Simple Approach for Evaluation of Abrasive Mixing Efficiency For

Article Simple Approach for Evaluation of Abrasive Mixing Efﬁciency for Abrasive Waterjet Rock Cutting

Yohan Cha 1, Tae-Min Oh 2 , Hyun-Joong Hwang 3 and Gye-Chun Cho 3,*

1 Deep Subsurface Research Center, Geologic Environment Division, Korea Institute of Geoscience and Mineral Resources (KIGAM), Daejeon 34132, Korea; [email protected] 2 Department of Civil Engineering, Pusan National University (PNU), Busan 46241, Korea; [email protected] 3 Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Korea; [email protected] * Correspondence: [email protected]

Abstract: The abrasive mixing variables, such as the abrasive and water flow rates and the focus geometry parameters, determine the profitability of an abrasive waterjet system. In this study, the mixing efficiency characteristics in abrasive waterjet rock cutting were investigated. To demonstrate comprehensively the efficiency reduction due to collision during abrasive mixing, the chance of collision was expressed as the distance between the abrasive particles in the focus. The mixing efficiency was then assessed by utilizing the empirical relationship between the experimental results and the developed model. Based on the particle density and the velocity, the closer particles showed higher chances of collision, thus yielding a reduced cutting performance. Using the distance between particles model, the optimum abrasive flow rate and the cutting performance of abrasive waterjet

systems can be estimated. This developed model can be used for the design selection of abrasive ﬂow rate and systems for the cost-effective use of abrasive waterjets.

Citation: Cha, Y.; Oh, T.-M.; Hwang, H.-J.; Cho, G.-C. Simple Keywords: abrasive waterjet; abrasive mixing; mixing efficiency; rock cutting; abrasive flow rate Approach for Evaluation of Abrasive Mixing Efficiency for Abrasive Waterjet Rock Cutting. Appl. Sci. 2021, 11, 1543. https://doi.org/10.3390/ 1. Introduction app11041543 Abrasive waterjets are used to cut target materials using high-velocity abrasives by accelerating them with high-pressure and high-velocity water. Although such waterjets Academic Editor: José Correia are mainly used for cutting metals and ceramics, their geotechnical applications such Received: 12 January 2021 as rock cutting, excavation, asphalt resurfacing, pavement adhesion improvement, and Accepted: 5 February 2021 road stripe removal have recently increased [1–3]. In particular, since the cutting force Published: 8 February 2021 can be controlled, it is possible to demolish concrete without damaging the internal steel reinforcement [4,5]. In addition, abrasive waterjets are suitable for urban construction Publisher’s Note: MDPI stays neutral because of their low noise and vibration compared to those in mechanical excavation (e.g., with regard to jurisdictional claims in rock blasting and breaking). However, the cost of the abrasive reaches approximately published maps and institutional affil- 60% of the total operation cost [6]. Efforts have been made to evaluate the effects of the iations. abrasive waterjet parameters on the cost [7–9] and to recycle abrasives after recollection and drying [10,11]. The abrasive cost can be reduced by using the optimum abrasive flow rate (AFR) and by using a system design with a high mixing efficiency. The optimum AFR has been estimated at the maximum cutting rate. If 80% of the maximum cutting rate is Copyright: © 2021 by the authors. achieved with 50% of the optimum AFR, the economical AFR is 50% of the optimum AFR. Licensee MDPI, Basel, Switzerland. The cutting energy of an abrasive waterjet is a function of the AFR, and the system This article is an open access article design determines its characteristics [12,13]. The optimum AFR is a function of the im- distributed under the terms and pact frequency, the terminal velocity, and the kinetic energy of the abrasive through the conditions of the Creative Commons relationship between the mixing efficiency and energy [14,15]. Meanwhile, optimizing Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ the focus shape improves the mixing efficiency, where the momentum of the water is 4.0/). transferred to the abrasive [16]. In principle, the length and diameter of the focus at which

Appl. Sci. 2021, 11, 1543. https://doi.org/10.3390/app11041543 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 1543 2 of 15

the abrasive is mixed are the main mixing and acceleration variables that affect the cutting energy and the optimum AFR [17–19]. In other words, an economical system design is achieved through an optimal combination of focus length, focus diameter, and orifice diameter [20]. Therefore, evaluation of the mixing variables is required to enable mixing efficiency adjustment. The focus keeps the abrasive in the water jet during acceleration, but abrasive friction with the inner surface of the focus and abrasive particle collisions reduce the mixing efficiency [19]. Abrasive experience breakdown (i.e., disintegration) is caused by friction and collisions [21]. Abrasive breakdown increases with increasing pump pressure, AFR, or focus length [11]. These relations indicate that the abrasives are involved in the damping mechanics depending on the abrasive density in the focus [19,22]. The water flow rate (WFR)–AFR ratio and the focus geometry (e.g., diameter and length) affect the abrasive breakdown, thereby reducing the abrasive mixing efficiency. Thus, AFR and waterjet system optimization begins by determining the characteristics of the mixing variables (e.g., WFR, AFR, and focus geometry). The orifice diameter and pump pressure determine the WFR [23], and the focus diameter and WFR are major factors affecting the cutting quality [17]. The larger the focus diameter, the larger the jet diameter, which affects the groove frequency and striation on the cutting surface, resulting in large changes in the surface profile amplitude [24]. Analysis of variance on the effects of the focus and orifice has shown that 98% of the cutting width and roughness results from the focus diameter and the AFR [18,25,26]. The abrasive mixing efficiency is the characteristic that is the most strongly affected by the AFR, and the larger the focus diameter, the higher the optimum AFR [15]. Moreover, the larger focus diameter reduces the energy that accelerates the abrasive [27]. As a result of evaluating the overall effects of the WFR and focus diameter, the economically optimum ratio between the orifice and focus diameter was determined to be three times [28]. Table1 summarizes the literature on the effects of the focus geometry, the AFR, and the WFR on an abrasive waterjet.

Table 1. Previous studies of the effects of the mixing variables on an abrasive waterjet.

Variables Findings Reference The focus geometry affects the momentum Focus geometry [16] transfer efficiency. The optimal mixing efficiency is Focus geometry–abrasive flow determined in the space in which [7] rate (AFR) mixing occurs. The optimum AFR is a function of pressure, Focus geometry–AFR and the focus diameter affects the [22] power density. The economically optimum focus diameter Flow rate–focus geometry [28] is three times the orifice diameter. The focus diameter affects the surface Focus–orifice geometry roughness and determines 98% of the [25] cutting width. The surface profile amplitude and surface Focus geometry roughness mainly depend on the [11] focus diameter. The focus diameter is the dominant Focus geometry parameter affecting the cutting width and [18] the surface roughness. A larger ratio between the diameters of the Focus geometry–AFR focus and abrasive particles reduces the [27] acceleration energy. The AFR affects the mixing efficiency more Focus and orifice geometry–AFR strongly than the focus and the water flow [5] rate (WFR). Appl.Appl. Sci. Sci. 2021 2021, 11, 11, x, xFOR FOR PEER PEER REVIEW REVIEW 3 3of of 15 15 Appl. Sci. 2021, 11, 1543 3 of 15

TheThe mixing mixing efficiency efficiency of of an an abrasive abrasive is is determined determined by by the the abrasive–abrasive abrasive–abrasive and and abrasive–focusabrasive–focusThe mixing inner inner efficiency surface surface ofcollisions. collisions. an abrasive However, However, is determined studies studies that bythat thecomprehensively comprehensively abrasive–abrasive demon- demon- and strateabrasive–focusstrate the the characteristics characteristics inner surface of of the the collisions. mixing mixing effi effi However,ciencyciency considering considering studies that the the comprehensively WFR, WFR, the the AFR, AFR, and demon-and the the focusstratefocus diameter thediameter characteristics simultaneously simultaneously of the are mixing are scarce. scarce. efficiency Under Under ideal considering ideal mixing mixing theand and WFR, acceleration, acceleration, the AFR, the the and abra- abra- the sivefocussive is is diametercompletely completely simultaneously surrounded surrounded by areby water scarce.water an an Underdd accelerated accelerated ideal mixing without without and interference. acceleration,interference. Figure theFigure abra- 1 1 providessiveprovides is completely a aconceptual conceptual surrounded illustrati illustration byon of waterof abrasive abrasive and acceleration, acceleratedacceleration, withoutshowing showing interference.the the characteristics characteristicsFigure of 1of theprovidesthe mixing mixing a variables. conceptualvariables. The The illustration AFR AFR and and ofthe the abrasive focus focus diameter diameter acceleration, affect affect showingthe the chance chance the of of characteristicscollision collision as as a a functionoffunction the mixing of of the the abrasive variables. abrasive density density The AFR in in the the and focu focu thes.s. focusThe The WFR WFR diameter and and the the affect AFR AFR the determine determine chance the of the collision mixing mixing timeastime a for function for which whichof the the the abrasive abrasive abrasive remains remains density in in inthe the the focus focus focus. because because The WFR it it determines determines and the AFRthe the abrasive determineabrasive veloc- veloc- the ity.mixingity. The The longer time longer for the the which abrasive abrasive the abrasive remains remains remainsin in the the focus, focus, in the the focusthe higher higher because the the collision itcollision determines frequency. frequency. the abrasive Thus, Thus, thevelocity.the effects effects The of of the longer the mixing mixing theabrasive variables variables remains on on the the inge ge theometricometric focus, behavior behavior the higher of of thethe the collisionabrasive abrasive frequency.were were ex- ex- pressed.Thus,pressed. the The The effects distance distance of the between between mixing particles variablesparticles (DBP) on(DBP) the in in geometric the the focus focus can behavior can represent represent of the the the abrasive mixing mixing wereeffi- effi- ciencyexpressed.ciency as as a afunction Thefunction distance of of the the betweenco collisionllision particleschance. chance. Figure (DBP)Figure 2in 2shows shows the focus the the characteristics cancharacteristics represent of the of the the mixing DBP DBP efficiency as a function of the collision chance. Figure2 shows the characteristics of the DBP andand mixing mixing variables. variables. As As the the AFR AFR increases, increases, the the DBP DBP decreases decreases (e.g., (e.g., DBP1–DBP2). DBP1–DBP2). As As the the and mixing variables. As the AFR increases, the DBP decreases (e.g., DBP1–DBP2). As the focusfocus diameter diameter or or WFR WFR increases, increases, the the DBP DBP increases increases (e.g., (e.g., DBP2–DBP3 DBP2–DBP3 and and DBP3–DBP4). DBP3–DBP4). focus diameter or WFR increases, the DBP increases (e.g., DBP2–DBP3 and DBP3–DBP4). AA high high DBP DBP means means that that the the abrasive abrasive particles particles are are far far from from each each other, other, so so the the chance chance of of A high DBP means that the abrasive particles are far from each other, so the chance of collisioncollision is is low low and and the the mixing mixing efficiency efficiency is is high. high. As As such, such, the the intuitive intuitive characteristics characteristics of of collision is low and the mixing efficiency is high. As such, the intuitive characteristics of thethe abrasive abrasive mixing mixing efficiency efficiency can can be be expresse expressedd in in terms terms of of the the DBP. DBP. In In addition, addition, because because the abrasive mixing efficiency can be expressed in terms of the DBP. In addition, because thethe concept concept of of space space occupied occupied by by a asingle single particle particle was was introduced introduced to to define define the the space space in in the concept of space occupied by a single particle was introduced to define the space in whichwhich the the abrasives abrasives independently independently remain, remain, not not only only abrasive–abrasive abrasive–abrasive collisions collisions but but also also which the abrasives independently remain, not only abrasive–abrasive collisions but also abrasive–focusabrasive–focus collisio collisionsns can can be be considered. considered. abrasive–focus collisions can be considered.

AbrasiveAbrasive particle particle

SpaceSpace of of occupied occupied by by abrasiveabrasive particle particle

DistanceDistance between between abrasive abrasive particles particles (DBP) (DBP)

Figure 1. Conceptual illustration of the distance between particles (DBP) in the focus. FigureFigure 1. 1. Conceptual Conceptual illustration illustration of of the the distance distance between between particles particles (DBP) (DBP) in in the the focus. focus.

: 0.76: 0.76 mm mm : 0.76: 0.76 mm mm : 3.0: 3.0 g/s g/s : 7.0: 7.0 g/s g/s : 10.67: 10.67 g/s g/s : 10.67: 10.67 g/s g/s

: 1.02: 1.02 mm mm : 1.02: 1.02 mm mm : 7.0: 7.0 g/s g/s : 7.0: 7.0 g/s g/s : 10.67: 10.67 g/s g/s : 29.50: 29.50 g/s g/s

FigureFigureFigure 2. 2.2. DBPDBP DBP according accordingaccording to toto AFR, AFR,AFR, WFR, WFR,WFR, and and and focus focus focus diameter. diameter. diameter.

InInIn this thisthis study, study,study, the the the mixing mixing mixing variables, variables,variables, developed developeddeveloped using usingusing a a atheoretical theoreticaltheoretical model modelmodel of ofof the thethe DBP, DBP,DBP, werewerewere experimentally experimentallyexperimentally evaluated, evaluated, and and and the the the mixing mixing mixing efficiency efficiency efﬁciency was was was estimated estimated estimated based based based on on the on the re- the re- lationshiprelationshiplationship between between between the the the rock rock rock cutting cutting cutting rate rate rate and and and the the the DBP. DBP. DBP. Through Through a aaparametric parametricparametric study studystudy of ofof the the the developeddevelopeddeveloped model, model,model, the thethe effects effectseffects of ofof the thethe mixing mixingmixing vari variablesvariablesables were werewere reviewed reviewedreviewed across across across wide wide wide ranges ranges ranges of of of thethethe variables. variables.variables. The TheThe developed developeddeveloped model modelmodel using usingusing the thethe DBP DBPDBP can cancan intuitively intuitivelyintuitively represent representrepresent the thethe effects effectseffects of of of thethethe mixing mixingmixing variables. variables.variables. Using UsingUsing this thisthis model, model,model, the thethe cutting cuttingcutting performance performanceperformance can cancan be bebe predicted predictedpredicted through throughthrough Appl. Sci. 2021, 11, 1543 4 of 15

the relationship between the input energy and the mixing efﬁciency, and the optimum AFR can be estimated. This approach can be employed for AFR and waterjet system design selection for low cost and high efﬁciency.

2. Abrasive in the Focus 2.1. Number of Abrasive Particles in the Focus Based on Bernoulli’s principle, high-pressure water produced by a pump reaches a high velocity as it exits the narrow inlet of an oriﬁce [29]. Considering the energy loss as the resistance constant (K), which is affected by the oriﬁce size and shape, the initial water velocity (vw,0) can be expressed as follows [23,30]: s 2g v = (1 − K) p , (1) w,o γ w

where g is the gravitational acceleration, γ is the unit weight of water, and pw is the pressure of the pump-generated water. The resistance constant is about 0.25 and 0.5 for square and sharp edges, respectively, and ranges from 0.04 to 0.28 for rounded orifices. A typical orifice has a square edge shape [31,32]. After the high-velocity water is generated, the abrasive is injected. When the abrasive accelerates, the momentum of the water is transferred to the abrasives according to the momentum balance and continuity equation for water and solid flow, which increases the momentum of the abrasives accordingly [23,33]. This approach, which utilizes water and abrasive mass, can be applied to calculate the equation for inputting energy. The exit velocity (velocity at which the abrasive leaves the focus after mixing) and acceleration of the abrasive are affected by the size and density of the abrasive. However, for easy and general calculation, the terminal velocity (vt) at which the velocities of the water and the abrasive become theoretically equal is considered as the abrasive velocity and is calculated as follows [34–36]: . . . . . mwvw,oto = mwvwto + mavato = mw + ma vtto, (2)

where vw and va are the velocities of the water and the abrasive, respectively; to is the . operating time (i.e., time for which the abrasive is injected); mw is the water flow rate; and . ma is the AFR. The abrasive is accelerated and mixed in the mixing chamber or the focus. During mixing, the mixing efficiency varies depending on the mixing variables such as the water . . flow rate (mw), the AFR (ma), and the focus geometry. The mixing efficiency is expressed by the momentum transfer parameter (ηt), which is the most influenced by the AFR (Cha et al. 2020). The terminal velocity (vt) considering the momentum transfer parameter is expressed as [31,37]: vw,o vt = ηt . . . (3) 1 + ma/mw Theoretically, the velocities of the abrasive and water reach the terminal velocity during the early phase of acceleration. The abrasive travels most of the time inside the focus at terminal velocity. The mixing time (t f ) is the time for which the abrasive is accelerated and mixed in the focus. In other words, the time taken for the abrasive to exit the focus can be expressed as follows in terms of the focus length (l f ) and terminal velocity:

t f = l f /vt. (4)

Substituting Equation (3) into Equation (4), the total number of abrasive particles in unit time (Np, f ) can be obtained as: . . . . Np, f = mat f /mp= ma 1 + ma/mw l f /mpηtvw,o, (5) Appl. Sci. 2021, 11, 1543 5 of 15

where, mp is the mass of a single abrasive particle.

2.2. Distance between Abrasive Particles in the Focus In the focus, the abrasive particles move and are independently located in each space, as shown in Figure1. The volume occupied by a single particle ( Vs) depends on the volume inside the focus (Vf ocus) and the total number of abrasives within the focus (Np, f ):

π . V = V /N = d2 l m /m t , (6) s f ocus p, f 4 f f p a f

where d f is the focus diameter and l f is the focus length. Assuming that the abrasives are uniformly distributed and are spherical-shaped with a volume Vs, they are all circumscribed. Thus, the distance between the centers of Vs is the diameter of Vs. Considering the abrasive diameter (dp), the DBP becomes:

p3 DBP = Vs·6/π − dp. (7)

Substituting Equation (6) into Equation (7), the DBP in the focus is represented by the AFR, the focus geometry, the abrasive diameter and mass, and the terminal velocity, as follows: r 3 . DBP = 3 d2 l m /m t − d . (8) 2 f f p a f p The main energy of the abrasive waterjet for material removal comes from the accelerated abrasive particles. In terms of the AFR and the terminal velocity, the kinetic energy of the abrasive per second (Ea) can be expressed as:

1 . E = m v2. (9) a 2 a t

Based on kinetic energy per unit second (Ea) and cutting depth (D), the energy required to cut a unit depth (ε) as a function of the DBP is assumed to be:

β Ea/D = ε = α(DBP) , (10)

where α and β are an empirically determined constant and an exponent, respectively, and ε is the energy required to cut one unit depth of rock. The cutting depth is affected by the mixing efficiency. As shown in Equation (8), it is possible to demonstrate the relationship between the kinetic energy removal and the experimental results in consideration of the focus geometry, the AFR, and the WFR. This model can also be expressed by the pump pressure using Equation (1). Through this study, the mixing variables can be expressed comprehensively. This model defines the energy in unit time, rather than the mixing time t f , to intuitively show the mass and velocity of the abrasive. At a high AFR, the kinetic energy decreases as the terminal velocity decreases, whereas t f increases. However, a limitation of this approach is that the ideal case of abrasive-fluid behavior is assumed in the developed model. The mixing is greatly affected by the turbulent flow, the viscosity of the fluid, and the physical properties of the abrasive as well as collision and friction. Therefore, the empirical relationship between the simple theoretical approach and the experimental results is shown in this study. Eventually, further studies regarding the effect of geometrical and frictional loss of both water and abrasive within the focus are required to overcome this limitation.

3. Experimental Program 3.1. Test Cases and Setup To evaluate the mixing efﬁciency and cutting performance according to the DBP in the focus, the main variables used were the focus diameter, the oriﬁce diameter, and the AFR. The focus diameter and the terminal velocity of the abrasive were controlled. To Appl. Sci. 2021, 11, 1543 6 of 15

evaluate these variables considering the applicability, commercial product sizes, which Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 15 are popular in the waterjet industry, were used. Figure3 shows the oriﬁces and focuses used in the experiment. Sapphire oriﬁces with diameters of 0.15, 0.254, and 0.33 mm and WFRs of 10.67, 29.50, and 50.00 mL/s were utilized at a pressure of 320 MPa. For each WFR,tungsten tungsten carbide carbide focuses focuses with withdiameters diameters of 0.76, of 0.76,0.91, 0.91,and 1.02 and 1.02mm mmwere were employed. employed. The TheAFR AFR was wascontrolled controlled by Venturi by Venturi suction suction according according to the to WFR the WFRand pressure and pressure from froma mini- a minimummum of 3.3 of g/s 3.3 to g/s a maximum to a maximum of 29 ofg/s. 29 Table g/s. Table2 lists2 thelists details the details of the of experimental the experimental cases, cases,including including the WFR, the WFR,the AFR, the and AFR, the and focus the focusdiameter. diameter.

FigureFigure 3.3. OriﬁcesOrifices andand focuses.focuses.

Table 2. Abrasive waterjet operation conditions and experimental cases. Table 2. Abrasive waterjet operation conditions and experimental cases. Orifice Diameter (mm) 0.15 0.254 0.33 Oriﬁce Diameter (mm) 0.15 0.254 0.33 WFR (mL/s) WFR (mL/s) 10.67 10.67 29.50 29.50 50.00 50.00 Focus diameter (mm) 0.76 1.02 0.76 0.91 1.02 0.76 0.91 1.02 Focus diameter (mm) 0.76 1.02 0.76 0.91 1.02 0.76 0.91 1.02 3.3 3.3 4.0 4.5 4.0 4.4 4.5 5.2 3.3 3.3 4.0 4.5 4.0 4.4 4.5 5.2 4.4 4.44.5 4.57.1 7.17.5 7.55.0 5.07.4 7.4 10.2 10.2 10.5 5.6 5.611.0 11.012.0 12.08.6 8.612.5 12.5 13.3 13.3 15.3 AFR (g/s) AFR (g/s) 6.6 6.6 13.0 13.015.5 15.514.0 14.0 20.0 20.0 1.3 1.3 18.8 7.9 20.0 18.0 24.0 24.0 24.4 23.2 7.9 20.0 18.0 24.0 24.0 24.4 23.2 10.0 29.0 25.0 10.0 29.0 25.0 Pump pressure (MPa) 320 Pump pressure (MPa) 320

AnAn abrasiveabrasive injectioninjection waterjetwaterjet (AWJ)(AWJ) consistsconsists ofof aa pumppump sectionsection andand anan abrasiveabrasive waterjetwaterjet system.system. FigureFigure4 4 depicts depicts the the intensiﬁer intensifier pump pump and and the the abrasive abrasive waterjet waterjet system system usedused inin the experiment. The The pump pump capacity capacity us useded was was 50 50 HP HP of ofhydraulic hydraulic power power and and the thewaterjet waterjet could could produce produce 6 mL/s 6 mL/s of water of water with with a maximum a maximum pressure pressure of 420 of 420MPa. MPa. In a Insus- a suspensionpension waterjet waterjet (slurry (slurry waterjet; waterjet; SWJ), SWJ), a hi agh-pressure high-pressure pump pump applies applies pressure pressure to a mix- to a mixtureture of water of water and and abrasive, abrasive, whereas whereas in inan an AWJ, AWJ, which which was was used used in in this study forfor rockrock cutting,cutting, thethe pump pump applies applies pressure pressure only only to to the the water water and and the the water water accelerates accelerates the abrasive.the abra- AWJssive. AWJs are more are suitablemore suitable for civil for engineering civil engineering purposes, purposes, such assuch rock as excavation,rock excavation, because be- thecause system the system is relatively is relatively simple simple and occupies and occupies a small a space. small space. 3.2. Abrasive and Rock Specimen

Figure5a shows the abrasive, pyrope-type Indian garnet. It is composed of Mg 3Al2(SiO4)3, and its speciﬁc gravity is 3.79. The abrasive particles remaining on the 80th sieve were utilized to ensure a uniform abrasive diameter, as the DBP is affected by the abrasive diameter (Equation (8)). The particle size distribution indicates a mean particle size of 0.18 mm, as shown in Figure5b [38,39]. Table3 provides the details of the abrasive. Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 15

Appl.Appl. Sci. Sci.2021 2021, 11, 11, 1543, x FOR PEER REVIEW 77 ofof 15 15

Figure 4. Experimental setup for abrasive waterjet rock cutting with pump and abrasive waterjet system.

3.2. Abrasive and Rock Specimen Figure 5a shows the abrasive, pyrope-type Indian garnet. It is composed of Mg3Al2(SiO4)3, and its specific gravity is 3.79. The abrasive particles remaining on the 80th sieve were utilized to ensure a uniform abrasive diameter, as the DBP is affected by the Figure 4. Experimental setup for abrasive waterjet rock cutting with pump and abrasive waterjet system. Figure 4. Experimentalabrasive setup diameter for abrasive (Equation waterjet rock(8)). cuThetting particle with pump size anddistribution abrasive waterjetindicates system. a mean particle size of 0.18 mm, as shown in Figure 5b [38,39]. Table 3 provides the details of the abrasive. 3.2. Abrasive and Rock Specimen 100

Figure 5a shows the abrasive,80 pyrope-type Indian garnet. It is composed of Mg3Al2(SiO4)3, and its specific gravity is 3.79. The abrasive particles remaining on the 80th sieve were utilized to ensure a uniform60 abrasive diameter, as the DBP is affected by the

abrasive diameter (Equation (8)). The40 particle size distribution indicates a mean particle size of 0.18 mm, as shown in Figure 5b [38,39]. Table 3 provides the details of the abrasive. 20100

Particle distribution rate [%] rate distribution Particle 80 0.425 0.25 0.18 0.15 0.075 60 Particle size [mm]

(a) 40 (b) FigureFigure 5.5. AbrasivesAbrasives forfor thethe experiment:experiment: ((aa)) IndiaIndia garnet;garnet; ((bb)) particleparticle sizesize distribution.distribution. 20

TableTable 3.3. Abrasive properties. 0

Abrasive properties. [%] rate distribution Particle 0.425 0.25 0.18 0.15 0.075 AbrasiveAbrasive Type type Source Source Specific Specific GravParticle Gravityity size [mm] Mean Mean Particle Particle Size Size (mm) Pyrope garnet Mg3Al2(SiO4)3 India 3.79 0.18 (Mesh size 80) (a) Pyrope garnet Mg3Al2(SiO4)3 India 3.79(b) 0.18 (Mesh size 80) Figure 5. AbrasivesThe rock for thespecimens experiment: to be(a) cutIndia were garnet; granite (b) particle with dimensionssize distribution. of 150 150 300 mm. × × As graniteThe rock is the specimens most common to be cut rock were in granite Korea, withit was dimensions selected considering of 150 150 the 300practical mm. Table 3. Abrasive properties. Asapplications granite is of the waterjet most common rock cutting. rock inThe Korea, granite it wasspecimens selected were considering quarried the at practicalthe same applications of waterjet rock cutting. The granite specimens were quarried at the same depth to ensureAbrasive uniform type physical properties. Source Specific As shown Grav in ityTable Mean 4, the Particle uniaxial Size compres- (mm) depth to ensure uniform physical properties. As shown in Table4, the uniaxial compressive sive strength (UCS) of the granite specimen was 236 MPa, the shear strength was 23 MPa, strengthPyrope (UCS) garnet of the Mg granite3Al2(SiO specimen4)3 India was 236 MPa, 3.79 the shear strength 0.18 (Mesh was 23 size MPa, 80) and and the tensile strength was 12 MPa; hence, it was classified as very strong rock [40,41]. the tensile strength was 12 MPa; hence, it was classified as very strong rock [40,41]. The The specific gravity was 2.65, the absorption ratio related to voids was 0.27%, and the specificThe gravity rock specimens was 2.65, theto be absorption cut were ratiogranite related with todimensions voids was of 0.27%, 150 and150 the 300 shore mm. shore hardness was 114. hardnessAs granite was is 114.the most common rock in Korea, it was selected considering the practical applications of waterjet rock cutting. The granite specimens were quarried at the same Tabledepth 4. toGranite ensure specimen uniform properties. physical properties. As shown in Table 4, the uniaxial compressive strength (UCS) of the granite specimen was 236 MPa, the shear strength was 23 MPa, Shear Tensile andUCS the tensile strength was 12 MPa; hence, it was classifiedAbsorption as very strong rockShore [40,41]. Strength Strength Specific Gravity (MPa) Ratio (%) Hardness The specific gravity(MPa) was 2.65,(MPa) the absorption ratio related to voids was 0.27%, and the shore hardness was 114. 236 23 12 2.65 0.27 114

3.3. Test Procedure The orifice was located inside the waterjet assembly for withstanding the high-pressure fluid, whereas the focus was simply attached outside. For efficiency, the experiment was Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 15

Table 4. Granite specimen properties.

UCS Shear Strength Tensile Strength Absorption Ratio Shore Hardness Specific Gravity (MPa) (MPa) (MPa) (%) 236 23 12 2.65 0.27 114

3.3. Test Procedure The orifice was located inside the waterjet assembly for withstanding the high-pressure fluid, whereas the focus was simply attached outside. For efficiency, the experiment Appl. Sci. 2021, 11, 1543 was streamlined in the order of AFR control, focus change, and orifice8 of change. 15 The AFR error was minimized by measuring the time required to consume 200 g of abrasive at least three times. The rock was cut once in one direction from the outside to the inside of the streamlinedgranite specimen, in the order as ofshown AFR control, in Figure focus 6. change, The jetting and orifice system change. could The AFRmove error at speeds of 1.2– was minimized by measuring the time required to consume 200 g of abrasive at least three times.16 mm/s, The rock and was the cut traversing once in one directionspeed in from this the experiment outside to the was inside 8.4 of themm/s granite considering back specimen,thrust due as shownto high-pressure in Figure6. The water jetting flow. system The could cutti moveng atpaths speeds in of each 1.2–16 case mm/s, were kept at least and70 mm the traversing apart to speedprevent in this them experiment from influencing was 8.4 mm/s or considering overlapping back with thrust each due other. to The stand- high-pressureoff distance water(SOD) flow. is the The distance cutting paths between in each the case end were of kept the atfocus least and 70 mm the apart specimen. It is the to prevent them from influencing or overlapping with each other. The standoff distance (SOD)variable is the that distance has the between greatest the end influence of the focus on andthe thecutting specimen. width It isand the variabledispersion that of the abrasive hasparticles; the greatest in this influence experiment, on the cutting SOD widthwas set and very dispersion close to of the10 mm abrasive to minimize particles; its effect. The incutting this experiment, depth was SOD measured was set very by closecalibrated to 10 mm calipers to minimize at five its points effect. The from cutting the start to the end depthalong was the measuredcutting path, by calibrated and three calipers measured at five points values from excluding the start tothe the maximum end along and minimum theones cutting were path, averaged. and three measured values excluding the maximum and minimum ones were averaged. Focus

Focus diameter ( ) Abrasive waterjet

Cutting depth ( )

Target rock

FigureFigure 6. 6.Schematic Schematic drawing drawing of abrasive of abrasive waterjet waterjet rock cutting. rock cutting. 4. Experimental Results and Analysis 4.1.4. Experimental Kinetic Energy and Results Rock Cutting and PerformanceAnalysis 4.1. FigureKinetic7 depicts Energy a graniteand Rock specimen Cutting after Performance the experiment, along with the focus geometry, the water ﬂow rate, and the AFR. The cutting surface indicates the cutting path, and the cuttingFigure side 7 depicts shows the a cuttinggranite depth specimen at the cuttingafter the start experiment, point according along to thewith test the focus geom- conditions.etry, the water Because flow the rate, SOD isand short, the the AFR. cutting The width cutting is similar surface to thatindicates shown the on the cutting path, and cuttingthe cutting surface. side Figure shows8 presents the thecutting cutting depth depth accordingat the cutting to the AFRstart at point each WFR according for to the test eachconditions. focus diameter, Because showing the SOD that theis short, larger thethe focus cutti diameter,ng width the is deeper similar the to cutting that shown on the depth. The maximum cutting depth occurs at a large focus diameter, and the optimum AFR . (cuttingma,o), which surface. is the AFRFigure at the 8 pres maximuments the cutting cutting depth, depth increases according with the focusto the diameter. AFR at each WFR for Witheach the focus same diameter, input energy showing conditions th (e.g.,at the water larger pressure, the focus water, anddiameter, AFR), the the mixing deeper the cutting variablesdepth. The affect maximum the cutting depth.cutting depth occurs at a large focus diameter, and the optimum AFR (𝑚 ,), which is the AFR at the maximum cutting depth, increases with the focus Appl.Appl. Sci. Sci. 2021 2021, ,11 11, ,x x FOR FOR PEER PEER REVIEW REVIEW 99 ofof 1515

Appl. Sci. 2021, 11, 1543 9 of 15 diameter.diameter. With With the the same same input input energy energy condit conditionsions (e.g., (e.g., water water pressure, pressure, water, water, and and AFR), AFR), thethe mixing mixing variables variables affect affect the the cutting cutting depth. depth.

FigureFigure 7. 7.7. Cutting Cutting results results with withwith different differentdifferent focus focusfocus diamet diametersdiametersers and andand WFRs WFRsWFRs on onon the thethe granite granitegranite specimens. specimens.specimens.

FigureFigure 8. 8. Rock RockRock cutting cuttingcutting depth depthdepth as as a a function function of of focus focus diameter diameter at at WFRs WFRs of of ( a(a) )10.67, 10.67, ( b(b) )29.5, 29.5, and and ( c(c) ) 50.00 50.00 mL/s. mL/s.

FigureFigure 9 99 shows showsshows thethethe DBP DBPDBP as asas a aa function functionfunction of ofof the thethe AFR AFRAFR by byby focus focusfocus diameter. diameter.diameter. This ThisThis figure figurefigure demonstratesdemonstrates that that the the larger larger the the focus focus diamet diameter,diameter,er, the thethe farther fartherfarther the thethe abrasives abrasivesabrasives are areare from fromfrom each eacheach other, resulting in a higher DBP. An increase in the AFR increases the number of abrasive other,other, resulting resulting in in a a higher higher DBP. DBP. An An increase increase in in the the AFR AFR increases increases the the number number of of abrasive abrasive particles in the focus and decreases the DBP (Figure9a). As derived from Equations (4) and particlesparticles in in the the focus focus and and decreases decreases the the DBP DBP (Figure (Figure 9a). 9a). As As derived derived from from Equations Equations (4) (4) (7), the WFR does not significantly affect the DBP. This finding is explained by the fact that andand (7), (7), the the WFR WFR does does not not significantly significantly affect affect the the DBP. DBP. This This finding finding is is explained explained by by the the all the abrasive particles are simultaneously accelerated and move at the same velocity. A factfact thatthat all all thethe abrasiveabrasive particlesparticles areare simultaneously simultaneously acceleratedaccelerated andand movemove at at thethe samesame higher WFR results in a faster initial water velocity (Equation (3)), so the abrasive–water velocity.velocity. A A higher higher WFR WFR results results in in a a faster faster initial initial water water velocity velocity (Equation (Equation (3)), (3)), so so the the abra- abra- mixture moves faster, but t decreases. Whether the abrasive particles move quickly or sive–watersive–water mixture mixture moves moves faster, faster,f but but 𝑡𝑡 decreases. decreases. Whether Whether the the abrasive abrasive particles particles move move slowly, the DBP remains the same. Because the WFR affects the terminal velocity of the quicklyquickly or or slowly, slowly, the the DBP DBP remains remains the the same. same. Because Because the the WFR WFR affects affects the the terminal terminal veloc- veloc- abrasive, a larger WFR has a larger kinetic energy for cutting, even with the same AFR. The ityity of of the the abrasive, abrasive, a a larger larger WFR WFR has has a a larger larger kinetic kinetic energy energy for for cutting, cutting, even even with with the the same same complex effect of the mixing variables can be expressed as the energy required to cut one unit depth of rock (ε) from the experimental results, as shown in Figure8. Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 15 Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 15

AFR. The complex effect of the mixing variables can be expressed as the energy required Appl. Sci. 2021, 11, 1543 AFR. The complex effect of the mixing variables can be expressed as the energy required10 of 15 to cut one unit depth of rock (𝜀) from the experimental results, as shown in Figure 8. to cut one unit depth of rock (𝜀) from the experimental results, as shown in Figure 8.

(a) (b) (a) (b) Figure 9. DBP as functions of the (a) focus diameter and (b) WFR. FigureFigure 9. 9.DBP DBP as as functions functions of of the the (a )(a focus) focus diameter diameter and and (b )(b WFR.) WFR.

Figure 10 presents the kinetic energy per unit second (𝐸) calculated using Equation FigureFigure 10 10 presents presents the the kinetic kineti energyc energy per per unit unit second second (Ea) ( calculated𝐸) calculated using using Equation Equation (9) (9) under the experimental conditions of this study. A high water flow rate corresponds under(9) under the experimental the experimental conditions conditions of this of study. this study. A high Awater high water ﬂow rate flow corresponds rate corresponds to a to a large acceleration momentum, yielding a high kinetic energy because of the high ter- largeto a accelerationlarge acceleration momentum, momentum, yielding yielding a high a kinetic high kinetic energy energy because because of the highof the terminal high terminal velocity. Increasing the AFR causes an increase in the total abrasive mass but de- velocity.minal velocity. Increasing Increasing the AFR causesthe AFR an causes increase an in increase the total in abrasive the total mass abrasive but decreases mass but the decreases the terminal velocity (Equation (2)). As such, the kinetic energy for rock cutting terminalcreases velocitythe terminal (Equation velocity (2)). (Equation As such, the (2)). kinetic As such, energy the for kinetic rock cuttingenergy andfor rock the effects cutting and the effects of the abrasive mixing variables resulting from the experiment were eval- ofand the the abrasive effects mixing of the abrasive variables mixing resulting variab fromles the resulting experiment from the were experiment evaluated were using eval- a uated using a function of the DBP. functionuated using of the a DBP. function of the DBP.

FigureFigure 10. AbrasiveAbrasive waterjet waterjet kinetic kinetic energy per second as functions of the AFR and the WFR. Figure 10. Abrasive waterjet kinetic energy per second as functions of the AFR and the WFR. 4.2. DBP and Cutting Characteristics 4.2. DBP and Cutting Characteristics 4.2.One DBP of and the Cutting objectives Characteristics of this study was to identify a comprehensive relationship among One of the objectives of this study was to identify a comprehensive relationship variablesOne affecting of the theobjectives mixing of and this acceleration, study was such to identify as the water a comprehensive flow rate, the relationship AFR, and among variables affecting the mixing and acceleration, such as the water flow rate, the theamong focus diameter.variables Theaffecting energy the required mixing toand cut acceleration, a unit depth such according as the to water the DBP flow is shownrate, the AFR, and the focus diameter. The energy required to cut a unit depth according to the inAFR, Figure and 11 .the These focus results diameter. indicate The both energy mixing requ efficiencyired to cut and a unit cutting depth performance. according Theto the DBP is shown in Figure 11. These results indicate both mixing efficiency and cutting per- DBPDBP and is shown energy in required Figure 11. to cutThese a unit results depth indicate are correlated both mixing with efficiencyα = 11.595 and± cutting0.63 and performance. The DBP and energy required to cut a unit depth are correlated with 𝛼= β formance.= −0.738 ± The0.02 DBPin Equation and energy (10). Therequired coefficient to cut of a determination unit depth are is 0.9606, correlated which with shows 𝛼= 11.595 0.63 and 𝛽 = −0.738 0.02 in Equation (10). The coefficient of determination a high11.595 goodness 0.63 and of fit 𝛽 and = can −0.738 be expressed 0.02 in Equation as follows: (10). The coefficient of determination is 0.9606, which shows a high goodness of fit and can be expressed as follows: is 0.9606, which shows a high goodness of fit and can−0.738 be expressed as follows: Ea/D = ε = 11.595·DBP . (11) Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 15

Appl. Sci. 2021, 11, 1543 11 of 15 . 𝐸⁄𝐷 =𝜀=11.595∙DBP . (11)

40 y = 11.595x-0.738

[J/mm] 35 R² = 0.9606 30

25 = 50.00 mL/s 20 = 10.67 mL/s 15

10 = 29.50 mL/s 5

Energy required to cut unit depth, 0 00.511.52 DBP [mm] FigureFigure 11. 11. EnergyEnergy required required to to cut cut a a unit unit depth depth versus versus DBP. DBP. . The colored mark in Figure 11 indicates the optimum AFR (m ) based on the WFR. The colored mark in Figure 11 indicates the optimum AFR (𝑚 ,a,o) based on the WFR. InIn general, general, the the larger larger the the WFR, WFR, the the larger larger the the amount amount of of energy energy required required to to cut cut 1 1 mm, mm, whichwhich means means that that the the cutting cutting efficiency efficiency is is high higherer at at a a lower lower WFR. WFR. Based Based on on this this result, result, the the DBPDBP corresponding corresponding to to the the optimum optimum AFR AFR at at a a high high WFR WFR was was estimated estimated to to be be less less than than approximatelyapproximately 0.6 0.6 mm. mm. Using Using the the relationship relationship between between the the energy energy required required to cut cut one one unit unit depthdepth and and the the DBP, DBP, the optimum AFR can be determined according according to to the the WFR WFR and and the the waterjetwaterjet system. system. ByBy substituting substituting the the energy energy required required to to cut cut one one unit unit depth depth from from Equation Equation (10) (10) and and the the DBP from Equation (7) into Equation (11), the cut depth can be expressed as follows: DBP from Equation (7) into Equation (11), the cut depth can be expressed as follows: . r !.0.738 2 ma vt 3 3 2 . D =𝐷= 𝑑d𝑙l𝑚m/𝑚m 𝑡t −𝑑− d . .(12) (12) 2·11.595∙. 2 f f p a f p This approach represents the characteristics of mixing variables such as the focus ge- This approach represents the characteristics of mixing variables such as the focus ometry, the abrasive particle diameter, the mass of a single abrasive particle, the AFR, and geometry, the abrasive particle diameter, the mass of a single abrasive particle, the AFR, the water flow rate based on the terminal velocity. The cutting depth can be estimated and the water flow rate based on the terminal velocity. The cutting depth can be estimated under a wide range of waterjet system conditions by measuring 𝛼 and 𝛽 from three or under a wide range of waterjet system conditions by measuring α and β from three or more morepoints points representing representing the relationship the relationship between between the energy the required energy required to cut one to unit cut depth one unit and depththe DBP. and the DBP. FigureFigure 1212 presentspresents the the results results of of a a parametr parametricic study, study, depicting depicting the the cutting cutting depth depth with with aa given given water water flow flow rate rate according according to to the the focus focus diameter diameter at at an an AFR AFR of of10 10 g/s g/s (according (according to Equationto Equation (12)). (12)). The Themixing mixing variables variables were wereevaluated evaluated over a over wide a range, wide range,and it was and found it was thatfound the that larger the the larger focus the diameter, focus diameter, the greater the greater the cut thedepth. cut depth.The effect The of effect the focus of the diam- focus eterdiameter is larger is largerat higher at higherwater flow water rates. flow It rates. can also It canbe demonstrated also be demonstrated that a high that WFR a high re- sultsWFR in results a large in reduction a large reduction in efficiency in efficiency because of because the focus of thediameter. focus diameter.Increasing Increasing the water flowthe water rate from flow 50 rate mL/s from to 50 90 mL/s mL/s to(i.e., 90 mL/sby 80%) (i.e., increases by 80%) the increases cutting the depth cutting by 20%. depth Alt- by hough20%. Although the maximum the maximum cutting depth cutting occurs depth at 90 occurs mL/s, at 50 90 mL/s mL/s, of 50flow mL/s rate ofis flowconsidered rate is toconsidered be economical. to be economical.Thus, it is possible Thus, itto is design possible an toeconomical design an water economical flow rate water using flow Equa- rate tionusing (12). Equation (12). Figure 13 illustrates the results of a parametric study, showing the effect of the focus diameter and the AFR on the cutting depth at a WFR of 50 mL/s. For a focus diameter less than 0.9 mm, the optimum AFR is close to 5 g/s (dashed dotted line), but for a larger diameter focus, the optimum AFR is close to 10 g/s (dashed line). This approach demonstrates the change in the optimum AFR according to the focus diameter. An increase in AFR increases the impact frequency (up arrow) while reducing the terminal velocity and cutting energy (down arrow), as indicated by Equation (3). In addition, a high AFR yields numerous collisions between abrasive particles, reducing the cutting efficiency. This Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 15

Appl. Sci. 2021, 11, 1543 12 of 15

Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 15 parametric study demonstrates that the DBP in the focus enables optimal design of the focus diameter, the AFR, and the WFR.

Figure 12. Effect of the focus diameter on the cutting depth owing to WFR.

Figure 13 illustrates the results of a parametric study, showing the effect of the focus diameter and the AFR on the cutting depth at a WFR of 50 mL/s. For a focus diameter less than 0.9 mm, the optimum AFR is close to 5 g/s (dashed dotted line), but for a larger diameter focus, the optimum AFR is close to 10 g/s (dashed line). This approach demonstrates the change in the optimum AFR according to the focus diameter. An increase in AFR increases the impact frequency (up arrow) while reducing the terminal velocity and cutting energy (down arrow), as indicated by Equation (3). In addition, a high AFR yields numerous collisions between abrasive particles, reducing the cutting efficiency. This parametric study demonstrates that the DBP in the focus enables optimal design of the focus diameter,FigureFigure 12.12. Effectthe AFR, ofof thethe and focusfocus the diameterdiameter WFR. onon thethe cuttingcutting depthdepth owingowing toto WFR.WFR.

FigureFigure 13. EffectsEffects of of the the focus diameter on th thee cutting depth owing to the AFR. 5. Conclusions 5. Conclusions The purpose of this study was to evaluate the mixing characteristics of abrasive waterjetThe rockpurpose cutting. of this The study dependence was to evaluate of the rock the cutting mixing performance characteristics on of the abrasive mixing waterjetvariables, rock such cutting. as the The abrasive dependence flow rate, of thethe waterrock cutting flow rate, performance and the focus on the geometry, mixing var- was iables,experimentally such as the tested. abrasive To evaluate flow rate, the the mixing water variables flow rate, comprehensively, and the focus geometry, the chances was of abrasive–abrasive and abrasive–focus inner surface collisions were defined in terms of the DBP in the focus. In addition, as the cutting energy and experimental results were expressed as functions of the DBP, the characteristics of the mixing variables were assessed. The results of this study can be used for geotechnical application such as improved abrasive waterjet rock excavation. The following are the major findings and contributions of this study: Figure 13. Effects of the focus diameter on the cutting depth owing to the AFR. • The abrasive mixing characteristics were derived based on the mixing variables, such 5. Conclusionsas the WFR, the AFR, and the focus geometry. To express the complex effects of the variables, a mixing efficiency model was developed considering the chance of collision The purpose of this study was to evaluate the mixing characteristics of abrasive waterjet rock cutting. The dependence of the rock cutting performance on the mixing variables, such as the abrasive flow rate, the water flow rate, and the focus geometry, was Appl. Sci. 2021, 11, 1543 13 of 15

by defining the concept of the DBP, which represents the abrasive density in the focus. This model comprehensively considers the WFR, the AFR, the focus diameter and length, and the abrasive mass and diameter. • As a result of the effects of the mixing variables, the cutting depth was experimentally obtained. It was found that using a large focus diameter or water flow rate, a higher mixing efficiency yields a greater cutting depth. The AFR affects the terminal velocity and mixing efficiency, so excessive input reduces the cutting depth. The higher the mixing efficiency, the higher the optimum AFR. • By expressing the relationships between the experimental results and input energy as functions of the DBP, the effects of the mixing efficiency on abrasive waterjet rock cutting could be calculated. The smaller the DBP, the higher the energy required to cut a unit depth of the rock. In other words, a high energy input was required owing to the low mixing efficiency. • The developed model demonstrated that, with a high water flow rate, the cutting efficiency was low compared to the energy input. In addition, high water flow rates indicated the optimum AFR at small DBPs. At water flow rates of over 50 mL/s, the optimum AFR was estimated to occur with a DBP of less than 0.6 mm. • The cutting depth model was proposed based on the empirical relationship between the experimental cutting results and the mixing variables. The mixing efficiency was considered by indicating the chance of collision in terms of the DBP. The proposed model also demonstrates the effects of the mixing variables. In particular, the changes in the optimum AFR according to changes in the focus diameter are revealed. By using this model, it is possible to estimate the optimum AFR and cutting performance with minimal experimentation.

Author Contributions: Conceptualization, G.-C.C. and Y.C.; formal analysis, Y.C. and T.-M.O.; investigation, Y.C. and H.-J.H.; writing–original draft preparation, Y.C. and H.-J.H.; writing—review and editing, Y.C. and T.-M.O.; supervision, G.-C.C. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Acknowledgments: This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2017R1A5A1014883) and the Basic Research Project of the Korea Institute of Geoscience and Mineral Resources (KIGAM, GP 2020-010) funded by the Ministry of Science and ICT, Korea. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Abbreviations

D Cutting depth (mm) d f Focus diameter (mm) dp Abrasive particle diameter (mm) Ea Kinetic energy per unit second (J/s) g Gravitational acceleration K Resistance constant l f Focus length (mm) . ma Abrasive flow rate (AFR) (g/s) . ma,o Optimum abrasive flow rate (g/s) mp Mass of single particle (g) . mw Water flow rate (WFR) (mL/s) Np, f Total number of particles in focus pw Pressure of pump generated water (MPa) to Operating time (i.e., time for which the abrasive is injected) (s) Appl. Sci. 2021, 11, 1543 14 of 15

t f Time abrasive mixing in focus (s) 3 Vf ocus Volume inside focus (mm ) 3 Vs Volume occupied by a single particle (mm ) va Velocity of abrasive particle (m/s) vt Terminal velocity (m/s) vw Velocity of water (m/s) vw,o Velocity of initial water in the orifice section (m/s) α Constant in the relationship between cutting depth and energy (mm) β Exponent in the relationship between cutting depth and energy γ Unit weight of water (kg/mm3) ε Energy required to cut unit depth (J/mm) ηt Momentum transfer parameter ε Energy required to cut one unit depth of rock (J/mm) AFR Abrasive flow rate (g/s) DBP Distance between particles (mm) SOD Standoff distance (mm) WFR Water flow rate (mL/s)

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