Nonlinear thermal conductivity in gases

. The nonlinear diﬀusion equation corresponds to the diﬀusion processes which can occur with a ﬁnite velocity. A.J. Janavičius proposed nonlinear equation which des-cribes more exactly the diﬀusion of impurities in Si crystals in many interesting practical applications. The heat transfer in gases is also based on diﬀusion of gas molecules from hot regions to the coldest ones with a ﬁnite velocity by random Brownian motions. In this case the heat transfer can be considered using similar nonlinear thermal diﬀusivity equation. The approximate analytical solution of this nonlinear equation can be used for the experimental analysis of thermal conductivity coeﬃcients using temperature proﬁles dependence on diﬀerent temperatures and pressures in gases.


Introduction
We discussed the thermodiffusion in semiconductors [5] excited by ultraviolet or X-rays [2] and metals heated by lasers [8]. The mathematical methods derived in papers [5,2,8,4] for the formulation and solution of the nonlinear heat transfer equations in gases have been used. These results can be important for engineering applications. We assume that the process of heat spreading is similar to other diffusion processes, and in the nonlinear case, can be described by nonlinear flow density [4]. In this case the frequency of the jumps [7] depends upon the molecule coordinates, concentration and temperature.
The coefficient of thermal conductivity of gases can be expressed in the following way [9] Here λ is mean values of a free path, v -mean velocities of molecular movement, nnumber of molecules per unit volume, k -Boltzmann constant, T -temperature of gases, µ -molar mass, R -gas constant, d -diameter of a molecule. The equation of thermal conductivity of gases [9] for one-dimensional case can be obtained from the heat flow continuity equation [9] ∂T ∂t where p = nkT , c p is the specific heat capacity of gases at constant pressure p, ρ is density of gases, D m -coefficient of thermal diffusion. The equation (2) can be rewritten in more a convenient form The nonlinear heat conduction equation [10] can be rewritten by introducing nonlinear equation for energy density E.
The complicated approximate analytical solution [10] E(x, t) of the equation (5) cannot be experimentally measured. In our case (4) temperatures T (x, t) can be measured directly and compared to the theoretical calculations.

Nonlinear heat diffusion equation in one-dimensional case
The solution of (4) can be obtained by introducing similarity variable [4] ξ and function f (ξ) which depends on environment temperature T e and constant D m0 .

The approximate analytical solution
We restrict the approximation (8) by polynomials. From the formula (10) at n = 0, 1, 2 we obtain the following system of equations: where in (13) was used approximation a 4 = 0 and a 0 = 1 for boundary condition f (0) = 1.
The solution f (z) (8) must satisfy equation (9) and boundary conditions at maximum penetration point of heat when T (ξ = ξ 0 ) = T e . The approximate expression of temperature T s at heat source is By using the approximate solution f (z) = a 0 + a 1 z + a 2 z 2 , a 0 = 1, we obtain the following system of equations Taking in the care that values of ∆T Te are sufficiently small, we can find from (17), (18), (19) the following approximate solutions (9) with constants presented in Table 1.
In Table 1 the approximate solution (8) of the equation (9) for temperature differences ∆T 1 = T s − T e = 0.05T e , ∆T 2 = 0.1T e and ∆T 3 = 0.2T e are presented for source temperature T s and environmental temperature T e .
The constants a 1 , a 2 practically do not change significantly at different ∆T Te what means that approximate solution is sufficiently exact for practical calculations of temperatures. The constant ξ 0 defining the maximum heat penetration depth x 0 (20)  which is directly proportional to square root from temperature of environment. The obtained solution of the equation (9) presented in Table 1 and experimental heat penetration depths can be used for defining the heat thermal diffusion coefficients D m0 T e [m 2 s −1 ]. In this way the dependence of D m0 on temperature at constant pressure p (2) can be obtained. We can find a more exact solution by the system of equations (11), (12) and (13), when a 4 = 0 and the boundary condition (15) is as follows The results for more exact solutions F (z) are presented in Table 2. For graphically representation in Fig. 1 we introduced the new functions f i and F i instead (f − 1) and (F − 1). We obtained that profiles f i and F i for obtained approximate solutions f (z) and F (z) presented in Table 1 and Table 2 practically coincide, when at heat source we used (f − 1) ≈ (F − 1) = (T s − T e )/T e = 0.2.

Results and conclusions
A similar task and approach was considered for nonlinear diffusion [4,1,6] in gases. In this case the definition of diffusion coefficients, which can be used at average values of frequencies of molecule jumps in the frontier region of diffusion profiles, was improved. For practical calculation of temperature profiles, the coefficients a 1 , a 2 , a 3 at ∆T /T e = 0.1 sufficient exact (21) and average values of v for approximate D m0 (2), the evaluation can be used. For definition of D m0 values dependencies on temperatures and pressures, the values 1 , a 2 , a 3 presented in Table 2 at ∆T /T e = 0.05 can be used. The results presented in Table 1 and Table 2 show that heat penetration depths (19) are approximately proportional to ∆T /T e values.