Determination of the engine power for quenching of glass by forced convection: simplified model and experimental validation of residual stress levels
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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 power1 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.
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 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
4 Cooling air velocity
5 Engine power of the tempering oven
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 |
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
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.
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