Reference ID: MET-9B74 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Microwave volumetric heat generation rate, Qv, quantifies the instantaneous conversion of electromagnetic energy into sensible heat inside a dielectric material. In process engineering, it underpins the design of microwave dryers, reactors, pasteurisers, and thawing units where rapid, selective, and penetration-limited heating is required. Knowing Qv allows engineers to predict temperature rise, optimise cavity geometry, prevent thermal runaway, and scale from laboratory to industrial ISM-band systems (915 MHz or 2.45 GHz).
Methodology & Formulas
Constitutive relation
The time-averaged power dissipated per unit volume is governed by the dielectric loss factor:
\[
Q_{v} = 2\pi f\varepsilon_{0}\varepsilon_{\text{r}}^{\prime\prime}E_{\text{rms}}^{2}
\]
where
Symbol
Description
Unit
\(f\)
microwave frequency
Hz
\(\varepsilon_{0}\)
permittivity of free space
F m-1
\(\varepsilon_{\text{r}}^{\prime\prime}\)
dielectric loss factor (dimensionless)
-
\(E_{\text{rms}}\)
root-mean-square electric field strength
V m-1
Unit conversion
To report in commonly used units:
\[
Q_{v}\ [\text{W cm}^{-3}] = \frac{Q_{v}\ [\text{W m}^{-3}]}{10^{6}}
\]
Validity regime
Parameter
Recommended Range
Consequence if Outside
\(\varepsilon_{\text{r}}^{\prime\prime}\)
5–30
Low heating efficiency or excessive absorption leading to surface overheating
\(E_{\text{rms}}\)
\(\le 2 \times 10^{4}\ \text{V m}^{-1}\)
Risk of air breakdown/arcing in cavity
\(f\)
0.915 GHz or 2.45 GHz
Non-ISM band; potential interference and non-compliance
Microwave heat generation rate, Qv, is the volumetric power absorbed by the material per unit volume. Typical units are W m-3. It is calculated from:
Qv = 2πfε0ε″r|E|2
f = frequency (Hz)
ε0 = 8.854 × 10-12 F m-1
ε″r = dielectric loss factor (dimensionless)
|E| = local electric field magnitude (V m-1)
Measure these temperature- and moisture-dependent properties at the exact operating frequency:
ε′r (dielectric constant) for field penetration depth
ε″r (dielectric loss factor) for direct heat generation
tan δ = ε″r/ε′r (loss tangent) for quick comparison
ρ (density) and Cp (specific heat) for transient thermal modeling
Both ε′r and ε″r rise with temperature for most polar materials, increasing Qv and creating thermal runaway risk. Compensation strategies:
Use feedback control with infrared or fiber-optic temperature probes
Program magnetron power vs. time based on pre-mapped ε″r(T) curves
Mix in low-loss ceramics or salts to flatten tan δ above the target temperature
Maintain similitude of the microwave field pattern and power density:
Keep the same power density (W kg-1) by scaling total power linearly with throughput
Preserve penetration depth by choosing similar frequency and bed thickness
Use multimode cavity mode-stirrers or conveyor speed to equalize residence time distribution
Validate with thermometric probes or inline microwave radiometry
Worked Example – Estimating Microwave Heat Generation Rate in a Food Slurry
A small-scale microwave pasteurisation line heats a pumpable vegetable purée at 2.45 GHz. The process engineer needs to know the volumetric heat generation rate to size the cooling section downstream. Laboratory measurements give the dielectric loss factor and the RMS electric field inside the product.
Frequency, \(f\) = 2.45 GHz = 2.45 × 109 Hz
Dielectric loss factor, \(\varepsilon''\) = 18
RMS electric field, \(E_{\text{rms}}\) = 15 kV m-1 = 1.5 × 104 V m-1
Vacuum permittivity, \(\varepsilon_0\) = 8.854 × 10-12 F m-1
Write the volumetric heat generation equation for microwave heating:
\[
Q_v = 2\pi f \varepsilon_0 \varepsilon'' E_{\text{rms}}^2
\]
Insert the known values:
\[
Q_v = 2\pi (2.45 \times 10^9)(8.854 \times 10^{-12})(18)(1.5 \times 10^4)^2
\]
