Introduction & Context

The heat penetration factor, denoted as f_h, is a critical parameter in thermal process engineering used to quantify the rate of temperature change at the slowest heating point (cold spot) of a food container during sterilization; for a detailed methodology see our heat penetration factor (f h) determination guide. In the context of retort processing, f_h represents the time in minutes required for the temperature difference between the retort medium and the product to decrease by one log cycle.

Understanding the effect of product viscosity on f_h is essential for food safety and quality. As viscosity increases, the internal heat transfer mechanism shifts from natural convection to conduction. This transition significantly alters the heating rate and the lag factor (j), which directly dictates the required process time to achieve commercial sterility. This calculation is standard practice for validating thermal processes for liquid, semi-solid, and particulate-laden food products.

Methodology & Formulas

The thermal behavior of a product is governed by the Rayleigh number (Ra), which relates buoyancy forces to viscous and thermal diffusion forces. The transition between heating regimes is determined by the following physical relationships:

The Rayleigh number is calculated as:

\[ Ra = \frac{g \cdot \beta \cdot \Delta T \cdot L^{3} \cdot \rho^{2} \cdot c_{p}}{\mu \cdot k} \]

where L is the characteristic length (typically the container height H for vertical cans). The regime boundaries are defined by a lower threshold Ra_low = 10² and an upper threshold Ra_high = 10⁴.

For products in the conduction regime, the theoretical f_h for a finite cylinder is derived from the individual thermal resistances of the infinite cylinder (radial) and infinite slab (axial) components:

\[ \frac{1}{f_{h,\text{conduction}}} = \frac{1}{f_{h,\text{cylinder}}} + \frac{1}{f_{h,\text{slab}}} \] \[ f_{h,\text{cylinder}} = \frac{0.399 \cdot R^{2}}{\alpha} \] \[ f_{h,\text{slab}} = \frac{0.933 \cdot \left(\frac{H}{2}\right)^{2}}{\alpha} \]

For products in the convection regime, f_h is typically determined experimentally from heat penetration tests, as it depends strongly on container geometry, headspace, and product rheology. Empirical values are used in the Ball formula below.

The sterilization time required to reach a target temperature at the cold spot is determined using the Ball formula:

\[ t = f_{h} \cdot \left(\log_{10}(j \cdot I) - \log_{10}(T_{\text{retort}} - T_{\text{target}})\right) \]

where I = T_retort − T_initial is the initial temperature difference.

Regime	Rayleigh Number (Ra)	Heating Characteristic
Convection	Ra > 10⁴	Rapid heat transfer; low f_h; j ≈ 1.0–1.4
Transitional	10² ≤ Ra ≤ 10⁴	Mixed mechanism; f_h sensitive to viscosity
Conduction	Ra < 10²	Slow heat transfer; high f_h; j ≈ 1.8–2.2

Note: The transition weight (w) used to interpolate between convection and conduction regimes is calculated based on the logarithmic scale of the Rayleigh number:

\[ w = \frac{\log_{10}(Ra) - \log_{10}(Ra_{\text{low}})}{\log_{10}(Ra_{\text{high}}) - \log_{10}(Ra_{\text{low}})} \]

where Ra_low = 100 and Ra_high = 10,000. For Ra ≤ Ra_low, w = 0 (pure conduction); for Ra ≥ Ra_high, w = 1 (pure convection). The weighted parameters are then:

\[ f_{h} = f_{h,\text{conduction}} \cdot (1 - w) + f_{h,\text{convection}} \cdot w \] \[ j = j_{\text{conduction}} \cdot (1 - w) + j_{\text{convection}} \cdot w \]

Higher viscosity significantly reduces the rate of heat penetration by shifting the primary mode of heat transfer from convection to conduction. As the fluid becomes more resistant to flow, the following effects occur:

The velocity of internal currents decreases, minimizing convective mixing.
The cold point of the container shifts toward the geometric center.
The total come-up time required to reach the target lethality increases.

When moving to a more viscous product, process engineers must re-validate the thermal process to ensure safety. Key adjustments include:

Increasing the process temperature or hold time to compensate for slower heat transfer.
Re-evaluating the container headspace to account for changes in product expansion.
Conducting heat penetration studies using thermocouples placed at the new predicted cold point.

Starch-based thickeners are highly sensitive to shear and temperature, which directly influences the final viscosity and, consequently, the thermal process safety. If the viscosity is not controlled:

Inconsistent hydration of starch can lead to localized cold spots within the container.
Variations in viscosity may result in under-processing if the product is thicker than the validated baseline.
Batch-to-batch variability can compromise the integrity of the established process authority filing.

Worked Example: Effect of Product Viscosity on Heat Penetration

Scenario: A still retort processes two liquid products in 307×409 cans. Product A is a low-viscosity broth with dynamic viscosity μ_A = 0.002 Pa·s. Product B is a thick cream soup with μ_B = 5000 Pa·s. The retort temperature is T_retort = 121.1 °C, initial product temperature is T_initial = 20.0 °C, and the target cold-spot temperature is T_target = 115.0 °C.

Knowns:

Gravity: g = 9.81 m/s²
Thermal expansion coefficient: β = 0.0004 K⁻¹
Thermal conductivity: k = 0.60 W/(m·K)
Density: ρ = 1020 kg/m³
Specific heat: c_p = 3800 J/(kg·K)
Thermal diffusivity: α = 1.55 × 10⁻⁷ m²/s
Can diameter: D = 0.0873 m
Can height: H = 0.1143 m
Can radius: R = 0.04365 m
Initial temperature difference: I = T_retort − T_initial = 101.1 K
Bulk temperature difference for Rayleigh: ΔT = 31.0 K
Product A viscosity: μ_A = 0.002 Pa·s
Product B viscosity: μ_B = 5000.0 Pa·s

Step-by-Step Calculation:

Calculate Rayleigh numbers. The Rayleigh number Ra = \(\frac{g \cdot \beta \cdot \Delta T \cdot H^{3} \cdot \rho^{2} \cdot c_{p}}{\mu \cdot k}\).
For Product A: Ra_A = 5.98 × 10⁸ (dimensionless).
For Product B: Ra_B = 239.4 (dimensionless).
Determine heating regime. The transition bounds are Ra_low = 100 and Ra_high = 10,000. Since Ra_A ≫ 10,000, Product A is in the pure convection regime. For Product B, Ra_B = 239.4 lies within the transition zone, so a weighted mixing model is used.
Assign heat penetration parameters.
For convection (experimentally determined): f_h,conv = 7.5 min and j_conv = 1.3.
For conduction (theoretical finite cylinder): f_h,cond = 65.42 min, j_conduction = 2.0.
Transition weight w_B = 0.1896 (from log-scale interpolation, ≈ 0.190).
Compute f_h and j for each product.
Product A (pure convection): f_h,A = 7.5 min, j_A = 1.3.
Product B (transitional): f_h,B = f_h,cond · (1 − w_B) + f_h,conv · w_B = 54.44 min, j_B = j_conduction · (1 − w_B) + j_conv · w_B = 1.867.
Calculate sterilization time (Ball formula).
t_sterile = f_h · [log₁₀(j · I) − log₁₀(T_retort − T_target)].
Product A: t_A = 10.0 min.
Product B: t_B = 81.2 min.

Final Answer:

Product A (broth): heat penetration factor f_h = 7.5 min, lag factor j = 1.3, sterilization time t = 10.0 min.
Product B (cream soup): heat penetration factor f_h = 54.4 min, lag factor j = 1.87, sterilization time t = 81.2 min.

"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