This modelisation concerns the heating of water at 2.45 GHz for which the loss factor e" varies as 230/ T in the temperature range 25 < T < 75° C, thus the heat transfer absorption coefficient a varies as ß/T where ß 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 :
2ßz/To = - ln (1- d) - [ ln (1- d) + d] DT/To (1)
where To is the inlet temperature, DT is the total temperature variation from the inlet to the outlet, d is the ratio DT(z)/ DT where DT(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 :
2ßz*/To = - ln (1- d) (2)
where z* is the approximate position at d ratio.
Substracting eqn. (2) from eqn. (1) and dividing by eqn.(2) yields :
Dz/z* = [1 + d/ln(1- d)] DT/To (3)
where Dz = z-z*. Dz/z* is the relative error of the approximate calculation.
For smaller values of d, Dz/z* ~ (d/2) DT/To and varies linearly with d.
For values of d near to 1, Dz/z* ~ DT/To 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 d, eqn.(1) can be written as :
V(z)/v = - ln (1- d) - [ln (1- d) + d] DT/To (4)
where by definition v is the volume of a rectangular liquid slab at a point distance To/2ß away from the inlet.
The last equation will be valid for a liquid slab for which the height varies along the heat transfer.