Abstract
The correlations for the dynamics of single bubbles in nonequilibrium turbulent flows, as derived in the previous chapter, are employed to obtain integral relations for the calculation of the intensity of vaporization in flashing flows of superheated fluid, which take into account the simultaneous accumulation and growth of bubbles carried by the flow. The cases of heterogeneous nucleation on the channel walls and homogeneous nucleation in the liquid volume are considered. A formula for the rate of channel surface nucleation sites in high-speed flow of superheated fluid is derived. The resulting integral relations were found to be reducible to a fairly simple system of ordinary differential equations that is suitable to numerical analysis. It is interesting to note that from the mathematical point of view such an approach is an “inverse” problem of the recovery of a system of differential equations from its available solution by quadratures. Similar relations for the rate of vapour condensation in nonequilibrium flows of a subcooled liquid with continuous vapour feed over the channel length are derived. This case, for example, is realized in surface boiling in channels of high-performance heat transfer systems. We discuss the difficulties associated with numerical realization of the equations obtained. Simplifying assumptions are considered. It is shown that the assumption on the homogeneity of the distribution of bubbles over sizes enables one to find an analytic solution of the problem of condensation of vapour in an adiabatic flow due to a local supply of vapour into a flow of subcooled liquid. Partial solutions for several degenerate cases were obtained (the cases of highly subcooled liquid and the zero relative enthalpy of a two-phase mixture at the inlet of the adiabatic condensation region). A comparison of the analytic solution with the results of “exact” numerical solutions and with the available experimental data (pertaining to surface boiling of subcooled liquid in channels with stepwise heat law, as well as for direct vapour injection into a flow of subcooled liquid) enables us to infer that there is a good agreement between the theoretical and experimental values of the void fraction (the confidence interval is close to +0.06 with probability 0.95).
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Notes
- 1.
Below we shall see that for small subcooling of liquid and for small distances from the channel entrance a more simple situation is possible, when \(\xi_{0}\) is equal to zero, i.e. the section \(\xi_{0}\) corresponds to the channel inlet.
- 2.
Note that for the case of condensation \(P(z) < 0\).
- 3.
Labuntsov et al. (1976) seem to be the first to obtain an approximate solution of the problem under consideration. The final expression involved two empirical constants, which were separately specified for each regime from two test points. By this experimental comparison the model adopted was shown to qualitatively agree with the experimental data.
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Avdeev, A.A. (2016). Phase Transitions in Nonequilibrium Bubble Flows. In: Bubble Systems. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-29288-5_6
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