Heterogeneous condensation on insoluble spherical particles: Modeling and parametric study
Introduction
The condensation from vapor to droplet is usually divided into three steps (Wagner, 1982): nucleation, growth and coagulation. For a single droplet, only the former two steps should be considered. The condensation can be activated by two ways, namely heterogeneous condensation and homogeneous condensation. The heterogeneous condensation which takes place on particles is much common in daily life, because the energy barrier is lower than that for homogeneous condensation which occurs in the interior of a uniform substance (Kozisek et al., 2004). Heterogeneous condensation plays an important role in many fields such as atmospheric physics (Mason, 1971, Maattanen et al., 2007), crystal study (Liu et al., 1997, Chow et al., 2002) and gas cleaning technology (Heidenreich and Ebert, 1995, Yan et al., 2011). Especially in the separation of submicron particle from gas, the heterogeneous condensation as a preconditioning technique can improve the particle removal efficiency substantially (Schaber, 1995, Johannessen et al., 1997, Heidenreich et al., 2000, Ehrig et al., 2002).
Study on heterogeneous condensation has a very long history. Volmer (1929) examined the heterogeneous nucleation on an insoluble flat surface in detail and proposed the classical heterogeneous nucleation theory. Then Twomey (1959) confirmed Volmer's theory through experiments. Fletcher (1958) established a theory by extending Volmer's theory to spherical particles. In the study of droplet growth, Wagner (1982) developed a first-order theory of droplet growth and reviewed the results of various experimental investigations. Gyarmathy (1982) combined the growth models in the continuum limit and in the free molecular limit into a new growth model, which is valid for the arbitrary droplet radius.
In spite of the continuous advance and the wide application for the study of heterogeneous condensation, the whole process of heterogeneous condensation on insoluble spherical particles has not been systematically considered yet. One difficult is how to bridge nucleation to growth by including the transition process. The transition process in heterogeneous condensation is very complex for simulation, because the spherical cap shape makes it difficult to calculate the growth rate of embryos, and the co-existence of the former embryos growth and the new embryos appearance complicates the process further. In the literature, there are some models to deal with the whole process of heterogeneous condensation approximately. Liu et al. (1997) proposed a two-dimensional kinetic process to simulate the whole heterogeneous condensation of crystals. Heiler (1999) constructed a three-dimensional model of heterogeneous condensation on particles, in which the formation and growth of embryos are assumed to occur on a flat surface to simplify the calculation. However, this model still has some limitations especially when embryo's radius is not much smaller than particle's. The transition stage was actually considered by Tammaro et al. (2012), but the number of embryos formed and the velocity of their growth were very large in their experimental conditions so that the transition stage was neglected. As a result, it was assumed by Tammaro et al. (2012) that heterogeneous condensation on particles will be activated once the condition reaches the critical saturation, and will continue the same way as the growth of a homogenous liquid droplet whose size is equal to the particle's. As already claimed by Tammaro et al. (2012), this model may fail to describe the first instants of embryos growth when nucleation rate or the wetting degree of particle is small.
In this paper, we will develop a full model to include the whole process of heterogeneous condensation on insoluble spherical particles in a closed and adiabatic system. By dividing the whole process of the heterogeneous condensation into three stages: nucleation, transition and growth, the new model bridges the gap between the nucleation and the growth in the heterogeneous condensation. Instead of the classical criteria, two new criteria for condensation ability are proposed based on the new model. We shall evaluate the effects of particle properties and initial conditions on the whole process of heterogeneous condensation by comparing new results with the classical model predictions. In Section 2, the new model is described in detail. The numerical method of our model is introduced in Section 3. Validation of the new model, and two new criteria for condensation ability are presented in Section 4. In Section 5, the effects of particle properties and initial conditions on condensation are then evaluated by our model. A brief summary of our model and results is given in Section 6.
Section snippets
Physical model
We firstly present the assumptions in our physical model. Then the three stages of the heterogeneous condensation are introduced in details. At last, the classical theories of heterogenous nucleation and droplet growth, which are used in our model, are described briefly.
Numerical methods
In this section, we shall derive the governing equation based on the population balance equation (McCoy, 2002, Su et al., 2007) (PBE) firstly. Then the embryos' and thermodynamic properties (embryos' number, embryos' mass, ambient temperature, saturation and pressure) are given. A second-order discrete method is introduced at last.
Validation and new criteria
In this section, we first validate our model by comparison with the experimental data of Tammaro et al. (2012). Then two new criteria for condensation ability are proposed.
Parametric study
The effects of particle properties (the particle wetting degree m and the particle size R) and initial conditions (the saturation S0 and the temperature T0) on the heterogeneous condensation are evaluated by using our model and numerical method. Other initial parameters are set as follows: pressure , and the concentration of particles is 105 cm−3 i.e. the finite volume containing one particle is .
Conclusions
A full model is developed to describe the whole heterogeneous condensation on insoluble spherical particles in a closed and adiabatic system. This model includes three stages: nucleation, transition and growth, and it also considers the conservation of mass and energy of system. The population balance equation is used as the governing equation to describe the evolution of the embryos’ size distribution in time. We derive the equation of state for mixture gas, including the embryo's properties
Nomenclature
interface area between embryo and solid particle (m2) interface area between embryo and vapor (m2) heterogeneous nucleation area (m2) surface area of particle (m2) specific heat of water (J kg−1 K−1) Cg condensation coefficient of the cluster containing g molecules (s−1) Cp specific heat at constant pressure (J kg−1K−1) CV specific heat at constant volume (J kg−1K−1) minimum size of the particle where the heterogeneous condensation occurs (m) particle diameter at the inlet of the growth
Acknowledgments
This work was sponsored by the Natural Science Foundation of China (11172292), by the Hefei Physical Science and Technology Center (2012FXCX005), by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, by the CAS Special Grant for Postgraduate Research, Innovation and Practice and by the USTC Special Grant for Postgraduate Research, Innovation and Practice.
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