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Glass Structures & Engineering

, Volume 4, Issue 1, pp 117–125 | Cite as

Determination of the engine power for quenching of glass by forced convection: simplified model and experimental validation of residual stress levels

  • N. PourmoghaddamEmail author
  • J. Schneider
Research paper
  • 125 Downloads

Abstract

This work presents a simplified model for the determination of the engine power as a function of the residual stress using quench parameters by forced convection and introduces the necessary empirical equations of integral heat and mass transfer coefficients suggested by Martin (1977). For a residual stress dependent production of thermally tempered glasses, float glasses were thermally tempered due to heat treatment of the glass panes with different heat transfer coefficients. In the method presented, quench parameters for determining the engine power required to reach the target residual stresses are taken into account. The plausibility of the model is checked on the basis of experimental data.

Keywords

Tempered glass Residual stress Quenching Heat transfer coefficient Cooling air velocity Engine power 

1 Introduction

Thermally tempered glass and heat strengthened glass are produced by heating and rapidly cooling float glass panes. For this purpose, glass plates are fed on rolls into an oven where they are heated to approximately \(100~^{\circ }\)C above the glass transition temperature. Afterwards they are moved further into a quench area where they are quenched i. e. rapidly cooled down by air jets from both sides to an ambient temperature. This process is illustrated in Fig. 1. When heated in the oven, the glass is in a viscous state and when quenched, the glass plate hardens through the thickness. First the both surfaces cool and harden. The temperature difference between the surface and the centre plane grows until a maximum is reached and then the centre plane also cools down and cools faster than the surface. The area between the two surfaces of the plate contracts and puts the surfaces into a permanent compressive state. Due to the equilibration of the surface compressive stresses tensile stresses result in the mid-plane of the glass plate. The residual stress distribution is parabolic through the thickness as it is shown in Fig. 2. The parabolic stress distribution is in equilibrium and symmetric about the mid-plane. The surface stress is approximately twice the tensile stress in magnitude \((2\sigma _m =\sigma _s)\). The zero stress level is at a depth of approximately 20% of the thickness t.

The residual stresses are strongly process-related and vary under different boundary conditions such as the cooling rate, the nozzle arrangement, the nozzle diameter, the distance between the nozzles and the glass surface as well as the roller distances. The initial temperature as well as the cooling rate in particular have a significant influence on the residual stress development during the tempering process, see e.g. Narayanaswamy and Gardon (1969) and Aronen and Karvinen (2017).
Fig. 1

Sketch of the process line for tempering flat glass

Fig. 2

Residual stress distribution through the thickness in the far field area of a tempered glass plate

Table 1

Series for the tempering process

Series

Specimens per series

Specimens per run

Runs per series

Dimension [mm]

Thickness [mm]

ScA

24

3

8

\(360\times 1100\)

4

ScB

24

3

8

\(360\times 1100\)

8

ScC

24

3

8

\(360\times 1100\)

12

The heat transfer coefficient, which describes the cooling property of a surface and governs the cooling process is very difficult to determine experimentally. For this reason, many authors who have numerically simulated the tempering process have always focused on the analytical or complex numerical determination of the heat transfer coefficient. The tempering process has been numerically calculated by several authors, e.g. Narayanaswamy and Gardon (1969), Laufs and Sedlacek (1999a), Laufs and Sedlacek (1999b), Schneider (2001), Daudeville et al. (2002), Bernard et al. (2005), Nielsen et al. (2010), Aronen (2012), Pourmoghaddam et al. (2016) and Pourmoghaddam and Schneider (2018a). This work presents equations, which have been compiled by Martin (1977) from experimental data for the determination of the heat transfer coefficient knowing the process boundaries as described above. This can also be used for the determination of the desired residual stresses in glass panes from the tempering process.

It is important for the manufacturer to be able to calculate the residual stress before the process for an effective utilization of the tempering oven and also for an accurate residual stress result. Our motivation was to produce glass specimens with systematically varying residual stresses for a subsequent fragmentation analysis (Pourmoghaddam and Schneider 2018b).

In order to study the correlation between the residual stress, particle count, particle weight and particle size, glass plates with three different thicknesses of 4, 8 and 12 mm were tempered with heat treatment conditions for a predetermined residual stress range. For the different heat treatment of the specimens the engine power of the tempering oven was estimated and varied by recalculating the air velocity and the air pressure from the iteratively determined heat transfer coefficients needed for yielding the target residual stress. After the tempering process, the residual stress in the glass specimens were measured, using a scattered light polariscope (SCALP).

2 Specimen preparation for the thermal tempering

For the fracture tests we produced three series of tempered glass specimens of size 360 mm \(\times 1100\) mm with three different thicknesses of 4, 8 and 12 mm (see Table 1). There were 24 specimens in each series divided into eight groups of three specimens for each run of the tempering process. Hence, there were eight runs of the tempering process per series for achieving eight different residual stresses in each series. In order to choose the target residual stresses for the heat treatment of the series, the elastic strain energy level for the start of the crack branching of glass according to Fineberg (2006) was considered. However, this level of energy represents the start of local crack branching by one rapidly moving tensile crack. The question was how this level would influence the fracture pattern of a tempered glass. Therefore, in the first step the target residual stresses were chosen above as well as below the strain energy level of \(35~\hbox {J/m}^2\). As it is shown in Fig. 3 , eight target residual mid-plane tensile stresses were chosen from 10 to 60 MPa for the 8 and 12 mm thick specimens and from 10 to 45 MPa for the 4 mm thick specimens.
Fig. 3

Eight target mid-plane tensile stresses for the heat treatment of the series with three different thicknesses 4, 8 and 12 mm considering the level of strain energy for the start of crack branching according to Fineberg (2006)

Fig. 4

Procedure for the determination of the engine power of the tempering oven

For the heat treatment of the series, the engine power of the thermal tempering oven was varied. Thus, the engine power needed for the corresponding residual stress was determined considering the procedure sketched in Fig. 4. The engine power for the cooling section of the thermal tempering oven was recalculated by the determination of the required heat transfer coefficient h, which in turn leads to a certain cooling air velocity w and subsequently a required air pressure P. The engine power was then calculated considering the experience values for the relation between the adjusted engine power and the resulting air pressure P.

3 Heat transfer coefficient

The heat transfer coefficient, which is required for yielding the target residual stresses (Fig. 3), was calculated by means of a Finite Element simulation of the tempering process using an infinite 2D-axisymetric model. A FE-Model for the simulation of the thermal tempering process and the numerical calculation of the residual stresses has been discussed in detail by several authors, see e.g. Laufs and Sedlacek (1999a, b), Aronen (2012), Nielsen (2009), Pourmoghaddam et al. (2016) and Pourmoghaddam and Schneider (2018a). The viscoelastic material behaviour of glass in the glass transition range is considered in the FE-Model. The structural relaxation is taken into account using the model of Narayanaswamy (1971). For the simulation of the tempering process the FE-Model of the glass plate was given an initial temperature of \(T_0=650\,^{\circ }\)C and cooled down to the ambient temperature of \(T_\infty =25\,^{\circ }\)C. The time increments needed for converged results of the tempering process for different glass thicknesses were given in Pourmoghaddam and Schneider (2018a). The resulting time-temperature relationship was put in terms of load steps on a structural mechanical model to calculate the stress response due to the tempering process. The different heat transfer coefficients were determined for yielding a residual surface compressive stress as shown in Fig. 5.
Fig. 5

Heat transfer coefficient \([\hbox {W}/\hbox {m}^2\hbox {K}]\) versus surface compressive stress [MPa] from FE simulations

Fig. 6

Left: Nozzle configuration: a single round nozzle (SRN), b single slotted nozzle (SSN), c array of round nozzles (ARN), d array of slotted nozzles (ASN) from VDI Heating atlas (Schabel and Martin 2013), right: sketch of the impinging jet flow, inner diameter of a nozzle D and distance H between the nozzles and the surface of the glass plate

Fig. 7

a Calculated cooling air velocity [m/s] versus heat transfer coefficient [W/\(\hbox {m}^2\)K], b calculated cooling air velocity [m/s] versus surface compressive stress [MPa], calculations in a and b based on the geometry values \(H=50\) mm, \(D=5\) mm and \(L_T=5\) mm (ARN\(\Delta \)) and the material values of the cooling medium air W/mK, \(\hbox {m}^2\)/s and \(\hbox {m}^2\)/s for an arithmetically averaged temperature \(T_m\) from the temperatures \(T_N=25\,^{\circ }\)C and \(T_S = 650\,^{\circ }\)C

4 Cooling air velocity

The various cooling air velocities w were calculated using the empirical equations of integral heat and mass transfer coefficients suggested by Martin (1977). The Nusselt-number Nu, the Reynolds number Re and the Prandtl-number Pr are material dependent, dimensionless values and can be written as:
$$\begin{aligned} Nu&= \dfrac{hD}{\lambda } \end{aligned}$$
(1)
$$\begin{aligned} Re&= \dfrac{wD}{\nu } \end{aligned}$$
(2)
$$\begin{aligned} \ Pr&= \frac{\nu }{a} \end{aligned}$$
(3)
where D is the inner diameter of a nozzle, h the heat transfer coefficient and w is the cooling air velocity. The material values of the cooling medium, air, are the thermal conductivity \(\lambda \), the kinematic viscosity \(\nu \) and the thermal diffusivity a at an arithmetically averaged fluid temperature \(T_m=(T_N + T_S)/2\) between the temperature at nozzles outlet \(T_N\) and the temperature at the surface of the glass plate \(T_S\). The empirical equations for the integral heat and mass transfer coefficients for an array of round triangular arranged nozzles (ARN\(\Delta \)), as shown in Fig. 6, in terms of the practical application is suggested by Martin (1977) as:
$$\begin{aligned} Nu_{ARN}= & {} G\cdot {Re^{2/3}}\cdot {Pr^{0.42}} \end{aligned}$$
(4)
$$\begin{aligned} \quad G= & {} \frac{d^{*}\cdot {(1-2.2d^{*})}}{1+0.2\cdot {(h^{*}-6)}\cdot {d^{*}}} \nonumber \\&\quad \cdot {\left[ 1+\left( \frac{10\cdot {h^{*}}\cdot {d^{*}}}{6}\right) ^6\right] ^{-0.05}} \end{aligned}$$
(5)
with the validity range for G:
$$\begin{aligned} 0.004\le & {} (d^{*}{2}=f)\le & {} 0.04, \\ 2\le & {} (h^{*}=H/D)\le & {} 12, \\ 2000\le & {} Re\le & {} 100,000 \end{aligned}$$
In the case of triangular arranged array of round nozzles (\(\hbox {ARN}\Delta \)) the relative nozzle area f can be written as:
$$\begin{aligned} f&= \frac{\pi }{2\sqrt{3}}\cdot {\frac{D^{2}}{L_{T}^{2}}} \end{aligned}$$
(6)
$$\begin{aligned} d^{*}&=\sqrt{f}=0.9523 \frac{D}{L_{T}} \end{aligned}$$
(7)
where \(L_T\) is the distance between the nozzles and H is the distance between the nozzles and the surface of the glass plate (see Fig. 6). The cooling air velocity was recalculated using the Eqs. (1)–(7). Inserting the Eqs. (1), (2) and (3) in Eq. (4) we obtain for the heat transfer coefficient:
$$\begin{aligned} h = \frac{G\lambda }{D} \cdot \left( \frac{wD}{\nu }\right) ^{2/3} \cdot \left( \frac{\nu }{a}\right) ^{0.42} \end{aligned}$$
(8)
In Fig. 7a), the heat transfer coefficient depending on the calculated cooling air velocity w is shown. Thereby, using Eq. (8) the value for the air velocity w was varied iteratively until the numerically determined heat transfer coefficient h was yielded. The cooling air velocities in correlation of the target residual surface compressive stresses are shown in Fig. 7b. The calculations were carried out with the nozzle values \(L_T=5\) mm, \(H=50\) mm and \(D=5\) mm for the cooling section of the thermal tempering oven.

5 Engine power of the tempering oven

The glass plates were tempered in groups of three using the thermal tempering oven of the company Semcoglas Holding GmbH (December 2016). Once the cooling air velocity w for different target residual stress states is determined, the air pressure P can be calculated using Eq. (9) for the total pressure with the air velocity pressure and the hydrostatic air pressure:
$$\begin{aligned} P = \frac{1}{2} \rho w^{2} + \rho g h_{j} \end{aligned}$$
(9)
where \(h_j\) is the distance between the pressure chamber and the air jets, \(\rho \) the air density and g is the gravity.
Table 2

Target residual mid-plane tensile stresses \(\sigma _{m}\) [MPa] and the corresponding heat transfer coefficients h [W/\(\hbox {m}^2\)K], cooling air velocities w [m/s], air pressure P [Pa] and the required engine power EP [%] as the percentage of the total power, \(h_j=0.65\) m, \(\rho =1.184~\hbox {kg/m}^{3}\) (atmospheric pressure and \(25~^{\circ }\)C)

t = 4 mm

t = 8 mm

t = 12 mm

\(\sigma _m\)

h

w

P

EP

\(\sigma _m\)

h

w

P

EP

\(\sigma _m\)

h

w

P

EP

[MPa]

[\(\hbox {W}/\hbox {m}^2\hbox {K}\)]

[m/s]

[Pa]

[%]

[MPa]

[\(\hbox {W}/\hbox {m}^2\hbox {K}\)]

[m/s]

[Pa]

[%]

[MPa]

[\(\hbox {W}/\hbox {m}^2\hbox {K}\)]

[m/s]

[Pa]

[%]

10

76.8

9

55.5

6.6

10

35.5

5

22.9

4.0

10

25.0

4

16.1

3.3

15

115.2

16

159.1

12.1

15

51.8

9

54.4

6.6

15

35.0

6

30.3

4.7

20

153.6

25

377.5

19.7

20

67.7

13

112.3

9.9

20

45.6

9

57.7

6.8

25

192.0

35

732.7

28.8

25

86.4

19

225.8

14.7

25

56.6

13

103.0

9.4

30

240.0

49

1428.9

42.1

30

106.1

26

410.8

20.7

30

68.5

17

178.6

12.9

35

278.4

61

2210.4

54.0

40

149.8

44

1138.1

37.0

40

94.2

27

452.0

21.9

40

335.0

78

3609.3

71.3

50

203.5

70

2891.8

62.9

50

123.8

41

1012.4

34.6

45

393.6

103

6288.1

97.8

60

264.0

102

6215.1

97.2

60

157.4

59

2068.3

52.0

Fig. 8

Engine power EP in percentage of the total power vs. air pressure P [Pa]; points show experience values of the tempering oven and solid curve show the trend line including the corresponding function and the coefficient of determination \(R^2\)

After the determination of the required air pressure for each target residual stress considering the cooling air velocity and the heat transfer coefficient, the air pressure was matched to the empirical function of the thermal tempering oven in terms of the correlation between the engine power EP as the percentage of the total power of the thermal tempering oven and the resulting air pressure in the height of the air jets. The empirical function of the thermal tempering oven is shown in Fig. 8. The eight target residual mid-plane tensile stresses, which were plotted in Fig. 3 and the corresponding heat transfer coefficients, cooling air velocities and subsequently the required engine powers are summed up in Table 2 for the three thicknesses of 4, 8 and 12 mm. For the determination of the air pressure an average atmospheric pressure of 1013.25 Pa and the temperature of \(25^{\circ }\)C was assumed (\(\rho =1.184\,\hbox {kg/m}^{3}\)).

6 Stress measurements

In order to check the actual stress state in the specimens the residual stresses were measured after the tempering process using a scattered light polariscope (SCALP) developed by GlasStress Ltd. The residual stresses in vertical and horizontal direction in thirteen measurement points were measured at both surfaces of the specimens, see Fig. 9. The anisotropy of the residual compressive surface stresses in the area of the measurement points was quite low with a coefficient of variation around 3%. In Fig. 10, the optical path difference under polarized light is shown for three different thicknesses. It was observed that the anisotropy and the inaccuracies in yielding the target residual stresses increased for the 12 mm thick plates. In Fig. 11a, the target residual stresses in comparison to the average of the measured mid-plane tensile stresses in the measurement points after the tempering process is shown.
Fig. 9

Positions and directions of the residual stress measurements

Fig. 10

Optical path difference under polarized light, at\(=3.8\) mm, \(\sigma _s=-\,91.3\) MPa (rel. deviation of 1.28%), bt\(=7.9\) mm, \(\sigma _s=-\,91.1\) MPa (rel. deviation of 1.15%), ct\(=12.0\) mm, \(\sigma _s=-\,97.4\) MPa (rel. deviation of 2.01%)

Fig. 11

a Elastic strain energy U [J/\(\hbox {m}^2\)] vs. mid-plane tensile stress [MPa]; coloured points show the average residual mid-plane stress from thirteen measurement points; hollow circles show the target residual mid-plane stress from Fig. 3; \(U=35~\hbox {J}/\hbox {m}^2\) is the start of the crack branching (Fineberg 2006), b mid-plane tensile stress [MPa] versus engine power [%] of the tempering oven; coloured points show the measured residual stress after the tempering process,solid lines are trend lines including the corresponding function and the coefficient of determination \(\hbox {R}^2\)

Due to the lack of accuracy of the thermal tempering oven for the low engine power range, it was not possible to reach the lowest target residual stress states, especially for the 12 mm thick plates, see Fig. 11a. However, considering the objective of yielding different residual stresses for a reasonable fragmentation analysis, the heat treatment of the specimen series was both necessary and successful.

In Fig. 11b the correlation between the residual mid-plane stress and the engine power of the tempering oven is shown. The values given on the x-axis for the engine power are the required fan powers of the motor relative to the maximum power. It was observed that the accuracy of the residual stresses decreased with thicker glass plates and at lower cooling rates. Especially the inaccuracy due to the thickness was expected. For thicker glass plates, a very low cooling rate is sufficient for the development of high residual stresses. The inaccuracy due to the lower cooling rates could be due to the inaccuracy of the fan power. The determined engine power for achieving low residual stresses could not be set exactly.

7 Summary and conclusion

This work comprehensively demonstrated a simplified model for the determination of the engine power as a function of residual stresses for quenching of glass by forced convection. For a residual stress dependent production of thermally tempered glasses, float glasses were thermally tempered due to heat treatment of the glass panes with different heat transfer coefficients. After specifying the target residual stresses the corresponding heat transfer coefficients were calculated by means of FE simulations. The value of the heat transfer coefficient was adjusted iteratively until the targeted residual stress was reached. After the determination of the corresponding heat transfer coefficient the cooling air velocity was recalculated using the empirical equations suggested by Martin (1977). Knowing the empirical function of the tempering oven engine power, the fan power was set according to the determined nozzle air velocity. The experiments in this work were only carried out on one tempering line, thus factors that may differ in different tempering lines were not taken into account.

With the help of SCALP measurements it could be shown that this approach is promising. In about 70% of the glass panes, a smaller deviation than 5% to the target residual stress was observed. Inaccuracies of the residual stress results for thicker plates (12 mm) at lower engine powers than 20% were expected. Due to the glass cooling property, based on the viscoelastic material behavior at temperatures above glass transition temperature and the temperature dependency of the glass structure, in thicker plates very high residual stresses occur at very low cooling rates.

It is important for the manufacturer to be able to determine and set the engine power according to the residual stress. This increases or optimizes the effectiveness of the tempering oven and production. It was observed that the accuracy of the residual stresses decreased with thicker glass plates and at lower cooling rates. However, at the typical industrial cooling rates and the engine power range of 30–100% the process parameters for the quenching of glass plates can be determined from this simplified calculations.

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Aronen, A., Karvinen, R.: Effect of glass temperature before cooling and cooling rate on residual stresses in tempering. In: Glass Structures and Engineering, no. Rantala 2015 (2017)Google Scholar
  2. Aronen, A.: Modelling of Deformations and Stresses in Glass Tempering. Ph.D. dissertation (2012)Google Scholar
  3. Bernard, F., Daudeville, L., René, G.: Prediction of residual stresses near joints in tempered glass plates| Calcul des contraintes résiduelles dans les zones d’assemblage de plaques en verre trempé. Mec. Ind. 6(1), 55–64 (2005)Google Scholar
  4. Daudeville, L., Bernard, F., Gy, R.: Residual stresses near holes in tempered glass plates. Mater. Sci. Forum 404–407, 43–48 (2002)CrossRefGoogle Scholar
  5. Fineberg, J.: The dynamics of rapidly moving tensile cracks in brittle amorphous material. In: Shukla, A. (ed.) Dynamic Fracture Mechanics, pp. 104–146. World Scientific, Singapore (2006).  https://doi.org/10.1142/9789812773326_0003 CrossRefGoogle Scholar
  6. Laufs, W., Sedlacek, G.: Stress distribution in thermally tempered glass panes near the edges, corners and holes. Part 1. Temperature distributions during the tempering process of glass panes. Glass Sci. Technol. Frankf. Main 72(1), 7–14 (1999)Google Scholar
  7. Laufs, W., Sedlacek, G.: Stress distribution in thermally tempered glass panes near the edges, corners and holes: Part 2. Distribution of thermal stresses. Glass Sci. Technol. Frankf. Main 72(2), 42–48 (1999)Google Scholar
  8. Martin, H.: Heat and Mass Transfer Between Impinging Gas Jets and Solid Surfaces. In: Hartnett J. P. & Irvine T. F. (eds.) Advances in Heat Transfer, pp. 1–60. Elsevier, New York (1977).  https://doi.org/10.1016/S0065-2717(08)70221-1
  9. Narayanaswamy, O.S.: A model of structural relaxation in glass. J. Am. Ceram. Soc. 54(10), 491–498 (1971)CrossRefGoogle Scholar
  10. Narayanaswamy, O.S., Gardon, R.: Calculation of Residual stresses in glass. J. Am. Ceram. Soc. 52(10), 554–558 (1969)CrossRefGoogle Scholar
  11. Nielsen, J.H.: Tempered Glass: Bolted Connections and Related Problems. Ph.D. dissertation, Technical University of Denmark (2009)Google Scholar
  12. Nielsen, J.H., Olesen, J.F., Poulsen, P.N., Stang, H.: Finite element implementation of a glass tempering model in three dimensions. Comput. Struct. 88(17–18), 963–972 (2010)CrossRefGoogle Scholar
  13. Pourmoghaddam, N., Nielsen, J. H., Schneider, J.: Numerical simulation of residual stresses at holes near edges and corners in tempered glass: a parametric study. In: Engineered transparency international conference at Glasstec (2016)Google Scholar
  14. Pourmoghaddam, N., Schneider, J.: Finite-element analysis of the residual stresses in tempered glass plates with holes or cut-outs. Glass Struct. Eng. 3(1), 17–37 (2018).  https://doi.org/10.1007/s40940-018-0055-z CrossRefGoogle Scholar
  15. Pourmoghaddam, N., Schneider, J.: Experimental investigation into the fragment size of tempered glass. Glass Struct. Eng. 3(2), 167–181 (2018).  https://doi.org/10.1007/s40940-018-0062-0 CrossRefGoogle Scholar
  16. Schabel, W., Martin, H.: Prallströmung. In: VDI-Wärmeatlas, pp. 841–847. Springer, Heidelberg (2013)Google Scholar
  17. Schneider, J.: Festigkeit und Bemessung punktgelagerter Gläser und stoßbeanspruchter Gläser. Ph.D. dissertation, Technische Universität Darmstadt (2001)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Technical University of DarmstadtDarmstadtGermany

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