Thermal radiation effects in phase-change energy-storage systems

Energy Vol. 21, No. 12, pp. 1277-1286, 1996

Pergamon

PII: S0~60-5442(96)00034-5

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All fights reserved 0360-5442/96 $15.00+ 0.00

THERMAL RADIATION EFFECTS IN PHASE-CHANGE ENERGYSTORAGE SYSTEMS B. YIMER Department of Mechanical Engineering,Universityof Kansas, 3013 LearnedHall, Lawrence, KA 660452234, U.S.A. (Received 28 August 1995)

Alrstraet--A numerical model was developed to determine the transient temperature distribution, solid/liquid interface location, and energy-storage capacity of a semi-transparent phase-change medium. The medium is bounded between two concentric cylinders and internal energy transfer occurs simultaneously by conduction and thermal radiation. The radiation transport equation was coupled with the energy equation; both enthalpy and temperature were employed as dependent variables. The spherical harmonic approximation (P-N approximation) was used to obtain solutions for the radiative heat flux. The coupled conservation of energy and moment differential equations were solved by using iterative numerical finite-difference schemes with appropriate thermal and radiant boundary and interface conditions. The numerical model was used to study the effects of radiation on solidification (melting), transient temperature distribution and energy-storage capacity of an absorbing, emitting, and isotropically scattering, semi-transparent, gray medium contained in a cylindrical annulus. The results increase our understanding of internal energy transfer and show the effects of optical properties, conduction/radiation parameter, and geometric dimensions and should lead to better designs and optimization of phase-change energy-storage systems. Copyright © 1996 Elsevier Science Ltd.

INTRODUCTION Phase-change thermal energy-storage (TES) systems have received significant attention in recent years. These systems are appealing since large amounts of thermal energy can be stored or released under near-isothermal conditions at reduced operating pressures and at high heat capacities, which will allow a volume reduction of storage units. Energy-transfer problems have been studied theoretically and experimentally for more than a century. Nearly all have dealt with opaque materials and, hence, the contribution of thermal radiation within the media has been ignored. Some investigators 1-6 have proved during the past 15 years that for the range of parameters encountered in the melting and solidification of many optical materials, such as the weakly absorbing semi-transparent and partially diathermous solids, internal radiant transfer has a significant effect and neglecting it leads to considerable error in predicting temperature distribution, interface position and energy flux. Other applications of this nature arise in areas such as melting or freezing of a solid, the growing of large synthetic crystals and vapor films, the burning of solid propellants, the heating and cooling of spacecraft, aircraft windows, nuclear fuel elements, and energy conversion in different solar devices. The analysis of combined energy transfer by conduction and radiation in participating materials is sufficiently complex so that numerical solutions are almost always required. As a result of the nonlinearity of these problems, some advanced and approximate analytical techniques have been used to obtain closed-form solutions for a limited range of geometries and conditions. Although several investigators have dealt with the restricted one-dimensional problem, including combined conductive and radiative transfer, minimal study has been devoted for the two-dimensional case for concentric cylindrical geometry. It is well known that computations involving radiative fluxes with participating media are lengthy. Difficulties arise for the following reasons: Quadruple integrals must be computed with respect to (a) physical distance, (b) optical thickness, (c) solid angle, and (d) wave length in order to obtain the local radiative flux. Even with the total band absorptance introduced, integrals with respect to (b) and (c) remain. Even though melting and solidification of materials by heat transfer has been of importance in many technical fields and a subject of interest for over a century, considerable effort has been devoted to the problem including combined conductive and radiative transfer only over the past 15 years. Abrams and 1277

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B. Yimer

Viskanta ~,2 first investigated the effects of energy transfer by using an explicit finite-difference method for the range of dimensionless parameters governing phase changes encountered in the melting and solidification of semi-transparent crystals. Consideration was limited to the one-dimensional radiative and conductive energy transfer in a region of finite thickness and of infinite lateral extent with physical assumptions, such as the absence of natural convection, the absence of scattering, isotropic media with uniform index of refraction for each phase, diffuse, parallel, and planar interfaces and boundaries. It has been concluded that radiation can significantly affect the dynamics of the melting and solidification of many optical materials, and that a prior neglect of radiation can cause the temperature profile within the liquid to assume a shape which promotes unstable interfacial growth, a finding which is contrary to the idea that radiation always exerts a "stabilizing influence". Habib 3,4 employed the heat-balance integral method (an approximation) to study the effect radiative heat transfer on the solidification rate and on the temperature distribution on the solid phase of a semi-transparent planar and infinite cylindrical medium. The heat balance integral method of approximate analytical solution was developed by Goodman. ~ Habib used the trial temperature profile as the combination of polynomial and logarithmic functions. The constants in the trial function were determined from the boundary and interfacial conditions. The significance of the radiative contribution to the process of phase-change on the solidification rate and on the temperature distribution in the solid phase had been presented with the results for the cases of absorbing, opaque, and nonparticipating media. A more general melting and solidification one-dimensional model was proposed by Chan and Cho 6 which accounted for the existence of a two-phase zone in which partial phase-change can occur. The two-phase zone was attributed to internal melting or solidification (as opposed to surface melting and solidification) included by internal thermal radiation. Diaz and Viskanta 7 developed an analytical model for predicting radiative and thermal conditions during radiation induced phase-change, as well as liquid/solid interface displacement with time. Energy equations are written separately for the two phases and required to meet simultaneous temperature and energy balance considerations at a common boundary, the liquid/solid interface. Experimental simulations were conducted by using a high intensity tungsten filament lamp to melt both horizontal and vertical slabs of a low fusion temperature material 01octadecane). Ratzel and Howell 8 used Pl and P3 approximations to analyze combined conduction and radiation heat transfer in a gray planar medium. Using the pl differential approximation, Hegeneyi and Bayazitoglu9.1° analyzed an axially symmetric radiation field for a gray medium within a finite cylindrical enclosure. In a related work they extended the approximation to a P5 analysis. Menguc and Viskanta 1; formulated a solution to the radiative transfer equation for an axisymmetric cylindrical enclosure using Pl and P3 approximations. Harris investigated the interaction of conduction and radiation in a gray cylindrical medium using p~ and P3 approximations. ANALYTICALAPPROACH In developing the governing equations, a method that uses the enthalpy along with the temperature as dependent variables is employed. In this approach the energy equation is applied once over the complete domain covering both phases. The location of the solid/liquid interface is eliminated from the formulation and is obtained as one of the results of the solution after the temperature distribution is found. The equivalence between the enthalpy model and the conventional form of the energy conservation equations where temperature is the sole dependent variable is shown by Shamsundar and Sparrow. ~3 The development assumes no energy source, no pressure variation and no external work done on the control volume. Convection within the fluid is neglected which implies that density is invariant and eddy effects are neglected. Applying the law of conservation of energy to a control volume, the net rate of energy increase of a control volume is equated to the net rate at which heat is conducted and radiated into the control volume through its surface area. For a ring element that contains a semi-transparent phase-change medium, the describing equation in dimensionless form is

K,6,r(dI-I/d~) = [1 + 8z/2T)IK, (O0/OT)I,+ 8,~2- [1 - ( S r / 2 r ) l K 2 (a0/Oz)l,_ 8,~z + {- [ 1 + (8"rl2"r)lFrl.~ + 8,-,2+ [1 - (8'r/2"r)lF~I,_ s,.,2}K, × (S/N). The dimensionless temperature and enthalpy, for various values of enthalpy, are related by

( 1)

Thermal radiation effects in phase-change energy-storage systems

1279

0 = ( ~ f l K t x , ) H for H < 0 (solid region), 0 = 0 for 0 < H < 1 (during phase change), O= ( c z K s / K a , ) ( H - 1) f o r H > 1 (liquid region).

(2) (3) (4)

The dimensionless variables are defined as follows:

:1"(

H= llpsSv)lll[p(h-h,)/h,l]dv,

0= C , ( T - T f ) / h , l = (K, I c t , p , ) ( T - T f ) l h s l ,

z = r(a + to) = r~, (~= m132t, F, = q,/o,T*, N = K,~/4o'T3,, S = K, T J 4 a , p , h ~ ,

(5)

where H = dimensionless enthalpy, 0 = dimensionless temperature, ~-= dimensionless radial optical distance, ~ = dimensionless time, F~--dimensionless radiative heat flux, N = Stark number (conductionradiation parameter), S = dimensionless parameter. NUMERICAL SOLUTION The describing equations, along with initial and boundary conditions, are first written in finite difference form. Details of the finite difference representation are shown in Ref. 14. For the various range of values of dimensionless enthalpy, the finite-difference representations are A~'-lH~i = Y H ~ i - ' +b~i-'O~i+,+a~i-lOtff_,+([iF~r,i+,+piF~r,i+giF~r,i_,)x(S/N)xKs,

= (/'/7'~/DTd-~) for HT' < 0 ,

(6)

(7)

Yl~i = Yl-l':i - ' + bT/- loft+, + a'['- '87/_ t + (fiiF~,,+, + P~F~,~+ g~F'7,~,~_t) x ( S / N ) x Ks,

(8)

07',-= 0 for 0 -
(9)

A':'- IH': = YH':i - 1 + b':- ~Ot:~+~ + a m- ~ff/_ ~ + A'/' - Y + ~F~:+ ~ + PiF~r.i + giF~r,i_ 1) X ( S / N ) X Ks,

(10)

~=(HT/-

I)/D?,.-~ for/-/~ > I,

[3(1 - A)(8~') 2 + 2 IriS., = (1 + 8~'/2r)d~o.,÷ t + (1 - 8~/2~')~o.,_ t + 1 2 ~ 1 - A ) ( S z ) 2 ~ ,

F~.i = -(1/6~'8~') x (d/o.i-

Illo,i-1),

(If)

(12)

(13)

where A'F-' ={K~(8~')2/8~+ (l/DT/-~) x [(1 + (8~'/2~')K'~1.7') + (1 - (8~'/2~')/~2.7 I)]}, Y = K~(8"r)2/&~DT: - ~ = (KT:- 1otfla'7/- 'K,),P, = -(8~')212~",bT/- ~ = [I + [8~'I27)]K~L7~, a m - ~ = [ I - (8"r/27)]K~2.7 ~,f~ = - (8~'/2)x [I + (8~'/2~')],g~ = (8~'/2)x [I -(8~'/2~')].

The Gauss--Seidel iterative method with successive over-relaxation is used to solve the non-linear simultaneous difference equations. This iterative method requires little computer memory. When the enthalpy,

1280

B. Yimer

temperature, radiative intensity, and radiative heat flux have been calculated, the position of the solid/liquid interface is determined from the enthalpy distribution. For the P-1 approximation, the moment differential equations when combined with Marshak type boundary conditions result in ¢/o - Ei(d~old~') = 4¢r~bwi at rt,

(14)

'{Y0+ E,(d~o/d~') = 4~'~b~i at interface 1,

(15)

where Ei = (2/3) × [(2 - ~ ~i)/~ ~], I = 1,1 . For boundary nodal points, Eqn. (1) must be modified to satisfy the appropriate boundary conditions. For the case where constant heat flux is extracted from an opaque inner surface, Eqn. (1) at the boundary element becomes Ks 8~-(dH/d~) = { K t [1 + (8~'12~')](001a~')}1~ l . a~/2 - Ste x K s - ( S I N ) x Ks[ 1 + (a~'/2~')]F~l. , + a./2,

(16) where Ste = q"/(asp~hst[3) ~= Stefan number. For the case where heat is extracted by convection from an opaque inner surface, Eqn. (1) is written as Ks 8"r(dHl d~) = {K~ [I + ( &r/2r) ]aOl Or}[,, + ~.:/2- B,O,,,K, - B, OoK, - (SIN) x Ks[ I + (8~'12~')]F,1.,+ a~/2.

(17) where B1=hc/~SKs=Biot number, O w = [ C ( T w - T r ) ] / h s t = d i m e n s i o n l e s s Oo = [ Cs( TF - To) ]/hsj = dimensionless secondary fluid temperature.

inner wall temperature,

NUMERICAL RESULTS

Since there are no known exact or approximate analytical solutions to this problem, an overall energy balance was used to assess the accuracy of the numerical solutions. At any time the change in internal energy of the system must be equal to the total energy extracted at the surface up to the corresponding time. For the case where the inner surface is subject to constant heat flux boundary condition, Fig. 1 shows excellent agreement between the internal energy change of the system and the heat extracted. The numerical model was then used to study the effects of internal radiation on the solidification (melting), transient temperature distribution, and energy storage capacity of an absorbing, emitting, and isotropically scattering semi-transparent gray medium contained in a cylindrical annulus. The study was performed for the case where heat at the inner surface is extracted at a constant rate and also by a secondary fluid that was maintained at constant temperature and the outer surface was insulated. The system initially was at the fusion temperature and the effects of internal radiation was taken into account in the solidified region only. The phase-change medium was assumed to be optically thick (a = 4.572 m-l). For the case where constant heat is extracted at the inner surface, Figs. 2 and 3 show the effect of scattering on the transient temperature distribution and solid/liquid interface location respectively. For the same case, the effect of emissivity on the interface location and the temperature distribution is shown in Figs. 4 and 5, respectively. For constant heat extraction, the effect of the Stark number (conduction-radiation parameter) on the rate of solidification is presented in Fig. 6. For the case where heat at the inner surface is extracted by a secondary fluid, the effects of scattering, emissivity, and the Stark number on heat extracted, temperature distribution and solid/liquid interface location are presented in Figs. 7-10. Overall, the results show that the effects of optical properties on the rate of solidification and heat-transfer process at low Stark numbers are significant. It is interesting to note that at high Stark numbers, the heat-transfer process and the rate of solidification are dominated by conduction.

Thermal radiation effects in phase-change energy-storage systems

[

30.0

~ HEAT EXTRACTED -....... INTERNAL ENERGY CHANGE

[ • ~ 24-.0

18.0

~

12.0

O

6.0

0.0

,

0.0

I

,

0.1

I

,

I

,

0.2 0.3 DIMENSIONLESS TIME

I

,

0.4-

0.5

Fig. 1. Dimensionless constant heat extracted vs dimensionless time; r#ro= 0.5, h,, = 400 kJ/kg, E wl = e w2 = 0.6, K, = 7.0 W / m K, a = 4.572/m, ¢0 = 0.0, p = 500 k g / m 3, a , = 0.003 m2/hr, Stefan number = 20.

0.0

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-0.2

DIMENSIONLESS 11ME -

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0.5

--0.4

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:

~

, . ~ . A ~

........

CONOUCTION ONLY

mOD~

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-0.6

f

1.0

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t

O 0 e Z = ~ C I E N T - 0.3

0.6962

I i [ r 1.4 1.6 DIMENSION~SSRADIUS

I 1.8

=

2.0

Fig. 2. Transient temperature distribution vs dimensionless radius for constant heat extraction; r#ro = 0.5, h,, = 400 kJ/kg, a = 4.572/m, ~ wJ = E ,,2 = 0.6, K, = 7.0 W / m K, ¢0 = 0.0, p = 500 k g / m 3, a , ffi 0.003 m2/hr, Stefan number -- 20. EGY 2]:Z2-P

1281

1282

B. Yimer 2.0 1.9 -

1.8

Scattering coefficient = 0.3

........ Conduction only N = 0.6962

to

1.7

;~

1.6

i

1.5 1.4 1.3 1.2 1.1

: 0.10 0.15 0.200.25 0.30 0.35 0.40 0.45 0.50 0.55 Dimensionless time Fig. 3. Solid/liquid interface location vs dimensionless time for constant heat extraction; rdro=0.5, hst=4OOkJIkg, ~s-~ ~ 2 = 0 , 6 , Ks=7.0WImK, a--4.5721m, ¢o=0.0, p = 5 0 0 k g / m 3, a~=m2/hr, Stefan number = 20.

2.0

1.9 ~ ~_ 1.8

Emissivity = 1.0 ........ Emissivity = 0.6 . . . . . Emissivity = 0.2

. . . . .

Conduction only

°° 1.7 -~

-~

..~:

,%t ° .°.'/" • ..'d / ,*'/ ,s

1.6

o,:/

./

~.S,°j

o

1.4

~

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1.2 1.1 1.0

I

I

0.10

i

I

l

I

I

0.20 0.30 Dimensionless time

I

0.40

l

0.50

Fig. 4. Solid/liquid interface location vs dimensionless time for constant heat extraction; rdro=0.5, h~l = 400 kJ/kg, v w~= ¢ w2= 0.6, Ks = 7.0 W/m K, a = 4.572/m, o0 = 0.0, p = 500 kg/m 3, ~ ffi m2/hr, Stefan number = 20.

Thermal radiation effects in phase-change energy-storage systems O.0

/ / . , , s -~ . . j . . . . - ~ /

// /

•

/

/"

/ • / / ~o'/ /

~,/o

--0.2

DIMENSIONLESS TIME

0.3

DIMENSIONLESS llME = 0.5

Ik$';'/.:i/

,77!11"

-o.,l

i,"I

,.,

,.o

"~

~/'/ l

1// --0.5 I

:--0.6

. I

........

.....

~=0.6 I~ilSSlVlIY- 0.2

....

CONDUCTION

I

1.0

,

1.2

I

I

ONLY

I

I

I

I

1.4 1.6 1.8 DIMENSIONLESSRADIUS

2.0

Fig. 5. Transient temperature distribution vs dimensionless radius for constant heat extraction; r t / r o = 0.5, hs~ = 400 kJ/kg, a = 4.572/m, e ~,t = E w2= 0.6, K, = 7.0 W / m K, ca = 0.0, p = 500 k g / m ~, as = 0.003 m2/hr, Stefan n u m b e r = 20.

2.0

1.9 " 1.8

N -- 0 . 6 9 6 2 , K s = 7.0 W / m K ". . . . . . . N = 6.533, K s -- 7 0 . 0 W / m K N = 18.67, K s = 2 0 0 . 0 W / m K ....

C o n d u c t i o n o n l y , K s = 7.0 W / m K

1.7

.j:i

_

; 1.5

•"

0 o E

1.3

1.2

1.1 1"~'5'~0

•

I

,

I

,

I

,

I

I

I

,

I

I

I

,

I

,

I

,

0.10 0.15 0.200.25 0.30 0.35 0.40 0.45 0.50 0.55 Dimensionless time

Fig. 6. Solid/liquid interface location vs dimensionless time for constant heat extraction; r d r , = 0 . 5 , h,~ = 400 kJ/kg, E wn = E,,2 = 0.6, Ks = 7.0 W / m K, a = 4.572/m, ca = 0.0, p = 500 k g / m 3, as = m2/hr, Stefan n u m b e r = 20.

1283

1284

B. Yimer 16.0

12.8

,ooooO°o°ooOO.°°°°°°°°°'" g.6

~

ooo°oOO ooooOO o.

U,~

o°°°

oz

6.4.

-'/Z

°oOOO°oood o°°°°°

uJ

3.2

7 / I

0.0

. . . . . . . CONDUCTION ONLY N = 0.692

t

I

O.O

t

0.1

I

,

,

I

I

0.2 0.3 DIMENSIONLESS TIME

i

0.4

0.5

Fig. 7. Dimensionless heat extracted by convection vs dimensionless time; B i o t number = 50.0, dimensionless fluid temp. = 5.0, r~lro = 0.5, h,~ = 400 l d / k g , ~ w, = ¢ ~ = 0.6, K, = 7.0 W / m K, a = 4.572/m, = = 0.0, p = 500 k g / m 3, a s = 0.003 m2/hr.

0

oodoooooOO

//

=

/ ss time = 0.25

~

~

Dimensionless time = 0.5

o =

.o

-0.2 -

.~..

,, Scattering coefficient = 0.3

r~

........ Conduction only N = 0.6962

"0'311.0

I

1!2

,

I 1.4

t

I

I

1.6 Dimensionless radius

I

1.8

,

2.0

Fig. 8. Transient temperature distribution vs dimensionless radius for heat extraction by convection; Biot number = 50.0, dimensionless fluid temp. = 5.0, r~/ro = 0.5, hsl = 400 kJ/kg, a = 4.572/m, E w, = ~ w2= 0.6, K, = 7.0 W / m K, w = 0.0, p = 500 kg/m 3, a+ = 0.003 m2/hr.

Thermal radiation effects in phase-change energy-storage systems 2.0 1.9

" 1.6

,.,

~

1.7

--~

1.6

~Q

1.5

~

1.4.

o ~.

~

SCATTERING COEFFICIENT = 0.3

........ CONDUCTION ONLY N = 0.6962

-

:7 7 u~ h,

1.3 1.2 1.1 1.0

l 0.0

I 0.1

I

I I I r 0.2 0.3 DIMENSIONLESS TIME

I 0.4-

, 0.5

Fig. 9. Solid/liquid interface location vs dimensionless time for heat extraction by convection; Blot number = 50.0, dimensionless fluid temp. = 5.0, rl/ro = 0.5, hst = 400 l d / k g , E ~ = ~ w2 = 0.6, Ks = 7.0 W / m K, a = 4.572/m, ~o = 0.0, p = 500 k g / m 3, a~ = 0.003 m2/hr.

16.0

J

EMISSIVITY = 1.0 ........ EMISSIVITY ffi 0.6 .... EMISSIVITY = 0.2

12.8

.....

~,~..'." ~..',:'"" .~.~;;" // CONDUCTION ONLY ~ " / s'"'" °-,~

,.,

9.6

.~,

=

o~

P~

.t"

/

s~/"

6.4.

¢,'/

3.2

0.0

, 0.0

I 0.1

I

I , I , 0.2 0.3 DIMENSIONLESS TIME

I 0.4-

, 0.5

Fig. 10. Dimensionless heat extracted by convection vs dimensionless time; B i o t number = 50.0, dimensionless fluid t e m p . = 5 . 0 , r~lro=0.5, h , ¢ = 4 0 0 1 U / k g , E , ~ = a w 2 = 0 . 6 , K , = 7.0 W / m K, a=4.S721m, o~=0.0, p = 500 k g / m 3, % = 0.003 m2/hr.

1285

1286

B. Yimer CONCLUSION

The energy equation, when both enthalpy and temperature are dependent variables, was coupled to the radiation-transport equation and solved numerically to study the effects of optical parameters and Stark number on the heat-transfer process and the rate of solidification of a phase-change energy-storage system. The results show that for low Stark numbers, the contribution of internal radiation to the rate of solidification and heat-transfer process is significant. At high Stark numbers, however, the process is dominated by conduction. REFERENCES 1. M. Abrams, "Energy Transfer During Melting and Solidification of Semi-Transparent Crystals", Ph.D. Thesis, Purdue University, Lafayette, Indiana (1971). 2. M. Abrams and R. Viskanta, J. Heat Transfer 96, 184 (1974). 3. I. S. Habib, Int. J. Heat and Mass Transfer 14, 2162 (1971). 4. I.S. Habib, J. Heat Transfer 37 (1973). 5. T.R. Goodman, Trans ASME 335 (1958). 6. S.H. Chan and D. H. Cho, Int. J. Heat and Mass Transfer 26, 621 (1983). 7. L.A. Diaz, "Radiative Induced Melting of Semi-Transparent Phase-Change Material", Ph.D. Thesis, Purdue University, Lafayette, Indiana (1983). 8. A.C. Ratzel and J.R. Howell, Trans. ASME J. Heat Transfer 104, 388 (1982). 9. J. Higenyi and Y. Bayazitoglu, J. Heat Transfer 102, 719 (1980). 10. J. Higenyi and Y. Bayazitoglu, AIAA Journal 18, 723 (1980). l l . M.P. Memguc and R. Viskanta, J. Heat Transfer 108, 271 (1986). 12. J.A. Harris, Trans. ASME J. Heat Transfer 111, 194 (1989). 13. N. Shamsunder and E. M. Sparrow, J. Heat Transfer 333 (1975). 14. K. S. Kim, "Transient Conduction and Radiation in a Semi-Transparent Phase Change Medium in an Annulus", Ph.D. Thesis, University of Kansas, Lawrence, Kansas (1986). NOMENCLATURE a = Absorption coefficient, m-~ C = Specific heat, J/kg K Fr = Dimensionless radial radiative heat flux, q/~rT~ h = Specific enthalpy, J/kg H~ = Latent heat of fusion, J/kg hc = Convective heat transfer coefficient, kJ/hr.m 2 K H = Dimensionless enthalpy K = Thermal conductivity, W/m K K~ = Thermal conductivity at element surface 1, W/m K K2 = Thermal conductivity at element surface 2, W/m K N = Conduction/radiation parameter, Ks/3/4¢r~ q, = Radiative heat flux, W/m 2 r = Radius, m S = Dimensionless parameter, K~, T, 14a, p, hst t = time, s T = temperature, K v = volume, m3

a = Thermal diffusivity, m2/s /3 = Extinction coefficient, (a + ¢0) ~ = Surface emissivity ~ = Dimensionless time /9= Dimensionless temperature p = Density, kg/m 3 ¢r = Stefan-Boltzmann constant, W/m2 K4 ~"= Radial optical coordinate • = Dimensionless Plank function t9 = Dimensionless intensity of radiation ¢o= Scattering coefficient, m-~ Subscripts f = Fusion i = Spatial location I = Inner l = Liquid 0 = Outer s = Solid phase at fusion temperature w = Wall