Reference ID: MET-6471 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Thermal process scale-up is a critical procedure in food engineering and pharmaceutical manufacturing, ensuring that the sterilization or pasteurization of a product remains consistent when transitioning from laboratory‑scale containers to industrial‑scale production; for a comprehensive guide on this transition, see our detailed overview of lab‑to‑production process scaling. The primary objective is to maintain heat penetration similarity, ensuring that the cold spot of the product reaches the required lethal effect (\(F_{0}\)) to guarantee safety and quality.
This calculation is essential for determining the heating rate index (\(f_{h}\)) and the lag factor (\(j\)), which characterize how quickly a product reaches the retort temperature. By utilizing these parameters, engineers can predict the required process time for larger containers or different geometries, preventing over-processing (which degrades quality) or under-processing (which poses safety risks).
Methodology & Formulas
The methodology relies on Ball theory, which models heat conduction in finite geometries. The process begins by determining the effective thermal diffusivity (\(\alpha\)) of the product using laboratory data, then applying that property to the production-scale geometry.
First, the effective thermal diffusivity is derived from the laboratory-scale cylindrical container (assuming radial conduction dominates for a tall can):
Where \(a_{\text{lab}}\) is the characteristic radius of the laboratory container, \(f_{h,\text{lab}}\) is the heating rate index, and \(\lambda_{\text{cylinder}}\) is the first eigenvalue for an infinite cylinder (\(\lambda_{\text{cylinder}} = 5.783\), corresponding to the square of the first root of the Bessel function \(J_{0}(x)=0\)). Ensure consistent units when applying this formula (e.g., convert \(f_{h}\) to seconds if \(\alpha\) is desired in m²/s).
Once \(\alpha\) is determined, the heating rate index for the production-scale container (modeled as an infinite slab, such as a pouch heated from both sides) is calculated as follows:
Where \(a_{\text{prod}}\) is the half-thickness of the production pouch and \(\lambda_{\text{slab}}\) is the first eigenvalue for an infinite slab (\(\lambda_{\text{slab}} = (\pi/2)^{2} \approx 2.467\)).
The geometric eigenvalue \(\lambda\) arises from the analytical solution to the transient heat conduction equation for regular geometries. For conduction-heated products with negligible surface resistance (Bi \(\to \infty\)), the values are:
Infinite cylinder: \(\lambda_{\text{cylinder}} = 5.783\) (square of the first root of \(J_{0}(x) = 0\), i.e., \(2.4048^{2}\)).
Infinite slab: \(\lambda_{\text{slab}} = 2.467\) (square of \(\pi/2\)).
Sphere: \(\lambda_{\text{sphere}} = 9.870\) (square of \(\pi\)).
These eigenvalues directly relate the heating rate index \(f_{h}\) to the product's thermal diffusivity and characteristic dimension. Using the correct eigenvalue for the geometry is essential for accurate scale-up.
A finite cylinder (can) experiences heat conduction from both the radial and axial directions. The total heating rate is governed by:
If the can's height-to-diameter ratio is large (typically > 3:1), the axial contribution is small and the radial infinite-cylinder approximation is acceptable. For squat cans, however, neglecting axial heat transfer will overestimate \(\alpha\) and lead to an under-conservative (unsafe) production process time. Always verify that the lab container geometry justifies the radial-dominant assumption, or use the full finite-cylinder solution.
The most frequent error is assuming that the heating rate index \(f_{h}\) scales linearly with the square of the characteristic dimension without verifying that the product's thermal diffusivity \(\alpha\) remains constant. In reality, scale-up pitfalls include:
Changes in product formulation or rheology that alter \(\alpha\) between lab and production batches.
Different headspace or fill levels that shift the cold spot location in the production container.
Applying the infinite-cylinder eigenvalue to a container that does not satisfy the geometric assumptions (e.g., using 5.783 for a very short can).
Neglecting the lag factor \(j\), which can differ between geometries and affect the come-up time correction.
Always validate scaled processes with experimental heat penetration tests using thermocouples at the confirmed cold spot.
Worked Example: Scaling a Canned Corn Product from Lab to Pouch
A food process development team has validated a thermal process for a conduction-heated corn product in a 1-kg cylindrical can. The team now wishes to scale this process to a production-scale pouch while maintaining heat penetration similarity. The target is to achieve the same heating rate parameter \(f_{h}\) (or an equivalent rapid heating) and a compatible lag factor \(j\) by choosing an appropriate geometry and verifying the calculated thermal diffusivity.
Production pouch: thickness = 25.0 mm, so half-thickness \(a_{\text{prod}} = 0.0125\ \text{m}\)
Constants: eigenvalue for infinite cylinder \(\lambda_{\text{cylinder}} = 5.783\), eigenvalue for infinite slab \(\lambda_{\text{slab}} = 2.467\), natural logarithm of 10 \(\ln(10) \approx 2.303\)
Step-by-Step Calculation:
Compute effective thermal diffusivity \(\alpha\) from the lab can data.
Using the Ball theory for a conduction-heated finite cylinder (radial component dominating):
\[
\alpha = \frac{a_{\text{lab}}^{2} \cdot \ln(10)}{f_{h,\text{lab}} \cdot \lambda_{\text{cylinder}}}
\]
Substitute the known numbers (expressing \(f_{h,\text{lab}}\) in seconds for unit consistency):
\[
\alpha = \frac{(0.030\ \text{m})^{2} \cdot 2.303}{(25.0\ \text{min} \times 60\ \text{s/min}) \cdot 5.783}
= \frac{0.0009 \cdot 2.303}{1500\ \text{s} \cdot 5.783}
= \frac{0.0020727}{8674.5\ \text{s}}
\]
\[
\alpha = 2.389 \times 10^{-7}\ \text{m}^{2}/\text{s}
\]
Rounded to four significant figures: \(\alpha = 2.389 \times 10^{-7}\ \text{m}^{2}/\text{s}\).
Predict the heating rate \(f_{h,\text{prod}}\) for the production pouch.
For an infinite slab (pouch) heated from both sides:
\[
f_{h,\text{prod}} = \frac{a_{\text{prod}}^{2}}{\alpha} \cdot \frac{\ln(10)}{\lambda_{\text{slab}}}
\]
Insert the values:
\[
f_{h,\text{prod}} = \frac{(0.0125\ \text{m})^{2}}{2.389 \times 10^{-7}\ \text{m}^{2}/\text{s}} \cdot \frac{2.303}{2.467}
\]
Compute the thermal resistance term and the geometric factor:
\[
\frac{a_{\text{prod}}^{2}}{\alpha} = \frac{0.00015625}{2.389 \times 10^{-7}} \approx 653.9\ \text{s}
\]
\[
\frac{\ln(10)}{\lambda_{\text{slab}}} = \frac{2.303}{2.467} \approx 0.9335
\]
\[
f_{h,\text{prod}} \approx 653.9\ \text{s} \cdot 0.9335 = 610.4\ \text{s}
\]
Convert to minutes:
\[
f_{h,\text{prod}} = \frac{610.4\ \text{s}}{60\ \text{s/min}} = 10.17\ \text{min}
\]
Rounded to four significant figures: \(f_{h,\text{prod}} = 10.17\ \text{min}\).
Validity checks on the calculated values.
- Effective thermal diffusivity \(\alpha = 2.389 \times 10^{-7}\ \text{m}^{2}/\text{s}\) lies within the empirical range for conduction-heated foods (0.8–2.5 \(\times 10^{-7}\ \text{m}^{2}/\text{s}\)).
- Production pouch heating rate \(f_{h,\text{prod}} = 10.17\ \text{min}\) is within the typical pouch range (5–30 min).
- Lab lag factor \(j_{\text{lab}} = 1.5\) is within the valid range for conduction-heated cylindrical cans (1.2–2.5).
All values pass the empirical range checks, confirming the calculation is physically realistic.
Final Answer:
The predicted heating rate for the production pouch is \(f_{h,\text{prod}} = 10.17\ \text{min}\). This shorter heating time (compared to the lab's 25.0 min) indicates that the cold point in the pouch will heat up faster, permitting a reduced process time while maintaining the target lethality \(F_{0}\). The lab-derived thermal diffusivity (\(\alpha = 2.389 \times 10^{-7}\ \text{m}^{2}/\text{s}\)) and the production geometry yield a consistent scale-up. A mandatory validation with thermocouples at the pouch cold spot must still be performed to confirm the actual \(f_{h}\) and \(j\) values before commercial implementation.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
