Mathematical Modeling of Collisional Heat Generation and Convective Heat Transfer Problem for Single Spherical Body in Oscillating Boundaries

Abstract

The application of high-energy ball milling in the field of advanced materials processing, such as mechanochemical alloying and ammonia synthesis, has been gaining increasing attention beyond its traditional use in material crushing. It is important to recognize the role of thermodynamics in high-energy processes, including heat generation from collisions, as well as ongoing investigations into grinding ball behavior. This study aims to develop a mathematical model for the numerical analysis of a spherical ball in a shaker mill, taking into account its dynamics, contact mechanics, thermodynamics, and heat transfer. The complexity of the problem for mathematical modeling is reduced by limiting the motion to one-dimensional translation and representing the vibration of the vial wall in a shaker mill as rigid boundaries that move in a linear fashion. A nonlinear viscoelastic contact model is employed to construct a heat generation model. An equation of internal energy evolution is derived that incorporates a velocity-dependent heat convection model. In coupled field modeling, equations of motion for high-energy impact phenomena are derived from energy-based Hamiltonian mechanics rather than vector-based Newtonian mechanics. The numerical integration of the governing equations is performed at the system level to analyze the general heating characteristics during collisions and the effect of various operational parameters, such as the oscillation frequency and amplitude of the vial. The results of the numerical analysis provide essential performance metrics, including steady-state temperature and time constant for the characteristics of temperature evolution for a high-energy shaker milling process with a computation accuracy of 0.1%. The novelty of this modeling study is that it is the first to obtain such a high accuracy numerical solution for the temperature evolution associated with a shaker mill process.

1. Introduction

The mechanochemical alloying of metals [1,2,3,4] and the synthesis of ammonia through nitrogen fixation and hydrogenation [5,6] can be performed by high-energy ball milling. These processes not only grind materials with high impact energies [7,8,9,10] but also generate heat and activation energy that can initiate chemical reactions [11,12]. In particular, high-energy ball milling-based ammonia production is gaining popularity as a substitute for the century-old Haber–Bosch process due to its higher yields in low-temperature and low-pressure conditions [13]. To comprehend these processes, it is crucial to gain insight into the dynamics and thermodynamics of milling balls. However, monitoring the position, velocity, and temperature of them in real time to analyze thermomechanical states remains challenging [14]. Only quasi-real-time experiments have reported intermittent temperature measurements by pausing the milling process [15,16]. To overcome this experimental limitation, computer simulation techniques have been developed and used to analyze the complex behavior of particles in the milling process [17,18,19,20]. The discrete element method (DEM) is a numerical method that can analyze contact, collision, and flow problems of particulate materials. Recently, DEM has been extended to encompass the analysis of interparticle contact heat conduction [21,22] and the investigation of fluid-structure interaction (FSI) [23,24,25,26]. However, there is still a lack of research on models for collision-induced heating and convective heat transfer to the ambient environment, both of which are essential for high-energy ball milling analysis [27,28,29]. This paper aims to develop a collision heating model and a convective heat transfer model that can explain the thermomechanical behavior of high-speed collisions of milling balls. Furthermore, the paper seeks to formulate multiphysical governing equations that account for both particle dynamics and thermodynamics. The mathematical model presented in this work provides the advantage of computing particle temperatures and heat fluxes that previous DEM modeling studies have yet to predict.

This study simplifies the shaker mill process and models the physics of a spherical milling ball as a first-order nonlinear differential equation system. The heat generation model is derived from the energy dissipation term of the nonlinear viscoelastic contact model. The energy balance equation takes into account a heat convection model that depends on the translational velocity of the spherical body. This paper employs energy-based Hamiltonian mechanics instead of traditional Newtonian mechanics to compute state variables in multiphysical studies. Hamilton’s canonical equations and nonholonomic constraints are used to obtain governing equations, which are then solved through numerical integration. The results of the numerical analysis are compared with the experimental data to assess the suitability and limitations of the model.

The remainder of this paper is divided into multiple sections. First, a high-speed collision problem with one degree of freedom is introduced. Next, mathematical models that consider viscoelastic contact, heat generation, and heat convection are developed using the Hamiltonian mechanics scheme for this problem. The following section presents the results of the numerical analysis and provides a detailed analysis of the findings. The last section summarizes the most important conclusions drawn from this study.

2. Problem Definition

This research aims to develop a suitable mathematical model to investigate the thermomechanical characteristics of a spherical ball in a shaker mill. The specific mill of interest in this study is the 8000M Mixer/Mill, manufactured by Spex CertiPrep in Metuchen, NJ, USA, which is a widely used shaker mill for research purposes [15,30,31].

Table 1. Specifications of a shaker mill (Spex 8000M Mixer/Mill).

Specification	Symbol	Value	Unit
Radius of milling ball	R	12.7	mm
Length of cylindrical vial	$l_v$	58	mm
Oscillation amplitude of vial	$X_v$	25	mm
Oscillation frequency range	$\omega$	90–130	rad/s

summarizes the fundamental specifications of the milling apparatus. Its kinematic mechanism allows the vial to move in an arc trajectory and oscillate simultaneously in the yaw direction, producing a synchronized three-dimensional roto-translational motion [30,31]. This study simplifies such a kinematically complex high-energy milling process into a one-dimensional problem by assuming that the cylindrical vial oscillates sinusoidally in its longitudinal direction and the milling ball only translates in the same direction. The kinematic state of the milling ball is mainly determined by its contact with the vial, where gravity is ignored.

Figure 1 shows the schematic of a spherical ball in a sinusoidally oscillating vial. The state of the milling ball can be uniquely determined by its kinematic and thermodynamic variables. The position and linear momentum of the ball, as described by Hamiltonian mechanics, are the kinematic variables, represented by x and p, respectively. The internal energy U is the thermodynamic state variable that is counterbalanced by the heat generated by collisions and the energy dissipated through convective heat transfer. In Figure 1, the only input to the system is the motion of oscillating boundaries $x_{v,L}$ and $x_{v,R}$ of the vial, which can be expressed in sinusoidal form as

$$x_v\left(t\right)=x_{v,L}=x_{v,R}=X_v\sin \omega t,$$

(1)

where the excitation function of the vial, $x_v\left(t\right)$, presents the harmonic osciallation with an amplitude of $X_v$ and an angular frequency of $\omega$ as a function of time t. Equation (1) has the functional form of a time-dependent holonomic, or rheonomic, configuration constraint. Therefore, the work performed by the associated constraint force does not vanish. Furthermore, because the constraint force acts only during contact, this constraint is a type of inequality constraint.

This study utilizes the concept of contact overlap or penetration depth between solid interfaces, which is commonly employed in discrete element method (DEM) and finite element method (FEM) contact algorithms, to calculate the collision and repulsion between the vial and the milling ball. When contact overlap occurs, a viscoelastic contact force is activated to prevent and repel penetration. This contact force is typically expressed as a function of the contact overlap and its time derivative. An exaggerated representation of the penetration depth of the milling ball is shown in Figure 1 to highlight its interaction with the left boundary.

3. Mathematical Modeling

3.1. Particle Contact Model

The contact between the ball and the vial can be identified by utilizing the Boolean operators defined by

$$\begin{array}{ccc}\hfill {\alpha }_L& =& {〈R-(x-x_{v,L})〉}^0,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill {\alpha }_R& =& {〈R-(x_{v,R}-x)〉}^0,\hfill \end{array}$$

(3)

where $<{>}^0$ is the singular bracket function of the zeroth order, which is equivalent to the Heaviside step function, and $x_{v,L}$ and $x_{v,R}$ are the positions of the left and right boundaries of the vial, respectively. Thus, ${\alpha }_L$ and ${\alpha }_R$ will be one if the sphere is in contact with the left and right boundaries, respectively. It is impossible for both ${\alpha }_L$ and ${\alpha }_R$ to both be nonzero at the same time, since the system setup does not allow the ball to make contact with both edges at the same time, as illustrated in Figure 1.

The contact overlap $\xi$ can be expressed analytically as

$$\xi ={\xi }_L+{\xi }_R={〈R-(x-x_{v,L})〉}^1+{〈R-(x_{v,R}-x)〉}^1,$$

(4)

where $<{>}^1$ denotes the singular bracket function of the first order that returns the bracketed expression itself only if it is positive but yields zero otherwise. Thus, ${\xi }_L$ and ${\xi }_R$ calculate the contact overlap with the left and right edges, respectively. It should be noted that ${\xi }_L$ and ${\xi }_R$ cannot be simultaneously nonzero, similar to ${\alpha }_L$ and ${\alpha }_R$ in Equations (2) and (3).

The contact overlap velocity $\dot{\xi }$ is defined by the time derivative of $\xi$ in Equation (4). It can be expressed in combination with the velocity terms and Boolean contact operators as

$$\dot{\xi }={\alpha }_L\left(-\dot{x}+{\dot{x}}_v\right)+{\alpha }_R\left(-{\dot{x}}_v+\dot{x}\right),$$

(5)

where $\dot{x}$ is the velocity of the milling ball and ${\dot{x}}_v$ is the vial’s translational velocity. ${\dot{x}}_v$ is the time derivative of the position function in Equation (1), which can be written as

$${\dot{x}}_v=\omega X_v\cos \omega t.$$

(6)

The specific particle contact model of interest in this study is the two-term nonlinear viscoelastic force model [32,33,34], which decomposes the normal contact force $F_c$ into a conservative elastic force $F_e$ and a non-conservative damping force $F_d$ in an explicit form as

$$F_c=F_e+F_d=\frac{4}{3}\sqrt{R{E_{\mathrm{eff}}}^2}{\xi }^{\frac{3}{2}}{n}_w-\sqrt{\frac{20}{3}}\left(\frac{\ln e}{{\ln }^{2}e+{\pi }^2}\right){\left(m^{2}R{E_{\mathrm{eff}}}^2\xi \right)}^{\frac{1}{4}}\dot{\xi }{n}_w,$$

(7)

where $n_w$ is the outward normal direction to the wall in contact, R is the sphere radius, e is the coefficient of restitution, and m is the ball mass. The effective elastic modulus $E_{\mathrm{eff}}$ for the ball and the flat-ended vial in contact with each other is defined by

$$\frac{1}{E_{\mathrm{eff}}}=\frac{1-{{\nu }_b}^2}{E_b}+\frac{1-{{\nu }_v}^2}{E_v},$$

(8)

where $E_b$ and $E_v$ are the elastic moduli, and ${\nu }_b$ and ${\nu }_v$ are Poisson’s ratio of ball and vial, respectively. In Equation (7), the normal direction $n_w$ can be analytically expressed with the Boolean contact operators ${\alpha }_L$ and ${\alpha }_R$ as

$$n_w={\alpha }_L-{\alpha }_R,$$

(9)

so that the contact force direction at the left and right boundaries should be opposite each other.

Equation (7) includes the elastic force $F_e$, which is the Hertizan contact force [35] for a sphere of radius R in contact with a flat surface of infinite radius of curvature. Additionally, the damping term $F_d$ takes into account the energy dissipation associated with the coefficient of restitution e [32].

3.2. Collisional Heating Model

When an object experiences an inelastic collision with a coefficient of restitution that is less than one, the initial kinetic energy is not conserved and instead decreases. This reduction is due to energy-dissipative mechanisms such as plastic deformation, contact surface friction, and material damping. In the milling process shown in Figure 1, the plastic deformation of the milling ball is negligible, and friction in the shear direction can be ignored in this one-dimensional normal impact problem. This research hypothesizes that the material damping effect is the primary cause of energy dissipation during a collision, and the corresponding non-conservative force is equivalent to $F_d$ in Equation (7). Consequently, the power dissipation due to the damping force can be expressed as

$${\dot{U}}^{F_d}=F_{d}v,$$

(10)

where $v=\dot{x}$ is the instantaneous velocity of the milling ball.

If the ratio of the internal heat generation rate to the power dissipation rate is denoted by $\eta$, the irreversible heat generation due to collision ${\dot{U}}^{\mathrm{irr}}$ can be expressed as

$${\dot{U}}^{\mathrm{irr}}=\eta {\dot{U}}^{F_d}=-\eta \sqrt{\frac{20}{3}}\left(\frac{\ln e}{{\ln }^{2}e+{\pi }^2}\right){\left(m^{2}R{E_{\mathrm{eff}}}^2\xi \right)}^{\frac{1}{4}}\dot{\xi }{n}_{w}v,$$

(11)

where $\eta$ is treated as an adjustable numerical parameter in this study since no measured value has been published.

3.3. Heat Convection Model

When a sphere is heated and transported through a fluid, the thermal energy released through convection can be modeled using the convection heat transfer coefficient. This coefficient is usually expressed as a dimensionless Nusselt number, which is related to the Reynolds number and the Prandtl number [36,37]. The Nusselt number ($\mathit{Nu}$), Reynolds number ($\mathit{Re}$), and Prandtl number ($\mathit{Pr}$) for a sphere are defined as

$$\mathit{Nu}=\frac{2\mathit{Rh}}{k},\mathit{Re}=\frac{2R\rho v}{\mu },\mathit{Pr}=\frac{C_p\mu }{k},$$

(12)

where h, k, $\mu$, and $C_p$ are the convection heat transfer coefficient, thermal conductivity, dynamic viscosity, and specific heat at constant pressure of air, respectively.

The Nusselt number model for a sphere, originally developed by Whitaker [38], can be written in a functional form as

$$\mathit{Nu}=2+\left(0.4R{e}^{\frac{1}{2}}+0.06R{e}^{\frac{2}{3}}\right)P{r}^{0.4}{\left(\frac{\mu }{{\mu }_r}\right)}^{-\frac{1}{4}},$$

(13)

where the Reynolds number $\mathit{Re}$ is explicitly included to calculate the velocity-dependent convection. The temperature-dependent dynamic viscosity of air in Equation (13) can be expressed by the Sutherland formula [39] as

$$\frac{\mu }{{\mu }_r}={\left(\frac{\theta }{{\theta }_r}\right)}^{\frac{3}{2}}\left(\frac{{\theta }_r+S_{\mu }}{\theta +S_{\mu }}\right),$$

(14)

where $\theta$ is the temperature of the milling ball, and ${\theta }_r$ is the ambient temperature considered to be constant in this study. In the above Equation (14), ${\mu }_r$ is the dynamic viscosity at ${\theta }_r$, and $S_{\mu }$ is the Sutherland constant for the viscosity.

It is important to note that Whitaker’s Nusselt number assumes airflow over a stationary sphere at a constant upstream velocity in an unbounded space. However, in the case of a shaker mill problem depicted in Figure 1, the air is confined within a bounded domain, resulting in less efficient convection heat transfer due to thermal energy accumulation in the surrounding air. To address this problem, a multiplicative calibration factor $\beta$ was implemented to adjust the velocity-dependent Nusselt number and its corresponding heat convection coefficient. Therefore, the effective convective heat transfer ${\dot{U}}^{\mathrm{conv}}$ can be calculated as

$${\dot{U}}^{\mathrm{conv}}=4\pi R^2\beta h(\theta -{\theta }_a),$$

(15)

where ${\theta }_a$ is the ambient air temperature, and the convection heat coeffieicnt h is determined from Equations (12)–(14) at each computational time step.

3.4. Internal Energy Time Evolution

The internal energy evolution equation, which is equivalent to the rate form of the first law of thermodynamics, can be expressed as

$$\dot{U}={\dot{U}}^{\mathrm{irr}}-{\dot{U}}^{\mathrm{conv}}.$$

(16)

This rate of change in internal energy can be expressed in terms of the rate of change in temperature using the definition of heat capacity [39] as

$$\dot{U}=m{C}_v\dot{\theta },$$

(17)

where $C_v$ is the specific heat capacity of the ball material at constant volume and $\dot{\theta }$ is the time derivative of the milling ball temperature.

This study assumes that the volume of the milling ball remains constant throughout the isochoric process. As a result, the term contributing to internal energy by volume change is not present in Equation (17). Therefore, the momentum evolution equation in Hamilton’s equations does not require incorporation of the constraint force caused by the volume change. In addition, when solving the governing equation numerically, there is no need for an additional equation of state to link the internal energy, pressure, and density of the solid.

3.5. Hamilton’s Equation of Motion

The configuration space of the 1D collision problem depicted in Figure 1 is characterized by three generalized coordinates: position, linear momentum, and internal energy. Accordingly, the thermomechanical Hamiltonian [40] of the spherical ball in the shaker mill is the sum of its kinetic, potential, and thermodynamic internal energies.

$$H(x,p,U)=V\left(x\right)+T\left(p\right)+U,$$

(18)

where $V\left(x\right)$ is the elastic potential energy stored during contact, $T\left(p\right)$ is the kinetic energy in terms of a linear momentum p, and U is the internal energy.

The kinetic co-energy $T^{\ast }$ and the kinetic energy T, which have a mutual Legendre transformation relationship [40], can be expressed as

$$T=\frac{1}{2}\frac{p^2}{m},T^{\ast }=\frac{1}{2}m{v}^2,$$

(19)

where m and v are the mass and velocity of the milling ball, and p is the linear momentum defined by $p=\partial T^{\ast }/\partial v=\mathit{mv}$.

The Hamilton’s canonical equations derived from the thermomechanical Hamiltonian [41] in Equation (18) are

$$\begin{array}{ccc}\hfill \dot{x}& =& \frac{\partial H}{\partial p},\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill \dot{p}& =& -\frac{\partial H}{\partial x}+Q_x,\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill \frac{\partial H}{\partial U}& =& Q_U,\hfill \end{array}$$

(22)

where $Q_x$ is the generalized non-conservative force associated with the variation of the position x. Therefore, $Q_x$ satisfies the following variation relations for non-conservative mechanical work and power.

$$\delta W^{\mathrm{nc}}=Q_x\delta x,\delta {\dot{W}}^{\mathrm{nc}}=Q_x\delta v,$$

(23)

where $\delta$ is the variation operator. It is worth mentioning that $\delta {\dot{W}}^{\mathrm{nc}}$ is equal to $\delta {\dot{W}}^{F_d}$ since the only non-conservative force that affects the milling ball is the contact damping force $F_d$. Therefore, the generalized force $Q_x$ can be derived from Equation (10) as

$$Q_x=F_d.$$

(24)

The partial derivative of H with respect to the generalized coordinate U in Equation (18) yields $Q_U=1$ [42]. Hamilton’s canonical Equation (22) with respect to U results in a trivial relationship between H and U. Consequently, a non-trivial time evolution equation of U should be obtained from the constraint Equation (16) instead.

From Equations (20), (21) and (16), the set of governing equations can be expressed in the form of Hamilton’s equations for kinematic variables x and p combined with a constraint equation concerning the internal energy U as

$$\begin{array}{ccc}\hfill \dot{x}& =& \frac{1}{m}p,\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill \dot{p}& =& F_e+F_d,\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill \dot{U}& =& \frac{\eta F_d}{m}p-4\pi R^2\beta h(\theta -{\theta }_0).\hfill \end{array}$$

(27)

The governing Equations (25) to (27) are first-order nonlinear ordinary differential equations for the thermomechanical state variables x, p and U. To obtain a computational solution, the governing equation must be integrated numerically with a given initial condition. Furthermore, the temperature $\theta$, the output variable of interest, can be calculated by integrating Equation (17) as

$$\theta ={\int }_0^t\frac{\dot{U}}{m{C}_v}\mathit{dt}.$$

(28)

Equation (28) implies that the temperature $\theta$ is not necessarily a generalized coordinate, but rather a redundant state variable dependent on other coordinates in Hamiltonian mechanics.

4. Numerical Analysis Results and Discussion

An in-house code was developed using Matlab (version 2023a by MathWorks Inc. in Natick, MI, USA) to numerically solve the governing Equations (25) through (27). Figure 2 shows the flowchart representation of the computational code. The first step involves loading input data, including simulation parameters, operating parameters of the vial, material constants, and initial conditions required for numerical analysis. Next, government equations are constructed by calculating the contact force, collisional heat generation, and convective heat transfer at each computational time step. Solutions for state variables are obtained through numerical integration, and this iterative process is repeated until the simulation time is reached. After conducting the computation, a comprehensive regression analysis is performed, and various data plots are created to facilitate a thorough understanding of the data.

Prior to numerical integration, the differential equation must be expressed in a way that can be solved numerically by providing all the necessary material constants and model parameters. The material properties of the steel milling ball, which are necessary to calculate the contact force and the internal energy, are presented in

Table 2. Material properties of steel.

Property	Symbol	Value	Unit
Density	${\rho }_s$	7800	$\mathrm{kg}/{\mathrm{m}}^3$
Modulus of elasticity	E	200	GPa
Poission’s ratio	$\nu$	0.3	-
Specific heat capacity	$C_v$	461	$\mathrm{J}/\mathrm{kg}·\mathrm{K}$

.

The material constants of air, a surrounding fluid, are shown in

Table 3. Physical properties of air.

Property	Symbol 1	Value	Unit
Density	$\rho$	1.225	$\mathrm{kg}/{\mathrm{m}}^3$
Dynamic viscosity	${\mu }_r$	1.716 × 10−5	Pa·s
Thermal conductivity	k	0.0241	W/m·K
Specific heat capacity	$C_p$	1003.5	$\mathrm{J}/\mathrm{kg}·\mathrm{K}$
Sutherland constant	$S_{\mu }$	111	K

¹ The subscript r indicates that the characteristic is evaluated at a reference temperature of 273 K (0 °C). Unless otherwise specified, the properties are determined at the ambient temperature of 293 K (20 °C).

. These values are required to calculate the convective heat transfer coefficient and the temperature changes. The dimensionless numbers in Equation (12) were determined with reference to an ambient temperature of 20 °C. On the other hand, the Sutherland formula in Equation (14) takes a reference temperature of 0 °C.

In

Table 4. Model parameters for numerical analysis.

Parameter	Symbol	Value	Unit
Simulation time	t	60	min
Coefficient of restitution	e	0.7	-
Collision heat generation ratio	$\eta$	0.0815	-
Convection calibration factor	$\beta$	0.155	-

, the simulation time t and the three model parameters for the contact force, heat generation, and convection heat transfer models are shown with their respective values. The simulation period was set to 60 min, which is in agreement with the measurement duration of the experimental data published in [15]. It has been reported that almost 90% of the energy dissipated by plastic deformation is transformed into heat in metallic materials [43]. However, since the contact model used in this research only takes into account viscoelastic properties and does not consider plastic deformation, a relatively low collision heat generation rate of 8.15% was selected. Furthermore, the calibration factor for convection heat transfer was determined to be 15.4%, ensuring that the numerical error in the steady-state temperature and time constant of the milling ball is less than 0.1% compared to the experimental values.

The system of first-order ordinary differential Equations (25) to (27) cannot be solved analytically due to its nonlinearity, mainly caused by $F_d$, $F_e$, and h. Therefore, the Bogacki–Shampine scheme [44] was employed for numerical integration. Specifically, the Matlab in-house code utilized with the ode23 function, which implements an adaptive integration step size algorithm based on the Bogacki–Shampine method. This algorithm reduces the time step size in the contact state, which is characterized by stiff behavior due to abrupt contact forces, and increases the time step in the non-contact flight state, which has a non-stiff temperature variation. This strategy offers both accuracy and efficiency in solving this shaker mill problem.

The collision of a spherical object can experience numerical instability if the computational time step exceeds the critical Rayleigh time step [45,46], which is defined in terms of material constants as

$$t_c=\frac{\pi R}{0.1631\nu +0.8766}\sqrt{\frac{\rho }{G}},$$

(29)

where G is the shear modulus of the milling ball. The critical Rayleigh time is associated with the time it takes for the shear elastic wave to travel along the surface of a spherical solid. If the size of the iteration time step is excessively large, the degree of contact overlap may be overestimated, resulting in the particle penetrating the boundary walls. Even without penetration, the elastic contact force, which increases as a power of 3/2 in Equation (7), will be excessively computed. This produces an unphysical circumstance in which the coefficient of restitution becomes greater than one, due to the incorrect calculation of excessive post-impact linear momentum. In the numerical problem studied, the critical Rayleigh time was calculated as 6.74 μs. However, during collision, the integration time sizes used were at least 10% lower than this value to ensure a stable numerical solution. Specifically, the numerical integration process yielded 4,265,455 time steps of solution data. The Intel^® Xeon^® Gold 6136 CPU with a clock rate of 3.0 GHz, manufactured by Intel in Santa Clara, CA, USA, was used to perform the calculations serially, and the entire process took 59.3 min.

Figure 3 shows the position and velocity graphs of the milling ball during the first 0.5 s, representing its kinematic state. As expected from collision dynamics theory, Figure 3 depicts a steplike velocity and ramplike position behavior resulting from impulsive loading. Both the position and velocity have upper and lower limits, avoiding numerical instability such as solution divergence due to excessive contact force estimation.

The temperature of the milling ball is a thermodynamic state variable that can be determined by using numerical solutions for the internal energy U, as expressed in Equation (17). Both the experimental data [15] and the numerical solutions in this study demonstrate that the temperature rises in a concave upward manner and asymptotically approaches a steady-state value. This behavior can be represented by an exponential curve with a constant term, which can be expressed in a close form as

$$\theta \left(t\right)=\left({\theta }_0-{\theta }_{\infty }\right)e^{-\frac{t}{\tau }}+{\theta }_{\infty },$$

(30)

where ${\theta }_0$ and ${\theta }_{\infty }$ are the initial and steady–state temperatures of the milling ball, respectively, and $\tau$ is the time constant for the exponentially decaying term. It should be noted that ${\theta }_{\infty }$ and $\tau$ can be regarded as merit figures for a high-energy milling system. The nonlinear least squares technique was used to determine these two parameters. Specifically, the Matlab built-in function fit was employed for nonlinear regression in the in-house code.

Figure 4 shows the change in the temperature of the milling ball over time, which was obtained from the internal energy solution using Equation (28). Figure 4a displays numerical data for temperature over a 60-minute period, along with the fitted curve obtained by nonlinear regression. A curve fitting process was performed on the 4,265,455 data points, but only 20 numerical values were displayed on the graph for a clear visualization. The temperature–time graph in Figure 4b is zoomed into a 100 ms section at 10 min, allowing a closer look at the temperature fluctuations. When the milling ball comes into contact with the vial, a sudden temperature rise is observed due to the heat generated from the collision. On the other hand, when the ball is in flight without any contact, a less stiff temperature decrease appears due to convective cooling. As shown in Figure 4a, the spherical ball inside the shaker mill undergoes a gradual temperature rise. This is caused by the collisional heating that takes place during its contact with the vial, as well as the convective cooling that occurs during its non-contact flight. These two processes alternate with each other, contributing to the overall increase in temperature. As the milling process progresses, the rate of collisional heating remains relatively stable due to the fixed frequency of operation, which approximately maintains the frequency of heat-generating collisions. However, as the temperature of the ball rises, the difference between it and the surrounding air also increases. This increasing temperature difference leads to a proportional increase in the rate of heat dissipation through convection, as indicated by the second term in Equation (27). The temperature–time graph eventually reaches a steady-state level when the convective heating rate is equal to the collisional heating rate.

The capacity of the shaker mill to generate heat through collisions can be better understood by examining the rate of internal energy gain instead of the time constant $\tau$. Furthermore, the initial rate of internal energy evolution can be determined by taking the initial time derivative of the milling ball temperature from Equation (16). The initial temperature evolution rate ${\dot{\theta }}_0$ can be expressed in terms of two regression parameters, ${\theta }_{\infty }$ and $\tau$, as

$${\dot{\theta }}_0=\frac{{\theta }_{\infty }-{\theta }_0}{\tau }.$$

(31)

The comparison of the regression analysis of the experimental data of [15] with the results of the numerical analysis obtained in this study for a 12.7 mm single milling ball in a Spex 8000M Mixer/Mill is presented in

Table 5. Comparison of computational results with experimental data.

Performance Metrics 1	Experiment	Computation	Error
Steady-state temperature ${\theta }_{\infty }$ (°C)	48.0	48.04	0.09%
Time constant $\tau$ (min)	16.7	16.80	0.06%
Initial temperature evolution rate ${\dot{\theta }}_0$ (°C/min)	1.6766	1.6692	−0.64%

¹ While ${\theta }_{\infty }$ and $\tau$ were obtained through the data regression to the nonliear Equation (30), ${\dot{\theta }}_0$ was determined by Equation (31) with the regression parameters.

. The numerical errors for the steady-state temperature and the time constant, in terms of the regression parameters of Equation (30), are 0.09% and 0.06%, respectively. These errors are within the 0.1% error range, indicating the high reliability of the modeling methodology used in this study. Furthermore, it was estimated that the accuracy of the initial temperature evolution rate calculated by Equation (31) was estimated to be within 1.0%.

The high-speed shaker mill utilizes the oscillation frequency and amplitude of the vial as control input variables. However, experimental investigations of the impact of operational changes on the heating performance of the milling ball could be time consuming and costly. Mathematical modeling is a more efficient approach to understanding the high-energy ball milling process, as it can analyze and predict how system performance variables respond to changes in operational parameters. In this study, a sensitivity analysis was conducted on the oscillation frequency and amplitude of the vial by varying the input conditions within the range of ±50% while maintaining the same model parameters as in Table 4. Two relative changes, ${\lambda }_w$ and ${\lambda }_{X_v}$, were established compared to the reference frequency of 90 rad/s and the amplitude of 25 mm, respectively. Numerical analysis was performed in eight cases, where the two values were −0.5, −0.25, +0.25, and +0.5.

Table 6. Sensitivity analysis results with variations in vial’s oscillation frequency $\omega$.

${\mathit{\lambda }}_{\mathit{\omega }}$	$\mathit{\omega }(\mathbf{rad}/\mathbf{s})$	${\mathit{\theta }}_{\mathit{\infty }}(°\mathbf{C})$	$\mathit{\tau }$ (min)	${\dot{\mathit{\theta }}}_0(°\mathbf{C}/\mathbf{min})$
0.50	45.0	24.94	23.69	0.2086
0.75	67.0	33.62	19.32	0.7052
1.00	90.0	48.04	16.80	1.6692
1.25	112.5	69.42	15.20	3.2517
1.50	135.0	98.59	14.07	5.5858

and

Table 7. Sensitivity analysis results with variations in vial’s oscillation amplitude $X_v$.

${\mathit{\lambda }}_{{\mathit{X}}_{\mathit{v}}}$	${\mathit{X}}_{\mathit{v}}$ (mm)	${\mathit{\theta }}_{\mathit{\infty }}(°\mathbf{C})$	$\mathit{\tau }$ (min)	${\dot{\mathit{\theta }}}_0(°\mathbf{C}/\mathbf{min})$
0.50	12.50	32.57	17.86	0.7035
0.75	18.75	38.99	17.20	1.1043
1.00	25.00	48.04	16.80	1.6692
1.25	31.25	51.51	15.16	2.0778
1.50	37.50	60.74	14.44	2.8224

show the results of the sensitivity analysis for changes in frequency and amplitude, in addition to the baseline data where ${\lambda }_w={\lambda }_{X_v}=1.0$.

Figure 5a,b display the temperature–time curves for the variations in ${\lambda }_{\omega }$ and ${\lambda }_{X_v}$, respectively. The regression parameters in Table 6 and Table 7 were used to create these curves. As ${\lambda }_{\omega }$ and ${\lambda }_{X_v}$ increase, both the rate of change in the initial temperature and the steady-state temperature tend to increase. This is because an increase in either ${\lambda }_{\omega }$ or ${\lambda }_{X_v}$ directly raises the kinetic energy of the vial. As a consequence of increasing the oscillation frequency and amplitude of the vial, the impact energy transmitted to the milling ball through contact also increases. It can be observed from the bandwidth of the curves in Figure 5 that the temperature rise is more sensitive to frequency changes than the amplitude.

Figure 6 and Figure 7 show the relative sensitivity of two performance indices, that is, steady-state temperature ${\theta }_{\infty }$ and initial temperature evolution rate ${\dot{\theta }}_0$, to changes in ${\lambda }_{\omega }$ or ${\lambda }_{X_v}$, respectively. It was found that increasing ${\lambda }_{\omega }$ by 50% causes ${\theta }_{\infty }$ and ${\dot{\theta }}_0$ to magnify by 205% and 335%, respectively. Similarly, increasing ${\lambda }_{X_v}$ by 50% leads to amplification by 126% and 169% in ${\theta }_{\infty }$ and ${\dot{\theta }}_0$, respectively. When developing a high-energy shaker mill, it is more advantageous to choose frequency as the operating parameter instead of oscillation amplitude, as it provides greater sensitivity and a wider range of milling energy for control.

5. Conclusions

This research aims to construct a mathematical model for advanced manufacturing processes that involve high-energy ball mills, such as mechanochemical alloying or novel ammonia synthesis techniques. The specific objective is to acquire a numerical solution for the dynamic and thermodynamic state variables of the single-degree-of-freedom milling ball that is exposed to frequent impacts within the harmonically oscillating shaker mill. Hamiltonian mechanics was employed to model the multiphysics system. Generalized coordinates, such as the position, linear momentum, and internal energy of the milling ball, were chosen to represent the configuration space. Hamilton’s canonical equations provide the basis for obtaining equations of position and linear momentum. The energy conservation law was used to obtain the equation for the internal energy in the rate form of the non-holonomic constraint. In order to fully establish the governing equations, a contact force model, a collision heating model, and a convective heat transfer model were incorporated, along with the necessary calibration factors to account for each model. A series of numerical analyses were conducted to investigate the influence of varying the operating parameters of the shaker mill on the thermomechanical behavior of the milling ball. The sensitivity analysis revealed that the frequency of the oscillation had a greater impact on the temperature variations than the amplitude. It was concluded that the oscillation frequency control strategy is more suitable than the amplitude control method to operate the shaker mill. The uniqueness of this modeling study is that it first achieved the highest accuracy numerical solution with 0.1% error for temperature evolution in a shaker mill process. However, the mathematical model presented in this paper only applies to one-dimensional milling ball motion in a shaker mill, with limitations for interpreting more complex three-dimensional motion and frictional heating phenomena. Therefore, future studies should focus on creating a more sophisticated mathematical model that considers the milling ball’s three-dimensional kinematics, including its translation and rotation, and a more comprehensive range of heating mechanisms, such as the heat produced by Coulomb and rolling friction. This future research direction will accelerate the development of advanced computational code to predict and optimize high-energy milling processes more accurately.