Reports Res. Lab. Asahi Glass Co., Ltd., 55（2005）
UDC：666.1.038：681.5.017
6. The New Index for High Surface Comperssive Stress
on Short Quenching
Takeshi Naraki* and Yasumasa Kato**
Glass tempering process with limited time quenching, we call short quenching, is
required for some kind of glass manufacturing. In manufacturing process of CRT panel,
short quenching is suitable for tempering process because annealing process for
removing irrelevant stress is required just after tempering process for shortening
manufacturing time. In case of short quenching, we find that there is an optimum initial
temperature for highest surface compressive stress by numerical tempering simulation.
Quenching from the optimum initial temperature gives highest surface compressive
stress. However, the degree of the optimum initial temperature is susceptible to the glass
thickness, the heat transfer coefficient of quenching and the quenching time. In this
paper, the new index which shows the optimum initial temperature on short quenching
with proper physical meanings is proposed. The new index allows us to find the
optimum initial temperature on short quenching without complex numerical simulations.
1. Introduction
Short quenching means quenching with limited
time. Short quenching is required for some kind of
glass manufacturing process, for example CRT
panel manufacturing process. The thickness of
CRT panels has become still thinner for weight
reduction in these days. The glass surface is
required higher compressive stress for higher vacuum stress of thinner panels. Short quenching is
suitable when we introduce tempering process to
manufacturing process of CRT panel in order to
get high compressive stress because annealing
process for removing irrelevant stress is required
just after tempering process for shortening manufacturing time. Quenching time is not limited in the
conventional full tempering process of safety glass.
In case of the tempering process with short
quenching, we find that there is an optimum initial
temperature of quenching for highest surface compressive stress by tempering simulation. Short
quenching is ineffective if the initial glass temperature of quenching is too high or too low. The opti*Display Company
mum initial temperature for short quenching is difficult to find because the degree of the optimum
initial temperature is susceptible to the glass thickness, the heat transfer coefficient of quenching and
the quenching time. In this paper we propose the
new index to find the optimum initial temperature
for short quenching for highest surface compressive residual stress with proper physical meanings.
The new index can be applied for any glass thickness, any heat transfer coefficient of short quenching and any quenching time.
2. Method
Numerical program based on the finite element
method is used for heat transfer and viscoelastic
analyses. 3D element with 8 nodes is used in heat
transfer analyses and shell element is used in vis(1)
coelastic analyses with Narayanaswamy’
s model .
Narayanaswamy expressed stress relaxation and
structural relaxation as equation [1] and [2] under
assuming thermal rheological simplicity.
** Research Center
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旭硝子研究報告 55（2005）
[stress relaxation]
Glass surface
G(t )

=
G(0) 
3

t 

si  

［1］

ξ 

vi  

［2］
∑W exp − τ
si
i =1
[structural relaxation]
M(ξ )
Glass midplane

=
M(0) 
Fig. 1 The analysis model assuming an infinite
planar glass plate.
Heat transfer analysis condition
Specific heat of glass
1.7442
W/mK
1046.51
J/KgK
Initial temperature (all point of glass)
800
℃
25
W/m2K
Heat transfer coefficient of natural convection
Viscoelastic analysis condition
Young's modulus
Poisson's ratio
73745.3
0.23
MPa
−
Mass density
2.78E−06
Kg/mm
Thermal expansion coeficient of solid
9.80E−06
1/℃
Thermal expansion coeficient of liquid
2.94E−05
1/℃
498600
J/mol
Activation energy
vi
i =1
The analysis model assumed an infinite glass
sheet as shown in Fig. 1. The analysis conditions
are shown in Table 1. The effects of heat transfer
coefficient, quenching time and glass thickness are
evaluated. Table 2 shows the analysis conditions of
case studies. In the heat transfer analyses, initial
glass temperature is set to 800℃ and uniform. The
analysis procedure is set as follows. In case of
analyses for short quenching, at first the glass
sheet is cooled by natural convection for set time
and next the glass sheet is quenched for set time
and at last the glass sheet is cooled by natural convection again. In case of analyses for full quenching, at first the glass sheet is cooled by natural convection for set time and next quenched enough to
reach elastic region at all point of the glass sheet.
The pattern diagrams of the heat transfer coefficient are shown in right side of Fig. 2.
Table 1 Analysis Conditions Used in Numerical
Program.
Thermal conductivity of glass
3
∑W exp − τ
3
Shear relaxation time ts1
196.7051
s
Shear relaxation time ts2
32.1332
s
Shear relaxation time ts3
3.4587
s
Weight Ws1
0.395
−
Weight Ws2
0.3757
−
3.1 Optimum initial viscosity for short quenching
Weight Ws3
0.2292
−
Structural relaxation time tv1
116.1225
s
Structural relaxation time tv2
436.0362
s
Structural relaxation time tv3
Figure 2 shows the results of residual stress of
glass surface as a function of initial viscosity of
quenching. In case of full quenching, residual stress
of glass surface increases as the initial viscosity
decrease. In the field of automotive glass or architectural glass, it is well known that higher initial
(2)
temperature is required for high residual stress .
In the range of low initial viscosity of quenching,
the result of residual stress of full quenching does
not change so mach. However, in case of 30 sec-
2056.251
s
Weight Wv1
0.2824
−
Weight Wv2
0.3189
−
Weight Wv3
0.3987
−
Table 2
Case 1
(Effect of heat transfer coefficient)
Case 2
(Effect of cooling time)
Case 3
(Effect of glass thickness)
3. Results by Numerical Analyses
Analysis Conditions for Case Studies.
Heat transfer coefficient (h)
Quenching time (t)
Glass thickness (2d)
W/m2K
sec
mm
50
30
10
110
30
10
200
30
10
110
10
10
110
30
10
110
50
10
110
30
5
110
30
10
110
30
15
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Reports Res. Lab. Asahi Glass Co., Ltd., 55（2005）
Residual stress of glass surface [MPa]
−140
−130
Full quenching
−120
−110
h
−100
−90
−80
t
−70
−60
−50
30 seconds quenching
−40
−30
−20
h
−10
0
16
15
14
13
12
11
10
9
8
7
6
t
Initial glass viscosity log η [log poise]
Full quenching
30 seconds quenching
Fig. 2 Calculated residual stress of glass surface as a function of initial viscosity at glass surface.
2
(Glass thickness 2d=10mm, heat transfer coefficient h=110W/m K)
3.2 Effect of quenching condition and glass
thickness
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
16 15 14 13 12 11 10 9 8 7 6
Initial viscosity [log poise]
10 seconds quenching
30 seconds quenching
50 seconds quenching
Fig. 3 Effect of quenching time on
calculated residual stress of
glass surface.
2
(2d=10mm, h=110W/ m K)
Residual stress of glass surface [MPa]
Residual stress of glass surface [MPa]
Figure 3, 4 and 5 show the relationship between
the quenching initial viscosity and residual stress
of the glass surface. Figure 4 shows that higher
heat transfer coefficient lowers the optimum initial
viscosity which gives highest residual stress.
Figure 3 shows that longer quenching time also
lowers the optimum initial viscosity. On the other
hand, Fig. 5 shows that the glass thickness raises
the optimum initial viscosity. The optimum initial
viscosity for glass thickness 5mm is 9 log poise and
for glass thickness 15mm is 11.5 log poise.
4. Discussions
4.1 Transient stress in short quenching
The calculated results show that there is an optimum initial viscosity for highest surface compressive stress on short quenching. In this section, we
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
16 15 14 13 12 11 10 9 8 7 6
Initial viscosity [log poise]
050W/m22K
110W/m K
200W/m2K
Fig. 4 Effect of heat transfer
coefficient on calculated
residual stress of glass
surface.
(2d=10mm, t=30sec)
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Residual stress of glass surface [MPa]
onds quenching (short quenching), the maximum of
residual stress is obtained at the log h=10.5 log
poise. This means there is an optimum initial viscosity for high surface compressive stress in short
quenching.
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
16 15 14 13 12 11 10 9 8 7 6
Initial viscosity [log poise]
5mm
10mm
15mm
Fig. 5 Effect of glass thickness on
calculated residual stress of
glass surface.
2
(t=30sec, h=110W/ m K)
Temperature [℃]
旭硝子研究報告 55（2005）
800
750
700
650
600
550
500
450
400
350
300
250
200
Glass surface
Glass midplane
Condition 1
Condition 2
Condition 3
Stress of glaass surface [MPa]
0
50
100
150
200
250
300
350
Time [s]
400
450
500
−80
−70
−60
−50
−40
−30
−20
−10
0
10
20
30
550
600
Condition 2
Condition 1
Condition 3
0
50
100
150
200
250
300
350
Time [s]
400
450
500
550
600
Fig. 6 Calculated transient glass temperature and transient stress of glass surface.
show how the optimum initial temperature
appears. Figure 6 shows the calculated results of
the glass temperature and surface stress. In these
calculated results, highest surface compressive
stress is obtained in condition 2. The glass temperature and the transient stress in short quenching
show characteristic pattern as shown in Fig. 7. In
the stage of the first natural convection before
quenching (stage a), there is no stress generation
since the glass temperature is too high. As the
glass temperature goes down, compressive stress
appears at the glass surface. This compressive
stress is generated when the temperature difference between the glass surface and midplane
(2)
becomes small . When quenching starts (stage b),
temporary tensile stress appears. After that, the
surface stress goes to compression (stage c). After
quenching, the temperature of the glass surface
temporarily goes up because the heat flux of heat
conduction from the glass midplane becomes larger
than that of heat transfer from the glass surface by
natural convection. This is called "reheating".
During the reheating, the surface stress goes to
compression because the glass surface temporarily
expand by temperature goes up (stage d). If the
glass surface temperature is viscoelastic region at
the end of the reheating, the compressive stress
decreases because of stress relaxation (stage e). As
the temperature of the glass surface goes down,
the compressive stress increases (stage f) according
to the same mechanism of the [stage a]. Thus the
transient stress of short quenching is quite different from that of conventional full quenching in
automotive glass or architectural glass. Three
stress histories in Fig. 6 are well understandable as
compared with characteristic stress history in Fig.
7. The stress history of condition 1 shows that
there is no stress generation in the [stage a] and
the [stage b] because the glass temperature is so
high to relax stress immediately. The initial compressive stress generation is explained by the
mechanism of the [stage c] and the [stage d]. The
reduction of the compressive stress after quenching is caused by large stress relaxation in the
[stage e] because the glass temperature is so high
to relax stress immediately. The stress history of
condition 2 shows that the stress relaxation in
[stage b] is still large, so initial tensile stress is
small. The stress relaxation of [stage e] is small
because the glass temperature is not so high compared with that of condition 1. The stress history of
condition 3 shows that the stress relaxation from
[stage b] to [stage f] is negligibly small. From these
calculated results and discussion, both large stress
relaxation at the start of quenching and small
stress relaxation in reheating are important for
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Reports Res. Lab. Asahi Glass Co., Ltd., 55（2005）
Temperature
of glass surface
Reheating
Quenching
Natural convection
Natural convection
Compression
Compression
e
Stress of
glass surface
Optimum initial viscosity log η [log poise]
13
a
f
d
0
b
Tension
c
Tension
Time
11
10
9
8
7
Fig. 7 Characteristic history of temperature and
stress of glass surface in short quenching.
0
0.2
0.4
0.6
0.8
Biot number (h･d /λ)
Fig. 8 Calculated optimum initial viscosity as a
function of conventional index Biot number.
short quenching to obtain high surface compressive
stress.
4.2 The new index for optimum initial viscosity
on short quenching
The calculated results from case studies in section 3.2 show the optimum initial viscosity on various short quenching. However, for new quenching
condition or new glass thickness, these results dose
not tell us the optimum initial viscosity on the
short quenching because the optimum initial viscosity on short quenching shifts depending on the
quenching conditions and the glass thickness. The
Biot number as shown in equation [3] has been
used conventionally as an index to discuss quench(2)
ing condition .
h ⋅d
h
=
λ
λ /d
Heat transfer
=
Heat conduction
12
Biot number =
[3]
Where h is the heat transfer coefficient of quenching, d is half of the glass thickness, and l is thermal
conductivity of the glass. In case of short quenching, however, the Biot number does not make
sense as shown in Fig. 8. The Biot number is not
useful to find the optimum initial viscosity on short
quenching because the Biot number does not
include parameter relating quenching time which
affects on residual stress of short quenching. In
short quenching, not instant heat transfer of
quenching as shown in the numerator of right
hand of equation [3] but total heat transfer of
quenching is important. Furthermore, in short
quenching, not heat conduction as shown in the
denominator of right hand of equation [3] but heat
capacity of glass bulk is important. To make a new
index on short quenching which gives us the optimum initial viscosity on short quenching, we propose a new index which means the ratio of total
heat transfer of quenching and heat capacity of
glass bulk. The new index called "Ratrac number"
for short quenching of sheet glass includes heat
transfer coefficient, quenching time and glass thickness as shown in equation [4]. "RATRAC" stands
for "RAtio of heat TRAnsfer and heat Capacity".
h ⋅t
h ⋅t ⋅ A
=
ρ ⋅ C p ⋅ d ρ ⋅ C p ⋅V
Heat transfer from glass surface
=
Heat capacity of glass
Ratrac number =
[4]
Where h is the heat transfer coefficient of quenching, t is the quenching time, r is mass density of
the glass, Cp is specific heat of the glass, d is half of
the glass thickness, A is area of the heat transfer
surface of the glass and V is half volume of the
glass. The index means the ratio of heat transfer
from the glass surface to atmosphere and heat
capacity of the glass. Figure 9 shows relationship
between Ratrac number and the optimum initial
viscosity for highest compressive stress. Figure 9
shows good correlation between Ratrac number
and the optimum initial viscosity on short quenching. Therefore we can see the optimum initial viscosity form Ratrac number. For example, when we
introduce short quenching which heat transfer is
2
90W/m K and which quenching time is 25sec for
3
CRT glass which mass density is 2780kg/m and
which specific heat is 1047J/kgK and which glass
thickness 6 mm, Ratrac number is calculated as
0.13. From this Ratrac number, the optimum initial
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旭硝子研究報告 55（2005）
5. Conclusion
Optimum initial viscosity log η [log poise]
13
12
11
10
9
8
7
0.0
0.1
0.2
0.3
0.4
0.5
Ratrac number (ht /ρCp d )
Fig. 9 Calculated optimum initial viscosity as a
function of new index “Ratrac number”.
Results from viscoelastic analyses show that
there is an optimum initial temperature for highest
surface compressive stress in case of short quenching. However this optimum initial temperature is
difficult to find because the optimum initial temperature shifts depending on the quenching condition
and the glass thickness. To find the optimum initial
temperature easily without complex numerical simulation, we propose new index called Ratrac number which shows an optimum initial temperature
from the quenching condition and the glass thickness. The index physically means the ratio of heat
transfer from the glass surface and heat capacity of
the glass. We can estimate the optimum initial temperature for highest surface compressive stress on
short quenching with Ratrac number.
−Acknowledgement−
viscosity for short quenching is 11.1 log poise as
shown in Fig. 9. Finally, from viscosity-temperature
curve, Ratrac number gives us the optimum initial
temperature for highest surface compressive stress
on short quenching.
We would like to thank Dr.Narayanaswamy for instructive discussion.
−References−
(1) Narayanaswamy O. S., J. Am. Ceram. Soc. 61 [3-4] 146152 (1978).
(2) Gardon R., Thermal tempering of glass, Glass science
and technology, Vol 5 (1980).
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