
5 ×104 –5 × 105
0.583
0.148
0.0208
0.471
0.633
0.814
Using thermophysical property values for water and Galinstan fluids listed in Table 1,
and d = 5 mm as the type K thermocouple diameter, we calculated h as a function of Re.
Results are plotted in Figure 16 for the range of 103 < Re < 2 × 104 , which spans the range
of Re numbers in the experiment. Note that h increased monotonically with Re for both
fluids, with a steeper slope for the case of Galinstan. Therefore, thermocouple time constant
τ decreased with increasing Re.
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Figure 16. Correlations between h and Re for water and Galinstan fluids.
Note that for the same value of Re, the value of h was higher for Galinstan. This
indicated that thermocouple τ in Galinstan was smaller than that in water. From the
discussion above, RMSE for Galinstan was higher than that for water. This is consistent
with the hypothesis that a larger value of RMSE is correlated with a smaller value of
thermocouple τ.
7. Conclusions
We investigated the performance of LSTM and TL-LSTM in monitoring and estimating
the thermocouple time series in water and Galinstan fluids at room temperature and
ambient pressure. LSTMs for predicting the time series of a thermocouple in each loop
were constructed by training on prior history of the same sensor, while TL-LSTMs were
constructed by training on the history of a correlated sensor. For each fluid, a similar
performance was observed for both LSTM and TL-LSTM models. The benchmarking LSTM
performance indicated that monitoring can be performed in real time.
Using linear correlations, we estimated that RMSE in LSTM prediction is smaller than
the uncertainty in temperature measurement within the range, with the upper bound of
Re~105 . During startup and shutdown (low power operation), typical values for fluid
flow in core coolant channels PWR and LMFBR are Re ~ 103 and Re~102 , respectively [39].
During full-power operation, the values for fluid flow in the same core coolant channels in
PWR and LMFBR are Re~105 and Re~104 , respectively [39].
Based on the findings of this paper, LSTM-based monitoring can provide reliable
real-time estimates of a thermocouple for typical operational ranges of a nuclear reactor.
However, further studies are needed to validate these predictions. In particular, LSTM
analysis should be performed on data obtained from measurements in water and liquid
sodium at temperatures, pressures, and flow rates similar to those in nuclear reactors.
Future work will also benchmark the performance of LSTMs (prediction accuracy and
runtime) against other ML algorithms, such as Auto Regressive Integrated Moving Average
(ARIMA), Generative Adversarial Networks (GAN), and Bayesian Networks. We will
also investigate transfer learning (TL) for a high-temperature thermal hydraulic system
by pre-training the model on a room-temperature fluid flow. As the operation of a roomtemperature thermal hydraulic facility is considerably less expensive than the operation of
a high-temperature facility, TL offers the possibility of cost saving for the development of
models for monitoring nuclear systems.
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Author Contributions: Conceptualization, A.H. and S.P.; methodology, S.P. and A.H.; software, S.P.
and V.A.; validation, S.P., V.A. and A.H.; formal analysis, S.P. and A.H.; investigation, S.P.; resources,
D.D.L., A.C. and L.H.T.; data curation, M.T.W. and D.D.L.; writing—original draft preparation, A.H.
and S.P.; writing—review and editing, A.H. and S.P.; visualization, S.P.; supervision, A.H., L.H.T. and
A.C.; project administration, A.H.; funding acquisition, A.H., M.T.W. and A.C. All authors have read
and agreed to the published version of the manuscript.
Funding: This work was supported by the U.S. Department of Energy, Advanced Research Projects
Agency—Energy (ARPA-E) under contract DE-AC02-06CH11357.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: No applicable.
Data Availability Statement: The data presented in this study are available on request from the
corresponding author.
Conflicts of Interest: The authors declare no conflict of interest.
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