Introduction - Microwave-Assisted Concrete Technology: Production, Demolition and Recycling - page 29

One way to account for the reflections at the material-ambient interface
is to multiply the incident microwave power by the transmissivity coeffi-
cient as defined by Equation 1.21:

() =− ∂ ()
∂

Ix
x

PL x ×= c 2 β I e

− 2 β x

× c (1.36)

0

In addition, to account for the distribution of microwave power at the
material interface, the incident power should be multiplied by an appro-
priate function that simulates the power distribution associated with the
dominant microwave mode. For instance, by considering the TE 10 mode as
the dominant microwave mode in typical rectangular waveguides used in
industrial microwave heating, Lambert's law should be modified to account
for the sinusoidal power distribution associated with this mode.
We previously showed that the radiative microwave power dissipated per
unit volume of the dielectric material is directly proportional to the square
of the electric field intensity's norm (Equation 1.33). Keeping this in mind,
we may conclude that the power dissipated in a dielectric material sub-
jected directly to microwaves with a TE 10 dominant mode should follow a
sine 2 distribution. Therefore, prior to using Lambert's law for estimating
the variation in microwave power at the TE 10 mode, the original form of
Lambert's law may be modified as

ay
a

−

() =

Ix Psin

2

× e

− 2 β x

(1.37)

π

2

where 2 × a is the waveguide width, and P is the peak of the sine 2 function.
The peak value of the sine 2 distribution may be obtained by equating the
area under the sine 2 distribution to the area under a uniform (rectangular)
power distribution with the intensity of I 0 as assumed in the original form
of Lambert's law:

a
=×

ay
a

−

∫

2 aI 2 Psin

2

π dy PI = (1.38)

2

0 0

2

0

Hence, the final power dissipation function accounting for the power
distribution associated with the TE 10 mode as well as the reflection at the
material interface with ambient may be considered as

() =− ∂ ()
∂

Ix
x

ay
a

−

2 −2β x

PL x ×= ××× c 22 β I c in π × e (1.39)

0

2