Building a shear crushing specific energy consumption prediction model for manufactured sand aggregate

Abstract

In the field of concrete materials in the construction industry, manufactured sand gradually replaces natural sand in order to cope with the shortage of natural sand resources, thus leading to the problem of energy consumption in producing manufactured sand. In this paper, single particle shear crushing test of three kinds of manufactured sand aggregates of granite, limestone and diabase were taken as the research object. The energy consumption in single particle shear crushing was measured and the fractal theory was introduced to analyze the mass distribution of crushed particles. With the help of crushing specific energy consumption, a fractal prediction model of crushing energy consumption was established, and R-R model was used to correct the model. Finally, this paper verified the effectiveness of the corrected model by a series of tests that compare actual measured values, predictive values from the uncorrected model and predictive values from the corrected model. The method is well worth considering and learning from when it comes to energy saving in producing manufactured sand.

Keywords

Manufactured sand shear crushing energy consumption prediction model crushing specific energy consumption

1. Introduction

Infrastructure construction in China is thriving with the development of the economy, leading to a sharp increase in the use of concrete, an important base material of which is sand aggregate. Since natural sand is a non-renewable resource, many parts of China have been short of natural sand. Therefore, manufactured sand, with its advantages, has gradually become a substitute for natural sand in the development of the industry. However, in the crushing process of producing manufactured sand, energy consumption is a problem that cannot be neglected.

Currently in the field of crushing, energy saving and consumption reduction remains a significant goal. Traditional “three major crushing theories” are partial because they are merely based on the particle sizes before and after crushing and cannot scientifically determine the energy consumption of crushing [13]. In the 1970s, Mandelbrot [17] proposed the fractal theory. The fractal theory is studied on the unsmooth and irregular geometry of the nonlinear system. It doesn’t change as the system zooms in or out. The fractals are self-similar objects that allow the fractal fragment size distribution to be predicted at any size [18]. A large number of scholars have studied the fractal theory of the particle crushing. Mandelbrot [17] established a fractal model of two-dimensional spatial particle distribution. Tyler et al. [20] established a fractal model of particle size distribution in 3d space. Xu [19] has established the model of the rockfill sample from the numerical simulation and test. Zhao et al. [21] and Tian et al. [22] applied fractal theory to engineering materials from the perspective of theory and experiment. And they established the model of fractal dimension represented by mass.

Dynamic compression tests were carried out on layered rocks with different bedding dips under different impact velocities, and fractional dimension was used to quantitatively characterize the fragmentation distribution characteristics, which indicated the energy utilization rate [1]. Zhang et al. studied the fracture and energy partitioning of rock specimens under various loading rates [2]. Li et al. [3] analyzed the proportion of absorption energy to input energy of bedding sandstone which was crushed under impact loading. It was regarded that the maximum energy absorption ratio was reached when bedding plane was parallel to loading direction. However, the above-mentioned approaches are applied to the field of rock mining and tunnel excavation. Xu [4] put forward that shear crushing strength could be deduced by using fractal model for particle breakage and size effect of single particle crushing strength, but its relationship with energy consumption was not mentioned. In Crushing tests were carried out on granite and the relationship between particle crushing energy consumption and fractional dimension was determined through the use of fractal theory [5]. The feasibility of the method was proved by verification tests, however, the variables concerned were in a complex relationship and fitting based on more than one hundred times of crushing tests was necessary.

On the basis of single particle shear-crushing tests of manufactured sand aggregates, granite, limestone and diabase’s crushing energy consumption and mass distribution of broken particles were measured respectively. Then fractal model and crushing specific energy consumption were introduced to build crushing energy consumption fractal prediction model. Measured values and predictive values were compared to verify the feasibility of the plan. The method proposed in this paper is easy to be copied and can be applied to other manufactured sand aggregates. It is well worth considering and learning from when it comes to energy saving in producing manufactured sand.

2. Experimental procedure

2.1 Test bed for sandstone crushing

Sandstone particle crushing test bed, shown in Fig. 1a, was used to carry out single particle crushing experiment. The vertical impact of the equipment is driven by pneumatic control system. The upper plate and the lower plate will stop squeezing after they contact with the sheared stone. In the horizontal direction, the screw is driven by the motor to move the lower plate. There is a force sensor in each of the horizontal and vertical directions, and is connected with the data collecting sheet to collect force data. The maximum transverse shear force of the test equipment is 2 KN and the maximum longitudinal extrusion pressure is 18 KN. The actual machine is shown in Fig. 1b.

Figure 1.

Test bed for sandstone particle crushing.

Initialization: enter manual mode after booting the test bed, then choose Mode II. First press Mode II for 3 seconds and then 5 seconds. When the screw moves to the leftmost point and the detecting sheet to the limit sensor in the middle, the initialization is completed. Selecting automatic mode: when all parts return to their initial positions, choose Automation and then press Start. In Mode II, the screw moves and powers the crushing.

Figure 2.

Diagram of particle breakage test.

As shown in the Fig. 2, there are two main cases of particle breakage in the shear process.In the first case, the particles break into two pieces, and in the second case, the particles break into several pieces. The characteristic diameter of a single particle was recorded in the test bed before the test began. And the data of shear force was collected automatically. When the device began to work, the upper panel did not move, and the lower panel loadde horizontally at the speed of 0.1 mm/s. After the particle was completely broken, the test bed stopped, the experimental data was recorded and the broken particle was collected.

2.2 Analysis and measurement of energy consumption

The shearing bed runs at a constant velocity $v$ . Force-time curve $F-t$ showing the change of shear forces can be obtained by connecting data collecting sheet with the sensor in crushing bed. As the shear force data collected by data collecting sheet are discrete data, suppose that the frequency of data collecting sheet is $f$ and the number of shear force data points collected in one test is $n$ . Then shear crushing energy consumption $W$ can be calculated by Discrete Differential Element Method [15, 16]:

$\displaystyle W=\Delta s\times\sum\limits_{i=1}^{n}{F(i)}=v\times\frac{1}{f}% \times\sum\limits_{i=1}^{n}{F(i)}$ (1)

Above, $\Delta s$ is displacement in a frequency time, whose unit is $m$ ; the unit of crushing energy consumption $W$ is $J$ ; and the unit of shearing bed velocity $v$ is $m/s$ .

3. Results and discussion

3.1 Principles of fractal model of crushing specific energy consumption

3.1.1 Crushing specific energy consumption

According to statistics, most crushing machines have an energy utilization rate of less than 30% [6], meaning that much of energy consumed by sand making factories has been wasted. For this reason, energy consumption is an indicator to evaluate crushers’ operating quality [7]. Crushing specific energy consumption is used to evaluate manufactured sand aggregate firmness and to determine the method and the strength to be applied to crush sandstone. It means per unit mass energy consumption for crushing manufactured sand aggregate:

$\displaystyle q=\frac{W}{M_{T}}$ (2)

Above, $M_{T}$ is total mass of manufactured sand aggregate, whose unit is $g$ . $W$ means the energy consumed for crushing, whose unit is $J$ .

3.1.2 Manufactured sand aggregate fractal model

In the 1980s, Mandelbrot [8] established Fractal Theory. The theory mainly deal with problems that Euclidean geometry cannot measure and describe. It is used to characterize that complicated or discrete geometries do not change even the whole system has been expanded or reduced. This property is referred to as “self-similarity” and “scale-invariance”. Manufactured sand aggregate, having and not having been crushed, has the property from the perspective of microcosmic morphology, so it can be analyzed by fractal model.

Manufactured sand aggregate having been sheared, the particles were sorted by sieves of mesh diameters from small to large $d_{i}$ ( $i=1,2,3,\ldots,N$ , unit: $m m$ ). The mass of the particles passing through sieve is $M_{1}(d_{i})$ , and that of those not passing through sieve is $M_{2}(d_{i})$ , for each $d_{i}$ :

$\displaystyle M_{T}=M_{1}(d_{i})+M_{2}(d_{i})$ (3)

In three-dimensional space, as mesh diameter increases, the number of particles passing through the sieve declines. The volume of the remaining particles that do not pass through the sieve can be expressed as:

$\displaystyle V_{2}(d_{i})=C_{v}[1-(d_{i}/\lambda_{i})^{3-D}]$ (4)

Here, $V_{2}(d_{i})$ is the volume of the particles that fail to pass through the sieve; $D$ is fractional dimension, usually used to characterize particle size fraction mass distribution and evaluate crushing effect [9]; and $C_{v}$ and $\lambda_{i}$ are constant coefficients related to particle shape. $\rho$ – particle density remains unchanged all the time, hence the mass of the particles that fail to pass through the sieve:

$\displaystyle M_{2}(d_{i})=\rho V_{2}(d_{i})=\rho C_{v}[1-(d_{i}/\lambda_{i})^% {3-D}]$ (5)

and the whole particle mass can be defined as:

$\displaystyle M_{T}=\rho C_{v}[1-(0/\lambda_{i})^{3-D}]=\rho C_{v}$ (6)

The largest particle diameter after crushing is assumed as $d_{\max}$ , hence $\lambda_{i}=d_{\max}$ . Put it in Eq. (5), then:

$\displaystyle\frac{M_{1}(d_{i})}{M_{T}}=\frac{M_{T}-M_{2}(d_{i})}{M_{T}}=\left% (\frac{d_{i}}{d_{\max}}\right)^{3-D}$ (7)

Take a logarithm of each side:

$\displaystyle\lg\left[{\frac{M_{1}(d_{i})}{M_{T}}}\right]=(3-D)\lg\left(\frac{% d_{i}}{d_{\max}}\right)$ (8)

From Eq. (8), we know that $\lg\left(\frac{d_{i}}{d_{max}}\right)$ and $\lg\left[\frac{M_{1}(d_{i})}{M_{T}}\right]$ are in a linear relationship, with the slope $k$ of $3-D$ , hence fractional dimension of particle distribution [10]:

$\displaystyle D=3-k$ (9)

3.1.3 Fractal model of crushing specific energy consumption

The mass distribution function of manufactured sand of size fraction $d_{i}$ after crushing is assumed as $m(x)$ [11], then fraction mass distribution after crushing is:

$\displaystyle m(d_{i})=Cx_{\max}^{3-D}$ (10)

Here, $m(d_{i})$ is the mass of particles of size fraction $d_{i}$ ; $C$ is correlation coefficient of physical properties of manufactured sand aggregate; and $x_{\max}^{3-D}$ is the largest particle diameter of corresponding size fraction, whose unit is $m m$ .

Then total mass $M_{T}$ can be expressed as:

$\displaystyle M_{T}=Cd_{\max}^{3-D}$ (11)

From Eqs (2) and (11), crushing specific energy consumption fractal prediction model can be deduced:

$\displaystyle q=\frac{W}{M_{T}}=\frac{W}{Cd_{\max}^{3-D}}$ (12)

Equation (11) shows the relationship in crushing specific energy consumption $q$ , fractional dimension $D$ and crushing energy consumption $W$ .

In the above model, sieve size fraction $d_{i}$ is selected before. Equation (10) is selected to the largest particle diameter $x_{\max}^{3-D}$ of size fraction $d_{i}$ . However, in the same range of fraction there exist particles of other sizes. Especially when the particles are broken into powders, whose size cannot be measured, mass error appears. Therefore, for mass distribution function, mesh diameter $\overline{{d}_{i}}$ of R-R model, appropriate for coarse fraction, from [12] was applied to correct $m(d_{i})$ . Equation (10) was corrected as follows:

$\displaystyle m(d_{i})=M_{d}{\{}1-\exp[-(d_{i}/\overline{d_{i}})^{3-D}]{\}}$ (13)

Here, $m(d_{i})$ is mass of particles of size fraction $d_{i}$ ; $\overline{d_{i}}$ is a correlative quantity of average size of crushed particles, that is, the diameter of the sieve mesh, the accumulated mass of particles of fraction ranges passing through which reaches half of the total mass of the rock fragments; and $M_{d}$ is correlation coefficient of rock total mass.

Then fractal model for predicting crushing specific energy consumption in Eq. (12) can be corrected as:

$\displaystyle q=\frac{W}{M_{d}{\{1-\exp}(d_{\max}/\overline{d_{i}})^{3-D}]\}}$ (14)

This corrected fractal model for predicting crushing specific energy consumption can not only reflect the relationship between crushing specific energy consumption $q$ , fractional dimension $D$ and crushing energy consumption $W$ but also its relationship with correlative quantity of average size of crushed particles.

3.2 Building fractal model of crushing specific energy consumption

3.2.1 Fitting of fractional dimension

The following were the parameters of the three particles in this experiment. The density of granite is 2.79 kg/m ${}^{3}$ and the elastic modulus is 1.3 $\times$ 10 ${}^{6}$ kg/cm ${}^{3}$ . The density of limestone is 2.68 kg/m ${}^{3}$ and the elastic modulus is 1.7 $\times$ 10 ${}^{6}$ kg/cm ${}^{3}$ . The density of diabase is 2.91 kg/m ${}^{3}$ and the elastic modulus is 5.7 $\times$ 10 ${}^{6}$ kg/cm ${}^{3}$ .

Thirty-group specimens of granite, limestone and diabase with particle diameters of 35–65 mm were selected respectively to carry out single particle crushing test on shear bed. After the aggregates were crushed, sieves with mesh diameters of 2, 5, 8, 11 and 30 mm (diameters of particles to be crushed were all below 30 mm) were used to sort and record mass distribution of different size fractions. The results are shown in Table 1.

Table 1
Size fraction mass distribution

Specimen type	Size fraction	30-11	11-8	8-5	5-2	2-0	Total
Granite	Mass (g)	877.81	103.08	132.79	24.29	2.03	1140
	Mass percentage (%)	77.1%	9%	11.6%	2.1%	0.2%	100%
Limestone	Mass (g)	616.21	71.16	78.78	25.70	0	791.85
	Mass percentage (%)	77.9%	9%	9.9%	3.2%	0	100%
Diabase	Mass (g)	572.7	57.79	33.1	27.51	3.95	695.05
	Mass percentage (%)	82.2%	8.4%	4.8%	4.0%	0.6%	100%

Statistical calculation was carried out according to size fraction mass distribution from Eq. (8) and Table 1. With horizontal coordinate $\lg\left(\frac{d_{i}}{d_{\max}}\right)$ and vertical coordinate $\lg\left[\frac{M_{1}(d_{i})}{M_{T}}\right]$ , curves were constructed using a series of data points of three manufactured sand aggregates. The fitting results are shown in Fig. 2.

Figure 3.

Fractional dimension fitting curves of three manufactured sand aggregates.

From the fitting results in Fig. 3, it is known that the fractional dimension fitting curve of granite is expressed as $f_{1}(x)=2.384x+0.1772$ ; the R-squared $R_{1}^{2}=0.9568$ ; and the curve slop $k_{1}=2.384$ , hence fractional dimension of granite $D_{1}=3-k_{1}=3-2.384=0.616$ . The fractional dimension fitting curve of limestone is expressed as $f_{2}(x)=1.834x-0.1148$ ; the R-squared $R_{2}^{2}=0.9662$ ; and the curve slop $k_{2}=1.834$ , hence fractional dimension of limestone $D_{2}=3-k_{2}=3-1.834=1.166$ . The fractional dimension fitting curve of diabase is expressed as $f_{3}(x)=1.892x-1.068$ ; the R-squared $R_{3}^{2}=0.9949$ ; and the curve slop $k_{3}=1.892$ , hence fractional dimension of diabase $D_{3}=3-k_{3}=3-1.892=1.108$ .

3.2.2 Fitting of coefficient

In Eq. (14) – fractal model for analyzing crushing specific energy consumption, there was an undetermined coefficient $M_{d}$ . Thirty-group specimens of granite, limestone and diabase (10 groups for each) with particle diameters of 35–65 mm were selected and their masses were measured. Then shear crushing tests were carried out on these specimens. During the process of the tests, data collecting sheet collected shear force data to construct the force-time curve $F-t$ . Then with the help of Eq. (1), shear crushing energy consumption $W$ was obtained.

According to Eq. (2), the crushing specific energy consumption $q$ of 30-group specimens can be obtained. The largest diameter $d_{\max}$ of particles crushed in each group was measured by vernier caliper and the average diameter of particles crushed $\overline{d_{i}}$ in each group respectively was obtained by calculating mass percentage of crushed particles. The results are shown in Table 2.

Table 2
Coefficient $M_{d}$ fitting data

Specimen type	Test no.	1	2	3	4	5	6	7	8	9	10
Granite	W (J)	130.86	160.35	230.24	195.45	223.35	144.56	160.45	120.25	375.50	47.72
	$d_{\max}$	22.57	23.16	27.48	26.3	24.46	26.14	19.32	17.08	27.27	12.74
	q (J/g)	1.88	2.20	2.11	1.97	2.69	1.48	3.37	3.41	3.49	2.72
	$\overline{d_{i}}$	30	30	30	30	30	30	30	30	30	30
Limestone	W (J)	8.66	27.84	40.28	30.20	20.03	12.15	29.08	16.54	37.21	19.07
	$d_{\max}$	15.9	25.49	21.71	23.48	18.11	11.38	21.93	15.51	25.08	14.87
	q (J/g)	0.51	0.69	1.34	0.87	0.92	1.32	0.95	1.02	0.95	1.27
	$\overline{d_{i}}$	30	30	30	30	30	30	30	30	30	30
Diabase	W (J)	35.20	64.38	26.90	48.52	14.02	46.55	46.84	21.05	33.20	71.85
	$d_{\max}$	15.41	19.88	14.22	18.76	15.46	12.78	17.44	15.43	19.65	18.90
	q (J/g)	1.78	2.01	1.58	1.69	0.70	1.20	1.87	1.06	1.06	2.47
	$\overline{d_{i}}$	30	30	30	30	30	30	30	30	30	30

Figure 4.

$F-t$ curve collected from three aggregates.

Figure 5.

Fitting of constant C of three aggregates respectively.

In Session 4.1 we have fractional dimension of each type of aggregate. And from Eq. (12) we know:

$\displaystyle M_{d}=\frac{W}{q\{1-\exp(d_{\max}/\overline{d_{i}})^{3-D}]\}}$ (15)

From Eq. (13) we know that energy consumption $W$ is in a linear relationship with $q\{1-\exp\linebreak(d_{\max}/\overline{d_{i}})^{3-D}]\}$ . With horizontal coordinate $q\{1-\exp(d_{\max}/\overline{d_{i}})^{3-D}]\}$ and vertical coordinate $W$ , curves were constructed using a series of data points of three manufactured sand aggregates from Table 2. The fitting results are shown in Fig. 4.

From Fig. 5 it is known that $M_{d}$ fitting curve of granite is expressed as $f_{4}(x)=-14.68x-2.22$ ; the R-squared $R_{4}^{2}=0.9978$ ; and the curve slop $k_{4}=-14.68$ , hence the constant of model for analyzing crushing specific energy consumption of granite $M_{d}=k_{4}=-14.68$ . $M_{d}$ fitting curve of limestone is expressed as $f_{5}(x)=-11.38x-2.037$ ; the R-squared $R_{5}^{2}=0.9918$ ; and the curve slop $k_{5}=11.38$ , hence the constant of model for analyzing crushing specific energy consumption of limestone $M_{d}=k_{5}=-11.38$ . $M_{d}$ fitting curve of diabase is expressed as $f_{6}(x)=-13.13x-0.4165$ ; the R-squared $R_{6}^{2}=0.9999$ ; and the curve slop $k_{6}=-13.13$ , hence the constant of model for analyzing crushing specific energy consumption of diabase $M_{d}=k_{6}=0.1119$ .

To sum up, corrected fractal prediction model of crushing specific energy consumption for granite, limestone and diabase can be expressed as Eqs (16)–(18) respectively:

$\displaystyle q_{(\text{Granite})}=\frac{W}{-14.68\times\{1-\exp(d_{\max}/% \overline{d_{i}})^{2.384}]\}}$ (16) $\displaystyle q_{(\text{Limestone})}=\frac{W}{-11.38\times\{1-\exp(d_{\max}/% \overline{d_{i}})^{1.834}]\}}$ (17) $\displaystyle q_{(\text{Diabase})}=\frac{W}{-13.13\times\{1-\exp(d_{\max}/% \overline{d_{i}})^{1.892}]\}}$ (18)

In the model for these three aggregates, the average diameters of crushed particle $\overline{d_{i}}$ have something to do with mesh diameters.

By the same method, constant $C$ fitting for the uncorrected model in Eq. (12) can be done and then the uncorrected fractal prediction model of crushing specific energy consumption can be obtained:

$\displaystyle q_{(\text{Granite})}=\frac{W}{0.04066d_{\max}^{2.384}}$ (19) $\displaystyle q_{(\text{Limestone})}=\frac{W}{0.1063d_{\max}^{1.834}}$ (20) $\displaystyle q_{(\text{Diabase})}=\frac{W}{0.1119d_{\max}^{1.892}}$ (21)

3.3 Validation of the accuracy of model

To verify the accuracy of corrected fractal prediction model of crushing specific energy consumption in Eqs (13)–(15), another 5 groups of specimens of each type of aggregate respectively were selected as verification data set, shown in Table 3.

Table 3
Verification data set

Specimen type	Test no.	1	2	3	4	5
Granite	W (J)	137.55	172.53	245.54	218.93	184.38
	$d_{\max}$	19.23	21.25	25.69	23.69	22.85
	M (g)	50.38	62.98	94.23	87.03	76.18
	Test no.	6	7	8	9	10
Limestone	W (J)	62.82	38.37	39.01	58.51	32.29
	$d_{\max}$	32.55	23.52	22.12	31.26	18.11
	M (g)	59.26	40.23	41.53	56.56	34.25
	Test no.	11	12	13	14	15
Diabase	W (J)	21.91	31.85	26.01	27	57.03
	$d_{\max}$	12.36	18.25	14.88	14.76	22.35
	M (g)	14.10	25.56	19.03	18.55	41.85

Table 4

Error value

Type	Granite	Limestone	Diabase	Average error
Error value (uncorrected vs actual)	0.0731	0.1873	0.0872	0.1158
Error value (corrected vs actual)	0.0413	0.1115	0.0391	0.0639

Put the verification data into Eq. (2) to obtain actual crushing specific energy consumption $q_{s}$ . Put the verification data into uncorrected model Eq. (20) and corrected model Eqs (16)–(18) to obtain uncorrected crushing specific energy consumption $q_{y}$ and corrected crushing specific energy consumption $q^{\prime}_{y}$ respectively in prediction model. Compare the three groups of crushing specific energy consumption values and the results are shown in Fig. 6.

Figure 6.

Crushing specific energy consumption predictive values vs actual measured values.

According to error formula $\frac{|q_{s}-q_{y}|}{q_{s}}$ , error values of uncorrected prediction model $q_{y}$ and corrected prediction model $q^{\prime}_{y}$ to actual crushing specific energy consumption value were obtained, shown in Table 4.

The average error of the uncorrected model to actual value $q_{s}$ is 0.1158 and the average error of the corrected model to actual value $q_{s}$ is 0.0639, indicating that the corrected model has a better effect than the uncorrected one. In these two models, limestone $q$ errors are higher than those of the other two aggregates, which may be because of outliers of the size fraction 0–2 and the mass 0 in fractional dimension $D_{2}$ fitting. Although there are still errors in the corrected model, compared to actual value, they are within a reasonable range, meaning that the corrected model for analyzing and predicting crushing specific energy consumption is scientifically rational and has important application value [13].

4. Conclusion

By single particle shear crushing tests and based on the introduction of crushing specific energy consumption and fractal model theory, this paper proposed a corrected fractal prediction model of crushing specific energy consumption. The relationship between crushing specific energy consumption, fractional dimension, the largest diameter of particles crushed, the average diameter of particles crushed and the mass of particles crushed was established. It is worth considering and is of certain application value in terms of manufactured sand shear crushing technology of different aggregates. In our future studies, power consumption model of shear crushing machine will be included in order to provide a more scientific design for shear crushing machine.

Footnotes

Acknowledgments

The authors acknowledge the National Natural Science Foundation of China (Grant: 51905100) and the Young and middle-aged teachers project of Fujian province (Grant: JAT190429).

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