Evaluation of master equations for the droplet size distribution in condensing flow

Abstract

The kinetic equation ͑KE͒ and its first-and second-order approximations, the general dynamic equation ͑GDE͒, and the Fokker-Planck equation ͑FPE͒, respectively, have been evaluated based on ͑a͒ their equilibrium distributions, ͑b͒ a nucleation pulse experiment, and ͑c͒ an expanding nozzle flow. Large differences are observed between the equilibrium distributions of the FPE and KE, whereas the GDE does not have an equilibrium distribution at all. For the nucleation pulse experiment, good agreement is found between the KE, FPE, and GDE due to quasisteady nucleation. For the condensing nozzle flow, the difference between the GDE and the KE distributions is large, although the relevant flow variables show fair agreement. A sensitivity study of the KE solution with respect to uncertainties in ͑a͒ the surface tension model, ͑b͒ the sticking probability, and ͑c͒ the equilibrium distribution revealed that both the sticking probability and the equilibrium distribution have a significant influence on the predicted condensation onset. Furthermore, it is found that the proposed Wölk and Strey-corrected Courtney equilibrium distribution yields the best agreement with the reported measurements.

Efficient solution methods for N-component condensation

“…(3.13) as demonstrated in Ref. [66]. We can circumvent this problem by using a more convenient form of Eq.…”

“…(3.11) is referred to as the generalized NFPE [50] for L → ∞. The rhs terms become smaller as the order increases [66], therefore to good approximation we can truncate the summation in Eq. (3.12) at the second-order term yielding the NFPE…”

“…Recent studies made an attempt to overcome these deficiencies by replacing the δ-function in Eq. (3.18) by a boundary condition at a size larger than the critical size [22,66]. We will extensively address the location of the source point n 0 in Chapter 6.…”

“…An alternative for the NBD equations is the N-component Fokker-Planck Equation (NFPE) which can be derived by continuation of ρ(n, t) to non-integer values of n. A Taylor series expansion of the right-hand side (rhs) of system (3.1) [66,85] leads to…”

“…Moreover, a rapid change in external conditions resulting in an increase of the critical size, leads to evaporation of all clusters. Recent studies made an attempt to overcome these deficiencies of the GDE by replacing the δ-function by a boundary condition at a certain size larger than the critical one [22,46,66]. The GDE is aimed at the description of large supercritical clusters and does not supply the cluster size distribution in the subcritical region.…”