Ohmic Heating Lethality Distribution

Reference ID: MET-D0FF | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Ohmic heating is a thermal processing technique where electrical current is passed through a food product, utilizing its internal electrical resistance to generate heat volumetrically. In Process Engineering, this method is highly valued for its rapid, uniform heating capabilities compared to conventional surface‑heat‑transfer methods. It is primarily used in the pasteurization and sterilization of viscous or particulate‑laden liquid foods, such as liquid egg, where maintaining product quality and ensuring microbial safety are critical; for detailed guidance on determining the required energy input, see our ohmic heating power calculation guide.

The Ohmic Heating Lethality Distribution calculation is essential for verifying that the temperature profile across the flow channel remains within specified limits. Because electrical conductivity is temperature‑dependent, non‑uniform flow velocities can lead to localized cold spots or overheating. This reference sheet provides the methodology to quantify the temperature variance at the exit of a continuous ohmic heater, ensuring the process meets safety and quality standards, and links to the detailed Ohmic Heating Lethality Calculation for further guidance.

Methodology & Formulas

The system is modeled as a steady-state, laminar flow between parallel plates. The temperature distribution is governed by the energy balance, accounting for volumetric heat generation due to the electric field.

The average velocity u_avg and the centerline velocity u_max are defined by the volumetric flow rate \(\dot{V}\), width W, and gap d:

\[ u_{avg} = \frac{\dot{V}}{W \cdot d} \] \[ u_{max} = 1.5 \cdot u_{avg} \]

The electric field E is determined by the applied voltage V_applied across the gap d:

\[ E = \frac{V_{applied}}{d} \]

The temperature evolution along a streamline is derived from the energy balance, where the temperature‑dependent conductivity is referenced to the inlet temperature: σ(T) = σ₀[1 + α(T − T_inlet)]. The temperature T at a given residence time τ is calculated using the following exponential growth model, which details the ohmic heating temperature rise:

\[ T(\tau) = \left( T_{inlet} + \frac{A}{B} \right) \cdot e^{B \cdot \tau} - \frac{A}{B} \]

Where the constants A and B are defined by the physical properties of the fluid:

\[ A = \frac{\sigma_{0} \cdot E^{2} \cdot (1 - \alpha \cdot T_{inlet})}{\rho \cdot c_{p}} \] \[ B = \frac{\sigma_{0} \cdot \alpha \cdot E^{2}}{\rho \cdot c_{p}} \]

The residence time τ for any specific streamline is defined by the length of the heater L and the local velocity u:

\[ \tau = \frac{L}{u} \]

The Fourier number Fo quantifies the relative importance of thermal conduction across the gap during the residence time:

\[ Fo = \frac{k \cdot \tau}{\rho \cdot c_{p} \cdot (d/2)^{2}} \]

Parameter	Regime / Condition	Threshold
Reynolds Number (Re)	Laminar Flow	1 ≤ Re ≤ 2000
Electric Field (E)	Safe Operating Range	1000 ≤ E ≤ 5000 V/m
Electrical Conductivity (σ₀)	Aqueous Food Range	0.1 ≤ σ₀ ≤ 1.0 S/m
Fourier Number (Fo)	Thermal Mixing	Fo > 0.5 (for low variance)

The lethality distribution is highly sensitive to the electrical conductivity mismatch between the solid particles and the carrier fluid. Process engineers should consider the following factors:

Particles with lower electrical conductivity than the fluid will experience slower heating rates, potentially creating cold spots.
Larger particle diameters increase the internal thermal resistance, leading to significant temperature gradients from the center to the surface.
The residence time distribution of larger particles often deviates from the fluid flow, requiring validation of the slowest-moving particle trajectory.

To maintain consistent lethality across the product stream, process engineers must control and monitor these critical variables:

Electrical field strength measured in V/cm.
The ratio of electrical conductivity between the solid and liquid phases.
Flow rate and system pressure to prevent localized boiling or arcing.
The frequency of the alternating current to minimize electrochemical effects at the electrode interface.

The carrier fluid acts as the primary medium for energy transfer. If the fluid conductivity is not properly matched to the target particles, the following risks occur:

If the fluid is too conductive, it may shunt the current, causing the particles to remain under-processed.
If the fluid is too resistive, the system may experience voltage spikes or power limitations.
Changes in fluid composition during processing, such as salt migration or concentration shifts, can alter the heating profile mid-run, necessitating real-time conductivity feedback loops.

Worked Example: Ohmic Heating Temperature Uniformity Analysis

Scenario: A continuous ohmic heater with parallel plate electrodes processes liquid egg. We analyze the temperature rise and spatial uniformity across the gap to ensure that the maximum temperature difference at the exit remains below a target of 2°C. This example demonstrates the calculation using the simplified slug-flow energy balance with temperature-dependent electrical conductivity.

Knowns

Gap distance: \( d = 0.05 \text{ m} \)
Electrode length: \( L = 2.0 \text{ m} \)
Electrode width: \( W = 0.5 \text{ m} \)
Density: \( \rho = 1020.0 \text{ kg/m}^3 \)
Specific heat: \( c_{p} = 3800.0 \text{ J/(kg·K)} \)
Thermal conductivity: \( k = 0.56 \text{ W/(m·K)} \)
Volumetric flow rate: \( \dot{V} = 0.0125 \text{ m}^3/\text{s} \)
Conductivity at inlet temperature (20°C): \( \sigma_{0} = 0.5 \text{ S/m} \)
Conductivity temperature coefficient: \( \alpha = 0.02 \text{ 1/°C} \)
Applied RMS voltage: \( V_{applied} = 250.0 \text{ V} \)
Inlet temperature: \( T_{inlet} = 20.0 \text{ °C} \)
Dynamic viscosity: \( \mu = 0.05 \text{ Pa·s} \)

Step-by-Step Calculation

Average velocity and centerline velocity: \[ u_{avg} = \frac{\dot{V}}{W \cdot d} = \frac{0.0125}{0.5 \cdot 0.05} = 0.5 \text{ m/s} \] \[ u_{max} = 1.5 \cdot u_{avg} = 1.5 \cdot 0.5 = 0.75 \text{ m/s} \]
Electric field strength: \[ E = \frac{V_{applied}}{d} = \frac{250.0}{0.05} = 5000.0 \text{ V/m} \]
Reynolds number (check laminar regime): \[ Re = \frac{\rho \cdot u_{avg} \cdot d}{\mu} = \frac{1020.0 \cdot 0.5 \cdot 0.05}{0.05} = 510.0 \] The flow is laminar (Re < 2000) as required.
Residence times for key streamlines: \[ \tau_{avg} = \frac{L}{u_{avg}} = \frac{2.0}{0.5} = 4.0 \text{ s} \] \[ \tau_{center} = \frac{L}{u_{max}} = \frac{2.0}{0.75} = 2.667 \text{ s} \] For the slow streamline near the wall (10% of gap from wall): \[ y_{offset} = 0.1 \cdot d = 0.005 \text{ m},\quad y_{from\ center} = \frac{d}{2} - y_{offset} = 0.02 \text{ m} \] \[ u_{slow} = u_{max} \left[1 - \left(\frac{2 \cdot y_{from\ center}}{d}\right)^2\right] = 0.75 \left[1 - \left(\frac{0.04}{0.05}\right)^2\right] = 0.27 \text{ m/s} \] \[ \tau_{slow} = \frac{L}{u_{slow}} = \frac{2.0}{0.27} = 7.407 \text{ s} \]
Heat generation coefficients for temperature-dependent conductivity: The energy balance for a fluid element moving with velocity \(u\) yields \(dT/dt = A + B T\), with: \[ A = \frac{\sigma_{0} \cdot E^{2} \cdot (1 - \alpha \cdot T_{inlet})}{\rho \cdot c_{p}} = \frac{0.5 \cdot (5000.0)^{2} \cdot (1 - 0.02 \cdot 20.0)}{1020.0 \cdot 3800.0} = 1.935 \text{ K/s} \] \[ B = \frac{\sigma_{0} \cdot \alpha \cdot E^{2}}{\rho \cdot c_{p}} = \frac{0.5 \cdot 0.02 \cdot (5000.0)^{2}}{1020.0 \cdot 3800.0} = 0.0645 \text{ 1/s} \]
Fourier number (check conduction mixing): \[ Fo = \frac{k \cdot \tau_{avg}}{\rho \cdot c_{p} \cdot (d/2)^{2}} = \frac{0.56 \cdot 4.0}{1020.0 \cdot 3800.0 \cdot (0.025)^{2}} = 0.00092 \] Since Fo << 0.5, thermal conduction across the gap is negligible during the residence time, and significant temperature spread is expected.
Exit temperatures for each streamline: The analytical solution is \(T(t) = (T_{inlet} + A/B) \cdot \exp(B \cdot t) - A/B\). \[ T_{center} = \left(20.0 + \frac{1.935}{0.0645}\right) \cdot \exp(0.0645 \cdot 2.667) - \frac{1.935}{0.0645} = 29.384 \text{ °C} \] \[ T_{avg} = \left(20.0 + \frac{1.935}{0.0645}\right) \cdot \exp(0.0645 \cdot 4.0) - \frac{1.935}{0.0645} = 34.717 \text{ °C} \] \[ T_{slow} = \left(20.0 + \frac{1.935}{0.0645}\right) \cdot \exp(0.0645 \cdot 7.407) - \frac{1.935}{0.0645} = 50.624 \text{ °C} \]
Temperature spread: \[ \Delta T = |T_{center} - T_{slow}| = |29.384 - 50.624| = 21.240 \text{ °C} \]

Final Answer

Under the given operating conditions, the maximum temperature difference across the gap at the exit is \(\Delta T = 21.240 \text{ °C}\). This exceeds the desired target of 2°C, indicating that the heater geometry or flow parameters must be adjusted (e.g., reduce gap, increase flow rate, or enhance radial mixing) to achieve sufficient temperature uniformity.

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