Experimental Quantification of Local Pressure Loss at a 90 ° Bend in Low-Pressure Dilute-Phase Pneumatic Conveying of Coarse Particles

Focusing on the insufficient estimation of the local pressure loss at a 90° horizontal-vertical bend in low-pressure pneumatic conveying of coarse particles, experiments are conducted in a 80mm inner diameter test bend by using polyethylene particles having an equivalent spherical diameter of 4.00mm.-e influences of the local pressure loss versus the gas flow Reynolds number, the solid-gas ratio, and the bending radius ratio are investigated. Based on the additional pressure theory of Barth, an empirical formula estimating the local pressure loss is obtained using dimensional and nonlinear regression analysis. Summarizing the experiments and literature, the results expound on the local gas flow pressure loss coefficient decreases with increasing Reynolds number, and first decreases and then increases with increasing bending radius ratios from 0.5 to 7. -e additional solid flow pressure loss coefficient decreases with the increasing Reynolds number and bending radius ratio in the dilute phase, and linearly increases with increasing solid-gas ratio. Compared with the estimated values with the experimental values, the calculated standard deviation is below 4.11%, indicating that the empirical formula can be used to predict local pressure loss at the bend in the low-pressure dilute-phase pneumatic conveying.


Introduction
To improve the flexibility of the pneumatic conveying system, a bend is usually regarded as an important component in the process route.However, a bend makes the flow situation complicated and causes a sharp pressure loss.Especially in engineering, pressure loss is typically used as a key parameter to guide and design pneumatic conveying systems.Hence, to reasonably estimate the local pressure loss through a bend is very significant for pneumatic conveying systems.
Considering the universality of the 90 °bend of a circular cross section, it has been applied and studied by numerous researchers.Cornish and Charity [1] listed all the important parameters (e.g., gas density and viscosity, particle density and size, bend curvature radius and diameter, the conveying velocity, and mass flow rate) of the local pressure loss at a 90 °bend for a given pneumatic conveying system and found that the local pressure loss is higher for a short bend curvature radius and linearly increases with the increasing solid-gas ratio (i.e., the mass flow rate of solids to gas).Ghosh and Kalyanaraman [2] studied the local pressure loss in dilute-phase (solid-gas ratio <5.3) pneumatic conveyance of coarse particles (e.g., wheat) for a horizontal-horizontal bend.e results show that the additional pressure loss coefficient is constant for all conveying velocities and is a linear function of the solid-gas ratio.Singh and Wolfe [3] considered the angle of bend deflection (i.e., change in flow direction), the changes of local pressure loss versus the conveying velocity, the mass flow rate of solids, and the bend curvature radius were investigated in the pneumatic conveying.An empirical formula for local pressure loss was deduced, but the impact of gravity was neglected.Mason and Smith [4] and Rossetti [5] reported that the local pressure loss includes gas flow and additional pressure loss of solids.ey took the local pressure loss of gas-only flow for a constant, and the additional pressure loss of solids is closely correlated with the particle terminal velocity.Westman et al. [6], based on the additional pressure theory of Barth [7], found that larger curvature radius bends produce lower pressure loss in vacuum pneumatic conveying systems.An empirical formula for local pressure loss was derived from the gasvelocity at the bend exit.Yu and Wang [8] fully considered the bend pressure loss upstream and downstream, where the gas-solid flow was impacted by the bend.ey found that the local pressure loss increases with increasing solidgas ratio in dense-phase conveying of powder and derived an empirical formula for estimating local pressure loss.However, its applicability is limited due to only one bending radius ratio (i.e., the bend curvature radius to diameter).Pan and Chi [9] investigated the effects of local pressure loss for different angles and short curvature radius bends.e results show that the local pressure loss (gas only) decreases with the increasing bend angle and curvature radius, and it proportionally increases with the increasing solid-gas ratio.Moreover, the gas-solid flow was difficult to fully develop in the short straight pipe after the bend.Pan [10] proposed an accurate way of estimating the local pressure loss at a bend (here, the straight pipe and bend are separately dealt with) in high-pressure conveying of fly ash.Based on numerous experiments, a semiempirical formula was set up to predict additional pressure loss using mathematical and dimensional analyses and starting from the bend exit conditions.After that, Pan and Wypych [11] also considered the compressibility of gas flow due to conveying pressure and mixed particles.On the basis of Barth's [7] and Ito's [12] researches, the semiempirical formula [10] of local additional pressure loss was corrected by the bend exit conditions (e.g., average gas density, velocity, and solid-gas ratio).Furthermore, Das and Meloy [13] found that the local pressure loss through a doublecoupled bend conveying solid material is not equivalent to the cumulative effect of two separated bends.
e local pressure loss in a double-coupled double bend is less than twice of that in a single bend.Liang et al. [14] considered the influences of the different materials, bend curvature radii, and locations (i.e., horizontal-horizontal, horizontalvertical, and vertical-horizontal bends) in high-pressure dense-phase of pulverized coal conveying.ey found that the pressure loss of a horizontal-vertical bend is the largest than the horizontal-horizontal and vertical-horizontal bends, and the local pressure loss increases with increasing coal size.Accordingly, the corresponding empirical formulas of different bend locations were obtained to estimate the additional pressure loss using Barth's theory [7] and multivariate regression analysis.
Recently, in order to study the applicability of the present empirical formulas, Naveen et al. [15] compared the values calculated by the formulas [3,5,6,10,11,13] with experimental values in conveying fly ash, and the results indicate that only the empirical formulas of Pan [10] and Pan and Wypych [11] could be applied to estimate the local pressure loss at bends in a dilute-phase regime.However, the formulas in [10,11] were only verified in the pneumatic conveying of powders (e.g., fly ash and pulverized coal), and the flow parameters (e.g., gas density, gas velocity, and mass flow rate) of gas-solid flow at the bend exit are difficult to obtain, causing the inconvenience in the design of dilutephase pneumatic conveying systems in the beginning.In addition, previous literature is almost aimed at the powders [3-5, 7-10, 12-14] or high-pressure pneumatic conveying [9,10,[13][14][15].From the above, the local pressure loss estimates for a bend in low-pressure dilute-phase pneumatic conveying (e.g., air supply from roots blower) of coarse particles are very insufficient.
In this study, based on a low-pressure dilute-phase pneumatic conveying system, the influences of local pressure loss arising from the change of superficial conveying gas velocity, particle mass flow rate, and bending radius ratio are investigated in a horizontal-vertical 90 °bend.Meanwhile, to provide theoretical support for designing the low-pressure dilute-phase pneumatic conveying systems as conveying the coarse particles, an empirical formula of the local pressure loss is derived using dimensional and nonlinear regression analysis.

Experimental Apparatus
e experimental apparatus of the low-pressure pneumatic conveying system is shown in Figure 1. e system consists of a fan, feed bin, rotary value, separator, bag filter, and several section pipes.e test pipeline frame has a horizontal length L � 4.0 m and a vertical height H � 2.5 m, connected by two 90 °bends and a short straight pipe.e pipes are made of organic glass and have the same inner diameter D � 80 mm ± 5.85%.When conveying, air from the fan blows away the particles fed by the rotary value into the test pipeline, and then, the gas-solid mixture is separated by the separator at the pipeline exit.Meanwhile, the gas flow rate and pressure are measured by the orifice meter and four pressure sensors (P 1 , P 2 , P 3 , and P 4 ), and the particle mass flow rate is controlled by the rotation speed of the rotary value.
Figure 2 shows three different bend radii of R � 300 mm, 400 mm, and 500 mm.ey are named as R300, R400, and R500 and corresponded to ratios of curvature radius and inner diameter R/D � 3.75, 5, and 6.25, respectively.Figure 3 shows the experimental particle made of polyethylene with an equivalent spherical diameter of d s � 4.00 mm ± 5.16%.

The Local Pressure Loss
3.1.e Local Pressure Loss of Gas Flow. Figure 4 shows the local gas flow pressure loss (Δp eg ) versus the different conveying velocities (U g ) and the bend radii (R).Clearly, the local pressure loss increases with increasing superficial conveying velocity and bend radius.Researchers [16][17][18][19][20][21] found that the local pressure loss is mainly generated by the flow separation emerging at the concave inner wall and the 2 International Journal of Chemical Engineering secondary ow arising from the radial pressure gradient at bend.erefore, with increasing super cial conveying velocity, the ow separation and secondary ow are enhanced by the increased inertial force and radial pressure gradient in the bend [18,20].And in the case of the same gas velocity, the ow separation and the secondary ow can be weakened by the large bend radius; however, the wall friction is the dominant factor contributing to pressure loss in the bend [21].e more pressure loss is generated for the more ow distance.

e Local
Pressure Loss of Gas-Solid Flow. Figure 5 shows the local gas-solid ow pressure loss (Δp e ) for the di erent bend radii at G s 0.3 and 0.5 kg/s.It is obvious that the local gas-solid ow pressure loss (Δp e ) decreases rst and then increases with increasing conveying velocity.ere exists a gas velocity of the minimum local pressure loss, de ned as the optimum conveying velocity.As proposed by Zenz [22], the optimum conveying velocity is the critical point between the dense-phase (the gas velocity is lower than the optimum conveying velocity) and dilute-phase (the gas velocity is larger than the optimum conveying velocity) in the same mass ow rate and bend radius.In a dense-phase, higher pressure loss is arisen from the dune ow, slug ow, and other unstable ows in the pipe; serious friction is occurred between the massive particles and the pipe; and the bend can even be blocked by particles lacking kinetic energy in the pattern of dune ow or slug ow.Whereas, in dilute-phase, strati ed ow, suspended ow, and other stable ows are distributed in the pipe.e particle kinetic energy is enhanced by increasing conveying velocity, causing violent and frequent collisions among particle and other particles or pipe walls.Hence, very larger conveying velocity can result in larger pressure loss [23,24].International Journal of Chemical Engineering 3 In addition, Figure 5 also obviously shows the local gassolid ow pressure loss increases with increasing mass ow rate (G s ) in the same bend.
e reason is that with the increase of the mass ow rate, the increase of the solid-gas ratio consumes massive gas kinetic energy [24,25].In the same mass ow rate, the local gas-solid ow pressure loss increases with increasing bend radius in the lower gas velocity region (U g < 18.51 m/s for G s 0.3 kg/s; and U g < 19.64 m/s for G s 0.5 kg/s), but decreases with increasing bend radius in the larger gas velocity region (U g > 18.51 m/s for G s 0.3 kg/s; and U g > 19.64 m/s for G s 0.5 kg/s).ese phenomena are similar to the conclusions of Pan [10].Because friction pressure loss plays the leading role in lower gas velocity, the pressure loss is increased by the long sliding distance in the larger bend radius means.And the gravity pressure loss is also increased by the higher height of the larger radius means at the horizontalvertical bend.However, at higher gas velocity, the inertial centrifugal force of the gas-solid ow is decreased by the larger bend radius for the same mass ow rate and bend radius, and collisions between particles and walls are reduced for the slow vector change of gas-solid ow and lessened ow separation [24][25][26][27].

Model of Pressure Loss
To set up a formula for estimating the local pressure loss through a horizontal-vertical bend, according to the additional pressure theory of Barth [7], the local gas-solid ow pressure loss through bend Δp e consists of the pressure loss of gas-phase Δp eg and additional pressure loss of solid-phase Δp es , which can be expressed as follows: where ζ eg is the local pressure loss coe cient of gas ow, ζ es is the additional pressure loss coe cient of solid ow, ρ g is the density of gas, and U g (U g Q g /A) is the super cial conveying velocity, in which Q g is the volume ow rate of gas ow and A is the conveying pipe sectional area.Since the local gas ow pressure loss coe cient ζ eg is mainly impacted by the gas velocity and bend radius, the additional solid ow pressure loss coe cient ζ es , besides the above factors, is also impacted by the particle mass ow rate.Hence, they can be expressed as follows: And after nondimensional disposal, it follows the following equation: where m s G s /ρ g U g A is the solid-gas ratio and Re ρ g U g D/μ g is the Reynolds number in which μ g denotes the gas viscosity.and increases with the increasing bending radius ratio.According to previous researchers [6,12,28,29], the local pressure loss coe cient (only air ow) ζ eg tends to be stable with increasing Reynolds number.erefore, the ζ eg can be inversely correlated proportional to the Reynolds number.

e Local Pressure Loss Coe cient of Gas Flow.
To investigate the change in ζ eg with increasing bending radius ratio from 0.5 to 7 in detail, the researches [9,12,30] that have the similar test conditions are considered, as shown in Figure 7.It is obvious that ζ eg is quickly decreased and then increased at Reynolds numbers Re 0.82 × 10 5 , 1.00 × 10 5 , and 1.18 × 10 5 .is result indicates that the relationship between ζ eg and the radius ratio R/D can be correlated to the Nike function [31] (Figure 8).
Accordingly, the local pressure loss coe cient (only air ow) ζ eg is deduced by using the nonlinear regression analysis as follows: In this study, nonlinear regression and analysis of variance (ANOVA) are used to analyse the reliability of the model for the local pressure loss coe cient (only air ow), as shown in Table 1.e R 2 0.921 ≈ 1 and adj-R 2 0.904 indicate that the tted model has high accuracy in the 95% con dence interval.Meanwhile, compared tted with experiment values, as shown in Figure 9, the relative deviation stays between −2.69% and +4.54%, and the calculated standard deviation is 1.52%, proving the reliability of this tted model.

e Additional Pressure Loss Coe cient of Solid Flow.
Figure 5 illustrates that lower gas velocity is bad for pneumatic conveying in unstable ow.Aiming at industrial processes, the gas velocity is usually designed larger than the optimum conveying velocity to prevent pipeline blockage.Accordingly, Figure 10 shows the relationship between the additional solid ow pressure loss coe cient (ζ es ) and the Reynolds number (Re), the bending radius ratio (R/D), and the solid-gas ratio (m s ) in dilute-phase pneumatic conveying.As shown in Figure 10(a), it is evident that ζ es decreases with increasing Re and R/D (3.75∼6.25).As shown in Figure 10(b), ζ es is increased in proportion to the increasing solid-gas ratio m s .

Experiment Pan & Chi [9]
Ito [12] Iwasaki [24] Re (×10 5 ) 0.82 1.00 1.18   International Journal of Chemical Engineering Accordingly, the additional solid ow pressure loss coe cient of ζ es could be deduced by nonlinear regression analysis: In this study, nonlinear regression and ANOVA are used to analyse the reliability for the additional solid ow pressure loss coe cient, as shown in Table 2. e R 2 0.907 ≈ 1 and adj-R 2 0.892 indicate that this tting model has high accuracy in the 95% con dence interval.Meanwhile, the comparison between the tted and experiment values are shown in Figure 9, the relative deviation stays between −9.992% and +7.754%, and the calculated standard deviation is 4.52%, which could prove the reliability of this model.

e Veri cation of Prediction Model for Pressure Loss.
To verify the model reliability, the gas ow pressure loss values from experiment and prediction at a bend are compared in Figure 11(a) and the pressure loss values of gas-solid ow are compared in Figure 11(b) in the case of dilute-phase pneumatic conveying.e results show that the relative deviation of gas ow pressure loss stays between −5% and +5%, and the calculated standard deviation is 3.52%.Meanwhile, the relative deviation of gas-solid ow pressure loss stays between +10% and −10%, and the calculated standard deviation is 4.11%, which indicate that the local pressure loss is more accurately predicted by the empirical formulas in this paper.

Conclusions
To accurately estimate the local pressure loss in low-pressure dilute-phase pneumatic conveying of coarse particles, in this   6 International Journal of Chemical Engineering study, the in uences arising from di erent super cial conveying velocities, particles mass ow rates, and bending radius ratios are investigated in a horizontal-vertical 90 °bend.
Considering the gravity pressure loss, the empirical formula of the local pressure loss is derived using dimensional and nonlinear regression analysis.e signi cant conclusions are as follows: (1) In the experiment, the local pressure loss of gas ow increases with increasing conveying velocity and bend curvature radius (2) For the short straight pipe, a long radius bend causes the large local pressure loss at lower conveying velocity and low local pressure loss at higher conveying velocity for gas-solid ow (3) Synthesized from the present work and previous researches, the local gas ow pressure loss coe cient decreases with increasing Reynolds number, rst decreases and then increases with the increasing bending radius ratio from 0.5 to 7 (4) e additional pressure loss coe cient of solid ow decreases with the increasing Reynolds number and bending radius ratio and increases in proportion with the increasing gas-solid ratio (5) From the comparison of the estimated local pressure loss values with experimental values in a horizontal-   International Journal of Chemical Engineering vertical 90 °bend, the relative deviation is ±10% and the calculated standard deviation is below 4.11%

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Figure 6 Figure 4 :Figure 5 :
Figure 4: e local pressure loss of air ow versus conveying velocities and bend radii.

Figure 7 :
Figure 7: Relationships between local pressure loss coe cient (only air ow) and bend radii for the di erent Re.

Figure 8 : 5 R
Figure 8: Comparison of local pressure loss coe cient (only air ow) between the experimental values and the tted values.

Figure 6 :
Figure 6: Relationships between local pressure loss coe cient (only air ow) and Reynolds number for the di erent bend radii.

Figure 9 :
Figure 9: Relationships between the additional pressure loss coe cient of solid ow and (a) Reynolds number and (b) solid-gas ratio.

Figure 10 :
Figure 10: Comparison of the additional pressure loss coe cient of solid ow between experimental values and predicted values.

Figure 11 :
Figure 11: e deviation between the experimental value and the predicted value of the local pressure loss of gas-solid ow.

Table 1 :
Analysis of nonlinear regression.

Table 2 :
Analysis of nonlinear regression.