Paper Melissari Elsevier

8/7/2019 Paper Melissari Elsevier

1/29

Development of a Heat Transfer

Dimensionless Correlation for a Wide Range

of Prandtl Numbers

Blas Melissari Cassanello

Stavros A. Argyropoulos

Department of Materials Science and Engineering, University of Toronto

Abstract

In this paper a computational approach is employed to derive a dimensionless heat

transfer correlation for forced convection in spheres. This correlation is applicableto a wide range of uid systems. The lower end of this range includes the Prandtl

number for liquid Sodium (P r 0:003), whereas the upper end includes the Prandtl

number for water (P r 10).

N u = 2 + 0:47Re1=2P r0:36

3 103 P r 101; 102 Re 5 104

The model predictions derived from this research were validated extensively. First,

the model was tested in l iquid Aluminum, and subsequently the model was comparedwith existing experimental data involving water. Both verication procedures have

shown very good agreement between experimental results and model predictions.

Key words: liquid metals, heat transfer coecient, modeling, experimental, forcedconvection, natural convection

NomenclatureA Area, m2

c Heat capacity, J/(kg.oC)C1; C2;::: Correlation coecientsCV Control VolumeD Diameter, mg Gravity, m/s2

Gr Grashof number, g::2:SPH:D3=2

G Buoyancy parameter, Gr=Re2

I Performance Indexk Thermal conductivity, W/(m.oC)LF Liquid Fraction, m/m

Preprint submitted to Elsevier Science 6 Septemb er 2004


2/29

LH Latent Heat, J/kgMT Melting time, sn1; n2; ::. Correlation exponentsNu Nusselt number, h:D=kPr Prandtl number, :c=kR Resistance, (m2.oC)/W

Re Reynolds number, :u:D=S Source termSPH Superheat, (T1 Tm) in oCT Temperature, oCt time, st time step, sU Uncertaintyu Velocity, m/sV Volume, m3

x Control volume size, m

Greek symbols Thermal diusivity, m2=s

Thermal expansion coe., 1=oC Dynamic viscosity, kg=m:s Density, kg=m3

Standard deviation

Subscripts0 Initial condition1 Liquid condition far from spherem Melting point (liquidus)S Solid condition (solidus)L Wall conditionFC Forced ConvectionINT Interface

i Indexcorr Correlationmodel Modeling Predictionexper Experimental Result

1 Introduction

There is a derth of dimensionless convective heat transfer correlations appli-cable to uids such as liquid metals. Knowledge of heat transfer rates from

particles at high ux levels has become increasingly important to the designof energy transfer systems and metallurgical processes in general.

Theoretical as well as experimental approaches have been carried out to ana-lyze the heat transfer in liquid metals. Hsu [1] and Sideman [2] have derivedequations for heat transfer from a sphere to a liquid metal by assuming po-tential ow around the sphere. Kreith et al [3] performed an experimentalinvestigation of rotating metallic spheres in liquid mercury and suggested a

2


3/29

correlation for forced convection. Witte [4] performed experiments on heattransfer from a non-melting sphere to liquid sodium and obtained an equationrelating the Nusselt number to Reynolds and Prandtl numbers. Argyropoulosand Mikrovas [5] [6] immersed spheres in liquid Aluminum and Steel and foundcorrelations for forced and natural convection based on the measurement of

the melting times of the spheres.

Many investigators studied the heat transfer characteristics of solid spheresto uids with Prandtl number around unity (PrAir h 0:7;P rWater h 10).McAdams [7] compiled numerous experimental results and correlated all ofthem into a single empirical correlation valid for Air and Water. Yuge per-formed pioneering experimental work on the heat transfer from a sphere toair under mixed convection [8], suggesting procedures for predicting the Nus-selt number. Vliet and Leppert [9] developed correlations for spheres in water.Hieber et al [10] studied the spherical system analytically, but their study waslimited to small Reynolds numbers.

Some researchers studied the melting dynamics of ice spheres in water at dif-ferent convective regimes. Vanier and Tien [11] performed experiments on themelting of a submerged ice sphere in water, calculating the melting rate byweighing the sphere. Solomon [12] obtained a solution for the melting of asphere in convection as a function of the average diameter and heat ux. Es-kandari [13] reported on a series of experiments to study the forced convectionheat transfer from a owing stream of water to an ice sphere. Anselmo [14][15] undertook an extensive theoretical and numerical analysis of melting offull and partially submerged ice spheres in a pool of water. Aziz et al [16] and

Hao et al [17] performed measurements of the heat transfer coecient in thewater system by measuring the melting time of ice spheres in forced convec-tion. Mukherjee et al [18], McLeod et al [19] and Hao et al [20] conductedvisualization studies of ice spheres melting in water under mixed convectionregime.

In terms of mixed convection around a sphere in liquid metals, the work of Kre-ith and his associates is worth mentioning [21] [3]. By performing experimentalmeasurements of rotating spheres in media as diverse as Air and Mercury, theyconcluded that if the buoyancy parameter (G= Gr=Re2) is less than 0:3, thennatural convection is negligible, i.e.: its eect is lower than 5% as far as the

heat transfer is concerned. This value agrees with the theoretical derivationby Sparrow et al [22]. Numerical model predictions regarding the inuence ofnatural convection on the total melting time of spheres has been publishedby Melissari and Argyropoulos in [23]. They concluded that for values of thebuoyancy parameter lower than the range G = 0:5 1:0, the total meltingtime is not aected by natural convection eects.

Table 1 summarizes the correlations for heat transfer around spheres and their

3


4/29

Table 1Nusselt number for Forced convection around spheres

Authors Applicability Nusselt number correlated

Hsu [1] P r < 1; Re 2 105 0:921(Re:Pr)1=2

Sideman [2] P r < 1; Re 2 105 1:13(Re:Pr)1

=2

Kreith [3] P r = 102; 7 104 Re 106 0:178Re0:375

Witte [4] P r = 103; 3 104 Re 2 105 2 + 0 :386(Re:P r)1=2

Argyrop. [5] 102 P r 101; Re 3 104 2 + 1:114Re0:557P r0:914

Whitaker [24] P r = 0:7; Re 8 104 2 + [0:4Re1=2 + 0:06Re0:67]P r0:4

Vliet and L.[9] P r 10; Re 105 (2:7 + 0:12Re0:66)P r1=2

Aziz et al [16] P r 10; 3 103 Re 3 104 0:991Re0:527P r0:043

Hao et al [17] P r 10; Re 3 103 1:015Re0:48P r0:23

McAdams [7] P r 0:7; Re 2 105 2 + 0:6Re1=2P r

1=3

range of applicability. As seen from this summary, all the correlations corre-spond to either a single or to a narrow range of Prandtl numbers. The presentpaper introduces a methodology which allows the development of dimension-less heat transfer correlations for forced convection around spheres applicablein a wide range of Prandtl number uids.

2 Mathematical Considerations

2.1 Calculating the Nusselt Number From the Melting Time

By applying a heat balance to the sphere, it is possible to relate the totalmelting time to the Nusselt number. It is assumed that the sphere is sub-

ject to uniform melting, meaning that we will calculate the Nusselt numberaveraged over the entire surface as opposed to the localized coecient. Theinstantaneous heat balance is shown in equation 1.

h:A:SPH:dt=:LH:dV (1)

where h=Nu:kL

D

4


5/29

In this context, D represents the diameter as a variable and D0 is the initialdiameter of the sphere. The sensible portion of the heat supplied to the sphereis not included because it is considered that the shell is formed at the expenseof heating the sphere up to its melting point. The volume, surface area andvolume dierential of a sphere are as follows:

V =1

6D3 (2)

A = D2

=)dV =1

2D2dD =

1

2A:dD

The Nusselt number can be expressed in terms of the diameter as Nu =C1Re

1=2 = C2D1=2 for forced convection. At this stage it would be desirable

to express the exponent in a parametric way since it would be valuable tond the parameter from the equation rather than setting it xed at a certainvalue. Hence these relationships become: Nu = C1RenFC = C2DnFC

Substituting Nu and dV in the dierential heat balance (1), and by rearrange-ment we obtain:

h:SPH:dt =1

2:LH:dD (3)

=)C2:kL:SPH:dt =1

2:LH:D(1nFC):dD

The LHS of the equation can be integrated in time, whereas the RHS ofthe equation has to be integrated between the initial diameter (D0) and themaximum diameter reached by the sphere when the shell is formed (Dmax),and then from Dmax to D = 0. This yields the following:

C2:kL:SPHZMT0

dt =1

2:LH

ZDmaxD0

D(1nFC)dD+Z0Dmax

D(1nFC)dD

!

=)C2:kL:SPH:MT=1

2

:LH:(2D(2nFC)max D

(2nFC)0 )

2 nFC

Substituting Nu = C2:DnFC and rearranging terms, we obtain a relationship

between the melting time and the Nusselt number as a function of the initialdiameter D0 as seen in equation 4.

Nu =

2DmaxD0

(2nFC) 1

!:LH:D20

2(2 nFC):kL:SPH:MT(4)

5


6/29

Assuming that the diameter can be obtained from the mass of the sphereusing a relationship of the form m_ D3; we can express the Nusselt numberas equation 5.

Nu=MTF:LH:D2

0

2(2 nFC):kL:SPH:MT(5)

where MTF = 2m1=3(2nFC)F 1 ; with mF =

mmaxm0

The melting time factor, MTF, aects the total time of the immersed sphereand is caused by the shell formation upon immersion. It is calculated basedon the mass increase of the sphere. The mass factor, mF, is calculated as theratio between the maximum mass of the sphere (mmax) and the initial mass(m0). In the subsequent sections these factors will be estimated. For a value

ofnFC =1

2 , the Nusselt number for forced convection is equation 6.

Nu=MTF:LH:D20

3kL:SPH:MT(6)

where MTF = 2m1=2F 1 (7)

2.2 Mathematical Modeling of the Melting Sphere

The problem of a melting sphere is modeled as a three-dimensional system inCartesian co-ordinates of uid ow and heat transfer coupled by the presenceof natural convection. The model was developed using the SIMPLER algo-rithm in (x;y;z) coordinates. A detailed explanation of the method can befound in reference [25].

Continuity equation:@ux

@x+@uy

@y+@uz

@z= 0 (8)

Flow equation in the xdirection:

@ux@t

+ ux@ux@x

+ uy@ux@y

+ uz@ux@z

!=

@p

@x+@

@x@ux@x

+@

@y@ux@y

+@

@z@ux@z(9)

Flow equation in the ydirection:

6


7/29

x

zzy

g

2.D 6.D

4.D

g 2.D

u

oo

u

T

T

Fig. 1. Domain used for the numerical simulations.

@uy

@t+ ux@u

y

@x+ uy @u

y

@y+ uz @u

y

@z

!= @p

@y+ @@x@u

y

@x+ @@y@u

y

@y+ @@z@u

y

@z(10)

Flow equation in the zdirection:

@uz@t

+ ux@uz@x

+ uy@uz@y

+ uz@uz@z

!= (11)

= @p

@z + @@x@uz

@x + @@y@uz

@y + @@z @uz@z + :g::(T T

1)

Energy equation:

:c

@H

@t+ ux

@H

@x+ uy

@H

@y+ uz

@H

@z

!=@

@xk@T

@x+@

@yk@T

@y+@

@zk@T

@z+S (12)

A schematic of the computational domain can be seen in Figure 1.

2.3 Parameters of the Numerical Method

2.3.1 Boundary Conditions

Inlet(x = 0), Bottom(z = 0), Top(z = 4D) and Back(y = 2D) :

7


8/29

ux = u; uy = uz = 0; T= T1

Outlet (x = 6:D): @ux=@x = @uy=@x = @uz=@x = @T=@x = 0

Front (symmetry plane y = 0) : uy = 0;@ux=@y = @uz=@y = @T=@y = 0

2.3.2 Initial Condition

Fluid: Liquid at T= T1

Sphere: Solid at T= T0; with T0 Tm T1:

2.3.3 Domain Size

The domain size used depends on the diameter of the addition. A minimum

size of 6 diameters in the direction of the velocity (x) and 4 diameters in thedirections perpendicular to the velocity (y; z) was found to be large enoughto obtain a stable and accurate solution. The thermo-physical properties ofAluminum and AZ91 used in the model are shown in Tables 2 and 3.

2.3.4 Enmeshment

A structured mesh compatible with the SIMPLER algorithm was employed.Due to the three-dimensional character of the simulation, a cube of side length1:5D is meshed around the sphere with a control volume (CV) size of D=28

while the rest of the domain is meshed using a CV of size D=7. Figure 2 showsa schematic of the non-uniform mesh used for this model. This results in 74CVs in the x-direction, 60 in the y-direction and 30 in the z-direction, totalling1:3 105 CVs.

2.3.5 Time Step

Due to the high thermal gradients during the initial transient, two dierenttime intervals are used: an initial value of t = 104s is used during the rstsecond and t = 103s is used for the rest of the simulation.

2.3.6 Other Parameters

As will be detailed in the experimental section of this paper, the spheres areintroduced into the bath by means of a 6 mm diameter stainless steel tube.For this reason, the holder is introduced in the model. The thermophysicalproperties of the stainless steel used can be found in Table 4. An interfacial

8


9/29

Table 2Thermophysical properties of Aluminum in SI units [27]

kS kL cS cL T m LH

220 92 1100 1000 2400 1:2 103 1:3 104 660 3:95 105

Table 3Thermophysical properties of AZ91 [27]

kS kL cS cL T S Tm LH

60 80 1200 1400 1750 1:4 103 1:2 104 437 600 3:7 105

Table 4Thermophysical properties of Stainless Steel in SI units [28]

kS cS

15 500 8000

heat resistance value of 104

m2

K=W is used around the sphere due to theoxide layer formed [26].

Fig. 2. Meshing around the sphere for the three-dimensional model.

2.3.7 Phase Change Treatment

The Heat Integration Algorithm is used in the Aluminum system due to thefact that the phase change is modeled as isothermal. When a pure materialis melting/solidifying, the heat supplied/extracted is consumed entirely forthe phase change without a change in its temperature. In the model, when a

9


10/29

control volume begins melting/solidifying, its temperature is set at the meltingpoint until the amount of heat supplied/extracted accounts for the latent heatof fusion.

Due to the large freezing range of the AZ91 Magnesium alloy, the numerical

integration algorithm is not suitable. The enthalpy method is implemented viathe introduction of a source term, S; in the energy equation. The expressionfor the source term can be seen in equation 13.

S= :LH

c

@LF

@t+ ~u r LF

!(13)

2.4 Results of the Numerical Model

The melting evolution of a D = 5cm sphere in a SPH = 60oC Aluminumbath and u = 0:2m=s can be seen in Figures 3 through 5. These eld plots arerepresented at the symmetry plane y = 0: It is worth noticing that due to thedierence in thermal conductivity between the holder and the sphere, there isa marked discontinuity on the isotherms.

X0.06 0.08 0.1 0.12 0.14 0.16

-0.04

-0.02

0

0.02

0.04

6

66

6

5

5

5

4

4

4

4

3

3

3

3

2

2

2

11

1

1

8

5

t= 0.1su= 0.2m/s SPH=60.0C D= 5.0cm

02 23 Jul2 004 sph3D

300300

500

500

500

660

660

660

660

680

680

680

680

700

700

700

X

Z

0.06 0.08 0.1 0.12 0.14

0.08

0.1

0.12

u= 0.2m/s SPH=60.0C D= 5.0cm t= 1su= 0.2m/s SPH=60.0C D= 5.0cm

Fig. 3. Numerical model result, 5cm Aluminum sphere melting in a 60oC superheatbath at a velocity of 0.2m/s, t=1s.

10


11/29

X0.06 0.08 0.1 0.12 0.14 0.16

-0.04

-0.02

0

0.02

0.04

6

6

66

5

5

5

4

4

4

4

3

3

3

3

2

2

2

1

11

1

8

5

t= 0.1su= 0.2m/s SPH=60.0C D= 5.0cm

02 23 Jul2 004 sph3D

660

660

660

660

680

680

680

680

700

700

700

700

X

Z

0.06 0.08 0.1 0.12 0.14

0.08

0.1

0.12


Fig. 4. Numerical model result, 5cm Aluminum sphere melting in a 60oC superheatbath at a velocity of 0.2m/s, t=3s.

The total melting times of 3cm and 7cm Aluminum spheres as a function ofthe bath velocity are shown in Figures 6 and 7 respectively. For u = 0cm=s,

the solution represents the pure natural convection solution.

Figure 8 shows the term (MT:SPHAZ91) as a function of the bath superheatfor 3cm and 5cm AZ91 spheres. The importance of this plot is that it canbe used to obtain the melting time for 3cm and 5cm AZ91 spheres in anycondition of bath temperature and velocity.

2.4.1 Melting Time Factor

By running the model for dierent sphere initial temperature, T0; it is possible

to obtain the melting time factor by nding the ratio between the melting timeat a given T0 and at T0 = Tm: Figure 9 shows the melting time factor MTFfor Aluminum spheres under dierent conditions of diameter, superheat andvelocity, for T0 = 20

oC and T0 = 450oC. The values ofMTF do not vary

signicantly with the diameter and superheat, but change from MTF = 1:5in natural convection to MTF = 1:3 for u = 0:4m=s, mainly due to the factthat the melting is non-uniform for the higher values of velocity in forced ow.If the initial temperature is closer to the melting point of the material, the

11


12/29

X0.06 0.08 0.1 0.12 0.14 0.16

-0.04

-0.02

0

0.02

0.04

6

6

66

5

5

5

4

4

4

4

3

3

3

3

2

2

2

1

11

1

8

5

t= 0.1su= 0.2m/s SPH=60.0C D= 5.0cm

02 23 Jul2 004 sph3D

660

660

660

68

0

680

680700

700

X

Z

0.06 0.08 0.1 0.12 0.14

0.08

0.1

0.12


Fig. 5. Numerical model result, 5cm Aluminum sphere melting in a 60oC superheat

bath at a velocity of 0.2m/s, t=10s.

SPH (C)

u(m/s)

u(m/s)

20

20

40

40

60

60

80

80

100

100

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

3cmsphere

5cmsphere

3cmsphere

>7cmsphere

7cmsphere

G1 5 9 13 17

0.7 0.7

0.8 0.8

0.9 0.9

1 1

MTCF

/MTFC

MTAID

/MTFC

0 500 1000 1500 2000 2500 30000.0 0.0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0 1.0

MTFC/MTPCMTAID/MT PCMTCF/MTPC

t (s)

m

/m0

0 5 10 15 20 25 30

0.8

1.0

1.2

1.4

1.6R

INT= 0

RINT

= 3 x10-5

RINT

= 1 x10-4

RINT

= 3 x10-4

Aluminum spheres, D=5cm, SPH=3 0C, NC

RINT

(m2C/W)

10-6

10-5

10-4

10-3

1 1

1.1 1.

1.2 1.

1.3 1.

1.4 1.

1.5 1.

1.6 1.

1.7 1.

MT / MTR=0

mmax

/ m0

Aluminum spheres, D=5cm, SPH=30C, NC

D (m)

minu(m/s)

0.03

0.03

0.04

0.04

0.05

0.05

0.06

0.06

0.07

0.07

0.10 0.10

0.15 0.15

0.20 0.20

0.25 0.25

0.30 0.30

SPH=30C

SPH=90C

Aluminum spheres, G < 0.1

u (m/s)

MT(s)

0

0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0 0

5 5

10 10

15 15

20 20

25 25

30 30

D=3cm; SPH=30C

D=3cm; SPH=90C

Aluminum spheres. T0=20C, R INT=1e-4 m2K/W

Fig. 6. Melting times of 3cm Aluminum spheres

12


13/29

SPH (C)

u(m/s)

u(m/s)

20

20

40

40

60

60

80

80

100

100

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

3cmsph

ere

5cmsphere

3cmsph

ere

>7cmsphere

7cmsphere

G1 5 9 13 17

0.7 0.7

0.8 0.8

0.9 0.9

1 1

MTCF

/MTFC

MTAID

/MTFC

0 500 1000 1500 2000 2500 30000.0 0.0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0 1.0

MTFC/MTPCMTAID/MT PC

MTCF/MTPC

t (s)

m

/m0

0 5 10 15 20 25 30

0.8

1.0

1.2

1.4

1.6R

INT= 0

RINT

= 3 x10-5

RINT = 1 x10-4

RINT

= 3 x10-4

Aluminum spheres, D=5cm, SPH=3 0C, NC

RINT

(m2C/W)

10-6

10-5

10-4

10-3

1 1

1.1 1.

1.2 1.

1.3 1.

1.4 1.

1.5 1.

1.6 1.

1.7 1.

MT / MTR=0

mmax

/ m0

Aluminum spheres, D=5cm, SPH=30C, NC

D (m)

minu(m/s)

0.03

0.03

0.04

0.04

0.05

0.05

0.06

0.06

0.07

0.07

0.10 0.10

0.15 0.15

0.20 0.20

0.25 0.25

0.30 0.30

SPH=30C

SPH=90C

Aluminum spheres, G < 0.1

u (m/s)

MT(s)

0

0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0 0

20 20

40 40

60 60

80 80

100 100

D=7cm; SPH=30C

D=7cm; SPH=90C

Aluminum spheres. T0=20C, R INT=1e-4 m2K/W

Fig. 7. Melting times of 7cm Aluminum spheres

u (m/s)

MT(s)

0 0.05 0.1 0.15 0.2 0.25 0.30

2

4

6

8

10

12

14

16

18

T = 630CT = 660C

T (C)

LF

450 475 500 525 550 575 600.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

.

u (m/s)

MT.S

PHAZ91

0 0.05 0.1 0.15 0.20 0

200 200

400 400

600 600

800 800

1000 1000

1200 1200

1400 1400

3cm model

5cm model

AZ91 Results

(sC)

Fig. 8. Melting times of AZ91 spheres (for all temperatures).

13


14/29

shell increase is much lower and subsequently the melting time factor wouldbe very close to unity. In any case, there is no more than a 20% increase inthe melting time for T0 = 450

oC; with little variation in diameter, superheat,or even velocity.

Fig. 9. Melting time factor, M TF, for dierent operating parameters in the Alu-

minum system (numerical model).

The maximum mass increase observed in the numerical simulations was of the

order of50%;meaning we obtain a mass factormF = 1:5. Using the expressionfor the melting time factor, MTF = 2m

1=2F 1, we observe that if we use the

measured values of mass increase, we obtain a value that is in agreement withthe results of the numerical simulation: MTF = 2(1:5)

1=2 1 = 1:45.

Some specic measurements were carried out where the spheres were extractedprior to the complete melting. In this way, the shell formed around the spherecould be measured and the melting time factor could be estimated. The max-imum mass increase observed experimentally was of the order of 40% to 60%.

It is convenient to minimize the error involved in the estimation of these

factors. A possible way is to perform experiments with heated spheres thus re-ducing the mass increase, as observed numerically in Figure 9. Ideally, a solidsphere at its melting point will not form a shell. However, it is highly imprac-tical to immerse such spheres mainly due to their poor mechanical integrity.Nevertheless, some experiments were carried out with Aluminum spheres pre-heated at T0 = 450

oC simply to determine its experimental feasibility. Thesewere performed using 3cm and 5cm spheres in 60oC superheat bath in thevelocity range 0m=s 0:3m=s.

14


15/29

3 Validation of the Numerical Model

In order to validate the numerical model developed, a series of tests are carriedout to run under the characteristics of a set of established solutions found in

the literature.

Paterson Point Heat Source Analytical Solution. First, a one dimen-sional heat diusion/melting system is developed in spherical co-ordinates tosolve for Patersons point heat source analytical solution [29]. The conditionschosen emulate the very high heat transfer rates found in liquid metals. Thistests the Heat Integration Algorithm as well as the ability of the code to handlea large heat transfer rate under a hypothetical heat source. A more detailedexplanation about this validation can be found in [30].

Experimental Results in Melting Gallium. Second, a two-dimensional

convective/melting system in rectangular coordinates is developed in orderto compare it with the experimental study of the melting of Gallium in anenclosure by Gau and Viskanta [31]. This tests the Heat Integration Algorithmas well as the natural convection induced ow in a liquid metal (completevalidation results can be found in [30]).

Experimental Results in Melting Ice. Third, the three-dimensional modeldeveloped for liquid metals is set to run using the thermophysical properties ofwater to simulate the melting of ice spheres in liquid water. The ow patternsaround the spheres obtained by the model are compared with images fromvisualization studies in water performed by Hao et al in [20]. The melting

times in forced convection obtained experimentally in the ice/water system byAziz et al [16] and Hao et al [17] are compared with results of our numericalmodel.

Finally, the model will be compared with experimental results in Aluminumand AZ91 carried out in the present work.

3.1 Melting of Ice Spheres in Water

Aziz et al [16] and Hao et al [17] performed numerous investigations in thewater/ice system. A series of experiments were conducted to measure forcedconvection heat transfer rate from submerged ice spheres in water.

The three-dimensional model was used to obtain the melting times of icespheres under pure forced convection. The thermophysical properties of waterused are shown in Table 5. The meshing and time step scheme used is similar

15


16/29

Table 5Thermophysical properties of water in SI units [28]

kS kL cS cL T m LH

2:2 0:6 2100 4200 1000 1:2 103 4:0 104 0 3:4 105

to the one used for the Aluminum system.

There was no mass increase observed in any of the spheres, due to the factthat their initial temperature was only 10oC below zero (melting point). Theexpression to calculate the Nusselt number is equation 14 (a particular caseof equation 6).

=) Nu =:LH:D20

3kL:SPH:MT(14)

A series of runs were performed using a 3:6cm ice sphere initially at 10oCimmersed in a water bath with superheats ranging between 10oC and 30oCand velocities between 0:01m=s and 0:1m=s. The calculated dimensionless heattransfer coecient was obtained from equation 14 and was plotted along withthe experimental results by Aziz et al (Figure 10) and Hao et al (Figure 11).

The thin solid line in Figure 11 represents the results obtained by Aziz et al in1995 and were included in the original paper by Hao et al for comparison [17].The deviation in the Nusselt number reported on both investigations is of theorder of 20%: Good agreement is observed between the numerical model andthe experimental results from both experimental investigations.

The ice melting system is implemented using the three-dimensional model tostudy the mixed convection regime ow around the sphere. Flow visualizationimages of the work by Hao et al in 2001 [20] are compared with ow plotsof our numerical model at dierent convective conditions. Figures 12 and 13

show the ow around a 3:6cm in diameter ice sphere immersed in water at30oCand an imposed velocity of0:01m=s as a result of the visualization studyand our numerical model, respectively. It is worth noticing the strong eectof the natural convection induced ow (G = 50). The ow pattern in mixedconvection predicted by the model is almost identical to the one observedexperimentally.

16


17/29

Fig. 10. Comparison between the numerical model and the experimental results by

Aziz et al [16].

4 Experimental Measurements In Liquid Metals

4.1 Experimental Work involving the Revolving Liquid Metal Tank

The apparatus used to immerse spheres consists of a cylindrical Stainless Steeltank that can rotate inside a heavily insulated electrical resistance furnace.The Revolving Liquid Metal Tank (RLMT) is connected to a DC motor ca-pable of controlling the rotating speed to a resolution of1 RPM. The interiordiameter of the RLMT is 380 mm, the height is 200 mm and it has a usablecapacity of 20 lit (50 kg of Aluminum, approximately). Figure 14 shows a

schematic of the RLMT inside the electrical resistances furnace. Figures 15and 16 show the RLMT in the furnace and the complete setup, respectively.

A 14HP Bodine Electric Company DC motor with velocity control is assem-

bled and connected to the RLMT shaft via a belt and a wheel. To adapt todierent velocity ranges, the rotation ratio could be modied by changing thebelt/wheel assembly. The centre of the perforation on the lid is located at aradius of approximately 12cm and one side of the lid is secured to the struc-

17


18/29

Fig. 11. Comparison between the numerical model and the experimental results by

Hao et al [17].

Fig. 12. Ice melting in water, D = 3:6cm; u = 0:01m=s [20](Re = 2 102; Gr = 2 106; G = 50)

18


19/29

X0.04 0.05 0.06 0.07 0.08 0.09 0.1

-0.02

-0.01

0

0.01

0.02

0.03

2

2

2

2

2

1

1

1

1

1

8

6

4

2

0

-2

-4

-6

-8

-1

-1-1

-1

-1

Ice at -20C in wateru= 0.01 m/s SPH= 3C D= 3.6cm t= 1 .0s

02 25 Nov 2003 sph3D

-15

-15

-15

-15

-10

-10

-10

-100

0

0

0

0

15

15

15

15

15

25

25

25

25

25

25

X

Z

0.05 0.06 0.07 0.08 0.09 0.1

0.05

0.06

0.07

0.08

0.09

t= 3.0s0.05m/sIce at -20C in water

u= 0.01m/s SPH=30C D= 3.6cm

Fig. 13. Ice melting in water. Numerical model result.

(Re = 2 102; Gr = 2 106; G = 50)

ture to avoid any shifts. Before beginning a set of experiments, the radius ofrotation is measured (this radius is denoted as rrot). The rotation speed ofthe tank is measured (in revolutions per minute, RPM) and the velocity u iscalculated using equation 15.

u =2

60 rrot RPM (15)

For the experiments carried out in AZ91, a protective atmosphere is used,consisting of a mixture of CO2 and 0.5% SF6. A 1=8 " Stainless Steel tubewas connected through the lid at approximately the centre of the RLMTand connected to the protective gas cylinder. The regular ow rate for theexperiments was 2 lit/min, although the system was calibrated to a maximumof 10 lit/min of gas to be used in an emergency situation. The gas inlet can

be seen in Figure 14.

The melting time of the immersed spheres is measured by means of the changein electrical resistance between the tip of a wire inside the sphere and the bath.Figure 17 shows a schematic of the sphere/holder assembly. A 5=32 " hole isdrilled to the centre and threaded up to 1

4" from the centre of the sphere.

The insulated wire runs inside the holder, leaving the tip in the centre. A 5VDC is supplied to the internal wire and the bath is grounded by the holder

19


20/29

metal bath

insulation

sphere

rotating shaft

180

200

90

roll bearing

lid

25

120

190

220

400

370

250

50

70 insulation

Notes:

- Not to scale

- All distances in mm

CL

gas inlet

Fig. 14. Schematic of the apparatus used to immerse spheres

when the sphere is immersed. The voltage potential between the wire and thebath is measured; when the sphere melts, the voltage drops to the short circuitvoltage (0V). The immersion of the sphere is also recorded in a similar way, bymeasuring the potential dierence between the sphere holder and the groundedbath. Figure 18 shows a schematic of the electrical circuit used to measure themelting time of the immersed sphere, as well as the voltage evolution duringan immersion.

The Aluminum spheres were manufactured by Brampton Foundries using sandcast commercially pure Aluminum; the AZ91 spheres were machined. Therewas no visibly observed porosity in the spheres cut. The 1

4" Stainless Steel

holder is painted with a layer of Boron Nitride to prevent dissolution, as wellas to prevent the Aluminum from adhering to the holder, particularly to thethreaded portion. The spheres and holder assembly can be seen in Figures 19and 20. The RLMT is coated with a layer of refractory paint (901 Alumina

20


21/29

Fig. 15. Photograph of RLMT inside the furnace.

Fig. 16. Photograph of the complete apparatus.

ceramic from Centronics) to avoid exposure of the Stainless Steel to the liquidmetal.

The output voltages and the bath temperature are recorded using a National

21


22/29

Sphere

holder s.s.

1/4" diam.

internal

insulated

wire

Fig. 17. Schematic of the sphere/holder assembly.

Vh Vs

immersion

(bath grounded)

to internal

wire

Power supply 5V

R

sphere

immersed

sphere

melted

0

5

time

R

VsVh

Fig. 18. Schematic of the circuit used to measure the melting time of the spheres.

Instruments 6034E data acquisition PCI board via an 8 channel isolated inputmodule with internal temperature compensation. The bath thermocouple is

type K and the sampling rate is set to 5 measurements per second. A Labviewcode is implemented to acquire the data from the PCI board. Figure 21 showsa typical result for a 5cm Aluminum sphere in a bath with a superheat of 60oCand a velocity ofu1 = 0:33m=s. The holder voltage Vh drops instantaneouslyas the sphere is immersed (A). Due to humidity and oil residues inside thecavity and the holder assembly, there is a drop in the spheres voltage Vs beforethe end of the melting (point B). However, when the liquid front reaches thecentre of the sphere, a sudden drop occurs (point C, at t = 15s).

22


23/29

Fig. 19. Photograph of 5cm and 7cm Aluminum spheres.

Fig. 20. Photograph of 3cm Aluminum sphere and holder detail.

4.2 Experimental Results in Liquid Metals

Using the relationship between the Nusselt number and the melting time of asphere, all the experimental results are grouped in a single graph. The resultsfor the preheated Aluminum spheres will also be included in the graph (themelting time factor used is MTF = 1:1; as obtained in the numerical model).Also, the results for AZ91 will be plotted in the same graph; the melting timefactor used is MTF = 1:5 because AZ91 and Aluminum show very similarshell formations due to the fact that their thermophysical properties are very

23


24/29

t(s)

Voltage

(V)

0 5 10 15 20 25 30 35 40 45

0

1

2

3

4

Vh

V0

V+

V-

B

C

D

A

t (hr)

T(C)

0 2 4 6640 640

660 660

680 680

700 700

Aluminum

t (s)

Voltage

T

0 5 10 15

0

1

2

3

4

5

700

705

710

715

720

725

TVhVs

A

B

C

Al 5cm, 0.33m/s

Fig. 21. Typical data acquired for the one-probe sphere.

similar. Moreover, the Prandtl numbers of both systems are also of the sameorder of magnitude (PrAl = 0:015 and P rAZ91 = 0:024), hence the Nusseltnumber expected should also be similar for the same convective condition(Nu = f(Re;Gr;Pr)).

All experiments carried out under forced convective conditions are groupedtogether in Figure 22. This Figure includes experimental results for 3cm, 5cm

and 7cm Aluminum spheres. In these experiments, the spheres were at roomtemperature prior to immersion. Experimental results in which the Aluminumspheres where preheated at T0 = 450

oC are also shown in Figure 22. Finally,experimental results for the AZ91 Magnesium alloy are also depicted in thesame Figure.

5 Predicting the Nusselt Number for Other Material Systems

The model has been validated with experimental results in Aluminum andAZ91 (Pr 102) as well as with experiments in the water/ice system (Pr 101) obtained from the literature. The Nusselt number can then be predictedfor various uids having dierent Prandtl numbers. This procedure was carriedout by running the numerical model for several conditions corresponding todierent Prandtl number uids. With the Nusselt number obtained for valuesof the Reynolds and Prandtl numbers, a correlation of the form Nu = 2 +C:RenReP rnPr is sought. The range of values of the dimensionless parameters

24


25/29

5

10 D=3cm

D=5cm

D=7cm

correlation

Natural Convection; Numerical Model, Al spheres; T0=20C; MT

F=1.5

+2

-2

5

0

5 D=3cm

D=5cm

D=7cmAZ91

corr N C

Natural Convection, Experimental Results.

Nu-2

103

104

100

100

101

101

102

102

Pr=3x10-3

Pr=1x10-2

Pr=1x10-1

Pr=1x101

Forced convection; Numerical Model

G

(Nuexper-

NucorrFC

)/Nuexper,%

10-2

10-1

100

101-60

-40

-20

0

20

40

Mixed Convection. Experimental results

+2

-2

u

5

10

15

20

25

30

35

40

D=3cmD=5cm

D=7cm

correlation

Forced Convection; Numerical Model, Al spheres; T0=20C; MT

F=1.5

u-

10-2

10-1

100

101

100

100

101

101

102

102

Re=3x102

Re=3x103

Re=3x104


Re

Nu

103 104 105

10

15

20

25

30

Al, D=3cm

Al, D=5cm

Al, D=7cmAl, T

0= 450C

AZ91

Numerical Prediction

Forced Convection. Experimental results

Fig. 22. Numerical model predictions and experimental results in Aluminum andAZ91 in forced convection (P r 102).

studied are 3 103 Pr 101 and 3 102 Re 3 104.

The sphere is set to be solid at the same thermal properties of the liquidand at an initial temperature equal to its melting point (T0 = Tm) in orderto avoid the shell formation and the subsequent error in estimating the massincrease and the melting time factor, MTF. For each condition, the meltingtime is obtained and the Nusselt number is calculated using equation 6 withMTF = 1; as done for the water/ice system.

Figure 23 shows the Nusselt number as a function of the Prandtl number forthe values of the Reynolds number studied. The slopes of the curves are almostidentical among each other in the log-log plot, meaning that the exponent of

the Prandtl number (nPr) is independent of the Reynolds number. Figure 24shows the relationship between the Nusselt and the Reynolds number for fourof the total of eight values of the Prandtl number studied. The exponent ofthe Reynolds number (nRe) appears to be independent of the Prandtl number.

By performing a regression analysis of the data from the results of the model,an expression for the dimensionless heat transfer coecient is obtained in therange 3 103 P r 101 and 3 102 Re 5 104 for the 24 points

25


26/29

Re

Nu-2

103

103

104

104

100

10

101

10

102 10

Pr=3x10-3

Pr=1x10-2

Pr=1x10-1

Pr=1x101


+2

-2

Gr10

610

710

8

5

10

15 D=3cm

D=5cm

D=7cm

AZ91corr NC


Gr

Nu

107

108

5

10 D=3cm

D=5cm

D=7cmcorrelation

Natural Convection; Numerical Model, Al spheres; T0=20C;MTF=1.5

G

(Nuexper-

NucorrFC)/Nuexper,%

10-2

10-1

100

101-60

-40

-20

0

20

40

Mixed Convection. E xperimental results

+2

-2

Re

Nu

103

104

105

5

10

15

20

25

30

35

40

D=3cm

D=5cmD=7cm

correlation


F=1.5

Re

Nu

103

104

10

15

20

25

Al, D=3cm

Al, D=5cm

Al, D=7cm

Al, T0 = 450C

AZ91

corr FC n =

Model

Forced Convection. Experimental results

Pr

Nu-2

10-2

10-2

10-1

10-1

100

100

101

101

100

100

101

101

102

102

Re=3x102

Re=3x103

Re=3x104


Fig. 23. Nusselt number as a function of the Prandtl number for forced c onvectionon spheres.

considered. The problem is as follows: nd the constants fC; nRe; nPrg 0such that I is minimized. The performance index I is given by equation 16.The resultant correlation is equation 17, with a standard deviation corr = 4%:

I =NXi=1

(Numodel;i (2 +C:RenRei P r

nPri ))

2 (16)

Nu = 2 + 0:47Re0:5Pr0:36 (17)

Equation 17 is used to predict the dependence of the Nusselt number on the

Reynolds number for the experimental results shown in Figure 22. In thisFigure, the straight solid line shows predictions based on equation 17. Asseen, there is a good agreement between the various experimental results andpredictions based on equation 17. The observed deviation on the lower rangeof Reynolds numbers is partly attributed to buoyancy eects.

Uncertainty analysis on the Nusselt number estimation from equation 17 hasshown that the error involved is of the order of 20%.

26


27/29

G

(Nuexper-

NucorrFC)/Nuexper,%

10-2

10-1

100

101-60 -

-40 -

-20 -

0 0

20 2

40 4

Mixed Convection. Experimental results

+2

-2

Re

Nu

103

104

105

5

10

15

20

25

30

35

40

D=3cm

D=5cm

D=7cmcorrelation


F=1.5

Pr

u-

10-2

10-2

10-1

10-1

100

100

101

101

100

100

101

101

102

102

Re=3x102

Re=3x103

Re=3x104


Re

Nu

103

104

10

15

20

25

Al, D=3cm

Al, D=5cm

Al, D=7cm

Al, T0

= 450C

AZ91

corr FC n =

Model

orce onvect on. xper menta resu ts

Gr10

710

8

5

10 D=3cm

D=5cm

D=7cmcorrelation

Natural Convection; Numerical Model, Al spheres; T0=20C;MTF=1.5

+2

-2

Gr10

610

710

8

5

0

5 D=3cm

D=5cm

D=7cm

AZ91corr NC


Re

Nu-2

103

103

104

104

100

100

101

101

102

102

Pr=3x10-3

Pr=1x10-2

Pr=1x10-1

Pr=1x10

1


Fig. 24. Nusselt number as a function of the Reynolds number for forced convectionon spheres.

6 Conclusions

A mathematical model was developed to describe the various transport phe-nomena involved when a melting sphere is immersed in a moving uid. Thismodel was validated with various experimental results involving liquid metalsand water.

Based on this model, a dimensionless correlation for convective heat transferto spheres was developed.

Nu = 2 + 0:47Re1=2Pr0:36 (18)

3 103 Pr 101; 102 Re 5 104

This correlation has applicability in uids with wide range of Prandtl numbers,and it was compared with experimental results derived in liquid Aluminumand Water. The comparisons have shown good agreement between predictionsfrom the derived correlation and experimental results.

27


28/29

References

[1] C. Hsu, Heat transfer to liquid metals owing past spheres and elliptical rod

bundles, International Journal of Heat and Mass Transfer, Vol 8 (1965) 303315.

[2] S. Sideman, The equivalence of the penetration theory and potential owtheories, Industrial and Engineering Chemistry Research, Vol 58 (2) (1966)

5458.

[3] F. Kreith, L. Roberts, J. Sullivan, S. Sinha, Convection heat transfer and

ow phenomena of rotating spheres, International Journal of Heat and MassTransfer, Vol 6 (1963) 881895.

[4] L. Witte, An experimental investigation of forced convection heat transfer from

a sphere to liquid sodium, ASME Journal of Heat Transfer, Vol 90 (1968) 912.

[5] S. Argyropoulos, A. Mikrovas, An experimental investigation on natural and

forced convection in liquid metals, International Journal of Heat and MassTransfer, Vol 39 (1995) 547561.

[6] S. Argyropoulos, A. Mikrovas, D.A.Doutre, Dimensionless correlations for

forced convection in liquid metals: Part i. single-phase ow, Metallurgical andMaterials Transaction, Vol 32B (2001) 239246.

[7] McAdams, Heat Transmission, 3rd edition, McGraw-Hill, 1954.

[8] T. Yuge, Experiments on heat transfer from spheres including combined natural

and forced convection, ASME Journal of Heat Transfer, Vol 82 (1960) 214220.

[9] G. Vliet, G. Leppert, Forced convection heat transfer from an isothermal sphere

to water, ASME Journal of Heat Transfer, Vol 83 (2) (1961) 163175.

[10] C. Hieber, B. Gebhart, Mixed convection from a sphere at small reynolds andgrashof numbers, Journal of Fluid Mechanics, Vol 38 (1969) 137159.

[11] C. Vanier, C. Tien, Free convection melting of ice spheres, AIChE Journal, Vol

16 (1970) 7682.

[12] A. Solomon, On the melting time of a simple body with a convection boundarycondition, Letters in Heat and Mass Transfer, Vol 7 (1980) 183188.

[13] V. Eskandari, G. Jakubowski, T. Keith, Heat transfer from spherical ice in

owing water, ASME 82-HT-58.

[14] A. Anselmo, V. Prasad, J. Koziol, Melting of a sphere when dropped in a pool

of melt with applications to partially-immersed silicon pellets, Heat Transfer inMetals and Containerless Processing and Manufacturing, ASME HTD, Vol 162

(1991) 7582.

[15] A. Anselmo, V. Prasad, J. Koziol, K. Gupta, Numerical and experimental studyof a solid-pellet feed continuous czochralski growth process for silicon single

crystals, Journal of Crystal Growth, Vol 131 (1993) 247264.

28


29/29

[16] S. Aziz, W. Janna, G. Jakubowski, A comparison of correlations for forcedconvection heat transfer from a submerged melting ice sphere to owing water,

ASME Heat Transfer Division, Vol 334-3 (1995) 329334.

[17] Y. Hao, Y. Tao, Heat transfer characteristics in convective melting of a solid

particle in a uid, ASME Heat Transfer Division, Vol 364-2 (1999) 207212.

[18] M. Mukherjee, J. Shih, V. Prasad, A visualizations study of melting of an icesphere in a pool of water, ASME 94-WA/HT-14.

[19] P. McLeod, D. Riley, R. Sparks, Melting of a sphere in hot uid, Journal ofFluid Mechanics, Vol 327 (1996) 393409.

[20] Y. Hao, Y. Tao, Melting of a solid sphere under forced and mixed convection:Flow characteristics, Journal of Heat Transfer, Vol 123, No 5 (2001) 937950.

[21] R. Nordlie, F. Kreith, Convection heat transfer from a rotating sphere,

International Developments in Heat Transfer, ASME (1961) 461467.

[22] E. Sparrow, R. Eichhorn, J. Gregg, Combined forced and free convection in aboundary layer, Physics of Fluids, Vol 2, No 3 (1959) 319328.

[23] B. Melissari, S. Argyropoulos, The identication of transition convective regimes

in liquid metals using a computational approach, Progress in ComputationalFluid Dynamics, an International Journal, Vol 4, No 2, Inderscience Publishers

(2004) 6977.

[24] S. Whitaker, Forced convection correlations, AIChE Journal, Vol 18 (2) (1972)

361371.

[25] S. Patankar, Numerical Heat transfer and Fluid Flow, McGraw-Hill, 1980.

[26] S. Argyropoulos, N. Goudie, M. Trovant, The estimation of thermal resistance

at various interfaces, Minerals, Metals and Materials Society/AIME, Fluid Flow

Phenomena in Metals Processing (USA) (1999) 535542.

[27] E. Turkdogan, P hysical Chemistry of High Temperature Technology, AcademicPress, 1980.

[28] P. R. H. C. Touloukian, Y.S., P. Klemens, Thermophysical Properties of Matter,The TPRC Data Series, Vol 1, IFI/Plenum Press., 1970.

[29] S. Paterson, Propagation of a boundary of fusion, Proceedings GlasgowMathematical Association 1 (1952) 4247.

[30] B. Melissari, The Sphere Melting Technique: Mathematical Modelling,

Experimental Measurements and Applications, Ph.D. Thesis, University of

Toronto, 2004.

[31] C. Gau, R. Viskanta, Melting and solidication of a pure metal on a verticalwall, ASME Journal of Heat Transfer, Vol 108 (1986) 174181.

Date post:	08-Apr-2018
Category:	Documents
Upload:	blas-melissari
View:	218 times
Download:	1 times

Paper Melissari Elsevier

Documents