I'm working in the plasma physics and trying to calculate the thermal conduction time or speed.
The plasma treated has cylindrical symmetry surrounded by the dielectric material.
Right now I need to calculate the time the energy or heat takes to move from the axis of the plasma to the wall with the given thermal conductivity.
I guess the time is function of not only conductivity but also temperature difference between two points thus let's assumed that the axis temperature is Te while dielectric is at room temperature..
This time is controlled by the ratio of thermal conductivity to the density and specific heat (thermal diffusivity). It is roughly proportional to the square of distance to the thermal diffusivity. The analytical solution for unstationary hear propogation in cylindrical coordinates can be found Carslaw, H. S.; Jaeger, J. C. (1959), Conduction of Heat in Solids (2nd ed.), Oxford University Press.
It depends on temperature, plasma radiates a lot over a few electronvolts accordingly Stefan Boltzmann equation
P=k*A*rho*T^4 where rho=5.84e-8 watts/K^4/m^2
k=thermal emissivity that is something lower than 1.
It depends also on pressure, at extreme high pressure there is a delay between plasma heating and radiation
You have Bremsstrahlung radiation also. You can see Stefan Boltzmann equation and thermal Bremsstrahlung in the excel table, formulas number 040 and 043 in the following link:
Data Useful formula and excel tables for plasma physics, plasma t...
Thanks for people and there is one more question. Estimation using thermal diffusivity as Evgene pointed out indicates that the conduction time is proportional to square of the diffusion length. I originally thought it would be linear. Is this because of the fact that thermal diffusion process actually angularly spreads like radiation does, instead of going straight? It may make sense as the thermal carriers randomly collide.
This is because the diffusion-advection equation (in 1D) has the form dT/dt ~ diffusivity d2T/dx2 + source terms (e.g. radiation term) - ( advection term = velocity* dT/dx )
where x can be undimentionalized as x* = x/sqrt (diffusivity*t) where
d = sqrt (diffusivity*t) would be characteristic conduction length during time t
That might be fine if the flow is laminar and parallel to the axis of the cylinder
If plasma is turbulent (check the Reynolds number ) or convective,
additional corrections would be needed
Oh, thanks. Thus..you suggested that the as long as the Reynolds number for plasma is small enough, I can apply thermal diffusion for conduction time estimation right? Could you give me more comments that how it is justified?
Adding one more comment: I concluded that I can use the Laser Flash Method ( thermal diffusivity measurement method) to determine characteristic thermal conduction time. In this method, there is a relationship of half conduction time (time it takes to reach half maximum value) to the thermal diffusivity and sample thickness. This formula also shows that the conduction time is proportional to the square of the thickness, as Evgene Burov mentioned above. Although this is for plane sample of finite thickness and infinite area, it can be also used for cylindrical radial heat flow since both are in 1-dimensional nature.
Donggyu, why are you interested in plasma conduction?
I ask this because plasma is usually confined by using magnetic fields, so it does not transmit energy to anything, so it is more interesting the plasma radiation
What is the temperature of your plasma?
Our plasma is gas-discharged one and electron temperature would be order of 1 eV. Thus energy loss of the radiation is really neglected and magnetic field is weak such that pinch effect is ignored, I thick.
Donggyu, the radiation could be neglected if the gas density is low.
I would recommend you to write the plasma data in the excel table that I said before in the equations number 40 and 43 in order to know the plasma radiation.
I agree with you that the magnetic field must be low
Book: Laser and electro-optics, Christopher C. Davis, 2nd edition, section 10.2, page 252
t~(D^2) /(cl)
D:diameter of tube
c:mean velocity of gas particles
l:average distance traveled by a gas particles between collisions.
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