Determination of Pressure Drop Correlation for Air Flow through Packed Bed of Sinter Particles in Terms of Euler Number

Abstract

In order to clearly understand the air flow resistance characteristics in vertical tanks for sinter waste heat recovery in the steel industry, experimental research on the air flow pressure drop (FPD) performance in a sinter bed layer (BL) was conducted. Based on a self-made experimental device, the measurement values of air FPD for different experimental conditions were determined firstly, and then the concept of Euler number (Eu) in heat exchangers was introduced into the study of air FPD in BL; the change rules of Eu under different particle diameters were analyzed. Finally, the air FPD correlation in sinter BL was obtained and described in the form of Eu, and the error analysis of obtained air FPD correlation was performed. The results show that, the air FPD increases as a second power relationship with the increase in air superficial velocity when the particle diameter is constant. The decrease amplitude of Eu gradually dwindles when increasing the Reynolds number (Re), and the decrease in the Eu shows a reciprocal relationship with the Re. As the bed geometry factor increases, the FPD coefficient, A, decreases as an exponential relationship, while the FPD coefficient, B, increases as a first power relationship. The obtained air FPD correlation in the form of Eu in the experiment is well compatible with the measurement values, and the mean deviation of obtained correlation is 4.67%, showing good originality.

1. Introduction

Packed beds are widely used in the metallurgical industries as the heat exchangers and high temperature gas-cooled reactors [1,2,3], which have gained popularity because of the convenience of operation and low cost. A lot of researchers have been commissioned to investigate the mechanisms of fluid flow and heat transfer in packed beds, and it is still in progress [4].

The sinter vertical tank (SVT) is high-efficient recovery equipment of sinter waste heat [5,6], which is aimed to overcome the shortcomings of existing waste heat recovery equipment [7]. In essence, the bed layer (BL) with sinter particles in SVT is a kind of random packed bed, and the air flow pressure drop (FPD) in SVT affects the required power of the air blower, and then affects the economic feasibility of SVT. Therefore, it is of great significance to study the air FPD characteristics in SVT for improving the sinter waste heat recovery rate and analyzing the economic feasibility of SVT.

Understanding the fluid FPD performance in a packed bed is very important as it affects the required power of pump and blower in actual applications. There have been many pieces of research on the fluid FPD performance in particle-packed beds in the last few decades [8,9,10,11,12,13,14,15,16,17,18]. Montillet [8] experimentally studied the fluid FPD behavior in a packed bed with spheres under different Reynolds number (Re) and bed geometry factor (D/d_p), and the values of fluid FPD in the central region of the packed bed were measured. Yang et al. [10] investigated the fluid FPD behavior in structured packed beds by experimental and numerical methods. Bu et al. [12] numerically investigated the fluid FPD behavior in structured packed beds with particles by using different treatment methods. Tian et al. [14] experimentally investigated the FPD characteristics in structured BL with spheres and determined the quantitative relationship of FPD with flow velocity. Halkarni et al. [16] studied the effect of D/d_p on FPD in randomly packed beds with uniform-sized spherical particles, and the measured friction factor compared reasonably with the available correlations in the literature. Toit et al. [18] investigated the FPD behavior in packed beds and performed an evaluation of the friction factors as a function of the modified Re for the FPD through randomly and structured packed beds consisting only of uniform-sized spheres.

Except for the above studies of fluid FPD performance in packed beds, the studies of FPD correlation in particle packed beds have also been conducted by lots of researchers [19,20,21,22,23,24,25,26,27,28,29,30]. The well-known Ergun’s equation [19] applied to describe the fluid FPD in packed beds is shown in Equation (1).

$$\frac{\Delta P}{H}=150\frac{\mu {\left(1-\epsilon \right)}^2}{{\epsilon }^3{d}_{\mathrm{p}}{}^2}u+1.75\frac{\rho \left(1-\epsilon \right)}{{\epsilon }^3{d}_{\mathrm{p}}}{u}^2$$

(1)

The first and second terms in Equation (1) are the laminar and turbulent components, respectively, and the corresponding coefficients (150, 1.75) are the viscous and inertial resistance coefficients. Due to the difference of particle size and shape in packed beds, the equation coefficients (150, 1.75) have been modified for predicting the FPD in various particle packed beds by some other researchers. Macdonald et al. [20] experimentally gave the modified coefficients of 180 and 1.8. Meanwhile, Comiti and Renaud [21] and Ozahi et al. [22] also experimentally presented their modified coefficients, namely the coefficients (141, 1.63) and coefficients (160, 1.61). Dukhan et al. [23] and Amiri et al. [24] also experimentally obtained the algebraic values of modified coefficients, which are all the interval range values. In addition to this, there is another indirect expression for describing the fluid FPD in particle packed beds, namely the friction factor, the definition of which is given as follows.

$$f_{\mathrm{p}}=\frac{\Delta P{d}_{\mathrm{p}}}{H\rho u^2}$$

(2)

Based on the definition of f_p proposed by Kürten et al. [25] and Hicks [26], they are valid in the range of 0.1 ≤ Re ≤ 4000 and 500 ≤ Re ≤ 60,000. Meanwhile, the correlations of f_p are obtained by Tallmadge [27] and Lee and Ogawa [28] with a large range of 0.1 ≤ Re ≤ 100,000 and 1 ≤ Re ≤ 100,000. In all of the above correlations, f_p are defined as a function of Re and BL voidage (ε) only. Furthermore, Montillet et al. [29] and Özahi et al. [30] experimentally obtained the correlations of f_p with the range of 10 ≤ Re ≤ 2500 and 708 ≤ Re ≤ 7772, and except for Re and ε, the D/d_p is also considered in their correlations.

The above studies on packed beds mostly focused on the FPD performance and correlation correction of FPD in BL with uniform-sized particles. Nevertheless, there was a great difference in the fluid FPD for different particle size and shape in packed beds. In addition, the existing fluid FPD correlations in particle-packed beds were mainly in the form of Ergun’s equation and f_p, while the FPD correlation in the form of Euler number (Eu) only appeared in the study of FPD in heat exchangers [31,32,33]. Based on an authors’ literature survey, there is almost no research to investigate the fluid FPD correlation in the form of Eu in particle-packed beds.

To sum up, the concept of Eu in heat exchangers was introduced into the fluid FPD research in particle BL, and the air FPD performance in sinter BL was restudied and analyzed through the Eu method in this paper by using the experimental data shown in our previous research [34]. Firstly, the measurement values of air FPD for different experimental conditions were obtained, according to the experimental data shown in our previous research [34], and then the change relationships between Eu and Re under different particle diameters were determined. Finally, the FPD correlation in the form of Eu in sinter BL was defined, and the applicability of FPD correlation obtained under different experimental conditions was also analyzed. The FPD correlation in the form of Eu related to air FPD characteristics will provide a theoretical reference in sinter BL for industrial applications.

2. Experimental Device and Process

The self-made experimental device shown in our previous research [34] was presented in Figure 1. The experimental device was mainly composed of four parts: an air blower to drive the air flow through the sinter BL, a throttle valve to adjust the air flow rate, an orifice plate flowmeter to display the specific size of the air flow rate, and a vertical device with three pressure measuring holes to measure the air FPD at different BL heights. The inner diameter of the vertical device was 430 mm, and three pressure measuring holes installed at different vertical locations (400 mm, 700 mm and 1200 mm) were applied to measure the air FPD in sinter BL.

Because of the inhomogeneity of sinter particles, the standard test sieves of different sizes were applied to sieve out the all sizes of sinter particles, and three kinds of sieved particle diameters with 14 mm, 24 mm and 35 mm were used in the experiment, shown in

Table 1. Related parameters of sinter particles.

d (mm)	Φ	dp (mm)	D/dp
14	0.69	9.66	44.5
24	0.72	17.28	24.9
35	0.89	31.15	13.8

. The particle sphericity (Φ) was obtained through the gas flow technique [35]. The particle equivalent diameter (d_p) was the product of sieved particle diameter and particle sphericity, and the D/d_p was the ratio of inner diameter of experimental cylinder to particle equivalent diameter. The rated flow of the air blower was 1650 m³/h, and the experimental conditions were conducted with the air flow rate range of 200~1600 m³/h under the normal condition of T₀ = 293.15 K.

The experimental values of air FPD per unit height, ΔP(r)/H, along the radial direction of sinter BL were calculated and determined for different BL heights by taking the differences of pressure values at measurement points firstly, and then the average values of FPD per unit height (ΔP/H) along the radial direction for different BL heights were obtained under different particle diameters and air flow rates. Subsequently, according to the average values along the radial direction for different BL heights, the changes of mean ΔP/H with the air superficial velocity for different particle diameters were determined, and the relationships between Eu and Re under different particle diameters were also obtained. Finally, the FPD correlation in the form of Eu was defined through the method of data regression analysis, and the error analysis of obtained FPD correlation was also performed.

3. Results and Discussion

3.1. Measurement of Air FPD

Along the radial direction of sinter BL, the locations of six measurement points are identified, and the r/R for different locations are 0, 0.2, 0.4, 0.6, 0.8 and 0.96, respectively. Among the r/R, r is the actual distance between the measured point and the center of BL, and R is the radius of BL. The purpose of this measurement operation is to overcome the influences of particle inhomogeneity and the wall effect of BL on the air FPD. The measurement values of ΔP(r)/H at six measurement points is used to calculate the average value of ΔP/H along the radial direction for a given experimental condition, and according to the experimental data of air FPD shown in our previous research [34], the changes of ΔP/H with the air flow rate under different BL heights and particle diameters are shown in Figure 2. It can be seen from Figure 2, the ΔP/H increases with the increase in air flow rate, and the increase amplitude of ΔP/H is getting larger and larger. This may be explained that the increase in air flow rate results in the air superficial velocity, and based on Equation (1), the ΔP/H in sinter BL is proportional to the quadratic power of superficial velocity, which leads to the larger increase amplitude of ΔP/H shown in Figure 2.

In addition, Figure 2 also shows that the values of ΔP/H are different for various BL heights when the particle diameter is constant, the ΔP/H for the BL height of 300 mm (400–700 mm) is the smallest, and the ΔP/H for the BL height of 800 mm (400–1200 mm) is middle, while the ΔP/H for the BL height of 500 mm (700–1200 mm) is the largest. The result may be because of the difference of packing structure of sinter particles in BL, the particle size at the bottom of BL is relatively larger than that at the top of BL, which results in the smaller ε at the top of BL, so the ΔP/H at the top of BL is relatively larger.

Based on the above measurement values of ΔP/H for different BL heights, the changes of mean ΔP/H with air superficial velocity under three particle diameters are listed in Figure 3, which is taken from our previous research [34]. As known from Figure 3, the increase in ΔP/H shows a second power relationship as the air superficial velocity increases, and the ΔP/H decreases as the particle diameter increases when the air superficial velocity is constant. This may be explained by considering that the ε increases as the particle diameter increases, and both the viscous and inertial resistances of air flow through the BL will decrease, which leads to the decrease in ΔP/H.

3.2. Analysis of Eu–Re Relationship

As known from the above literatures [19,20,21,22,23,24], the general correlation of Ergun’s equation can be described as follows.

$$\frac{\Delta P}{H}=k_1\frac{\mu {\left(1-\epsilon \right)}^2}{{\epsilon }^3{d}_{\mathrm{p}}{}^2}u+k_2\frac{\rho \left(1-\epsilon \right)}{{\epsilon }^3{d}_{\mathrm{p}}}{u}^2$$

(3)

where k₁ is the coefficient of viscosity resistance, and k₂ is the coefficient of inertia resistance.

According to the definition of Eu, the expression of Eu is given below, which reflects the relative relationship of FPD in flow field with dynamic pressure.

$$Eu=\frac{\Delta P}{\rho u^2}$$

(4)

Through the combination of Equations (3) and (4), the Eu used to describe the fluid FPD in particle BL is determined below.

$$Eu=H\left[k_1\frac{\mu {\left(1-\epsilon \right)}^2}{\rho {\epsilon }^3{d}_{\mathrm{p}}{}^{2}u}+k_2\frac{\left(1-\epsilon \right)}{{\epsilon }^3{d}_{\mathrm{p}}}\right]$$

(5)

$$Eu=\frac{H}{D}\left[k_1\frac{{\left(1-\epsilon \right)}^2}{{\epsilon }^3}\frac{D}{d_{\mathrm{p}}}\frac{1}{Re}+k_2\frac{\left(1-\epsilon \right)}{{\epsilon }^3}\frac{D}{d_{\mathrm{p}}}\right]$$

(6)

For a given experimental condition, the ε and D/d_p, as well as the coefficients k₁ and k₂ are constant, so Equation (6) can be modified to the following form.

$$Eu=\frac{H}{D}\left(\frac{A}{Re}+B\right)$$

(7)

where A and B are the correlation coefficients, which can be written below, respectively.

$$A=k_1\frac{{\left(1-\epsilon \right)}^2}{{\epsilon }^3}\frac{D}{d_{\mathrm{p}}}$$

(8)

$$B=k_2\frac{\left(1-\epsilon \right)}{{\epsilon }^3}\frac{D}{d_{\mathrm{p}}}$$

(9)

Based on the measurement values of ΔP/H under the different particle diameters mentioned in Figure 3, the changes of Eu with Re under three particle diameters for the BL height of 800 mm are listed in Figure 4. As known from Figure 4, the Eu gradually decreases as the Re increases for the three particle diameters, and the larger is the Re, the smaller is the decrease amplitude of Eu. Through the data fitting, the quantitative relationships of Eu with Re for the three particle diameters are also obtained and listed in Figure 4, and the Eu decreases as a reciprocal relationship with the increase in Re, which is consistent with the analysis of Equation (7).

3.3. Determination of Pressure Drop Correlation

According to the literature [19,20,21,22,23,24,25,26,27,28,29,30] mentioned above, some FPD correlations in the form of Ergun’s equation and f_p have been determined to calculate the ΔP/H in particle-packed beds. In order to verify the applicability of the FPD correlations mentioned in the above literatures, four correlations in the form of f_p [25,26,27,28] are selected and listed in

Table 2. Correlations for fluid FPD in packed bed with particles.

Authors/Reference	Correlation	Range of Validity
Kürten et al. [25]	$f_p=\left[\frac{25{\left(1-\epsilon \right)}^2}{4{\epsilon }^3}\right]\left[21R{e}^{-1}+6R{e}^{-0.5}+0.28\right]$	$0.1\le Re\le 4000$
Hicks [26]	$f_p=6.8\frac{{\left(1-\epsilon \right)}^{1.2}}{{\epsilon }^3}R{e}^{-0.2}$	$500\le Re\le 60000$
Tallmadge [27]	$f_p=150\frac{{\left(1-\epsilon \right)}^2}{{\epsilon }^{3}Re}+4.2\frac{{\left(1-\epsilon \right)}^{1.1666}}{{\epsilon }^3}R{e}^{-1/6}$	$0.1\le Re\le 100000$
Sug Lee and Ogawa [28]	$f_p=\frac{1}{2}\left[\frac{12.5{\left(1-\epsilon \right)}^2}{{\epsilon }^3}\right]\left[29.32R{e}^{-1}+1.56R{e}^{-n}+0.1\right]$ with $n=0.352+0.1\epsilon +0.275{\epsilon }^2$	$1\le Re\le 100000$

, and the ε for the packed beds with particle diameters of 14 mm, 24 mm and 35 mm are 0.44, 0.49 and 0.53, respectively [35]. The comparisons of calculation values of ΔP/H calculated by the FPD correlations cited in the literature [25,26,27,28] and measurement values of ΔP/H obtained under different experimental conditions are shown in Figure 5. It could be easily obtained from Figure 5, although the correlation cited in the literature [28] fits the measurement values of ΔP/H for d = 14 mm with a small deviation, and the correlations cited in the literature [26,27] also fit the measurement values of ΔP/H for d = 35 mm with a small deviation, but according to the whole measurement values of ΔP/H, all of the above selected correlations cited in the literature [25,26,27,28] are not suitable for calculating the ΔP/H in sinter BL.

Based on our previous research results [36], the ε is the function of D/d_p. Therefore, the correlation coefficients, A and B, are also considered as the function of D/d_p. The quantitative relationships of coefficients A and B with the D/d_p are shown in Figure 6. As known from Figure 6, the coefficient, A, decreases as an exponential relationship with the increasing D/d_p, while the coefficient, B, increases as a first power relationship. Furthermore, the specific relationships of coefficients, A and B, with the D/d_p are given below through the data fitting.

$$A=\left[1.96+3.35{\mathrm{e}}^{(-0.053D/d_{\mathrm{p}})}\right]\times {10}^4$$

(10)

$$B=3.51D/d_{\mathrm{p}}-16.1$$

(11)

To sum up, the pressure drop correlation in the form of Eu can be determined and given below through the combination of Equation (7) with the Equations (10) and (11).

$$Eu=\frac{H}{D}\left[\frac{1.96\times {10}^4+3.35\times {10}^4{\mathrm{e}}^{(-0.053D/d_{\mathrm{p}})}}{Re}+3.51D/d_{\mathrm{p}}-16.1\right]$$

(12)

The comparisons of calculation values of Eu calculated by Equation (12) and experimental values of Eu obtained under different experimental conditions are shown in Figure 7.

As easily seen from Figure 7, the calculation values of Eu match the experimental values of Eu better for various experimental cases. The average deviation of Equation (12) is 4.67%, and the maximum deviation of which is 11.51%, which means that Equation (12) provides a better prediction correlation to calculate the ΔP/H in sinter BL. Furthermore, compared with the above FPD correlations mentioned in the literature [19,25,26,27,28], the component of Equation (12) is relatively simple—only including the effortless calculation factors, namely the height–diameter ratio (H/D), Re and D/d_p—and shows good originality. In addition, compared with the modified Ergun’s correlation in our previous result [34], the average deviation of Equation (12) is basically the same as that of modified Ergun’s correlation, but when the inner diameter of the experimental cylinder (D) is larger, the applicability of modified Ergun’s correlation in our previous result [34] is worse. However, the inner diameter of experimental cylinder has very little effect on the result of the pressure drop correlation in Equation (12). That is why we refit the FPD correlation in particle BL by the Eu method, the FPD correlation in the form of Euler number overcomes this problem better.

4. Conclusions

The air FPD performance in sinter BL under different particle diameters is investigated by the experimental method, and the major research results are given below.

(1)

The air FPD, ΔP/H, increases as a second power relationship with increasing air superficial velocity when the particle diameter is constant, and the ΔP/H decreases as the particle diameter increases for a given air superficial velocity.

(2)

The Eu decreases as the Re increases for a given particle diameter, and the decrease in the Eu shows a reciprocal relationship with the Re. The coefficient, A, decreases as an exponential relationship with the increasing D/d_p, while the coefficient, B, increases as a first power relationship.

(3)

The air FPD correlation in the form of Eu is obtained experimentally. Compared with the FPD correlations cited in the previous literature, the obtained correlation in the form of Eu match the whole experimental values better, and the average deviation of the obtained correlation is only 4.67%, which shows good originality.

The research result of the present work will provide a theoretical reference for the determination of required air blower power in SVT for industrial applications.