Archivo de la categoría: Fluids

Basis of Microwave Heating

Microwave oven has become very popular in recent years, and has become an essential appliance in any kitchen. However, microwave heating seems an esoteric, almost magical, issue for many people who have the oven at their home. In this post we are going to explain the basis of microwave heating, not only for the food heating, but also for industrial heating and HDW (hot domestic water).

In 1946, a British researcher from the Raytheon Corporation, Mr. Percy Spencer, working on RADAR applications, discovered that a candy bar in his pocket was melted. He was testing a magnetron and began experimenting, confining the EM field inside a metal cavity. He tested first with corn and then with a chicken egg. This latter one exploded.

He verified that a high intensity EM field affected food due to the presence of water inside. Water is a bad propagator of radio waves, because it has a high dielectric constant and losses. Being a polar molecule, when a variable EM field is applied, the dipole tends to be oriented in the direction of the field, and that makes the water molecule is agitated, increasing its temperature. The popular belief is that this only happens at 2,4 GHz, but it actually happens throughout the microwave band. This frequency is used by the ovens because it is a frequency within a free emission band known as ISM (short for Industrial, Scientific and Medical). However, there are heating processes at 915MHz and another frequencies..

First, the water, like almost all dielectrics, has under normal conditions a complex dielectric constant ε=ε’−jε”. When this complex dielectric constant is introduced into the Maxwell equations, the complex term means a dielectric conductivity, by the next expression

\sigma = \omega \epsilon" \epsilon_0

This conductivity is not produced by the mobility of electrons, but by the mobility of the polar molecules of water. Therefore, it is higher as the frequency is increased.

On the other hand, the presence of this conductivity limits the microwaves penetration in the water, attenuating the EM intensity with distance. It is related to the depth of penetration, expressed by

\delta_p=\dfrac {\lambda \sqrt{\epsilon'}}{2 \pi \epsilon"}

and therefore at higher frequency, lower penetration depth. If the intensity of the electric field is |E|, and, by the Ohm’s law, the volumetric power is given by

Q=\omega \epsilon" \epsilon_0 |E|^2

This volumetric power will affect a specific region of the water, causing heating.

On the other hand, there is a heat transfer effect due to thermal conductivity, such that the surface heat flux is

\dfrac {dQ_s}{dt}=-k \displaystyle \int_s {\vec \nabla T d \vec S}

Applying the divergence theorem, the variation of heat per unit volume will be

\dfrac {dQ_V}{dt}=-k \nabla^2 T

This flow distributes the temperature inside the volumetric element, lossing energy, and therefore its sign is negative.


Under macroscopic conditions, the energy per unit volume that must be applied to water increasing its temperature is given by

E_v=\rho_m c_e \Delta T

with ρM the water density and ce its specific heat, being ΔT the increasing of the temperature. Speaking in terms of power, we will have to

Q=\rho_m c_e \dfrac{dT}{dt}

where it must be calculate the global time variation of the temperature, and being a fluid that can be in movement, it must be applied the material derivative, an operator that includes the time variation and the convection. Applying this operator we may get

\dfrac{dT}{dt}=\dfrac{\partial T}{\partial t}+\vec v \vec \nabla T

and the volumetric power is given by

Q=\rho_m c_e \left(\dfrac{\partial T}{\partial t}+\vec v \vec \nabla T \right)-k\nabla^2 T

which is the expression that governs the water heating when a volumetric density of EM power Q is applied.

On the other hand, fluid movement is governed by the Navier-Stokes equations, through

\rho_M \dfrac {\partial \vec v}{\partial t}=-\vec \nabla P+\mu \nabla^2 \vec v + \rho_M \vec g

Where P is the volumetric pressure, μ the fluid viscosity and g the gravitational field.


In the case of a hot domestic water system, there would be two possibilities of heating:

  1. Through a closed circuit system moving a water flow, due to its very low viscosity (10-3 Pa·s).
  2. Using a vessel with rest water and accumulating the heat to transmit it to another areas.

In the first case, the volumetric power necessary to heat a closed circuit system must solve both with the thermal variation and the Navier-Stokes equations, and its efficiency is greater than in the second one, where the expression of the thermal increase is given by

Q+k\nabla^2 T=\rho_m c_e \dfrac{\partial T}{\partial t}

This equations can be solved using the FEM method, as we saw in the post about the simulation.

In any case, although both methods are possible, the first method will always be cheaper than the second, since the second can only be applied to raise the temperature of another fluid in motion and will need more energy due to the losses due to that transfering of heat..


Normally, any material that has losses by dielectric constant can be capable of being heated using microwaves, if these losses do not raise the electrical conductivity to values that cancel the electric field (in a perfect conductor, the electric field is zero). If we write the expression obtained in terms of electric field we get

\omega \epsilon" \epsilon_0 |E|^2+k\nabla^2 T=\rho_m c_e \left(\dfrac{\partial T}{\partial t}+\vec v \vec \nabla T \right)

and therefore, we can obtain a relationship between ε” and the increase of temperature at a given electric field |E|.


The human body is another dielectric which contains mostly by water. Therefore, the effect of the EM radiation on our body should cause heating. Let’s study what would be the field that would increase our temperature above 50o C in one minute, reducing the expressions to

\omega \epsilon" \epsilon_0 |E|^2=\rho_m c_e \dfrac{\Delta T}{\Delta t}

Taking ε”=4,5 (water at 2,4 GHz), knowing that the average human density is 1100 kg/m3 and its specific heat, 14,23 kJ/kg o C, it is got the next

|E|=\sqrt {\dfrac {1100 \cdot 14230 \cdot \left(\dfrac{50-33}{60} \right)}{2 \pi \cdot 2,4 \cdot 10^9 \cdot 4,5 \cdot 8,85 \cdot 10^{-12}}}=3,1 kV/m

and a WIFI router emits with less than 2 V/m field strength at 1 m. of distance. Therefore, a WIFI router will not cause heating in our body or even if we are close by it..

And a mobile phone? These devices are already powerful … Well, at its emission peak either, since at most it will emit with 12 V/m, and we need 3100 V/m, about 260 times more. So the mobile does not warm our ear either. And keeping in mind the depth of penetration, as much the EM radiation gets to penetrate about 2 cm, attenuating the field strength in half and power to the fourth part, due to the dielectric conductivity of our body. That without keeping in mind that each of our tissues has a different attenuation capacity depending on its composition and structure.


This post tries to explain the microwave heating phenomenon based on the ones that produce this heating, and its possible industrial applications, apart from those already known as the popular oven that almost every kitchen already has as part of its home appliance furniture. One of the most immediate applications is in the HDW, although applications have also been achieved in other industrial areas. And although the microwaves produce that heating, the necessary field strengths are very far from the radiation we receive from mobile communications.


  1. Menéndez, J.A., Moreno, A.H. “Aplicaciones industriales del calentamiento con energía microondas”. Latacunga, Ecuador: Editorial Universidad Técnica de Cotopaxi, 2017, Primera Edición, pp 315. ISBN: 978-9978395-34-9
  2. D. Salvi, Dorin Boldor, J. Ortego, G. M. Aita & C. M. Sabliov “Numerical Modeling of Continuous Flow Microwave Heating: A Critical Comparison of COMSOL and ANSYS”, Journal of Microwave Power and Electromagnetic Energy, 2016, 44:4, 187-197, DOI: 10.1080/08327823.2010.11689787