Supplementary Materials
LIST OF TABLES
Table S1. The true and retrieved input parameters from numerically generated temperature
profiles predicted from the combined conduction and radiation heat transfer model for
different data sets of a, b, 1 ,  2 , and Tf,R.
-1-
LIST OF FIGURES
Figure S1. (a) Spectral absorption coefficient of gray and low-iron glass reported in the literature
[23] as well as the corresponding band coefficients used to predict steady-state
temperature profile in the glassmelts; (b) the predicted steady-state temperature profiles
in the gray and low-iron glassmelts slab for various cut-off wavelength of cut = 5 m
and 8 m and for furnace temperature Tf = 1400oC.
Figure S2. Estimated spectral normal emissivity of the crucible wall based on the optical
properties of various soda-lime silicate glass reported in the literature [23] and the optical
properties of alumina (the crucible was made of ~97% alumina) reported in the literature
[30].
Figure S3. Spectral absorption coefficient of (a) float/clear glass and (b) green glass reported in
Ref. [23] as well as the corresponding eight bands approximation of absorption
coefficient used in the present study.
Figure S4. Comparison between the predicted steady-state temperature profiles and the results
reported by Lee and Viskanta [24], (a) float/clear glass, and (b) green glass.
Figure S5. Spectral absorption coefficient of gray and low-iron glass reported in Ref. [23] and the
corresponding bandwise Rosseland mean two-band absorption coefficient (  R,1 ,  R ,2 )
defined by Equation (12).
Figure S6. Comparison of the temperature profiles predicted by using either the spectral
absorption coefficient [23] or the two-band Rosseland mean absorption coefficient
approximation for (a) gray and (b) low-iron soda-lime silicate glass for Tf = 1400oC.
-2-
Figure S7. Comparison between experimental data and numerical temperature profiles predicted
using the retrieved properties reported in Table 1 for gray soda-lime silicate glassmelt
(composition 1) at furnace temperature Tf of (a) 1350oC, (b) 1400oC, (c) 1450oC and (d)
1500oC.
Figure S8. Comparison between experimental data and numerical temperature profiles predicted
using the retrieved properties reported in Table 2 for green soda-lime silicate glassmelt
(composition 2) at furnace temperature Tf of (a) 1300oC, (b) 1350oC, (c) 1400oC, (d)
1450oC, and (e) 1500oC.
Figure S9. Comparison between experimental data and numerical temperature profiles predicted
using the retrieved properties reported in Table 3 for clear soda-lime silicate glassmelt
(composition 3) at furnace temperature Tf of (a) 1300oC, (b) 1350oC, (c) 1400oC, (d)
1450oC, and (e) 1500oC.
Figure S10. Comparison between experimental data and numerical temperature profiles predicted
using the retrieved properties reported in Table 4 for low-iron soda-lime silicate
glassmelt (composition 4) at furnace temperature Tf of (a) 1350oC, (b) 1400oC, and (c)
1450oC.
-3-
Table S1. The true and retrieved input parameters from numerically generated temperature
profiles predicted from the combined conduction and radiation heat transfer model for different
data sets of a, b, 1 ,  2 , and Tf,R.
Parameter
Tf,R (oC)
Case 1
Exact
Retrieved
a (W/m∙K)
 1 (m-1)
 2 (m-1)
Case 3
Case 4
1400
1400
1.20
1.70
1400
1400
1400
Exact
Retrieved
b (W/m∙K2)
Case 2
1.14
1.35
1.35
6.24×10-4
Exact
Retrieved
4.39×10-4
5.20×10-4
7.20×10-4
2.70×10-4
Exact
20
40
150
250
Retrieved
18.8
40.2
150.3
251.3
413.5
442.7
Exact
Retrieved
450
499.1
-4-
407.5

Absorption coefficient,m 
(a) 106
10
5
10
4
10
3
10
2
10
1
10
0
10
Gray, spectral
Gray, band
Low-iron, spectral
Low-iron, band
-1
0
1
2
3
4
5
6
7
8
wavelength, (m)
(b) 1395
Low-iron, cut = 5 m
Low-iron, cut = 8 m
Gray, cut = 5 m
Gray, cut = 8 m
o
Temperature, T( C)
1390
1385
1380
o
Tf = 1400 C
1375
0.0
0.4
0.8
1.2
1.6
2.0
Location, x (cm)
Figure S1. (a) Spectral absorption coefficient of gray and low-iron glass reported in the literature
[23] as well as the corresponding band coefficients used to predict steady-state temperature
profile in the glassmelts; (b) the predicted steady-state temperature profiles in the gray and lowiron glassmelts slab for cut-off wavelength of cut = 5 m and 8 m and for furnace temperature
Tf = 1400oC.
-5-
The effect of cut-off wavelength cut was investigated by comparing the predicted steadystate temperature profile in a plane-parallel slab of thickness of 2 cm for cut equal to either 5 m
or 8 m . Two types of glassmelt were investigated namely gray and low-iron glassmelts (the
extreme cases). The spectral absorption coefficient of these glasses were taken from the literature
[23], Figure S9(a) shows the corresponding band absorption coefficients used to predict the
temperature profile. The true thermal conductivity of both glasses was taken as kc(T) = 1.14 +
6.24×10-4T [22], the furnace temperature Tf was set as 1400oC. The total heat flux across the gray
and low-iron glassmelts was taken arbitrarily as 15,000 W/m2 and 12,000 W/m2, respectively.
Figure S9(b) shows the predicted steady-state temperature profile in the glassmelts. For both gray
and low-iron glassmelts, the maximum difference in the temperature profiles predicted for cut-off
wavelength cut = 5 m and 8 m was less than 1oC. These results indicate that choosing cut =
5 m , as performed in the literature [22, 24-25] gives acceptable temperature profile predictions.
-6-
Spectral normal emissivity, n,
1.000
lowiron
clear
green
gray
0.998
0.996
0.994
0.992
0.990
0
1
2
3
Wavelength, (m)
4
5
Figure S2. Estimated spectral normal emissivity of the crucible wall based on the optical
properties of various soda-lime silicate glass reported in the literature [23] and the optical
properties of alumina (the crucible was made of ~97% alumina) reported in the literature [30].
-7-
10
4
Spectral absorption coefficient [23]
Band model approximation
-1
Absorption coefficient,(m )
(a)
10
3
10
2
10
1
Float(clear) glass
0
10
4
10
3
10
2
2
3
4
Wavelength, (m)
5
Spectral absorption coefficient [23]
Band model approximation
-1
Absorption coefficient,(m )
(b)
1
Green glass
10
1
0
1
2
3
4
Wavelength, (m)
5
Figure S3. Spectral absorption coefficient of (a) float/clear glass and (b) green glass reported in
Ref. [23] as well as the corresponding eight bands approximation of absorption coefficient used in
the present study.
-8-
(a) 1800
Temperature, T (K)
1600
1400
1200
Float/clear glass
H =0.01 m, Lee et al. [24]
H =0.01 m, Present study
H =0.1 m, Lee et al. [24]
H =0.1 m, Present study
H =1.0 m, Lee et al. [24]
H =1.0 m, Present study
1000
800
600
400
0.0
0.2
0.4
0.6
Location, x/L
0.8
1.0
(b) 1800
Temperature, T (K)
1600
1400
1200
Green glass
H =0.01 m, Lee et al. [24]
H =0.01 m, Present study
H =0.1 m, Lee et al. [24]
H =0.1 m, Present study
H =1.0 m, Lee et al. [24]
H =1.0 m, Present study
1000
800
600
400
0.0
0.2
0.4
0.6
Location, x/L
0.8
1.0
Figure S4. Comparison between the predicted steady-state temperature profiles and the results
reported by Lee and Viskanta [24], (a) float/clear glass, and (b) green glass.
-9-
4
10
3
10
2
-1
Absorption coefficient,  (m )
10
2.8 m
gray, 
10
gray, R
1
low-iron, 
10
low-iron, R
0
0
1
2
3
4
5
wavelength, (m)
Figure S5. Spectral absorption coefficient of gray and low-iron glass reported in Ref. [23] and the
corresponding bandwise Rosseland mean two-band absorption coefficient (  R,1 ,  R ,2 ) defined by
Equation (12).
- 10 -
(a)
1400
Spectral calculation
Two bands (R1, R2)
o
Temperature, T( C)
1380
1360
1340
1320
1300
0
(b)
2
o
6
8
10
Location, x (cm)
12
14
1400
Spectral calculation
Two bands (R1,R2)
1395
Temperature, T ( C)
4
1390
1385
1380
1375
1370
1365
1360
0
2
4
6
8
10 12
Location, x (cm)
14
16
Figure S6. Comparison of the temperature profiles predicted by using either the spectral
absorption coefficient [23] or the two-band Rosseland mean absorption coefficient approximation
for (a) gray and (b) low-iron soda-lime silicate glass for Tf = 1400oC.
- 11 -
(a) 1360
(b) 1400
Experimetal data
Predicted data
1340
1320
Predicted data
1360
Temperature, T( C)
1300
o
o
Temperature, T( C)
Experimetal data
1380
1280
1260
1240
1220
o
Tf = 1350 C
1200
1340
1320
1300
1280
o
Tf = 1400 C
1260
1180
1240
0
2
4
6
8
10
12
14
0
2
4
Location, x (cm)
10
12
14
(d) 1520
Experimetal data
Predicted data
1440
Experimetal data
1500
Predicted data
1480
o
Temperature, T( C)
1420
o
8
Location, x (cm)
(c) 1460
Temperature, T( C)
6
1400
1380
1360
1340
o
Tf = 1450 C
1320
1460
1440
1420
1400
o
Tf = 1500 C
1380
1300
1360
0
2
4
6
8
10
12
14
Location, x (cm)
0
2
4
6
8
10
12
14
Location, x (cm)
Figure S7. Comparison between experimental data and numerical temperature profiles predicted
using the retrieved properties reported in Table 1 for gray soda-lime silicate glassmelt
(composition 1) at furnace temperature Tf of (a) 1350oC, (b) 1400oC, (c) 1450oC and (d) 1500oC.
- 12 -
(a) 1280
(b) 1340
Experimetal data
Predicted data
1260
o
Temperature, T( C)
o
Predicted data
1300
1240
Temperature, T( C)
Experimetal data
1320
1220
1200
1180
1160
o
Tf = 1300 C
1140
1280
1260
1240
1220
1200
o
Tf = 1350 C
1180
1120
1160
0
2
4
6
8
10
12
14
16
0
2
4
Location, x (cm)
10
12
14
16
(d) 1440
Experimetal data
Predicted data
1360
Experimetal data
1420
Predicted data
1400
o
Temperature, T( C)
1340
o
8
Location, x (cm)
(c) 1380
Temperature, T( C)
6
1320
1300
1280
1260
o
Tf = 1400 C
1240
1380
1360
1340
1320
o
Tf = 1450 C
1300
1220
1280
0
2
4
6
8
10
12
14
16
Location, x (cm)
0
2
4
6
8
10
12
14
16
Location, x (cm)
Figure S8. Comparison between experimental data and numerical temperature profiles predicted
using the retrieved properties reported in Table 2 for green soda-lime silicate glassmelt
(composition 2) at furnace temperature Tf of (a) 1300oC, (b) 1350oC, (c) 1400oC, (d) 1450oC, and
(e) 1500oC.
- 13 -
(e) 1480
Experimetal data
Predicted data
1440
o
Temperature, T( C)
1460
1420
1400
1380
o
Tf = 1500 C
1360
1340
0
2
4
6
8
10
12
Location, x (cm)
Figure S8. (Continued)
- 14 -
14
16
(a) 1260
(b) 1320
Experimental data
Predicted data
Experimental data
Predicted data
1310
Temperature, T( C)
1240
1300
o
o
Temperature, T( C)
1250
1230
1220
1210
o
Tf = 1300 C
1290
1280
1270
o
Tf = 1350 C
1260
1200
1250
0
2
4
6
8
10
12
14
16
0
2
4
Location, x (cm)
8
10
12
14
16
Location, x (cm)
(c) 1380
(d) 1430
Experimental data
Predicted data
1370
Experimental data
Predicted data
1420
1410
o
Temperature, T( C)
1360
o
Temperature, T( C)
6
1350
1340
1330
o
Tf = 1400 C
1320
1310
1400
1390
1380
o
Tf = 1450 C
1370
1360
0
2
4
6
8
10
12
14
16
Location, x (cm)
0
2
4
6
8
10
12
14
16
Location, x (cm)
Figure S9. Comparison between experimental data and numerical temperature profiles predicted
using the retrieved properties reported in Table 3 for clear soda-lime silicate glassmelt
(composition 3) at furnace temperature Tf of (a) 1300oC, (b) 1350oC, (c) 1400oC, (d) 1450oC, and
(e) 1500oC.
- 15 -
(e) 1490
Experimental data
Predicted data
1470
o
Temperature, T( C)
1480
1460
1450
1440
o
Tf = 1500 C
1430
1420
0
2
4
6
8
10
12
Location, x (cm)
Figure S9. (Continued)
- 16 -
14
16
(b) 1390
(a) 1340
Experimental data
Predicted data
Experimental data
Predicted data
1380
Temperature, T( C)
1320
o
o
Temperature, T( C)
1330
1310
1300
o
Tf = 1350 C
1290
1280
1370
1360
1350
o
1340
Tf = 1400 C
1330
0
2
4
6
8
10
12
14
16
0
2
4
Location, x (cm)
6
8
10
12
14
16
Location, x (cm)
(c) 1440
Experimental data
Predicted data
o
Temperature, T( C)
1430
1420
1410
1400
o
Tf = 1450 C
1390
0
2
4
6
8
10
12
14
16
Location, x (cm)
Figure S10. Comparison between experimental data and numerical temperature profiles predicted
using the retrieved properties reported in Table 4 for low-iron soda-lime silicate glassmelt
(composition 4) at furnace temperature Tf of (a) 1350oC, (b) 1400oC, and (c) 1450oC.
- 17 -
Estimate of the uncertainties for the retrieved properties shown in Figure (9b):
Figure 9(b) shows the average values of 1,av ,  2,av , and the associated uncertainties 1 ,  2 for
the four types of soda-lime silicate glassmelts. The uncertainties 1 and  2 were estimated as
 1 NF
 i  
 i , j  i,av
 NF  1 j 1


1/2
2
 , with i= 1 or 2

where NF is the number of datasets collected at different furnace temperatures and  i , j is the
associated band absorption coefficient retrieved. Here, NF = 6 for furnace temperature Tf ranging
from 1300oC to 1550oC by 50 oC increments.
- 18 -