An Improved Model for Predicting the Drag Coefficient and Terminal Settling Velocity of Natural Sands in Newtonian Fluid

Abstract

The drag coefficient C_D plays an important role in studying the interaction forces between individual particles and fluid. Due to the irregular particle shape of natural sands, studying the sedimentation characteristics and drag coefficient model of irregular particles is of great significance in explaining natural phenomena, predicting the sedimentation process, and calculating the interphase forces between individual particles and fluid. In this paper, firstly, an experimental system for measuring the settling velocity was built, the settling velocity of 67 tests of spheres with different particle Reynolds number Re_s in the Newtonian fluid were obtained, and the C_D–Re_s correlation of sphere settling in the Newtonian fluid was established. The proposed C_D–Re_s correlation was in good agreement with the existing classical C_D–Re_s correlations, which proves the reliability of the experimental system and data processing method. Existing literature shows that the previous models are only suitable for irregular-shaped particles with three-dimensional shape-described parameters. However, the three-dimensional shape information of sand particles can only be obtained through accurate laboratory measurements, and it is often impossible to calculate accurately. By introducing the two-dimensional shape-described parameter (circularity c), using image analysis technology, the two-dimensional shape information of natural sands was obtained. The settling velocity of 221 tests of natural sands in the Newtonian fluid was obtained experimentally. It is found that the sand particles’ drag force exerted by the fluid is more significant than its equivalent sphere. With the increase in the particle Reynolds number, the shape irregularity’s influence on sand particle drag coefficient is more significant, and the C_D–Re_s correlation of natural sand was proposed by fitting. The established C_D–Re_s correlation has good prediction accuracy and can better predict the drag coefficient and terminal settling velocity of natural sand with irregular shapes.

1. Introduction

The drag coefficient C_D is essential in studying the interaction forces between individual particles and fluid, so it is widely used in the chemical industry, medicine, geotechnical, geology, petroleum engineering, and other fields [1,2,3,4,5]. Many scholars have studied drag coefficient and the settlement characteristics of different types of particles interacting with liquid. As for the free settlement characteristics of a single sphere, the shape is simple, the influence of shape factors is not involved, and the control variables are relatively simple, and relevant studies have been profound and thorough [6,7,8]. However, the particle shape of natural sands is irregular. Therefore, studying the sedimentation characteristics and establishing the drag coefficient model of irregular particles is of great significance in explaining natural phenomena, predicting the sedimentation process, and calculating the interphase forces between individual particles and fluid.

Drag force is the force exerted by the fluid on the solid particle with a relative velocity. This force is opposite to the relative velocity’s direction, also called the relative motion resistance [9]. The drag coefficient is also called the fluid resistance coefficient, which represents the interaction strength between particles and fluid; it is one of the core parameters which describe the interaction between particles and fluid [10]. The drag force is not a constant, but varies as a function of flow velocity, density, shape, windward area, and other factors, which are difficult to express in a unified expression. For the convenience of research, the calculation of drag force is usually reduced to the calculation of drag coefficient.

The free settlement method is the conventional experimental method used to measure the drag coefficient. Due to the shape of the spheres being regular and the structure being simple, the drag mechanism of the spheres was studied earlier [11,12,13,14,15]. In 1851, Stokes first proposed the drag force of a sphere when the fluid flows around the sphere in a steady flow with a small particle Reynolds number Re_s. Stokes then presented the “Stokes’ formula”, which introduced the drag coefficient, and obtained the relationship between the drag coefficient and the particle Reynolds number when the particle Reynolds number is small. On this basis, many experts and scholars have thoroughly investigated the C_D–Re_s correlation.

However, compared with the regular spherical particles, the shape of the irregular particles is complex. The particle shape is the most critical factor that leads to the difference between the drag characteristics of spherical particles and irregular particles [14,16,17]. Many experts and scholars have proposed different methods to express the shape of irregular particles. Dioguardi and Mele [18] use the ratio of sphericity to roundness to describe the shape of particles. Bagheri and Bonadonna [19] use elongation (the ratio of the length of the intermediate axis to that of the longest axis) and flatness (the ratio of the length of the shortest axis to that of the intermediate axis) to express the shape of particles by measuring the three axes of the particles. Hottovy and Sylvester [20] compared irregular particles with spherical particles of the same volume. They found that when Re_s is less than 10, the drag force on irregular particles is the same as on spherical particles of the same volume. When the particle Reynolds number is more significant than 100, the drag force on irregular particles is more significant than that on spherical particles of the same volume.

Existing literature shows that the previous models are only suitable for irregular-shaped particles with three-dimensional shape-described parameters. However, the three-dimensional shape information of sand particles can only be obtained through accurate laboratory measurements, and it is often impossible to calculate accurately. It is not likely to be popular or even applicable in engineering if it relies on complex measurement methods to describe irregular-shaped particles [21,22]. In this study, the terminal settling velocities of 67 tests of spheres and 221 tests of irregular-shaped sand particles in the Newtonian fluid were obtained experimentally by setting up an experimental platform for particle sedimentation. Using image analysis technology, the circularity c, a two-dimensional shape description parameter, was introduced to describe the shape characteristics. We proposed the use of C_D–Re_s correlations of sphere and natural sands settling in the Newtonian fluid. The established C_D–Re_s correlation has good prediction accuracy and can better predict the drag coefficient and terminal settling velocity of natural sand with irregular shapes.

2. Experiment Setup

2.1. Principle

Particle sedimentation occurs in many natural and industrial processes. When a single particle settles freely in a static viscous fluid, it will be dominated by three vertical forces: buoyancy F_B, gravity F_G, and drag force F_D. In laminar and turbulent states, the particle sedimentation process is shown in Figure 1.

The force exerted by the fluid when a single particle moves in the fluid, called drag force F_D, is the core parameter used to describe the momentum and energy transfer between the fluid and particles, and is one of the critical characteristic parameters used for studying the interaction between fluid and particles. It can be expressed by the following equation:

$$F_{\mathrm{D}}=\frac{1}{8}{\rho }_l{C}_{\mathrm{D}}\pi d_{\mathrm{e}\mathrm{q}}^2{(V_s-V_l)}^2$$

(1)

where ρ_l is the fluid density; V_s and V_l are the velocities of particles and fluid; d_eq is the equivalent diameter; and C_D is the drag coefficient.

It can be seen from Equation (1) that, in the settling process, the drag force F_D is directly proportional to the square of particle settling velocity. In the initial sedimentation stage, particles begin to sink due to gravity. Due to the initial settling velocity being small and the drag force F_D also being small, at this time, the floating weight F_W (the difference between gravity F_G and buoyancy F_B) is greater than the drag force F_D, resulting in the continuous increase in the settling velocity. With the rise of settling velocity, the drag force F_D on the particles continues to increase until the force acting on the particles reaches equilibrium; that is, the drag force F_D, buoyancy F_B, and gravity F_G on the particles are in equilibrium. At this time, the particles settle at a uniform speed, that is, the terminal settling velocity [23,24,25], which can be expressed as follows:

$$V_{\mathrm{t}\mathrm{s}}=\sqrt{\frac{4}{3}\frac{g{d}_{\mathrm{e}\mathrm{q}}}{C_{\mathrm{D}}}\frac{{\rho }_{\mathrm{s}}-{\rho }_{\mathrm{l}}}{{\rho }_{\mathrm{l}}}}$$

(2)

A key parameter needs to be used in the calculation of terminal settling velocity V_ts, that is, the drag coefficient C_D, which is expressed as follows:

$$C_{\mathrm{D}}=\frac{4}{3}\frac{g{d}_{\mathrm{eq}}}{V_{\mathrm{ts}}^2}\frac{{\rho }_{\mathrm{s}}-{\rho }_{\mathrm{l}}}{{\rho }_{\mathrm{l}}}$$

(3)

The drag coefficient C_D is a dimensionless quantity and is the main parameter required to describe the settling characteristics. It is mainly affected by the particle Reynolds number Re_s and particle shape. The particle Reynolds number (Re_s) is another significant dimensionless quantity that needs to be determined to describe the characteristics of particle sedimentation, which can be defined as follows:

$$R{e}_{\mathrm{s}}=\frac{{\rho }_{\mathrm{l}}{V}_{\mathrm{t}\mathrm{s}}{d}_{\mathrm{e}\mathrm{q}}}{\mu }$$

(4)

where μ is the fluid viscosity.

2.2. Procedure

The drag coefficient is usually calculated by measuring the terminal settling velocity of particles through experiments. In this study, the settlement cylinder of a high-transparent polyvinyl chloride (PVC) with a height of 1.5 m and an inner diameter of 0.1 m was used to measure the settling velocity. The settlement test device is shown in Figure 2. A high-speed camera (Revealer 2F04C) was adopted to catch the sedimentation process of particles. The camera’s maximum resolution is 2320 × 1720, which can be photographed in a high frame rate format of 190 frames per second. To improve the contrast between the particles in the image and the background, we chose a photographic light-absorbing cloth arranged on the wall behind the experimental circular tube, and a high-frequency lighting device (Jinbei EF-200 LED) was applied to improve the clarity [2]. The camera was directly connected to the computer to store image data. The GetData graph digitizer (version 2.26) was used to analyze the captured particle sedimentation image to calculate the terminal settling velocity V_ts.

2.3. Materials

In this study, the various particles used in the settlement experiment were stainless steel sphere, zirconia sphere, glass sphere and white quartz sand particles. The physical parameters of particles used in the experiment are shown in

Table 1. Main physical parameters of particles used in the experiment.

Granular Material	Equivalent Diameter deq (mm)	Density ρs (kg/m3)
Stainless steel	1.0–5.0	7930
Zirconia	1.0–1.5	6080
Glass	1.0–5.0	2500
Natural quartz sand	1.9–4.5	2650

.

The fluid medium used in the experiment was a glycerol aqueous solution with different concentrations, which was mixed and stirred by glycerol and water in a particular proportion. At the same time, different concentrations of solutions were configured to ensure that the experiment can be within the larger particle Reynolds number coverage. The fluid rheological properties were tested with an advanced rheometer and the fluid viscosity was obtained by fitting. The rheometer’s temperature control system controlled the temperature during measurement to be consistent with the experimental temperature. The schematic diagram of an Anton Paar modular compact rheometer MCR 92 is shown in Figure 3.

The rheological tests were carried out on different solutions to measure the rheological properties. To make the settled particles have extensive coverage of particle Reynolds number, we selected six aqueous glycerol solutions with different concentrations. The viscosity values were obtained by a MCR 92 rheometer, as shown in

Table 2. Rheological test results of glycerol aqueous solution used in the experiment.

Concentration (wt%)	Temperature (°C)	Density ρl (kg/m3)	Viscosity μ (Pa.s)
100	24.9	1260	0.865
95	24.3	1250	0.368
90	24.2	1230	0.165
80	25.2	1210	0.044
50	25.2	1116	0.00527
0	24.7	998.2	0.00104

.

2.4. Measurement of Circularity

Unlike the spheres, for irregular-shaped particles, the drag force is different for particles with different shapes and the settlement velocity is also different. Many researchers have established C_D–Re_s correlations for irregular-shaped particles, which contain detailed parameters describing the shape characteristics of particles (such as sphericity and Corey shape factor). Among these shape description parameters, sphericity is the most widely used. However, the three-dimensional shape information of sand particles can only be obtained through accurate laboratory measurements, and it is often impossible to calculate accurately. Compared with sphericity, circularity is a two-dimensional shape parameter, which is sensitive to the irregularity of particle contour and is easy to measure, which can be defined as:

$$c=\frac{4\pi A_{\mathrm{p}}}{P_{\mathrm{p}}{}^2}$$

(5)

where A_P is the surface area of the maximum projection surface and P_P is the perimeter of maximum projection.

C_D–Re_s correlations can be established by introducing the two-dimensional shape information of particles, and the three-dimensional shape parameters do not need to be considered. The circularity of particles was measured by image analysis and processing technology. The identification of circularity mainly includes the following two steps:

(1)

Image conversion: convert the original RGB digital image captured by the camera into an 8-bit gray-scale image. Examples of original and converted images of sand particles with different shapes are shown in Figure 4a,b.

(2)

Particle edge recognition: the gray threshold method was adopted to recognize the edge of particles. The principle of selection was to make the edge of the identified particle coincide with the edge of the original particle as much as possible. Typical images of different particle edge recognitions are shown in Figure 4c,d, in which the black line is the recognized particle edge.

Using 221 tests of effective irregular-shaped sand settlement experimental data obtained in the experiment, the equivalent diameter d_eq and circularity c were statistically analyzed, and the results are shown in Figure 5. The circularity c of most sand particles is concentrated between 0.7 and 0.8.

2.5. Data Processing

The position of each experimental equipment was determined to ensure its stable placement. Then, we used the high-frequency lighting equipment, adjusted the lens focal length and aperture size of the high-speed camera, and changed the brightness of the lighting equipment to achieve the best photo shooting effect. The specific process of the experiment was as follows:

(1)

The particles were immersed into the test liquid to ensure the surface of the particles was thoroughly wetted and the surface gas could escape. Then, we used tweezers to gently place the test particles in the center of the circular tube to make them settle freely along the center.

(2)

The high-speed camera was used to catch the settling trajectory of particles and multi-frame images of particle trajectory at different times were obtained.

(3)

The GetData graph digitizer (version 2.26) software was used to obtain image information, the particle settling trajectory was detected, and the particle terminal settling velocity was obtained later.

As shown in Figure 6, it is an example of a group of single-particle sedimentation experiments.

The digitizing software GetData graph digitizer (version 2.26) was used to analyze and process the position of particles at different times to determine the coordinates of the lowest point of particles in each image frame. The ratio of the displacement value Δs of the particles between the two images and the time interval Δt of the terminal settling velocity V_ts are given by:

$$V_{\mathrm{t}\mathrm{s}}=\frac{\Delta s}{\Delta t}=\frac{s_2-s_1}{t_2-t_1}$$

(6)

where Δs is the vertical displacement distance of the particles at times t₁ and t₂ and Δt is the time interval between two images at t₁ and t₂.

Three statistical parameters are applied to evaluate the prediction accuracy of the proposed C_D–Re_s correlation: mean relative error (MRE), maximal MRE, and root mean-squared logarithmic error (RMSLE), which can be expressed as:

$$MRE=\frac{1}{N}{\displaystyle \sum _{i=1}^N\frac{\left|C_{\mathrm{D},\mathrm{c}\mathrm{a}\mathrm{l},\mathrm{i}}-C_{\mathrm{D},\mathrm{m}\mathrm{e}\mathrm{a},\mathrm{i}}\right|}{C_{\mathrm{D},\mathrm{m}\mathrm{e}\mathrm{a},\mathrm{i}}}}$$

(7)

$$RMSLE=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(\ln C_{\mathrm{D},\mathrm{c}\mathrm{a}\mathrm{l},\mathrm{i}}-\ln C_{\mathrm{D},\mathrm{m}\mathrm{e}\mathrm{a},\mathrm{i}}\right)}^2}}$$

(8)

The terminal settling velocity was obtained through the experiments, and the relationship between C_D and Re_s was fitted. Based on the proposed C_D–Re_s correlation, Newton’s iterative method can be used to calculate C_D and V_ts. The calculation flow chart of the Newton iteration method is shown in Figure 7.

3. Results and Discussion

3.1. Drag Coefficient and Terminal Settling Velocity Analysis of Sphere

At present, many scholars have carried out a large number of experimental studies on the drag characteristics of spheres and obtained a large amount of experimental data. Classic correlations such as the Haider and Levenspiel model [7], Brown and Lawler model [8], and Cheng model [6] are expressed as follows:

Haider and Levenspiel model:

$$C_{\mathrm{D}}=\frac{24}{R{e}_{\mathrm{s}}}(1+0.186R{e}_{\mathrm{s}}{}^{0.6459})+\frac{0.4251}{1+6880.95/R{e}_{\mathrm{s}}}$$

(9)

Brown and Lawler model:

$$C_{\mathrm{D}}=\frac{24}{R{e}_{\mathrm{s}}}(1+0.15R{e}_{\mathrm{s}}{}^{0.681})+\frac{0.407}{1+8710/R{e}_{\mathrm{s}}}$$

(10)

Cheng model:

$$C_{\mathrm{D}}=\frac{24}{R{e}_{\mathrm{s}}}{(1+0.27R{e}_{\mathrm{s}})}^{0.43}+0.47\left[1-\exp (-0.04R{e}_{\mathrm{s}}{}^{0.38})\right]$$

(11)

In this study, we first carried out a sedimentation experiment on spherical particles, compared the drag coefficient C_D of 67 tests of spherical particles measured in the experiment with the particle Reynolds number Re_s, and plotted the relationship between the two in the form of classical logarithmic coordinates, as shown in Figure 8. The drag coefficient decreases gradually with the increase in the particle Reynolds number. When the particle Reynolds number is small, the distribution of the drag coefficient is consistent with Stokes’ formula. With the continuous growth of the particle Reynolds number, the drag coefficient decreases, but the decreasing rate gradually slows down.

The form of fitting formula refers to the existing classic expression ‘form of drag coefficient’ and carries out parameter fitting for 67 tests of experimental points. It is found that the expression form of the drag coefficient proposed by Cheng has the best goodness of fit, and its expression is:

$$C_{\mathrm{D}}=\frac{24}{R{e}_{\mathrm{s}}}{(1+AR{e}_{\mathrm{s}})}^B+C\left[1-\exp \left(DR{e}_{\mathrm{s}}{}^E\right)\right]$$

(12)

By the fitting and regressing of 67 tests of the experimental results, a C_D–Re_s correlation of the sphere in the Newtonian fluid is established as follows:

$$C_{\mathrm{D}}=\frac{24}{R{e}_{\mathrm{s}}}{(1+0.59R{e}_{\mathrm{s}})}^{0.28}+0.3\left[1-\exp \left(-0.76R{e}_{\mathrm{s}}{}^{0.26}\right)\right]$$

(13)

The drag coefficients of 67 tests of the spheres measured in the experiment are compared with the C_D–Re_s correlations mentioned in Equations (9)–(11) and (13), as shown in Figure 9, and the error statistics are carried out as shown in

Table 3. Error statistics of the drag coefficient of spherical particle.

Reference	Prediction Error
	MRE	RMSLE	The Maximal MRE
Brown and Lawler model	3.55%	0.045	17.32%
Cheng model	3.79%	0.048	15.77%
Haider and Levenspiel model	4.04%	0.053	18.75%
The present model	2.29%	0.032	9.51%

. The drag coefficient prediction model of spheres in Newtonian fluid proposed in this paper and the Brown and Lawler, Cheng, and Haider and Levenspiel models have a good fitting correlation with the experimental results. According to the error analysis in Table 3, the MREs between the predicted C_D of the four C_D–Re_s correlations and the experimental value are 3.55%, 3.79%, 4.04%, and 2.29%, respectively. The drag coefficient prediction accuracy of the sphere settling in Newtonian fluid proposed in this paper is the highest, and the values of the three statistical parameters are the lowest. In general, the prediction accuracy of the four C_D–Re_s correlations is very high, and the error is relatively small compared with the experimental data, which also proves the reliability of the experimental system and data processing method.

Based on the established C_D–Re_s correlation, the Newton iteration method is used to calculate the drag coefficient C_D and terminal settling velocity V_ts of the sphere settling in the Newtonian fluid. The model-predicted results are compared with the experimental results, as shown in Figure 10. It can be seen from the figure that the predicted value of the drag coefficient C_D is in good agreement with the experimental value, and the MRE between the experimental value and the calculated value of the correlation is 3.71%. The MRE between the terminal settling velocity V_ts and the experimental value is 1.88%. The proposed model can better predict the drag coefficient C_D and terminal settling velocity V_ts of the spheres in Newtonian fluid.

3.2. Drag Coefficient and Terminal Settling Velocity Analysis of Irregular-Shaped Sand Particles

In order to obtain the C_D–Re_s correlation of a single irregular-shaped sand particle settling in the Newtonian fluid, the particle settlement experiment was carried out, and the terminal settling velocity V_ts and drag coefficient C_D of the 221 tests of irregular-shaped sand particles with different particle sizes and shapes were obtained, as shown in Figure 11. Based on the C_D–Re_s correlation of the sphere proposed in the previous section, by introducing the circularity c, we intend to establish a C_D–Re_s correlation suitable for irregular-shaped sand particles. For any particle Reynolds number, the irregularity of the particle surface will increase the flow resistance and aggravate the flow separation phenomenon. Hence, sand particles’ drag force exerted by the fluid is more significant than its equivalent sphere. Due to the settling trajectory of the irregular-shaped particle being relatively complex, the jumping, rotating, swinging, and other behaviors of particles are also relatively strong in the settling process. The terminal settling velocity of irregular-shaped sand particles will be reduced compared to its equivalent sphere. Therefore, the ratio of drag coefficient C_D of the sand particle and its equivalent sphere is slightly greater than 1. With the increase in the particle Reynolds number, the influence of shape irregularity on sand particle drag coefficient is more significant. At a high Reynolds number, C_D/C_D,sph value will be higher. For example, when the particle Reynolds number reaches about 1000, the drag coefficient ratio C_D/C_D,sph can reach about 3–5.

In order to find out the best functional relationship between the shape description parameter (circularity c) and the particle Reynolds number Re_s, the drag coefficient C_D was measured using 221 tests of irregular-shaped sand particle settling experiments, and the prediction values of its equivalent sphere were normalized. Through analysis and fitting, we found that the function f (c) of circularity when used as the function of the natural logarithm of C_D/C_D,sph can obtain a better goodness of fit. Therefore, the relationship between circularity c and C_D/C_D,sph can be defined as follows:

$$C_{\mathrm{D}}=C_{\mathrm{D},\mathrm{s}\mathrm{p}\mathrm{h}}\exp \left[f\left(c\right)\right]$$

(14)

For extreme cases, such as c = 1, the drag coefficient C_D of sand particles should be equal to the drag coefficient of its equivalent sphere. That is, when c = 1, f (c) = 0. To ensure that C_D/C_D,sph = 1 is satisfied when the particles are spherical, we choose the relationship of f (c) to be determined by Equation 15, which can be defined as follows:

$$f(c)=\alpha R{e}_{\mathrm{s}}^{\beta }{(1-c)}^{\lambda }$$

(15)

The drag coefficient expressions of sand particles are investigated in data fitting through nonlinear regression using the Levenberg–Marquardt algorithm, which can be written as:

$$C_{\mathrm{D}}=C_{\mathrm{D},\mathrm{s}\mathrm{p}\mathrm{h}}\exp \left[1.45R{e}_{\mathrm{s}}{}^{0.07}{(1-c)}^{0.13}\right](0.02<R{e}_{\mathrm{s}}<1130)$$

(16)

The model-predicted drag coefficient C_D,sph exp f(c) of irregular-shaped sand particle calculated in Equation (16) and the experimentally measured drag coefficient are compared, as seen in Figure 12. This figure shows that the established drag coefficient prediction model with the particle shape description parameter (circularity c) has a good prediction accuracy (MRE = 9.72%).

So far, as shown in Equations (13) and (16), the C_D–Re_s correlation of irregular-shaped sand particles in the Newtonian fluid is established. According to the proposed model, the Newton iteration method is used to calculate the drag coefficient of irregular-shaped sand particles. The model-predicted results are compared with the experimental measured results, as shown in Figure 13a. It can be seen from the figure that the MRE between the model-predicted and the experimentally measured drag coefficients is 13.54%, and the data points are scattered to a certain extent. Still, the data are well distributed near the best fitting line, with high prediction accuracy. According to the proposed model, the terminal settling velocity V_ts of irregular-shaped sand particles is calculated by the Newton iteration method and compared with the experimental measurement results, as shown in Figure 13b. This figure shows that the MRE between the model-predicted and experimentally measured terminal settling velocities is 11.56%. The proposed model can better predict the settling velocity of irregular sand particles in a Newtonian fluid.

4. Conclusions

In this study, the terminal settling velocities of 67 tests of spheres and 221 tests of irregular-shaped sand particles in the Newtonian fluid were obtained experimentally by setting up an experimental platform for particle sedimentation. Using image analysis technology, the circularity c, a two-dimensional shape description parameter, was introduced to describe the shape characteristics, and C_D–Re_s correlations suitable for spheres and natural sands were established, which is important for further predictions of hydraulic behavior in the chemical industry, medicine, geotechnical, geology, petroleum engineering, and other fields. The following conclusions can be drawn:

(1)

Based on 67 sets of experimental data, we establish a C_D–Re_s correlation to predict the drag coefficient of spheres settling in the Newtonian fluid. The predicted results agree with the published classical formula and experimental results. The mean relative errors of the drag coefficient and settling velocity between the predicted and experimental results are 3.71%, and 1.88%, respectively, which proves the accuracy of the experiment and data processing.

(2)

The ratio of the drag coefficient C_D of the sand particles and its equivalent sphere is greater than 1. Natural sand particles’ drag force exerted by the fluid is more significant than its equivalent sphere. At a high Reynolds number, the C_D/C_D,sph value will be higher.

(3)

Based on 221 tests of experimental data, we establish a C_D–Re_s correlation to predict the drag coefficient of irregular-shaped sand particles settling in the Newtonian fluid. by introducing circularity c to describe the shape of the particles. The predicted results are in good agreement with the experimental results. The mean relative errors of the predicted drag coefficient and terminal settling velocity between the predicted and experimental results are 13.54% and 11.56%, respectively.

(4)

For future research, to expand the range of particle Reynolds number, the computational fluid dynamics (CFD) method and the deep learning (DL) method will be selected to improve the drag coefficient correlations of irregular-shaped particles.