Keywords

1 Introduction

Cement concrete is an artificial stone made of cement, water, sand, stone and additives according to a certain proportion, which is widely used in China's infrastructure construction industry [1]. Mixing is a process to make the overall dispersion of concrete mixture more uniform, and it is a key procedure in concrete production [2]. After mixing ordinary concrete, it is found that the cement slurry in the homogeneous concrete is observed macroscopically. Under the microscope, it is found that 10–30% of the cement agglomerates will agglomerate together [3], forming tiny flocculent aggregates and dry powder aggregates, which are not evenly dispersed in the water, affecting the overall uniformity of the final concrete, and having a very negative impact on the strength, durability and workability of the concrete [4].

The mixing process plays a role in breaking up agglomerates and dispersing particles, which is essential for improving the fluidity of the slurry [5, 6]. The fluidity of the slurry depends on the microstructure of the slurry, and cement agglomeration and dispersion directly affect the flow of the slurry [7,8,9,10]. Mixing speed is the main operating parameter imposed during the mixing process and has an important influence in the agglomeration and dispersion of cement agglomerates. Therefore, it is important to study the effect of speed on cement agglomerates to improve the strength of concrete.

Numerical simulation is an effective means to study the collision, crushing and dispersion of materials in the mixing process. On the basis of molecular dynamics, Cundall proposed the discrete element method (DEM) as an analysis method of discrete materials in 1971 [11]. DEM is a common method to study the flow behaviour of particles. The macroscopic flow behaviour of mud and cement agglomerates is reproduced by reconstructing the internal meso-structure [12, 13]. Compared with computational fluid dynamics (CFD), which is difficult to reproduce the movement process of particles, DEM has certain advantages in dealing with the interaction between leaves and particles under mixed conditions and characterizing the agglomeration and dispersion process of particles [14, 15]. Therefore, the aim of this article is to model cement agglomerates using the discrete element software EDEM and the discrete cell method to study the flow behaviour of particles during the mixing process more clearly. By simulating different rotational speeds, the important effects of rotational speed on cement agglomerate crushing and on concrete strength enhancement can be deeply investigated.

2 Establishment of Discrete Element Simulation Model

In order to simulate the crushing of cement aggregates in the mixer, it is necessary to model the mixer and cement aggregates respectively. When the mixer is modelled, the solid mixer is simplified, and the mixing device part is retained and modelled in the three-dimensional modelling software SolidWorks. Using the API (Application Programming Interface) secondary development interface provided by EDEM, the particle replacement and modelling of cement aggregates are carried out. Mixer model and cement aggregates model constitute the basis of the mixing model in this article. Set different mixer speeds to study the crushing of cement aggregates at different speeds.

2.1 EDEM and Discrete Element Method

EDEM is a general discrete element solution software, which consists of three parts: pre-processing module (Creator), solving module (Simulator) and post-processing analysis module (Analyst). The pre-processing mainly completes the model import and setting, the definition of particle properties, the establishment and setting of particle model and particle factory. The solver mainly completes the simulation of the motion process of the geometric model. The post-processing analysis module mainly completes the analysis and processing of the calculation results, and completes the graphical and visual expression of the results.

In the study of particle flow problems, the discrete element method usually divides the studied area into spherical particles, and then based on Newton's second law, calculates the displacement, velocity, acceleration, force and other variables of the particles in each time step, and then enters the next operation process through contact judgment. As shown in Fig. 1. There are usually two different types of processing methods for soft sphere and hard sphere models. The hard sphere model completely ignores the size of the particle contact force and the deformation details of the particle surface, and the contact process is simplified to an instantaneous collision process. The cement group studied in this article has the surface deformation of the particles, so the soft ball model is selected. The soft sphere model simplifies the contact process between particles into the damping motion of the spring oscillator, and its motion equation is calculated according to Eq. (1).

$$m\ddot{x} + \eta \ddot{x} + kx = 0$$
(1)
Fig. 1
A and B are schematics of soft and hard sphere models, respectively. In A, the circuit has a resistor k and capacitor eta. In B, arrows from four different sized discs point to an explosion symbol.

Soft sphere model and hard sphere model

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In the Eq. (1), \(x\) is the displacement deviating from the equilibrium position; \(m\) is the mass of vibrator; \(\eta\) and \(k\) are spring damping coefficient and elastic coefficient respectively.

2.2 Cond Model

In the EDEM software, the cement particles in a cement agglomerate are connected to each other by means of bonds. The bonds are subjected to tangential and normal forces. When the critical value is reached, the bonds breaks, and the particles are treated as rigid bodies in subsequent calculations. When the particles have not been bonded, the calculation of the particles is calculated by the standard contact model. When the bonding occurs, the ratio of the normal force and moment of the particles is 0, and the following equations are used to calculate the moment and normal force of the particles in each time step.

$$\delta F_{n} = - \upsilon_{n} S_{n} A\delta t$$
(2)
$$\delta F_{t} = - \upsilon_{t} S_{t} A\delta t$$
(3)
$$\delta M_{n} = - \omega_{n} S_{t} J\delta t$$
(4)
$$\delta M_{t} = - \omega_{t} S_{n} \frac{J}{2}\delta t$$
(5)
$$A = \pi R_{B}^{2}$$
(6)
$$J = \frac{1}{2}\pi R_{B}^{4}$$
(7)

In the above equation, \(S_{n}\), \(S_{t}\) are the normal stiffness and tangential stiffness of the particles, respectively; \(F_{n}\), \(F_{t}\) are the normal force and tangential force of the particles, respectively; \(\delta t\) is the time step; \(\upsilon_{n}\), \(\upsilon_{t}\) are the normal and tangential velocity of particles, respectively; \(\omega_{n}\), \(\omega_{t}\) are the normal and tangential angular velocity of particles, respectively; \(M_{n}\), \(M_{t}\) are the normal and tangential torque of the particles, respectively; \(R_{B}\) is the radius of the bonds.

The critical values of normal and tangential shear stresses can be calculated by the following equation:

$$\sigma_{\max } < \frac{{ - F_{t} }}{A} + \frac{{2M_{t} }}{J}R_{B}$$
(8)
$$\tau_{\max } < \frac{{ - F_{t} }}{A} + \frac{{2M_{n} }}{J}R_{B}$$
(9)

The bonds parameters between particles and particles are shown in Table 1. The replacement time in API compilation is 0.4 s, and the particles are bonded immediately after replacement. Therefore, the bond time is also set to 0.4 s.

Table 1 Bond parameter
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In the simulation process, the constructed cement aggregates should be as close as possible to the actual situation to ensure the accuracy of the simulation. Therefore, it is one of the factors to ensure the simulation results to reasonably determine the intrinsic parameters, basic contact parameters and contact model parameters of aggregates, cement, cement agglomerates and mixing equipment geometric model. In order to simplify the calculation and solution process, two kinds of spherical particles are used to represent single cement particle and cement agglomerates respectively. The particle radius is 3 mm and 20 mm respectively, and the bonds radius between cement particles in the cement agglomerate is 4 mm. Aggregates particles are continuous random distribution with particle size distribution of 15–40 mm. The intrinsic parameters of the material are shown in Table 2, and the attribute parameters between the materials are shown in Table 3.

Table 2 Material intrinsic parameter
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Table 3 Material construction contact attribute property
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2.3 Cement Aggregates Modeling

The cement slurry in the mixing process involves two kinds of particle phases that need to be mixed, one is a single cement, and the other is a cement aggregates. EDEM software is often used to build a geometric model of cement agglomerates by covering and dispersing a single cement particle in the geometric model of cement agglomerates, as shown in Fig. 2. The average particle size of cement aggregates was selected as 20 mm. The steps of EDEM to construct cement aggregates model are as follows:

Fig. 2
A set of 4 illustrations, A, B, C, and D, of the modeling of cement aggregates represented by circular particles through four different stages.

Cement aggregates modeling

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  1. (1)

    The spherical model of cement agglomerates was drawn in SolidWorks, with a radius of 20 mm, saved in IGS format and imported into EDEM software, as shown in Fig. 2a.

  2. (2)

    New cement particle, set the spherical filling particle size of 3 mm;

  3. (3)

    Set up a particle factory to statically produce filled particles, wait for the filled particles to completely cover the cement aggregates ball shell, and pause after standing, as shown in Fig. 2b.

  4. (4)

    The cube container material is changed to virtual, and the cement aggregates material is changed to reality. After preservation, the simulation is continued. The filling particles outside the shell of cement agglomerates will gradually disperse out of the calculation domain and disappear under their own gravity and the interaction between particles, as shown in Fig. 2c. The filling particles inside the shell are retained, which is the cement aggregates, as shown in Fig. 2d.

When EDEM software is used for simulation calculation, the software cannot directly produce cement aggregates and needs to run API plug-ins to replace them. As shown in Fig. 3a, in the simulation of the process of crushing cement agglomerates by mixing equipment, the particles before replacement are first generated from the particle factory, that is, a complete cement agglomerate. Under the action of the API plug-in, EDEM removes the previously generated replaced particles and then replaces them with cement aggregates composed of many small cement particles, as shown in Fig. 3b. At the same time, a bond (blue cylinder) is generated between small cement particles, as shown in Fig. 3c. The cement ball collides with the aggregates and the machine during the mixing process, and the bonds breaks, as shown in Fig. 3d.

Fig. 3
A set of four schematics featuring the transformation of a solid sphere into a complex network structure, illustrating a process or change over stages.

The generation and fracture of cement agglomerate bonds

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2.4 The Establishment of Mixer Model

As shown in Fig. 4, the double horizontal shaft forced mixer is the main model in the mixing equipment at present. In this article, this model is used as a vessels machine for mixing cement aggregates and aggregates. As shown in Fig. 5, this article simplifies the mixer, only considering the mixing device, leaving the mixing cylinder, mixing shaft and mixing blade. The number of blades installed on the single stirring shaft is 5, and the blade installation angle \(\alpha\) is 37°. The mixing blades are arranged in staggered arrangement to avoid interference, and the arrangement is positive and negative. The two shaft speeds are the same, and the steering is opposite. Save it as IGS format and import it into EDEM.

Fig. 4
A photo of an old, dirty industrial machine in a cluttered workshop.

Double horizontal shaft mixer

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Fig. 5
Three schematics of three different views of a mixer model with various components and mechanisms.

Mixer model

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3 The Effect of Rotational Speed on Cement Aggregates

This article mainly studies the influence of stirring speed on the crushing process of cement agglomerates. The rotation speed of traditional cement mixing equipment is usually below 100 rpm. The rotation speed of mixing equipment studied in this article is set to 50 rpm, 60 rpm and 70 rpm respectively. The crushing situation of cement agglomerates under three rotation speeds is analyzed, and the retention of bonds between cement agglomerates under three rotation speeds are comprehensively analyzed to explore the influence of stirring speed on the crushing process of cement agglomerates. In order to control the variables and ensure the accuracy of the test comparison scheme, the number of cement agglomerates generated in each comparison test is set to be the same. Considering that the cement agglomerates are replaced by small particles, the number of cement agglomerates is fixed, and the number of bonds between small particles are also the same.

Figure 6 shows the simulation of the crushing process of cement aggregates at different rotational speeds. The dark yellow arrow indicates the movement direction and collision of each aggregate during the mixing process. The colored balls represent the cement aggregates produced during the mixing process, and the balls with different colors represent the magnitude of the force. It can be seen that there is a large amount of collision between cement particles, aggregates, and machines under the action of mixing. The bonds of cement aggregates in the middle and bottom areas of the mixer break, and the cement aggregates are broken and stirred more evenly. At the same time, the number of broken cement aggregates increases with the increase of the rotational speed of the mixer.

Fig. 6
A simulation diagram demonstrates the crushing process at different rotations per minute (50 r p m, 60 r p m, 70 r p m) and times (1 s, 8 s, 15 s), visualizing the dispersion of particles over time.

Simulation of crushing process diagram

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During the mixing process, there is a large amount of collision between cement particles and aggregates, as well as between the mixer and the mixing shaft, which is affected by various forces in different directions. In the process of cement particle crushing, there are both normal and tangential forces. In the process of cement particle crushing, there are both internal stress caused by normal force exceeding the strength, which leads to fragmentation, and shear stress caused by tangential force exceeding the cutting strength, which leads to fragmentation. As shown in Figs. 7 and 8, the average normal force and average tangential force acting on the cement paste at different rotational speeds are respectively shown.

Fig. 7
A line graph plots normal force versus time. 50 r p m, (0, 0.00), (35, 0.25), (50, 0.26). 60 r p m, (0, 0.00), (35, 0.35), (50, 0.35). 70 r p m, (0, 0.00), (35, 0.35), (50, 0.43). Values are estimated.

Average normal force on cement particles at different rotational speeds

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Fig. 8
A line graph plots tangential force versus time. 50 r p m, (0, 0.00), (35, 0.25), (50, 0.27). 60 r p m, (0, 0.00), (35, 0.35), (50, 0.35). 70 r p m, (0, 0.00), (30, 0.30), (50, 0.40). Values are estimated.

Average tangential force on cement particles at different rotational speeds

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With the operation of the mixer, the average normal force and average tangential force on the cement agglomerate at each rotational speed increased slowly with time. At 0–25 s, the difference between the normal force and tangential force at each rotational speed is not big, and both of them are growing slowly and steadily. There is a clear difference between the normal force and tangential force applied at 50 rpm and the other two rotational speeds after 25 s. At 25–50 s, the normal force and tangential force at 50 rpm rotational speed are no longer growing upward, but keeping a slight fluctuating stability. The normal force and tangential force at 60 and 70 rpm are keeping a slow and steady growth. The normal force at 60 rpm is smaller than that at 70 rpm and the tangential force is larger than that at 70 rpm.

As shown in Fig. 9 is the breakage of cement bonds with time for cement agglomerates at 50 rpm, 60 and 70 rpm mixing speeds, respectively. The number of cement bond breakage over time shows a rapid rise and then a slow rise and finally stabilizes. The number of bond breakage at all three speeds rises rapidly in the early stage, reaches the upper limit of the number of breakage and remains unchanged until the mixing stops. The number of breakages at 50 rpm gradually rises from 0 to 20 s, and reaches the upper limit at 20 s and remains unchanged until the end of the mixing process. The number of breakages at 60 and 70 rpm rises rapidly from 0 to 15 s, and reaches the upper limit at 15–50 s until the end of the mixing process. At 60 rpm and 70 rpm, the fracture number rises rapidly in 0–15 s, and reaches the upper limit in 15–50 s and remains stable. The faster the mixing speed, the faster the pre-breakage number grows, and the first to reach the upper limit of breakage.

Fig. 9
A line graph plots the number of broken bonds versus time. 50 r p m, (0, 0), (20, 4000), (50, 4300). 60 r p m, (0, 0), (20, 4200), (50, 4300). 70 r p m, (0, 0), (10, 4000), (50, 4300). Values are estimated.

Bonds changes with time at different speeds

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In this article, the number of bonds fracture in cement aggregates before and after mixing are used as a method to evaluate the crushing of cement aggregates, and the parameter of fracture percentage is introduced. The more the number of bonds fracture after mixing, the better the crushing effect of cement aggregates, the more conducive to the improvement of the microscopic uniformity of cement slurry, and the better the mixing effect. Therefore, by comparing the changes in the number of bonds before and after the completion of mixing, the crushing effect of cement aggregates and the mixing effect can be roughly judged.

Since the difference in the number of breaks at the later stages of the three rotational speeds was not significant, it was not possible to compare the advantages and disadvantages. Therefore, the point where the fracture number did not reach the upper limit was chosen as the end point to compare the percentage of fracture of the stirring. This article selects 15 s as the final point for comparison, as shown in Table 4. With the increase of stirring speed, the crushing effect of cement aggregates is correspondingly enhanced. The fracture rate of cement aggregates bonds in the mixer at 60 and 70 rpm is more than 90%, and the fracture rate of cement aggregates in the mixer at 50 rpm is less than 80%. Therefore, when the cement mixing operation is carried out, the stirring time should be maintained at least 15 s when the rotation speed is above 60 rpm. When the rotation speed is below 60 rpm, the stirring time should be prolonged accordingly, and the stirring strength should be improved.

Table 4 Bonds fracture at 15 s
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4 Conclusion

This article uses discrete element software to simulate the process of cement agglomerates crushing at different speeds in a mixer, and the following conclusions are obtained:

  1. (1)

    The normal and tangential forces on the cement aggregates increased slowly with time at all three speeds. In the early stage, the growth rate is similar and the difference is not significant; in the middle and late stage, the force at 50 rpm no longer grows, and the force at 60 and 70 rpm is still growing slowly.

  2. (2)

    When the speed of the mixer is constant, the fracture number of bonds between cement will show a rapid increase followed by a slow increase with time steps, and finally the speed tends to stabilize.

  3. (3)

    When the mixer was run for 15 s, the cement agglomerates bonds breaking rate reached more than 90% in the mixers with 60 and 70 rpm speeds, and the cement agglomerates breaking rate was below 80% in the mixer with 50 rpm speed.