This modelisation concerns the heating of water at 2.45 GHz for which the loss factor 
varies as 
in the temperature range 25 < 
< 75° C, thus the heat transfer absorption coefficient 
varies as
![\[ \frac{\beta }{T} \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-1bb3252a200c6eb5f6a8f41d325db125_l3.png "Rendered by QuickLaTeX.com")
where 
is a physical constant parameter [4], [7]. Using the energy conservation equation in conjunction with the exponential decay of microwave power, a simple treatment (private notes) gives the equation :
![\[ 2\frac{\beta{z} }{T_o} = - \ln ( 1 -\delta ) - [ \ln (1 -\delta) + \delta ] \frac{\Delta T}{T_o} \hspace{10 mm} (1) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-8908cc4fa960940cb5e303b84f546b3e_l3.png "Rendered by QuickLaTeX.com")
where 
is the inlet temperature, 
is the total temperature variation from the inlet to the outlet, 
is the ratio 
/ where is the temperature variation at a point distance z away from the inlet. Ignoring the absorption coefficient variation along the heat transfer, an approximate calculation can be obtained and gives the well-known equation :
![\[ 2\frac{\beta{z^*} }{T_o} = - \ln ( 1 -\delta ) \hspace{10 mm} (2) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-980151f85f7f2bb05d7a349c152d4031_l3.png "Rendered by QuickLaTeX.com")
where 
is the approximate position at ratio. Substracting eqn. (2) from eqn. (1) and dividing by eqn.(2) yields :
![\[ \frac{\Delta z}{z^*} = [ 1 + \frac{\delta}{\ln (1 -\delta)} ] \frac{\Delta T}{T_o} \hspace{10 mm} (3)\]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-624025f8a9d31df2a1b3a45b9a8559d5_l3.png "Rendered by QuickLaTeX.com")
where 
, 
is the relative error of the approximate calculation.
For smaller values of 
,
![\[ \frac{\Delta z}{z^*} {\sim}\frac{\delta}{2}\frac{\Delta T}{T_o}\]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-fbab1b6178a2dc8473e7f4c4202a2504_l3.png "Rendered by QuickLaTeX.com")
and varies linearly with .
For values of near to 1,
![\[ \frac{\Delta z}{z^*} {\sim}\frac{\Delta T}{T_o}\]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-2a5cd7a9b0966e255934425b2fd6e667_l3.png "Rendered by QuickLaTeX.com")
which is the maximum relative error.
For useful numerical calculations eqn.(2) and eqn.(3) must be ploted. Introducing the necessary volume of water 
which leads to the required ratio , eqn.(1) can be written as :
![\[ \frac{V(z) }{v} = - \ln ( 1 -\delta ) - [ \ln (1 -\delta) + \delta ] \frac{\Delta T}{T_o} \hspace{10 mm} (4) \]](http://bouarfamahi.com/wp-content/ql-cache/quicklatex.com-175c7fdba89b7e00d02e9612eee1abae_l3.png "Rendered by QuickLaTeX.com")
where by definition 
is the volume of a rectangular liquid slab at a point distance 
away from the inlet. The last equation will be valid for a liquid slab for which the height varies along the heat transfer.
