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.