A New Development of Cross-Correlation-Based Flow Estimation Validated and Optimized by CFD Simulation

Abstract

The accurate measurement of mass flow rates is important in nuclear power plants. Flow meters have been invented and widely applied in several industries; however, the operating environment in advanced nuclear power plants is especially harsh due to high temperatures, high radiation, and potentially corrosive conditions. Traditional flow meters are largely limited to deployment at the outlet of pumps, on pipes, or in limited geometries. Cross-correlation function (CCF) flow estimation, on the other hand, can estimate the flow velocity indirectly without any specific instruments for flow measurement and in any geometry of the flow region. CCF flow estimation relies on redundant instruments, typically temperature sensors, in series in the direction of flow. One challenge for CCF flow estimation is that the accuracy of the flow measurement is mainly determined by inherent, common local process variation across the sensors, which may be small compared to the uncorrelated measurement noise. To differentiate the process variations from the uncorrelated noise, this research implements periodic fluid injection at a different temperature than the bulk fluid before the temperature sensors to amplify process variation. The feasibility and accuracy of this method are investigated through flow loop experiments and Computational Fluid Dynamics (CFD) simulations. This paper focuses on a CFD simulation model to verify the previous experimental results and optimize CCF flow estimation with different configurations. The optimization study is carried out to perform a grid search on the optimal location of the sensor pair under different flow rates. The CFD results show that the optimal sensor spacing depends on the flow rate being measured and provides guidance for sensor location implementation under various anticipated flow rates.

1. Introduction

There are many implementation challenges for instrumentation in advanced reactor designs [1]. One main challenge is accurately measuring the reactor coolant flow because the reactor power rate is estimated from the coolant flow and temperature. Accurate reactor power rate monitoring can enable operations at higher powers due to lower uncertainties in measurements. Even marginal power increases can translate to large revenue increases over the reactor’s lifetime. However, many traditional flow measurement techniques may not be appropriate for advanced reactor designs. Advanced reactors that have pool-type designs do not have primary piping as a means to enhance safety by avoiding loss of coolant accidents caused by the primary pipe breaking. However, traditional flow meters, such as Venturi and turbine flow meters, measure the flow rate through direct interfacing with the pipeline, which is not possible or difficult to implement in pool-type reactors such as sodium-cooled fast reactors (SFR) and molten salt reactors (MSR). Further issues with existing measurement techniques are the high temperature and corrosive nature of advanced reactor coolants, including liquid metals, molten salts, and high-temperature gasses. It is expected that reactor components interfacing with these coolants have to be operated under such harsh conditions for a long time without repair or replacement. For example, when the main coolant flow is a liquid metal such as sodium or lead–bismuth, which is high density with other chemical erosive issues [2], traditional flow meters cannot be operated safely for continuous, long periods of time.

To allow for flexible measurement installation and operation under harsh environments, cross-correlation function (CCF) flow estimation is introduced as an inferential flow measurement methodology that can be applied in operating and advanced reactors to resolve issues that prevent direct flow measurement [3]. Previously, Por et al. [3] studied CCF flow estimation in a pool-type reactor, establishing the basis for it. Due to the fluid dynamic scaling, the measurement is not restricted to water; it can be applied to liquid metals [2], gases [4], and even non-Newtonian fluids such as surrogate fluids or blood [5]. It can also be extended to two-phase flows, including liquid–gas flow (oil mixed with nitrogen gas) [6], solid–gas swirl flow [7], or liquid–solid flow (water mixed with solid ceramic nodules) [8]. The CCF method for flow measurement was first studied by Bentley et al. in 1966 [9], and CCF flow estimation was applied in nuclear fields in the 1980s [10,11]. CCF flow estimation infers the flow velocity by estimating the delay time between two adjacent sensors using the cross-correlation function and using knowledge of the relative sensor placement. The two adjacent sensors are installed in series in the direction of bulk flow and are not restricted by the pipeline structure. This means that the two sensors can be applied in pool-type or integral reactors. Further, the only restriction for the selection of the two sensors is a preference for a common sensor type to enable better cross-correlation estimation. This allows for the sensor selection to focus on sensors that are more reliable under the harsh operating conditions experienced in advanced reactors. Several sensor signals can be used as the source signals for CCF flow estimation, including temperature, neutron noise [12], voltage [13], or capacitance [6]. The temperature signal is widely selected for CCF flow estimation because the profile of the temperature fluctuation can reflect the profile of the flow rate [14]. Moreover, temperature sensors, such as thermocouples, which are robust and reliable in most flow estimation situations, can be operated under the expected harsh conditions and are typically required process instrumentation in reactor designs. Therefore, temperature sensors are selected for this study with a focus on thermocouple measurements.

For these reasons, CCF flow estimation is being proposed and studied for potential implementation in advanced reactor concepts. Although CCF flow estimation is not restricted by geometry and pipes and can be operated under harsh conditions, the accuracy of this approach is impacted by signal resolution, the ratio of the correlated process noise to the uncorrelated measurement noise, and the sensor response time [15,16]. Based on NRC efforts [17] to address the response time issue, prior work has applied the same brand and type of temperature sensors to alleviate potential response time issues in the experimental flow loop. This is due to the response time being directly impacted by dissimilar temperature sensor selection. A thermocouple (TC) is selected due to its small response time. The bare-wire TC generally has a smaller response time compared with Resistance Temperature Detectors (RTDs) [18]. The signal resolution is related to the sampling frequency and heavily depends on the sensor manufacturing properties, which do not improve much after being selected and are not worth focusing on. The last issue of uncorrelated noise compared to correlated noise was addressed in previous efforts by Gao et al. [19] using periodic fluid injection. Using this method, the authors showed how manually adding the correlated signals was able to improve the CCF method. The methodology is intended to support monitoring and situational awareness in advanced reactors and integral reactors; however, concerns remain regarding the implementation of periodic injection in a nuclear primary system. The proposed magnitude of injection is not sufficient to affect bulk temperature and is meant to only introduce local perturbations. A small amount of flow could be diverted from one side (i.e., inlet or outlet) of a heat exchanger or steam generator to the other side (i.e., outlet or inlet) to introduce a local temperature perturbation without affecting the inventory of the coolant. The purpose of this work is to demonstrate the efficacy and optimization of injection-enhanced CCF flow estimation. Specific implementation in a reactor system will depend on the particulars of that system. Further, the proposed method could be used at any location with redundant sensors that provide time-delayed signals relative to each other. In fact, it may be preferable to implement this solution in multiple locations (e.g., cold leg and hot leg) to support more complete flow monitoring.

This paper follows up on previous experimental results and focuses on verifying those results and optimizing the injection configuration using Computational Fluid Dynamics (CFD) simulations. The conditions considered in this work and the previous experimental work are not directly relevant to NPPs, but the findings can be extrapolated to such systems. In the experimental flow loop, a series of flow measurements were conducted with different setups for the sensor and injection pipe locations. However, these configurations provide only limited temperature data at specific locations. CFD simulations can provide a larger data set to investigate the sensitivities of the CCF measurement method. To analyze and optimize the performance of CCF flow estimation under different flow rates of the mean flow, it is necessary to collect the temperature and velocity distribution to observe how the injected flow spreads out in the pipe and to determine the best locations for the sensors. This application of CFD aims to simulate the experimental conditions to collect the temperature and velocity in ANSYS Fluent, v19.3.0 [20]. There are two main studies conducted in this paper. The first study aims to verify and validate the ability to simulate the flow (velocity and pressure fields) and heat transfer (temperature fields) behavior of the experiments in the CFD simulation. The second and most important study is the optimization of sensor pair locations in the mean flow based on flow-rate estimation using the CCF method. The optimization study is conducted with a range of flow rates for the mean flow and injected flow. The resulting information from the second CFD simulations will enable a more detailed study of the impact of sensor location on the accuracy of the CCF method. Section 2 briefly introduces the CCF-based flow estimation methodology. The subsequent computational domain section discusses the CFD simulation model built in Fluent, including the geometry, boundary and initial conditions, meshing, and computational setup. Next, is Section 4, which mainly discusses the two main studies: the validation experiment and the optimal temperature sensor location investigation. The last section provides the conclusions.

2. Cross-Correlation Function-Based Flow Estimation

CCF flow estimation relies on small fluctuations in the measured signal due to normal process variation [21]. Temperature measurements are commonly used for CCF flow estimation because temperature sensors are reliable, economical, and widely used in industry, and temperature flows display normal local variation. A general schematic of a piping system where CCF flow estimation can be performed is shown in Figure 1, and examples of relevant measured signals are shown in Figure 2.

The two signals, $x\left(t\right)$ and $y\left(t\right)$, measure a common perturbation with a time delay, $t_d$, between them. This transport time can be estimated by performing the CCF on the two signals, $x\left(t\right)$ and $y\left(t\right)$, as shown in Equation (1):

$$R_{xy}\left(\tau \right)={\displaystyle \frac{1}{T}}{\int }_0^{T}x\left(t\right)y(t+\tau )dt$$

(1)

where $\tau$ is the delay between the two signals. When $R_{xy}\left(\tau \right)$ is at a maximum, $\tau$ is equal to the transport time $t_d$. The target mass flow rate $\dot{m}$ is calculated by the system geometry, the known distance d between the two sensors, and the time delay $t_d$:

$$\dot{m}={\displaystyle \frac{\rho dA}{t_d}}$$

(2)

where $\rho$ is the flow density and A is the cross-sectional area.

The accuracy of CCF-based flow estimation is primarily driven by the magnitude and persistence of the process variations, which may be small relative to the measurement noise. In previous work, periodic injection of coolant to increase the process variation signal was explored in an experimental facility [19,21]. This work focuses on the development of a CFD model to investigate the relationship between the sensors’ locations, the rate of fluid injection, and the accuracy of CCF flow estimation.

3. Computational Methods

This section introduces the computational domain of the CFD simulation model, including the geometry, boundary and initial conditions, meshing, and computational solver framework. These simulations were performed using ANSYS FLUENT and the ANSYS meshing/geometry tools.

3.1. Computational Domain

The computational domain was created in the ANSYS design modeler and is a section of the pipeline where the sensors are implemented. Figure 3 shows the geometry that was simulated in ANSYS Fluent with the flow direction, temperature sensor region, and injection pipe indicated. As shown in Figure 3, the pipe diameter and length are 0.0762 m (3 inches) and 0.5 m (1.97 inches), respectively, based on the dimensions of the experimental configuration with a reduced pipe length. The fluid flows from left to right, past the injection pipe and the two temperature sensors. The injection pipe is located 0.1 m away from the inlet of the domain with a diameter of 0.01 m (0.39 inches) and extends 0.013 m (0.5 inches) from the wall into the main pipe. The injection pipe geometry was created using a cut-out cylinder in the domain and is marked with a white rectangle in Figure 3. The temperature sensors were not explicitly modeled due to their low impact on the flow distribution in the mean flow with and without injection, as well as the small size of the temperature sensors and their position in the wake of the larger injection pipe [22]. By not explicitly modeling temperature sensors, we can acquire temperature traces at any point in the wake of the injected fluid. This speeds up data collection for sensor location optimization and lowers the mesh cell count needed to resolve the sensor itself. Prior experimental work investigated the impact of sensor response time and sampling frequency [19]. The authors suggested approaches to correct for mismatches in sensor response time. Because these implementation issues have been previously addressed, we did not consider focusing on the impact of sensor placement and injection pressure in this study.

3.2. Boundary and Initial Conditions

The boundary conditions for the CFD simulations are shown on the computational domain in Figure 4, including the inlet injection. For the main flow inlet, a fully developed velocity profile was used as the boundary condition at the entrance of the pipe. This fully developed velocity profile for different main flow rates was achieved through a series of precursor steady-state simulations with a longer pipe of the same diameter. A uniform distributed flow was applied at the inlet, and its outlet fully developed velocity profile was collected and exported to the main simulations. Using the fully developed velocity profile at the inlet, allowed us to avoid the entrance effect of the uniform velocity boundary condition. This allowed for shortening the inlet region, which, in turn, reduced the mesh count/computational requirements for each simulation. In total, there were 12 different main flow rates used in the CFD simulations. They were divided into three groups for three different meshing refinement levels. The main inlet boundary conditions for the low-flow-rate meshing model were 39.52 GPM (0.00249 m³/s), 65.13 GPM (0.00411 m³/s), 75.82 GPM (0.00478 m³/s), and 93.53 GPM (0.00590 m³/s). The main inlet boundary conditions for the medium-flow-rate meshing model were 101.95 GPM (0.00643 m³/s), 114.2 GPM (0.00720 m³/s), 123.84 GPM (0.00781 m³/s), and 133.51 GPM (0.00842 m³/s). The main inlet boundary conditions for the high-flow-rate meshing model were 145.83 GPM (0.00920 m³/s), 156.42 GPM (0.00987 m³/s), 167.61 GPM (0.01057 m³/s), and 178.3 GPM (0.01125 m³/s). The main flow Reynolds number ranged from 5.8 × ${10}^4$ to $2.6 \times {10}^5$, so all the cases were with a turbulent flow regime. The temperature of the main flow rate was set to 308 K (∼95 F).

For the injection pipe boundary conditions, water was injected into the main flow periodically with time-dependent velocities and temperatures. The injected temperature and velocity followed a sharply ramped step function at the beginning and end of the injection period, reflecting the actual physical injection function applied in the experimental flow loop. One injection period for the injection pipe boundary condition is shown in Figure 5 for both velocity and temperature. The total injection period was 4.0 s, including 1.0 s for fluid injection and a 3.0 s delay before the next injection. Figure 5 shows an initial three-second delay from the previous injection, followed by injection of the fluid for 1.0 s. In the experimental setup, three different injection flow rates were investigated: 0.920 m/s, 0.821 m/s, and 0.497 m/s. In the simulations, these three injection flow rates were applied separately and compared with each other under different main flow rates. This enabled a comparison of whether the injection flow rate was correlated with the optimal setup for flow estimation under different flow rates. All three injection flow rates used the same injected fluid temperature of 285 K (∼53 F). The injection temperature was 23 K lower than the temperature of the main flow to introduce additional perturbations and increase the signal-to-noise ratio, which benefits CCF flow estimation.

The remaining boundaries (walls) were set as no-slip wall boundary conditions. The computational domain was initialized based on the fully developed inlet condition at the given flow rate and with a constant initial temperature of 308 K.

3.3. Meshing

The meshes used were created in ANSYS ICEM [20], where the pipe was divided into three sub-regions—the inlet region, the injection region, and the temperature collection region—as shown in Figure 6. To mesh the main flow, the meshing was gradually transitioned along the radial direction with three separate sections, as shown in the top-right corner of Figure 6. The radial center of the main pipe has a square cross-section that transitions to a circular mesh, which expands and then contracts as the wall is reached. At the injection pipe, vortex shedding and a wake likely form as the mean flow passes, resulting in careful consideration of the meshing in this region. A detailed image of the meshing near the injection pipe is shown in Figure 7. Since an inserted cylinder represents the injection pipe here, hexahedral meshing was difficult to apply in this irregular geometry. Instead, tetrahedral meshing was applied specifically in this region, as shown in Figure 7. The temperature collection region mesh was created to be sufficiently resolved to accurately represent the temperature response in the sensors by resolving the flow.

3.4. Mesh Sensitivity Study

To ensure an appropriate set of meshes was used, a meshing sensitivity study was conducted to determine how the mesh element sizing and density impact the mean flow. The general process for mesh sensitivity studies is to create multiple meshes with different densities, ranging from coarse to fine. The boundary conditions and initial conditions are held constant for each mesh to ensure a proper comparison. Based on the results using a meaningful quantity of interest, the mesh with sufficient density to accurately resolve the behavior is selected. Generally, mesh comparisons are performed using a baseline comparison against a significantly finer mesh than the rest as a stand-in for an analytical solution.

This study used a total of four meshing models, ranging from the coarsest to the finest, labeled as M1, M2, M3, and M4, respectively. Detailed meshing parameters for these four meshing models are summarized in

Table 1. Mesh sensitivity study.

	M1	M2	M3	M4
Number of circular divisions	40
Number of radial divisions (outer cylinder)	14
Number of radial divisions (medium cylinder)	10
Element size for the inner rectangle (m)	$4\times {10}^{-4}$
Number of divisions for the inner rectangle	10
Inlet region length (m)	0.07
Injection region length (m)	0.06
Temperature collection region length (m)	0.37
Number of divisions in the inlet region in the axial direction	40	50	65	80
Number of divisions in the injection region in the axial direction	28	40	52	60
Number of divisions in temperature collection region in the axial direction	114	150	200	250

. To compare these four meshing models, a timed cooled injection scenario was used, involving two injections with a time delay. The overall transient lasted 10 s with an injection flow velocity of 0.545 m/s at 285 K (∼53 F). The mean flow rate for the study was varied to cover three groups spanning from 39.52 to 178.30 GPM at a set temperature of 308 K (∼95 F). To compare these four meshing models, the area-weighted average temperature and velocity traces at five different cross-sections and the temperature and velocity traces at two local positions were selected, as shown in Figure 8. Figure 8 shows the five cross-sections (indicated by the blue solid regions) and two local points (indicated by the blue stars). Both the temperature and velocity are plotted at different locations, including five locations equally distributed along the flow direction and at the two sensor locations. Figure 9 shows the area-weighted average temperature traces at the three different cross-sections shown in Figure 8. Figure 10 shows the two local temperature traces at the two star-marked locations shown in Figure 8.

Figure 9 shows the area-weighted average temperature of the cross-sections evenly distributed along the flow direction. The three average temperature comparisons exhibit almost identical trends, and the two local temperature comparisons are at 50.8 mm (2 in) and 152.4 mm (6 in) from the injection point. In Figure 10, some variation is observed during the injection period, with a temperature difference of around 0.2–0.4 K. This amount is insignificant in total magnitude when compared with the experimental temperature sensor uncertainty in the experimental flow loop. Comparing the different meshes, the mesh cell count does not significantly affect the temperature distribution along the mean flow. This is confirmed by the investigation of the velocity profiles across the four meshes, which helped in selecting the suitable mesh.

For the velocity comparison, the cross-sections and two local points were also selected, as shown in Figure 11 and Figure 12, respectively. Figure 11 and Figure 12 show that M1 and M2 generated a higher velocity than M3 and M4 do, and the average velocity data from M1 and M2 were not sufficiently resolved when compared with the M3 and M4 meshes. Excluding the injection period, the meshes with a higher cell count were able to better resolve the momentum behavior. Figure 12 shows that the overall local velocity is greater than the average value. The reason for this is that the velocity distribution was not uniform across the cross-section.

One example of the 3D velocity distribution of the fully developed turbulent flow is shown in Figure 13. Figure 13 shows that the velocity is not uniformly distributed across the cross-section. The endpoint of the temperature sensor location is close to the wall, where the local velocity is relatively smaller than the average velocity. Figure 12 supports this conclusion, indicating that the model with a larger mesh count or degrees of freedom exhibits lower velocity, just like the previous average velocity plots. To further examine how the M1, M2, and M3 meshes resolved the velocity compared with the M4 mesh, which has the highest density mesh, the absolute percentage error for both average and local velocities between the M4 mesh and the other three meshes was calculated, as shown in Figure 14 and Figure 15. Both Figure 14 and Figure 15 show differences between M1, M2, and M4, while the absolute percentage error between M3 and M4 is close to zero. This shows that M3 is sufficient to be used due to the minimal velocity differences between M3 and M4. Therefore, M3 was chosen as the base meshing model.

M3 in Table 1 was used as the base mesh to ensure high-quality temperature and velocity simulations, with additional refinement of the wall region mesh. The $u^{+}$ vs. $y^{+}$ plot was used as a reference to judge whether the wall region was simulated correctly. After several simulations, it was found that no single meshing model was sufficient for all the flow-rate cases with suitable $y^{+}$ values. The main reason for this is that the range of the mean flow was quite large, from around 40 GPM up to 170 GPM (approximately 2.5 × ${10}^{-3}$ m³/s to 1.1 × ${10}^{-2}$ m³/s), with Reynolds numbers ranging from 5.7 × ${10}^4$ to 2.5 × ${10}^5$. A single meshing schema might better resolve the flow at lower flow rates but would not work well for higher flow rates. To solve the problem caused by the wide flow-rate range and achieve better accuracy, the flow-rate cases were divided into three subgroups: low flow rate, medium flow rate, and high flow rate. Each group used a unique mesh designed for a specific range of flow rates. Details of the three different meshing models are summarized in

Table 2. Three different meshing models.

	Low-Flow-Rate Group (G1)	Medium-Flow-Rate Group (G2)	High-Flow-Rate Group (G3)
Flow rate (GPM)	39.52∼93.53	101.95∼133.51	145.83∼178.30
First cell height (m)	1.4 × ${10}^{-3}$	6.0 × ${10}^{-4}$	4.3 × ${10}^{-4}$
Nodes	406818	362198	374494
Elements	459890	408016	421176

. The first cell height was slightly different to guarantee that the $y^{+}$ values met the requirements. The $u^{+}$ vs. $y^{+}$ plots for all three groups are shown in Figure 16.

Figure 16 shows scatter plots of the results for the three groups, with different colors representing different flow rates. With the exception of the region near the buffer layer, the $u^{+}$ values follow the curve in most regions, which is expected when using the standard wall treatment. The three different meshing models provide relatively similar simulation performance, and all subsequent results are based on these meshes according to the mean flow rate. Based on this verification (mesh sensitivity) study, we expect that the following CFD simulation results will accurately represent the predicted CCF performance analysis.

3.5. Computational Solver Setup and Fluid Properties

The simulations were run for 5.0 s, with a fixed time step size of 0.001 s. The event scenario, as depicted in Figure 4, included an additional 1.0 s at the initial velocity and temperature values after the injection occurred. Each case was simulated without injection for the first 3.0 s. Water injection then occurred from 3.0 s to 4.0 s, followed by an additional 1.0 s of simulation. The numerical convergence criteria (residuals) were set to ${10}^{-3}$ for continuity, x velocity, y velocity, z velocity, turbulence kinetic energy (k), and dissipation rate ($ϵ$), and ${10}^{-6}$ for energy. The maximum number of iterations for each time step was set to 10, and individual numerical convergence was monitored.

Constant fluid properties were used for the water, with a density of 994.08 kg/m³, a dynamic viscosity of $7.2147 \times {10}^{-4}$ kg/m-s, and a thermal conductivity of 0.6231 W/m-K.

4. Results

The CFD model results were validated with data collected from an experimental flow loop. The loop was described by Gao et al. [19]. The validation comparison between the experiments and CFD simulation included twelve flow-rate cases applied in the experimental flow loop, ranging from 39.52 GPM to 178.30 GPM. The experimental validation was followed by an investigation of the optimal temperature sensor locations under different flow rates.

4.1. Experimental Validation

The CFD simulations require validation against relevant experimental data based on the same boundary and initial conditions. The CFD simulations can show how the temperature distribution of the entire flow region changed after the injection. Figure 17 shows the temperature distribution during the injection of colder liquid under the 178.3 GPM flow rate. The figure shows that the temperature field was perturbed by the injection. The closer the temperature sensor is to the injection pipe, the stronger the signal that can be collected.

The CFD temperature data at the two sensor locations from the experiment flow loop were collected and compared to the corresponding experimental data. Since the absolute values were slightly different, the relative perturbation in the temperature was the main feature to be compared. Therefore, the percentage of perturbation was calculated and used in the comparison based on the equation below:

$$\begin{array}{c}T=T_{raw}-T_{steady}\end{array}$$

(3)

$$\begin{array}{c}\Delta T_{\%}={\displaystyle \frac{T-T_{min}}{T_{max}-T_{min}}}\times 100\%\end{array}$$

(4)

where $T_{raw}$ is the raw temperature recorded in the experimental loop (or in the CFD simulations) and $T_{steady}$ is the pre-injection temperature when there is no fluid injection in the flow region. Equation (4) evaluates the temperature perturbation as a percentage of the maximum range. Figure 18, Figure 19 and Figure 20 show the temperature comparison at the sensor locations under three different flow rates. The temperature data from the CFD model were collected as close to the same locations as possible as those in the experimental flow loop.

The three temperature comparison figures (Figure 18, Figure 19 and Figure 20) show one injection within a 4.0 s range. TC1 is the sensor closest to the injection pipe (∼5 cm). TC2 is ∼15 cm from the injection pipe and ∼10 cm from TC1. Based on these three figures, the CFD simulations accurately simulated the temperature changes at the two sensor locations. The root mean squared error (RMSE) between the CFD simulations and the experimental data was used to quantify the accuracy of the simulations.

Table 3. Root mean squared error between experimental data and CFD simulations.

Flow Rate	TC1	TC2
Low (∼40 GPM, Figure 18)	0.173	0.248
Medium (∼133 GPM, Figure 19)	0.0853	0.202
High (∼170 GPM, Figure 20)	0.128	0.268

gives the RMSE for TC1 and TC2 under each flow condition. The RMSE values range from $0.0853$ to $0.268$, further supporting the strong agreement of the CFD simulations with the experimental data. A few observations can also be made: The temperature data from the experiment are quite noisy. One reason for the smoothness of the CFD results is that the average Navier–Stokes k-$ϵ$ turbulent model was applied, which resulted in small- and large-scale fluctuations unrelated to the injection being averaged out. Another minor reason is that the physical sensor geometry was not modeled in the CFD model, which did not introduce any local turbulence likely caused by passing the sensor’s physical body. Another issue is the timestamp of when the temperature started to change; the figures show that the experimental temperature drop occurred slightly later than the CFD results when the injection happened. The main reason is that the CFD simulations collected the temperature instantaneously (no minimum read time), while the temperature sensor installed in the flow loop had a response time issue, causing the temperature data to delay slightly and exhibiting expected sensor noise. For the lower flow rate, the percentage temperature change was not as large as in the higher-flow-rate cases. One possible reason is that the flow was relatively slow, and the cold injected water had been warmed up before being measured by the temperature sensor. This phenomenon was observed in both the experiments and CFD simulations. It was more obvious for TC2 because TC2 was farther away from the water injection source, resulting in a smaller temperature difference from the main flow. This can also be supported by the temperature distribution shown in Figure 17. One possible solution is to inject a much cooler flow to introduce a relatively more powerful perturbation. Another observation from the comparison figures is that the temperature of TC2 was relatively noisy. The main reason might be that the flow was more perturbed after passing the injection pipe and the first temperature sensor. This was expected because it was further out in the wake of the injected pipe perturbations.

After comparing the temperature between the CFD simulations and experiments, the velocity distribution was examined in the CFD simulations. The flow region after the injection pipe was a concern because the CCF flow estimation inferred the flow based on the temperature sensors installed in this region. Figure 21 shows the velocity distribution after the water injection under a low flow rate (39.52 GPM).

Figure 21 shows the velocity distribution of the cross-section at different locations along the flow direction under the flow rate after 0.5 s of injection. The pipe centerline is marked with a dotted dashed line (–·–), and the two sensor locations are marked with blue points and black arrows indicating the flow direction. Each examined cross-section is marked with a dashed line (–), and for each velocity distribution, the average main flow rate is marked with a dotted line ($···$). The selected cross-sections roughly cover the most important locations in the flow pipe, including the region before the injection pipe, the two exact sensor locations, and some other randomly chosen locations.

Before the injection pipe, the velocity profile has a U-shaped curve that is standard in pipe flow. It has a higher local velocity near the centerline and the highest gradient toward zero near the wall. The injected water disturbed the U-shaped velocity profile, as shown in Figure 21.

After passing the injection pipe, the flow was affected by the physical structure of the pipe, and the flow distribution shows that the upper flow rate sharply decreased and then later recovered. Then, the flow rate passed by the two sensor locations sequentially. The transit time was determined by the flow rate along the path from the first sensor to the second sensor (marked in light blue in Figure 19), but it was hard to examine how the flow rate changed along this path. Therefore, a more detailed flow-rate distribution along this specific light-blue path is shown in Figure 22. The blue scattered dots represent the flow rate discretely collected along the two sensor paths from the CFD simulations for the lowest flow rate. The red dashed line represents the reference average flow rate mentioned previously. The x-axis only covers the horizontal range between the two sensors. It shows that when the flow reached the first sensor, the flow rate was lower since it had not fully recovered from the disturbance caused by being blocked by the injection pipe. As the flow traveled forward, it recovered and returned to the normal flow rate, which is roughly equal to the reference flow rate in this case.

It was quite a different story when the main flow rate was high. Figure 23 and Figure 24 show the same flow-rate distribution plots when the main flow rate was high (178.9 GPM). By comparing the different flow-rate cases, it was found that both suffered from being blocked by the injection pipe. However, the low-flow-rate case exhibited around a 14% decrement, while the high-flow-rate exhibited around a 10% decrement. One possible reason might be that the high flow rate recovers more quickly than the low flow rate. When comparing Figure 22 with Figure 24, another issue was observed where the high-flow-rate case did not return to the average flow rate even after passing the second sensor. Although the flow rate did not decrease as much as in the lower-flow-rate case, the transit time between the two sensors was much shorter since the flow rate increased Therefore, it is reasonable to increase the distance between the sensors to achieve better flow estimation when the flow rate increases. This is one of the motivations for the later section: the optimal temperature sensor location investigation.

After validating the temperature and velocity quantities of interest for the low and high flow rates, the final step was to validate the flow rate estimated by the CCF method from the CFD simulations and compare it to the experimental estimation and the flow meter in the test facility. Figure 25 shows the CCF flow estimation based on the previous experimental results and the CFD simulation model. For the experimental data, a black “x” indicates the averaged CCF flow estimation value among five repetitions, with the error bar representing two standard deviations based on these repetitions. The CFD results are marked with a red “o”. The overall CFD results are lower than the true flow rate. The main reason is that the CFD results measured the local flow rate near the wall, while the true flow rate is the average flow rate marked with a dashed line in Figure 21 and Figure 23. The temperature was collected at the exact location as the endpoint of the two sensors near the wall in the experimental flow loop, which had a smaller local flow rate than the true flow rate. Additionally, Figure 25 shows that the CCF-estimated flow rate in the experiment was much lower than the true value. For the highest flow-rate case, even the two-standard-deviation bar does not reach the one-to-one line. As the main flow rate increased, the local turbulence became stronger. Comparing Figure 22 with Figure 24, the flow with a higher rate had less time to recover from being blocked by the injection pipe in the CFD simulations. It should be noted that the two sensor geometries were not modeled. Therefore, in the experiment, both the injection pipe and sensors may have introduced extra local perturbations. This was not regarded as a good source for the correlated process signal because such local disturbances rapidly changed the local velocity distribution. They disappeared quickly and did not travel along the flow path to be detected by the second sensor. This only happened during high flow rates, mainly because during lower flow rates, such local perturbations did not dominate the correlated process signal, and the flow had more time to recover from being blocked. Potential future work will involve measuring the flow rate at the same locations as the two sensors instead of measuring the averaged mean flow rate. Tighter controls on the flow-rate measurements will be helpful, especially when the flow rate is high.

4.2. Optimal Temperature Sensor Location Investigation

Figure 21 and Figure 23 suggest that the local velocity depends on how far the sensor tip is from the centerline. The CCF flow estimation measured the local velocity at the sensor locations instead of the bulk flow rate. Since the sensor locations were mainly close to the wall, this resulted in an underestimation of the bulk flow. The discrepancy between the CCF flow estimation and the bulk flow rate increased for larger bulk flows. This section aims to use these CFD simulations to answer questions about the impact of sensor location on flow-rate estimation and to identify the optimal sensor locations for different bulk flow rates. This is accomplished through a brute-force search of potential sensor locations that cover flow regions following the injection pipe. Figure 26 shows all the candidate sensor locations along the flow direction in the CFD simulations. For every cross-section along the flow direction, there are seven candidate locations. These locations are 0.01 m away from each other along the radial direction. These seven candidates form a group at each cross-section, and there are 39 groups uniformly distributed along the horizontal flow direction. Thus, there are a total of 273 locations (39 × 7) where the temperature was collected under different flow rates. Two locations were selected for each sensor pair, where the second sensor was restricted to being positioned further from the injection pipe than the first sensor in the flow direction. As a result, there were 36,309 possible sensor pair locations.

After collecting the temperature at each of the candidate sensor locations under a prescribed bulk flow and injection pressure, the CCF was applied to each possible pair to investigate the relationship between the pair locations and the performance of CCF flow estimation. For each main flow rate and injection pressure, the top 10 sensor location pairs, in terms of bulk flow estimation accuracy, were selected and plotted on a temperature distribution plot for visualization. We explored optimal sensor pair locations for low, medium, and high flow rates, as well as low and high injection pressures.

Figure 27 shows the top ten sensor pairs for CCF flow estimation plotted on the temperature contour of the main cross-section after 0.5 s of water injection at 80 psi. The background contour shows the temperature distribution after the colder water had been injected. Since the injected water was colder than the main flow, the major yellow region shown in the plot represents the cool region. With few exceptions, the distance between the two sensors among these top ten pairs did not vary significantly for a given flow rate, although the distance between sensors tended to increase as the bulk flow rate increased (

Table 4. Average distance (m) between two sensors based on the top 10 pairs under different mean flow rates and injection pressures.

		Mean Flow Rate (GPM)
		39.52	123.84	178.30
Injection pressure (psi)	80	0.07	0.12	0.22
	30	0.07	0.20	0.20

). Figure 27 presents the results under the highest injection pressure (with the largest injection flow rate of 0.920 m/s). The highest injection flow rate introduced strong perturbations and diffused over a broader region, especially when the mean flow rate was low (Figure 27a). This is one of the reasons why these top ten pairs were not clearly clustered within the cooler region along the pipe, indicated in yellow in the figure, which had a temperature of approximately 303 K. Additionally, the RANS equation could not simulate the local eddy motions and only modeled this behavior. The temperature fluctuations might have impacted a larger region due to diffusion and convection. Therefore, some sensor pairs outside the investigated temperature region might produce better CCF flow estimation performance.

To further understand the influence of fluid injection on CCF flow estimation performance, the same analysis was performed for injections at a low pressure of 30 psi. Figure 28 shows the results under the same bulk flow-rate conditions with the reduced injection pressure. Comparing Figure 27 and Figure 28, it was observed that the area of the temperature perturbation region depends on the injection pressure. Figure 27 shows that the temperature perturbation region is generally larger when the injection pressure is high. For the high injection pressure (80 psi, Figure 27), the top ten sensor pairs are mainly located within the temperature fluctuation region regardless of the bulk flow rate, while the top ten sensor pair locations are more dispersed under the low injection pressure (30 psi, Figure 28). The higher injection pressure of 80 psi is generally preferable because the location of sensors within the temperature perturbation region ensures a higher signal-to-noise ratio. The time delay calculated within this region was expected to be more stable and accurate. The notable exception to this was the low-flow-rate condition (39.52 GPM). The results with a high injection pressure (80 psi, Figure 27a) show a greater spread of optimal sensor pair locations than those with a low injection pressure (30 psi injection). This suggests that there may be a relationship between the bulk flow rate to be estimated and the optimal fluid injection pressure. This is an area for future research.

Figure 27 and Figure 28 only include the top ten performing pairs, which may obscure the more persistent relationship between the horizontal distance from the injection pipe to the sensors and the CCF flow estimation performance. One main observation from Table 4 is that the top ten pairs have a larger distance between the two sensors when the main flow rate is high. Given the mean flow rate, a larger distance results in a longer delay time between the two sensors. The delay time should be large enough to be detected and used for inferring the CCF flow estimation. Detailed examinations of the correlation between the two distances are conducted below.

Instead of selecting the top ten pairs, the 1000 smallest RMSE sensor pairs (out of over 36k pairs) were selected for each bulk flow rate and injection pressure combination. For each sensor pair, only the flow direction coordinate (regarded as x-direction) was considered. All 1000 pairs of the first sensor and second sensor x-locations were plotted in a 3D histogram, weighted by the inverse of the RMSE.

$$\begin{array}{c}{n}_{x_1,x_2,w}={\displaystyle \frac{1}{RMS{E}_{x_1,x_2}}}{n}_{x_1,x_2}\end{array}$$

(5)

where $x_1$ and $x_2$ represent the distances from the injection pipe of the first and second sensors; $n_{x_1,x_2}$ is the frequency of the $x_1$ and $x_2$ sensor locations among the top 1000 pairs; and $RMS{E}_{x_1,x_2}$ is the RMSE found from sensors at locations $x_1$ and $x_2$. Thus, $n_{x_1,x_2,w}$ will be large if $RMS{E}_{x_1,x_2}$ is small and/or if the $x_1$/$x_2$ location pair frequently appears in the top 1000 pairs. The same $x_1$/$x_2$ location pair may be selected multiple times because multiple y-axis locations are available. Figure 29 shows the weighted optimal locations for the top 1000 sensor pair locations for low (39.52 GPM, Figure 29a), medium (101.95 GPM, Figure 29b), and high (178.30 GPM, Figure 29c) flow conditions. In all cases, a high-pressure injection (80 psi) was used. A low-pressure injection (30 psi) showed similar trends but generally lower overall $n_{x_1,x_2,w}$ values due to typically higher RMSE values. Within each bulk flow rate, we observed strong linear trends, indicating preferred distances between the two sensors ($x_2-x_1$), but not a strong preference for the location of the sensors relative to the injection pipe. We also observed that there was no common “best” location for all three flow rates considered, which suggests that the sensor position should be strongly informed by the expected flow rate(s) in a particular system. Systems with expected flow rates that cover a wide range of values may benefit from an array of sensors that can be queried for CCF flow estimation as needed. Development of this sensor array approach and real-time flow estimation under variable flow rates is left to future work.

5. Conclusions

This paper developed a CFD simulation model to verify experimental data and investigate the impact of sensor location on CCF flow estimation. In the experimental loop, only a limited number of sensor position options were available. The CFD model offers greater flexibility in selecting the optimal locations for a sensor pair. The results show that sensors should be placed where the local temperature exhibits the greatest perturbation induced by the injected water; however, the location of this region of greatest perturbation is highly dependent on the bulk flow rate and the injection pressure used to introduce process variations. Another finding is that the optimal distance between the sensors depends on the flow velocity. When the flow velocity is higher, the distance between the two sensors should be larger so that the transport time between the two sensors is long enough to be accurately inferred by cross-correlation analysis.

Further study may be able to establish and demonstrate a relationship between the expected flow rate, sensor location, and injection pressure. Because the optimal injection pressure and sensor location are strongly dependent on the bulk flow rate, it would be challenging to employ this technique with a simple set of two temperature sensors. An array of sensors with variable injection pressure capability may provide the flexibility necessary to accurately estimate the flow rate under a variety of conditions. The design of such a sensor array system is an area for future work.