Float Glass, Container & Optical Glass Manufacturing, Adams-Williamson Annealing Lehr
Glass Annealing Lehr Cooling Rate & Birefringence Formula
Calculate Maximum Safe Annealing Lehr Cooling Rate hmax (°C/min), Critical Strain-Point Cooling Time (min), Polariscope Optical Retardation (nm/cm), and Anneal Quality Class.
Switch a float-glass annealing lehr from 8 mm architectural glass to 19 mm heavy plate without slowing cooling 6x and the ribbon snaps cleanly in half on the cutting table!
Whereas thermal tempering *intentionally* freezes high `>69` MPa surface compression so glass cannot be cut, **Float-Glass & Optical Annealing Lehrs** must cool glass slowly through its **`~550°C` Annealing Point down past its `~440°C` Strain Point (`Annealing Span Delta_T_{ann} ~= 110°C`)** so permanent residual stress `sigma_{perm}` stays low enough (`<= 2.5` MPa commercial float, `<= 1.0` MPa precision optical!) for clean diamond-wheel scoring without spontaneous running cracks! By **L. H. Adams & E. D. Williamson’s (`1920`) Thermo-Viscoelastic Annealing Law**, cooling a slab of **Thickness `d`** (`mm`) and **Thermal Diffusivity `alpha`** (`mm²/s`) at constant rate `h_c` (`°C/min`) induces a parabolic temperature lag `Delta_T = (h_c / 60) * d^2 / (8 * alpha)` that locks in permanent room-temperature stress `sigma_{perm} = (beta * d^2 * h_c) / (720 * alpha)`—limiting **Maximum Safe Cooling Rate `h_{max} = (720 * alpha * sigma_{allow}) / (beta * d^2)`** (`°C/min`, scaling inversely with **`d^2`**!) and **Polariscope Optical Retardation `R_{opt} = sigma_{allow} * 26.0`** (`nm/cm` for soda-lime Brewster constant `2.60` TPa⁻¹)!
Verify on an `8.0` mm Commercial Float-Glass Ribbon—Plate_Thickness `d` = 8.0 mm (B2), Max_Allowable_Stress `sigma_{allow}` = 2.50 MPa (B3), Glass_Thermal_Diffusivity `alpha` = 0.52 mm²/s (B4), Thermoelastic_Constant `beta` = 0.82 MPa/K (B5), and Critical_Annealing_Range `Delta_T_{ann}` = 110°C (B6): Max Safe Cooling Rate `h_{max}` equals `17.84` °C/min (E2), Critical Zone Transit Time equals `6.2` min (E3), Polariscope Retardation equals `65.0` nm/cm (E4), qualifying as `"COMMERCIAL FLOAT ANNEALED"` (E5)!
| Adams-Williamson Annealing Lehr KPI (alpha=0.52 mm²/s, beta=0.82 MPa/K, Span=110°C) | Formula | Verified Output |
|---|---|---|
| Max Safe Lehr Cooling Rate (°C/min) | (720 * alpha_mm2_s * Allow_MPa) / (beta * d_mm^2) | 17.84 °C/min (1.26 °C/min @ 19 mm optical!) |
| Critical Zone Cooling Time (min) | Annealing_Range_C / Max_Cooling_C_min | 6.2 min (87.0 min = 14x longer lehr dwell!) |
| Polariscope Birefringence (nm/cm) | Allowable_Stress_MPa * 26.0 (Brewster Const) | 65.0 nm/cm float (26.0 nm/cm precision lens) |
| Optical Annealing Grade | <= 30 nm/cm Fine Optical Retardation Gate | COMMERCIAL FLOAT ANNEALED |
Adams & Williamson (1920) Inverse-Square Thickness Lehr Cooling Formula
Validate `Thickness d` in `[1.0, 150.0]` mm, `Allowable_Stress sigma_{allow}` in `[0.1, 15.0]` MPa, `Diffusivity alpha` in `[0.2, 1.5]` mm²/s, `Thermoelastic beta` in `[0.3, 1.5]` MPa/K, and `Annealing_Span` in `[40, 200]`°C right on `E2` (checking `ISTEXT(E2)` on `E3..E4` so `E5` text stays decoupled)!
=LET(dm_m, B2, sa_m, B3, al_d, B4, bt_m, B5, dt_c, B6, ro_n, sa_m * 26, IF(OR(dm_m < 1, dm_m > 150, sa_m < 0.1, sa_m > 15, al_d < 0.2, al_d > 1.5, bt_m < 0.3, bt_m > 1.5, dt_c < 40, dt_c > 200), "Invalid Lehr Input", IF(ro_n <= 30, "PRECISION OPTICAL (<=30 nm/cm)", "COMMERCIAL FLOAT ANNEALED")))19 mm Heavy Optical Window Blank (`d = 19` mm, `sigma = 1.0` MPa) 14x Slowdown Check
Anneal a `19.0` mm (`3/4"`) optical window blank to `B3 = 1.0` MPa (`26.0` nm/cm birefringence in E4 -> `"PRECISION OPTICAL (<=30 nm/cm)"` in E5): because thickness `d^2` jumps `5.64x` AND stress target tightens `2.5x`, Safe Cooling Rate E2 plunges **`14.1x` from `17.84` down to `1.26` °C/min**, stretching critical lehr cooling E3 to **`87.0` minutes**!
=IF(ISTEXT(E2), E2, LET(hm_x, (720 * B4 * B3) / (B5 * (B2 ^ 2)), ROUND(B6 / hm_x, 1)))Try the example
Use the original values or change them to test your understanding.