UNSTEADY HEAT TRANSFER OF A MONATOMIC GAS BETWEEN TWO COAXIAL CIRCULAR CYLINDERS

We consider a kinetic-theory treatment of the cylindrical unsteady heat transfer. A model kinetic equation of the BGK (Bhatnager-Gross-Krook) type is solved using the method of moments with a two-sided distribution function. We study the relations between the different macroscopic properties of the gas as the temperature, density, and heat flux with both the radial distance r and the time t. Also we study the problem from the viewpoint of irreversible thermodynamics and estimate the entropy, entropy production, entropy flux, thermodynamic forces, kinetic coefficients, the change in internal energy, and verify Onsager’s relation for nonequilibrium thermodynamic properties of the system.


Introduction
The Couette problem with heat transfer is one of the important situations in gas dynamics, which involve the nature of a rarefied gas near a solid surface.From the kinetic viewpoint, the rarefied cylindrical Couette flow has been analyzed by many authors.One of the main methods of constructing the transfer theory at arbitrary Knudsen number consists of the use of moments obtained from Boltzmann equation.The idea behind the method of moments consists of transforming the boundary value problems from the microscopic form to the form of equations of the continuum in which the principle variables that define the state of the system are certain moments of the distribution function.The motion of a rarefied gas between two coaxial cylinders: one is fixed and the other rotates with constant angular velocity, was studied in [2], using the moments method for obtaining a suitable solution for any Knudsen number.The flow of a gas between two coaxial cylinders, the inside cylinder being at rest with temperature T i , while the outside cylinder rotates at a constant angular velocity with temperature T * , was studied in [12].A numerical solution to the problem of a cylinder rotating in a rarefied gas and a comparison with the approximate analytical solution are given in [9].The problem of flow over a right circular cylinder within the framework of the kinetic theory of gases is studied in [1].The heat transfer of a cylindrical Couette flow of a rarefied gas with porous surfaces was investigated in [5] in the framework of the kinetic theory of gases.In [3], the cylindrical Couette flow problem of rarefied gases was numerically analyzed.This estimate is based upon the characteristic equations, which are equivalent to the BGK (Bhatnager-Gross-Krook) model of Boltzmann equation.Over a wide range of Knudsen numbers, it was found that the BGK solutions show good agreement with the other numerical solutions and with the existing experimental data of density profiles and drag coefficients for light gases such as argon and air.The free cylindrical Couette flow of a rarefied gas with heat transfer, porous surfaces, and arbitrary reflection coefficient was discussed in [7], solving the moment equations with convenient boundary conditions concerning heat transfer, porosity, and reflection at the surfaces using the small parameters method.The behavior of the velocity, density, and temperature was predicted by Mahmoud [8], he studied steady motion of a rarefied gas between two coaxial cylinders: one is fixed and the other is rotating with angular velocity Ω.The free unsteady expansion of an ideal gas into a vacuum was discussed by Kraiko [6], starting with onedimensional isentropic unsteady gas flow, he derived an asymptotic expansion for the density, and considering only the first term.It was concluded that the density decreases as a negative power of the time.In [11], a problem of a steady radial gas flow between two infinitely long coaxial cylinders, with boundary conditions of evaporation (emission) and condensation (absorption) which is formulated for a nonlinear kinetic equation with a model operator of collisions was studied.This problem is solved by the finite difference method.Considerable attention is paid to the flow from the inner evaporation cylinder to the absolutely absorbent outer one.Sone et al. [13,14] studied the steady behaviour of the gas between two rotating cylinders in the basis of the kinetic theory from the continuum to the Knudsen limit.In this paper, we study the unsteady heat transfer of a gas using the moments and perturbation methods, and we study the problem from the standpoint of irreversible thermodynamics to estimate the macroparameters and verify Onsager's relations applied to the system.

The physical problem and mathematical formulation
Consider an axially symmetric problem of an unsteady heat transfer of a rarefied gas between two coaxial cylinders of infinite length and circular cross-section with the radii r = r 1 and r = r 2 , where r 1 < r 2 , under the conditions of evaporation from the surface of the inner cylinder and absorption on the surface of the outer cylinder.The gas is evaporated from the cylindrical surface r = r 1 with the parameters of saturated vapor.The axially symmetric state of a rarefied gas at the distance r from the symmetry axis OZ of the Cartesian coordinate system, is determined by a distribution function f(r, c z , c r , c θ , t) of molecules over velocities, where c z , c r , and c θ are the components of the molecular velocity in the axial, radial, and azimuthal directions, respectively.In the space of molecular velocities, we also use the cylindrical coordinates (c z , c n , ψ) related to the orthogonal coordinates (c z , c r , c θ ) by the formulas Here, c n is the component of molecular velocity that lies in the plane perpendicular to the symmetry axis so that c 2 n = c 2 r + c 2 θ and ψ is the angle between the vectors c n and r, where r is the radius vector of a point of the physical space in the cylindrical coordinate system (z, r, θ).Assuming that the distribution function satisfies the kinetic equation with the BGK approximate collision operator, we will solve the problem in a simplified statement.The kinetic equation can be written as follows: where f 0 is the local Maxwellian distribution function and τ is the relaxation time.Hence, we obtain the transfer equation in cylindrical coordinates in the form where Q i is a function of the velocity.The moment ) is given by 144 Unsteady heat transfer where cos α = r 1 /r.Since the particles between the two cylinders are colliding with each other via binary collisions, the cone of influence will be generated [12].Let n 1 , T 1 be the density and temperature in region one of the cone and n 2 , T 2 be the density and temperature in region two.It is assumed from the beginning that the gas flows in a coordinate system moving with the mean gas velocity, furthermore, the distribution function is divided as follows: and the local Maxwillian distribution function f 0 is All necessary macroparameters of the gas, such as the number density, temperature, pressure, and radial heat flux are expressed in terms of the distribution function in the usual way: On the surface of each cylinder, we specify the flow of particles from the cylinder or, analogously, the distribution function for molecular velocities directed into the domain of integration.On the outer cylinder A. M. Abourabia et al. 145 r = r 2 , we assume that this function is the Maxwellian distribution with the known macroparameters n s and T s .If we then take Q = c r , c 2 , c r c 2 , 1 and substitute in (2.3), using the normalized quantities we get the following equations in nondimensional form 12) The boundary conditions can be taken as follows: where here K n is the Knudsen number and λ s is the mean free path at the outer cylinder.

The nonequilibrium thermodynamic predictions of the problem
In order to study the irreversible thermodynamic properties of the system, we begin with the evaluation of the entropy per unit mass s.It is A. M. Abourabia et al. 151 written in nondimensional form as also we get the entropy flux in the radial direction The law of increase of entropy is written in the local form [4] as where σ is the entropy production, hence where Following the general theory of irreversible thermodynamics [10], we could estimate the thermodynamic forces corresponding to the parameters n s and T s at the boundary: , (3.9) According to Onsager theorem there are kinetic coefficients that relate the entropy production to the thermodynamic forces via the relationship hence, we get Also we get the nondiagonal coefficients from the relations These kinetic coefficients must satisfy Onsager relation such that the diagonal coefficients must be positive and the following inequality must hold true: The temperature gradient between the two cylinders causes a work done on the gas, which gains energy from the surroundings.According to the first and second laws of thermodynamics (3.18)

Discussion
This paper deals theoretically with a problem of actual interest in the field of evaporation and condensation processes.In all calculations and figures we take the ratio q = 0.25 and the parameter γ=0.15.Due to the monotone increase with time and monotone decrease with radial distance of the temperature (Figure 4.1), the gas is evaporated from the inner cylinder and in the course of time and radial distance tends to condensate at the outer cylinder, this behaviour agrees with the numerical results in [14, Figure 9-a] and [13, Figure 11-a], also at a constant Knudsen number ~0.6 the temperature behaves in the same manner with radial distance [13, Figure 12].The reverse process appears clearly in Figure 4.2 for the number density, which in the course of time reaches  its maximum and minimum values at the outer and inner walls, respectively, this agrees with the numerical results made by [11] in the same range of Knudsen number.As the temperature decreases from the inner to the outer cylinder, the radial heat flux vector qr behaves similarly.In spite of the fact that qr increases nonlinearly with time in a nonmonetary A. M. Abourabia et al. 155    the system from the viewpoint of thermodynamics for irreversible processes.As the system is adiabatic, the temporal rate of the entropy will be positive (see   relations we determined the thermodynamic forces X 1 and X 2 as functions of r and t, they are opposite to each other, the first one behaves similar to the temperature (see   to r and t. Figure 4.12 shows the validity of inequality (3.16) which is in good agreement with the general rules of irreversible thermodynamics.For the monatomic gas, the total energy is conserved.At the inner cylinder, where the temperature is maximum, the atoms gain their

( 2 . 36 )
Hence we get the following four algebraic equations: A. M. Abourabia et al. 149 n

Figure 4 . 1 .
Figure 4.1.Variation of the temperature with radial distance and time.

Figure 4 . 2 .
Figure 4.2.Variation of the density with radial distance and time.

Figure 4 . 3 .
Figure 4.3.Variation of the heat flux with radial distance and time.

Figure 4 . 4 .
Figure 4.4.Variation of the heat conductivity with radial distance and time.

Figure 4 . 5 .
Figure 4.5.Variation of the pressure with radial distance and time.

Figure 4 . 6 .
Figure 4.6.Variation of the entropy with radial distance and time.

Figure 4 . 6 )
, consequently there is a source of entropy or entropy production σ which is always a positive value with respect to the radial distance r and time t, but it is an increasing function of time and a decreasing function of radial distance (seeFigure 4.7).By Onsager A. M.Abourabia et al. 157

Figure 4 . 7 .
Figure 4.7.Variation of the entropy production with radial distance and time.

Figure 4 . 8 .
Figure 4.8.Variation of the thermodynamic force X 1 with radial distance and time.

Figure 4 . 8 )
, and the second behaves similar to the number density (see Figure 4.9).The diagonal coefficients are shown in Figures 4.10

2 Figure 4 . 9 .
Figure 4.9.Variation of the thermodynamic force X 2 with radial distance and time.

Figure 4 . 10 .
Figure 4.10.Variation of the kinetic coefficient L 11 with radial distance and time.

Figure 4 . 11 .
Figure 4.11.Variation of the kinetic coefficient L 22 with radial distance and time.