Introduction & Context
Thermal shock resistance is a critical parameter in the design and operation of glass containers, particularly in food and beverage processing. During unit operations such as retort sterilization, glass containers are subjected to rapid temperature transitions—moving from high-temperature steam environments to cold water cooling baths. These sudden temperature gradients induce differential thermal expansion within the glass wall, generating tensile stresses on the outer surface. If these stresses exceed the material's fracture strength, catastrophic failure occurs. This calculation is essential for process engineers to define safe operating windows for heating and cooling cycles, ensuring structural integrity while maintaining production throughput.
Methodology & Formulas
The analysis treats the glass wall as a thin slab, utilizing the Biot number (Bi) to characterize the heat transfer regime. When Bi exceeds 0.1, internal temperature gradients become significant, necessitating distributed (internal gradient) analysis rather than a lumped-capacitance approach. The empirical stress correlation used in this methodology is validated within the range 0.5 ≤ Bi ≤ 5.0.
The characteristic length (Lc) for a thin slab is defined as half the wall thickness (t):
\[ L_{c} = \frac{t}{2} \]
The Biot number is calculated using the convective heat transfer coefficient (h) and the thermal conductivity of the glass (k):
\[ \text{Bi} = \frac{h \cdot L_{c}}{k} \]
To determine the allowable step change in temperature (ΔTallow), we account for the material's Young's modulus (E), coefficient of thermal expansion (α), Poisson's ratio (ν), and the design tensile strength (σallow). The factor f(Bi) represents the dimensionless stress distribution based on the Biot number:
\[ \Delta T_{\text{allow}} = \frac{\sigma_{\text{allow}} \cdot (1 - \nu)}{E \cdot \alpha \cdot f(\text{Bi})} \]
For operational safety, a design allowable temperature change (ΔTdesign) is established by applying a safety factor (SF):
\[ \Delta T_{\text{design}} = \frac{\Delta T_{\text{allow}}}{\text{SF}} \]
| Parameter |
Condition / Regime |
Criteria |
| Biot Number (Bi) |
Distributed Analysis |
0.5 ≤ Bi ≤ 5.0 |
| Wall Thickness (t) |
Empirical Validity |
3.0 mm ≤ t ≤ 5.0 mm |
| Ramp Rate (R) |
Safety Guideline |
R ≤ 50 °C/min |
| Thermal Stress |
Failure Criterion |
ΔTactual ≤ ΔTdesign |
Worked Example: Thermal Shock Resistance of a Glass Container
A 4 mm thick soda-lime glass jar undergoes retort sterilization followed by sudden water cooling. The critical event is a step change in fluid temperature during cooling. Material properties and process parameters are taken from the design blueprint.
Knowns
- Thermal conductivity, \( k = 1.0\ \text{W/(m·K)} \)
- Density, \( \rho = 2500\ \text{kg/m}^3 \)
- Specific heat, \( c_p = 800\ \text{J/(kg·K)} \)
- Coefficient of thermal expansion, \( \alpha = 9 \times 10^{-6}\ \text{K}^{-1} \)
- Young's modulus, \( E = 70 \times 10^{9}\ \text{Pa} \)
- Poisson's ratio, \( \nu = 0.22 \)
- Allowable design tensile stress, \( \sigma_{\text{allow}} = 25 \times 10^{6}\ \text{Pa}\ (25\ \text{MPa}) \)
- Safety factor, \( \text{SF} = 1.5 \)
- Convective heat transfer coefficient (water cooling), \( h = 500\ \text{W/(m}^2\cdot\text{K)} \)
- Wall thickness, \( t = 4.0\ \text{mm} = 0.004\ \text{m} \)
- Actual temperature step change, \( \Delta T_{\text{actual}} = 33.392\ ^\circ\text{C} \)
- Actual ramp rate, \( R_{\text{actual}} = 40.0\ ^\circ\text{C/min} \)
Step-by-Step Calculation
- Compute the characteristic length (half-thickness for a thin slab):
\[
L_c = \frac{t}{2} = \frac{0.004}{2} = 0.002\ \text{m}
\]
- Calculate the Biot number:
\[
\text{Bi} = \frac{h \cdot L_c}{k} = \frac{500 \cdot 0.002}{1.0} = 1.0
\]
Since \( 0.5 \leq \text{Bi} \leq 5.0 \), the part qualifies for distributed analysis.
- Obtain the empirical factor \( f(\text{Bi}) \) for a plate approximation. For \( \text{Bi}=1.0 \), the factor is \( f = 0.6 \).
- Compute the numerator of the allowable step change formula:
\[
\sigma_{\text{allow}} \cdot (1 - \nu) = 25 \times 10^{6} \cdot (1 - 0.22) = 19\,500\,000\ \text{Pa}
\]
- Compute the denominator:
\[
E \cdot \alpha \cdot f = (70 \times 10^{9}) \cdot (9 \times 10^{-6}) \cdot 0.6 = 378\,000\ \text{Pa/K}
\]
- Determine the calculated allowable step change (exact fraction, then decimal):
\[
\Delta T_{\text{allow,calc}} = \frac{19\,500\,000}{378\,000} = \frac{3\,250}{63} \approx 51.587\ ^\circ\text{C}
\]
- Apply the safety factor. Using the exact value carries full precision through the division:
\[
\Delta T_{\text{design}} = \frac{\Delta T_{\text{allow,calc}}}{\text{SF}} = \frac{19\,500\,000}{378\,000 \cdot 1.5} = \frac{19\,500\,000}{567\,000} \approx 34.392\ ^\circ\text{C}
\]
- Compare the actual step change with the design limit. The actual change (\( \Delta T_{\text{actual}} = 33.392\ ^\circ\text{C} \)) is less than \( \Delta T_{\text{design}} = 34.392\ ^\circ\text{C} \), so the thermal shock criterion is satisfied.
Final Answer
The thermal shock resistance design PASSED. The allowable step change (including safety factor) is \( 34.392\ ^\circ\text{C} \), and the actual process step change is \( 33.392\ ^\circ\text{C} \). The Biot number (\( \text{Bi}=1.0 \)), wall thickness (\( t=4.0\ \text{mm} \)), and ramp rate (\( 40.0\ ^\circ\text{C/min} \)) all remain within the empirical guidelines.
