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

Written By

Yuichi Tsukaguchi, Kodai Fujita, Hideki Murakami and Roderick I.L. Guthrie

Submitted: December 9th, 2021 Reviewed: January 13th, 2022 Published: March 6th, 2022

DOI: 10.5772/intechopen.102655

IntechOpen
Casting Processes Edited by Thoguluva Vijayaram

From the Edited Volume

Casting Processes [Working Title]

Prof. Thoguluva Raghavan Vijayaram

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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.

Keywords

  • physical modelling
  • physical model experiment
  • similitude
  • similarity
  • viscosity
  • surface tension
  • density
  • Reynolds number
  • froude number
  • weber number
  • flow phenomena in mould

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.

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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.

Fr=VgLE1
Re=ρVLηE2
We=VρLσE3

where Vis characteristic velocity of fluid, Lis characteristic length, gis gravitational acceleration, ρis density of fluid, ηis viscosity of fluid, and σis surface tension of fluid.

Describing the prototype with subscript 0, flow velocity Vin the physical model is expressed as Eq. (4) in the case of the Fr similitude (Fr=V0gL0=VgL=VgλL0) [8].

V=λV0E4

where λis scale ratio of the physical model.

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

Re=ρ0V0L0η0=ρVLη=ρVλL0ηE5

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

We=V0ρ0L0σ0=VρLσ=VρλL0σE6

Scale ratio λfor the Fr and Re similitudes obtained by substituting Eq. (4) into Eq. (5) and scale ratio λfor 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).

λ=η/ρη0/ρ023=σ/ρσ0/ρ012E7

Eq. (7) can be rearranged to Eq. (8) by applying kinematic viscosityν=η/ρand a new parameter of ι=σ/ρ. ι1/2/ν2/3in 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).

ι1/2ν2/3=ι01/2ν02/3E8

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 bis 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 db/Lin Eq. (10) is same for the prototype and the physical model (dbL=λdb0λL0=db0L0).

Eö=gρρbdb2σE9
Eögρdb2σE10
gρdb2σ=V2ρLσgLV2dbL2=WeFrdbL2E11

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 gand 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.

Mo=gη4ρσ3E12
We6Re4Fr2=V6ρ3L3σ3η4ρ4V4L4gLV2=MoE13

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.

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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/m3]η [Pa·s]σ [N/m2]
×103×10−3
Fe1538 [42]15757.001 [42]4.732 [43]1.844 [43]
Hg−38.9 [44]2013.55 [44]1.568 [44]0.4802 [45]
Sn232 [45]232 ∼ 3326.971 ∼ 6.907 [46]2.302 ∼ 1.713 [46]0.5645 ∼ 0.5527 [46]
Cu1085 [47]1085 ∼ 11857.997 ∼ 7.915 [47]4.019 ∼ 3.417 [47]1.2576 ∼ 1.237 [48]
Zn420 [49]5006.488 [49]3.072 [49]0.7507 [50]
Al660 [51]7002.379 [52]1.295 [51]0.8694 [53]
Ga29.8 [49]706.053 [49]1.407 [49]0.7140 [45]
Bi271 [54]3209.986 [54]1.644 [54]0.3758 [54]
Sn-40Bi174 [54]2257.900 [54]2.219 [54]0.4551 [54]
Water010 ∼ 701.000 ∼ 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 (ι1/2/ν2/3), respectively. This ν2/3versus ι1/2chart 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 ι1/2/ν2/3=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 ν2/3=η/ρ2/3, as shown in Eq. (7). Please note that kinematic-viscosity νof 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].

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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 Db, and thus Dbcan be calculated using Eq. (14) [55], provided that the density of the gas bubble is negligibly small compared with that of the surrounding fluid.

Db=6Dnσρg3E14

where Dnis the internal diameter of the gas injection nozzle.

Table 2 shows the parameters of physical properties (νand 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[m2/s]ι[m3/s2]S-parameter = ι1/2/ν2/3Scale ratio λ = (ν/ν0)2/3
×10−7×10−4
Al7005.4413.6552871.0 (BASE)
Ga702.3251.1802870.57
Fe15756.7592.6352111.0 (BASE)
Cu11454.7161.5682110.77
Sn2882.7768.0452110.55
Zn5004.7341.1571771.0 (BASE)
Sn-40Bi2252.8095.7611770.71

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.

CD=maxmin24Reb1+0.15Reb0.68572Reb83EöEö+4E15

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=λV0). 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 λ=0.5, time progression in the physical model will shrink 0.50.71times 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=LV=λL0V0λ=λt0E16

Figure 6 shows the relationship between equivalent bubble diameter in molten Fe and relative rising velocity (RRV) of bubbles under variable similitude conditions. RRVis (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, U0is bubble rising velocity in the prototype (in molten Fe), Uis bubble rising velocity in the physical model experiment. Please note that the bubble diameter dbfor Uand bubble diameter db0for U0are not same (db=λdb0).

Figure 6.

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

RRV=UλU0λ=UU0λE17

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 (λ=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 (λ=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.

Ra=gβ0ΔT0L03ν0α0=gβΔTL3ναE18

where αis thermal diffusivity, βis thermal expansion coefficient, and ΔTis temperature drop across the convection layer.

The thermal convection flow phenomena will also be represented under the SMDM condition in the case that ΔTwas controlled to make the Rayleigh numbers equal expressed in Eq. (18).

Eq. (18) can be rearranged to Eq. (19) by substituting L=λL0, Eq. (7) that expresses the scale ratio of the SMDN condition and kinematic viscosity ν=η/ρ. Then theΔTratio 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 ΔTratio 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 ΔT0itself will not induce a lower accuracy in the experimental results.

Ra=gβ0ΔT0L03ν0α0=gβΔTλL03να=gβΔTνν02L03ναE19
ΔTΔT0=αβ0ν0α0βνE20

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.

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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.

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Nomenclature

VVelocity of fluid,
gGravitational acceleration
LCharacteristic length
ρDensity
ηViscosity
σSurface tension
νKinematic viscosity
iσ/ρ
λScale ratio of physical model
DDiameter
αThermal diffusivity
βThermal expansion coefficient
ΔTTemperature drop across convection layer
0Subscript for prototype
bSubscript for bubble
nSubscript for nozzle

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Written By

Yuichi Tsukaguchi, Kodai Fujita, Hideki Murakami and Roderick I.L. Guthrie

Submitted: December 9th, 2021 Reviewed: January 13th, 2022 Published: March 6th, 2022