Theoretical investigation of the cluster size distribution in nozzles and jets

A. L. Itkin, A. V. Rodionov

International Institute for Problems of the Asteroid Hazard, Moscow, Russia

Abstract Results of numerical calculations of a condensible gas flow in smooth nozzles, nozzles with a break of the contour and jets are presented. The unimolecular condensation theory (UCT) 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 through S and T. It turns out that for nozzles with the break of the contour behind a shock wave in each cross-section x = const concentrations of j-clusters are uniformly distributed over y for each j. This effect may be of great importance, for instance, for applying a cluster coating by precipitation.

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 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 twodimensional flow with condensation in the supersonic nozzle with a wall’s bend was executed.

For calculation of the gas mixture flow where one component is capable to condense it is necessary to close the 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]. With allowance for it UCT [2] permits to interpret a great number of experimental data which can’t be explained by the existing theories, and reach a good agreement between theoretical and experimental data in a wide range of the determined parameters of the flow for various substances [2, 3]. Furthermore, a convenient and economic algorithm for calculation of such a one-dimensional flow with condensation was proposed on the basis of UCT. The main advantage of this algorithm is that in order to calculate the parameters of the condensible flow one should update the gasdynamic equation system for incondensible gas by the only kinetic equation for supersaturation S. The other parameters such as the cluster concentrations or the populations of the cluster energetic levels can be determined from the analytic formulae given in [2]. A generalization of this approach for the multi-dimensional flows is given in [1].

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

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 jets with the special attention paid to the cluster size distribution function.

2 Model of the nozzle flow

Let us consider a stationary supersonic flow of the condensible 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{∂A}{∂x} + \frac{∂B}{∂y} + ν \frac{ρv}{y} C = 0  (2.1)

\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}} (2.2)

Here A = (ρu, ρu^{2} + P, ρuv, ρuh_{0}), B = (ρv, ρuv, ρv^{2} + P, ρvh_{0}), C = (1, u, v, h_{0}), x and y are the Cartesian axes, 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 enthalpy, 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 the plane or axisymmetric configuration, respectively.

To complete (2.1),(2.2) it is necessary to introduce the thermodynamic and caloric equations of state of the mixture and 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 [4] one can get

P = ρRT ∑_{j=1}^{N} \frac{α_{j}}{μ_{j}} ,  h = ∑_{j=1}^{N} ∑_{k=0}^{M} α_{j}(k)h_{j}(k)  (2.3)

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

j = 2 … N, k = 0 … M_{j}

Φ_{1}(k) = − ∑_{j,i,l} [I_{j}(l, k | i)(1 + δ_{j1}δ_{lk}) + ∑_{ν} 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 α_{j}(k) = jmx_{j}(k)/ρ, α_{j} = ∑_{k} α_{j}(k), 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 enthalpies of the corresponding components of the mixture, α_{j} are the mass fractions of the clusters of size j, x_{j}(k) are the populations of the energetic state k in the cluster of size j, R is the gas constant, C_{j−1}(l, i | k) is the rate constant of the j-cluster formation with energy E_{j}(k) from (j − 1)-cluster with energy E_{j−1}(l) by absorbing a monomer with energy E_{1}(i); E_{j}(k | l, i) is the rate constant of the reverse process; R_{j}(l, i | k, ν) is the relaxation rate constant for j-cluster from the energy state E_{j}(l) to state E_{j}(k) simultaneously with a monomer transition from the energy state E_{1}(i) to 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 j-cluster.

In derivation (2.2) the processes of formation (dissociation) of the 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 (2.2) ∑_{j} ∑_{k} α_{j}(k) = 1.

The system (2.1)-(2.3) 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” [2, 3]. This idea consists in a separation in (2.2) of slow and fast variables and searching for the steady state solutions in which the fast variables depend upon the time only indirectly through the slow variables. The result of this method application is that instead of (2.2) the system (2.1) should be 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}  (2.4)

The state equations should be now rewritten in the form: h_{0} ≈ (u^{2} + v^{2})/2+h_{1} −L(1−α_{1}), P ≈ ρRT α_{1}/μ_{1}. In (2.4) m = max(r, j_{∗}), r is a new physical parameter of the unimolecular theory (the certain expressions of r are given in [5] (see also another report of A.Itkin), K_{j} are the equilibrium constants of the j-cluster formation, n_{je} is the equilibrium cluster distribution function over the size [6], bj^{2/3} is the dimensionless surface energy of j-cluster,S = α_{1}/α_{1e} is supersaturation, h_{1} ≈ C_{p}T is the vapor enthalpy, j_{∗} is the classical critical size.

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 [2, 4] using the values of T and α_{1} obtained from (2.1)-(2.2). Thus, the algorithm proposed in [2, 4] allows to manage a radical simplification of the initial system and therefore gives the possibility to simulate the multidimensional flows with condensation using low-powered computers.

3 Numerical method

A finite-difference method proposed in [7] is used for the numerical prediction of steady and unsteady supersonic flows of the nonequilibrium gas. This method is a modification of the well-known Godunov’s one. It is based on the piece wise-linear distribution of parameters in 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 calculation of the nonequilibrium flow were described in [7]. 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.

4 Simulated results

Below the simulated test results of the H_{2}O condensible vapor flow in the supersonic part of the plane nozzle 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, 01m, the pressure and the temperature at the critical section - P_{k} = 0, 2MP a, T_{k} = 400K, the Mach number at the critical section is M_{k} = 1, 05. For each section x = const a number of the calculated point distributed over y was equal to 30.

The isolines of pressure P (Fig. 1), temperature T (Fig.2), the total cluster mass fraction α_{c} (Fig. 3) and the mean cluster size j_{m} (Fig.4) are given in the figures (see notations in Table 1). 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 the shock wave gives a resumption of the vapor condensation. It turns out that behind the shock wave in each cross-section x = const concentrations of j-cluster are almost uniformly distributed over y. It is caused by a special character of the flow because in this area all streamlines are located between the shock wave and the nozzle contour so that the flow picture is close to the spherical source flow. This effect may be of great importance, for instance, for applying the cluster coating by precipitation.

In Fig.5, 6 the distribution of the cluster mass fractions over j and y is presented for two nozzle cross-sections x = 0, 5cm (Fig.5) and x = 3cm (Fig.6). In the first cross-section large S are realized in the wall-adjacent area because of the intensive stream expansion in this region in contrast to the axis-adjacent area where condensation is still weak enough. The typical form of CSDF for such a situation can be seen in Fig.5. Since the active condensation occurs and j_{∗} becomes less than r, the maximum in CSDF

Table 1. The correspondence of the parameter value to the isoline number in Figures.

The isoline number T,_{K} P,_{MP A} α_{c}  j_{m} 1  400 0.200  0.034 36.2 4  385 0.173  0.029 31.0 6  374 0.155  0.025 27.5 9  359 0.128  0.020 22.3 11  349 0.110  0.018 18.8 14  333 0.082  0.011 13.6 16  323 0.064  0.007 10.1 18  313 0.046  0.004 6.9

6 Investigation of the cluster size distribution in nozzles and jets appears around j = r. In particular, it is observed in the wall-adjacent zone. At small S the value of j_{∗} exceeds R that results in removing the maximum of CSDF in the area of small j (the central area in Fig.5). At the same time at the nozzle axis there are almost no condensation as well as clusters. The similar plot of CSDF is given in Fig.6 however in this case a zone of the active condensation covers all y. Once again the maximum of CSDF is located at j = r. Furthermore, one can see a narrow zone located near the wall where there are no clusters. Such a behaviour is caused by the shock wave crossed the cross-section in this zone and led to evaporation of the clusters formed before.

Doing in the same manner we can calculate condensible free jets exhausted into the vacuum as well as various kinds of flows in channels and tubes.

Bibliography

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2. Itkin, A.L., Kolesnichenko, E.G. and Gorbachev, Y.E. (1991). Preprints of Ioffe Phys.Tech.Inst.Acad.Sci. USSR, Quasy - stationary state method in the theory of condensation. 1488, Asymptotical methods in the theory of condensation (unimolecular model). 1490, Toward the theory of the nucleation inside the thermodiffusion chamber. 1503, The analyses of the nucleation kinetics experimental data from the point of unimolecular condensation theory view. 1523.

3. Itkin, A.L. and Kolesnichenko, E.G. (1992). New equations in condensation kinetics. Soviet Journal of Technical Physics, 37, No.3, 261–266.

4. Itkin, A.L. and Kolesnichenko, E.G. (1990). Calculation of condensible gas flows by means of the unimolecular condensation theory. Fluid Dynamics. 25, No.5, 765–774.

5. Itkin, A.L., and Kolesnichenko, E.G. (1993). Calculation of dissociation rate constants for vibrational-excited clusters and an estimation of the main parameter of unimolecular condensation theory. Soviet Journal of Chemical Physics. 10, No.11, 2403–2418.

6. Bogdanov, A.V., ..., Itkin, A.L., Kolesnichenko, E.G....(1992). Analytical investigation of equilibrium and quasisteady solutions for a quasichemical condensation model. Soviet Journal of Chemical Physics. 9 No.5, 1113–1129.

7. Rodionov, A.V. (1987). Monotonic scheme of the second order accuracy for uniform calculation of nonequilibrium flows. Soviet Journal of Numerical Mathematics and Mathematical Physics. 27, No.2, 175 8 Investigation of the cluster size distribution in nozzles and jets

app.37, Taganskaya 44, Moscow, 109147, Russia, e-mail: itkin@momipao.msk.su