Figure 1 presents the experimental setup used to investigate PHE channel flows. Siemens Sinamics V20 controlled centrifugal pumps (Multisteel, type RF 32/160) as to fine-tuning of the Reynolds number in the measurement section. Mass flow rate is measured employing a Rosemount 8700M magnetic flowmeter (inaccuracy is less than 0.5% of the registered flow rate). A mesh flow conditioner is placed 40 pipe diameters upstream of the test section. Following Laws et al. [23] , a nearly developed flow at the PHE channel entrance is expected. The flow is then conducted back to the water reservoir.

Omega transducers, type PX409 (0.08%FS uncertainty), allowed absolute and differential pressure measurements (not shown in Figure 1 ). The absolute pressure is monitored at the test section entrance by Omega transducers with the aid of pressure taps. Omega 100Ω platinum RTD monitors water temperature in the flow loop. National Instruments acquisition systems were applied to collect measuring data, monitored through LabVIEW software. Downward and upward flows have been tested to confirm that gravity would not affect adiabatic flows within PHE channels. Table 1 summarizes sensors’ uncertainty specification.

Water flows were measured within PHE channels by Particle Tracking Velocimetry (PTV) with a fast camera (SpeedSense type). It has 12-bit grayscale CMOS sensor and a resolution of 1280 × 1280 pixels. Nikon 18 - 200 mm f/3.5 - 5.6 lens was used to capture almost instantaneous particle positions in a 211 × 695 mm² rectangular area of the test section. The camera operated at 1000 Hz. The experimental settings can be summarized as: pixel sensor resolution equal to 10 µm²; focal length of 100 mm; exposure time of 1 ms; distance from the lens to the object of 0.95 m. In order to reduce reflections to the fast camera, a lighting system consisting of LEDs was positioned around the channel.

Two pairs of acrylic sheets with the thickness equal to one inch were machined to generate PHE channels. Each pair of sheets was machined with different corrugation angles. Several bolts press a pair of sheets against each other and a gasket which circumvents them as to assure ideal sealing. The primary dimensions of the frontal view of a single plate which composes the PHE channel are presented in Figure 2(a) . The origin of the Cartesian coordinate system is at the frontal view center. Gravity is antiparallel to the y-axis. The variable β stands for chevron angle: the angle formed by the corrugation and the y-axis. The chevron angle is well-known to affect thermal and hydraulic behaviors of GPHEs as showed in the study of Yang, Jacobi and Liu [9] , who presented increasing heat transfer and friction factor coefficients with increasing β-values.

With the motivation of investigating PHE hydraulic features affected by different β-values, three chevron arrangements were tested: 30˚/30˚, 30˚/60˚ and 60˚/60˚. The acrylic sheets were built in a way that allowed the use of different corrugation angles to form a channel. 30˚/30˚ configuration typifies channels with a minimal pressure decrease (i.e. L-channels), whereas 60˚/60˚ arrangement characterizes channels with substantial pressure drop (i.e. H-channels). Following the flow pattern chart of Sarraf et al. [24] and for channel Reynolds numbers surpassing 500, furrow or spiral flow patterns manifest in L-channels, whereas longitudinal wavy or cross-flow patterns arise in H-channels. Flows in PHE channels encompass both configurations with 30˚/60˚ arrangement, with a prevalence of the furrow pattern under the specified boundary conditions.

Cross-section complex form is shown with the aid of Figure 2(b) : note the presence of several irregular areas along the y-axis. Cross-section values along y-axis, denoted as Asec, are presented in Figure 2(c) in a line segment which comprises the channel inlet and outlet centers; see the variable L_v in Figure 2(a) . Dissipation processes are very intense in the inlet and outlet distribution zones: y < −200 mm and y > 200 mm in Figure 2(c) . Flow experiences deceleration and acceleration with high velocity components. Within the heat transfer zone, the periodic cross-section variation encompasses components of low fluid velocity. Channel primary geometric characteristics are summarized in Table 2 .

Gravel seeding particles were added to the water as flow tracers. To assure these particles could represent main flow structures, time and length scales of particles and fluid were compared as well as the effect of gravity on particles, and the effect of fluid acceleration on particle motion. The incorporation of these particles into the water has been a consistent practice in our prior studies, as documented by Beckedorff et al. [21] [22] .

Following Albrecht et al. [25] , relaxation time for particles, τ_p, in stationary flow is given by:

${\tau }_p=\left(d_p{}^2{\rho }_p/18\mu \right)\left(1+0.5{\rho }_f/{\rho }_p\right)$(1)

where d_p is the particle diameter, ρ is the mass density and μ is the dynamic viscosity. The subscripts p and f stand for particle and fluid, in that order. The relaxation time of the gravel particle with a diameter of 0.15 mm is 2.6 ms. Prior values are then compared to dissipative fluid time and length scales. By applying viscous wall units and adopting Kolmorogov scales as the fluid ones, one finds ${\tau }_k=\nu /u_{\tau }^2$and l_k = ν/u_τ, respectively, where ν is the kinematic viscosity and u_τ, the wall shear velocity. The latter is defined as u_τ = (τ_w/ρ_f)^1/2 where τ_w is the wall shear stress. An approximation of the magnitude order can be achieved by utilizing τ_w = μv_B/t_ave,, where t_ave is the average channel height and equals to 3 mm, and v_B is the bulk velocity at the PHE center (plane y = 0). For v_B = 605 mm·s⁻¹, the estimated values for τ_w, u_τ, l_k, and τ_k are as follows: 0.20 Pa, 14.23 mm·s⁻¹, 0.0705 mm, and 4.9 ms, respectively. Time- and length-scale ratios for particles and fluid are now presented with the Stokes number, St = τ_p/τ_k ≈ 0.5, and with the ratio d_p/l_k, ≈ 2. It is concluded that the main flow scales can be captured by applied gravel particles.

In addition, the effect of gravity on particles can be measured by comparing the final velocity of the gravel particles with the bulk flow velocity, while the effect of fluid acceleration on particle motion can be demonstrated by comparing the ratio of pressure gradient forces (including additional mass forces) to buoyancy forces. For actual boundary conditions, both effects are negligible, see references [21] [22] .

The methodology for identifying separate particle trajectories through PTV resembles the approaches employed by Oliveira et al. [26] - [28] in pipe flow and Beckedorff et al. [21] [22] in a PSHE channel. Tracer trajectories were acquired with the aid of Davis PTV code from La Vision. References [29] [30] provided detailed information on the algorithm used for tracking. Processing calibration images establish a correlation between pixel data and the physical dimensions of the calibration plate. The latter is made of plastic, 0.5 mm thick, and it contains circular voids regularly spaced in 20 mm in both horizontal and vertical dimensions. The plate surface area has been cut to match the xy-plane flow area of Figure 2(a) , i.e. the projection of the frontal view area where PHE channel flow occurs. A third-order polynomial establishes a correlation between pixel data and physical dimensions of the calibration plate, achieving an RMS error of less than 0.5 pixels, or approximately 5 µm. Once the calibration procedure is completed, flow measurement images generate files containing the time and space positions of separate particle trajectories.

Figure 3 shows photographs of the PTV method. Figure 3(a) displays calibration plate image at the channel center, while Figure 3(b) illustrates the identified voids (in green). Note that the calibration recording is registered at a single position, limiting the trajectory analysis to the channel frontal view plane. Figure 3(c) shows channel flow image: note that light noise is significant. Imaging filters enhance the contrast between the object and the background as to facilitate identification of particle tracks. Particles centers’ are identified with the aid of a Gaussian fit. Space and time references of each particle identified are exported to the analysis method; see the integration of individual tracer trajectories within a limited time window in Figure 3(d) .

Figure 4 shows the applied PTV mesh. Figure 4(a) shows separate bins projected on the channel frontal view (left), and geometry of a measurement bin (right). Figure 4(b) shows discrete bins projected in the frontal view of the channel inlet or outlet (left), and geometry of a measurement bin at the same positions (right). False trajectories generated during the PTV procedure are discarded by a length filter and a displacement outlier-check at the channel periphery [21] [22] . False trajectories were mostly created by light reflections at the channel edges. Afterwards, velocity vectors in the plane of view are attained by the differentiation of the validated trajectories with regard to time. Over 100,000 velocity vectors were obtained with twenty individual measurement sets of 2 s. Finally, ensemble-averaging of velocity vectors at grid points around of Figure 4 is performed.

The grid volume at the center of the channel is 3217 mm³, and at the inlet/outlet, 533 mm³. PTV measurements were obtained with the Reynolds number, based on the PHE hydraulic diameter, equal to 3,450. A camera frequency of 1000 Hz was applied. Flow measurement analysis at the channel inlet and outlet undergo conversion from Cartesian to cylindrical coordinates, with the origin at the center of the inlet and outlet manifolds. Radial velocities, denoted as vrad, are collected in bins of Figure 4(b) .

Method validation occurred with mass flow rate integration in different channel cross-sections. Integration was obtained with the product of the mean axial velocity and the cross-sectional area of each grid. With the mass density of water at nearly atmospheric condition, one can find the PTV mass flow rate in different y-planes. Finally, PTV accuracy was guaranteed by comparisons with the electromagnetic flowmeter reading at 15 cross-sections. These results match within experimental error.

To calculate the frictional pressure drop (∆P_ch), it was adopted a procedure similar to that employed by references [20] [31] . Needles were positioned at the channel entry and exit by creating small holes through the gasket sealing. Friction factor can be calculated according to:

$f=\frac{2\Delta P_{ch}{D}_h\rho }{4L_v{\left(\frac{{\stackrel{\dot{}}{m}}_{ch}}{A_{\sec }}\right)}^2}$(2)

$D_h=\frac{2b}{\varphi }$ (3)

where ${\stackrel{\dot{}}{m}}_{ch}$ denotes the channel mass flow rate, and D_h is the hydraulic diameter. The variable ϕ represents the enlargement factor, defined as the ratio of the effective to the projected plate area [1] . Following the approach of reference [16] , ϕ can be computed as:

$\varphi =1/6\left(1+\sqrt{1+x^2}+4\sqrt{1+x^2/2}\right)$(4)

where x represents the corrugation number, $\pi b/P_c$. Following Beckedorff et al. [32] , the maximum uncertainty for friction factors was ±5.5%, calculated in accordance with:

$u\left(f\right)=\left(\frac{\rho D_h\Delta P{A}_{ch}^2}{2D{L}_v{\stackrel{\dot{}}{m}}_{ch}^2}\right){\left\{{\left(\frac{u\left(\Delta P\right)}{\Delta P}\right)}^2+{\left(-2\frac{u\left({\stackrel{\dot{}}{m}}_{ch}\right)}{{\stackrel{\dot{}}{m}}_{ch}}\right)}^2\right\}}^{0.5}$(5)

Figure 5(a) shows the effect of chevron angle arrangements (30˚/30˚, 30˚/60˚ and 60˚/60˚) on the channel pressure drop (ΔP_ch) and Figure 5(b) , on the Fanning friction factor. Circles, diamonds and triangles represent L/L, L/H and H/H arrangements, in that order. The Reynolds numbers, Re, based on the cross-section area, Asec, ranged approximately from 1175 to 8325:

$\mathrm{Re}=\frac{m_{ch}{D}_h}{A_{\sec }\mu }$ (6)

Kakaç et al. [1] defined flow regime as turbulent for channel Reynolds numbers over 400. The same flow regime is considered according to Bond [33] and Gusew and Stuke [34] .

For the same Reynolds number, channel pressure drop increases with increasing chevron angles owing to the increased flow disturbance in longitudinal wavy flows as compared to furrow patterns; see Sarraf et al. [24] , for example. Note that the pressure drop in mixed channels cannot be approximated by averaging the chevron angle arrangement: it is slightly higher than the one presented for L-channels. Despite the occurrence in mixed channels of both patterns (furrow and longitudinal wavy), furrow pattern prevails in the present conditions.

Table 3 summarizes friction factor correlations for the actual experiments. R²-values over 0.98 assure the quality of each fit.

Figure 6 shows the mean axial (a) and lateral (b) velocity components for chevron angle arrangements of 30˚/30˚, and with channel Reynolds number equal to 3450. Despite the differences in fluid particle trajectories with respect to the channel chevron angle, the projection of the mean velocity components onto the channel frontal view is comparable. Therefore, for the sake of brevity, results were only presented for one chevron angle configuration. With the aim of comparing the effect of rectangular and circular corrugated plate geometries on the mean velocity field, Figure 6 also presents the mean axial (c) and lateral (d) velocity components for a Plate and Shell Heat Exchanger (PSHE) channel with the same Reynolds number and with chevron angle arrangements of 45˚/45˚, as investigated by Beckedorff et al. [22] . The PSHE plate external diameter is 295 mm, and its heat transfer area is nearly half of the PHE area.

The asymmetry in the mean axial velocity concerning the plane x = 0 is evident in Figure 6(a) . In the left region of the channel (−0.2 m < y < 0.2 m; x < 0), the flow resistance is reduced due to the shorter exit length: note that yellow color indicates velocities with higher magnitude in regard to orange/red colors. Fouling preferably occurs in low-velocity areas (−0.2 m < y < 0.2 m; x > 0.05; see dashed line in Figure 6(a) . The distribution regions exhibit the highest axial velocities (y < −0.2 m; y > 0.2 m) due to mass conservation, as seen in the blue regions of Figure 6(a) . The lateral bulk approaches zero in the central region (−0.2 m < y < 0.2 m), as illustrated in Figure 6(b) . Lateral velocity is nearly anti-symmetric concerning the plane y = 0; compare yellow/red colors for y > 0.2 m to light and dark blue areas for y < −0.2 m.

Comparison of Figures 6a and 6c reveals that the PHE mean velocity field is more homogenous in the heat transfer area in regard to the PSHE velocity field. Average axial velocity components are meaningful in the PSHE central area (−0.08 m < y < 0.08 m; −0.05 m < x < 0.05 m), see light and dark blue colors in Figure 6(c) . The short distance between ports promotes high velocities. Fouling preferably occurs near the external circumference in the PSHE channel, where axial bulk velocities approach zero. The peak velocity magnitude in the PSHE center point (y = 0; x = 0) is 1.5 v_B, whereas this value is 1.15 v_B in the PHE center point. Figure 6(d) reveals that the lateral velocity is nearly anti-symmetric concerning planes y = 0 and x = 0. Maximum lateral velocity is nearly 0.45 v_B for the PHE channel, and roughly 0.63 v_B for the PSHE channel.

Figure 7(a) and Figure 7(b) present profiles of the mean axial and lateral velocity components, in that order, in five PHE y-positions: three in the heat transfer area (−0.13, 0 and 0.13) and two in the distribution regions (−0.26 and 0.26); see Figure 7(c) . The velocity is made non-dimensional with the bulk velocity, v_B, at the PHE center (y = 0).

The asymmetry of the mean axial velocity field is demonstrated with the aid of Figure 7(a) . At the channel mid-plane, the peak axial velocity is 1.15 v_B and occurs at x/Lw = −0.11, whereas the relative horizontal velocity is nearly zero. At the distribution regions, plane y = ±0.26, the maximum velocity reaches 2.4 v_B at the entrance, and 2 v_B at the exit. Error-bars (standard errors with confidence intervals of 95%) give an indication of the variation of velocity over the channel height. Note that the magnitude of the lateral velocity is higher at the outlet distribution, Figure 7(b) . Significant velocity components in the distribution areas (as shown in planes y = ±0.26) will play important role in the PHE dissipation process. In accordance to reference [13] , the pressure drops at the channel inlet and outlet with the distribution zones can account for up to 50% of the total pressure drop in the PHE.

Figure 8(a) shows the dimensionless mean radial velocities at the inlet and exit ports. Cylindrical coordinate systems with their origins at the ports’ centers are used, as indicated in the top/left of Figure 8(a) and in Figure 4(b) . Outlet and inlet results are plotted in clockwise and in counter clockwise directions, in that order. Figure 8(b) shows linear streamlines in radial direction at the channel inlet with the join of separate tracer trajectories in a limited time span, while Figure 8(c) highlights the recirculation zones formed at the channel outlet with the aid of air bubbles added to the main stream.

The radial velocity can reach up to 1.4 v_B at the PHE exit, and it can exceed 1.7 v_B at the entrance. The PHE channel outlet exhibits flow recirculation ( Figure 8(c) ), influencing the radial velocity distribution. The annular cross-section surrounding the PHE channel outlet is smaller than the exit pipe cross-section, resulting in flow deceleration and increased static pressure. This pressure rise induces boundary layer separation near the channel outlet, leading to the formation of recirculation zones. Similar behavior was found at the exit of PSHE channel by reference [21] . On the fictional closed curves, the average measured fluid velocity is 0.7 m·s⁻¹. Conversely, at the channel inlet, the flow accelerates while static pressure decreases, allowing flow trajectories to enter the PHE channel along roughly linear streamlines in radial direction ( Figure 8(b) ).

Flow recirculation characterizes the channel outlet as observed in Figure 8(c) . The recirculation zones affect therefore the turbulent pipe flow in the manifold outlet. To quantify the recirculation intensity, Γ, around a fictional recirculation boundary, one can apply:

$\Gamma =2\pi uR$(7)

where R is the recirculation length-scale (half of the hydraulic diameter of the circulation area), roughly 9 mm, and u is the average measured fluid velocity on the fictional closed curves, just about 0.7 m·s⁻¹. In the given conditions, the magnitude of Γ is equal to 0.04 m²·s⁻¹.

With the aim of visualizing swirling flow structures at the channel outlet, bubbles have been added to the main PHE flow. Figure 9 shows a photograph of the turbulent pipe flow with air bubbles added to the main stream at the exit manifold.

The necessary length to obtain a fully developed turbulent pipe flow at the channel outlet is estimated following the experimental work of Rocklage-Marliani et al. [35] who investigated swirling turbulent pipe flows with Laser-Doppler measurements and a swirl generator. The latters investigated pipe flow statistics in several cross-sections downstream the generator. Following Rocklage-Marliani et al. [35] , statistics were compared with different rotation numbers (Ro) defined as:

$Ro=\Omega D/2v_B$(8)

where Ω is the angular velocity. A rough estimation of Ω length-scale for the present experiment is 12.7 s⁻¹. By taking v_B as 0.15 m·s⁻¹, one estimates the rotation number as 2.4.

Escue and Cui [36] studied numerical swirling turbulent pipe flows by numerical simulations and validated with the data of Rocklage-Marliani et al. [35] . The formers observed that swirling pipe flows with Ro = 2 did not achieve a fully-developed state for a distance of about 14D. In present boundary conditions, statistics of fully turbulent pipe flows are only expected for distances over 20D.