Development of Viscometer Based on Single Sphere Pulling Method for Application to High Temperature Melts

Abstract

To evaluate the viscosities of molten slags that are suitable for capillary refining, a viscometer has been developed based on the single sphere pulling method that enables high accuracy viscosity measurement of high temperature melts. Viscosity measurements using the sphere pulling method generally carry experimental uncertainties attributed to shear stress on the wire suspending the sphere, which is not taken into account in the conventional Stokes’ law when deriving the liquid viscosity from the viscous force applied to a moving sphere in a liquid. In this study, we first considered a modified Stokes’ law equation that describes the balance between the external force and the sum of forces applied to the sphere and the suspension wire including shear stress, ascending force and the surface tension of a liquid applied to the wire, and then determined the liquid viscosity from movement of a single sphere by measuring the external force when the sphere passed through a set position. Second, we designed a new viscometer that controls the liquid container’s velocity using an electrical actuator and measures viscous force on a single sphere statically suspended from an electrical valance. We confirmed that relative sphere velocity in the liquid reaches terminal velocity immediately. These treatments improved the viscosity evaluation accuracy using the single sphere pulling method, and this viscometer enables the viscosity of the standard reference material (SRM2) for high temperatures to be measured to within ±5% relative errors from the recommended values, which is adequate for viscosity measurement of high temperature melts.

1. Introduction

Our motivation for development of a viscometer for high temperature melts is related to improvement of the capillary refining method for iron and steelmaking processes that has been proposed and attempted by one of the authors.^1,2,3,4) In iron and steel production, dephosphorization and desulfurization processes are conducted to remove phosphorus and sulfur contaminants from liquid iron and transfer them into molten slag as phosphate and sulfide ions, respectively. In current dephosphorization and desulfurization processes, large amounts of lime are added to the molten slag to promote the dephosphorization and desulfurization reactions; however, most of the lime remain in the solid state during these processes. In this case, the phosphorus and sulfur in the molten slag are precipitated at the interface between the slag and the solid CaO as tricalcium phosphate (3CaO.P₂O₅) and calcium sulfide (CaS); however, the reactions are controlled by the mass transfer of these impurities in solid CaO, which means that the solid CaO usage is lower than expected during the above processes.

To improve the solid CaO usage efficiency in the dephosphorization and desulfurization processes, “capillary refining” focuses on the porous structure in solid CaO that is simultaneously formed when calcium carbonate pellets or calcium hydroxide particles are sintered at high temperatures for CO₂ or H₂O emission.^1,2,3,4) When molten slag is present between molten iron and solid CaO with a porous structure, then the slag immediately penetrates into the porous structure because of its good wettability with solid CaO, and the impurity elements in the slag such as phosphorus and sulfur can be trapped there as calcium phosphate and calcium sulfide, respectively. Because this capillary refining process is controlled by the capillary phenomenon of liquid slag penetration into the porous solid CaO structure, much higher phosphorus and sulfur entrapment reaction rates are expected, which will lead to much higher solid CaO usage than in conventional processes.

Because the process is based on the capillary behavior of liquid slag, the optimum conditions for capillary refining are strongly related to the viscosity and surface tension of molten slag. In the simplest case, the liquid penetration distance into a single long pore is described as a function of time by the following equation:   

$$L=\sqrt{\frac{d_0\cdot \sigma }{\eta }}\cdot \sqrt{t}\mathrm{\hspace{0.17em} };$$(1)

where L denotes the liquid penetration distance, t is the penetration time, d₀ is the pore diameter, and η and σ are the viscosity and surface tension of the liquid, respectively. Equation (1) indicates that the liquid penetration distance is inversely proportional to the square root of the viscosity of the liquid when the other parameters are fixed. Therefore, preparation of molten slag with low viscosity is important when performing capillary refining because such a slag can immediately penetrate and transfer phosphorus and sulfur impurities into the pore structure of solid CaO.

The viscosity of molten slag is known to be significantly influenced by the slag composition as well as temperature. Many model-based and experimental studies^5,6,7) have been conducted to evaluate the viscosities of multicomponent slags; however, the composition dependence of the slag viscosity has not yet been understood completely. In addition, when capillary refining is applied to the dephosphorization process, the molten slag is in equilibrium with the solid CaO or is saturated with phosphorus to form tricalcium phosphate; therefore, the viscosity information is particularly necessary for molten slag that is equilibrated with these solid phases. For the design of a slag composition that is suitable for capillary refining, it is consequently necessary to prepare a viscometer that enables the viscosities of molten slags in equilibrium with the solid phases to be accurately measured and evaluated.

Various types of viscometers have been designed to measure the viscosities of high temperature melts, including those based on the rotational cylinder/crucible method, the sphere pulling method (or the counter-balance sphere method), the oscillating method, and the capillary method.^5,6,7) The rotational cylinder/crucible method determines the liquid viscosity by measuring the torque applied to a cylindrical material inserted in a liquid when the cylinder or the crucible is rotated at a controlled speed, and the method has mostly been used to determine a wide range of liquid viscosities with high accuracies. However, for accurate evaluation of the viscosity of molten slag equilibrated with a particular solid phase, it is appropriate to apply a method that determines liquid viscosity from the simple and static motion of the substances immersed in the liquid. On this basis, the present study focused on development of a viscometer using the sphere pulling method, where liquid viscosity is determined from the viscous force applied to a sphere that is statically moving in the liquid using Stokes’ law, which describes the balance between the force required to pull up the sphere and the viscous force on the sphere when it achieved terminal velocity.^6,7) It is expected that this method can be applied to viscosity measurement of molten slag equilibrated with a solid phase such as CaO by using a crucible made from the equilibrating solid material, and a sphere made from a material that will not react with the slag.

However, the sphere pulling method was originally accompanied by experimental uncertainty due to the shear stress applied to the wire used to suspend the sphere, which was not taken into account in the conventional Stokes’ law. In addition, this method conventionally required a long vertical distance into the liquid for a moving sphere to achieve terminal velocity. Therefore, this paper describes several improvements to the measurement accuracy for determination of liquid viscosity in terms of both methodology and apparatus, and also describes the application of the apparatus based on the single sphere pulling method that was fabricated in this study to viscosity measurement of high temperature melts. First, a modified Stokes’ law equation has been considered that describes the balance between the external force and the sum of forces applied to the sphere with its suspension wire, including the shear stress and the ascending forces on the wire, and the modified equation was applied directly to determine the viscosity of a liquid from the movement of a single sphere by measuring the external force when the sphere was at a fixed position in the liquid. Second, new experimental apparatus has been designed where a sphere is statically suspended from the top and is immersed in a liquid sample, and a viscous force is applied to the sphere by moving the liquid container with a constant velocity, while in conventional apparatus, the viscous force was produced by pulling the sphere in the liquid with a constant force. It has been confirmed that the apparatus presented here allows the relative movement of the sphere in the liquid to achieve terminal velocity immediately, and contributes to improved viscosity measurement accuracy.

2. Methodology for Determination of Liquid Viscosity by the Single Sphere Pulling Method

In general, the sphere pulling method simply determines the liquid viscosity from the viscous force on a sphere moving vertically in the liquid when an external force is applied to pull the sphere up. The relationship between the viscous force, the ascending force applied to the sphere and the external force is described by Stokes’ law as follows, when the forces are balanced with each other:   

$$W=3\pi fD\mu \cdot v+\frac{\pi }{6}g{D}^3\left({\rho }_s-{\rho }_l\right)\mathrm{\hspace{0.17em} };$$(2)

where W is the external force, μ represents the viscosity coefficient (or simply the viscosity), v is the terminal velocity of the sphere, D is the sphere diameter, f represents a constant that is dependent on the container diameter and the sphere diameter, g is the gravitational constant, and ρ_s and ρ_l are the densities of the sphere and the liquid, respectively.

However, the conventional sphere pulling method had been accompanied by a degree of experimental uncertainty because of the additional shear force applied to the wire used to suspend the sphere, which was not accounted for by the above equation. In addition, the surface tension of a liquid applied vertically to the suspension wire at a gas–liquid–wire triple interface should be also taken into account to express the practical external force to be balanced.

Figure 1 shows a schematic diagram of the forces applied to the moving sphere and the wire in a liquid. When the shear force, the ascending force applied to the wire used to suspend the sphere, and the vertical force on the wire due to the surface tension of the liquid are taken into account, the following equation can be derived:   

$$\begin{array}{c}W=\left(3\pi fD+\frac{2\pi b}{d-b}h\right)\mu \cdot v+\frac{\pi }{6}g{D}^3\left({\rho }_s-{\rho }_l\right)\\ +\frac{\pi {\rho }_w{b}^{2}g}{4}{h}_0-\frac{\pi {\rho }_l{b}^{2}g}{4}h+2\pi r\sigma cos\theta \mathrm{\hspace{0.17em} };\end{array}$$(3)

where b denotes the diameter of the wire used to suspend the sphere, d represents the internal diameter of the liquid container, h₀ and h are the initial and dynamic immersed depth of the suspending wire respectively, ρ_w is the density of the suspension wire, r is the radius of curvature of the meniscus at the gas–liquid–wire triple interface, θ is the contact angle between the liquid and the suspension wire, and σ represents the surface tension of the liquid. The second and fourth terms on the right hand side of Eq. (3) correspond to the shear stress and the ascending force applied to the suspension wire, respectively, which are both proportional to the immersed length of the suspension wire. In addition, the fifth term on the right side of Eq. (3) indicates the contribution of the surface tension of a liquid applied to the suspension wire at a gas–liquid–wire triple interface to express the total external force required for balance.

Fig. 1.

Schematic diagram of forces applied to the moving sphere and the suspension wire in a liquid used in this study.

Previously, several attempts were made to remove the effects of the additional forces applied to the suspension wire by measuring the total external forces on different spheres of different diameters when these spheres are moving with identical terminal velocities, and by evaluating the differences between them.^6,7) In this case, the relationship between the external force and the viscosity of the liquid is described by the following equation:   

$$\mathrm{\Delta }W=3\pi \mathrm{\Delta }\left(fD\right)\mu \cdot v+\frac{\pi }{6}g\left(D_2^3-D_1^3\right)\left({\rho }_s-{\rho }_l\right)\mathrm{\hspace{0.17em} };$$(4)

where D₁ and D₂ are the diameters of the different spheres. This equation indicates that the difference of the external forces (ΔW) would be high when the difference between the sphere diameters (D₁ and D₂) is large, which enables the liquid viscosity to be evaluated accurately. However, our preliminary experiments using two different sphere sizes in a high temperature melt indicated that scatter in the measured external forces for each sphere due to thermal convection flow in the melt affects the viscosity (μ) determination accuracy significantly. In particular, when a wire without a sphere was used (D₁ = 0), the magnitude of the scatter in the measured external force was comparable to or even larger than the external force itself. Therefore, an alternative methodology is required to improve the accuracy of viscosity determination by the sphere pulling method when applied to high temperature melts.

As one approach to determination of liquid viscosity based on the sphere pulling method, this study attempts to apply Eq. (3) directly to the movement of a single sphere and its suspension wire by assuming a fixed position for the sphere in the liquid where the external force is measured. When the immersed depth of the suspension wire, h, is fixed, Eq. (3) can be expressed as follows:   

$$W=A\mu \cdot v+C\mathrm{\hspace{0.17em} };$$(5)

where A and C are constants at a fixed value of h. Equation (5) indicates that when the external force (W) is measured at a fixed sphere position in the liquid (h), the force shows a linear relationship with the terminal velocity of the moving sphere (v), where its gradient corresponds to the value of Aμ. Therefore, when the value of A has been determined, the viscosity of the liquid (μ) can be then evaluated by dividing the gradient of the linear relationship by the A value.

The coefficient A in Eq. (5) is a constant that is dependent on the diameters of the sphere, the suspension wire, and the liquid container when the immersion position of the sphere is fixed, as expressed by the following equation:   

$$A=3\pi fD+\frac{2\pi b}{d-b}h\mathrm{\hspace{0.17em} };$$(6)

The value of A can be determined from the gradients (Aμ) of the relationship between the terminal velocity and the total external force using standard solutions where the viscosity values are known.

3. Experimental Apparatus and Procedure

The experimental apparatus for measurement of liquid viscosity has been designed as follows, based on the single sphere pulling method described above.

The conventional apparatus pulled up a sphere in the liquid with a fixed external force to apply a viscous force to the sphere, and let the sphere’s movement achieve terminal velocity based on the balance between the external force and the viscous force to determine the liquid viscosity using Eq. (2). However, this apparatus required a long vertical distance in the liquid container to enable the moving sphere to achieve terminal velocity, which made it difficult to apply the sphere pulling method to accurately evaluate liquid viscosity in high temperature melts in particular.

Figure 2 shows a schematic diagram of the experimental apparatus that was designed in this study. This apparatus consists of an electrical balance on top, a sphere suspended from the bottom of the electrical balance by a wire, a liquid container, a stage for the container that is connected to an electrical actuator that is settled at the bottom of the furnace, so that it can be moved vertically down with a constant velocity. Therefore, unlike the conventional apparatus, the presented equipment operates to move the liquid container with a constant velocity to apply a viscous force to the sphere immersed in the liquid, and we expect the equipment to allow the relative velocity of the sphere in the liquid to reach terminal velocity immediately. When the sphere diameter is suitably smaller than the inner diameter of the container, the terminal relative velocity of the sphere in the liquid is then assumed to be in accordance with the controlled velocity of the container. The external force detected from the electrical balance is then equivalent to the total force applied to the sphere and to the suspension wire in the liquid, and is dependent on both the terminal velocity and the immersion depth of the sphere in the liquid because the shear force applied to the suspension wire is described as a function of the immersed length of the wire. Therefore, the external force is measured as a function of the terminal velocity when the sphere passes across a particular position in the liquid, and the liquid viscosity is then determined from the gradient of the linear relationship between the external force and the terminal velocity, according to Eq. (5).

Fig. 2.

Schematic diagram of the experimental apparatus for the single sphere pulling method that was designed in this study.

Figure 3 shows a schematic diagram of a sphere with a suspension wire and the liquid container used in this study. A 12.7-mm-diameter sphere made from iron (99.5% purity) was used to apply the viscous force in the liquid, and the sphere was suspended from the top of the furnace using a pure iron wire with a diameter of 1 mm. A liquid container with dimensions of 50 mm (outer diameter, OD) × 40 mm (inner diameter, ID) × 60 mm (height, H) was also made from pure iron (99.5% purity). The sphere and the container were mechanically prepared from corresponding iron rod materials supplied by Nilaco Corp., Japan. Iron was selected as the material for contact with the liquid sample on the basis of a reliable report from the Round-Robin project,⁸⁾ which previously evaluated the accuracy of viscosity measurements of high temperature melts, and which recommended iron as a suitable contact material, along with molybdenum and platinum, for precise viscosity measurement of the high temperature standard reference material (SRM2). In terms of the sizes of the sphere and the container, we confirmed that the terminal relative velocity of the sphere in the liquid is close to the velocity of the container movement as controlled by an actuator, based on theoretical support from the continuity equation and the results of our preliminary experiments using liquid containers with different inner diameters, a sphere with a fixed diameter and a standard reference viscosity solution at room temperature. In terms of tensile strength of the iron suspension wire at high temperatures, it is well known that it shapely decreases at temperatures higher than 723 K according to the measured data of the tensile strength of the cast iron at high temperatures.⁹⁾ However, it could be assumed that the tensile strength of the iron wire is still adequately higher than the shear force on the wire when immersed in the SRM2 melt, because it is predicted as in the order of 10⁻³ Pa at 1673 K using the expression of the shear force on the wire in Eq. (3) and the literature viscosity data of the SRM2 melt.⁸⁾

Fig. 3.

Schematic diagram of the sphere, the suspension wire and the liquid container used in this study.

First, we conducted a calibration procedure for our equipment using several standard reference materials with predetermined viscosities at room temperature to determine the value of the coefficient A in Eq. (5) for the sphere at a fixed position in the liquid. Several centistokes standard viscous liquids made from dimethylpolysiloxane were selected as the reference materials, and their viscosities were predetermined using capillary-type kinetic viscometers within uncertainties of ±1%. Each liquid was set in the container, and a sphere was then immersed in the liquid. Because the sphere was suspended by a wire from the bottom of the electrical balance at the top of furnace, the total force applied to the sphere and to the wire was recorded when the container was moved vertically using an electrical actuator.

Second, the suitability of the proposed methodology and apparatus for high temperature viscosity measurements was examined using the standard reference material (SRM2) for high temperatures that was proposed in the Round-Robin project.⁸⁾ The chemical composition of the standard reference material for high temperatures that was used in this study is summarized in Table 1. Specially graded reagents of silicon dioxide, aluminum oxide, sodium carbonate, potassium carbonate, lithium carbonate, and calcium carbonate of corresponding weights were mixed, premelted in air at 1673 K, and quenched by pouring the melt on a copper plate. 120 grams of the prepared SRM2 glass was stored in an iron container, and then set at the center of the furnace together with the sphere suspended by the wire. After the furnace was evacuated and filled with Ar gas (99.999% purity, dehydrated by silica gel and magnesium perchlorate, and deoxidized in a preheated secondary furnace containing magnesium tips), the furnace temperature was raised to 1673 K using a MoSi₂ heater, and a high temperature SRM2 melt was prepared in the container. We preliminary confirmed that when the furnace was heated at 1673 K, the vertical soaking range in the length of 50 mm was obtained in the middle of the furnace within ±3 K in deviation. Then, the depth of the SRM2 liquid in the container was about 45 mm, so that it was included in the soaking area in the furnace. The liquid viscosity was then evaluated in the following way using a series of decreasing temperatures: first, the position of the liquid surface was determined when the sphere was placed in contact with the liquid surface by raising the liquid container using an actuator. Second, the sphere was immersed in the liquid, and then the total force applied to the sphere and the suspension wire was recorded by an electrical balance when the container was moved down at a predetermined velocity. When the liquid viscosity was determined according to Eq. (5) and the coefficient A, the effect of the thermal expansion of the contact materials on the value of A was taken into account.

Table 1. Chemical composition of standard reference material (SRM2) used in this study for viscosity measurements at high temperatures (mass%).

	Chemical composition (mass%)
	SiO2	Al2O3	Li2O	K2O	Na2O	CaO	T. Fe
As designed	63.9	14.5	20.7	0.1	0.4	0.4	0.0
From chemical analysis after measurements	63.6	14.1	19.9	0.2	0.6	0.4	0.8

4. Results and Discussion

4.1. Calibration of the Apparatus

Figure 4 shows a typical change in the detected external force as a function of the immersion depth of the sphere in the liquid when the container is moved, where a standard viscous liquid made from silicone oil (viscosity of 1.06 Pa·s) was used at room temperature. This external force increases immediately from zero and achieves a specific value over a short distance, and later shows linear change relative to the distance moved with small fluctuations. This result indicates that the relative velocity of the sphere in the liquid immediately achieves terminal velocity, and the total force applied to the sphere and to the wire in the terminated state depends on the position of the sphere, which is subject to the shear force and the ascending force applied to the suspension wire. As decreasing the immersed depth of the sphere, h in Eq. (3), the magnitudes of these forces both decrease proportionally. However, the shear force and the ascending force work oppositely to each other on the wire, and the magnitude of the ascending force is adequately higher than the shear force in the present experimental condition. Therefore, the external force, W in Eq. (3), increases when the immersed depth of the sphere decreases. To examine the relationship between the external force and the terminal relative velocity of the sphere, the external force should be determined when the sphere is at a particular position.

Fig. 4.

Change of external force as a function of the immersion depth of the sphere in the liquid when the container is moved (v = 1 mm/s), where a standard viscous liquid (viscosity: 1.06 Pa·s) was used as a sample at room temperature.

Figure 5 represents the relationships between the external force, corresponding to the total force applied to the sphere with the suspension wire, and the relative velocity of the sphere in several silicone oils with different viscosities, where the force was measured when the immersion depth of the sphere was 15 mm, because that position was included in the terminated state as shown in Fig. 4. The relationships were clearly shown to be linear, and the gradient increased as the liquid’s viscosity increased. These tendencies correspond to the phenomenon described by Eq. (5), and the gradient of the above relationship (Aμ) should therefore be directly proportional to the liquid viscosity. Figure 6 shows the relationship between the gradient Aμ and the predetermined viscosity values for various standard viscous liquids, where another linear relation can be clearly observed. Consequently, the coefficient A in Eq. (5) has been determined at room temperature (RT) as follows:   

$$A=0.363\left(m\right)at\mathrm{\hspace{0.17em} }h=15\left(mm\right),\mathrm{\hspace{0.17em} }at\mathrm{\hspace{0.17em} }RT$$(7)

Fig. 5.

Relationship between the detected external force and the relative velocity of a sphere in a liquid, where various standard viscous liquids with different viscosities are used as liquid samples at room temperature. The viscosities of the silicone oils were predetermined using a capillary-type viscometer. The external force was detected when the sphere immersion depth was 15 mm.

Fig. 6.

Relationship between the gradient Aμ obtained in Fig. 5 and the predetermined viscosity values for the different silicone oils used as standard viscous liquids.

The results of the values of A in Eq. (5) that were determined for various immersion depths of the sphere are summarized in Table 2, indicating almost constant values against the sphere immersion depth.

Table 2. Coefficient A in Eq. (5) that describes the balance between the external force and the sum of the viscous forces on the sphere and the suspension wire (W = Aμ·v + C) determined at room temperature as a function of the immersion depth of the sphere in liquid.

Immersion depth of sphere in liquid, h (mm)	Coefficient A (m)
10	0.361
11	0.362
12	0.362
13	0.363
14	0.363
15	0.363

In addition, the uncertainty of the A value, which corresponds directly to the uncertainty of the viscosity value determined using the proposed method, has been evaluated in the following way. The deviation of the A values was calculated by assuming uncertainties of ±0.1 mm for the sphere diameter (D), the inner diameter of the container (d), the diameter of the wire suspending the sphere (b), and the immersion depth of the sphere in the liquid (h), because these parameters are directly related to the determination of the viscosity value in the proposed method. Equation (6) was used to calculate the value of A, where a semi-empirical equation by Faxen,¹⁰⁾ as expressed by Eq. (8), was taken into account to estimate the f constant as a function of the sphere diameter and the inner diameter of the container:   

$$f={\left\{1-2.104\times \left(\frac{D}{d}\right)+2.09\times {\left(\frac{D}{d}\right)}^3-0.95\times {\left(\frac{D}{d}\right)}^5\right\}}^{-1};$$(8)

Table 3 summarizes the calculated results for the uncertainties of the A values when deviations of ±0.1 mm were assumed from the predetermined sizes of the materials used in this study. As a result, it was shown that the uncertainty of the sphere diameter (D) mostly affects the integrated uncertainty of the A value in Eq. (5) and consequently that of the viscosity value. However, it was also concluded that the integrated uncertainty of the A value is less than 1% under the given experimental conditions, and precise determination of the liquid viscosity can therefore be expected.

Table 3. Estimated uncertainties of coefficient A in Eq. (5) when an uncertainty of ±0.1 mm is assumed in measurements of the sphere diameter (D), the inner diameter of the container (d), the diameter of the wire suspending the sphere (b) and the immersion depth of the sphere in the liquid (h) (Initially, D =12.7 mm, d = 40 mm, b = 1 mm, and h = 15 mm).

Uncertainty of coefficient A when uncertainty of ±0.1 mm is assumed (%)
Sphere diameter (D)	Inner diameter of container (d)	Diameter of wire suspending the sphere (b)	Immersion depth of sphere in liquid (h)
0.78	0.0017	0.068	0.0044

Subsequently, we examined the viscosities of some liquids based on glycerin - water solutions using the proposed apparatus and Eqs. (5) and (7) at room temperature, and compared the results with those predetermined using a capillary-type viscometer, as shown in Table 4. Good agreements were obtained between the viscosities measured by the proposed apparatus and those measured by capillary-type viscometers. Therefore, it has been shown that the liquid viscosity can be accurately evaluated using the equation derived in this study, which takes account of the shear force and the ascending force applied to the wire to express the total force applied to the immersed material, and the apparatus designed in this study based on the single sphere pulling method.

Table 4. Comparison between the viscosity values of glycerin-water solutions determined by a capillary-type viscometer and by the single sphere pulling method.

	Viscosity, μ/Pa·s		Relative deviation between μ1 and μ2, $\left|\frac{{\mu }_2-{\mu }_1}{{\mu }_1}\right|$ × 100 (%)
	Determined using capillary-type viscometer, μ1	Determined using single sphere pulling method, μ2
Glycerin 90 - 10 vol.% water solution	0.337	0.330	2.26
Glycerin 95 - 5 vol.% water solution	0.403	0.416	3.24

4.2. Viscosity Measurement of High Temperature Melt Using Single Sphere Pulling Method

To verify the suitability of the proposed method and apparatus for viscosity measurement of high temperature melts, we attempted accurate evaluation of the viscosities of a high temperature SRM2 melt using the single sphere pulling method. For viscosity measurements at high temperatures, the effects of the thermal expansions of the sphere, the suspension wire and the liquid container on the coefficient A in Eq. (5) were taken into account using the thermal expansion coefficient of iron.¹¹⁾

Figure 7 shows the results of the detected external force applied to the sphere and the wire as a function of the immersion depth of the sphere in the liquid when the container was moved at a fixed velocity (v = 1 mm/s), where a high temperature SRM2 melt at 1623 K was used as the liquid sample. As expected for a standard viscous liquid at room temperature (Fig. 4), the external force increased immediately from zero to reach a specific value, and then increased linearly with the distance moved. Several runs were conducted under identical conditions, and it was confirmed that good agreement was obtained between the profiles of the external force with small scatters, which were partially caused by possible convection flow in the liquid.

Fig. 7.

Change of external force as a function of the immersion depth of the sphere in the liquid when the container is moved (v = 1 mm/s), where a standard viscosity reference melt (SRM2) at 1623 K was used as the liquid sample.

Figure 8 represents the relationships between the external force and the relative velocity of the sphere in the SRM2 melt at various temperatures, where the force was measured when the immersion depth of the sphere in the liquid was 15 mm. It was confirmed that this sphere position in the liquid was included in the terminated state, as shown in Fig. 7. For the individual sphere velocity, the container movements and the external force measurements were conducted several times, and the average values are shown in the figure. A linear relationship can be clearly observed between the external force applied to the sphere and the relative velocity of the sphere at each temperature, and the gradient decreased with increasing temperature, which corresponds to the general tendency that liquid viscosity should decrease as temperature increases. In addition, the intercept of the linear relationship decreased with increasing temperature, indicating that the density of the SRM2 melt increases when the temperature increases, according to Eq. (3).

Fig. 8.

Relationships between the detected forces on the sphere and the relative velocity of the sphere in the SRM2 melt at various temperatures, where the force was recorded when the sphere immersion depth in the liquid was 15 mm.

The viscosity of the SRM2 melt was then determined from the gradients of the linear relationships shown in Fig. 8, and using coefficient A after adjusting for the effects of the thermal expansions of the iron sphere, the wire and the container. Figure 9 shows the measured results for the viscosity of the SRM2 melt as a function of inverse temperature, where the viscosity values are plotted on a logarithmic scale. The scatter of the measured viscosity was calculated by comparing the results for the viscosity values determined at different sphere immersion depths in the melt (h is in the range between 10 and 15 mm). When compared with the recommended values, the viscosities that were determined in this study show good agreement, and most of the values are within deviations of ±5%. Some of the results presented here deviated negatively from the recommended values, and this may be partially attributed to iron contamination of the SRM2 melt during the viscosity measurements, as discussed below.

Fig. 9.

Comparison between the viscosities of the SRM2 melt determined in this study and the recommended values as a function of inverse temperature, where the viscosity values are plotted on a logarithmic scale.

Table 1 includes the results of a chemical analysis of the SRM2 melt performed by inductively coupled plasma spectroscopy after the viscosity measurements, and a small amount of iron contamination was detected in the sample. Because the iron oxide in the molten silicate slag generally behaves as a network modifier and reduces the slag viscosity when the molten slag is equilibrated with metallic iron, the iron contamination described above may have been partially responsible for the negative deviations of the presented viscosity results from the recommended values.

Consequently, the suitability of the proposed viscosity measurement method and apparatus based on the single sphere pulling method has been successfully confirmed.

5. Conclusions

A new liquid viscosity measurement methodology based on the single sphere pulling method has been proposed. First, a modified equation for Stokes’ law has been considered that describes the balance between the external force and the sum of forces applied to the sphere with its suspension wire when moving in a liquid, including the viscous force on the sphere, along with the shear force, the ascending force and an additional force related to the surface tension of the liquid applied to the suspension wire. Then, the viscosity of a liquid has been determined based on the movement of a single sphere in the liquid by measuring the external force applied to the sphere and its suspension wire when it passes through a fixed position. In addition, a uniquely styled apparatus has been designed to perform viscosity measurements by the single sphere pulling method described above, where the liquid container is moved at a constant velocity to apply a viscous force to the immersed sphere so that the relative velocity of the sphere in the liquid can immediately achieve terminal velocity. Using the proposed methodology and apparatus, it was confirmed that liquid viscosity can be evaluated to within deviations of ±2–3% at room temperature.

Measurement of the viscosity of a high temperature melt was also attempted using the proposed method. It was confirmed that the viscosity of the high temperature standard reference viscous melt can be determined to within deviations of ±5% when compared with reliable literature values, which indicated that the proposed methodology and apparatus are suitable for accurate viscosity measurements of high temperature melts.

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