Introduction

Polymeric hollow fiber heat exchangers (PHFHEs) comprise hundreds or thousands of small polymeric tubes. These tubes' outer diameters are typically in the 0.8–1.6 mm range. Because of their small characteristic dimension, the flow regime is laminar and therefore, the internal Nu is constant, and the internal pressure drop increases linearly as the flow rate increases [1]. Due to a large heat exchange surface in a compact volume, these heat exchangers achieve favorable ratio of performance and mass (or size). Most PHFHEs are of the shell and tube type, but Krasny et al. [2] presented a rectangular PHFHE for the air–liquid application. The study targets the rectangular PHFHE as an alternative to automotive radiators. It was found that these devices can achieve thermal performance comparable to conventional metal heat exchangers (HEs) and can be used as a replacement for metal finned tube HEs in the automotive industry. Kroulikova et al. [3] presented a full-scale automotive radiator sized to match the original finned aluminum radiator. The PHFHE alternative demonstrated about 20% higher thermal performance for the same airflow in comparison with the aluminum HE. The polymeric radiator featured higher air pressure loss, but both radiators had similar pressure loss on the liquid side. The experimental work was later accompanied by successful driving tests with PHFHE implemented as a front radiator in an actual vehicle.

PHFHEs can also be successfully used in condensation applications [4]. Since the polymer surface tends to be hydrophobic, drop-wise (rather than film-wise) condensation improves heat transfer [5]. The smooth surface of polymers provides an advantage in the form of a reduced fouling tendency. The natural vibration of flexible fibers helps defouling and prevents large deposits [6]. In combination with condensation, an effective self-cleaning device can be achieved [7].

Predicting the pressure drop and heat transfer performance is crucial to designing the rectangular PHFHE efficiently. Both parameters can be determined using well-established formulas [8,9,10,11]. However, these models are based on measurements with relatively large rigid tubes and do not correlate well with airside pressure drop measured at PHFHE, as reported in [12]. Flexible HEs interact with the flow through intrinsic high-amplitude vibrations, which affect heat transfer and pressure conditions. Four types of flow-induced vibration (FIV) mechanisms are recognized in the crossflow over a bank of tubes: turbulent buffeting, acoustic resonance, vortex-shedding and fluid-elastic instability [13]. The topic of FIV has been of interest to many researchers over decades, mainly to ensure the structural stability of the heat exchanger [14,15,16,17,18]. Considerably fewer studies focused on FIV of HEs from the aspect of heat transfer. A study by Akhtar et al. [19] suggested that FIV can also lead to reduced heat transfer efficiency besides the risk of mechanical failures. The natural convection heat transfer coefficient of a vibrating flat plane was experimentally studied in [20]. Characteristics of a crossflow through flexible Teflon tubes HE were investigated in [21]. Authors reported that a HE made of flexible tubes with a 4.5 mm outer diameter had a higher pressure drop and outside heat transfer coefficient. A numerical study [22] used a quasi-static method, where the first tube in the row was placed in 20 various asymmetrical positions. Averaged pressure drop and heat transfer coefficient were higher than in a symmetrical rigid tube case. However, it is questionable how much the results are relevant to the vibration of tubes in an actual HE. Authors of the study [23] reported an increased drag coefficient of a flexible cylinder (1.6–2.4) in a crossflow (Re of order 105) in comparison with a rigid one (1.2). The effect of prestressing a long cylinder in a crossflow was investigated in the study [24], where values such as amplitude, lift force and frequency spectrum were monitored. Recent studies [25, 26] focused on the forced vibration effect on heat transfer and pressure drop in a double heat pipe and an aluminum car radiator, respectively. An increase in both measured characteristics was reported.

This study aims to find if FIV affects the drop in airside pressure of PHFHE. This is done by tensioning the heat exchangers. When the fibers in HE are tensioned, the frequency and amplitude of the fiber vibrations changes. This study uses an in-house developed and manufactured experimental stand capable of controlling the tension of the polymeric fibers that form the core of a crossflow polymeric heat exchanger. Two specifically designed heat exchangers were tested in the calorimetric tunnel with pressure drop and heat transfer performance measured with various values of the tension force applied on the fibers.

The pressure drop in the tube bundle in crossflow

When a fluid flows across a bank of tubes, it experiences resistance due to the presence of the tubes. This resistance leads to a pressure drop as the fluid encounters and flows around the tubes. The pressure drop is influenced by several factors, including the geometry of the tubes (such as their diameter, shape and spacing), the fluid properties (such as viscosity and density) and the flow conditions (such as velocity and Reynolds number). The geometry of a bank of tubes in crossflow can be categorized based on the arrangement and orientation of the tubes. The two most common tube arrangements are staggered and inline, as shown in Fig. 1. The longitudinal, transversal and diagonal tube pitches, together with the diameter of the tubes, characterize the geometry of the tube bundle.

Fig. 1
figure 1

Inline (left) and staggered (right) tube arrangements with longitudinal (\({S}_{{\text{L}}}\)), transversal (\({S}_{{\text{T}}}\)) and diagonal (\({S}_{{\text{D}}}\)) pitches

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The study of pressure drop began in the early twentieth century, with notable contributions from Grimison [9] and Jakob [8]. However, both models faced criticism and are no longer used today. Kays and London later enhanced Grimison's model [27]. Zhukauskas [10] conducted extensive work, resulting in a model widely referenced in various handbooks [28, 29]. Unfortunately, both models only provide pressure drop coefficients in tables or graphs for commonly used pitches (1.25, 1.5, 2.0, 2.5).

Additionally, Zhukauskas' model lacks data for typical Reynolds numbers of PHFHE. The VDI-Heat Atlas [30] contains a model based primarily on Gnieliski, which does not suffer from these limitations. The pressure drop is computed using the coefficients, which differ for the inline and staggered arrangement. They are given explicitly as a function of the arrangement of non-dimensional pitches and Reynolds number, see [30]. In recent years, microtubes have become more popular, so models for tubes with small diameters have started to appear. It is mentioned in [31] that most of the experiments for the previously mentioned models were measured before 1980. Therefore, it can be assumed that the pipe diameters used were somewhat higher. The study also says that for microtubes, those equations overestimate the value of pressure drop. In order to determine the new model, they used CFD simulations for a bundle of pipes in an offset design. The diameter of the tubes ranged from 0.5 to 2 mm. The number of rows ranged from 2 to 40, the spacing from 1.2 mm to 4.0 and air velocity from 0.5 to 7.0 ms−1.

A study [32] reported an overestimation of the pressure drop for small-diameter tubes by the Zhukauskas model study. Pressure drops were measured on a heat exchanger with 484 tubes with an outer 0.8 mm diameter. This study shows lower pressure drops than those calculated by the model, namely by more than 15%. The authors attribute the difference between the measured values and the model to the small diameter not covered by the Zhukauskas model. From years of experience in the authors' laboratory, several measurements of the pressure drop of PHFHE were gathered. Those results were compared to the VDI-HA [30] and Bacelar's [31] models. These comparisons can be seen in Fig. 2. The figure shows that models and measurements in some cases differ by over 20% (dashed lines). Generally, the VDI-HA model tends to underestimate, and the Bacelar model, on the other hand, overestimates the pressure drop compared to the experimentally measured values. The discrepancy between the model and measurements is even more significant for lower Reynolds numbers.

Fig. 2
figure 2

Comparison of measured airside pressure drop of PHFHEs with theoretical models VDI-HA and Bacelar (right) with a detail on the lower pressure values showing even higher inconsistency between standard models and measured data (left). A relative error of 20% between measured and modeled data is displayed as dashed lines

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Description of experiment

Heat exchangers

Two staggered PHFHEs were made for the measurement. The PHFHEs were made with high precision to minimize the effect of randomness in fibers arrangement. Therefore, holes in a fiberglass laminate G10 tube sheet were drilled using CNC technology. The fibers extrusion from polyamide 612 (PA 612) was controlled so that the outer diameter did not exceed 105% of the nominal diameter. Then, the fibers were placed one by one in tube sheets and fixed with an epoxy resin. The two HEs differ only in the diagonal pitch. The parameters of HEs can be found in Table 1.

Table 1 Parameters of tested HEs RG025 and RG026
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Tensioning device

The heat exchangers were tensioned with the device, as shown in Fig. 3. A manual hydraulic pump was used to lift the hydraulic pistons at the sides of the device. Below each piston was a strain gauge force transducer. The force was monitored at a frequency of 10 Hz. Before measuring each set of air velocities, the exchanger was stretched with a given force. Due to the stress relaxation in the fibers and the lateral force caused by the airflow, the force fluctuated during the experiment. However, the individual measurements were taken at time intervals of 10 min, at which the force did not change by more than 3%.

Fig. 3
figure 3

a Tensioning device with PHFHE and schemes b of the tension device during experiments and c of the calorimetric test facility

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It was also necessary to prepare and plan the measurements to determine the forces that would be applied to the exchangers. The force of 9000 N was set as a maximal tensile force value in order to not exceed the yield stress limit and to prevent plastic displacement of fibers. The inaccuracy of the fiber extrusion is ± 5% for the inner and outer fiber diameter. Therefore, during the calculation, the worst-case scenario was considered (inner diameter larger by 5%, outer diameter smaller by 5%). During experiments, five values of the tension forces were applied on HEs, and the case with no load was applied. Table 2 shows tension force and tensile stress values calculated for one fiber, assuming that the outer and inner diameters are nominal.

Table 2 Applied tension loading parameters
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Calorimetric test facility

Pressure drop and heat transfer rate measurements were carried out at the calorimetric test facility at Hanon Systems Autopal s.r.o. The scheme of the experimental setup is displayed in Fig. 3c. The calorimeter is equipped to measure automotive radiators' heat performance and pressure drop. For this reason, a mixture of 50% water and 50% ethylene glycol (coolant) is used as the medium in the fibers. The temperatures of the coolant inlet (Th,i) and outlet (Th,o), the flow rate and the pressure drop (ph,i, ph,o) are measured. The airside temperature is measured at the inlet (Tc,i) and outlet (Tc,o) of the heat exchanger, and the airflow rate and pressure drop (pc,i, pc,o) are monitored. The air is preheated to the initial temperature by an electric heater, and the fan controls air velocity. The inlet and outlet temperatures of both air and the coolant, as well as the pressure drops and flow rates, are monitored at a frequency of 1 Hz. The mass flow rates of both media, air velocity and the heat transfer rate are then calculated by the software included in the calorimeter based on the volumetric flow rates and continuously monitored barometric pressure and humidity. According to automotive standards, the calorimeter has an error of 5% in measuring heat transfer rate and pressure drop. The calorimeter parameters set for the experiment are in Table 3. The air velocity range was set according to the conventional testing range used for automotive car radiators at the testing facility. Reynolds number is defined for the flow across the bank of tubes according to [29] as

$${\text{Re}}_{{{\text{D}},\max }} = \frac{{V_{\max } D}}{\nu },$$
(1)

where \({V}_{{\text{max}}}\) is the average air velocity in the minimal flow cross section, \(D\) is the fiber diameter, and \(\nu\) is the air kinematic viscosity, respectively. The \({{\text{Re}}}_{{\text{D}},{\text{max}}}\) was in the range of 240–970 during experiments.

Table 3 Setting of calorimeter tunnel parameters for the experiment
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Results and discussion

Effect on heat transfer rate

Table 4 contains the measured heat transfer rate dependent on the stretching force and air velocity for the heat exchanger RG025 and RG026. The heat transfer of the RG025 increased by 11% on average for a stretching force of 9000 N compared to the unstretched heat exchanger (0 N). However, the increase in heat transfer rate as a function of strain is not linear. For a significant increase in performance, it would be sufficient to stretch the heat exchanger only to a maximum force of 3000 N, after which the increase in performance is relatively small. Similarly, the RG026 heat exchanger shows a trend of increasing heat transfer rate with increase in stretching of the fibers in the exchanger. Again, the most significant increase in thermal performance is up to 3000 N.

Table 4 Heat transfer rate for each combination of air velocity and stretching force for heat exchangers RG025 and RG026
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Figure 4 shows the results of the heat transfer rate as a function of the tension force and the air velocity for both heat exchangers. The cross-sectional flow area increased as the HE was stretched and the fibers were elongated. The air velocity in front of the heat exchanger was adjusted for each measurement point so that a higher stretching force resulted in a higher mass flow. Therefore, the plots in Fig. 4 have air mass flow on the x-axis instead of air velocity. Both HEs achieved very similar thermal performance. The different diagonal pitches of HEs do not significantly affect the heat transfer rate.

Fig. 4
figure 4

Heat transfer rate of RG025 and RG026 dependent on air mass flow for HE tension forces in range 0–9000 N

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The difference in heat transfer rate occurs because tension makes a fiber stiffer and less prone to get away from the main flow direction and create a bypass. The formation of bypasses leads to a lower heat transfer rate. This difference is primarily noticeable at lower tension force values (up to 3000 N). As the fibers get stiff enough to remain in their original position, the amplitude and frequency change with increase in tension, but this effect is practically negligible. Here, it is necessary to point out a difference with the study [22], where several irregular fibers positions were analyzed, and a heat transfer rate increase was reported. The first difference is that the inline structure was considered, whereas both tested HEs had staggered structures. The second difference was that the asymmetry was modeled artificially and only for the first row, which is substantially different from the natural formation of bypasses due to the flexibility of fibers.

Effect on pressure drop

Figure 5 shows the measured pressure drop as a function of the air velocity. The quadratic dependence of pressure drop on air velocity can be seen. This is valid for all applied values of the tension force. Comparing HEs to each other shows that the RG026 has an overall lower pressure drop than RG025. This concurs with the conventional prediction models [8,9,10,11]; larger space between rows of tubes leads to lower pressure drop. All measured values are displayed in Table 5.

Fig. 5
figure 5

Airside pressure drop for different tension forces dependent on the air velocity

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Table 5 Airside pressure drop for each combination of air velocity and stretching force for heat exchangers RG025 and RG026
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The effect of tension force can be seen in detail in Fig. 6, which shows the pressure drop values depending on the applied force for each air velocity. Also, a 5% error bar is shown. The difference between pressure drops for different tension forces is negligible for the two lower velocities. For the two higher velocities, especially for 8 ms−1, we can initially see the increase in pressure drop—the highest value is achieved primarily at 3000 N. Since the experiments were done with fully functioning HEs, the fibers were elongated due to the warm coolant flowing inside the fibers. The second cause of elongation was the stress relaxation of fibers. Stress relaxation is the partial transition of elastic strain to plastic, which means that fibers remain elongated after unloading. The fibers' elongation caused regular structure decomposition and created bypasses, which reduced the pressure drop at the unloaded state. A subtle trend of pressure drop decrease with increase in tension can be seen at higher tension force values. This trend can be explained by the contraction of fibers cross section due to transversal strain. The elongation of fibers at 9000 N tension force was, on average, 14 mm after subtraction of elongation due to the temperature expansion. It means the fiber diameter was reduced by about 2% from the initial value (considering the Poisson's ratio of 0.4), which reduces the airside pressure drop approximately by 5% (due to expanding the crossflow area and decreasing the maximal air velocity inside the tube bundle). Nevertheless, the overall difference between the pressure drop values for different stretching forces lies in a 5% error interval, especially for RG026.

Fig. 6
figure 6

Airside pressure drop for different air velocities dependent on the tension force

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It is interesting to compare the results of the present study with the finding of [21], where pressure drop and heat transfer rate were measured on the crossflow HE made of flexible Teflon tubes. It was observed that the pressure drop and heat transfer rate slightly increased for the flow of Reynolds numbers comparable to the present study compared to rigid tubes. However, a more significant increase was reported for higher Re, up to 6000. Flow with higher Re is not applicable in the case of PHFHE as it would require excessive air velocity due to the smaller characteristic dimension. The differences with the present study were tube arrangement (inline), the tube diameter (4.5 mm) and probably the tube wall thickness (not reported). The study identified vibrations as the main cause of the increase in both observed quantities. In the case of hollow fibers with a considerably smaller diameter (1 mm) and very thin wall (0.1 mm), the main differences are attributed to the change in fiber arrangement.

Conclusions

This work aimed to assess the influence of tension force on the two main characteristics of crossflow PHFHE—heat transfer rate and pressure drop. It was found that for the maximal tension force of 9000 N, the heat transfer rate increased by about 11% on average compared to the unloaded state. This increase occurs mainly due to eliminating bypasses that are relatively easily formed in a not tensioned HE. In the case of pressure drop, firstly, an increase was observed, again due to the elimination of bypasses, followed by a slightly decreasing trend with higher tension applied to fibers. The contraction of the cross section of the fibers caused the decrease. However, the pressure drop differences were generally smaller than the measurement uncertainty range of 5%. Thus, they are negligible for engineering practice. This work used fully functioning hollow fiber HEs. Results are, therefore, relevant for the various applications of PHFHEs. The fibers' length can change due to thermal expansion or stress relaxation, which considerably influences the fiber arrangement and behavior, particularly in the unloaded state. It can be concluded that the phenomenon of flow-induced vibrations (FIV) was negligible compared to the role of fiber compliance, that was decreasing with increase in tension. HE made of flexible fibers without applied tension is prone to the formation of bypasses and irregularities in the structure that causes a slightly lower heat transfer rate and pressure drop. Based on the findings, the empirical model for predicting crucial characteristics of the crossflow PHFHE is one step closer, which will enable the fast design of prototypes in various applications.