Introduction & Context

Ohmic heating is an advanced thermal processing technique where electrical current is passed directly through a food product, utilizing its internal electrical resistance to generate heat volumetrically. Unlike conventional heat exchangers that rely on conductive or convective heat transfer from a surface, ohmic heating allows for rapid, uniform temperature increases, particularly in particulate-liquid mixtures.

In Process Engineering, calculating the lethality (F₀) is critical to ensure food safety and regulatory compliance. Because ohmic heating provides a rapid come-up time, the lethality accumulated during the heating phase is significant and cannot be ignored. This calculation is used to determine the total microbial destruction achieved during both the dynamic heating phase and the subsequent isothermal holding phase, ensuring the process meets the required sterilization targets for pathogens such as Clostridium botulinum.

Methodology & Formulas

The total lethality of the process is the sum of the lethality accumulated during the heating phase (F_0,heat) and the holding phase (F_0,hold). The calculation assumes a linear temperature rise during the ohmic heating period.

The instantaneous lethality rate L(t) at any temperature T is defined as:

\[ L(t) = 10^{\frac{T(t) - T_{\mathrm{ref}}}{z}} \]

For the heating phase, where temperature increases linearly from T₀ to T_f over time t_h, the integrated lethality is calculated as:

\[ F_{0,\mathrm{heat}} = \frac{z}{\ln(10)} \cdot \frac{t_h}{T_f - T_0} \left( 10^{\frac{T_f - T_{\mathrm{ref}}}{z}} - 10^{\frac{T_0 - T_{\mathrm{ref}}}{z}} \right) \]

For the holding phase, where the product is maintained at a constant temperature T_f for duration t_hold, the lethality is:

\[ F_{0,\mathrm{hold}} = t_{\mathrm{hold}} \cdot 10^{\frac{T_f - T_{\mathrm{ref}}}{z}} \]

The total process lethality is the summation of these two components:

\[ F_0 = F_{0,\mathrm{heat}} + F_{0,\mathrm{hold}} \]

Parameter	Description	Typical Empirical Range
z	Temperature sensitivity coefficient	7.0 – 12.0 °C
t_h	Heating residence time	5.0 – 15.0 s
T_f	Target sterilization temperature	< 140.0 °C (Equipment limit)
T_ref	Reference temperature	121.1 °C

The lethality value is calculated by integrating the time-temperature profile of the coldest point within the food product. For ohmic heating, the process engineer must account for the following factors:

The electrical conductivity of the food matrix, which changes as temperature increases.
The specific heat capacity and density of the product components.
The uniform internal heat generation rate resulting from the applied voltage gradient.
The kinetic parameters of the target microorganism, specifically the D-value and z-value.

Electrical conductivity is the critical variable in ohmic heating because it dictates the rate of heat generation. If the conductivity of the solid particles differs significantly from the liquid carrier, the heating rate will be non-uniform. Engineers must ensure:

Conductivity measurements are taken across the entire target temperature range.
The system accounts for potential changes in ionic mobility during the heating phase.
The power input is adjusted to maintain the required temperature at the slowest heating component.

Validating lethality in multi-phase systems requires verifying that the solid particles reach the target temperature for the required duration. Recommended steps include:

Using fiber-optic temperature sensors to monitor the core temperature of representative solid particles.
Performing heat distribution studies to identify cold spots within the ohmic chamber.
Comparing theoretical lethality models against experimental data obtained from inoculated pack studies.
Ensuring the residence time distribution (RTD) is accounted for in continuous flow systems.

Worked Example: Ohmic Heating Lethality Calculation for Particulate Soup

Scenario: A continuous ohmic heater processes a particulate soup contaminated with Clostridium botulinum spores. The soup enters at 20 °C and is heated linearly to 130 °C over 10 s, then held at 130 °C for 30 s. The target sterility is a 12D reduction (F₀ ≥ 3 min) using D_121.1 = 0.25 min and z = 10 °C.

Knowns:

Initial temperature, \( T_0 = 20.0 \,^{\circ}\mathrm{C} \)
Final temperature, \( T_f = 130.0 \,^{\circ}\mathrm{C} \)
Heating time, \( t_h = 10.0\ \text{s} \)
Holding time, \( t_{\mathrm{hold}} = 30.0\ \text{s} \)
Reference temperature, \( T_{\mathrm{ref}} = 121.1 \,^{\circ}\mathrm{C} \)
z-value, \( z = 10.0 \,^{\circ}\mathrm{C} \)
D-value at 121.1 °C, \( D_{121.1} = 0.25\ \text{min} \)
Sterility target, \( F_0 \geq 3.0\ \text{min} \) (12D reduction)

Step-by-Step Calculation:

Convert times to minutes:
\( t_h = \frac{10.0\ \text{s}}{60} = 0.167\ \text{min} \)
\( t_{\mathrm{hold}} = \frac{30.0\ \text{s}}{60} = 0.500\ \text{min} \)
Compute the exponential factors:
\( \displaystyle \frac{T_f - T_{\mathrm{ref}}}{z} = \frac{130.0 - 121.1}{10.0} = 0.890 \)
\( \displaystyle \frac{T_0 - T_{\mathrm{ref}}}{z} = \frac{20.0 - 121.1}{10.0} = -10.110 \)
Lethality during the heating phase (F_0,heat = 0.051 min):
The analytical integration for a linear temperature rise gives:
\[ F_{0,\mathrm{heat}} = \frac{z}{\ln(10)} \cdot \frac{t_h}{T_f - T_0} \left( 10^{\frac{T_f - T_{\mathrm{ref}}}{z}} - 10^{\frac{T_0 - T_{\mathrm{ref}}}{z}} \right) \] \[ F_{0,\mathrm{heat}} = \frac{10.0}{2.3026} \cdot \frac{0.167}{110.0} \left( 10^{0.890} - 10^{-10.110} \right) = 4.3429 \cdot 0.001518 \cdot (7.7625 - 7.76\times10^{-11}) \] \[ F_{0,\mathrm{heat}} \approx 0.051\ \text{min} \]
Lethality during the holding phase (F_0,hold = 3.881 min):
\[ F_{0,\mathrm{hold}} = t_{\mathrm{hold}} \cdot 10^{\frac{T_f - T_{\mathrm{ref}}}{z}} = 0.500 \times 10^{0.890} = 3.881\ \text{min} \]
Total lethality (F_0,total = 3.932 min):
\[ F_0 = F_{0,\mathrm{heat}} + F_{0,\mathrm{hold}} = 0.051 + 3.881 = 3.932\ \text{min} \]
Verification against sterility target (achieved):
Since \( 3.932\ \text{min} \geq 3.0\ \text{min} \), the process achieves the required 12D reduction.

Final Answer:
The total process lethality is \( F_0 = 3.932\ \text{min} \), which exceeds the commercial sterility target of 3.0 min. The ohmic heater and holding tube design is adequate for C. botulinum control.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

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