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AERODYNAMICS
E. V. STUPOCHENKO
ON THE TEMPERATURE JUMP IN POLYATOMIC GASES
(Presented by Academician G. I. Petrov, January 18, 1965)
1. If a gas and the surface of a solid body bounding it are in thermodynamic equilibrium, then the influence of the surface on the distribution function of the gas molecules disappears outside the region of direct interaction of the molecules with the surface. Within this region only the density of the number of molecules changes. The situation is different for a nonuniformly heated gas. In the case of a monatomic gas, the influence of the wall is felt at distances of the order of the mean free path \(l_0\), distorting in this region the temperature field, which beyond it may be regarded as a linear function of distance.
Extrapolation of the linear course of the temperature directly to the surface leads to a temperature value different from the true temperature of the surface. The difference \(\delta T\) of these temperature values—the temperature jump—is, for not too large gradients, proportional to the derivative \(\partial T / \partial x\) in the region of the linear course of the temperature,
\[ \delta T = g \, \partial T / \partial x, \tag{1} \]
where the positive direction of the \(x\)-axis, normal to the surface, is taken to be the direction from the wall into the gas \((^1)\). The quantity \(g\) (of dimension length)—the temperature-jump coefficient—is of the order of \(l_0\) and depends on the accommodation coefficient \((^{1,2})\), which characterizes the degree of equalization of the (gas-kinetic) temperature of the molecules reflected from the wall and the temperature of the wall.
Usually these results are extended to polyatomic gases \((^2)\). In doing so, an essential premise, sometimes made implicitly, is the characterization of the exchange of energy between the gas molecules and the wall by means of some effective accommodation coefficient, common to all degrees of freedom. In this case the theory contains only one quantity of dimension length—the mean free path \(l_0\)—which determines the order of magnitude of \(g\).
However, the situation changes substantially if the accommodation coefficient for some type of internal energy differs appreciably from the accommodation coefficient for translational energy. In this case, in the immediate vicinity of the wall the equilibrium distribution of energy between the internal and translational degrees of freedom is disturbed. If by \(\tau\) we denote the relaxation time (the establishment of statistical equilibrium between the internal and translational degrees of freedom) and by \(D^*\) the diffusion coefficient determining the transfer of internal energy, then the length \(l \sim \sqrt{D^*\tau}\) gives the width of the zone of nonequilibrium energy distribution. In \((^3)\) the influence of retarded exchange of translational and vibrational energy on the values of thermal conductivity obtained by the heated-filament method was considered. Below we consider the influence of the zone \(l\) on the boundary conditions, formulated as a temperature jump, assuming \(l \ll L\), where \(L\) is the characteristic size of the temperature field.
- Let us assume that the \(f\) degrees of freedom of a molecule can be divided into two groups: \(f=f_1+f_2\). For the group \(f_1\), which includes, for example, the translational and rotational degrees of freedom, a relaxation time of the order of the free-path time \(\tau_0\) is characteristic, with
\[ \tau_0 \ll \tau, \tag{2} \]
where \(\tau\) is the time for equilibrium to be established between the \(f_1\) and \(f_2\) degrees of freedom. By \(f_2\) one may understand, for example, the vibrational degrees of freedom. Let the accommodation coefficient for the group \(f_2\) be smaller than for the group \(f_1\).
The region near the wall can be divided into three zones. In zone I, whose width \(a\) is of order \(l_0\), the (gas-kinetic) temperatures \(T_1\) and \(T\) of the first and second groups of degrees of freedom, respectively, vary in a complicated manner, and Fourier’s equation for the heat flux is not applicable. The conditions at the boundary of this zone can be written in the form:
\[ \delta_1 T_1 = g_1 \partial T_1/\partial x \big|_{x\sim a}, \tag{3} \]
\[ \delta_2 T = g_2 \partial T/\partial x \big|_{x\sim a}, \tag{3'} \]
where \(\delta_1 T_1\), \(\delta_2 T\), \(g_1\), \(g_2\) are the temperature jumps, defined in the usual way, and their coefficients for the groups \(f_1\) and \(f_2\) of degrees of freedom, respectively.
The quantity \(g_1\) can be related in the usual way (see, for example, \((^2)\)) to the accommodation coefficient \(a_1\) for the \(f_1\) degrees of freedom, since in zone I, by virtue of (2), the energy exchange between the groups \(f_1\) and \(f_2\) may be neglected. The quantity \(g_2\) depends in principle on the accommodation coefficients \(a_1\) and \(a_2\) for both groups, since the transfer of energy of the \(f_2\) degrees of freedom is also determined by the distribution of energy over the \(f_1\) degrees of freedom. In our treatment we shall characterize the processes in zone I directly by the quantities \(g_1\) and \(g_2\), since the main interest lies in elucidating the role of zone II.
- In this zone, whose width \(l \gg l_0\), there occurs a relatively slow process of equalization of the temperatures \(T_1\) and \(T\). The heat-flux density \(\mathbf q\) can be represented in the form
\[ \mathbf q = \mathbf q_1 + \mathbf q_2, \tag{4} \]
where \(\mathbf q_i = n\overline{\varepsilon_i \mathbf C}\); \(n\) is the number density of molecules; \(\varepsilon_i\) is the energy of the \(f_i\) degrees of freedom; \(\mathbf C\) is the thermal velocity; the bar denotes averaging with the distribution function at the given point.
In the relations (one-dimensional problem)
\[ q_1 = -\lambda_1 \frac{\partial T_1}{\partial x}, \quad q_2 = -\lambda_2 \frac{\partial T}{\partial x}. \tag{5} \]
\(\lambda_1\) is the coefficient of thermal conductivity of a gas whose molecules would possess only the degrees of freedom of the group \(f_1\). For the transfer of the energy \(\varepsilon_2\) one may adopt a diffusion mechanism \((^4)\) and set
\[ \lambda_2 = D^* n c_v^{(2)}. \tag{6} \]
Here \(c_v^{(i)}\) is the heat capacity of the \(f_i\) degrees of freedom at constant volume, referred to one molecule; \(D^*\) is a certain effective diffusion coefficient, differing from the ordinary self-diffusion coefficient \(D\) by taking into account a possible “relay” transfer of the energy \(\varepsilon_2\), and depending on \(T_1\) and \(T\).
In the stationary state, \(T_1\) and \(T\) outside zone I satisfy the equations
\[ \lambda_1 \frac{d^2 T_1}{dx^2} + A \frac{T-T_1}{\tau} = 0, \tag{7} \]
\[ \lambda_2 \frac{d^2 T}{dx^2} - A \frac{T-T_1}{\tau} = 0, \tag{7'} \]
where \(\lambda_1\) and \(\lambda_2\) are taken to be constant. The second term in (7) represents the energy transferred from the \(f_2\) degrees of freedom to the group \(f_1\) per unit volume per unit time.
The quantity \(A\) is equal to
\[ A = n c_v^{(1)} c_v^{(2)} / c_v^{(0)}, \tag{8} \]
where \(c_v^{(0)} = c_v^{(1)} + c_v^{(2)}\). From (7) and \((7')\), taking into account that \(T - T_1 \to 0\) as \(x \to \infty\), we find
\[ T - T_1 = C e^{-x/\alpha}; \tag{9} \]
\[ \alpha^2 = \frac{\lambda_1 \lambda_2}{A\lambda}\tau; \tag{10} \]
\[ T_1 = C \frac{\lambda_2}{\lambda} e^{-x/\alpha} + Bx + E, \tag{11} \]
where the constants \(B, C, E\) are determined from the boundary conditions. The length \(\alpha\) gives the refined width of zone II, which, with increasing \(x\), passes into the region of a constant temperature gradient (common to all degrees of freedom)
\[ \left.\frac{dT_1}{dx}\right|_{x\to\infty} = B. \tag{12} \]
Extrapolating the asymptotic behavior of the temperature \(T_1\)
\[ T_1 = Bx + E \tag{13} \]
to the values \(x \to 0\), we obtain
\[ T_{\rm st} + \delta = E, \tag{14} \]
where \(T_{\rm st}\) is the wall temperature, and \(\delta\) is the effective temperature jump. In view of the inequality \(1 - e^{-a/\alpha} \ll 1\), in equation (3) one may substitute the expression \(\delta_1 T_1 = T_1(0) - T_{\rm st}\) and replace the derivative \(dT_1/dx|_{x\sim a}\) by the quantity \(dT_1/dx|_{x=0}\); here \(T_1(0)\) and \(dT_1/dx|_{x=0}\) are calculated from (11). In this way, on the basis of (14), we obtain
\[ - C \frac{\lambda_2}{\lambda} + \delta = g_1 C \frac{\lambda_2}{\lambda}\frac{1}{\alpha} + g_1 B. \tag{15} \]
Similarly, from \((3')\), (9), (11), and (14) we obtain
\[ C \frac{\lambda_1}{\lambda} + \delta = - g_2 C \frac{\lambda_1}{\lambda}\frac{1}{\alpha} + g_2 B. \tag{16} \]
Eliminating \(C\) from (15) and (16), we obtain
\[ \delta = g\, dT_1/dx \quad \left(dT_1/dx \equiv \left.dT_1/dx\right|_{x\to\infty}\right), \tag{17} \]
where the coefficient \(g\) of the (effective) temperature jump is equal to
\[ g = \alpha \left[ \frac{\lambda_1}{\lambda}\frac{g_1}{\alpha} + \frac{g_1 g_2}{\alpha^2} + \frac{\lambda_2}{\lambda}\frac{g_2}{\alpha} \right] \Bigg/ \left[ 1 + \frac{\lambda_1}{\lambda}\frac{g_2}{\alpha} + \frac{\lambda_2}{\lambda}\frac{g_1}{\alpha} \right]. \tag{18} \]
- The quantity \(g_1\) is usually of the order of the free-path length \(l_0\). An estimate of the order of magnitude of \(\alpha\) gives
\[ \alpha \sim l_0 \sqrt{\tau/\tau_0}, \tag{19} \]
where \(\tau_0\) is the free-flight time. Usually \(\tau/\tau_0 \gg 1\); for example, for the equalization of the energies of the translational and rotational degrees of freedom, a time of the order \(10^2\)–\(10^5\tau_0\) is required. Therefore one may take \(g_1 \ll \alpha\).
If \(g_1\) and \(g_2\) are quantities of the same order, then approximately we have
\[ g=\frac{\lambda_1}{\lambda}g_1+\frac{\lambda_2}{\lambda}g_2. \tag{20} \]
For \(g_2\sim a\) (which is possible for a sufficiently small accommodation coefficient for the \(f_2\) degrees of freedom) we obtain
\[ g=\frac{\lambda_2 g_2}{\lambda a+\lambda_1 g_2}\,a. \tag{21} \]
For \(g_2\to\infty\) (the internal degrees of freedom \(f_2\) are practically excluded from direct heat exchange between the gas and the wall)
\[ g=g_1+\frac{\lambda_2}{\lambda}(a+g_1). \tag{22} \]
- Usually the condition for applicability of the concept of a temperature jump (without taking into account the influence of relaxation zone II considered here) is the inequality \(l_0\ll L\), where \(L\) is the characteristic size of the temperature field. In this case the quantity \(g\) in equality (1) does not depend on \(L\) and is inversely proportional to the gas pressure. When, as the dimensions of the apparatus or the gas pressure \(p\) decrease, \(l_0\) becomes comparable with \(L\), an additional dependence of the (formally introduced) temperature jump on \(L\) and \(p\) arises.
Result (18) was obtained under the assumption \(L\gg a\gg l_0\). Therefore analogous anomalies in the behavior of the temperature jump in polyatomic gases may appear already at considerably higher pressures and dimensions \(L\). This circumstance could be regarded as experimental confirmation of the influence of the relaxation zone.
In conclusion we note that the results presented can be extended to the case when the role of zone II is played by the region in which chemical equilibrium (in a gas mixture), disturbed by the wall, is restored.
Moscow State University
named after M. V. Lomonosov
Received
6 I 1965
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- S. Chapman, T. Cowling, The Mathematical Theory of Non-Uniform Gases, IL, 1960.