# Gas Flow Characteristics through Irregular Particle Bed with the Vertical Confined Wall for Waste Heat Recovery

Authors: Sizong Zhang; Zhi Wen; Xunliang Liu; Yi Xing; Hui Zhang
Journal: International Journal of Photoenergy
 (2022)

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## Abstract

The vertical waste heat recovery technology of the sinter in the iron and steel industry will be a great driving force for China to realize the “double Carbon” in the near future. For promoting the application of the new technology, the influence of the confined wall on the pressure distribution and pressure drops of the gas flow in the sinter bed was experimentally studied. For the irregular sinter with the rough surface, the gas pressure near the wall is higher than that at the center. Moreover, the radial distribution of the dimensionless pressure is nearly the same at different gas velocities. Therefore, whatever flow state the gas is in, the wall effect on irregular sinters only reduces the pressure drop of the gas flow, which is different from that on spherical particles. The vertical wall limits the randomness and uniformity of the particle accumulation, which is further intensified with the increase of the particle irregularity and particle size. Therefore, the confined wall causes a greater difference in the gas pressure between the wall and the center. With the particle size increasing from 5~10 mm to 55~60 mm, the ratio of the gas pressure between the wall and the center increases from 1.03 to 1.26. If the wall effect is ignored, the pressure drop of the gas flow would be overestimated by 16.01% on average, whereas the correlation of the wall correction can well predict the pressure drop with the mean error and maximum error of 2.74% and 9.48%, respectively.


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## Body

## 1. Introduction

To realize the development system of the green economy, China has put forward the targets of the Carbon Peak in 2030 and Carbon Neutrality in 2060. At present, there are mainly two technical routes to solve this problem. On the one hand, it is the large-scale application of the new energy, such as the solar energy. On the other hand, it is the energy conservation and emission reduction of the traditional industry, such as the iron and steel industry [1–3]. The latest statistics show that the energy consumption of the iron and steel industry accounts for about 20% of the whole industry. In all processes of the iron and steel industry, the energy consumption of the sintering process ranks second, which is about 19% higher than the international level. Therefore, the efficient recovery of the waste heat of the sinter is the most promising method to reduce the energy consumption of the iron and steel industry. This will be a great driving force for China to realize the “double Carbon” in the near future.To improve the rate of the energy-saving and emission reduction, the sinter vertical tank cooling (SVTC) process has been newly proposed by imitating the coke dry quenching (CDQ) process [4]. Compared with the existing annular cooling process, the SVTC process can increase the recovery rate of the sensible heat from 30% to 80% and reduce the air leakage rate from 35~55% to nearly 0%. Before the industrial application, it is necessary to study the feasibility of the SVTC process in the laboratory scale, that is, gas-solid heat transfer characteristics and gas flow characteristics [5, 6]. Gas flow characteristics not only affect the gas-solid heat transfer but also determine the energy consumption of fans, thus affecting the feasibility and economy of the process [5, 7–9]. Therefore, it is of great significance to reliably predict the pressure drop of the gas in the sinter vertical tank.Since the shape of the sinter is very irregular and its particle size is large, the sinter vertical tank is essentially the packed bed with irregular and large particles [9, 10]. Therefore, some related researches have been carried out to accurately predict the pressure drop of the gas in the sinter bed in recent five years [5–13]. Results show that the pressure drop of the gas increases linearly with the height of the sinter layer increasing, increases in a quadratic relationship with the gas velocity increasing, and declines exponentially with the equivalent particle size increasing. Moreover, considering the influences of the particle size [5, 12], particle size distribution [8, 13], voidage [7], particle shape [10, 12], and wall effect [6, 11], the empirical correlations of the pressure drop are obtained by modifying the Ergun equation. However, the application of the Ergun equation is based on the assumption of the uniform packed bed, which ignores the wall effect of finite packed beds in practice [14–22]. It can only accurately describe infinite packed beds or packed beds with the large ratio of the bed diameter to particle diameter (D/dp>50:1). Owing to the existence of the confined wall in real beds, particles can only be packed finitely near the wall. This would increase the wetted surface area and the local voidage near the wall. Compared to the predicted value obtained the by the Ergun equation for the same average voidage, the combined effect may lead to the increase or decrease in the pressure drop of the gas flow [20].For studying the wall effect, a lot of effort has been done. However, different conclusions have been drawn [14–30]. Some researchers have found that the wall effect can increase the pressure drop of the gas flow compared with an identical infinite bed [14], while others have found that this would cause a decline [23]. However, most studies indicate that the effect of the confined wall depends not only on the value of D/dp, but also on the Reynolds number or the gas flow regime [16, 18, 20, 22, 24, 25]. Whatever the Reynolds number is, the wall effect can be ignored when D/dp is higher than 50, whereas when D/dp is lower than 50, the combined influences of the wall friction and the voidage would have opposite results on the pressure drop at different Reynolds numbers. At the low Reynolds number (i.e., laminar flow), the friction effect of the wall plays a prominent role, leading to the increase in the pressure drop of the gas, while the voidage effect is dominant at the high Reynolds number (i.e., turbulent flow), causing a decline in the pressure drop of the gas.However, most of previous studies on the wall effect are focused on spherical particles. To our best knowledge, there are few studies on irregular particles due to the complexity of the particle shape [31–33]. Although Feng et al. [6] and Tian et al. [11] considered the wall effect to modify the equation of the pressure drop, the influences of the confined wall on the pressure distribution and pressure drops of the gas flow in the sinter bed are still unclear. Therefore, this paper studies experimentally the wall effect on the gas flow characteristics in the bed with irregular sinters under the background of the SVTC process. Furthermore, the accuracy of two correlations without and with the wall correction in predicting the pressure drop is compared.


## 2. Experimental Method and Data Processing

### 2.1. Experimental Apparatus

Figure1 is a schematic diagram of the experimental apparatus for measuring the gas pressure in the sinter bed. This apparatus is composed of the fan, the cylindrical packed bed, and the system of the measurement and acquisition. The frequency conversion blower is selected for the gas supply to precisely control the gas flow rate in the test. As the main part of the apparatus, the height (H) and inner diameter (D) of the packed bed are 1000 mm and 400 mm, respectively. The vortex flowmeter with the compensation of the temperature and pressure is applied to measure the gas flow rate under the standard condition. To acquire the information of the gas pressure inside the bed, a pressure transmitter with the length of 650 mm is used, and the measured data are collected by the paperless recorder. Basic parameters of above instruments are illustrated in Table 1.Figure 1
The sketch of the experimental apparatus for measuring the gas pressure in the sinter bed.Table 1
Basic parameters of experimental instruments.
InstrumentModeRangeAccuracyFrequency conversion blowerHRD 65FU-100/7.53720 m3·h-1, 9 kPa,100 Hz—Vortex flowmeterLUGB1315C-P3Z150~3000 m3·h-11.0%Pressure transmitterCGYL-2020~10 kPa0.5%Paperless recorderLD-300G4~20 mA, 8 channels0.2%The gas operation state in the experiment is designed according to that of the sinter vertical cooling process. The gas is first blown into the packed bed from the bottom, transported through the sinter layer, and then discharged to the outside of the bed from the outlet. To make the gas flow uniformly distributed on the cross section of the entrance, an air distributor with uniform openings is installed at the bottom. Six pressure taps with an interval of 200 mm are evenly arranged along the vertical direction. For analyzing the influence of the wall effect, thirteen measuring points with an interval of the 25 mm are set uniformly along the radial direction at each pressure tap. Then, these measured data are used to calculate the average gas pressure of the cross section.


### 2.2. Experimental Materials and Characterization Methods

The sinter particles are from HBIS Group Hansteel Company in China. All experiments are carried out within the representative particle size range of 5~60 mm [9, 34–36]. It is sieved into 11 kinds of particle sizes in the interval of 5 mm, as shown in Figure 2. It can be seen that the sinter shape is very irregular, and its surface is rough and porous.Figure 2
Photographs taken on 11 kinds of the sieved sinter samples within the ranges of (a) 5~10 mm, (b) 10~15 mm, (c) 15~20 mm, (d) 20~25 mm, (e) 25~30 mm, (f) 30~35 mm, (g) 35~40 mm, (h) 40~45 mm, (i) 45~50 mm, (j) 50~55 mm, and (k) 55~60 mm.To describe the sinter in detail, the particle characteristics are characterized in this paper. Firstly, the apparent densityρa for the sinter of each particle size is measured by the displacement method [8, 9, 11], as shown in the following equation:
(1)ρa=m1m3−m2−m4ρH2O,where ρH2O is the density of the water; m1 and m3 are the mass of the dry sinter and the wet sinter, respectively; m2 is the mass of the sinter and test basket in the water; m4 is the mass of the test basket in the water.Besides, the bulk densityρb is measured by the direct weighing method [8, 9, 11], as follows:
(2)ρb=G1−G2V,where G1 is the total mass of the sinter and test container; G2 and V are the mass and volume of the test container, respectively.The bed voidageε and shape factor ϕ of sinter particles are determined by Equation (3) [8–11] and Equation (4) [33, 37], respectively, as follows:
(3)ε=1−ρbρa,(4)ϕ=0.7295ε2−2.4963ε+1.8537.In addition, 50 sinters of each particle size are randomly selected, and the equivalent particle sizedp is obtained by the equal volume method [8–11], as follows:
(5)dp=6msπρa,3where ms is the mean mass of the single sinter.The particle characteristic parameters of the sinter obtained by above methods are shown in Table2. A wide range of the gas velocity is also designed for the test condition of each kind of the sinter.Table 2
Particle characteristic parameters and test conditions for 11 kinds of sinters.
Particle size (mm)Equivalent particlesizedp (mm)Ratio of bed to particlediameterD/dpShape factorϕApparent densityρa (kg·m-3)Bulk densityρb (kg·m-3)VoidageεGas velocityug (m·s-1)5~105.5671.620.6633847.611643.850.57270.4, 0.8, 1.2, 1.6, 2.010~1510.9036.690.6293726.411513.400.59380.4, 0.8, 1.2, 1.6, 2.0, 2.415~2014.4827.620.6123656.281448.690.60370.4, 0.8, 1.2, 1.6, 2.0, 2.4, 2.820~2519.2120.820.6073614.621419.460.60730.4, 0.8, 1.2, 1.6, 2.0, 2.4, 2.825~3023.2817.180.5993561.801382.780.61180.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.230~3526.8214.910.5913547.141359.180.61680.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.235~4030.8212.980.5863536.251342.540.62040.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.640~4534.9911.430.5813526.841327.550.62360.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.645~5038.4910.390.5723523.351306.080.62930.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.650~5543.279.240.5683523.091297.220.63180.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.655~6047.398.440.5613522.731282.110.63610.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.6


### 2.3. Data Processing

For infinite packed beds with spherical particles, the pressure drop of the gas flow is usually predicted by the Ergun equation, as follows [6, 7, 9, 12]:
(6)ΔPL=k11−ε2μg⋅ugε3dp2+k2ρg1−ε⋅ug2ε3dp,where k1 and k2 are the viscous loss coefficient and the inertial loss coefficient, respectively; ΔP, L, and ΔP/L are the pressure drop of the gas flow through the sinter layer, the height of the sinter layer, and the pressure drop per unit height, respectively; ug, ρg, and μg are the velocity, density, and dynamic viscosity of the gas, respectively.For irregular particles, the shape factorϕ is often introduced to modify the correlation of the pressure drop, as follows [10, 12]:
(7)ΔPL=k11−ε2μg⋅ugε3ϕdp2+k2ρg1−ε⋅ug2ε3ϕdp.To facilitate the linear fitting, the dimensionless parameters, i.e., friction pressure dropfp and particle Reynolds number Rep, are usually introduced, as follows [6, 7, 9, 12]:
(8)fp=ΔPLε3ϕdp21−ε2μgug,Rep=ρgugϕdpμg1−ε.Then, Equation (7) can be simplified to the dimensionless form expressed by
(9)fp=k1+k2Rep.For finite packed beds, the wall corrected factorM is introduced as follows [6, 11, 20, 22, 30]:
(10)M=1+2dp3D1−ε.Then, Equation (9) can be arranged as follows:
(11)fw=k1w+k2wRew,where fw is the wall-modified friction pressure drop, fw=fp/M2; Rew is the wall-modified particle Reynolds number, Rew=Rep/M [6, 11]; k1w and k2w are the wall-modified viscous loss coefficient and the wall-modified inertial loss coefficient, respectively.Moreover, Table3 shows the relative uncertainty of main parameters estimated by the error transfer formula [8–11]. For example, the particle Reynolds number Rep can be calculated as follows:
(12)δRepRep=δρgρg2+δugug2+δdpdp2+δϕϕ2+δμgμg2+δ1−ε1−ε21/2,where δ is the absolute uncertainty of the corresponding parameter.Table 3
Summary of the relative uncertainty of main parameters.
ParameterSymbolUncertainty/%ParameterSymbolUncertainty/%Equivalent particle sizedp3.07Pressure drop per unit heightΔP/L4.45Apparent densityρa1.42Particle Reynolds numberRep4.87Bulk densityρb2.15Friction pressure dropfp6.52Voidageε2.58Wall-modified Reynolds numberRew6.31Shape factorϕ2.58Wall-modified friction pressure dropfw7.65


## 2.1. Experimental Apparatus

Figure1 is a schematic diagram of the experimental apparatus for measuring the gas pressure in the sinter bed. This apparatus is composed of the fan, the cylindrical packed bed, and the system of the measurement and acquisition. The frequency conversion blower is selected for the gas supply to precisely control the gas flow rate in the test. As the main part of the apparatus, the height (H) and inner diameter (D) of the packed bed are 1000 mm and 400 mm, respectively. The vortex flowmeter with the compensation of the temperature and pressure is applied to measure the gas flow rate under the standard condition. To acquire the information of the gas pressure inside the bed, a pressure transmitter with the length of 650 mm is used, and the measured data are collected by the paperless recorder. Basic parameters of above instruments are illustrated in Table 1.Figure 1
The sketch of the experimental apparatus for measuring the gas pressure in the sinter bed.Table 1
Basic parameters of experimental instruments.
InstrumentModeRangeAccuracyFrequency conversion blowerHRD 65FU-100/7.53720 m3·h-1, 9 kPa,100 Hz—Vortex flowmeterLUGB1315C-P3Z150~3000 m3·h-11.0%Pressure transmitterCGYL-2020~10 kPa0.5%Paperless recorderLD-300G4~20 mA, 8 channels0.2%The gas operation state in the experiment is designed according to that of the sinter vertical cooling process. The gas is first blown into the packed bed from the bottom, transported through the sinter layer, and then discharged to the outside of the bed from the outlet. To make the gas flow uniformly distributed on the cross section of the entrance, an air distributor with uniform openings is installed at the bottom. Six pressure taps with an interval of 200 mm are evenly arranged along the vertical direction. For analyzing the influence of the wall effect, thirteen measuring points with an interval of the 25 mm are set uniformly along the radial direction at each pressure tap. Then, these measured data are used to calculate the average gas pressure of the cross section.


## 2.2. Experimental Materials and Characterization Methods

The sinter particles are from HBIS Group Hansteel Company in China. All experiments are carried out within the representative particle size range of 5~60 mm [9, 34–36]. It is sieved into 11 kinds of particle sizes in the interval of 5 mm, as shown in Figure 2. It can be seen that the sinter shape is very irregular, and its surface is rough and porous.Figure 2
Photographs taken on 11 kinds of the sieved sinter samples within the ranges of (a) 5~10 mm, (b) 10~15 mm, (c) 15~20 mm, (d) 20~25 mm, (e) 25~30 mm, (f) 30~35 mm, (g) 35~40 mm, (h) 40~45 mm, (i) 45~50 mm, (j) 50~55 mm, and (k) 55~60 mm.To describe the sinter in detail, the particle characteristics are characterized in this paper. Firstly, the apparent densityρa for the sinter of each particle size is measured by the displacement method [8, 9, 11], as shown in the following equation:
(1)ρa=m1m3−m2−m4ρH2O,where ρH2O is the density of the water; m1 and m3 are the mass of the dry sinter and the wet sinter, respectively; m2 is the mass of the sinter and test basket in the water; m4 is the mass of the test basket in the water.Besides, the bulk densityρb is measured by the direct weighing method [8, 9, 11], as follows:
(2)ρb=G1−G2V,where G1 is the total mass of the sinter and test container; G2 and V are the mass and volume of the test container, respectively.The bed voidageε and shape factor ϕ of sinter particles are determined by Equation (3) [8–11] and Equation (4) [33, 37], respectively, as follows:
(3)ε=1−ρbρa,(4)ϕ=0.7295ε2−2.4963ε+1.8537.In addition, 50 sinters of each particle size are randomly selected, and the equivalent particle sizedp is obtained by the equal volume method [8–11], as follows:
(5)dp=6msπρa,3where ms is the mean mass of the single sinter.The particle characteristic parameters of the sinter obtained by above methods are shown in Table2. A wide range of the gas velocity is also designed for the test condition of each kind of the sinter.Table 2
Particle characteristic parameters and test conditions for 11 kinds of sinters.
Particle size (mm)Equivalent particlesizedp (mm)Ratio of bed to particlediameterD/dpShape factorϕApparent densityρa (kg·m-3)Bulk densityρb (kg·m-3)VoidageεGas velocityug (m·s-1)5~105.5671.620.6633847.611643.850.57270.4, 0.8, 1.2, 1.6, 2.010~1510.9036.690.6293726.411513.400.59380.4, 0.8, 1.2, 1.6, 2.0, 2.415~2014.4827.620.6123656.281448.690.60370.4, 0.8, 1.2, 1.6, 2.0, 2.4, 2.820~2519.2120.820.6073614.621419.460.60730.4, 0.8, 1.2, 1.6, 2.0, 2.4, 2.825~3023.2817.180.5993561.801382.780.61180.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.230~3526.8214.910.5913547.141359.180.61680.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.235~4030.8212.980.5863536.251342.540.62040.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.640~4534.9911.430.5813526.841327.550.62360.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.645~5038.4910.390.5723523.351306.080.62930.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.650~5543.279.240.5683523.091297.220.63180.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.655~6047.398.440.5613522.731282.110.63610.4, 0.8, 1.2, 1.6, 2.0, 2.8, 3.6


## 2.3. Data Processing

For infinite packed beds with spherical particles, the pressure drop of the gas flow is usually predicted by the Ergun equation, as follows [6, 7, 9, 12]:
(6)ΔPL=k11−ε2μg⋅ugε3dp2+k2ρg1−ε⋅ug2ε3dp,where k1 and k2 are the viscous loss coefficient and the inertial loss coefficient, respectively; ΔP, L, and ΔP/L are the pressure drop of the gas flow through the sinter layer, the height of the sinter layer, and the pressure drop per unit height, respectively; ug, ρg, and μg are the velocity, density, and dynamic viscosity of the gas, respectively.For irregular particles, the shape factorϕ is often introduced to modify the correlation of the pressure drop, as follows [10, 12]:
(7)ΔPL=k11−ε2μg⋅ugε3ϕdp2+k2ρg1−ε⋅ug2ε3ϕdp.To facilitate the linear fitting, the dimensionless parameters, i.e., friction pressure dropfp and particle Reynolds number Rep, are usually introduced, as follows [6, 7, 9, 12]:
(8)fp=ΔPLε3ϕdp21−ε2μgug,Rep=ρgugϕdpμg1−ε.Then, Equation (7) can be simplified to the dimensionless form expressed by
(9)fp=k1+k2Rep.For finite packed beds, the wall corrected factorM is introduced as follows [6, 11, 20, 22, 30]:
(10)M=1+2dp3D1−ε.Then, Equation (9) can be arranged as follows:
(11)fw=k1w+k2wRew,where fw is the wall-modified friction pressure drop, fw=fp/M2; Rew is the wall-modified particle Reynolds number, Rew=Rep/M [6, 11]; k1w and k2w are the wall-modified viscous loss coefficient and the wall-modified inertial loss coefficient, respectively.Moreover, Table3 shows the relative uncertainty of main parameters estimated by the error transfer formula [8–11]. For example, the particle Reynolds number Rep can be calculated as follows:
(12)δRepRep=δρgρg2+δugug2+δdpdp2+δϕϕ2+δμgμg2+δ1−ε1−ε21/2,where δ is the absolute uncertainty of the corresponding parameter.Table 3
Summary of the relative uncertainty of main parameters.
ParameterSymbolUncertainty/%ParameterSymbolUncertainty/%Equivalent particle sizedp3.07Pressure drop per unit heightΔP/L4.45Apparent densityρa1.42Particle Reynolds numberRep4.87Bulk densityρb2.15Friction pressure dropfp6.52Voidageε2.58Wall-modified Reynolds numberRew6.31Shape factorϕ2.58Wall-modified friction pressure dropfw7.65


## 3. Experimental Results and Discussion

### 3.1. Effect of the Confined Wall on the Bulk Density and Bed Voidage

Figure3 shows the changes of the bulk density ρb and the bed voidage ε with the particle size under five repeated experiments. With the particle size increasing, ρb decreases and ε increases, respectively. This change trend is consistent with the published results [6, 9, 10]. It can be observed from Table 2 that the particle shape factor ϕ gradually reduces with the increase in the particle size. This means that the irregularity of sinters increases, which makes it easier to form bridges when the sinter particles are heaped up. Therefore, the voids between particles in the bed would increase with the increase of the particle size. Besides, it is also found that the reproducibility and uncertainty of ρb and ε are getting worse with D/dp decreasing, which is similar to results obtained by Tian et al. [11] and Raichura [17]. This indicates that the wall effect limits the randomness and uniformity of the particle accumulation in the bed. What's more, the influence of the confined wall becomes more significant with the increase in the particle irregularity and particle size. This further intensifies the nonrandomness and heterogeneity of the particle accumulation.Figure 3
Variation of the bulk densityρb and voidage ε of the sinter bed with the particle size under five repeated experiments.


### 3.2. Effect of the Confined Wall on the Gas Pressure Distribution

Figure4 illustrates the radial distribution of the gas pressure (P) at the bed height of L=0 mm under four kinds of particle sizes (i.e., 5~10 mm, 20~25 mm, 40~45 mm, and 55~60 mm). As the radial position (r) moves from the center to the wall, the gas pressure (P) gradually increases at different gas velocities. This indicates that the pressure drop of the gas near the wall is relatively low due to the wall effect for the finite bed of irregular sinters. It is different from the wall effect on spherical particles. The confined wall can increase the viscous loss term of the wall friction effect and decrease the inertial loss term of the voidage effect [16, 18, 20, 24, 25]. Only in the laminar flow, the viscous loss plays a leading role to increase the pressure drop of the gas, whereas under the turbulent flow, the inertial loss is dominant to reduce the pressure drop of the gas [18, 20, 24, 25]. The previous study [11] shows that the gas flow in the sinter bed is easy to reach the turbulent regime or transitional regime due to the irregularity of particles. It is mainly attributed to two aspects. On the one hand, the irregular particles lead to irregular gas channels in the bed, which is easy to destroy the stability of the gas flow. On the other hand, the irregular shape causes the uneven distribution of the gas channel. This would increase of the real velocity of the gas flow in the channel. In addition, the surface of sinter particles is very rough and full of concaves and convexes [9, 10]. Therefore, the friction effect of the confined wall is not necessarily greater than that of the sinter surface. Based on the above two points, the effect of the wall friction on the pressure drop is relatively weak in the irregular sinter bed. Moreover, the bed voidage increases gradually from the center to the wall [38–41]. Therefore, the pressure drop of the gas is relatively low in the region close to the wall. The gas pressure near the wall is larger than that of the interior, especially for the large particle at the high gas velocity.Figure 4
The radial distribution of the gas pressure (P) at the bed height of L=0 mm under four kinds of particle sizes of (a) 5~10 mm, (b) 20~25 mm, (c) 40~45 mm, and (d) 55~60 mm.
(a)(b)(c)(d)To further analyze the influence of the gas velocity, Figure5 illustrates the distribution of the dimensionless gas pressure (P/P0, P0 is the gas pressure at the center) with the dimensionless radial distance (r/R, R is the radius of the sinter bed). It is found that the radial dimensionless pressure distribution is nearly the same at different gas velocities. This means that the gas velocity does not change the wall effect on the pressure drop in the bed of the irregular particle. Therefore, whatever state the gas flow is in, the influence of the confined wall on the irregular particle only reduces the pressure drop in the bed.Figure 5
The radial distribution of the dimensionless gas pressure (P/P0) at the bed height of L=0 mm under four kinds of particle sizes of (a) 5~10 mm, (b) 20~25 mm, (c) 40~45 mm, and (d) 55~60 mm.
(a)(b)(c)(d)Figure6 illustrates the radial distribution of the average dimensionless gas pressure (P/P0¯) at different gas velocities under 11 kinds of particle sizes. It is observed that the values of P/P0¯ increase with the particle size increasing (i.e., the decrease of D/dp). With the particle size increasing from 5~10 mm to 55~60 mm, P/P0¯ near the wall increases from 1.03 to 1.26. This indicates that the confined wall has a more significant effect on large particles. With the increase in the particle size, the difference of the voidage between the wall and the center becomes larger [38], and the uniformity of the voidage distribution becomes worse [39, 40].Figure 6
The radial distribution of the average dimensionless gas pressure (P/P0¯) for different gas velocities at the bed height of L=0 mm under 11 kinds of particle sizes.


### 3.3. Effect of the Gas Velocity and Equivalent Particle Size on the Pressure Drop

Figure7 illustrates the variation of the pressure drop of the gas flow per unit height (ΔP/L) with the gas velocity (ug) under different particle sizes. Firstly, it is seen that ΔP/L increases with the increase of ug, fitting well with the quadratic function in the form of ΔP/L=a⋅ug2+b⋅ug. The correlation coefficients of all fitting curves are greater than 0.99985, as shown in Table 4. This is consistent with the results published [5, 7, 12]. Similar to the Ergun equation, the pressure drop of the gas is composed of the viscous loss linearly related to ug and the inertial loss of a quadratic relationship with ug. As ug increases, the boundary layer gradually disappears and the collision between the gas and particles is increasingly intensified [12, 42]. Therefore, the inertial loss becomes the dominant factor, resulting in a sharp increase in the pressure drop of the gas flow. Moreover, it can be observed from Table 4 that the coefficients of the quadratic function, a and b, decline with the particle size increasing, which is similar with other studies [5, 7, 12]. Also, the bed voidage increases with the increase of dp, as shown in Figure 3. This causes a reduction in the instability and specific surface area of the gas flow. Therefore, it makes in the inertial loss and viscous loss decrease [42].Figure 7
The variation of the pressure drop of the gas flow per unit height (ΔP/L) vs. the gas velocity (ug) under different particle sizes.Table 4
Coefficients of the fitting function between the pressure drop per unit height (ΔP/L) and gas velocity (ug) under 11 kinds of particle sizes.
Particle size (mm)ΔP/L=a⋅ug2+b⋅ugCorrelation coefficientR2ab5~102111.29659.220.9998510~15988.75389.390.9980815~20645.57307.560.9997920~25456.34251.170.9998025~30352.03226.560.9994130~35283.05193.260.9990835~40235.07164.720.9994140~45199.09141.370.9995145~50170.91123.260.9995250~55148.51109.600.9994755~60130.0198.090.99959


### 3.4. Analysis of the Empirical Correlation of the Pressure Drop

#### 3.4.1. Considering the Wall Effect

According to Equation (11), the relationship between the wall-modified friction pressure drop fw and the wall-modified particle Reynolds number Rew can be linearly fitted by the least square method. The obtained wall-modified viscous loss coefficient k1w and inertial loss coefficient k2w for each particle size are shown in Figure 8(a). In addition, Figure 8(a) indicates that k1w decreases and k2w increases with the increase of D/dp, respectively. This change trend is similar to that obtained by others [11, 17]. Therefore, k1w and k2w are determined as functions of D/dp,
(13)k1w=540+639e‐0.0581D/dp,k2w=2.73‐1.883e‐0.0733D/dp.Figure 8
Analysis of the wall-modified correlation of the pressure drop: (a) relationship betweenk1w and k2w vs. D/dp; (b) comparison between the experimental value and the predicted value of ΔP/L.
(a)(b)Then,fw is expressed as follows:
(14)fw=540+639e‐0.0581D/dp+2.73‐1.883e‐0.0733D/dpRew.The mean relative error (MRE) between the measured value and the predicted value of the pressure drop per unit height is calculated by(15)MRE%=1n∑j=1nΔP/Lcalj−ΔP/LexpjΔP/Lexpj×100,where j is the serial number of experimental data, j=1,2,⋯⋯,n; n is the number of experimental data; ΔP/Lexp and ΔP/Lcal are the measured value and the predicted value of the pressure drop per unit height, respectively.It is found that the predicted values by Equation (14) achieve a satisfied agreement with the measured values, as shown in Figure 8(b). The MRE and maximum error are 2.74% and 9.48%, respectively. Therefore, the correlation of the wall correction can well predict the pressure drop of the gas flow through the packed bed of irregular sinters.


#### 3.4.2. Ignoring the Wall Effect

For the comparison, the correlation of the pressure drop without the wall correction is also fitted by the least square method according to Equation (9), as shown in Figure 9(a). The correlation of the pressure drop between the friction factor fp and the particle Reynolds number Rep is expressed by
(16)fp=3357+2.388Rep.Figure 9
Analysis of the empirical correlation of the pressure drop without the wall correction: (a) relationship betweenfp and Rep; (b) comparison between the experimental value and the predicted value of ΔP/L.
(a)(b)It is found from Figure9(b) that most of the predicted values by Equation (16) are larger than the measured values. Since the wall effect leads to the decrease of the pressure drop, the pressure drop of the gas in the sinter bed will be overestimated when it is assumed to be an infinite packed bed. Therefore, the prediction accuracy of the correlation of the pressure drop without the wall correction will decline. The MRE and maximum error are as high as 16.01% and 139.84%, respectively.


## 3.1. Effect of the Confined Wall on the Bulk Density and Bed Voidage

Figure3 shows the changes of the bulk density ρb and the bed voidage ε with the particle size under five repeated experiments. With the particle size increasing, ρb decreases and ε increases, respectively. This change trend is consistent with the published results [6, 9, 10]. It can be observed from Table 2 that the particle shape factor ϕ gradually reduces with the increase in the particle size. This means that the irregularity of sinters increases, which makes it easier to form bridges when the sinter particles are heaped up. Therefore, the voids between particles in the bed would increase with the increase of the particle size. Besides, it is also found that the reproducibility and uncertainty of ρb and ε are getting worse with D/dp decreasing, which is similar to results obtained by Tian et al. [11] and Raichura [17]. This indicates that the wall effect limits the randomness and uniformity of the particle accumulation in the bed. What's more, the influence of the confined wall becomes more significant with the increase in the particle irregularity and particle size. This further intensifies the nonrandomness and heterogeneity of the particle accumulation.Figure 3
Variation of the bulk densityρb and voidage ε of the sinter bed with the particle size under five repeated experiments.


## 3.2. Effect of the Confined Wall on the Gas Pressure Distribution

Figure4 illustrates the radial distribution of the gas pressure (P) at the bed height of L=0 mm under four kinds of particle sizes (i.e., 5~10 mm, 20~25 mm, 40~45 mm, and 55~60 mm). As the radial position (r) moves from the center to the wall, the gas pressure (P) gradually increases at different gas velocities. This indicates that the pressure drop of the gas near the wall is relatively low due to the wall effect for the finite bed of irregular sinters. It is different from the wall effect on spherical particles. The confined wall can increase the viscous loss term of the wall friction effect and decrease the inertial loss term of the voidage effect [16, 18, 20, 24, 25]. Only in the laminar flow, the viscous loss plays a leading role to increase the pressure drop of the gas, whereas under the turbulent flow, the inertial loss is dominant to reduce the pressure drop of the gas [18, 20, 24, 25]. The previous study [11] shows that the gas flow in the sinter bed is easy to reach the turbulent regime or transitional regime due to the irregularity of particles. It is mainly attributed to two aspects. On the one hand, the irregular particles lead to irregular gas channels in the bed, which is easy to destroy the stability of the gas flow. On the other hand, the irregular shape causes the uneven distribution of the gas channel. This would increase of the real velocity of the gas flow in the channel. In addition, the surface of sinter particles is very rough and full of concaves and convexes [9, 10]. Therefore, the friction effect of the confined wall is not necessarily greater than that of the sinter surface. Based on the above two points, the effect of the wall friction on the pressure drop is relatively weak in the irregular sinter bed. Moreover, the bed voidage increases gradually from the center to the wall [38–41]. Therefore, the pressure drop of the gas is relatively low in the region close to the wall. The gas pressure near the wall is larger than that of the interior, especially for the large particle at the high gas velocity.Figure 4
The radial distribution of the gas pressure (P) at the bed height of L=0 mm under four kinds of particle sizes of (a) 5~10 mm, (b) 20~25 mm, (c) 40~45 mm, and (d) 55~60 mm.
(a)(b)(c)(d)To further analyze the influence of the gas velocity, Figure5 illustrates the distribution of the dimensionless gas pressure (P/P0, P0 is the gas pressure at the center) with the dimensionless radial distance (r/R, R is the radius of the sinter bed). It is found that the radial dimensionless pressure distribution is nearly the same at different gas velocities. This means that the gas velocity does not change the wall effect on the pressure drop in the bed of the irregular particle. Therefore, whatever state the gas flow is in, the influence of the confined wall on the irregular particle only reduces the pressure drop in the bed.Figure 5
The radial distribution of the dimensionless gas pressure (P/P0) at the bed height of L=0 mm under four kinds of particle sizes of (a) 5~10 mm, (b) 20~25 mm, (c) 40~45 mm, and (d) 55~60 mm.
(a)(b)(c)(d)Figure6 illustrates the radial distribution of the average dimensionless gas pressure (P/P0¯) at different gas velocities under 11 kinds of particle sizes. It is observed that the values of P/P0¯ increase with the particle size increasing (i.e., the decrease of D/dp). With the particle size increasing from 5~10 mm to 55~60 mm, P/P0¯ near the wall increases from 1.03 to 1.26. This indicates that the confined wall has a more significant effect on large particles. With the increase in the particle size, the difference of the voidage between the wall and the center becomes larger [38], and the uniformity of the voidage distribution becomes worse [39, 40].Figure 6
The radial distribution of the average dimensionless gas pressure (P/P0¯) for different gas velocities at the bed height of L=0 mm under 11 kinds of particle sizes.


## 3.3. Effect of the Gas Velocity and Equivalent Particle Size on the Pressure Drop

Figure7 illustrates the variation of the pressure drop of the gas flow per unit height (ΔP/L) with the gas velocity (ug) under different particle sizes. Firstly, it is seen that ΔP/L increases with the increase of ug, fitting well with the quadratic function in the form of ΔP/L=a⋅ug2+b⋅ug. The correlation coefficients of all fitting curves are greater than 0.99985, as shown in Table 4. This is consistent with the results published [5, 7, 12]. Similar to the Ergun equation, the pressure drop of the gas is composed of the viscous loss linearly related to ug and the inertial loss of a quadratic relationship with ug. As ug increases, the boundary layer gradually disappears and the collision between the gas and particles is increasingly intensified [12, 42]. Therefore, the inertial loss becomes the dominant factor, resulting in a sharp increase in the pressure drop of the gas flow. Moreover, it can be observed from Table 4 that the coefficients of the quadratic function, a and b, decline with the particle size increasing, which is similar with other studies [5, 7, 12]. Also, the bed voidage increases with the increase of dp, as shown in Figure 3. This causes a reduction in the instability and specific surface area of the gas flow. Therefore, it makes in the inertial loss and viscous loss decrease [42].Figure 7
The variation of the pressure drop of the gas flow per unit height (ΔP/L) vs. the gas velocity (ug) under different particle sizes.Table 4
Coefficients of the fitting function between the pressure drop per unit height (ΔP/L) and gas velocity (ug) under 11 kinds of particle sizes.
Particle size (mm)ΔP/L=a⋅ug2+b⋅ugCorrelation coefficientR2ab5~102111.29659.220.9998510~15988.75389.390.9980815~20645.57307.560.9997920~25456.34251.170.9998025~30352.03226.560.9994130~35283.05193.260.9990835~40235.07164.720.9994140~45199.09141.370.9995145~50170.91123.260.9995250~55148.51109.600.9994755~60130.0198.090.99959


## 3.4. Analysis of the Empirical Correlation of the Pressure Drop

### 3.4.1. Considering the Wall Effect

According to Equation (11), the relationship between the wall-modified friction pressure drop fw and the wall-modified particle Reynolds number Rew can be linearly fitted by the least square method. The obtained wall-modified viscous loss coefficient k1w and inertial loss coefficient k2w for each particle size are shown in Figure 8(a). In addition, Figure 8(a) indicates that k1w decreases and k2w increases with the increase of D/dp, respectively. This change trend is similar to that obtained by others [11, 17]. Therefore, k1w and k2w are determined as functions of D/dp,
(13)k1w=540+639e‐0.0581D/dp,k2w=2.73‐1.883e‐0.0733D/dp.Figure 8
Analysis of the wall-modified correlation of the pressure drop: (a) relationship betweenk1w and k2w vs. D/dp; (b) comparison between the experimental value and the predicted value of ΔP/L.
(a)(b)Then,fw is expressed as follows:
(14)fw=540+639e‐0.0581D/dp+2.73‐1.883e‐0.0733D/dpRew.The mean relative error (MRE) between the measured value and the predicted value of the pressure drop per unit height is calculated by(15)MRE%=1n∑j=1nΔP/Lcalj−ΔP/LexpjΔP/Lexpj×100,where j is the serial number of experimental data, j=1,2,⋯⋯,n; n is the number of experimental data; ΔP/Lexp and ΔP/Lcal are the measured value and the predicted value of the pressure drop per unit height, respectively.It is found that the predicted values by Equation (14) achieve a satisfied agreement with the measured values, as shown in Figure 8(b). The MRE and maximum error are 2.74% and 9.48%, respectively. Therefore, the correlation of the wall correction can well predict the pressure drop of the gas flow through the packed bed of irregular sinters.


### 3.4.2. Ignoring the Wall Effect

For the comparison, the correlation of the pressure drop without the wall correction is also fitted by the least square method according to Equation (9), as shown in Figure 9(a). The correlation of the pressure drop between the friction factor fp and the particle Reynolds number Rep is expressed by
(16)fp=3357+2.388Rep.Figure 9
Analysis of the empirical correlation of the pressure drop without the wall correction: (a) relationship betweenfp and Rep; (b) comparison between the experimental value and the predicted value of ΔP/L.
(a)(b)It is found from Figure9(b) that most of the predicted values by Equation (16) are larger than the measured values. Since the wall effect leads to the decrease of the pressure drop, the pressure drop of the gas in the sinter bed will be overestimated when it is assumed to be an infinite packed bed. Therefore, the prediction accuracy of the correlation of the pressure drop without the wall correction will decline. The MRE and maximum error are as high as 16.01% and 139.84%, respectively.


## 3.4.1. Considering the Wall Effect

According to Equation (11), the relationship between the wall-modified friction pressure drop fw and the wall-modified particle Reynolds number Rew can be linearly fitted by the least square method. The obtained wall-modified viscous loss coefficient k1w and inertial loss coefficient k2w for each particle size are shown in Figure 8(a). In addition, Figure 8(a) indicates that k1w decreases and k2w increases with the increase of D/dp, respectively. This change trend is similar to that obtained by others [11, 17]. Therefore, k1w and k2w are determined as functions of D/dp,
(13)k1w=540+639e‐0.0581D/dp,k2w=2.73‐1.883e‐0.0733D/dp.Figure 8
Analysis of the wall-modified correlation of the pressure drop: (a) relationship betweenk1w and k2w vs. D/dp; (b) comparison between the experimental value and the predicted value of ΔP/L.
(a)(b)Then,fw is expressed as follows:
(14)fw=540+639e‐0.0581D/dp+2.73‐1.883e‐0.0733D/dpRew.The mean relative error (MRE) between the measured value and the predicted value of the pressure drop per unit height is calculated by(15)MRE%=1n∑j=1nΔP/Lcalj−ΔP/LexpjΔP/Lexpj×100,where j is the serial number of experimental data, j=1,2,⋯⋯,n; n is the number of experimental data; ΔP/Lexp and ΔP/Lcal are the measured value and the predicted value of the pressure drop per unit height, respectively.It is found that the predicted values by Equation (14) achieve a satisfied agreement with the measured values, as shown in Figure 8(b). The MRE and maximum error are 2.74% and 9.48%, respectively. Therefore, the correlation of the wall correction can well predict the pressure drop of the gas flow through the packed bed of irregular sinters.


## 3.4.2. Ignoring the Wall Effect

For the comparison, the correlation of the pressure drop without the wall correction is also fitted by the least square method according to Equation (9), as shown in Figure 9(a). The correlation of the pressure drop between the friction factor fp and the particle Reynolds number Rep is expressed by
(16)fp=3357+2.388Rep.Figure 9
Analysis of the empirical correlation of the pressure drop without the wall correction: (a) relationship betweenfp and Rep; (b) comparison between the experimental value and the predicted value of ΔP/L.
(a)(b)It is found from Figure9(b) that most of the predicted values by Equation (16) are larger than the measured values. Since the wall effect leads to the decrease of the pressure drop, the pressure drop of the gas in the sinter bed will be overestimated when it is assumed to be an infinite packed bed. Therefore, the prediction accuracy of the correlation of the pressure drop without the wall correction will decline. The MRE and maximum error are as high as 16.01% and 139.84%, respectively.


## 4. Conclusions

The vertical waste heat recovery technology of the sinter will be a great driving force for China to realize the “double Carbon.” To promote the application of the new technology, the wall effect on the gas flow characteristic in the bed with irregular sinters was studied by the experimental method.(1)
The irregularity of the sinter increases with the particle size increasing, which causes an increase in the bed voidage. Besides, the vertical wall limits the randomness and uniformity of the particle accumulation. This effect is further intensified with the increase of the irregularity and particle size, which makes the reproducibility and uncertainty of the voidage become worse(2)
As the radial position moves from the center to the wall, the gas pressure increases gradually. Moreover, the radial distribution of the dimensionless pressure is nearly the same at different gas velocities. Therefore, the wall effect on irregular sinters only reduces the pressure drop of the gas no matter what state the gas flow is in, which is different from the wall effect on spherical particles. Besides, the wall effect is increasing significantly with the particle size increasing (i.e., the decrease ofD/dp). With the particle size increasing from 5~10 mm to 55~60 mm, the dimensionless gas pressure near the wall increases from 1.03 to 1.26(3)
When the wall effect is ignored, the pressure drop of the gas would be overestimated by 16.01% on average, whereas the empirical correlation of the wall correction can well predict the pressure drop of the gas in the packed bed of the irregular sinter with the mean error and maximum error of 2.74% and 9.48%, respectively



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*Source: 1890541-2022-06-26.xml*