Solution of the Condensation Equation

The condensation equation assuming continuum regime growth, unity accommodation coefficient, and constant gas-phase supersaturation can be written using (13.10) and (13.13) as 

This above partial differential equation (PDE) is to be solved subject to the following initial and boundary conditions  

Equation (13.16) provides simply the initial size distribution for the aerosol population, while (13.17) implies that there are no particles of zero diameter. 

FIGURE 13.2 Growth of particles of initial diameters 0.2, 0.5, and 2 pm assuming D, = 

Using the method of characteristics for partial differential equations, F(Dp,t) will be constant along the characteristic lines given by 

Returning to the solution of the condensation equation, the constant value of F Dp,t) along each characteristic curve can be determined by its value of f = 0, that is. 

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Gibbs found the solution of the fundamental Equation 9.1 only for the case of moderate surfaces, for which application of the classic capillary laws was not a problem. But, the importance of the world of nanoscale objects was not as pronounced during that period as now. The problem of surface curvature has become very important for the theory of capillary phenomena after Gibbs. R.C. Tolman, F.P. Buff, J.G. Kirkwood, S. Kondo, A.I. Rusanov, RA. Kralchevski, A.W. Neimann, and many other outstanding researchers devoted their work to this field. This problem is directly related to the development of the general theory of condensed state and molecular interactions in the systems of numerous particles. The methods of statistical mechanics, thermodynamics, and other approaches of modem molecular physics were applied [11,22,23],... 

Figure 6. Solutions of the gap equations and the charge neutrality condition (solid black line) in the /// vs //, plane. Two branches are shown states with diquark condensation on the upper right (A > 0) and normal quark matter states (A = 0) on the lower left. The plateau in between corresponds to a mixed phase. The lines for the /3-equilibium condition are also shown (solid and dashed straight lines) for different values of the (anti-)neutrino chemical potential. Matter under stellar conditions should fulfill both conditions and therefore for //,( = 0 a 2SC-normal quark matter mixed phase is preferable.

The quasi-steady-state theory has been applied particularly where a condensed phase exists whose volume changes slowly with time. This is true, for example, in the sublimation of ice or the condensation of water vapor from air on liquid droplets (M3, M4). In the condensation of water vapor onto a spherical drop of radius R(t), the concentration of water vapor in the surrounding atmosphere may be approximated by the well-known spherically symmetric solution of the Laplace equation ... 

Within the approximation of the effective mass, consideration of the field created by the condensed media is confined to substitution of the real electron mass by the effective mass. Precise calculation of the effective mass is equivalent to solution of the Schrodinger equation with the consideration of the field created by the medium, and, consequently, as noted before, is hardly possible. Thus, as far as the problem of electron tunneling is concerned, the effective mass must be considered as a phenomenological parameter. In the case of tunneling with the energy I of the order of 1-5 eV, the field created by the medium apparently increases considerably the probability of electron tunneling, and the effective mass of electron can be noticeably lower than the real mass. 

A common and important problem in theoretical chemistry and in condensed matter physics is the calculation of the rate of transitions, for example chemical reactions or diffusion events. In either case, the configuration of atoms is changed in some way during the transition. The interaction between the atoms can be obtained from an (approximate) solution of the Schrodinger equation describing the electrons, or from an otherwise determined potential energy function. Most often, it is sufficient to treat the motion of the atoms using classical mechanics,... 

The SR method can be applied to distillation columns, but the equations of the algorithm do not allow the solution of the condenser and the reboiler with the other stages in the column. Because only the energy balances are used as independent functions, reboiler and condenser duties, reflux ratio, and the boilup ratio have to be specified. This overspecifies the column and the solution cannot be found. The condenser and the reboiler can be solved as separate unit operations in a flowsheet as demonstrated by Fonyo et al. (39). The SR method is used in the ABSBR step of FLOWTRAN of Monsanto, St. Louis, Missouri, and also in both the public release version of ASPEN and in ASPENPlus of AspenTech, Cambridge, Massachusetts. 

Continuous Stirred Tank Reactors. Biesenberger (8) solved for the MWD with condensation polymerization in a CSTR, analogous to the treatment Denbigh (14) provided for the other two mechanisms. In this case, the variable residence time distribution leads to an extremely broad MWD with even the maximum weight fraction at the lowest molecular weight (monomer). The dispersion index approaches infinity as the condensation is driven to completion in a stirred tank reactor. A sequential analytical solution of the algebraic equations was obtained with a numerical evaluation of the consecutive equations. 

The temperature or pressure at which a vapor of known composition first begins to condense is given by solution of the appropriate equation,... 

The solution of the system equations for all unknown variables is straightforward. The idea gas equation of state applied to the fresh feed stream yields no- The specified overall CO conversion yields h from the equation 0.01 3 = (1 - 0.98)no Raoult s law at the condenser outlet combined with the calculated value of h yields /i6, and an overall carbon balance yields Balances on CO and CH3OH at the mixing point yield hi and hi, and an energy balance for the same subsystem yields Ta. An energy balance on the preheater then yields Qh> A methanol balance on the condenser yields hi, and then energy balances on the reactor and the condenser yield and Q, respectively. 

The results of an analytical solution of the differential equations for a turbulent condensate film on a vertical tube are reproduced in Fig. 4.12, [4.15]. Line A represents Nusselt s film condensation theory according to (4.39). 

The diffusion cloud chamber has been widely used in the study of nucleation kinetics it is compact and produces a well-defined, steady supersaturation field. The chamber is cylindrical in shape, perhaps 30 cm in diameter and 4 cm high. A heated pool of liquid at the bottom of the chamber evaporates into a stationary carrier gas, usually hydrogen or helium. The vapor diffuses to the top of the chamber, where it cools, condenses, and drains back into the pool at the bottom. Because the vapor is denser than the carrier gas, the gas density is greatest at the bottom of the chamber, and the system is stable with respect to convection. Both diffusion and heat transfer are one-dimensional, with transport occurring from the bottom to the top of the chamber. At some position in the chamber, the temperature and vapor concentrations reach levels corre.sponding to supersaturation. The variation in the properties of the system are calculated by a computer solution of the onedimensional equations for heat conduction and mass diffusion (Fig. 10.2). The saturation ratio is calculated from the computed local partial pressure and vapor pressure. 

The basic idea proposed by Mohanty et al. is that the counterion condensation prediction can be justified by the PB equation through a thermodynamic quantity called the adsorption excess [90a], In this approach, the surface charge density a is obtained from the solution of the PB equation ... 

Sitarski, M., and Nowakowski, B. (1979) Condensation rate of trace vapor on Kundsen aerosols from solution of the Boltzmann equation, J. Colloid Interface Sci. 72, 113-122. 

The model representing diffusion/reaction involved solution of the transport equations for each single pore simultaneously to give concentration profile in the pore network. The calculations related to capillary condensation were performed in the same way as for the Fickian model, described in Section lll.C. 

Thus, there is an equation for each component for each stage. This set of equations may be solved simultaneously by matrix methods, there being a tiidiagonal matrix, Nhy N (N + by N + I including the leboiler or iV + 2 by /V + 2 if there is also a partial condenser), shown in Fig. S.3-13. This approach to the simultaneous (not stagewise) solution of the MESH equations was introduced by Amundson and Pontinen. ... 

An approach that has met with success is the Adamson and Ling iterative method." This method converges to an acceptable solution much faster than any other iterative method although it is not a general iterative solution of the Fredholm equation since the initial approximation to /(y) is obtained by using the condensation approximation. Thus the method utilizes information concerning the physical form of the local isotherm functions (see Section 4 for details and applications). 

The various efforts to improve the effectiveness of the condensation-approximation method were made [5,6,9]. A more exact solution of the integral equation gives the asymptotically correct approximation method, developed by Hobson [118] for mobile adsorption and later refined by Cerofolini [66] for localized adsorption. In this treatment, the local isotherm is assumed to be a combination of a linear and a condensation isotherm. Hsu et al. [119] and Rudzinski et al. [109,110,120] adapted the Sommerfeld expansion method [121] to the solution of the integral equation in question. Although numerous modifications of the condensation-approximation method are known, aU improvements to this method make it more complicated and introduce additional numerical problems, but they do not change its... 

For transitions from a condensed phase into the vapor phase (the vapor phase is assumed to be perfect Vy — V nd Vy = R T/p) the solution of the CC equation results in... 

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