2.1 Cutting heat

The analytical model used to estimate the heat transfer between the single cutter and ice during the drilling process is illustrated in Figure 2. The cutting model of the drill bit shown in Figure 2a was simplified to obtain the cutting model of a single cutter, as shown in Figure 2b. The cutting heat primarily results from shear deformation and frictional heat between the ice chip and cutter (Komanduri and Hou, Reference Komanduri and Hou2001c):

(1)$$Q_{\rm c} = Q_{\rm s} + Q_{\rm f}{\rm , \;}$$

where Q _c, Q _s and Q _f denote the heat generated by the cutting process, shearing and friction, respectively.

Figure 2. Mechanical and heat transfer models of a single cutter: (a) cutting model of the drill bit, (b) simplified cutting model of a single cutter and (c) mechanical analysis of a single cutter. See main text for an explanation of the quantities shown.

Based on the mechanical analysis shown in Figure 2c (Shan and others, Reference Shan, Zhang, Shen and Zhang2019; Veiga and others, Reference Veiga, Arizmendi, Jiménez and Val2021), the calculation formula for each heat component can be expressed as follows:

(2)$$\eqalign{Q_{\rm s} &= F_{\rm s}V_{\rm s} \cr & = \displaystyle{{\pi n( r + R) \cos \alpha \,{\rm cos( }\phi + \delta -\alpha ) ( P_X\cos \delta -P_Y\sin \delta ) } \over {60\,{\rm cos}( \phi -\alpha {\rm ) cos( }\alpha -2\delta ) }}, \;}$$

(3)$$Q_{\rm f} = F_ 1V_ 1 = \displaystyle{{\pi n( {r + R} ) {\rm sin}\,\phi [ P_X{\rm sin}\,2\delta -2P_Y{{\rm ( sin}\,\delta ) }^2] } \over {120\,{\rm cos}( \phi -\alpha ) {\rm cos}( \alpha -2\delta ) }}, \;$$

where F _s and F ₁ respectively denote the shear force and friction in N; V _s and V ₁ respectively denote the shear velocity and moving speed of ice chips in m s⁻¹; P_Y and P_X respectively denote the axial and horizontal cutting forces in N; n denotes the rotation speed in r min⁻¹; R and r respectively denote the outer and inner diameters of the cutter in m; ϕ denotes the angle between the horizontal and shear plane (i.e. shear angle) in rad; α denotes the rake angle in rad; δ denotes the surface friction angle in rad.

2.2 Relationship between cutting heat and temperature

According to the correlation formula for ice cutting under dry conditions (Azuma and others, Reference Azuma, Tanabe and Motoyama2007), the temperature rise of the cutter can be expressed as

(4)$$\Delta T_{\rm s}{\rm} = \displaystyle{{ 0.754} \over {k_ 1\sqrt {V_ 0\varepsilon h/ 4\lambda _{\rm i}} }}R_ 1\displaystyle{{Q_{\rm s}} \over {2b}}, \;$$

(5)$${\rm \Delta }T_{\rm f}{\rm} = \displaystyle{{ 0.377R_ 2Q_{\rm f}} \over {k_{\rm i}b\sqrt {V_ 1l_{\rm f}/ 4\lambda _{\rm i}} }}, \;$$

where ΔT _s and ΔT _f respectively denote the temperature rise caused by shearing and friction in °C, k ₁ and k _i respectively denote the thermal conductivities of the cutter and ice in W (m °C) ⁻¹, b denotes the width of the cutter in m, l _f denotes the contact length between the ice chips and cutter in m, λ _i denotes the thermal diffusion coefficient of ice in m s⁻¹, R ₁ and R ₂ represent proportional coefficients of heat transfer to the cutter, h denotes the cutting depth in m; ε denotes chip deformation, which is dimensionless and the other parameters are as previously described.

Therefore, the temperature of the cutter T _c can be given by

(6)$$T_{\rm c} = T_{\rm i} = \Delta T + T_0 = \Delta T_{\rm s} + \Delta T_{\rm f} + T_0, \;$$

where T _i and T ₀ denote the temperature of ice chips and ambient temperature in °C, respectively.

2.3 Heat exchange and temperature of the cutter under drilling fluid circulation

Heat exchange occurs when drilling fluid flows on the surface of a cutter. This can be simplified as convective heat transfer on a 1-D swept flat plate. Assuming that the drilling fluid is an incompressible Newtonian fluid with no internal heat source and constant physical properties, heat dissipation caused by viscous friction appears negligible. According to Fourier's law of heat conduction and the differential equations of convective heat transfer, the heat transfer coefficient of the cutter surface, ψ, can be obtained as follows:

(7)$$\psi = 0.664\displaystyle{{k_{{\rm df}}} \over {l_{\rm c}}}\left({\displaystyle{{V_{{\rm df}}l_{\rm c}} \over {\nu_{{\rm df}}}}} \right)^{{\rm 1/2}}\left({\displaystyle{{\mu_{{\rm df}}c_{{\rm df}}} \over {k_{{\rm df}}}}} \right)^{{\rm 1/3}}, \;$$

where k _df denotes the thermal conductivity of the drilling fluid in W (m °C) ⁻¹, V _df denotes the flow rate of the drilling fluid in m s⁻¹, l _c denotes the length of the cutter in m, v _df denotes the kinematic viscosity of the drilling fluid in m² s⁻¹, μ _df denotes the dynamic viscosity of the drilling fluid in Pa s and c _df denotes the specific heat capacity of the drilling fluid in J (kg °C) ⁻¹.

Based on Newton's law of cooling, the amount of heat removed by the drilling fluid from the cutter per unit of time is described as follows:

(8)$$q_t = \psi ( {T_{\rm c}-T_{{\rm df}}} ) A.$$

The heat exchange area A between the drilling fluid and cutter is simplified as

(9)$$A{\rm} = l_{\rm c}b.$$

The heat exchange time t between the drilling fluid and cutter is described as

(10)$$t = \displaystyle{{l_{\rm c}} \over {V_{{\rm df}}}}.$$

The heat Q_t absorbed by the drilling fluid in t time is calculated as follows:

(11)$$Q_t = q_tt = 0.664\displaystyle{{k_{{\rm df}}} \over {l_{\rm c}}}\left({\displaystyle{{V_{{\rm df}}l_{\rm c}} \over {\nu_{{\rm df}}}}} \right)^{{\rm 1/2}}\left({\displaystyle{{\mu_{{\rm df}}c_{{\rm df}}} \over {k_{{\rm df}}}}} \right)^{{\rm 1/3}}( {T_{\rm c}-T_{{\rm df}}} ) l_{\rm c}b\displaystyle{{l_{\rm c}} \over {V_{{\rm df}}}}.$$

According to the heat transfer theory, the heat Q_t eliminated by the drilling fluid at t time is given as follows:

(12)$$Q_t = c_{{\rm df}}m_{{\rm df}}\Delta T_{{\rm df}}.$$

The mass of the drilling fluid on the cutter surface is as follows:

(13)$$m_{{\rm df}} = \rho _{{\rm df}}l_{\rm c}bd_{\rm s}, \;$$

where T _df denotes the drilling fluid temperature in °C, ρ _df denotes the density of the drilling fluid in kg m⁻³, d _s denotes the average thickness of the drilling fluid on the cutter surface in m and the other parameters are as previously described.

The average temperature rise of the drilling fluid on the cutter surface can be deduced from Eqns (11)–(13):

(14)$$\eqalign{& {\rm \Delta }T_{{\rm df}} =\cr& \displaystyle{{ 0{\rm.664( }k_{{\rm df}}/l_{\rm c}) {( {( V_{{\rm df}}l_{\rm c}) /\nu_{{\rm df}}} ) }^{{\rm 1/2}}{( {( \mu_{{\rm df}}c_{{\rm df}}) /k_{{\rm df}}} ) }^{{\rm 1/3}}( {T_{\rm c}-T_{{\rm df}}} ) ( l_{\rm c}/V_{{\rm df}}) } \over {c_{{\rm df}}\rho _{{\rm df}}d_{\rm s}}}}.$$

The average temperature of the drilling fluid on the cutter surface after heat absorption is expressed as

(15)$$\eqalign{& T_{{\rm df}} = T_{{\rm df}0} + {\rm \Delta }T_{{\rm df}} = T_{{\rm df}0} \cr& + \displaystyle{{ 0.664( k_{{\rm df}}/l_{\rm c}) {( {( V_{{\rm df}}l_{\rm c}) /\nu_{{\rm df}}} ) }^{{\rm 1/2}}{( {( \mu_{{\rm df}}c_{{\rm df}}) /k_{{\rm df}}} ) }^{{\rm 1/3}}( {T_{\rm c}-T_{{\rm df}}} ) ( l_{\rm c}/V_{{\rm df}}) } \over {c_{{\rm df}}\rho _{{\rm df}}d_{\rm s}}}, \;}$$

where T _df0 denotes the initial temperature of the drilling fluid on the cutter surface in °C.

The heat transfer process can be regarded as flat-wall steady-state heat conduction with steady heat transfer owing to the relative stability of the two heat sources under stable conditions. In addition, the temperature change during the cutting process primarily occurs along the thickness direction of the cutter instead of the length and width. Thus, it can be regarded as a 1-D heat conduction problem formulated with reasonably simplified boundary conditions based on a realistic cutting process. So the following equation can be given as

(16)$$\displaystyle{{\partial ^ 2t} \over {\partial x^ 2}} = 0.$$

Substituting conditions wherein temperature T ranges from T _df to T _c when x ranges from 0 to d into Eqn (16) yields

(17)$$T_x = T_{{\rm df}}-( T_{{\rm df}}-T_{\rm c}) \displaystyle{x \over d}, \;$$

where T_x denotes the temperature at an arbitrary point in the cutter in °C, x denotes the coordinate value of an arbitrary point in the cutter in m and d denotes the thickness of the cutter in m.

The average temperature of the cutter in the presence of the drilling fluid can be obtained by calculating the definite integral of Eqn (17):

(18)$$T_{{\rm ac}} = \displaystyle{{\mathop \smallint \nolimits_0^d T_x{\rm d}x} \over d} = \displaystyle{{\mathop \smallint \nolimits_0^d [ {T_{{\rm df}}-( T_{{\rm df}}-T_{\rm c}) ( x/d) } ] {\rm d}x} \over d} = \displaystyle{{T_{{\rm df}} + T_{\rm c}} \over 2}.$$

Substituting Eqn (6) and Eqn (15) into Eqn (18) yields

(19)$$\eqalign{&T_{{\rm ac}} =\cr& \displaystyle{ 1 \over 2}\left[{T_{{\rm df}0} + \displaystyle{{ 0.664( k_{{\rm df}}/l_{\rm c}) {( {( V_{{\rm df}}l_{\rm c}) /\nu_{{\rm df}}} ) }^{{\rm 1/2}}{( {( \mu_{{\rm df}}c_{{\rm df}}) /k_{{\rm df}}} ) }^{{\rm 1/3}}( {T_{\rm c}-T_{{\rm df}}} ) } \over {c_{{\rm df}}\rho_{{\rm df}}d_{\rm s}}}} \right.\cr&\left.\qquad+ T_0 + \displaystyle{{ 0.754} \over {k_1\sqrt {L_1} }}R_1\displaystyle{{Q_{\rm s}} \over { 2b}} + \displaystyle{{ 0.377R_ 2Q_{{\rm f}1}} \over {k_{\rm i}b\sqrt {V_1l_{\rm f}/ 4\lambda_{\rm i}} }} \right].}$$

The average temperature rise ΔT _c of the cutter in the presence of drilling fluid during drilling is expressed as

(20)$$\Delta T_{\rm c} = T_{{\rm ac}}-T_0.$$

The relevant parameters in Eqn (19) can be expressed as follows:

(21)$$R_1 = \displaystyle{{ 1.33\sqrt {\lambda _{\rm i}\varepsilon /V_0h} } \over { 1 + 1.33\sqrt {\lambda _{\rm i}\varepsilon /V_0h} }}, \;$$

(22)$$R_2 = \displaystyle{{ 0.377Q_{\rm f}/( bk_{\rm i}\sqrt {V_1l_{\rm f}/4\lambda _{\rm i}} ) + {\rm \Delta }T_{\rm s}} \over {( Q_{\rm f}/bk_{\rm c}) \Lambda + 0.377Q_{\rm f}/( bk_{\rm i}\sqrt {V_1l_{\rm f}/4\lambda _{\rm i}} ) }}, \;$$

(23)$${\rm \;}\varepsilon {\rm} = \displaystyle{{{\rm cos}\,\alpha } \over {{\rm sin}\,\phi \,{\rm cos}( {\phi -\alpha } ) }}, \;$$

(24)$$\phi {\rm} = \displaystyle{\pi \over 4}{\rm} + \alpha -\delta {\rm , \;}$$

(25)$$\eqalign{&\Lambda = \displaystyle{2 \over \pi }\left\{{{\rm sin}\,h^{- 1}\left({\displaystyle{b \over { 2l_{\rm f}}}} \right) + \displaystyle{b \over { 2l_{\rm f}}}{\rm sin}\,h^{- 1}\left({\displaystyle{{ 2l_{\rm f}} \over b}} \right) + \displaystyle{ 1 \over 3}{\left({\displaystyle{b \over { 2l_{\rm f}}}} \right)}^ 2}\right.\cr&\left.\qquad + \displaystyle{ 1 \over 3}\left({\displaystyle{{ 2l_{\rm f}} \over b}} \right)-\displaystyle{ 1 \over 3}\left({\displaystyle{b \over { 2l_{\rm f}}} + \displaystyle{{ 2l_{\rm f}} \over b}} \right){\left[{1 + {\left({\displaystyle{b \over { 2l_{\rm f}}}} \right)}^ 2} \right]}^{{\rm 1/2}} \right\}}, \;$$

(26)$$l_{\rm f} = \displaystyle{{h\,{\rm sin( }\phi + \delta -\alpha ) } \over {{\rm sin}\,\phi \,{\rm cos}\,\delta }}, \;$$

where Λ represents the shape factor of a moving plane heat source, and it is dimensionless; and the other parameters are as described above.

Through the mathematical description of the theoretical model, the intrinsic heat transfer mechanism between the drilling fluid and cutter during the process of ice cutting can be elucidated. According to the model, various factors such as the drilling parameters, drilling fluid parameters, environmental factors and geometric dimensions of the cutter can influence the cutting heat. The flow rate of the drilling fluid and its thermophysical parameters are closely related to the cooling process.

3.1 Experimental device

Figure 3 depicts a schematic of the experimental device used to monitor the temperature during rotary cutting. A single cutter fixed to the guideway can be moved horizontally and is controlled by an electric draw stem with a maximum stroke of 550 mm. The displacement and speed of the draw stem are continuously monitored using an MT pull-wire displacement sensor, which has a range of 0–1000 mm and an accuracy of 1 mm. The motor drives the rotation of the ice core connected to the motor via a flange. The rotation speed is monitored by the GM8905 rotation speed sensor with an accuracy of 0.1 r min⁻¹. The drilling fluid, circulated by a pump placed in a reservoir under the ice core, continuously cools the cuter at a flow rate of 20 L min⁻¹. The fluid carrying the ice chip falls back to the reservoir under gravity. The reservoir is equipped with a screen to filter the ice chips.

Figure 3. (a) Schematic and (b) image of the measurement apparatus: (1) motor; (2) drilling fluid; (3) reservoir; (4) guideway; (5) ice; (6) pipe; (7) cutter; (8) electric draw stem; (9) base frame; (10) ice chip; (11) screen; (12) pump.

To measure the cutting temperature, a PT100 thermoresistor is used, which has an accuracy of 0.1°C and a temperature range of −50 to 200°C. It is placed in a hole filled with silicone grease and close to the rake face of the cutter, which records temperature during the experimental procedure (Fig. 4). Similar applications of this silicone grease have been carried out in temperature measurement processes in other fields (Xu and others, Reference Xu2020), demonstrating that the impact on temperature monitoring can be disregarded. The diameter and depth of the hole are 3 mm and half of the cutter length respectively, and the wall thickness from the rake face is 0.5 mm. The inner and outer diameters of the cutter are 60 and 100 mm, respectively.

Figure 4. (a) Layout and (b) image of the cutter thermoresistor: (1) cutter; (2) sensor probe; (3) silicone grease; (4) insulation glue; (5) sensor wire.

The above parameters were monitored during the cutting process in real time.

3.2 Ice sample preparation

The ice sample investigated in this study was prepared from purified water in circular steel tubes. The steel tubes contained a central cylinder with steel sheets, which could maintain a coaxial connection between the ice core and the motor. The steel sheets could increase the bonding strength between the ice core and cylinder. The preparation procedure for the ice core is presented in Figure 5. To prepare the ice samples, purified water was poured into steel tubes sealed by rubbers, which were then placed in a cold room with an adjustable temperature range of 0 to −30°C and an increment of 0.1°C. Once the water was completely frozen, the steel tubes were removed from the cold room and placed in an outdoor environment with a temperature of 20°C for several minutes to allow the ice surface to melt. Subsequently, the ice samples were taken from the steel tubes, and the samples were then placed in the cold room again to cool them to the desired test temperature. This procedure allowed for easy removal of the ice samples from the steel tubes while maintaining their structural integrity. The diameters and lengths of the samples were 100 and 400 mm, respectively.

Figure 5. Preparation procedure of ice samples with 100 mm in diameter and 400 mm in length.

3.3 Testing procedure

The experiment was conducted in the cold room. After installing the ice core, we adjusted the rotation rate of the motor and the feeding speed of the electric draw stem to maintain it at a fixed value. Following this, we carried out a series of experiments on cutting ice cores with different cutters using the aforementioned device (Section 3.2) to validate the reliability of the theoretical model. It was necessary to maintain an interval of 30 min between each group of tests to lower the temperature of the cutter, which increased owing to an increase in the temperature during the cutting process. During the experiments, the drilling fluid was circulated on the surface of the cutter, and data acquisition software was used for real-time monitoring.

Aviation kerosene was utilized as the drilling fluid at a flow rate of 20 L min⁻¹. The thermophysical parameters of the drilling fluid, ice and cutters are listed in Table 1. A dash sign means that the property is not applicable to the material. A total of 55 test groups were executed, incorporating the factors detailed in Table 2, and each group of tests was repeated at least three times to obtain accurate results. The cutter parameters that we used in the theoretical and experimental analysis are shown in Table 3.

Table 1. Thermal properties of the cutter, ice and drilling fluid

Table 2. Experimental parameters

Table 3. Cutter parameters