Models for Nucleation and Condensation on Nanoparticles

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1 PFAU X Models for Nucleation and Condensation on Nanoparticles Stefan Radl, 1 Mario Schriefl 2,3 1 TU Graz, Institute of Process and Particle Engineering 2 TU Graz, Institute of Electronics Tristan Reinisch, 3,4 Alexander Bergmann, 3 Athanasios Mamakos 3 3 AVL List GmbH 4 CTR Carinthian Tech Research AG 1

2 Why Nano?

3 Why Nano? Condensation Particle Counters (CPCs) AVL List GmbH 3

4 Why Nano? Droplets nucleate and grow in a CPC on nanoparticles with fast dynamics, coupling of heat and mass transfer, droplets are polydisperse, and a size change typically 3 orders of magnitude Where do these phenomena take place? What is the final droplet size distribution? How can we troubleshoot CPCs? 4

5 Why Nano? Droplets nucleate and grow in a CPC on nanoparticles What is the challenge? poly- (alpha)- olefin (PAO) Mamakos et al., Aerosol Science and Technology, 47:11-21 (2013) 5

6 The Models

considered (thermodiffusion, heat of evaporation / condensation, etc.

7 The Models Multiphase Flow inside the CPC continuous phase (e.g., n-butanol vapor in air) modeled with the cpcfoamcompressible solver implemented in OpenFOAM all effects relevant for CPCs considered (thermodiffusion, heat of evaporation / condensation, etc.) (b) saturation ratio profile solution for the continuous phase disperse phase qmomcloud library solves the population balance equation for the droplets using a QMOM approach (univariate in droplet diameter) (a) Evaporator-Condenser system in a CPC 7

8 The Models Population Balance Equation (PBE) growth nucleation Disperse phase: characterized by the number density function (NDF) CPC: property ξ is the droplet size NDF = droplet size distribution PBE very difficult to solve directly for n(x,ξ,t) k-th moment of the NDF: solve the PBE for some lower order moments of the NDF transformation to a set of moment equations method of moments (MOM) the moment equation for the k-th moment is obtained by: multiply PBE with ξ k integrate over phase space 8

9 The Models Moment Transport Equations Moments approximate the NDF Moments have a physical meaning: m 0. number concentration [m -3 ] m 1. total droplet size p. vol. [m -2 ] L 32 =m 3 /m 2 Sauter mean diameter [m] If R term unclosed, reconstruct NDF with QMOM (set of N nodes and N weights) R terms determined by physical models 9

10 The Models Moment Transport Equations (Example) Unclosed for k = 0 and k = 1 k = 0,.., 2N 1 Solution for the disperse phase approximated by time evolution of first 2N moments (m 0, m 1,., m 2N-1 ) 10

11 The Models Quadrature Method of Moments Approach moments are approximated by N weights w α and N nodes ξ α weights and nodes are calculated with the first 2N moments (e.g., PD algorithm) Result the first 2N moments are reproduced exactly unknown moments in the source terms of the moment transport equations can be computed to close the system of equations!! 11

12 The Models Growth Models 0) The Basics m Sh D c cond m cond 2 2 l Sh D v 2 v 1 c l Mass concentration Key scaling of the growth rate in the case of mass transfer limitation 1/ inserted into the moment evolution equations yields in case we assume that: Dm Dt k km f k 2 12

13 The Models Growth Models 1) Simple Continuum-Regime Closure (thermal equilibrium, dilute vapor, large droplets) 2 Sh D v sat 1 1 p p T l R g, vap T Td vap vap d 2) Classical Closure (Abramzon & Sirignano, 1989; thermal equilibrium, large droplets) B m Sh D v gas 2 ln 1 y sat vap 1 y y sat vap l vap B m considers Stefan flow 13

14 The Models Growth Models 3) Free-Molecule-to-Continuum-Regime Closure (Fuchs and Sutugin, 1970; Ahn and Liu, 1990) f only approximate, because of correction factors! sat Sh D 1 1 pvap K v Kelvin pvap T d 2 F l R g, vap T T d K Kelvin 4 MW vap exp lrg, vap Td 1 Kn i Fi, Kni 4 / 3 Kni Kn 2 / i i F, vap sat Nu Sh 1 pvap KKelvin pvap Td Td T air Dv hevap R g, vap T Td Fg, Expensive: iterations are required to determine T d. 14

15 The Models Nucleation Model 1) Standard Heterogeneous Nucleation Model Particle Geometry & Contact Angle G f G * * het g hom J het J 0 het exp G B * het k T G 4 3 r 2 * * hom J A larger number d het p r * 2 R T ln S l g, vap 15

16 The Software

17 The Software Unfortunately, there is no publicly-available (x)mom implementation in OpenFOAM! So, let s start from scratch: cpcfoamcompressible as new solver qmomcloud for modeling droplets

18 The Software solvers Extra math, diffusion coefficients, liquid properties Markdown-based documentation Library of QMOM routines

19 Demo Time

20 Models for Nucleation and Condensation on Nanoparticles THANK YOU! Stefan Radl, 1 Mario Schriefl 2,3 1 TU Graz, Institute of Process and Particle Engineering 2 TU Graz, Institute of Electronics Tristan Reinisch, 3,4 Alexander Bergmann, 3 Athanasios Mamakos 3 3 AVL List GmbH 4 CTR Carinthian Tech Research AG PFAU X CTR and AVL List GmbH gratefully acknowledge the financial support of this project by the BMVIT and the BMWFW and the federal provinces of Carinthia and Styria within the COMET-Competence Centers for Excellent Technologies Program. 20

THE VERIFICATION OF THE TAYLOR-EXPANSION MOMENT METHOD IN SOLVING AEROSOL BREAKAGE

144 THERMAL SCIENCE, Year 1, Vol. 16, No. 5, pp. 144-148 THE VERIFICATION OF THE TAYLOR-EXPANSION MOMENT METHOD IN SOLVING AEROSOL BREAKAGE by Ming-Zhou YU a,b * and Kai ZHANG a a College of Science, China