The form of the local energy conservation equation which governs the microwave water heating process during stady state conditions can be written as :

where and are the microwave power and the temperature of water at a distance away from the inlet, , , and are the volumetric mass, flow rate and specific heat of water respectively. However the microwave power at a distance away from the inlet is given by, [4], [7] :

where and are the input microwave power and the heat transfert absorption coefficient respectively. For a WR 340 water heat transfer for which is the thickness of the liquid slab,

in the temperature range 25 < < 75° C where is the following physical constant :

= 7.35 230 dB.° C/m = 1.690 dB.° C/mm = 0.194 °C/mm where is in mm.

Integrating eqn.(1) gives :

where is totally absorbed along the heat transfer, and are the input and output temperatures of water respectively.

Differentiating eqn.(2) gives :

Substracting eqn.(4) from eqn.(1) yields :

eqn.(3) gives :

Substituing , eq.(5) yields :

Integrating eqn.(7) along the heat transfer gives :

where and:

is related to the process parameters by the formula :

Ignoring the absorption coefficient variation with gives the simplified equation :

where is the approximate position at ratio . Now let us consider a volume of water of thickness and height described as follows :

This design is well adapted to minimize microwave reflections on the inner pipe. The corresponding heat transfer absorption coefficient can be written as:

where is the height of the wave guide. By following the same procedure as for the rectangular slab, eqn.(7) becomes :

Integrating eqn.(11) along the heat transfer gives :

where is the necessary volume of water which leads to the required ratio . Therefore :

By definition is the volume of a rectangular liquid slab of eight at a point distance away from the inlet :

As the thickness is substantially smaller than the width of the heat transfer, varies linearly with [7], therefore as well as at a given ratio do not depend upon the shape of the inner pipe.

It will be noted that the volume is the appropriate parameter for describing the microwave heating process.