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Experiments were performed in a rectangular channel blocked by transverse lines of four inclined cylindrical rods.The pressure on the center rod surface and the pressure drop across the channel were measured by varying the rod’s inclination angle.Three different diameter rod assemblies were tested.The measurement results are analyzed using the principle of conservation of momentum and semi-empirical considerations.Several invariant sets of dimensionless parameters are generated that relate the pressure at critical locations of the system to the characteristic dimensions of the rod.The independence principle is found to hold for most Euler numbers characterizing pressure at different locations, i.e. if the pressure is dimensionless using the projection of the inlet velocity normal to the rod, the set is independent of the dip angle. The resulting semi-empirical correlation can be used for Design similar hydraulics.
Many heat and mass transfer devices consist of a set of modules, channels or cells through which fluids pass in more or less complex internal structures such as rods, buffers, inserts, etc.More recently, there has been renewed interest in gaining a better understanding of the mechanisms linking internal pressure distribution and forces on complex internals to the overall pressure drop of the module.Among other things, this interest has been fueled by innovations in materials science, the expansion of computational capabilities for numerical simulations, and the increasing miniaturization of devices.Recent experimental studies of pressure internal distribution and losses include channels roughened by various shaped ribs 1 , electrochemical reactor cells 2 , capillary constriction 3 and lattice frame materials 4 .
The most common internal structures are arguably cylindrical rods through unit modules, either bundled or isolated.In heat exchangers, this configuration is typical on the shell side.Shell side pressure drop is related to the design of heat exchangers such as steam generators, condensers and evaporators.In a recent study, Wang et al. 5 found reattachment and co-detachment flow states in a tandem configuration of rods.Liu et al.6 measured the pressure drop in rectangular channels with built-in double U-shaped tube bundles with different inclination angles and calibrated a numerical model simulating rod bundles with porous media.
As expected, there are a number of configuration factors that affect the hydraulic performance of a cylinder bank: type of arrangement (eg, staggered or in-line), relative dimensions (eg, pitch, diameter, length), and inclination angle, among others.Several authors focused on finding dimensionless criteria to guide designs to capture the combined effects of geometric parameters.In a recent experimental study, Kim et al. 7 proposed an effective porosity model using the length of the unit cell as a control parameter, using tandem and staggered arrays and Reynolds numbers between 103 and 104.Snarski8 studied how the power spectrum, from accelerometers and hydrophones attached to a cylinder in a water tunnel, varies with the inclination of the flow direction.Marino et al. 9 studied the wall pressure distribution around a cylindrical rod in yaw airflow.Mityakov et al. 10 plotted the velocity field after a yawed cylinder using stereo PIV.Alam et al. 11 conducted a comprehensive study of tandem cylinders, focusing on the effects of Reynolds number and geometric ratio on vortex shedding.They were able to identify five states, namely locking, intermittent locking, no locking, subharmonic locking and shear layer reattachment states.Recent numerical studies have pointed to the formation of vortex structures in flow through restricted yaw cylinders.
In general, the hydraulic performance of a unit cell is expected to depend on the configuration and geometry of the internal structure, usually quantified by empirical correlations of specific experimental measurements.In many devices composed of periodic components, flow patterns are repeated in each cell, and thus, information related to representative cells can be used to express the overall hydraulic behavior of the structure through multiscale models.In these symmetric cases, the degree of specificity with which general conservation principles are applied can often be reduced.A typical example is the discharge equation for an orifice plate 15.In the special case of inclined rods, whether in confined or open flow, an interesting criterion often cited in the literature and used by designers is the dominant hydraulic magnitude (e.g., pressure drop, force, vortex shedding frequency, etc.) ) to contact.) to the flow component perpendicular to the cylinder axis.This is often referred to as the independence principle and assumes that the flow dynamics are driven primarily by the inflow normal component and that the effect of the axial component aligned with the cylinder axis is negligible.Although there is no consensus in the literature on the validity range of this criterion, in many cases it provides useful estimates within the experimental uncertainties typical of empirical correlations.Recent studies on the validity of the independent principle include vortex-induced vibration16 and single-phase and two-phase averaged drag417.
In the present work, the results of the study of the internal pressure and pressure drop in a channel with a transverse line of four inclined cylindrical rods are presented.Measure three rod assemblies with different diameters, changing the angle of inclination.The overall goal is to investigate the mechanism by which the pressure distribution on the rod surface is related to the overall pressure drop in the channel.Experimental data are analyzed applying Bernoulli’s equation and the principle of conservation of momentum to evaluate the validity of the independence principle.Finally, dimensionless semi-empirical correlations are generated that can be used to design similar hydraulic devices.
The experimental setup consisted of a rectangular test section that received air flow provided by an axial fan.The test section contains a unit consisting of two parallel central rods and two half-rods embedded in the channel walls, as shown in Fig. 1e, all of the same diameter.Figures 1a–e show the detailed geometry and dimensions of each part of the experimental setup.Figure 3 shows the process setup.
a Inlet section (length in mm).Create b using Openscad 2021.01, openscad.org.Main test section (length in mm).Created with Openscad 2021.01, openscad.org c Cross-sectional view of the main test section (length in mm).Created using Openscad 2021.01, openscad.org d export section (length in mm).Created with Openscad 2021.01, exploded view of the tests section of openscad.org e.Created with Openscad 2021.01, openscad.org.
Three sets of rods of different diameters were tested.Table 1 lists the geometrical characteristics of each case.The rods are mounted on a protractor so that their angle relative to the flow direction can vary between 90° and 30° (Figures 1b and 3).All rods are made of stainless steel and they are centered to maintain the same gap distance between them.The relative position of the rods is fixed by two spacers located outside the test section.
The inlet flow rate of the test section was measured by a calibrated venturi, as shown in Figure 2, and monitored using a DP Cell Honeywell SCX.The fluid temperature at the outlet of the test section was measured with a PT100 thermometer and controlled at 45±1°C.To ensure a planar velocity distribution and reduce the level of turbulence at the entrance of the channel, the incoming water flow is forced through three metal screens.A settling distance of approximately 4 hydraulic diameters was used between the last screen and rod, and the length of the outlet was 11 hydraulic diameters.
Schematic diagram of the Venturi tube used to measure the inlet flow velocity (length in millimeters).Created with Openscad 2021.01, openscad.org.
Monitor the pressure on one of the faces of the center rod by means of a 0.5 mm pressure tap at the mid-plane of the test section.The tap diameter corresponds to a 5° angular span; therefore the angular accuracy is approximately 2°.The monitored rod can be rotated about its axis, as shown in Figure 3.The difference between the rod surface pressure and the pressure at the entrance to the test section is measured with a differential DP Cell Honeywell SCX series.This pressure difference is measured for each bar arrangement, varying flow velocity, inclination angle \(\alpha \) and azimuth angle \(\theta \).
flow settings.Channel walls are shown in gray.The flow flows from left to right and is blocked by the rod.Note that view “A” is perpendicular to the rod axis.The outer rods are semi-embedded in the lateral channel walls.A protractor is used to measure the angle of inclination \(\alpha \).Created with Openscad 2021.01, openscad.org.
The purpose of the experiment is to measure and interpret the pressure drop between the channel inlets and the pressure on the surface of the center rod, \(\theta\) and \(\alpha\) for different azimuths and dips.To summarize the results, the differential pressure will be expressed in dimensionless form as Euler’s number:
where \(\rho \) is the fluid density, \({u}_{i}\) is the mean inlet velocity, \({p}_{i}\) is the inlet pressure, and \({p }_{ w}\) is the pressure at a given point on the rod wall.The inlet velocity is fixed within three different ranges determined by the opening of the inlet valve.The resulting velocities range from 6 to 10 m/s, corresponding to the channel Reynolds number, \(Re\equiv {u}_{i}H/\nu \) (where \(H\) is the height of the channel, and \(\nu \) is the kinematic viscosity) between 40,000 and 67,000.The rod Reynolds number (\(Re\equiv {u}_{i}d/\nu \)) ranges from 2500 to 6500.The turbulence intensity estimated by the relative standard deviation of the signals recorded in the venturi is 5% on average.
Figure 4 shows the correlation of \({Eu}_{w}\) with the azimuth angle \(\theta \), parameterized by three dip angles, \(\alpha \) = 30°, 50° and 70° .The measurements are split in three graphs according to the diameter of the rod.It can be seen that within the experimental uncertainty, the obtained Euler numbers are independent of the flow rate.The general dependence on θ follows the usual trend of wall pressure around the perimeter of a circular obstacle.At flow-facing angles, ie, θ from 0 to 90°, the rod wall pressure decreases, reaching a minimum at 90°, which corresponds to the gap between the rods where the velocity is greatest due to flow area limitations.Subsequently, there is a pressure recovery of θ from 90° to 100°, after which the pressure remains uniform due to the separation of the rear boundary layer of the rod wall.Note that there is no change in the angle of minimum pressure, which suggests that possible disturbances from adjacent shear layers, such as Coanda effects, are secondary.
Variation of the Euler number of the wall around the rod for different inclination angles and rod diameters.Created with Gnuplot 5.4, www.gnuplot.info.
In the following, we analyze the results based on the assumption that the Euler numbers can be estimated only by geometric parameters, i.e. the feature length ratios \(d/g\) and \(d/H\) (where \(H\) is the channel’s height) and inclination \(\alpha \).A popular practical rule of thumb states that the fluid structural force on the yaw rod is determined by the projection of the inlet velocity perpendicular to the rod axis, \({u}_{n}={u}_{i}\mathrm {sin} \alpha \) .This is sometimes called the principle of independence.One of the goals of the following analysis is to examine whether this principle applies to our case, where flow and obstructions are confined within closed channels.
Let us consider the pressure measured at the front of the intermediate rod surface, i.e. θ = 0.According to Bernoulli’s equation, the pressure at this position\({p}_{o}\) satisfies:
where \({u}_{o}\) is the fluid velocity near the rod wall at θ = 0, and we assume relatively small irreversible losses.Note that the dynamic pressure is independent in the kinetic energy term.If \({u}_{o}\) is empty (i.e. stagnant condition), the Euler numbers should be unified.However, it can be observed in Figure 4 that at \(\theta =0\) the resulting \({Eu}_{w}\) is close to but not exactly equal to this value, especially for larger dip angles.This suggests that the velocity on the rod surface does not vanish at \(\theta =0\), which may be suppressed by the upward deflection of the current lines created by the rod tilt.Since the flow is confined to the top and bottom of the test section, this deflection should create a secondary recirculation, increasing the axial velocity at the bottom and decreasing the velocity at the top.Assuming that the magnitude of the above deflection is the projection of the inlet velocity on the shaft (ie \({u}_{i}\mathrm{cos}\alpha \)), the corresponding Euler number result is:
Figure 5 compares the equations.(3) It shows good agreement with the corresponding experimental data.The mean deviation was 25%, and the confidence level was 95%.Note that the equation.(3) In line with the principle of independence.Likewise, Figure 6 shows that the Euler number corresponds to the pressure on the rear surface of the rod, \({p}_{180}\), and at the exit of the test segment, \({p}_{e}\), Also follows a trend proportional to \({\mathrm{sin}}^{2}\alpha \) .In both cases, however, the coefficient depends on the rod diameter, which is reasonable since the latter determines the hindered area.This feature is similar to the pressure drop of an orifice plate, where the flow channel is partially reduced at specific locations.In this test section, the role of the orifice is played by the gap between the rods.In this case, the pressure drops substantially at the throttling and partially recovers as it expands backwards.Considering the restriction as a blockage perpendicular to the rod axis, the pressure drop between the front and rear of the rod can be written as 18:
where \({c}_{d}\) is a drag coefficient explaining the partial pressure recovery between θ = 90° and θ = 180°, and \({A}_{m}\) and \ ({A}_{f}\) is the minimum free cross-section per unit length perpendicular to the rod axis, and its relationship to the rod diameter is \({A}_{f}/{A}_{m}=\ Left (g+d\right)/g\).The corresponding Euler numbers are:
Wall Euler number at \(\theta =0\) as a function of dip.This curve corresponds to the equation.(3).Created with Gnuplot 5.4, www.gnuplot.info.
Wall Euler number changes, in \(\theta =18{0}^{o}\) (full sign) and exit (empty sign) with dip.These curves correspond to the principle of independence, i.e. \(Eu\propto {\mathrm{sin}}^{2}\alpha \).Created with Gnuplot 5.4, www.gnuplot.info.
Figure 7 shows the dependence of \({Eu}_{0-180}/{\mathrm{sin}}^{2}\alpha \) on \(d/g\), showing the extreme Good consistency.(5).The obtained drag coefficient is \({c}_{d}=1.28\pm 0.02\) with a confidence level of 67%.Likewise, the same graph also shows that the total pressure drop between the inlet and outlet of the test section follows a similar trend, but with different coefficients that take into account the pressure recovery in the back space between the bar and the outlet of the channel.The corresponding drag coefficient is \({c}_{d}=1.00\pm 0.05\) with a confidence level of 67%.
The drag coefficient is related to the \(d/g\) pressure drop fore and aft of the rod\(\left({Eu}_{0-180}\right)\) and the total pressure drop between the channel inlet and outlet.The grey area is the 67% confidence band for the correlation.Created with Gnuplot 5.4, www.gnuplot.info.
The minimum pressure \({p}_{90}\) on the rod surface at θ = 90° requires special handling.According to Bernoulli’s equation, along the current line through the gap between the bars, the pressure in the center\({p}_{g}\) and the velocity\({u}_{g}\) in the gap between the bars ( coincides with the midpoint of the channel) is related to the following factors:
The pressure \({p}_{g}\) can be related to the rod surface pressure at θ = 90° by integrating the pressure distribution over the gap separating the central rod between the midpoint and the wall (see Figure 8 ) . The balance of power gives 19:
where \(y\) is the coordinate normal to the rod surface from the center point of the gap between the central rods, and \(K\) is the curvature of the current line at position \(y\).For the analytical evaluation of the pressure on the rod surface, we assume that \({u}_{g}\) is uniform and \(K\left(y\right)\) is linear.These assumptions have been verified by numerical calculations.At the rod wall, the curvature is determined by the ellipse section of the rod at the angle \(\alpha \), i.e. \(K\left(g/2\right)=\left(2/d\right){\ mathrm{sin} }^{2}\alpha \) (see Figure 8).Then, regarding the curvature of the streamline vanishing at \(y=0\) due to symmetry, the curvature at the universal coordinate \(y\) is given by:
Feature cross-sectional view, front (left) and above (bottom).Created with Microsoft Word 2019,
On the other hand, by conservation of mass, the average velocity in a plane perpendicular to the flow at the measurement location \(\langle {u}_{g}\rangle \) is related to the inlet velocity:
where \({A}_{i}\) is the cross-sectional flow area at the channel inlet and \({A}_{g}\) is the cross-sectional flow area at the measurement location (see Fig. 8) respectively by :
Note that \({u}_{g}\) is not equal to \(\langle {u}_{g}\rangle \).In fact, Figure 9 depicts the speed ratio \({u}_{g}/\langle {u}_{g}\rangle \), calculated by the equation.(10)–(14), plotted according to the ratio \(d/g\).Despite some discreteness, a trend can be identified, which is approximated by a second-order polynomial:
The ratio of the maximum\({u}_{g}\) and average\(\langle {u}_{g}\rangle \) velocities of the channel center cross-section\(.\) The solid and dashed curves correspond to the equations.(5) and the variation range of the corresponding coefficients\(\pm 25\%\).Created with Gnuplot 5.4, www.gnuplot.info.
Figure 10 compares \({Eu}_{90}\) with the experimental results of the equation.(16).The mean relative deviation was 25%, and the confidence level was 95%.
The Wall Euler number at \(\theta ={90}^{o}\).This curve corresponds to the equation.(16).Created with Gnuplot 5.4, www.gnuplot.info.
The net force \({f}_{n}\) acting on the central rod perpendicular to its axis can be calculated by integrating the pressure on the rod surface as follows:
where the first coefficient is the rod length within the channel, and the integration is performed between 0 and 2π.
The projection of \({f}_{n}\) in the direction of the water flow should match the pressure between the inlet and outlet of the channel, unless friction parallel to the rod and smaller due to incomplete development of the later section The momentum flux is unbalanced. Therefore,
Figure 11 shows a graph of the equations.(20) showed good agreement for all experimental conditions.However, there is a slight 8% deviation on the right, which can be attributed and used as an estimate of the momentum imbalance between the channel inlet and outlet.
Channel power balance.The line corresponds to the equation.(20).The Pearson correlation coefficient was 0.97.Created with Gnuplot 5.4, www.gnuplot.info.
Varying the inclination angle of the rod, the pressure at the rod surface wall and the pressure drop in the channel with the transverse lines of the four inclined cylindrical rods were measured.Three different diameter rod assemblies were tested.In the tested Reynolds number range, between 2500 and 6500, the Euler number is independent of flow rate.The pressure on the central rod surface follows the usual trend observed in cylinders, being maximum at the front and minimum at the lateral gap between the rods, recovering at the back part due to boundary layer separation.
Experimental data are analyzed using momentum conservation considerations and semi-empirical evaluations to find invariant dimensionless numbers that relate Euler numbers to the characteristic dimensions of channels and rods.All geometrical features of blocking are fully represented by the ratio between the rod diameter and the gap between the rods (laterally) and the channel height (vertical).
The independence principle is found to hold for most Euler numbers characterizing pressure at different locations, i.e. if the pressure is dimensionless using the projection of the inlet velocity normal to the rod, the set is independent of the dip angle. In addition, the feature is related to the mass and momentum of the flow The conservation equations are consistent and support the above empirical principle.Only the rod surface pressure at the gap between rods deviates slightly from this principle.Dimensionless semi-empirical correlations are generated that can be used to design similar hydraulic devices.This classical approach is consistent with recently reported similar applications of the Bernoulli equation to hydraulics and hemodynamics20,21,22,23,24.
A particularly interesting result stems from the analysis of the pressure drop between the inlet and outlet of the test section.Within the experimental uncertainty, the resulting drag coefficient equals unity, which indicates the existence of the following invariant parameters:
Note the size \(\left(d/g+2\right)d/g\) in the denominator of the equation.(23) is the magnitude in parentheses in the equation.(4), otherwise it can be calculated with the minimum and free cross-section perpendicular to the rod, \({A}_{m}\) and \({A}_{f}\).This suggests that Reynolds numbers are assumed to remain within the range of the current study (40,000-67,000 for channels and 2500-6500 for rods).It is important to note that if there is a temperature difference inside the channel, it may affect the fluid density.In this case, the relative change in Euler number can be estimated by multiplying the thermal expansion coefficient by the maximum expected temperature difference.
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