This modelisation concerns the heating of water at 2.45 GHz for which the loss factor \epsilon{"} varies as 230/ T in the temperature range 25 < T < 75° C, thus the heat transfer absorption coefficient \alpha varies as 

    \[ \frac{\beta }{T} \]

where \beta 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 :

    \[ 2\frac{\beta{z} }{T_o} = - \ln ( 1 -\delta ) - [  \ln (1 -\delta) + \delta ] \frac{\Delta T}{T_o}  \hspace{10 mm} (1) \]

where T_o  is the inlet temperature,  \Delta T is the total temperature variation from the inlet to the outlet, \delta is the ratio \Delta T(z)/\Delta T where \Delta T(z) 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 :

    \[ 2\frac{\beta{z^*} }{T_o} = - \ln ( 1 -\delta )   \hspace{10 mm} (2) \]

where z^*  is the approximate position at \delta ratio. Substracting eqn. (2) from eqn. (1) and dividing by eqn.(2) yields :

    \[ \frac{\Delta z}{z^*} =  [  1 + \frac{\delta}{\ln (1 -\delta)} ] \frac{\Delta T}{T_o}  \hspace{10 mm} (3)\]

where \Delta z = z-z^*\Delta z/z^*  is the relative error of the approximate calculation.

For smaller values of \delta ,

    \[ \frac{\Delta z}{z^*} {\sim}\frac{\delta}{2}\frac{\Delta T}{T_o}\]

and varies linearly with \delta.

For values of \delta near to 1,

    \[ \frac{\Delta z}{z^*} {\sim}\frac{\Delta T}{T_o}\]

which is the maximum relative error.

For useful numerical calculations eqn.(2) and eqn.(3) must be ploted. Introducing the necessary volume of water V(z)  which leads to the required ratio \delta , eqn.(1) can be written as :

    \[ \frac{V(z) }{v} = - \ln ( 1 -\delta ) - [  \ln (1 -\delta) + \delta ] \frac{\Delta T}{T_o}  \hspace{10 mm} (4) \]

where by definition v is the volume of a rectangular liquid slab at a point distance {T_o}/{2\beta } away from the inlet. The last equation will be valid for a liquid slab for which the height varies along the heat transfer.

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