Cold Point Temperature Prediction

Reference ID: MET-33B9 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The Cold Point Temperature Prediction calculation is a fundamental procedure in thermal process engineering, specifically designed for the sterilization of conduction‑heated solid food products packaged in cylindrical containers, as explained by Newman's Law for unsteady heating of finite solids. In this context, the cold point refers to the geometric center of the can, where the temperature rise is slowest due to the maximum distance from the heat‑transfer surfaces.

This calculation is critical for ensuring food safety and regulatory compliance. By predicting the temperature at the cold point over time, engineers can determine the necessary retort residence time to achieve a target lethality (sterilization value), and for detailed guidance see our process time calculation for a target F0 value. This methodology is standard practice in the canning industry for products like tuna or meat pastes, where heat transfer occurs primarily through conduction rather than convection.

Methodology & Formulas

The calculation utilizes the Ball formula, which approximates the transient heat conduction in a finite cylinder. The process assumes a constant surface temperature equal to the retort temperature, which is valid when the Biot number (Bi) is sufficiently high (Bi > 40).

1. Thermal Diffusivity

The thermal diffusivity (\(\alpha\)) represents the rate at which heat propagates through the material:

\[ \alpha = \frac{k}{\rho \cdot c_{p}} \]

2. Heating Rate Index (\(f_{h}\))

The heating rate index represents the time required for the temperature difference between the retort and the product to change by one log cycle. It is calculated based on the geometry of the cylinder:

\[ f_{h} = \frac{\ln(10)}{\alpha \cdot \left( \left( \frac{\lambda_{1}}{R} \right)^{2} + \left( \frac{\lambda_{2}}{L} \right)^{2} \right)} \]

3. Lag Factor (\(j_{c}\))

The lag factor accounts for the initial period of heating where the temperature response is not yet linear on a semi-logarithmic scale:

\[ j_{c} = \left( \frac{2}{\lambda_{1} \cdot J_{1}(\lambda_{1})} \right) \cdot \left( \frac{2 \cdot \sin(\lambda_{2})}{\lambda_{2}} \right) \]

4. Center Temperature Prediction

The temperature at the cold point at any given time (\(t\)) is determined by the following relationship:

\[ T(t) = T_{r} - (T_{r} - T_{0}) \cdot j_{c} \cdot 10^{-\frac{t}{f_{h}}} \]

Validity and Regime Criteria

Parameter	Condition	Engineering Implication
Biot Number (Bi)	Bi > 40	Surface resistance is negligible; constant surface temperature assumption holds.
Fourier Number (Fo)	Fo_effective > 0.1	System has moved past the initial transient; first-term approximation is valid.
Time (t)	t > 0.1 · f_h	System is in the linear-log regime; Ball formula is applicable.

The Ball formula for conduction-heated cans assumes a constant retort temperature and negligible external surface resistance to heat transfer. Key conditions:

Constant retort temperature: The model uses a single, uniform retort temperature (\(T_{r}\)) for the entire process. Real retorts experience a come-up time; if this is significant, a corrected effective retort temperature or a time-integrated lethality approach should be used.
High Biot number (Bi > 40): This ensures the can surface reaches retort temperature almost instantaneously. For steam or water-immersion retorts this is generally valid; for air or superheated steam systems, surface resistance must be accounted for.
No ambient fluctuations: The model does not dynamically adjust for fluctuating steam pressure, cooling water temperature, or ambient air conditions during the hold period. Engineers must ensure the retort control system maintains a stable set-point.

To predict the cold point temperature in a conduction-heated cylindrical can, the following inputs are required:

Product thermal properties: Thermal conductivity (\(k\)), density (\(\rho\)), and specific heat capacity (\(c_{p}\)) – or alternatively the thermal diffusivity (\(\alpha\)) if measured directly.
Can geometry: Radius (\(R\)) and half-height (\(L\)) of the cylindrical container.
Process conditions: Initial product temperature (\(T_{0}\)), retort temperature (\(T_{r}\)), and the elapsed processing time (\(t\)).
Geometric eigenvalue constants: \(\lambda_{1} = 2.4048\) (first root for infinite cylinder) and \(\lambda_{2} = \pi/2\) (first root for infinite slab), along with \(J_{1}(\lambda_{1}) = 0.5191\).

If the predicted cold point temperature does not match experimental heat-penetration data, calibration should follow standard Ball-method procedures:

Experimental determination of \(f_{h}\) and \(j_{c}\): Conduct a heat-penetration test by placing a thermocouple at the cold point, recording time-temperature data, and plotting the log of the unaccomplished temperature difference \((T_{r} - T)\) versus time. The reciprocal slope of the linear portion gives the experimental \(f_{h}\), and the intercept ratio yields the experimental \(j_{c}\).
Replace theoretical with experimental values: Use the experimentally measured \(f_{h}\) and \(j_{c}\) directly in the Ball formula instead of the geometrically derived values. This accounts for real-world effects such as product heterogeneity, fill consistency, headspace, and container shape imperfections.
Verify temperature sensor placement: Ensure the thermocouple is precisely at the geometric center. Off-center placement is a common source of systematic error between modeled and measured cold-point temperatures.

Worked Example: Cold Point Temperature Prediction for a Conduction-Heated Can

Scenario: A cylindrical can (diameter 0.084 m, height 0.115 m) filled with solid tuna (conduction heating) is sterilized in a still retort. The retort is maintained at 121 °C and the initial product temperature is 20 °C. The cold point is at the geometric center of the can. We predict the center temperature after 40 minutes of processing, using the first-term log model (Ball formula). The Biot number is very large (Bi > 40), so surface resistance is negligible.

Knowns (Input Parameters and Units):

Can radius: \(R = 0.042\) m
Can half-height: \(L = 0.0575\) m
Thermal conductivity: \(k = 0.5\) W/(m·K)
Density: \(\rho = 1080.0\) kg/m³
Specific heat: \(c_{p} = 3600.0\) J/(kg·K)
Initial temperature: \(T_{0} = 20.0\) °C
Retort temperature: \(T_{r} = 121.0\) °C
Process time: \(t = 40.0\) min = 2400.0 s
First cylinder root: \(\lambda_{1} = 2.4048\)
First slab root: \(\lambda_{2} = \pi/2 \approx 1.571\)
Bessel function: \(J_{1}(\lambda_{1}) = 0.5191\)
Natural log of 10: \(\ln(10) \approx 2.303\)

Step-by-Step Calculation:

Thermal diffusivity (m²/s and m²/min):
\(\alpha = \dfrac{k}{\rho \cdot c_{p}} = \dfrac{0.5}{1080.0 \cdot 3600.0} = 1.286\times10^{-7}\) m²/s.
Convert to per minute: \(\alpha_{\text{min}} = \alpha \cdot 60 = 7.716\times10^{-6}\) m²/min.
Fourier numbers (cylindrical, slab, effective):
\(Fo_{\text{cyl}} = \dfrac{\alpha \cdot t}{R^{2}} = \dfrac{1.286\times10^{-7} \cdot 2400.0}{(0.042)^{2}} = 0.175\).
\(Fo_{\text{slab}} = \dfrac{\alpha \cdot t}{L^{2}} = \dfrac{1.286\times10^{-7} \cdot 2400.0}{(0.0575)^{2}} = 0.093\).
\(Fo_{\text{eff}} = \dfrac{\alpha \cdot t}{R^{2} + L^{2}} = \dfrac{1.286\times10^{-7} \cdot 2400.0}{0.001764 + 0.003306} = 0.061\).
Note: \(Fo_{\text{eff}} < 0.1\); the first-term model may have ~10% error, but we proceed as requested.
Rate factor \(f_{h}\) (min):
\(f_{h} = \dfrac{\ln(10)}{\alpha_{\text{min}} \left[ (\lambda_{1}/R)^{2} + (\lambda_{2}/L)^{2} \right]}\)
\((\lambda_{1}/R)^{2} = (2.4048 / 0.042)^{2} = 3278.38\) m^-2.
\((\lambda_{2}/L)^{2} = (1.571 / 0.0575)^{2} = 746.28\) m^-2.
Sum = 4024.66 m^-2.
\(f_{h} = \dfrac{2.303}{7.716\times10^{-6} \cdot 4024.66} = 74.15\) min.
Lag factor \(j_{c}\) (dimensionless):
\(j_{c,\text{cyl}} = \dfrac{2}{\lambda_{1} \cdot J_{1}(\lambda_{1})} = \dfrac{2}{2.4048 \cdot 0.5191} = 1.602\).
\(j_{c,\text{slab}} = \dfrac{2 \cdot \sin(\lambda_{2})}{\lambda_{2}} = \dfrac{2 \cdot \sin(1.571)}{1.571} = 1.273\).
\(j_{c} = j_{c,\text{cyl}} \cdot j_{c,\text{slab}} = 1.602 \cdot 1.273 = 2.039\).
Center temperature using Ball formula:
Temperature ratio: \(10^{-t / f_{h}} = 10^{-40.0 / 74.15} = 0.289\).
\(T_{c} = T_{r} - (T_{r} - T_{0}) \cdot j_{c} \cdot 10^{-t / f_{h}}\)
\(T_{c} = 121.0 - (121.0 - 20.0) \cdot 2.039 \cdot 0.289\)
\(T_{c} = 121.0 - 101.0 \cdot 0.589 = 121.0 - 59.49 = 61.51\) °C.

Final Answer: The predicted cold point (center) temperature after 40 minutes of retort processing is 61.5 °C.

Note: The low Fourier number (Fo_effective = 0.061) indicates that this first-term model may underpredict the actual temperature rise. For design purposes, either a multi-term solution or experimental validation is recommended.

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