INVESTIGATION OF A SHOCK WAVE INFLUENCE ON CONDENSATION PROCESS IN A NOZZLE

A. L. Itkin ^{1} , A. V. Rodionov ^{2} ,

International Institute for Problems of the Asteroid Hazard, 191187, Russia, St.-Petersburg, Nab. Kutuzova, 10

and

E.G. Kolesnichenko ^{3}

Moscow State University, Institute of Mechanics Russia, Moscow, Michurinsky pr. 1

Abstract: Results of numerical calculations of a condensable gas flow in a nozzle with a break of the contour are presented. The unimolecular condensation theory by Itkin and Kolesnichenko is used to describe condensation kinetics. The main advantage of the method is that it allows to solve hydrodynamic equations in conjunction with a single one for supersaturation S. Other cluster concentrations and populations of the energetic levels are analytically expressed via S and T. It turns out that behind a shock wave in each cross-section x = const concentrations of a j-cluster are uniformly distributed in y for each j. This effect may be of great importance, for instance, for applying a cluster coating by precipitation.

Key words: condensation, nozzle, break of contour, numerical simulation, analytical solutions

1 Introduction

For modern technologies it is very important to produce clusters of various substances and flows with the cluster size distribution function (CSDF) determined. At the present time there are a few manners of influencing the CSDF: ultraviolet radiation, charged particles, a carrier gas and some others. In the present report we consider a new approach for forming CSDF of the determined type by the action of a shock wave on the condensable flow. In order to study this problem the numerical simulation of a two-dimensional flow with condensation in the supersonic nozzle with a wall’s bend was executed.

For the calculation of the flow of the gas mixture where one of the gas components is able to condense it is necessary to close respective gasdynamic equations by the kinetic ones. A short review of the familiar kinetic methods is given in [1]. The theories described in [1] don’t take into account nonequilibrium of the cluster internal energy distribution, although this effect dramatically influences the type of kinetics equations for small clusters [2-6]. The account of it in the framework of the unimolecular condensation theory [2-6] allows to interpret a number of experimental data which can’t be explained by the known theories, and to reach a good agreement between theoretical and experimental data in the wide range of the initial parameters of the flow for various substances [2-6]. In addition to that a convenient and economic algorithm for the calculation of such a one-dimensional flow with condensation was proposed on the basis of the analytical solution of kinetic equations. The main advantage of this algorithm is that in order to calculate the parameters of the condensable flow one should update the gasdynamic equation system for uncondensable gas by the only kinetic equation for supersaturation S. The other parameters such as the cluster concentrations or the popularities of the cluster energetic levels can be determine from the analytic formulae given in [2-6]. A generalization of this approach for the multi-dimensional flows is given in [1].

1Professor

2Leading Research Associate

3Leading Research Associate

In the present work the above mentioned approach is used for the numerical simulation of the condensable vapor flow in the plane and axisymmetric nozzles and the special attention was paid to the cluster size distribution function.

2 MODEL OF THE NOZZLE FLOW.

Let us consider a stationary supersonic flow of the condensable vapor in a plane or axisymmetric nozzle. We assume as usual that the vapor is a mixture of ideal gases of molecules and clusters and that processes of translational and rotational relaxation of clusters are quick enough so that it is possible to introduce the rotational - translational temperature for the whole system T. The basic hydrodynamic equations for such a mixture can be written in the conservative form

\frac{∂ρu}{∂x} + \frac{∂ρv}{∂y} + ν \frac{ρv}{y} = 0  (1)

\frac{∂(ρu^{2} + P)}{∂x} + \frac{∂ρuv}{∂y} + ν \frac{ρuv}{y} = 0  (2)

\frac{∂ρuv}{∂x} + \frac{∂(ρv^{2} + P)}{∂y} + ν \frac{ρv^{2}}{y} = 0  (3)

\frac{∂ρuh_{0}}{∂x} + \frac{∂ρvh_{0}}{∂y} + ν \frac{ρvh_{0}}{y} = 0  (4)

\frac{∂ρuα_{j}(k)}{∂x} + \frac{∂ρvα_{j}(k)}{∂y} + ν \frac{ρvα_{j}(k)}{y} = ρΦ_{j}(k)  _{j=1...N, k=0...M_{j}}  (5)

Here x and y are the Cartesian axises, u and v are the velocities of the mixture in the direction x and y, respectively, ρ, T and P are the density, temperature and pressure of the mixture, h_{0} = (u^{2} + v^{2})/2 + h is the specific stagnation entalpy, α_{j}(k) are the mass fractions of the clusters consisting of j molecules with the total internal energy E_{j}(k), Φ_{j}(k) are the terms describing the nonequilibrium condensation, ν = 0 or 1 for plane or axisymmetric configuration, respectively.

To complete (1)-(5) it is necessary to introduce the thermodynamic and caloric equations of state of the mixture and to determine explicitly the kinetic terms Φ_{j}(k) describing an evolution of α_{j}(k) along the stream line due to condensation and relaxation of the vibrational energy of the clusters. In accordance with [3] one should get

P = ρRT ∑_{j=1}^{N} \frac{α_{j}}{μ_{j}} ,  h = ∑_{j=1}^{N} ∑_{k=0}^{M} α_{j}(k)h_{j}(k)  (6)

α_{j}(k) = \frac{jmx_{j}(k)}{ρ}, α_{j} = ∑_{k=0}^{M} α_{j}(k)

Φ_{j}(k) = ∑_{i=0}^{M_{1}}[j ∑_{l=0}^{M_{j−1}} I_{j−1}(l, i | k) − j ∑_{l=0}^{M_{j+1}} I_{j}(k, i | l)(1 − δ_{jN}) + ∑_{l=0}^{M_{j}} ∑_{ν=0}^{M_{1}} J_{j}(l, i | k, ν)],

j = 2 … N, k = 0 … M_{j}

Φ_{1}(k) = − ∑_{j=1}^{N−1} ∑_{i=0}^{M_{j+1}} ∑_{l=0}^{M_{j}} I_{j}(l, k | i)(1 + δ_{j1}δ_{lk}) − ∑_{j=1}^{N} ∑_{i,l=0}^{M_{j}} ∑_{ν=0}^{M_{1}} J_{j}(l, k | i, ν)(1 + δ_{j1}δ_{lk}),

k = 0 … M_{1}

I_{j−1}(l, i|k) = \frac{C_{j−1}(l, i|k)α_{j−1}^{∗}(l)α_{1}^{∗}(i)ρ}{m(j − 1)} − \frac{E_{j}(k|i, l)α_{j}^{∗}(k)}{j}

J_{j}(l, i|k, ν) = [R_{j}(l, i|k, ν)α_{j}^{∗}(l)α_{1}^{∗}(i) − R_{j}(k, ν|l, i)α_{j}^{∗}(k)α_{1}^{∗}(ν)] \frac{mρ}{j}

α_{j}^{∗}(k) = α_{j}(k)/s_{j}(k)

where μ_{j} = jμ_{1} is the molecular mass of the cluster consisting of j molecules, m is the monomer mass, h_{j}(k) are the specific entalpies of the corresponding components of the mixture, α_{j} are the mass fractions of the clusters with a size j, x_{j}(k) are the populations of the energetic state k in the cluster of a size j, R is the gas constant, C_{j−1}(l, i | k) is the rate constant of the j-cluster formation with the energy E_{j}(k) from the (j − 1)-cluster with the energy E_{j−1}(l) by absorbing a monomer with the energy E_{1}(i); E_{j}(k | l, i) is the rate constant of dissociation of the j-cluster with the energy E_{j}(k) to the (j − 1) cluster with the energy E_{j−1}(l) and a monomer with the energy E_{1}(i); R_{j}(l, i | k, ν) is the relaxation rate constant for the j-cluster from the energy state E_{j}(l) to the state E_{j}(k) simultaneously with a monomer transition from the energy state E_{1}(i) to the state E_{1}(ν); M_{j} is the number of the top energy state of the cluster of size j, s_{j}(k) is the statistic weight of the k-th state of the j-cluster.

In derivation (6) the processes of formation (dissociation) of a cluster by absorbing (emitting) dimer, trimer etc. as well as three particle collisions were neglected. But the law of conservation of the number of molecules in the system has been already taken into account, because the next relationship follows from (6)

∑_{j=1}^{N} ∑_{k=0}^{M} α_{j}(k) = 1  (7)

The system (1)-(6) is extremely difficult for solution because of the high dimension and the lack of reliable data for the rate constants. It can be simplified on the basis of the idea of the ”contracted description” [3,6]. This idea consists in a separation in (1)-(6) of the slow and fast variables and searching for the steady state solutions in which the fast variables depend over the time only indirectly through the slow variables. The result of this method application is that instead of (5) and (6) the system (1)-(4) is coupled with the only equation

\frac{∂ρuα_{1}}{∂x} + \frac{∂ρvα_{1}}{∂y} + ν \frac{ρvα_{1}}{y} = −\frac{S^{m+1}n_{m+1,e}K_{m+1}b}{9m^{4/3}}α_{1e},  m = max(r, j_{∗,cl})  (8)

h_{0} ≈ \frac{u^{2} + v^{2}}{2} + h_{1} − L(1 − α_{1}), P ≈ ρRT \frac{α_{1}}{μ_{1}}

Here r is the new physical parameter of the unimolecular theory (the particular expressions of the r are given in [7]), K_{j} are the equilibrium constant of the j-cluster formation, n_{je} is the equilibrium cluster distribution function over the size [8], bj^{2/3} is the undimensional surface energy of the j-cluster,S = α_{1}/α_{1e} is supersaturation, h_{1} ≈ C_{p}T is the vapor entalpy.

The other variables like

α_{j}(k), (j = 2 … N, k = 0 … M_{j})

and

α_{j} = ∑_{k} α_{j}(k), (j = 2 … N)

have to be determined by the expressions given in [4,5] using the values of T and α_{1} obtained from (1-4). Thus the algorithm proposed in [4,5] allows to manage the radical simplicity of the initial system and therefore gives the possibility to simulate the multi-dimensional flows with condensation using low-powered computers even like PC.

3 NUMERICAL METHOD.

A finite-differential method proposed in [9,10] is used for the numerical prediction of steady supersonic flows of the nonequilibrium gas. This method is a modification of the well-known Godunov’s one. The method is based on a piece wise-linear distribution of parameters in the each section of the nozzle x = const (x is the marching direction of the calculation) and has the second order of accuracy on the retention of monotonicity. One of the main features of the procedure used is the application of a predictor-corrector recomputation. This fact makes the algorithm comparatively simple and universal.

The main features of the method’s implementation for the calculation of the nonequilibrium flow were described in [9]. They include: 1) the freezing of the nonequilibrium processes during the Riemann problem calculation; 2) the semi-implicit approximation for the source terms of the kinetic equations.Such an approximation of the source terms does not change the structure of the finite-difference scheme and provides a reliable calculation of the flow parameters using the unified algorithm even for the substantially different relaxation times of the nonequilibrium processes. In such a case it is possible to perform the calculation with the step defined by the condition of stability of the gasdynamic equations. The implicit parts of these equations for each elementary finite-difference cell are resolved iteratively by the Newton procedure.

4 SIMULATED RESULTS.

Below the simulated results of H_{2}O vapor flow in the supersonic part of a plane nozzle under condensation are presented. The nozzle shape is specified with the fracture of the contour that leads to the generation of a shock wave coming from the fracture point. The initial data of the problem are the following: the critical section radius r_{k} = 0, 005m; the pressure and the temperature at the critical section −P_{k} = 60P a, T_{k} = 250K; the Mach number at the critical section sets as M_{k} = 1, 05. For each section x = const the number of the calculated point distributed in y was equal to 30.

The isolines of the pressure (Fig. 1), temperature (Fig.2) and the dimer concentration (Fig.3) are given in the figures. It is seen that condensation begins in the central part of the flow at x ≈ 3r_{k}. In spite of the further stream expansion the temperature in the condensation region remains about constant because of the latent heat of condensation.

As the result of a shock wave passage the gas temperature increases and the condensation processes slow down sharply. A new flow expansion behind shock wave gives resumption of vapor condensation. It turns out that behind a shock wave in each cross-section x = const concentrations of a j-cluster are uniformly distributed in y. This effect may be of great importance, for instance, for applying a cluster coating by precipitation.

5 CONCLUSIONS.

Numerical method for coupled solution of gasdynamic and kinetic equations is developed. It is modification of Godunov’s method of the second order of accuracy. To simulate condensation processes the unimolecular condensation theory based upon the analytical solutions of the condensation kinetic equations is used. The main advantage of the method is it allows to solve the hydrodynamic equations in conjunction with the only kinetic equation for S. The other cluster concentrations are analytically expressed via S and T.

6 ACKNOWLEDGEMENT.

We acknowledge support from grant No. 286 of the Russian Foundation for Basic Research program.

References

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