# The Theory and Demonstration of the Solid-Fluid Transformation of Ice Water

Authors: Yong-Yan Wang; Xi-Yan Fan; Nan Qin; Jian-Guang Li; Chuan-Qi Su
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, a mathematical model for describing the solid-fluid transformation of ice water is put forward based on the special geometry cases. The correctness of the obtained model is verified through comparison with numerical analysis and experiments. The good agreement indicates that the obtained model is available for the study of the solid-fluid transformation of ice water. The theory derived in this paper lays a foundation for the research of solid-fluid transformation phenomena of other materials and may have important applications in engineering areas such as rheology, creep, and instability of materials.


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## Body

## 1. Introduction

The phenomena of solid-fluid transformation exist in the processes of ice melting, metal melting, rheology, and so on [1–7]. On September 11, 2001, the secondary collapse of the world trade center in the terrorist attack is a typical problem of creep under the action of thermal flow. The metals and concrete supports of the building crept under the action of thermal flow after a period of time, which led to the strength loss and collapse. Although the rheology and creep of materials have been hot topics in the engineering science, the mathematical aspect of the solid-fluid transformation has not been pointed out clearly. Starting from the Newton’s law of cooling, the theory of the solid-fluid transformation of ice water will be derived in this paper. That can be a theoretical foundation for the solid-fluid transformation problem and may have important applications in the field of engineering science.


## 2. The Mathematical Model for the Ice Water’s Solid-Fluid Transformation

Firstly, we take the ice ball as a special example to facilitate the deduction. Assume the radius of the ice ball asr, then the surface area of the ball as 4πr2. Newton’s law of cooling states that the rate of heat loss of a body is directly proportional to the difference between its own temperature and the temperature of its surroundings [8]. Therefore, the heat transferred through the surface of the ball within Δt time can be expressed as follows:
(1)ΔQ=4πr2κT−T0Δt,where κ is the heat transfer coefficient, T0 is the temperature of ice ball, and T is the temperature of its surroundings. It should be noted that we assume T0 and T are both constant and T0<0°C<T. Within Δt time, there are ΔV volumes of ice melting into water:
(2)ΔV=4πr2Δr.Dividing the both sides of the above equation byΔt and taking the limit, we have
(3)limΔt→0ΔVΔt=4πr2limΔt→0ΔrΔt.The above equation can be rewritten as(4)dVdt=4πr2drdt.The density of ice is set to beρ; then the quality of melted ice is Δm=−ρΔV, where the negative sign signifies that the quality of ice ball decreases. The heat needed to convert T0 ice to 0°C ice is
(5)ΔQ1=cΔm0−T0=−ρcΔV0−T0=4πr2ρcΔrT0,where c denotes the specific heat of ice, namely, the heat needed to raise 1°C per unit mass of ice. The heat needed to melt 0°C ice to 0°C water can be calculated as follows:
(6)ΔQ2=γΔm=−ργΔV=−4πr2ργΔr,where γ is the latent heat per unit mass of ice. Thus, the total amount of heat is
(7)ΔQ=ΔQ1+ΔQ2=4πr2ρcT0−γΔr.By combining Equations (1) and (7), we have
(8)4πr2ρcT0−γΔr=4πr2κT−T0Δt.When∆t→0, the following differential equation could be derived:
(9)drdt=κT−T0ρcT0−γ.In order to integrate the above equation, we assume that the thermal parametersc and γ are both constant. Thus, we have
(10)r=κT−T0tρcT0−γ+C,with C as a constant of integration. When t=0,r=r0, thus, we have C=r0. When the ice ball is completely melted, i.e., r=0, the time of melting can be expressed as
(11)t=ρcT0−γκT−T0r0.The denominator in the above equation is positive becauseT>T0, while the numerator is negative because T0<0°C. Therefore, the time of melting t is positive.In fact, Equation (11) can also be obtained by the calculus relations. We assume that the thickness of ice ball melted within dt time is dr; then the mass of ice melted is dm=−ρdV=−4πr2ρdr. The heat of melting needed is
(12)dQ=βdm=−4πr2ρβdr,where β is the heat of melting needed per unit mass of ice, with the unit J/kg. Based on Newton’s law of cooling, the heat transferred through the surface of the ball within dt time is
(13)dQ=4πr2κT−T0dt.The following equation can be derived according to the conservation of energy:(14)dQ=−4πr2ρβdr=4πr2κT−T0dt.Then, the decreasing rate for the ice ball’s radius with timet is
(15)drdt=κT−T0ρβ.The reduced surface area of ice ball withindt time can be calculated as
(16)dS=4πr+dr2−r2=4π2rdr+dr2.By ignoring the two-order infinitesimal, Equation (16) could be simplified as
(17)dS=8πrdr.By dividing both sides of the above equation bydt, we can get the decreasing rate for the ice ball’s surface area with time t:
(18)dSdt=8πrdrdt.Substituting Equation (15) into Equation (18), the following equation can be obtained:
(19)dSdt=−8πrκT−T0ρβ.Although Equations (11), (15), and (19) are derived from the melting process of ice ball, they could apply to the general melting cases. For the cylinder, cone, and other shapes of ice, the corresponding equations can be derived in a similar way. In the following, we will take the ice cylinder as another example to derive the corresponding equations. Assume that the height of the ice cylinder is h and the radius is r, then the surface area of the cylinder is 2πr2+2πrh. The heat transferred through the surface of the cylinder within dt time can be expressed as
(20)dQ=κ2πr2+2πrhT−T0dt.Withindt time, there are dV volumes of ice melting into water:
(21)dV=2πrhdr+πr2dh.The heat needed to convertT0 ice to 0°C ice is
(22)dQ1=cdm0−T0=−ρcdV0−T0=ρc2πrhdr+πr2dhT0.The heat needed to melt 0°C ice to 0°C water can be calculated as(23)dQ2=γdm=−ργ2πrhdr+πr2dh.Thus, the total amount of heat is(24)dQ=dQ1+dQ2.By combining Equations (20), (22), (23), and (24), we have
(25)κ2πr2+2πrhT−T0dt=2πrhρcT0−γdr+πr2ρcT0−γdh.The above mathematical relationships can be used as a unified theory of the solid-fluid transformation of ice water. That can be a theoretical foundation for the solid-fluid transformation problem and may have important applications in the field of engineering science.


## 3. Experimental Results on Ice Water’s Solid-Fluid Transformation

Ice melting is an unsteady heat transfer process, including not only heat conduction, convection, and radiation but also phase transformation. As for the influence factors regarding ice melting, the external influence factors such as environment temperature, humidity, pressure, convective heat transfer coefficient, and thermal radiation field play important roles, while the internal factors such as the temperature, shape, and size of ice also have nonnegligible effects [9]. The melting experiments were carried out by testing ice of different sizes and shapes in a RPH-80 thermotank, in which the environment temperature was set to be 25°C. Specimens were prepared by pouring purified water into containers of different shapes and sizes, and then the containers were put in the freezer for 24 hours. The initial temperature of ice is set at -18°C. Table 1 shows the records of melting time of the ice with different shapes and sizes under the same experimental conditions.Table 1
The melting time for three kinds of shapes.
ShapeDiameter (mm)Height (mm)Time of melting (min)Cylinder25773596451001205515565179Circular truncated cone25 (Φ1), 40 (Φ2)9135 (Φ1), 50 (Φ2)11645 (Φ1), 60 (Φ2)10014455 (Φ1), 70 (Φ2)17565 (Φ1), 80 (Φ2)215Ball2542356345815510165125


## 4. Comparison between the Mathematical Models with Experimental Results

By processing the experimental data using Origin software, the linear fitting curves (as shown in Figure1) and nonlinear (quadratic function) fitting curves (as shown in Figure 2) for the three types of specimens can be obtained, respectively. It can be seen that the experimental data agree well with the melting time of ice predicted through Equation (11). The melting time and diameter have similar relationship for three types of specimens, which further shows that Equation (11) also works for the cases of the cylinder and circular truncated cone. The differences on slopes and curvatures of fitting curves show that the heat transfer coefficient is a comprehensive parameter, which changes with the shape and size of specimen. Two kinds of fitting are analyzed by the way of comparative error analysis, as shown in Figure 3, from which it can be seen that the nonlinear fitting error is smaller than the linear fitting case for the relationship between the melting time and the diameter of the ice ball cylinder and circular truncated cone. The heat transfer coefficients for the ice ball, cylinder, and circular truncated cone are unrelated to their diameters.Figure 1
The linear fit between the melting time and diameter: (a) ice cylinder, (b) ice circular truncated cone, and (c) ice ball.
(a)(b)(c)Figure 2
The nonlinear fit between the melting time and diameter: (a) ice cylinder, (b) ice circular truncated cone, and (c) ice ball.
(a)(b)(c)Figure 3
The comparative error analysis for the two kinds of fitting: (a) ice cylinder, (b) ice circular truncated cone, and (c) ice ball.
(a)(b)(c)


## 5. Numerical Analysis on the Solid-Fluid Transformation of Ice Water

In the following, ANSYS and fluent analyses will be employed to study the process of ice melting numerically [10, 11]. The mesh size used in the software is 3 mm, and the number of nodes for the ice ball is 1766, the ice cylinder is 5888, and the ice circular truncated cone is 8014.


### 5.1. ANSYS Numerical Simulation Analysis

Ice melting is a process related to the phase transformation. The latent heat must be taken into consideration when the phase transformation is involved, because the latent heat is defined from the concept of enthalpy, which will be taken as an attribute definition of material. The curve that enthalpy changes with temperature can be obtained. According to the definition, solid temperatureTs and fluid temperature Tl will divide the enthalpy curve into three areas, i.e., the solid area T<Ts, phase-transition area Ts<T<Tl, and fluid area Tl<T. The phases can be distinguished by the temperatures in different time. The temperature of ice for phase transformation is 0°C in the atmospheric circumstances. When the temperature is higher than 0°C, the material will melt from ice to water [10].The melting process of ice in the thermotank is mainly influenced by the heat convection and radiation. Typical parameters for the thermotank wall are shown in Table2. The temperature is set to be 25°C in the thermotank, and the initial temperature of ice is −18°C. Considering the heating principle and the heat transfer process of ice in thermotank, we set the mixed boundary condition on the surface of ice that contacts with air and selects the convective heat transfer coefficient as 20 W/(m2·K) [10].Table 2
Typical parameters for the thermotank wall.
Coefficient of thermal conductivity [W/(m·K)]Density (kg/m3)Modulus of elasticity (Pa)Poisson’s ratio (1)Specific heat capacity [J/(kg·K)]60.6478502e110.396From the numerical simulation results worked by ANSYS, we find that the temperature of ice ball reduces from the external to the inner on the melting nephogram at 60 min, which agrees perfectly with the experimental results in the thermotank. The temperature of the ice cylinder and circular truncated cone also reduces from the external to the inner part on the melting nephogram at 60 min, and the temperature at the edges and angles remains the highest, indicating that the melting process starts from the edges and angles to the inner gradually. The results of the numerical simulation also agree well with that of the corresponding experiments in the thermotank. Table3 shows the melting time in the numerical simulation and the thermotank experiment, respectively. It can be seen from Table 3 that the agreement degree between numerical and experimental results on the melting time for ice ball, cylinder, and circular truncated cone reaches 92.6%, 92.5%, and 95.3%, respectively. The melting time for the numerical simulation is shorter than that for the experiment, which can be attributed to the loss of heat during the experiment.Table 3
The comparison of melting time between ANSYS simulation and experimental results.
ShapeSize (mm)ANSYS (min)Experiment (min)Difference (%)BallDiameter4575817.4CylinderDiameter451111207.5Height100CircularTop45TruncatedRoot601371444.7ConeHeight100


### 5.2. Fluent Numerical Simulation Analysis

In order to further verify the correctness of the theoretical model and make a comparison with the results of ANSYS simulation, fluent analysis is employed to simulate the melting process of ice. The air medium in the box is set to be ideal gas, and environmental pressure is set as a standard atmospheric pressure. The latent heat of ice medium is 333146 J/kg, and the melting point of ice and freezing point of water are the same (273.15 K) [11].Fluent analysis uses the heat enthalpy technology to handle with the melting and solidification processes. The melting process is characterized by the changing of the liquid fraction without tracing on the interface [11]. The simulation results are shown in Figure 4 and Table 4, from which it can be seen that the melting processes start from the corners, then carry on from the outside to the inside. The simulation results by fluent analysis also agree well with experiments and are consistent with the results worked out by ANSYS.Figure 4
The simulation result for the ice ball by fluent analysis at 60 min.Table 4
The comparison of melting time between fluent simulation and experimental results.
ShapeSize (mm)Fluent (min)Experiment (min)Difference (%)BallDiameter4574818.6CylinderDiameter451151204.2Height100CircularTop45TruncatedRoot601401442.8ConeHeight100It can be seen from Table4 that the melting time for the ice ball, cylinder, and circular truncated cone by the Fluent simulation is less than that by the experiments and the rates of agreement reach 91.4%, 95.8%, and 97.2% for the ice ball, cylinder, and circular truncated cone, respectively. The reason for this phenomenon is that in the numerical simulation, the boundary conditions and material properties are in the ideal conditions and there is no heat loss.


## 5.1. ANSYS Numerical Simulation Analysis

Ice melting is a process related to the phase transformation. The latent heat must be taken into consideration when the phase transformation is involved, because the latent heat is defined from the concept of enthalpy, which will be taken as an attribute definition of material. The curve that enthalpy changes with temperature can be obtained. According to the definition, solid temperatureTs and fluid temperature Tl will divide the enthalpy curve into three areas, i.e., the solid area T<Ts, phase-transition area Ts<T<Tl, and fluid area Tl<T. The phases can be distinguished by the temperatures in different time. The temperature of ice for phase transformation is 0°C in the atmospheric circumstances. When the temperature is higher than 0°C, the material will melt from ice to water [10].The melting process of ice in the thermotank is mainly influenced by the heat convection and radiation. Typical parameters for the thermotank wall are shown in Table2. The temperature is set to be 25°C in the thermotank, and the initial temperature of ice is −18°C. Considering the heating principle and the heat transfer process of ice in thermotank, we set the mixed boundary condition on the surface of ice that contacts with air and selects the convective heat transfer coefficient as 20 W/(m2·K) [10].Table 2
Typical parameters for the thermotank wall.
Coefficient of thermal conductivity [W/(m·K)]Density (kg/m3)Modulus of elasticity (Pa)Poisson’s ratio (1)Specific heat capacity [J/(kg·K)]60.6478502e110.396From the numerical simulation results worked by ANSYS, we find that the temperature of ice ball reduces from the external to the inner on the melting nephogram at 60 min, which agrees perfectly with the experimental results in the thermotank. The temperature of the ice cylinder and circular truncated cone also reduces from the external to the inner part on the melting nephogram at 60 min, and the temperature at the edges and angles remains the highest, indicating that the melting process starts from the edges and angles to the inner gradually. The results of the numerical simulation also agree well with that of the corresponding experiments in the thermotank. Table3 shows the melting time in the numerical simulation and the thermotank experiment, respectively. It can be seen from Table 3 that the agreement degree between numerical and experimental results on the melting time for ice ball, cylinder, and circular truncated cone reaches 92.6%, 92.5%, and 95.3%, respectively. The melting time for the numerical simulation is shorter than that for the experiment, which can be attributed to the loss of heat during the experiment.Table 3
The comparison of melting time between ANSYS simulation and experimental results.
ShapeSize (mm)ANSYS (min)Experiment (min)Difference (%)BallDiameter4575817.4CylinderDiameter451111207.5Height100CircularTop45TruncatedRoot601371444.7ConeHeight100


## 5.2. Fluent Numerical Simulation Analysis

In order to further verify the correctness of the theoretical model and make a comparison with the results of ANSYS simulation, fluent analysis is employed to simulate the melting process of ice. The air medium in the box is set to be ideal gas, and environmental pressure is set as a standard atmospheric pressure. The latent heat of ice medium is 333146 J/kg, and the melting point of ice and freezing point of water are the same (273.15 K) [11].Fluent analysis uses the heat enthalpy technology to handle with the melting and solidification processes. The melting process is characterized by the changing of the liquid fraction without tracing on the interface [11]. The simulation results are shown in Figure 4 and Table 4, from which it can be seen that the melting processes start from the corners, then carry on from the outside to the inside. The simulation results by fluent analysis also agree well with experiments and are consistent with the results worked out by ANSYS.Figure 4
The simulation result for the ice ball by fluent analysis at 60 min.Table 4
The comparison of melting time between fluent simulation and experimental results.
ShapeSize (mm)Fluent (min)Experiment (min)Difference (%)BallDiameter4574818.6CylinderDiameter451151204.2Height100CircularTop45TruncatedRoot601401442.8ConeHeight100It can be seen from Table4 that the melting time for the ice ball, cylinder, and circular truncated cone by the Fluent simulation is less than that by the experiments and the rates of agreement reach 91.4%, 95.8%, and 97.2% for the ice ball, cylinder, and circular truncated cone, respectively. The reason for this phenomenon is that in the numerical simulation, the boundary conditions and material properties are in the ideal conditions and there is no heat loss.


## 6. Conclusion

In this paper, the theory of the solid-fluid transformation of ice water with special shapes such as the ball, cylinder, and circular truncated cone has been derived and the mathematical relationships of ice water’s solid-fluid transformation have been presented. The unified theory and formulas of ice water’s solid-fluid transformation are validated through simulation analysis and corresponding experiments. It is expected that the theory derived in this paper could lay a foundation for the research of solid-fluid transformation phenomena of other materials and may have important applications in engineering areas.



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