1. Introduction
Reliable knowledge of temperature fields in fluid flows is extremely important to monitor and better understand heat transport in flow processes. Temperature measurements also provide indirect information about the flow structures, flow regimes, and operating conditions in experimental setups or industrial facilities. The motivation for the present study arose from investigations of turbulent thermal convection in liquid metals and the interest in how the resulting flow structures influence the heat transport in the system [
1,
2,
3,
4,
5]. Both numerical and experimental works have shown that detailed temperature data in the bulk of the fluid provide crucial information about local heat transport, leading to a more accurate and advanced understanding of the studied system [
6]. Having access to a temperature map within a layer of turbulent fluid is very attractive, as it allows one to characterize the thermal structures and track their dynamics. Existing experiments for presenting internal thermal structures rely on measurement techniques that are either intrusive or based on optical images, such as shadowgraphy or, more recently, the use of thermochromic crystals [
7,
8,
9]. However, for some applications, typically with opaque fluids such as liquid metals, optical access is limited or impossible. Optical-fiber-based sensors provide an interesting alternative for this problem. Optical fibers are small, thermally stable, and do not carry electrical current; they are therefore particularly attractive for liquid metal applications, which often involve high temperatures, strong magnetic fields, and confined environments. Various methods to obtain temperature information from optical fibers have been developed. Among them, Fiber Bragg Gratings (FBGs) are a versatile solution, since they enable the realization of a large number of sensing points in a small space [
10]. Previous studies have shown that FBG sensors can successfully operate in a very large range of temperatures, from cryogenic conditions [
11] to very high temperatures [
12]. Temperature measurement based on FBGs has been used in many applications, taking advantage of their very fast response time compared to more widely used electric sensors, such as thermocouples or resistance thermometers [
13], as well as their high multiplexing potential [
14], allowing one to have access to an increased number of measuring points with a more compact system. The fast response time also reduces the overall acquisition time, hence allowing higher acquisition rates. For in-bulk measurements in the presence of flows, optical fibers’ small size (starting from around 100 μm in diameter) is particularly attractive. In our projected application of the thermal mapping of a fluid layer, the high multiplexing potential (up to 50 sensing points in one fiber) is also an advantage, allowing one to significantly reduce the logistical requirements of the experiment.
In previous studies, the temperature was mostly measured at only a few selected positions inside the fluid or by means of ring-shaped arrangements of up to 16 sensors on the inner side wall of the convection cell (see, for example, [
15,
16,
17]). In this way, estimations of temperature gradients or the rough reconstruction of large-scale flow structures were possible, but the resolution and dynamics of two-dimensional thermal structures could not be reconstructed from these data. The use of the aforementioned advantages of FBGs for mapping the temperature in liquid metal convection experiments in a laboratory setup and the availability of highly resolved temperature distributions inside the convection cell could be a game changer in this research field. However, the presence of the temperature sensors in the fluid volume must not significantly disturb the flow to be investigated. Optical fibers are thin and therefore very attractive for this purpose, but access with FBG sensors has a decisive disadvantage. In addition to temperature changes, these sensors are very sensitive to mechanical deformations due to tension and shape. The local flow on the fiber can also generate mechanical deformation and thus a significant signal component, which is very difficult to distinguish from the contribution of the temperature. Specific measures must therefore be taken to ensure an unimpeded determination of the temperature.
A common measure is to stiffen the fibers by coating or guiding them in a rigid housing like cladding tubes. For temperature measurements, various type of coated fibers have been developed, allowing one to increase the sensitivity and measuring range [
18,
19,
20,
21,
22]. Such coatings slightly increase the fiber’s diameter, typically from 50–100 μm to a few hundred micrometers. Furthermore, they are usually encased in a rigid housing (e.g., stainless steel tubing, typically >1 mm in diameter) to filter out strain effects from the flow [
23,
24,
25]. Since this is inevitably associated with a noticeable increase in the fiber diameter, we first looked for a possible solution using coated fibers without housing. We wished to pursue the approach of placing the fibers under a pre-tension that made them immobile so that a flow no longer caused any additional significant deformation. This paper aimed to investigate whether reliable and reproducible temperature measurements in the bulk of the fluid could be achieved by fixing the fibers under pre-tension. This procedure of the direct mounting of the optical fibers between the solid side walls for in-bulk measurements has, to our knowledge, not yet been investigated (see Yi et al. [
26] for measurements with a fiber mounted at the surface of the solid wall. In this paper, we present a systematic study of temperature measurements with FBGs in a dedicated experimental setup with quiescent fluids. Each FBG sensor was paired with a standard thermocouple whose measured values were used for validation. On the basis of the results obtained, we discuss to what extent the proposed procedure is suitable for practical applications with regard to temperature measurements in liquid flows.
2. Description of the Problem
Fiber Bragg Grating (FBG)-based sensors use the photo-sensitivity of the core of an optical fiber to indirectly infer information about physical quantities in the environment at each sensing point (see reviews in, e.g., Othonos [
27], Sahota et al. [
10]). By their very nature, FBG measurements are based on the deformation of the fiber, which is caused by either temperature changes or force effects. Therefore, it is crucial to exclude the measurements from effects that are not of interest. For temperature measurements, external encapsulation is usually used to protect the fiber from strain-driven deformation effects.
The present study was motivated by our investigations into thermal liquid metal convection (e.g., [
4,
5]). The direct measurement of the temperature field across a designated plane within the fluid volume would be an important step towards a better understanding of the heat transport through the large-scale flow and would provide important experimental data for the validation of numerical simulations. This goal could be achieved by spanning a network of optical fibers with a large number of FBG measurement points in the fluid. These fibers should be very thin so that the flow to be measured remains unaffected by the sensors. The sheathing or coating of the fibers would increase both their diameter and the flow resistance of the sensor network. Thus, there would be a risk that the sensors could distort the flow. Conversely, the thermally driven flow, whose velocities in turbulent liquid metal convection are typically a few centimeters per second [
4], would also affect the measured values by adding strain on the sensors. For this reason, the mounting of the fiber is of central importance in order to avoid distortions of the temperature measurements by the additional deformation, displacement, and vibration of the fiber caused by the fluid flow [
28]. We decided on an approach in which the fiber is subjected to tensile force of a defined intensity and is firmly glued in place. This kind of fiber mounting should ensure that the measuring positions are fixed and that the temperature measurements are not affected by the flow-induced strain.
To specify the strength of the tensile force to which the fiber is exposed, we present some theoretical considerations in order to estimate to what extent the fiber diameter influences disturbances of the temperature measurements in a flow. Optical fibers are cylindrical obstacles to the flow, and the corresponding effect can be quantified by means of the Reynolds number:
where
U is the flow velocity,
D is the fiber diameter, and
is the kinematic viscosity of the fluid.
For
, the wake created by the fiber is stable, local, and short-lived. For
, the wake generated is unstable, long-lived, and can propagate in the fluid [
29].
Table 1 lists
estimates for selected fluids of interest for laboratory experiments: air, water, and the liquid metal eutectic alloy GaInSn [
30,
31].
is calculated at the highest expected flow velocity of 10 cm/s and two different diameters, corresponding to a coated non-encapsulated fiber (
mm) and the smallest feasible encapsulation (
mm). For an encased fiber,
is above the stability threshold for both water and GaInSn, whereas the flow around a non-encapsulated fiber remains in a region in which only small disturbances occur for water and slightly larger ones for GaInSn. This confirms that noticeably fewer flow disturbances can be expected for both fluids considered here when coated, non-encapsulated fibers are used.
The Reynolds number can be used to compute the drag force
on the fiber through the drag coefficient
:
where
is the density of the fluid.
Empirical data [
32] (p. 17) show that
varies steeply for our expected range of parameters,
. The corresponding values of
for encased and coated fibers are given in
Table 1. By implementing a force balance in Equation (
2), the drag force can be used to estimate the tension that needs to be applied to the fiber during assembly to compensate for the flow effect. This can then be converted to a minimum equivalent mass
:
with
g, the value of the gravitational acceleration, taken as 9.81 m/s
2.
Estimations of the equivalent mass needed to keep the fiber immobile in the presence of a flow of 3 cm/s are given in
Table 1, providing a guide for the experimental assembly procedure.
3. Experimental Setup
To determine whether accurate temperature measurements are feasible with a non-encapsulated FBG fiber mounted under tension, a benchmark comparison was required. For this purpose, we built a dedicated table-top experiment where a standard thermocouple was installed at each position of an FBG sensor. The temperature readings from the FBG were compared point by point with the readings from the thermocouples.
The experimental setup consisted of a 24 cm high column with a square cross-section of 6 × 6 cm2. The column was made of 5 mm thick polymethyl methacrylate (PMMA) on all sides, fixed with screws to a 1 cm thick copper plate at the bottom. The top cover made of PMMA was glued to the side walls and was used to hold the optical fiber. The top of the column also had a 1 cm diameter hole, used to insert a heating element, allowing for local heating and response time testing. For the heating, we used a 8 W electrical heater (Weller ref. WP-80, Weller, Berlin, Germany).
The top and bottom of the column were drilled with 0.5 mm diameter holes for the fiber. The fiber was coated with Ormocer
® and contained 24 FBGs over a length of 23 cm, with a distance of 1 cm between the individual grids, spanning from the first measuring point at
to
cm. The fiber was mounted in the middle of the column with rigid epoxy glue (Weicon ref. Easy-Mix0S50, Weicon, Muenster, Germany). To ensure the required tension, one end of the fiber was first glued to the top of the column and left to cure. When the glue had fully solidified, a weight of 50 grams was attached at the bottom end of the fiber with a piece of tape, and the fiber was glued to the bottom plate. The weight was removed once the glue was fully solid. The attached weight was well above the minimum equivalent mass
needed to balance the expected drag force, as estimated in Equation (
3). During gluing, special attention was paid to ensure correct alignment between the FBGs and the thermocouple holes on the side wall. The setup, including all components, is shown in
Figure 1.
The thermocouples were inserted from the side wall and positioned so that the tip of each sensor was as close to the fiber as possible; see insert in
Figure 1. Thermocouples of type K were glued in place with flexible silicon (Weicon ref. Flex 310M). In total, there were 23 thermocouples in the side wall, each corresponding to an individual FBG, the last FBG being in the bottom plate.
The fiber was connected to a broadband spectrometer (FiSens ref. FiSpec FBGX152, FiSens, Braunschweig, Germany) with an emitting spectrum ranging from 795 nm to 885 nm. The FBGs were each assigned a reference wavelength by the manufacturer, from 822.5 nm for the first one at
to 880 nm, and spaced every 2.5 nm, ensuring that the spectrometer spectrum encompassed the FBGs’ reflections. The optical fiber’s temperature variation coefficient was
K
−1, according to the manufacturer. An example of the reflected spectrum is shown in
Figure 2. The wavelength data were read and converted into temperature with the proprietary software of Fisens (FiSens ref. BraggSens v. 1.81). The reflected wavelength for each FBG was extracted from the reflected light signal using Gaussian peak detection [
33]. We assigned fitting windows of 2.5 nm centered on the reference wavelength of the FBG and obtained the reflected wavelength from the fit results. The difference between the reference and measured wavelength allowed us to deduct the temperature difference thanks to the temperature coefficient; in our case, the displacement was typically below 0.1 nm.
To ensure the high precision of the thermocouples, they were connected to an external temperature reference (Klasmeier ref. Isotech TRU 937/50, Klasmeier, Fulda, Germany), which, along with custom calibration, ensured a precision of 0.03 K. Thermocouple measurements were read by a digital multimeter (Keithley ref. DAQ6510, Keithley, Cleveland, OH, USA) through multiplexer cards (Keithley ref. 7710) with an acquisition time of 1 s. Both types of temperature data, from the FBGs and thermocouples, were post-processed and compared using a Python script. An overview of the data acquisition process is shown in the flowchart in
Figure 3.
5. Discussion
In this study, we considered whether temperature measurements using FBG fibers in a flowing fluid are feasible. An essential question was whether it was possible to design the measurement system in such a way that the temperature data were not distorted by local deformations of the fiber due to transient flow effects. FBG sensors are inherently sensitive to deformation, and the influence of flow-driven mechanical forces needed to be taken into account for acquiring reliable temperature data. We pursued the idea of applying a defined pre-tension to the fiber, which stiffened it and thus prevented flow-induced deformation. This approach was tested for its suitability in a dedicated experimental setup where stagnant fluids such as the ternary metal alloy GaInSn and water were exposed to a temperature gradient and local heating, respectively. The results provided by the FBG sensors were verified by the temperature data obtained using standard thermocouples, which were installed in the immediate vicinity of the FBG positions. We used an optical fiber coated with Ormocer®, mounted directly without encapsulation, in order to achieve the thinnest possible cross-section of the fiber (∼200 μm), thus limiting the disturbances to the flow due to the presence of the sensor to a minimum. The assumption that omitting a sheath from the fiber would noticeably reduce the transient character of the wake and thus the changes in the flow to be measured was supported by theoretical predictions. Estimations of the drag force expected on the fiber allowed us to define a protocol for mounting the fiber so that it stayed immobile in the presence of an acting flow. In practice, the fiber was glued under a pre-tension that was significantly larger than the drag force exerted by the expected fluid flows.
We suggested a post-processing method based on ad hoc strain correction by taking one FBG as the strain reference and using its readings to correct the measurements from the other FBGs located within the same fiber. This method is considered to be straight-forward and has low effort and logistics requirements for implementation, as it employs one FBG sensor for correction that cannot be used for temperature measurements without adding specialized sensors. This approach was tested for reliability using two example cases, a stable thermal gradient in liquid metal (GaInSn) and a local time-dependent heat source in water. We demonstrated that in both cases, the optical fiber sensors were able to measure and follow temperature variations in both time and space, in good agreement with the benchmark provided by thermocouples. The measurement of absolute temperature values could be achieved by adding a stable reliable temperature sensor (e.g., thermocouple) near the FBG strain reference position. Due to the small size of the optical fiber sensors, their response time was faster than that of the benchmark sensors.
However, the agreement between the temperature values of the FBG sensors and the measurements of the thermocouples was only satisfactory over approximately half the length of the examined fiber, starting from the reference point for the tensile stress measurement. This discrepancy apparently increased with an increasing distance from the reference point. This is a clear indication that the proposed method cannot be used for arbitrarily long fibers and only for a finite number of neighboring FBGs.
We assume that this problem can be explained as follows. If the fiber is firmly glued between the side walls of the fluid container under pre-tension, not only is tension imposed, but the length of the fiber is also fixed. If this fiber is now placed in a temperature gradient, different thermal expansions occur along the fiber. If the fluid container is cooled at the bottom as in the first experiment, the FBGs measure not only the local thermal expansion, but also an additional tensile stress that results from the entire fiber contracting due to cooling. In our case, however, only the tensile stress determined by the upper reference sensor was compensated for, which was located at the warmest location. This effect was not sufficiently taken into account in the design of the measurement system. Remedial action can be taken through the following measures: (i) The fiber is placed under pre-tension, but only fixed to one wall. In this way, one would retain a pre-tension that avoids the falsification of the measured value by the flow and allow an additional change in the length of the fiber due to thermal expansion. However, this variant would be difficult to implement in practice, as the side on which the fiber is free to move would be difficult to seal against fluid leakage. (ii) Another possibility is to use more FBGs as reference sensors for a tension measurement. However, this would mean losing temperature measurement points. It may be possible to compensate for this by laying two fibers directly next to each other and measuring temperature and tensile stress alternately. (iii) A slight influence on the flow is accepted by encasing the fiber in a sheath. This approach should work best for small Rayleigh numbers and low flow velocities.
In summary, for the characterization and tracking of thermal structures where relative variations are paramount, a coated, non-encapsulated optical fiber mounted under tension with multiple FBG positions is a promising solution as it offers a fast response time and high sensitivity while providing a sensor with a very small diameter and high multiplexing capabilities for minimal flow disturbances. However, it must be taken into account that the effective compensation of the influence of the tensile stress is necessary to obtain accurate temperature data.
For future applications, careful considerations need to be made for each specific experimental system and problem of interest, as a compromise between the acceptable disturbance to the flow and the precision of temperature data has to be reached. Typically, for experiments where temperature measurement with high accuracy is targeted, an encased fiber is preferable. This still retains many of the advantages of optical fiber technology, such as multiplexing, passiveness, and a large temperature range. For turbulent thermal convection experiments, large-scale mapping in the bulk may be more advantageous with an encased fiber, provided that the housing is kept at a minimal size, since the absolute temperature can provide insightful information about heat transport. For local in-bulk measurements, typically focusing on structures like plumes and boundary layers, where the integrity of the structures is crucial, a coated fiber without encapsulation, as presented in this study, offers an interesting alternative.