Novel Physical Modelling under Multiple Dimensionless Numbers Similitudes for Precise Representation of Molten Metal Flow

Abstract

Physical model experiments, together with numerical model calculations, are essential for scientific investigations such as molten metal flow in casting processes. Considering the physical modelling of flow phenomena, a common method is used to construct a physical model with a reduced scale ratio and then, experiment is carried out under one or two dimensionless number(s) similitude(s). It is an ideal condition of the experiment to establish the simultaneous similitude of multiple dimensionless numbers (SMDN) concerned with the objective flow phenomena but was considered difficult or impossible to realize in practice. This chapter presents a breakthrough in this matter. A simple relationship between the physical properties of fluids and the scale ratio of the physical model is clearly expressed for the simultaneous similitude of the Froude, Reynolds, Weber, Galilei, capillary, Eötvös and Morton numbers. For establishing the physical modelling to represent molten Fe flow phenomena under the SMDN condition, the physical properties of some molten metals can be demonstrated to meet the required relationships. Furthermore, this novel concept is also applicable for other combinations of molten metals. Precise, safe, and easy physical model experiments will be conducted under the SMDN condition that exactly mimics industrial casting operations in higher-temperature systems.

1. Introduction

Molten metal flow in casting mould has great influence on the productivity and quality of the cast products. Figure 1 shows an image of flow phenomena in a continuous casting (CC) mould of steel production. There are various flow phenomena described in Figure 1; closed channel flow contains small vortex in submerged entry nozzle (SEN), injected argon (Ar) behaviour, free-surface flow, viscous bulk flow, and so on. It is difficult to observe or measure the velocity of high-temperature opaque molten metal flow. Consequently, physical model experiments, as well as numerical simulations, are widely carried out to estimate the flow phenomena in the mould.

Figure 1.

Image of molten steel flow in CC mould.

As for physical modelling, similitude is an essential matter for accurate representation. Prior dimensionless numbers to represent flow phenomena in the continuous casting mould are Froude number (Fr), Reynolds number (Re), and Weber number (We) [1, 2, 3, 4]. However, the simultaneous similitude of these three dimensionless numbers has been recognised as being difficult or impossible to realise [5, 6, 7]. As a matter of record, there has been no concept previously proposed for similitudes of multiple dimensionless numbers in physical modelling [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].

Standing on this point of view, physical modelling condition of simultaneous similitude of multiple dimensionless numbers (SMDN) has been studied and then established [37, 38] for the precise representation of the real flow phenomena with an appropriate relationship of scale ratio and physical properties of liquids.

2. Physical modelling condition to satisfy simultaneous similitude of multiple dimensionless numbers

Bulk flow is governed by inertial, gravitational, and viscous forces. Consequently, dimensionless numbers must correspond with are the Fr and the Re to represent the bulk flow. In addition, the We is important for the precise representation of the surface flow phenomena, such as droplet/bubble formations or ripple waves.

These three dimensionless numbers are defined in Eqs. (1), (2), and (3) [39], where the Fr is defined as the square root of (inertial force/gravitational force), the Re is defined as (inertial force/viscous force), and the We is defined as the square root of (inertial force/surface-tension force). Alternative definitions for the Fr (inertial force/gravitational force) and We (inertial force/surface-tension force) attain the same result.

$$\mathrm{Fr}=\frac{V}{\sqrt{g\bullet L}}$$E1

$$Re=\frac{\rho \bullet V\bullet L}{\eta }$$E2

$$\text{We}=V\sqrt{\frac{\rho \bullet L}{\sigma }}$$E3

where V is characteristic velocity of fluid, L is characteristic length, g is gravitational acceleration, ρ is density of fluid, η is viscosity of fluid, and σ is surface tension of fluid.

Describing the prototype with subscript 0, flow velocity V in the physical model is expressed as Eq. (4) in the case of the Fr similitude ($\because \mathit{Fr}=\frac{V_0}{\sqrt{g\bullet L_0}}=\frac{V}{\sqrt{g\bullet L}}=\frac{V}{\sqrt{g\bullet \lambda \bullet L_0}}$) [8].

$$V=\sqrt{\lambda }\bullet V_0$$E4

where $\lambda$ is scale ratio of the physical model.

Coincidence of the Re between a physical model and the prototype are expressed as Eq. (5).

$$Re=\frac{{\rho }_0\bullet V_0\bullet L_0}{{\eta }_0}=\frac{\rho \bullet V\bullet L}{\eta }=\frac{\rho \bullet V\bullet \lambda \bullet L_0}{\eta }$$E5

In the same manner, coincidence of the We between a physical model and the prototype is expressed as Eq. (6).

$$\text{We}={\mathrm{V}}_0\sqrt{\frac{{\rho }_0\bullet L_0}{{\sigma }_0}}=\mathrm{V}\sqrt{\frac{\rho \bullet L}{\sigma }}=\mathrm{V}\sqrt{\frac{\rho \bullet \lambda \bullet L_0}{\sigma }}$$E6

Scale ratio $\lambda$ for the Fr and Re similitudes obtained by substituting Eq. (4) into Eq. (5) and scale ratio $\lambda$ for the Fr and We similitudes obtained by substituting Eq. (4) into Eq. (6) should be equal in the condition of the Fr, Re, and We similitudes [37, 38]. Subsequently, Eq. (7) is obtained to express a relationship of physical properties for the simultaneous similitude of multiple dimensionless numbers (SMDN).

$$\lambda ={\left(\frac{\eta /\rho }{{\eta }_0/{\rho }_0}\right)}^{\frac{2}{3}}={\left(\frac{\sigma /\rho }{{\sigma }_0/{\rho }_0}\right)}^{\frac{1}{2}}$$E7

Eq. (7) can be rearranged to Eq. (8) by applying kinematic viscosity$\nu =\eta /\rho$ and a new parameter of $\iota =\sigma /\rho$. ${\iota }^{1/2}/{\nu }^{2/3}$ in Eq. (8) is a parameter (named the S-parameter) to indicate satisfaction of the SMDN condition [37]. Precise physical model experiments will be carried out employing fluids that satisfy the relationship of physical properties described in Eq. (8) with the appropriate scale ratio shown in Eq. (7).

$$\frac{{\iota }^{1/2}}{{\nu }^{2/3}}=\frac{{{\iota }_0}^{1/2}}{{{\nu }_0}^{2/3}}$$E8

As a result of the Fr, Re, and We similitudes satisfying Eqs. (7) and (8), the similitude conditions of Eötvös number (Eö) and the Morton number (Mo) are also established as described below.

As shown in Clift, Grace, and Weber’s chart that classifies the shape of the rising bubble [40], the Eö and Mo have a dominant influence on the bubble shape and the bubble rising velocity.

The Eö described in Eq. (9) [40], represents buoyancy force/surface-tension force. If the density of the gas is sufficiently smaller than that of the surrounding liquid, Eq. (9) can be converted to Eq. (10). Where subscript b is for the gas phase of the bubble.

The rearranged right side of Eq. (10) as shown in Eq. (11) indicates that the Eö similitude is established under the condition of the Fr and We similitudes [37, 38]. Here, the value of d_b/L in Eq. (10) is same for the prototype and the physical model ($\because \raisebox{1ex}{$d_b$}\!\left/ \!\raisebox{-1ex}{$L$}\right.=\raisebox{1ex}{$\lambda d_{b0}$}\!\left/ \!\raisebox{-1ex}{$\lambda L_0$}\right.=\raisebox{1ex}{$d_{b0}$}\!\left/ \!\raisebox{-1ex}{$L_0$}\right.$).

$$\mathrm{E}\mathrm{ö}=\frac{g\left(\rho -{\rho }_b\right)\bullet {d_b}^2}{\sigma }$$E9

$$\mathrm{E}\mathrm{ö}\approx \frac{g\bullet \rho \bullet {d_b}^2}{\sigma }$$E10

$$\frac{g\bullet \rho \bullet {d_b}^2}{\sigma }=\frac{V^2\bullet \rho \bullet L}{\sigma }\bullet \frac{g\bullet L}{V^2}\bullet {\left(\frac{d_b}{L}\right)}^2={\left(\frac{\text{We}}{\mathrm{Fr}}\bullet \frac{d_b}{L}\right)}^2$$E11

Provided that the density of the bubble is quite little compared with that of the surrounding fluid, Mo described in Eq. (12) [1, 41] consists of gravitational acceleration g and the physical properties of the fluid. Accordingly, the Mo similitude is undoubtedly established in the case that the physical model employs the same fluid as in the prototype. In contrast, the physical properties should satisfy a strict relationship for the Mo similitude in the case that the fluids are different for the physical model and the prototype. As described in Eq. (13), Mo is comprised of three dimensionless numbers; Fr, Re, and We [1, 41]. As such, Mo similitude is always established under the SMDN condition for the case of simultaneous dimensionless numbers equalities for the Fr, Re, and We.

$$\mathrm{Mo}=\frac{g\bullet {\eta }^4}{\rho \bullet {\sigma }^3}$$E12

$$\frac{{\text{We}}^6}{{Re}^4\bullet {\mathrm{Fr}}^2}=\frac{V^6\bullet {\rho }^3\bullet L^3}{{\sigma }^3}\bullet \frac{{\eta }^4}{{\rho }^4\bullet V^4\bullet L^4}\bullet \frac{g\bullet L}{V^2}=\mathrm{Mo}$$E13

Similarly, any other combinations of inertial, gravitational, viscous, and surface tension forces, such as gravitational/viscous forces (Galilei number) and viscous/surface tension forces (the capillary number), are identical for the physical model and the prototype, under the above mentioned SMDN condition.

3. Physical properties of molten metals and water

Precise physical model experiment to represent flow phenomena under the SMDN (similitude of multiple dimensionless numbers) condition could be realised in the case that physical properties of molten metal in the prototype and experimental fluid satisfy the strict relationship described in Eq. (8).

Table 1 shows the physical properties of molten metals and water reported by many researchers and scientists [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. Where “Sn-40Bi” in Table 1 means 60mass% Sn – 40mass% Bi alloy.

	Melting point [°C]	Temperature [°C]	ρ [kg/m^3]	η [Pa·s]	σ [N/m^2]
			×10^3	×10^−3
Fe	1538 [42]	1575	7.001 [42]	4.732 [43]	1.844 [43]
Hg	−38.9 [44]	20	13.55 [44]	1.568 [44]	0.4802 [45]
Sn	232 [45]	232 ∼ 332	6.971 ∼ 6.907 [46]	2.302 ∼ 1.713 [46]	0.5645 ∼ 0.5527 [46]
Cu	1085 [47]	1085 ∼ 1185	7.997 ∼ 7.915 [47]	4.019 ∼ 3.417 [47]	1.2576 ∼ 1.237 [48]
Zn	420 [49]	500	6.488 [49]	3.072 [49]	0.7507 [50]
Al	660 [51]	700	2.379 [52]	1.295 [51]	0.8694 [53]
Ga	29.8 [49]	70	6.053 [49]	1.407 [49]	0.7140 [45]
Bi	271 [54]	320	9.986 [54]	1.644 [54]	0.3758 [54]
Sn-40Bi	174 [54]	225	7.900 [54]	2.219 [54]	0.4551 [54]
Water	0	10 ∼ 70	1.000 ∼ 0.978 [44]	1.307 ∼ 0.404 [44]	0.0742 ∼ 0.0644 [44]

Table 1.

Physical properties of molten metals and water for Figure 2.

These physical properties listed in Table 1 are plotted in Figure 2, where the abscissa X (horizontal)-axis and the ordinate Y (vertical)-axis are the denominator and numerator of the S-parameter (${\iota }^{1/2}/{\nu }^{2/3}$), respectively. This ${\nu }^{2/3}$ versus ${\iota }^{1/2}$ chart has been named the SMDN chart. The SMDN chart indicates whether a combination of two liquids satisfies the relation in Eq. (8), or not. As indicated in Figure 2, Fe at 1575°C, Cu (1085–1185°C, closely at 1145°C), and Sn (232–332°C, closely at 288°C) are all located on the same S-parameter line of ${\iota }^{1/2}/{\nu }^{2/3}=211$. This indicates that the SMDN condition is established between these three molten metals at the appropriate temperatures and scale ratios.

Figure 2.

Physical properties of molten metals and water plotted on SMDN chart.

Scale ratios to satisfy the SMDN conditions are also indicated in the SMDN chart. For example, the ratio of the distance between the origin point and the Ga plot to the distance between the origin point and the Al plot is 0.57, as shown in Figure 3. As is obvious from Eq. (7), this value of 0.57 is the scale ratio of molten Ga models needed to represent molten Al flow satisfying the SMDN condition. In the same manner, molten Sn-40Bi models with a scale ratio of 0.71 can represent molten Zn flow under the SMDN condition. Naturally, the same rule is applicable for molten Fe, Cu, and Sn.

Figure 3.

Proper scale ratios indicated on SMDN chart.

In the case only the Fr and Re similitudes are required to represent the bulk flow, water models could represent flow phenomena of molten Fe, Cu, Sn, Al, Ga, Zn, Sn-40Bi, Bi, and Hg with proper scale ratios according to the ratios of ${\nu }^{2/3}={\left(\eta /\rho \right)}^{2/3}$, as shown in Eq. (7). Please note that kinematic-viscosity $\nu$ of water has a large temperature-dependency, as shown in Figure 2. Therefore, temperature of the water should be carefully considered to determine the scale ratio.

Moreover, the physical modelling concept of the SMDN is applicable not only for molten metal systems described in this chapter but also for many types of liquids [37].

4. Applications of physical modelling under multiple dimensionless numbers similitudes

As described above, the physical modelling concept of SMDN is applicable for flow phenomena influenced by followed four forces—inertial, gravitational, viscous, and surface tension forces. Some examples of flow phenomena representations under the SMDN condition are shown in this section.

4.1. Injected bubble size

In the case of bubble formation during gentle gas injection, three forces—gravity, surface tension, and buoyancy forces, are dominant to determine the bubble diameter D_b, and thus D_b can be calculated using Eq. (14) [55], provided that the density of the gas bubble is negligibly small compared with that of the surrounding fluid.

$$D_b=\sqrt[3]{\frac{6\bullet D_n\bullet \sigma }{\rho \bullet g}}$$E14

where D_n is the internal diameter of the gas injection nozzle.

Table 2 shows the parameters of physical properties (ν and ι), the S-parameters described in Eq. (8), and the adequate scale ratios obtained by Eq. (7) for several molten metals. The physical properties of these molten metals are shown in Table 1. As is obvious in Table 2, there are three groups with similar S-parameters—molten Al and Ga for the S-parameter of 287, molten Fe, Cu, and Sn for the S-parameter of 211, and molten Zn and Sn-40Bi for the S-parameter of 177. Scale ratios shown in Table 2 are defined against “BASE” molten metals for each S-parameter group. The “BASE” metals are—molten Al for the group of S-parameter = 287, molten Fe for the group of S-parameter = 211, and molten Zn for the group of S-parameter = 177.

	Temperature [°C]	ν [m^2/s]	ι [m^3/s^2]	S-parameter = ι^1/2/ν^2/3	Scale ratio λ = (ν/ν0)^2/3
		×10^−7	×10^−4
Al	700	5.441	3.655	287	1.0 (BASE)
Ga	70	2.325	1.180	287	0.57
Fe	1575	6.759	2.635	211	1.0 (BASE)
Cu	1145	4.716	1.568	211	0.77
Sn	288	2.776	8.045	211	0.55
Zn	500	4.734	1.157	177	1.0 (BASE)
Sn-40Bi	225	2.809	5.761	177	0.71

Table 2.

Physical properties and S-parameters for various molten metals.

Bubble diameters formed by injected inert gas under those conditions calculated by Eq. (14) for various nozzle diameters are shown in Figure 4. The abscissa X (horizontal)-axis of Figure 4 shows equivalent bubble diameters considering the scale ratios. As is clear from Figure 4, equivalent bubble diameters are plotted on the same point under the conditions with the same S-parameters. This is a simple comparative case to show the availability of physical model experiments respecting the SMDN condition.

Figure 4.

Generated bubble sizes for various S-parameter groups.

4.2. Rising velocity of bubbles

Inert gas injection is widely applied to casting processes to remove non-metallic inclusions. Meanwhile, it causes bubble defects of the cast products. Consequently, it is valuable to represent the two-phase flow of bubbles and molten metals in physical model experiments.

Complicated demeanour of the rising bubble is dominated by two dimensionless numbers—the Eö and Mo. They have dominant influences on the bubble shape as well as the bubble rising velocity [40]. Therefore, physical models under the concept of SMDN will accurately represent bubble rising demeanour.

In connection with the bubble deformation, the drag coefficient of a rising bubble changes in value complicatedly, which has been investigated by many researchers. Eq. (15) shows Tomiyama’s Equation [56], one of the trustworthy equations to express the drag coefficient of rising bubbles. However, this equation was obtained by the experiments with water and some organic liquids, so it should be confirmed if this equation could be applied to the molten metal systems or not.

$$C_D=max\left[\mathit{min}\left\{\frac{24}{{Re}_b}\left(1+0.15{{Re}_b}^{0.685}\right),\frac{72}{{Re}_b}\right\},\frac{8}{3}\frac{\mathrm{E}\mathrm{ö}}{\mathrm{E}\mathrm{ö}+4}\right]$$E15

Therefore, data of investigated terminal velocity of rising bubble in molten metals of reliable researches [38, 57] were compared with Tomiyama’s equation. As shown in Figure 5, Tomiyama’s equation well represents the bubble terminal velocity in molten Sn and Hg. The result shows that Tomiyama’s equation can be extensively applied to molten metal systems.

Figure 5.

Comparison of terminal velocity of rising bubbles in molten metals.

Accordingly, bubble rising velocity in molten metals could be calculated by Tomiyama’s equation. Before comparing the bubble rising velocity, it should be noted that in the case of the Fr similitude with the scale ratio of physical modelling, flow velocity is described in Eq. (4; $V=\sqrt{\lambda }\bullet V_0$). Consequently, time progression in the physical model will expand or contract as expressed in Eq. (16) [38]. For example, in the case of scale ratio $\lambda =0.5$, time progression in the physical model will shrink $\sqrt{0.5}\approx 0.71$ times as compared with the real-time progression. For example, in the case of the physical models under the Fr similitude and the scale ratio of 0.5, real flow phenomena generated in 100 seconds will be represented in 71 seconds.

$$t=\frac{L}{V}=\frac{\lambda \bullet L_0}{V_0\sqrt{\lambda }}=\sqrt{\lambda }\bullet t_0$$E16

Figure 6 shows the relationship between equivalent bubble diameter in molten Fe and relative rising velocity (RRV) of bubbles under variable similitude conditions. RRV is (rising velocity of the bubble in physical model experiment considering the scale ratio/rising velocity of the bubble in prototype) as expressed in Eq. (17). The time progression described in Eq. (16) is also taken into account in Eq. (17). Here, $U_0$ is bubble rising velocity in the prototype (in molten Fe), $U$ is bubble rising velocity in the physical model experiment. Please note that the bubble diameter $d_b$ for $U$ and bubble diameter $d_{b0}$ for $U_0$ are not same ($d_b=\lambda \bullet d_{b0}$).

Figure 6.

Comparison of relative terminal velocity of rising bubbles under a variety of similitude conditions.

$$\mathit{RRV}=\left(\raisebox{1ex}{$\frac{U}{\lambda }$}\!\left/ \!\raisebox{-1ex}{$U_0$}\right.\right)\sqrt{\lambda }=\frac{U}{U_0\bullet \sqrt{\mathrm{\lambda }}}$$E17

As shown in Figure 6, rising velocity of bubbles in molten Sn or molten Cu under the SMDN condition precisely represents rising velocity of bubbles in molten Fe. On the other hand, rising velocity of bubbles in water under the Fr and Re similitudes condition ($\lambda =1.0$) corresponds to that in molten Fe only in the region of the small bubble diameters. In contrast, rising velocity of bubbles in water under the Fr and We similitudes condition ($\lambda =0.52$) correspond to that in molten Fe only in the region of the large bubble diameters. The reason can be read from Eq. (15). Drag coefficient of bubbles are dominated by the Re in the region of the small bubble diameters, consequently, the water model under the Fr and Re similitudes condition is enough to represent the small-bubble rising velocity. In the region of the large bubble diameters, drag coefficient of bubbles is dominated by the Eö, and the Eö similitude is established under the Fr and We similitudes as described in Eqs. (10) and (11). Consequently, the water model under the Fr and We similitudes condition is adequate to represent the large-bubble rising velocity.

4.3. Other applications of physical modelling under multiple dimensionless numbers similitudes

Other applications of the SMDN concept include the following examples—in the case of the Mesler entrainment of a liquid drop to produce many fine bubbles, three-dimensionless numbers, the We, Fr, and capillary numbers, describe the entrainment phenomenon [58]. Consequently, physical model experiments under the SMDN condition will precisely represent the bubble sizes formed by the Mesler entrainment.

As expressed in the above examples, the SMDN concept of the physical modelling is applicable for the precise representation of all the flow phenomena dominantly affected by four forces—inertial force, gravitational force, viscous force, and surface-tension force. However, physical models under the SMDN condition could not precisely represent flow phenomena affected by other forces or flow phenomena of other types of fluids. For example, flow phenomena of compressible fluid flow, two-phase flow with solid-particles and liquid are outside the SMDN concept region in this chapter.

As for the representation of the thermal convection flow, the similitude of the Rayleigh number (Ra), shown in Eq. (18) for both the prototype and the physical model [59], is required.

$$\mathrm{Ra}=\frac{g\bullet {\beta }_0\bullet {\mathrm{\Delta }T}_0\bullet {L_0}^3}{{\nu }_0\bullet {\alpha }_0}=\frac{g\bullet \beta \bullet \mathrm{\Delta }T\bullet L^3}{\nu \bullet \alpha }$$E18

where α is thermal diffusivity, β is thermal expansion coefficient, and ΔT is temperature drop across the convection layer.

The thermal convection flow phenomena will also be represented under the SMDM condition in the case that $\mathrm{\Delta }T$ was controlled to make the Rayleigh numbers equal expressed in Eq. (18).

Eq. (18) can be rearranged to Eq. (19) by substituting $L=\lambda \bullet L_0$, Eq. (7) that expresses the scale ratio of the SMDN condition and kinematic viscosity $\nu =\eta /\rho$. Then the$\mathrm{\Delta }T$ ratio to satisfy the Rayleigh number similitude under the SMDN condition is obtained, as shown in Eq. (20) [37]. It should be noted that if the $\mathrm{\Delta }T$ ratio calculated by Eq. (20) was large, thermal effects on the physical properties of the experimental liquid could not be negligible. If temperature dependencies of the physical properties of two liquids for the prototype and the physical model are similar, a large $\mathrm{\Delta }{T}_0$ itself will not induce a lower accuracy in the experimental results.

$$\mathrm{Ra}=\frac{g\bullet {\beta }_0\bullet {\mathrm{\Delta }T}_0\bullet {L_0}^3}{{\nu }_0\bullet {\alpha }_0}=\frac{g\bullet \beta \bullet \mathrm{\Delta }T\bullet {\left(\lambda \bullet L_0\right)}^3}{\nu \bullet \alpha }=\frac{g\bullet \beta \bullet \mathrm{\Delta }T\bullet {\left(\raisebox{1ex}{$\nu $}\!\left/ \!\raisebox{-1ex}{${\nu }_0$}\right.\right)}^2\bullet {L_0}^3}{\nu \bullet \alpha }$$E19

$$\frac{\mathrm{\Delta }T}{{\mathrm{\Delta }T}_0}=\frac{\alpha \bullet {\beta }_0\bullet {\nu }_0}{{\alpha }_0\bullet \beta \bullet \nu }$$E20

Naturally, the SMDN concept of physical modelling is applicable to represent the turbulent flow. Minutes of turbulent flow, such as small eddies, vortexes, and swirls, will be precisely represented since such flow phenomena are governed by inertial, gravitational, and viscous forces. It should be noted that the thermal effects on the physical properties should be considered under heavy turbulent flows with the rising temperature of the experimental fluid. Besides, the wettability between the fluid and the wall of the flow channel will affect the result.

5. Conclusion

The novel physical modelling concept, involving the simultaneous similitude of multiple dimensionless numbers, the Froude, Reynolds, Weber, Galilei, capillary, Eötvös, and Morton numbers, have been proposed in this chapter indicating a simple relationship among density, viscosity, surface tension of fluids, and the scale ratio of physical model. This concept has been named SMDN (similitude of multiple dimensionless numbers). Subsequently, it was shown that some combinations of molten metals, such as Fe & Sn, Cu & Sn, Al & Ga, and Zn & Sn-Bi, could satisfy the strict relationship of physical properties required for the SMDN condition. As a matter of record, no one except for us has made the specific proposal of the multiple dimensionless numbers similitudes in the physical modelling of flow phenomena.

Physical modelling satisfying the SMDN condition can represent many kinds of flow phenomena influenced by inertial, gravitational, viscous, and surface tension forces as some examples of applications were shown in this chapter. This novel concept of physical modelling will enable precise, safe, and easy physical model experiments using low-temperature fluids to exactly represent flow phenomena in various high-temperature liquid operations within the pyro-metallurgical industries.

Nomenclature

V	Velocity of fluid,
g	Gravitational acceleration
L	Characteristic length
ρ	Density
η	Viscosity
σ	Surface tension
ν	Kinematic viscosity
i	σ/ρ
λ	Scale ratio of physical model
D	Diameter
α	Thermal diffusivity
β	Thermal expansion coefficient
ΔT	Temperature drop across convection layer
0	Subscript for prototype
b	Subscript for bubble
n	Subscript for nozzle