The form of the local energy conservation equation which governs the microwave water heating process during stady state conditions can be written as :
![\[\frac{dP(z)}{dz}=-{\rho}{\Phi}{c}\frac{dT(z)}{dz} \hspace{10mm} (1) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-6cc054a2820d869161628f1c4faf44ab_l3.png "Rendered by QuickLaTeX.com")
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] :
![\[ P(z)=P_{o}\hspace{1mm} exp\{-2 \int_0^z \alpha (z') \, dz' \}\hspace{10mm} (2)\]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-6cfb91999bb0c2ac355007eca6e05720_l3.png "Rendered by QuickLaTeX.com")
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,
![\[ \alpha (z') = \frac{\beta}{T(z')} \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-55588015d5c257aed797ddbf7551578b_l3.png "Rendered by QuickLaTeX.com")
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 :
![\[ P_o + \rho \Phi c T_o = P(z) + \rho \Phi c T(z) = \rho \Phi c T_f \hspace{10mm} (3) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-77779c72874954802f4001868d4fb65b_l3.png "Rendered by QuickLaTeX.com")
where is totally absorbed along the heat transfer, 
and 
are the input and output temperatures of water respectively.
Differentiating eqn.(2) gives :
![\[\frac{dP(z)}{dz}=-2\beta \frac{P(z)}{T(z)} \hspace{10mm} (4) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-486452fa9635de51e16330782aaf6cc0_l3.png "Rendered by QuickLaTeX.com")
Substracting eqn.(4) from eqn.(1) yields :
![\[{\rho}{\Phi}{c}\frac{dT(z)}{dz} = 2\beta \frac{P(z)}{T(z)} \hspace{10mm} (5) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-ffa75bdcacb59402c9bde5ecbec32904_l3.png "Rendered by QuickLaTeX.com")
eqn.(3) gives :
![\[ \frac{P(z)}{ T(z)} = {\rho}{\Phi}{c}\frac{T_f}{T(z)} - {\rho}{\Phi} c \hspace{10mm} (6) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-ef18c29aa0b8dc3347694998cd65cb94_l3.png "Rendered by QuickLaTeX.com")
Substituing 
, eq.(5) yields :
![\[ dz = \frac{dT(z)}{2\beta (\frac {T_f} {T(z)} - 1)} \hspace{10mm} (7) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-b13e1d932853eff3098f8b12a503a9a7_l3.png "Rendered by QuickLaTeX.com")
Integrating eqn.(7) along the heat transfer gives :
![\[ 2\frac{\beta{z} }{T_o} = - \ln ( 1 -\delta ) - [ \ln (1 -\delta) + \delta ] \frac{\Delta T}{T_o} \hspace{10 mm} (8) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-6a926d7042cc4b6ae7d4e05d1e956f8a_l3.png "Rendered by QuickLaTeX.com")
where 
and:
![\[ \delta = \frac{T(z) - T_o }{\Delta T} = \frac{\Delta T(z)}{ \Delta T} \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-9242bf91763886eaad4f639d023dfb47_l3.png "Rendered by QuickLaTeX.com")
is related to the process parameters by the formula :
![\[ \Delta T = \frac{P_o }{\rho \Phi c} \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-6254af43f75c8ac4ae05fa1b7721ee76_l3.png "Rendered by QuickLaTeX.com")
Ignoring the absorption coefficient variation with 
gives the simplified equation :
![\[ 2\frac{\beta{z^*} }{T_o} = - \ln ( 1 -\delta ) \hspace{10 mm} (9) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-73a6a07c9cafdf6fcec15e1bc8872bd5_l3.png "Rendered by QuickLaTeX.com")
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:
![\[ \alpha '(z')= \alpha (z') \frac{h(z')}{b} \hspace{10mm} (10)\]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-6a32bb8bbf65c8f017696ad1139207c3_l3.png "Rendered by QuickLaTeX.com")
where 
is the height of the wave guide. By following the same procedure as for the rectangular slab, eqn.(7) becomes :
![\[ \frac{h(z)}{b}dz = \frac{dT(z)}{2\beta (\frac {T_f} {T(z)} - 1)} \hspace{10mm} (11) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-8e5f1eb451283ef752bb051e45964c25_l3.png "Rendered by QuickLaTeX.com")
Integrating eqn.(11) along the heat transfer gives :
![\[ \frac{V(z) }{v} = - \ln ( 1 -\delta ) - [ \ln (1 -\delta) + \delta ] \frac{\Delta T}{T_o} \hspace{10 mm} (12) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-fd7b8d17a4bea4f15b04d9da62d1282f_l3.png "Rendered by QuickLaTeX.com")
where 
is the necessary volume of water which leads to the required ratio . Therefore :
![\[ V(z)=\int_0^z e h (z') \, dz' \}\hspace{10mm} \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-8b53c96d077757f6a8e30661f9303d5a_l3.png "Rendered by QuickLaTeX.com")
By definition 
is the volume of a rectangular liquid slab of eight 
at a point distance 
away from the inlet :
![\[ v =\frac{be T_o} { 2\beta } \hspace{10mm} (13) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-bba9fcb83de52654af5eeebe1b696762_l3.png "Rendered by QuickLaTeX.com")
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.
