Wall Pressures In a Channel Flow Obstructed By a Line of Inclined Rods: Invariant Groups and Semi-Empirical Correlations

An experiment is conducted in a rectangular channel obstructed by a transverse line of four inclined cylindrical rods. The wall pressure around the perimeter of a central rod and the pressure drop through the channel are measured varying the inclination angle of the rods. Three assemblies of rods with different diameters are tested. The measurements were analyzed applying momentum conservation principles and semi-empirical considerations. Several invariant dimensionless groups of parameters relating the pressure at key locations of the system with characteristic dimensions of the rods are produced. It was found that the independence principle holds for most of the Euler numbers characterizing the pressure at different locations, that is, the group is independent of the inclination angle provided that the inlet velocity projection normal to the rods is used to non-dimensionalize the pressure. The resulting semi-empirical correlations can be useful for designing similar hydraulic units.


Introduction
Many heat and mass transfer devices are composed by sets of modules, channels or cells through which a uid passes amid more or less complex internal structures, like rods, buffers, inserts, etc. Recently there has been a renewed interest for a deeper understanding of the mechanisms relating the internal pressure distribution and forces on complex internals with the overall pressure drop of the modules. This interest is brought about, amongst others, by innovations in material science, expansion of computational capacity for numerical simulations, and the increasing miniaturization of devices. Recent experimental studies of pressure internal distributions and losses include channels roughened by variously shaped ribs (Ruck et al., 2017), electrochemical reactors cells (Wu et al., 2020) and lattice-frame materials (Kim et al., 2004).
The most common internal structures are arguably cylindrical rods passing through the unit module, either as bundles or isolated. In heat exchangers this con guration is typical on the shell side. Shell-side pressure drops are relevant in the design of heat exchangers such as steam generators, condensers and evaporators. In a recent study, Wang et al. (2018) found reattachment and co-shedding ow regimes in tandem con guration of rods. Liu et al. (2017) measured the pressure drop in a rectangular channel with a built-in double-U-shaped tube bundle with different inclinations and calibrate a numerical model emulating the rod bundle with a porous medium.
As expected, there are numerous con guration factors that in uence the hydraulic performance of cylinder banks: type of arrangement (e.g., staggered or in-line arrays), relative dimensions (e.g., spacing, diameter, length), and inclination angle, among others. Several authors focused their efforts in nding dimensionless criteria to guide the design capturing the combining effects of the geometric parameters.
Among the more recent experimental studies, Kim et al. (2006) proposed a model of effective porosity taking the length of the unit cell as the control parameter using in-line and staggered arrays and Reynolds numbers between 10 3 and 10 4    presented a comprehensive study of tandem cylinders, focusing on the effects of the Reynolds number and geometric ratios on vortex shedding. They were able to identify ve regimes, namely lock-in, intermittent lock-in, no lock-in, subharmonic lock-in and shear-layer reattachment regimes. Recent numerical studies remarked the formation of vortex structures in a ow passing con ned yawed cylinders (Liang and Duan, 2019).
Generally, the hydraulic performance of unit cells is expected to depend on the con guration and geometry of the internal structures, often quanti ed through empirical correlations that spring from speci c experimental measurements. In many of these devices consisting of periodic assemblies, the ow patterns repeat in each unit cell, hence, the information related to a representative unit cell can be In the present work, the results of a study of the wall pressure and the pressure drop in a channel with a transverse line of four inclined cylindrical rods are presented. Three rod assemblies having different diameters are measured, varying the inclination angle. The experimental data is analyzed applying the Bernoulli equation and momentum conservation principles, to assess the validity of the independence principle. Finally, dimensionless semi-empirical correlations are produced that can be useful for designing similar hydraulic units.

Experimental Setup And Method
The experimental setup consists of a rectangular test section that receives an air ow provided by an axial blower. The test section hosts a cell composed of two parallel central rods and two half rods embedded in the channel walls as shown in Fig.1-e, all of them having the same diameter. Figs. 1-a to 1-e show the detailed geometry and dimensions of each part of the experimental setup.
Three sets of rods with different diameters are tested. Table 1 lists the geometric characteristics of each case. The rods are mounted on a protractor so that their angle respect to the ow direction can be varied between 90º and 30º (Figs. 1-b. and 3). All rods are made of stainless steel, and they are centered keeping the same gap distance between them. The relative position of the rods is xed by two spacers located outside the test section. The inlet ow rate to the test section is measured by means of a calibrated Venturi Tube, shown in Fig. 2, which is monitored with a DP Cell Honeywell SCX. The uid temperature at the exit of the test section is measured with a PT100 thermometer, and it is controlled at 45±1 C o . To ensure a planar velocity pro le and reduce turbulence levels at the channel inlet, the incoming ow is forced through three metallic screens. A settling distance of approximately 4 hydraulic diameters was taken between the last screen and the rods, whereas the outlet has a length of 11 hydraulic diameters.
The wall pressure one of one of the central rods is monitored through a 0.5-mm pressure tap at the middle plane of the test section. The pressure tap diameter corresponds to 5º angular span; hence the angular precision is approximately 2º. The monitored rod can be rotated around its axis, as can be seen in Fig. 3. The difference between the wall pressure and the pressure at the inlet of the test section was measured with a differential DP Cell Honeywell SCX Series. This pressure difference was measured for each arrangement of bars, varying the ow rate, the inclination angle, α, and the azimuthal angle, θ.

Results
The goal of the experiment is measuring and interpreting the pressure drop between the channel inlet and the wall pressure of the central bar, for different azimuthal and inclination angles, θ and α. In order to generalize the results, the pressure difference will be presented in dimensionless form as the Euler number: where ρ is the uid density, u i is the average inlet velocity, p i is the inlet pressure and and p w is the pressure at a given point of the wall. The inlet velocity was xed within three different range levels determined by the opening of the inlet valve. The resulting Reynolds numbers (Re ≡ u i d/ν) range between 2500 and 6500, where ν is the dynamic viscosity. In what follows, we present an analysis of the results based on the hypothesis that the Euler number can be estimated solely by geometric parameters, namely, the characteristic length-ratios d/g and d/H (where H is the height of the channel), and the inclination angle α. A popular practical rule of thumb states that the uid-structure forces on yawed rods are determined by the projection of the inlet velocity normal to the rods axis, u n = u i sinα. This is sometimes referred as the independence principle. One of the goals of the following analysis is to check whether this principle holds in our case, where ow and obstacles are con ned inside channel walls.
Let us consider the wall pressure measured at the front of the middle rod, i.e., θ = 0. According to the Bernoulli equation, the wall pressure at that position, p o , satis es: where u o is the uid velocity at adjacent to the wall at θ = 0, and ρ is the uid density, and we are assuming that the irreversible losses are relatively small. If u o were null (i.e., stagnation conditions) the Euler number should be unity. However, it can be observed in Fig. 4 that at θ = 0 the resulting Eu w are close but not exactly equal to that value, particularly for larger inclinations. This suggests that the velocity at the wall at θ = 0 does not vanish, which can be chocked up to the upward de ection of the current lines produced by the inclination of the rods. Since the ow is con ned at the top and the bottom of the test section, such de ection should generate a secondary recirculation that increases the axial velocity at the bottom and reduced the velocity at the top. Assuming that the magnitude of the mentioned de ection is the projection of the inlet velocity over the axis bar (i.e., u i cosα), the corresponding Euler  3) with the corresponding experimental data, showing good agreement. The average deviation is 25% with 95% con dence level. Notice that Eq. (3) is in agreement with the independence principle. Likewise, Figure 6 shows that the Euler numbers corresponding to the pressure wall at the back of the rod, p 180 , and at the exit of the test section, p e , also follows a trend proportional to sin 2 α. However, in both cases, the coe cient depends on the rods diameter, which is reasonable since the latter determines the obstructed area. This feature is similar to the pressure drop across an ori ce plate, where the ow passage is partially reduced at a certain location. In the present test section, the role of the ori ce is played by the gap between bars. In such cases, the pressure experiences a substantial drop at the restriction and is partially recovered in the backward expansion. Viewing the restriction as a blockage in the direction normal to the rods axis, the pressure drop between the front and the back of the rods can be written as (White, 2016): The resulting drag coe cient is c d = 1.28 ± 0.02 with 67% con dence. Likewise, the same graphic also shows that the total pressure drop between the inlet and the outlet of the test section follows a similar trend, but with a different coe cient accounting for the pressure recovery in the backspace between the bars and the channel exit. The corresponding drag coe cient is c d = 1.00 ± 0.05 with 67% con dence.

Wall pressure atθ = 90 o
The minimum wall pressure, p 90 , at θ = 90 o , requires a special treatment. According to the Bernoulli equation along a current line passing through the gap between bars, the pressure, p g , and the velocity,u g , at the center of the gap between bars (which coincides with the middle point of the channel) are related by: The pressure p g can be related to the wall pressure at θ = 90 o by integrating the pressure pro le across gap separating the central rods between the middle point and the wall (see Fig. 8). The balance of forces gives (Shappiro, 1953): dp dy = ρK(y)u 2 g (7) where y is a coordinate that goes from the center point of the gap between the central rods perpendicularly to the rod wall, and K is the curvature of the current line at position y. To produce an analytical assessment of the pressure at the wall, let us assume that u g is uniform and that K(y) is linear. These assumptions were veri ed with numerical calculations. At the wall the curvature is determined by the ellipsoidal section of the bar at angle α, namely, K(g/2) = (2/d)sin 2 α (see Fig. 8).
Then, regarding that due to the symmetry the curvature of the stream lines vanishes at y = 0, the curvature at a generic coordinate y is given by: Integrating Eq. (7) gives: Combining Eqs. (6) and (9), u g can be calculated as: On the other hand, by mass conservation, the average velocity 〈u g 〉over the plane perpendicular to the ow at the measurement position is related to the inlet velocity as: where A i is the cross-section ow area at the channel inlet, and A g is the cross-section ow area at the measurement position (see Fig. 8) given respectively by: It should be noticed that u g is not equal to 〈u g 〉. In effect, Figure 9 depicts the velocity ratio u g /〈u g 〉, calculated from Eqs. (10)- (14), plotted against the ratio d/g. Although there is some dispersion, it is possible to identify a trend, which was approximated by a second order polynomial:  (15) Combining Eqs. (10) and (15) Figure 10 compares the experimental results for Eu 90 with Eq. (16). The average relative deviation is 25 % wth a con dence level of 95%.

Net force acting on the rods
The net force, f n , acting on the central rod perpendicular to its axis can be calculated by integrating the pressure over the rod surface, as: where the rst coe cient is the rod length inside the channel and the integral is performed between 0 and 2π.
The projection of f n on the stream direction should match the pressure force between the channel inlet and outlet, barring friction forces parallel to the rods and minor momentum ux imbalances due to incomplete pro le development at the back. Accordingly, 3f n sinα = 3H(d + g) p i − p e (19) The coe cient 3 in Eq. (19) accounts for the three rods. In dimensionless form, we can then write:

Conclusions
The wall pressure and the pressure drop in a channel with a transverse line of four inclined cylindrical rods were measured varying the inclination angle of the rods. Three assemblies of rods with different diameters were tested. Within the range of Reynolds numbers of the test, between 2500 and 6500, the Euler numbers are independent of the ow rate. The wall pressure around the central rod follows the usual trend observed in cylinders, with the maximum at the front, the minimum at the lateral gap between bars, and the partial recovery at the back due to the detachment of the boundary layer.
The experimental data were analyzed using momentum conservation considerations and semi-empirical assessments, searching for invariant dimensionless numbers that relate the Euler numbers with the characteristic dimensions of the channel and rods. It was found that the independence principle holds for most of the Euler numbers characterizing the pressure at different locations, that is, the group is independent of the inclination angle provided that the inlet velocity projection normal to the rods is used to non-dimensionalize the pressure. Only the wall pressure at the gap between rods presents a slight deviation from this principle. Dimensionless semi-empirical correlations were produced that can be useful  Diagram of the Venturi tube used to measure de inlet ow rate (lengths in mm).

Figure 4
Variation of the wall Euler number around the rod for different inclinations angles and rod diameter.  Variation of the wall Euler number, at θ = 180° (full symbols) and the exit, (empty symbols) with the inclination angle. The curves corresponds to the independence principle, i.e., Eu ∝ sin 2 α.

Figure 7
Dependence of the drag coe cient with d/g for the pressure drop between the front and the back of the rod (Eu 0-180 ) ) and the total pressure drop between the channel inlet and outlet. The grey zones are the 67% con dence bands of the correlations.

Figure 8
Page 16/16 Diagram of the characteristic cross sections, frontal (left) and from above (below).

Figure 9
Ratio between the maximum u g and the average 〈u g 〉 velocities at the central cross section of the channel. The solid and dashed curves corresponds to Eq. (5) and the bounds varying ±25% of the corresponding coe cients.

Figure 11
Balance of forces of the channel. The line corresponds to Eq. (20). The Pearson correlation coe cient is 0.97.