Heating of ceramic flat layer with cooled and heat-insulated walls
P.V. Kozlov, V.M. Lelevkin
Kyrgyz-Russian Slavic University, Bishkek, Kyrgyzstan
ABSRRACT
The mathematical model of ceramic material heating in a microwave with
cooled and heat-insulated walls was developed. Calculation of thermal and
electromagnetic characteristics of flat plate made of aluminum oxide has been carried
out subject to time of heating, plate thickness, electromagnetic field power and
frequency.
INTRODUCTION
At the present time, tasks of the mathematical modeling of ceramic material
heating by microwave radiation intensively are developed [1-6]. Nonlinear
dependences of temperature changes of ceramic material made of aluminum oxide
and silicium nitride on electromagnetic radiation input power and frequency were
established. Nearly constant temperature distribution on a ceramic cross-section was
obtained. Possibility of stepped monotonous heating of ceramic plate and
implementation of stable and unstable thermal conditions are showed.
In this work the model of flat plate heating in a microwave with cooled and
heat-insulated walls is offered.
MODEL
Unsteady heating of flat ceramic plate in a microwave with cooled and heatinsulted walls with symmetrical supply of electromagnetic waves (fig.1) with the


components E 0, 0, E z  E ( x)  exp( it ) , H 0; H y  H ( x),0exp( it ) is considered.
The process description of ceramic item microwave heating is carried out on
the base of the solution of non-stationary equation of energy balance and wave
equation:
T
  T  1
*
cC p
 
   i  0 EE ,
t x  x  2
(1)
d 2E
 k 02  K E  0 ,
dx 2
(2)
where T – temperature, E, E* – complex and complex conjugate amplitude of
electrical field,  0 – dielectric constant , , c, Cp – heat conductivity, density,
thermal capacity,  k   r  i i – complex permeability,  r  Re  k ,  i  Im  k real and imaginary parts of complex permeability, k 0   / c - wave number in the
vacuum,  – circular frequency, с – speed of light, x, y, z – rectangular coordinate
system with center in the middle of a plate.
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Fig. 1. Scheme of ceramic plate heating with symmetrical supply of electromagnetic
field power: 2R, 2Rc – lateral dimensions of a microwave chamber and plate
thickness. Fields: I, III – gaseous medium of a microwave (region of conductive heat
removal to chamber’s walls), II – ceramic plate (region of electromagnetic field
energy dissipation).
In consequence of the symmetry of task rated operating conditions are
specified in the interval 0 ≤ x ≤ R. In the case of cooled walls of a microwave’s
chamber initial and boundary conditions are in the following form:
t = 0, 0 ≤ x ≤ R: T (x, 0) = TR, E(x, 0) = 0;
t > 0, x = 0:
x = R,
∂ T / ∂x = 0, ∂E/∂x = 0;
T = TR, E = ER.
On the surface (plate – gaseous medium) continuity of temperature and thermal flow
is supposed, and electrical field on the walls of chamber ER(t) is determined by input
power value of electromagnetic radiation Qn(t).
THE METHOD OF SOLUTION
As the analysis of stationary conditions and calculation of non-stationary task [6]
show, temperature drop on cross section of ceramic made of aluminum oxide
represents small value with respect to maximal temperature. Therefore, in order to
solve the wave equation (2) model approximation of uniformly distributed
coefficients can be used [6]. Obtained solution can be used at the model
approximation of stationary conditions to determine electrical field tensity in ceramic
material:
E( x, t )  E0 (t ) cos kx, k  k 0  K ,
(3)
.
In the center of a plate, E0(t) links to value of microwave radiation input power
Qп(t) and maximal temperature in the center of plate T0(t) (via dependence of
coefficients on T0) by the next ratio:
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E0 (t )  4 0 cQп (t ) / cos( kRc )  i  k sin( kRc ) .
(4)
Relations (3) and (4) define the source of energy release in the energy equation
(1) at specified law of temporal changes of electromagnetic radiation input power.
The heat balance equation (1) numerically is solved by the lines method, for this
equation digitization on spatial variable is performed by the volume control method.
The task of temperature determination comes to the solution of the simple differential
equations system by temporary variable in spatial grid nodes. In matrix form it can be
expressed as:
∂T / ∂t = [A] T,
(5)
where [А] is tridiagonal matrix. Numerical solution of equations (5) is carried out in
the computer system Matlab 7 with the function ode23t.
For reflection  and absorption η coefficients of electromagnetic power in
ceramic material the approximation of coefficient uniform distribution gives the next
dependences on temperature in the center of a plate. [1]:

(6)
For
cos kRc  i  K sin kRc
cos kRc  i  K sin kRc
2
,
  1  .
k 0 Rc  r tg  1
( tg   i /  r - tangent of angle of dielectric loss), absorption coefficient of a plate (6)
is defined by the next expression [6]:

the
2F
,
1 F
relatively
transparent
materials
tg  1 ,
sin 2
)
2
F
,   k 0 Rc  r .
1  ( r  1) sin 2 
 r tg(1 
(7)
The formula (7) shows, that dependence () suffers significant resonance quasiperiodic fluctuations with the period close to , and fluctuations amplitude practically
linear depends on the value k 0 Rc  r tg (fig.2). As a result of it non-monotonous
relation occurs between input power of electromagnetic field and temperature in the
center of a plate (fig.3 and fig.4). Maximums of the dependence Qп(T0) take a special
part at nonstationary ceramic heating, extracting parts of the heating process stability
(dQп/dT0>0 ) and instability (dQп/dT0<0 ).
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Fig. 2. Dependence of plate absorption coefficient on dimensionless parameter  /  .
Fig. 3. Reflection coefficient and density of input power as the functions of
electromagnetic radiation frequency f and temperature in the center of ceramic plate
for stationary states at microwave heating [6].
Fig. 4. Dependence of input power and reflection coefficient of electromagnetic field
on temperature in the center of ceramic plate.
RESULTS
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The calculation of microwave heating of a flat plate made of Al2O3 is carried
out at Qn = 80 kilowatt/m2, f= 30 GHz, Rc = 15 mm in a chamber with cooled walls R
=50 mm, filled with the air at the atmospheric pressure. Coefficients (, ρc, Cp , K) for
ceramic material made of aluminum oxide at the frequency f=30 GHz and for the air
are taken from the same sources as [6] .
As it follows from the calculation results (fig. 5), for this heating regime
increase of plate temperature up to Т0 = 1000°К during 15 hours progresses
approximately in proportion to time. Because of high thermal conductivity of
aluminum oxide the temperature profile on a plate cross-section Т(х, t) ≈ const. The
temperature of ambient air parabolically changes from Тс to the temperature of cooled
walls. Power dissipation of electromagnetic radiation insignificantly changes Qd ≈ 2
кВт/м2, reflection coefficient ρ ≈ 95%. Further, during 2 hours of heating (15 < t < 17
hours) reflection coefficient decreases to 30%, absorption coefficient increases
возрастает, Qd ~ 50 kilowatt/м2, the temperature of ceramic plate rapidly rises up to
1700°К. At t > 17 hours of heating speed of plate temperature rising gradually
decreases and during t ≈18,3 hours temperature reaches 2000°K.
During the heating process (fig. 5) such regimes can be divided: rapid heating
(max |Eo|(To), min (To)), slow heating – (min |Eo|(To), max (To)), quasi-stationary
(|Eo|(To)  const, (To)  const).
Time of plate heating tmax up to specified temperature 2000°К nonlinearly
depends on frequency and input power of electromagnetic field (fig. 6). At f=30 GHz
ceramic items’ heating time, as the function of electromagnetic radiation input power,
changes under near-exponential law. Dependence of heating time on frequency has
more complex oscillating character at fixed input power. It relates to non-monotonous
character of reflection and absorption coefficients of electromagnetic energy changes
in ceramic material (fig. 3). In the process of electromagnetic field’s frequency
increases the reduction of heating time of flat ceramic plate up to specified
temperature occurs.
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Fig. 5. Changes of T, η, E in a flat ceramic plate and in the air at heating in a
microwave with cooled walls.
Fig6. Influence of input power of electromagnetic field on ceramic plate heating.
Heat-insulted (adiabatic) chamber’s walls. The heating of ceramic items in
a chamber with heat-insulted walls, i.e. when conductive and radiation heat exchange
with environmental gaseous medium and cooled walls of a microwave’s chamber is
absent (thermal baths) х = R: ∂T / ∂x = 0, represents the great interest for practical
aims.
For the acceleration of heating process a chamber’s area occupied with air may
be excluded (Rc ≤ x ≤ R), and “adiabatic walls” on the boundary of ceramic plate (х =
Rc : ∂T / ∂x = 0) can be placed. «Adiabatic walls» must be made of dielectric material
with the following properties ( ≈ 0, εr = const, εi ≈ 0), which practically don’t take
heat from a plate in the heating process and don’t influence on electromagnetic waves
propagation.
As the calculation results shows (fig.7), temperature profile practically is
constant on cross-section at the flat ceramic plate heating in such “adiabatic” packing
or “thermal bath”. The plate heating occurs in 1,5 times faster ( t = 11,8 hours at
Qп=80 kilowatt/m2, f=30 GHz) than in a chamber with cooled walls. Character of
T(t), ρ(t), Qd(t) changes qualitatively remains the same as in the case of heating with
cooled walls (fig.7). Uniform volume heating of ceramic plate in T(x, t) ≈ T0(t) is
realized.
In a chamber with heat-insulted walls the dynamics of plate heating by
microwave radiation is defined by simple energy balance equation
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cC p
dT0 Qп
,

dt
2 Rc
(8)
As it follows from the fig. 7, ceramic plate temperature rises linearly by
heating time on sections of insignificant changes of absorption coefficient. Sharp
temperature rise is observed on the parts of resonance change of η(ξ) dependence.
Lets consider the next regimes of ceramic plate heating:
1. Heating with constant input power of electromagnetic radiation. In this case
the heating time of ceramic plate up to specified temperature is found by equation (8).
2R
t (T0 )  c
Qп
сC p
  dT .
T
T0
R
(9)
It follows from the (9), that heating time tmax up to specified temperature T0 depends
on field frequency and plate thickness by quasi-periodic form (fig.8).
2. Heating of ceramic plate with specified speed of linear temperature rise.
T0 (t )  TR  (Tm  TR )t / t m
Then, from the (8) temporal dependence of microwave radiation’s input power,
providing linear temperature rise in time through coefficients (ρc, Сp ,  ) is found:
Qп (t ) 
2 Rc (Tm  TR )  c C p
.
tm

(10)
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Fig. 7. Microwave heating of ceramic plate in a chamber with “adiabatic” walls in the
regime with specified input power Qп=80 kilowatt/m2 (on the top) and in the regime
of linear temperature rise (below).
Fig. 8. Dependence of heating time tmax at Qп=30 kilowatt/m2 on electromagnetic field
frequency (Rc = 15 mm) and plate thickness (f=30 GHz).
CONCLUSIONS
Simple one-dimensional mathematical model of the heating of flat ceramic
plate in a microwave with cooled and heat-insulted walls was worked out.
It was showed, that the heating of ceramic items is more effective to carry out
in a microwave with heat-insulted walls.
The heating time of ceramic items in a microwave (and counting time)
noticeably grows shorter at plate packing in heat-resistant material with low heat
conductivity (analog of “adiabatic walls” of a chamber) and “transparent” for
electromagnetic radiation material.
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