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Have also to check the heat(t) at the inner surface of copper (might be higher/faster than expected) as a rise of temperature of 10° on 100cm^3 can give as much "oomph" as 5° on 200cm^3 or 1° on 1000cm^3, so lack of thermal conductivity of air might not be the relevant factor to set the typical Tau. + some heat equations...
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Maybe if you are working on heat conduction aspects at the cavity's wall, this might help me if you have an idea of the Temp(t) of the inner skin. I know that copper is a very good thermal, conductor. The epoxy behind the (probably) much thinner copper of PCBs end plate not that much. The RF power is dissipated in the first µm depth skin. What gives instant temperature for the air molecules hitting the inner walls ? This could be a fast alternative to volumetric humid air heating.
Considering the copper thickness to be thermally insulated at z=0 and be subject to heat (energy per unit time, per unit area) "heatFlux" at z=thickness, the complete transient solution for
the temperature increase is (where "t" is time in seconds):
temperature=(heatFlux*thickness/thermalConductivity)*( (t/fourierTime) + ((1/2)*((z/thickness)^2) - (1/6)) + temperatureSum
where:
temperatureSum=(heatFlux*thickness/thermalConductivity)*(-(2/(Pi^2))*NSum[(((-1)^n)/(n^2))*(Exp[- (t/fourierTime)*((n*Pi)^2)])*Cos[n*Pi*(z /thickness)], {n, 1, Infinity}])
fourierTime = (thickness^2)/thermalDiffusivity
thermalDiffusivity = thermalConductivity/(density*heatcapacityperunitmass)
For copper, we have the following material properties:
density = 8940 (* kg / (m^3) *);
thermalConductivity = 390 (* W/(m * degK)*);
heatcapacityperunitmass = 385 (* J/(kg * degK)*);
therefore:
thermalDiffusivity = 0.00011331 (m^2)/s
fourierTime = (thickness^2)*8825.38
So we see that steady-state conditions occur very fast due to the very small thickness.
For example, for thickness = (1/32 inch) * (25.4/1000 m/inch)
fourierTime = 0.00556034 seconds
For example, for thickness = (1/4 inch) * (25.4/1000 m/inch)
fourierTime = 0.355862 seconds
Therefore the temperatureSum term is negligible for time responses in the order of seconds. The term ((1/2)*((z/thickness)^2) - (1/6)) is also negligible in comparison with (t/fourierTime), so essentially we are left with
temperature ~ (heatFlux*thickness/thermalConductivity)*(t/fourierTime)
~ heatFlux*t*/(density*heatcapacityperunitmass*thickness)
For copper,
density*heatcapacityperunitmass = heatcapacityperunitvolume = 8940*385 (* J/((m^3) * degK)*)
hence,
temperature ~ heatFlux*t*/((3.4419*10^6)*thickness)
where heatFlux has units of W/m^2, thickness in meters and t in seconds.
Calculation of the Heat Flux:
For the transverse electric mode TE012 (p. 18, Table 2. Tapered Cavity Testing: Summary of Results) of the "Anomalous Thrust..." paper by Brady et.al., the Input Power was 2.6 Watts.Adopting the latest measurement estimates by aero:
SmallDiameter = 0.1549 m
BigDiameter = 0.2793 m
and the input power gets converted into heat (see Greg Egan for discussion), and since the TE012 only the axial magnetic field is non-zero in contact with the big and small diameters, the heat flux is:
heatFlux = (2.6 W) /FluxedArea
where the FluxedArea is:
(Pi/4)*( SmallDiameter^2 + BigDiameter^2 ) / Factor
where
Factor = (ActualDiameter / DiameterOfAreaExperiencingHeatFlux)^2accounts for the fact that the magnetic flux in mode TE012 contacts only a fraction of the entire circular areas at the ends of the truncated cone.
Therefore,
heatFlux = (32.454 W/m^2) * Factor
Substituting this in our expression for temperature:
temperature ~ heatFlux*t*/((3.4419*10^6)*thickness)
~ (32.454 W/m^2) * Factor * t*/((3.4419*10^6)*thickness)
~ Factor * t*/((1.0605*10^5)*thickness)So, for thickness = 1/32 inches = 0.0007938 m and Factor = 3 (equivalent to a diameter being heated by the magnetic axial flux as 57.74% of the full diameter)
temperature = (t / 28.059 s) deg C
the temperature increase at 2.6 sec (the pendulum half period) is only 0.09 deg C, and at 30 sec is 1.07 deg CSo,
for thickness = 0.02 inches = 0.0005080 m and Factor = 3 (equivalent to a diameter being heated by the magnetic axial flux as 57.74% of the full diameter)
temperature = (t / 17.958 s) deg C
the temperature increase at 2.6 sec (the pendulum half period) is only 0.14 deg C, and at 30 sec is 1.67 deg C
Notice assumptions:
1) All input power is converted into heat
2) axial heat conduction away from heated zone is neglected
3) surface (at z=0) is assumed to be perfectly insulated (by the PCBoard composite material and the surrounding air)
4) diameter being heated by the magnetic axial flux assumed as 57.74% of the full diameter. For other values change "Factor" accordingly. In the worst case (assuming that the whole diameter is being heated, which is clearly not the case) Factor = 1, and hence the calculated temperatures are 1/3 of the value calculated.