Relation between Scale-Up and Life Cycle Assessment for Wet Grinding Process of Pumice

Abstract

This study examines the grinding process of pumice based on the dry and wet laboratory measurements, scale-up method, and life cycle assessment. This research’s main goal was to set up the relation between scale-up and life cycle assessment results for the wet grinding process with the help of mathematical equations. Within the first research works, basic grinding testing in a laboratory dry Bond mill was accomplished. This step allowed the description of the estimated particle size distribution, median particle size, specific grinding work, and grindability index number of pumice. The second step was the determination of power consumption and scale-up in a laboratory stirred media mill, and it involved the assessment of resources, primary energies, and environmental impacts of wet grinding using GaBi 8.0 software. According to the results, we obtain life cycle emission factors by introducing five coefficients for grinding in laboratory and industrial conditions. These constants depend on the external dimensions of the mill and can be expressed by a derived scale constant from the scale-up. Research results enable the industry to make a prognosis for industrial plants based on the integration between life cycle assessment and scale-up of the pilot grinding processes.

1. Introduction

Pumice is a natural volcanic rock with a high silica content, formed at a temperature of 500–600 °C after violent volcanic eruptions. Pumice is formed when very hot, high-pressure material suddenly flies out of a crater during magmatic explosive volcanic activity. The sudden cooling freezes the gas bubbles caused by the overpressure reduction in the solidified lava, and porous rock is formed [1]. Pumice is used in many industries, from mineral processing to mineral, glass, ceramic, building, fine surface treatment, abrasive material, paint, and cosmetics. Nowadays, fine grinding processes are actual research development areas. The main aims of fine grinding processes are reducing the particle size and increasing the specific surface area of the ground product. These goals can be achieved primarily by grinding in high-energy mills. Grinding fineness below 5 µm is required as an essential specification for industrial applications. Pumice-related research primarily focuses on using pumice in concrete applications in geopolymers and pumice-based alkali-activated composites, as well as soil improvement and cement replacement material [2,3,4,5]. A research study by Kabay et al. [6] investigated the effect of the partial replacement of pumice and fly ash with cement on the mechanical–physical and durability properties of concrete in Turkey. Liu et al.’s research [7] investigated pumice-replaced cement’s compressive strength and SEM morphologies. Planetary and ball mills are used for grinding the pumice, where smaller particle size increases the specific surface area. Pourghahramani and Azami [8] studied the mechanical activation of perlite using a planetary mill. They found that grinding results in a higher specific surface area for pumice than for perlite and granite. However, the use of stirred media mills for fine grinding is also known, and its effectiveness is confirmed by many previous research results [9,10,11,12,13]. From these antecedents, we developed the idea of examining the behavior of pumice in parallel in a dry laboratory ball mill and a wet laboratory stirred media mill.

Fine grinding processes are widely used in the mineral processing, chemical, pharmaceutical, and cement industries, as well as in the cosmetic, pigment, and food industries [14,15]. Much research on grinding processes is typically related to optimizing milling processes and is based on different modeling approaches. A significant view in fine grinding is the feasible final size of the materials. According to the research results of Karbstein et al. [16], particle size distribution will be impacted by the geometric and operating indicators, media, and feed material. Parker et al. [17] defined the impact of stirrer speed and milling bead loading on energy consumption. According to the product-related stress model of Kwade [18], the product quality and fineness of the grinding can be described by two facts: the stress event number and the stress intensity.

Stirred media mills are used for the mechanical grinding of ultrafine products and can be operated in continuous or batch mode. They can exist in many different sizes, and their specific energy consumption has been significantly reduced in recent years. Flach et al. [19] presented that the reduction in the specific energy consumption of these mills is due to the small size of the grinding media and the high energy intensity. However, it is also essential to explore the relationships between different parameters, such as the ground material’s density, the stirrer’s speed, and the mill’s major geometric dimensions. Modeling the processes in wet stirred and ball mills is challenging. It requires simultaneous modeling of the complex physics of interactions between the grinding media, the moving mixing elements, and the grinding materials [20,21]. According to Esteves et al. [22], in fine grinding applications, the energy efficiency of gravity-induced stirred mills is higher than that of traditional ball mills. Mannheim [9] came to a similar conclusion during her many years of research. The relationship between grinding efficiency and particle size was described by Kick (1885), Bond (1952), Walker and Shaw (1954), and Rittinger (1987). Based on our previous research results [9,23,24], the relationship between grinding fineness and specific grinding work can be mathematically described for all materials. Furthermore, the grinding fineness can be determined using the grindability index number and an exponent. The value of this angular coefficient for several reference materials is constant (0.2). Shin et al. [25] investigated the effect of the ball size on the grinding efficiency in a laboratory-scale wet ball mill. Rhymer et al. [26] investigated the sliding friction coefficients and the grinding media restitution using the discrete element method in a vertical stirred mill. Toraman et al. [27] examined the effect of ceramic grinding media and media wear on stirred milling. They concluded that the choice of grinding media significantly affects energy efficiency, internal wear, and operating costs. Ulusoy et al. [28,29,30] analyzed the characteristic properties of ground calcium carbonate, calcite, and talc particles with the same grinding fineness, and they achieved a larger final particle size in the ball mill than in the stirred media mill. In the case of ore mining tailings, in connection with comparing these two mills, Mannheim [9] also obtained a smaller final particle size after a grinding time of 20 min in a laboratory stirred media mill. The fact that stirred media mills result in significant energy savings thanks to the grinding media’s small size and the ability to mix at a higher speed is now basically known [31,32,33,34,35,36].

Researchers have been interested in describing scale-up models in stirred media mills for a long time. Many research studies [37,38,39,40] have been published on this topic from 1985 to the present day. In connection with our latest joint research work [24], we mathematically described the relationship between the energy model and the life cycle assessment. In this research work, we define the relationship between scale-up and LCA. This approach makes it possible to compare the technological solutions in terms of laboratory and industrial mills to reduce environmental burdens. According to Laso et al. [41], life cycle assessment is the most frequently used method for determining the impacts of products. Life cycle assessment is a technical environmental management method that enables the assessment of various products’ environmental and energetic burdens during their lifetime and can also be applied to grinding processes [24,42].

As part of our research, we simulated the life cycle of pumice grinding in a complex way in a laboratory stirred media mill based on cradle-to-gate assessment. Using these research results, an LCA-based model was determined and combined with scale-up. The scale-up model was investigated based on the laboratory measurements in a laboratory stirred media mill. Scale-up of different mills is essential to the industrial design of manufacturing grinding mills [43]. The scale-up can primarily be carried out empirically and with the help of discrete element simulation [44].

In our research work, the energy consumption of the stirred media mill for scale-up was examined by a method based on dimensional analysis. We assumed that the industrial stirred media mill and the laboratory stirred media mill have the same geometry. The stirrer speed of the industrial stirred media mill was basically given by knowing the rotational speed of the laboratory stirred media mill, so some parameters were kept at a constant value.

The first aim of this research work was to investigate the particle characteristics and the grindability of the pumice and to find a relationship between the median particle size and the specific grinding work in a mathematical way. A conventional dry Bond mill and a laboratory horizontal wet stirred media mill were used as laboratory equipment. The grindability of the pumice is represented by the Bond Work Index. The ground pumice particles’ size distribution can be described using nonlinear parameter estimation. The power consumption of the laboratory stirred media mill with grinding factors (stirrer speed, solid mass concentration, grinding time) was calculated using dimensional analysis. Besides the particle size distribution, the rheological behavior of the suspension and the grinding media wear are important indices of the grinding process. The dynamic viscosity of grinding media was already described in previous work [9].

The second goal was to estimate the primary energy and environmental burdens of the wet grinding process of pumice in a laboratory stirred media mill. The applied life cycle assessment includes the life cycle inventory, determining the functional unit, the system boundary, and the allocation method. GaBi 8.0 software was used to obtain primary energy values and environmental potentials for laboratory wet fine grinding of the pumice in a laboratory stirred media mill.

The third objective was to establish a relationship between the scale-up and the LCA results. The described mathematical integration between the scale-up and the life cycle assessment can be utilized to design industrial stirred media mills and wet grinding processes.

2. Materials and Methods

2.1. Material Preparation for Grinding

The examined pumice came from the Zemplén Mountains (in Hungary). The pumice was prepared by crushing in a closed cycle in a laboratory jaw crusher (less than 20 mm) and roll crusher (less than 3 mm). After a representative sampling, the particle size analysis of feed material was performed with a standard sieve series (sieve sizes: 2.5, 2.0, 1.6, 0.8, 0.4 and 0.1 mm).

2.2. Bond Method

The Bond test was measured in a laboratory dry Bond mill (volume: 700 cm³, mill diameter: 305 mm, mill length: 305 mm, speed: 70 rpm, ball balls’ weight: 20.1 kg) under specified conditions. The Bond Work Index determination, according to the standard Bond’s test, was performed on all these samples with comparative sieve sizes of 74, 105, and 150 microns. The maximum feed particle size of pumice is 3 mm for Bond grinding. The Bond Work Index (Wi_B) of ground pumice was estimated using the Bond formula under laboratory conditions, where G is the grindability coefficient and x_max is the grinding fineness (the sieve size is 100 μm for the laboratory measurements). Furthermore, x₈₀ and X₈₀ are particle sizes of 80% for the ground and feed materials [45]. Equation (1) describes this Bond Work Index under laboratory circumstances empirically [46].

$$W_{iB}=\frac{4.9}{{{x_{max}}^{0.23}·G}^{0.82}\left(\frac{1}{{\sqrt{x}}_{80}}-\frac{1}{{\sqrt{X}}_{80}}\right)}$$

(1)

The grindability coefficient was determined by laboratory measurements. This value was 0.00225 kg/min in the third grinding period after 66 g/rev.

2.3. Hardgrove Method

During the determination of the Hardgrove Grindability Index, 50–50 g pumice samples from the 0.63–1.25 mm and 50–100 μm particle size fractions were ground to 60 mill speeds. Then, the ground material was divided into 0.074 mm sieve fractions. In Equation (2), the Hardgrove Grindability Index (HGI) is determined by the standard Hardgrove formula, where m_H means the mass of particles smaller than 0.074 mm. According to Equation (3), the Bond Operating Index (Wi_B^H) is calculated from the HGI.

$$HGI=13+6.93·m_H$$

(2)

$$W_{{iB}^H}=\frac{435}{{HGI}^{0.82}}$$

(3)

2.4. Laboratory Wet Grinding Method

As experimental equipment, a wet laboratory stirred media mill (stirrer speed is 1440 1/min, mill filling ratio is 70%, beads filling ratio is 45.5%, and diameter of steel grinding balls is 3.175 mm) was applied. The grinding chamber of 700 mL is agitated by a rotor equipped with five mixing discs. During the laboratory measurements, the mill filling ratio (70% and 80%), the solid mass concentration (20% and 40%), and the time of the grinding (5, 10, and 20 min) were varied. A Sympatec Helos H0541 laser particle sizer was used to measure the grinding fineness of stirred media milling. The maximum feed particle size of pumice was 140 μm. The specific surface area was calculated using a Blaine and Griffin measurer.

2.5. Nonlinear Parameter Estimation and Dimensional Analysis

The particle size distribution of the pumice was depicted using a nonlinear parameter estimation method. According to the Gauss–Newton–Marquardt method, the characteristic particle size distribution can be accurately described. The relationship between the median particle size of pumice and the specific grinding work can be described mathematically. The dimensional analysis method was used to determine the power consumption of the stirred media mill for scale-up.

2.6. Life Cycle Assessment Method

The grinding processes can be assessed with the life cycle assessment method. The applied life cycle assessment method was based on the professional database and the construction extension database of GaBi 8.0 software. The life cycle system boundary ranged from the raw material extraction and the material processing (classification, different separation processes of the lightweight pumice, wastewater treatment, processing of secondary material input, transport to the manufacturing) to the wet grinding process. The loose, gravelly pumice stone is extracted from an open-cast mine. Following the degradation, a processing machine subdivides the raw pumice into different mineral size classes. Firstly, the machine disposes of all metallic matter using a magnet separation. Then a primary sieve separates the raw pumice into two groups, each with its processing sequence. The grains larger than 2 mm are fed into a sieve jigger in which the raw pumice is separated into two quality grades on an impulse-driven waterbed. The lighter, floating high-grade pumice is separated in a complex, 3-step process in which the light and the heavier parts with poor insulating properties are removed. In a separate processing of the fine material (=2 mm), the pumice with a high purity level in this fraction can be separated from the heavier sand. The water required for this process is sent to its reservoir, where it is clarified and recirculated. Foreign bodies from the separation processes, such as grit and sand, can be used in roadbuilding or can be used as abrasives.

Different inputs such as electricity production mix, tap water from surface water, and the prepared lightweight pumice are used for the laboratory wet grinding process. The life cycle dataset represents a cradle-to-gate inventory and is linked with wet grinding data to create life cycle inventories for the pumice product. Equipment and machinery are not relevant in this examination. The energy consumption was assigned as a function of the energetic content. Figure 1 presents the system boundary of the applied LCA method.

The dataset was modeled according to the European Standard EN 15804 for Sustainable Construction. This laboratory wet grinding process was assigned as a function of the mass of the ground pumice according to the allocation suggested by ISO 14044 [47,48]. All data sources come from the European Union (EU-28) and are related to energy and material balances. Background processes were modeled with the GaBi professional and GaBi supplementary construction database [49].

All material and energy resources, all emissions, and all environmental burdens were associated with the ground pumice product from the wet grinding process in the laboratory stirred media mill. Eleven environmental impacts—global warming excluding biogenic carbon, acidification, eutrophication, photochemical oxidant formation, human toxicity, abiotic depletion (fossil and elements), and different ecotoxicities—but mainly eight impact categories were estimated.

2.7. Scale-Up Method

Scale-up of the stirred media mills is possible if the industrial and the laboratory stirred media mills have similar geometry and use the same grinding media with the same filling ratios. Only a little professional literature is available on the scale-up of stirred media mills. The main question regarding the scale-up of stirred media mills is what conditions must be met to obtain the same grinding results with different-sized mills. Another question is how to operate differently stirred media mills to achieve optimal results. However, the peripheral speed must be constant. If one of these conditions is not met, the specific energy can be used for scaling, where the same specific energy gives the same particle size regardless of mill type or geometry. The intensity of the use of the grinding medium must be increased by raising the grinding medium size or the peripheral speed of the stirrer discs to obtain a constant grinding result with constant specific energy input. We must keep two of the three operational parameters (relative number of stress events, stress intensity, and specific energy) at a constant value for constant grinding results. Based on laboratory measurements, the scale-up can be designed for the laboratory stirred media mill.

3. Research Results

3.1. Results in Laboratory Dry Bond Mill

The main particle sizes of Bond grinding are X₅₀ = 0.33 mm and X₈₀ = 1.16 mm for the feed pumice and x₅₀ = 70.5 μm and x₈₀ = 88 μm for the ground material. The transmission weight of the sample in the Bond mill is 0.768 kg, and the proportion by weight of particles below 100 microns is 27.6%.

Using the Bond formula based on Equation (1), the calculated WiB of the pumice is 8.69 kWh/t. The calculated HGI value is 114.21 according to Equation (2). The Bond Operating Index was estimated from the HGI using Equation (3).

Table 1. Bond Work Index, Hardgrove Grindability Index, and Bond Operating Index.

Grinding Parameter	Value
Bond Work Index WiB, kWh·t−1	8.69
Hardgrove Grindability Index HGI	114.21
Bond Operating Index WiBH, kWh·t−1	8.94

presents the calculated grinding parameters.

3.2. Results in Laboratory Wet Stirred Media Mill

The median particle size of stirred media grinding is X₅₀ = 37.55 μm for the feed material. After wet grinding for 20 min in the laboratory stirred media mill, the median particle size of ground pumice is x₅₀ = 3.4 μm (x₈₀ = 10.2 μm). After grinding for 10 min, the median particle size is 4.46 μm; after 20 min, the value is 3.4 μm. According to these stirred media mill measurements, the mill filling ratio was 70%, and the solid mass concentration was 20%. Figure 2 shows the median particle size values depending on the time in the applied stirred media mill.

According to nonlinear parameter estimation, the particle size distribution of the pumice can be described by a Rosin–Rammler function. Equation (4) describes this particle size distribution, where the parameter “a” is the particle size at which 63.2% of the aggregate particles are finer (F (x = a) = 0.632) and exponent “n” is the standard deviation.

$$F\left(x\right)=100[1-{\mathit{exp}\left[-\left(\frac{x}{a}\right)\right]}^n$$

(4)

3.3. Median Particle Size and Specific Grinding Work

For wet stirred media grinding, the primary particle sizes are X₅₀ = 37.55 μm and X₈₀ = 80 μm for the feed pumice and x₅₀ = 3.4 μm and x₈₀ = 10.2 μm for the ground material at the grinding time of 20 min. The specific energy value can be determined from the stirred media mill power of 203.58 Nm/s by a filling ratio of 70% and a solid mass concentration of 20%.

Table 2. Median particle size, power consumption, and specific grinding work depending on the time in the laboratory stirred media mill.

	5 Min	10 Min	20 Min
Median particle size x50, µm	5.00	4.46	3.4
Power consumption of mill Pm, kW	0.196	0.196	0.203
Specific grinding work Ws, kWh·t−1	423	846	1758

Solid mass concentration: 20%, mill filling ratio: 70%.

shows the power and specific grinding work values in the laboratory wet mill. The operational Bond grindability factor can be estimated from the specific energy value. The operational Bond grindability factor value is 9656.6 kWh/t.

The relation between the median particle size (x₅₀) and the specific grinding work (W_s) can be described mathematically based on the grindability index number (GI) with Equation (5).

$$x_{50}=\frac{\mathrm{G}\mathrm{I}}{W_s^{0.198}}$$

(5)

The grindability index number describes the grindability in the stirred media mill, which is taken under the grinding conditions of n = 1440 min⁻¹, ϕ_m = 70%, and c_m = 20%. In this case, the value of GI is 15.32.

3.4. Scale-Up for Stirred Media Mill

When scaling up mills with a mixed medium, more than the specific energy input’s constant value is required to achieve a constant grinding result.

The grinding result does not change when the scale of stirred media mills is increased as long as the specific energy and the stress intensity or the relative number of stress events and the stress intensity are kept constant [38,39,40]. However, the laboratory stirred media mill must be able to be scaled up. The dimensional analysis method was used to determine the stirred media mill (P_m) power consumption for scale-up.

$$P_m=A·{d_d}^5·n^3·{\rho }_s{\left(\frac{{d_d}^{2}n{\rho }_s}{\eta }\right)}^{-m}{\left(\frac{d_d{n}^2}{g}\right)}^{-n}·{\left(\frac{w_k}{d_d}\right)}^c·{\left(\frac{D_m}{d_d}\right)}^e·{\left(\frac{d_b}{d_d}\right)}^f$$

(6)

In the relationship of Equation (6), the stirrer Reynolds number (Re) and the Froude number (Fr) can be identified if the geometric proportions are similar. Suppose the industrial stirred media mill and the laboratory stirred media mill have similar geometric proportions (w_d/d_d, D_m/d_d, and d_b/d_d). We can proceed by applying the scale-up model to the laboratory stirred media mill based on two viewpoints. The first aspect is that the power consumption per unit volume remains unchanged. In the case of turbulent flow, the power consumption of the stirred media mill refers to the industrial scale (with index “ind”) and the laboratory scale (with index “lab”). In the turbulent field, this first construction is the following:

$$P_{m\left(ind\right)}=P_{m\left(\mathrm{l}ab\right)}$$

(7)

$$\frac{A·{n_{\left(ind\right)}}^3{d_{d\left(\mathrm{i}nd\right)}}^5{\rho }_s}{{d_{d\left(ind\right)}}^3}=\frac{A·{n_{\left(lab\right)}}^3{d_{d\left(lab\right)}}^5{\rho }_s}{{d_{d\left(lab\right)}}^3}$$

(8)

The second viewpoint is that the circumferential speed of the stirrer is constant. The stirrer speed for the industrial stirred media mill can be given by knowing the laboratory stirrer speed if A and ρ are equal. Constant “k” is the scale for the industrial and laboratory sizes.

$$n_{\left(ind\right)}=n_{\left(lab\right)}{\left(\frac{d_{\left(lab\right)}}{d_d}\right)}^{\frac{2}{3}}=n_{\left(lab\right)}·k^{-\frac{2}{3}}=n_{\left(lab\right)}{\left(\frac{1}{k}\right)}^{\frac{2}{3}}$$

(9)

3.5. Life Cycle Model for Wet Grinding Process

In the built life cycle assessment plan, the energy input of wet grinding was Hungarian energy with a value of 6.33 MJ. The ground pumice as a product was transported by truck (Euro 6) with diesel oil for the use stage. The assumed transport distance was 100 km, and the transport utilization was 80%.

In addition to the main grinding, the created LCA process in GaBi software produces wastewater mass (0.136 kg) which goes into a municipal wastewater treatment plant. The functional unit was defined as 0.114 kg of ground pumice under laboratory conditions. The applied normalization references are the environmental impacts for European Union countries. The weighting method is thinkstep LCIA Survey 2012 with CML 2016 (Centrum voor Milieukunde Leiden, Leiden, The Netherlands), excluding biogenic carbon in Europe [50]. Results allow the structured expression of research results throughout the life cycle. Our life cycle model of the wet laboratory grinding represents the data in the European Union [51,52] and considers the life cycle from the mining of raw material through preparation processes to the grinding with material transport. The LCA software (GaBi 8.0 (version 10.6))with the latest database (2021) assures the development of the life cycle assessment and grinding process [52,53].

Table 3. Resources and emissions of the wet grinding process for pumice in kilograms.

Flow	Resources, Emissions (kg)	Energy (Net Cal. Value in MJ)
Primary energy from non-renewable resources		16.80
Primary energy from renewable resources		2.54
Energy resources	0.368
Material resources	763
Deposited goods	5.04
Emissions to air	17.40
Emissions to freshwater	996
Emissions to seawater	0.0434
Emissions to agricultural soil	6.64 × 10−7
Emissions to industrial soil	1.08 × 10−5
Flows/Primary energy in total	1781.85	19.34

Functional unit: 0.114 kg of ground pumice in one hour after grinding time of 20 min, with transport; impact assessment method: CML 2016.

summarizes primary energies, resources, and emissions for wet grinding of the pumice in kilograms in the EU. Primary energies from renewable and non-renewable resources are in net caloric values.

If the relative contribution of resources as a percentage is examined, the load on the environment is distributed as follows: material resources at 43%, energy resources at 0.021%, and deposited goods at 0.28%. According to the percentage values of emissions, it can be determined that the most significant change is observed in emissions to freshwater (56%). The largest share of resources and emissions comes from using electricity and preparing pumice for grinding.

Table 4. The absolute values of environmental impact categories in equivalents.

Impact Category	Value	Unit of Measure
Abiotic Depletion ADP elements, ADPE	1.32 × 10−7	kg Sb Equivalent
Abiotic Depletion ADP fossil, ADPF	7.54	MJ
Acidification Potential AP	9.21 × 10−4	kg SO2 Equivalent
Eutrophication Potential EP	1.86 × 10−4	kg Phosphate Equivalent
Freshwater Aquatic Ecot. Pot. FAETP inf.	2.27 × 10−3	kg DCB Equivalent
Global Warming Pot. GWP 100 years (exl. biogenic carbon)	0.646	kg CO2 Equivalent
Human Toxicity Potential HTP inf.	2.88 × 10−2	kg DCB Equivalent
Marine Aquatic Ecotox. Pot. MAETP inf.	109	kg DCB Equivalent
Photochem. Ozone Creat. Pot. POCP	9.28 × 10−5	kg Ethene Equivalent
Terrestrial Ecotox. Potential TETP. Inf.	1.29 × 10−3	kg DCB Equivalent
Ozone Depletion Pot. ODP steady state	6.41 × 10−15	kg R11 Equivalent

Functional unit: 0.116 kg of ground pumice, with transport. Impact assessment method: non-baseline CML 2016.

describes eleven impact categories in the wet grinding process in equivalents. Figure 3 summarizes the normalized and weighted values for the main examined eight categories.

According to the values of Figure 3, the relative contributions of impacts are distributed as follows: 80% from marine aquatic ecotoxicity, 7.3% from fossil abiotic depletion, 5.8% from global warming, 2% from human toxicity, and 1.7% from acidification.

3.6. Connection between Life Cycle Model and Scale-Up

To assess the life cycle of crushing pumice, a functional unit should be used to recalculate the impacts at the adopted reference point. A convenient functional unit for comminution processes may be the capacity Q_r expressed as mass m (or volume V) of comminuted material at a given time t, as used in the LCA in Section 3.5. Then, the results should be interpreted as that during the process with a capacity of 0.116 kg/h, e.g., 0.646 kg of CO₂ equivalent is emitted (according to Table 4). In practice, the W_x life cycle emission factor is the result of specific emissions for materials/processes located within the boundaries of the analyzed system. The W_x indicator can therefore be expressed as a general relationship:

$$W_x=\sum _{i=1}^n\left(X_i·P_i\right)$$

(10)

where X_i—unit emission factor for the material/process, in units, e.g., kg CO₂/kg of material kg/m³; P_i—number of units, e.g., volume or mass, of materials/processes within the analyzed system. For the researched example of pumice grinding in a wet laboratory mill, W_x(lab) can be determined as follows:

$$W_{x\left(lab\right)}=m_{w\left(lab\right)}·t_r·X_w+Q_{r\left(lab\right)}·t_r·X_p+E{C}_{\left(lab\right)}·X_E+m_{ww\left(lab\right)}·t_r·X_{ww}+m_{t\left(lab\right)}·X_t$$

(11)

where m_w(lab)—water mass in the grinding process for laboratory conditions, kg/h; X_w—unit emission factor for water consumption, e.g., kg CO₂/kg of water; Q_r(lab)—efficiency/capacity of grinding pumice in a laboratory mill, kg/h; t_r—grinding time, h; X_p—unit emission factor for the used pumice; EC_(lab)—energy consumption for grinding of the material, kWh; X_E—emission factor per unit of used electricity; m_ww(lab)—the mass of wastewater for grinding in a laboratory mill, kg/h; X_ww—unit emission factor for wastewater output; m_t(lab)—the mass of used fuel during the transport of the obtained product; X_t—unit emission factor for fuel consumption.

Similarly, the life cycle emission factor for grinding in industrial conditions W_x(ind) will take the following form:

$$W_{x\left(ind\right)}=m_{w\left(ind\right)}·t_r·X_w+Q_{r\left(ind\right)}·t_r·X_p+E{C}_{\left(ind\right)}·X_E+m_{ww\left(ind\right)}·t_r·X_{ww}+m_{t\left(ind\right)}·X_t$$

(12)

In Equation (12), the index (ind) denotes the respective unit values of the used materials during the grinding process in industrial conditions, knowing that

$$\frac{m_{w\left(ind\right)}}{m_{w\left(lab\right)}}=C\Rightarrow m_{w\left(ind\right)}=C·m_{w\left(lab\right)}$$

(13)

$$\frac{Q_{r\left(ind\right)}}{Q_{r\left(lab\right)}}=K\Rightarrow Q_{r\left(ind\right)}=K·Q_{r\left(lab\right)}$$

(14)

$$\frac{E{C}_{\left(ind\right)}}{E{C}_{\left(lab\right)}}=S\Rightarrow E{C}_{\left(ind\right)}=S·E{C}_{\left(lab\right)}$$

(15)

$$\frac{m_{ww\left(ind\right)}}{m_{ww\left(lab\right)}}=F\Rightarrow m_{ww\left(ind\right)}=F·m_{ww\left(lab\right)}$$

(16)

$$\frac{m_{t\left(ind\right)}}{m_{t\left(lab\right)}}=G\Rightarrow m_{t\left(ind\right)}=G·m_{t\left(lab\right)}$$

(17)

$$Q_r=\frac{m_p}{t_r}$$

(18)

$$EC=P_{m,t}·t_r$$

(19)

$$m_t=\frac{m_p}{W_t}·U_D·D$$

(20)

where C, K, S, F, G—constants representing the scale-up of the mill; m_p—the mass of the ground material; P_m,t—average total power consumption during grinding; W_t—load capacity of the transport; U_D—unit fuel consumption; and D—the distance the vehicle must travel to reach its destination. The expression m_p/W_t specifies the number of trips of a truck with load capacity W_t to the destination to transport the ground material and is rounded up to an integer. Therefore, W_x(ind) per mass of ground material in 1 h is as follows:

$$W_{x\left(ind\right)}=C·m_{w\left(lab\right)}·X_w+K·{m_p}_{\left(lab\right)1h}·X_p+S·P_{m,t\left(lab\right)}·X_E+F·m_{ww\left(lab\right)}·X_{ww}+G·\frac{m_{p\left(lab\right)1h}}{W_t}·U_D·D·X_t$$

(21)

In Equation (21), the coefficients C, K, S, F, and G depend on the external dimensions of the mill and can be expressed by the constant k derived in the previous section.

4. Discussion

This study determined the fundamental parameters describing the milling performance based on an experimental approach for dry and wet pumice grinding in laboratory Bond and stirred media mills. This work relates the following methods: particle size analysis with laser Doppler sizer, testing of grinding parameters of pumice, life cycle assessment, scale-up with dimensional analysis, and describing of mathematical relationships for connection between LCA and scale-up.

The results showed that the wet grinding in the stirred media mill is more energy-consuming than milling the pumice in a dry state in the Bond mill, which is suggested by the higher Bond Operating Index and Hardgrove Grindability Index for milling (Table 1 and Table 2). This is because, in the stirred mill, the batch material was characterized by a smaller median size than that found during dry milling in the Bond mill. As was proved in other studies [54,55,56,57], the smaller the particle size, the higher the energy consumption for grinding. The energy consumption also depends on the desired final particle size and the size reduction ratio [58]. The experiments conducted in a wet stirred media mill showed that increasing the grinding time decreases the median particle size of pumice with a simultaneous increase in the unit energy demand (Table 2 and Figure 2). A similar observation for the stirred ball mill was presented in the author’s previous study for limestone [24] and other types of mills, for instance, ball mills [58], beater mills [59,60], and hammer mills [61]. The analysis of the results presented in Figure 2 suggests that grinding of pumice in the analyzed laboratory stirred mill (solid mass concentration: 20%, mill filling ratio: 70%) becomes inefficient after 5 min, as the further increase in grinding time results in only a small median particle size decrease and a very high rise in the energy consumption (Table 2). For this grinding time, the limit median particle size is 5 microns. This value is essential for improving the performance and design of full-scale wet stirred mills. The limited residence time and limited particle size for inefficient comminution were previously reported for hammer mills [61] and ball mills [55]. It was also suggested that this milling inefficiency depends on fracture mechanisms used in the mills and the batch material’s mechanical properties [55].

The proposed scale-up model built based on Equations (6)–(9) is essential for improving stirred mill operational parameters and design. When the scale-up model allows for the accurate prediction of the operating parameters, it is possible to reduce the cost of the re-design of the full scale (industrial mill), and all the necessary improvements can be made on the laboratory prototype or during numerical modeling [62,63,64]. This reduces the time and cost of experiments for adjusting optimal operational parameters for grinding different materials. The life cycle assessment results show that marine ecotoxicity, fossil abiotic depletion, and global warming have higher environmental impacts. These results did not surprise us, as we obtained similar results during our previous research [24], where we performed LCA testing for the wet grinding of limestone. In parallel, for emissions, it can be decided that the main change is observed in freshwater with a value of 56%.

The main factors influencing the emissions occurring in the laboratory mill’s life cycle are the material resources and electricity used for grinding and sieving operations. The transport operations will also be significant for the industrial-scale stirred media mills due to diesel usage. It should be noted that the assessment was performed for the Hungarian electricity mix, and the use of other electricity mixes and other geographical scopes will affect the values of the research [65,66], with the lower environmental hazards connected with electricity usage for regions with high electricity production from renewable energy sources [67]. The transport-related environmental impacts depend strictly on the fuel consumption of the used means of transport, its capacity, and the distance that should be covered.

Only some professional literature is available regarding integrating LCA and scale-up, and these research works have been written in the last few years. These models have been described primarily in the chemical and agri-food sectors [68,69,70]. The environmental loads associated with the laboratory-, pilot-, and industrial-scale dimensions are influenced by fundamental changes correlated with many process parameters. Presenting an industrial process from laboratory parameters is very complex due to the difficulty of predicting the behavior of the industrial process during the scale-up. The risk of designing an environmental factor that does not correspond to the actual industrial system is very high. According to our results, it can be established that the relationship between the life cycle assessment and the scale-up can be given mathematically. Integration between the LCA and scale-up can be elaborated that helps to scale up wet grinding processes when only data from laboratory conditions are available.

5. Conclusions

This study sets up an original, not previously presented in the literature, integration with the help of comparative mathematical equations between life cycle assessment and scale-up. It thus helps to calculate the life cycle emission factors for wet grinding processes in laboratory and industrial conditions. In the described equations, several constants represent the scale-up of the stirred media mill. These constants primarily depend on input and output parameters such as water mass, energy consumption, ground material mass, transport fuel mass, wastewater mass, and grinding time in the wet grinding process for laboratory and industrial states. The particle size, the different preparation processes of the used pumice, the mill energy consumption, the geometric sizes of the applied mill, and the main grinding parameters strongly influence the environmental burdens of ground products across their life cycle.

From this study, some recommendations for the reduction in environmental impacts based on the experimental determination of operational parameters, the scale-up model, and the life cycle assessment can be formulated:

• The emissions related to the use of electricity can be reduced by grinding only be-low the time and to particle size at which the milling starts to be ineffective (in this study, for laboratory mill and pumice grinding, less than 5 min and particle size equal to 5 microns);
• The emissions related to the transport of materials can be reduced using low-fuel-consumption transport modes with a preference for low-emission and high-payload means of transport;
• The reduction in the waste and wastewater from grinding using closed-loop systems and efficient water purification systems;
• Development of the new, better design of stirred mills ensuring high throughput and low energy consumption.

These research results can help design industrial stirred media mills and enable the industry to make a prognosis for industrial plants based on the integration between life cycle assessment and scale-up of the pilot grinding processes.