Problems Microphysics ATSC 5006
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This 3 page Class Notes was uploaded by Tony Bode on Tuesday October 27, 2015. The Class Notes belongs to ATSC 5006 at University of Wyoming taught by Jefferson Snider in Fall. Since its upload, it has received 12 views. For similar materials see /class/230346/atsc-5006-university-of-wyoming in Atmospheric Science at University of Wyoming.
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Date Created: 10/27/15
Derivation of the K6hler Equation Consider a solution droplet consisting of water and dissolved solute This solution droplet has two components we will use the subscript l to indicate the solvent water and the subscript 2 to indicate the dissolved solute We will also refer to the solute as salt but this can be any material associated with an aerosol particle which dissolves in water an thus lowers the saturation vapor pressure over the resulting solution Such materials are said to be hygroscopic The water activity ie a1 em 1n esw corresponding to a watersalt solution can be expressed in terms of the mole amounts of water and salt 141 and n2 respectively and a vant Hoff factor 139 see for example McDonald 1953 a1 quot 1 A1 I ll l 712 Equation Al arranges to Equation A2 1 7 11 12 A2 a1 quot1 Assuming water and salt contribute to the volume of the droplet solution as pure components and assuming that the salt particle is spherical the mole amount of water carried by a solution droplet of diameter DW is n1D Dj A3 Here Dd is the diameter of the salt particle and p1 and M1 are the density and molecular mass of water Evoking the spherical assumption again the mole amount of salt is mfg pj A4 Combining A2 A3 and A4 the water activity can be described in terms of the vant Hoff factor and the two diameters 1 P2 M 1 D31 P1 Mz D3 a1 A5 li Assuming the second term in the denominator is small relative one and making a Taylor series expansion of the right side of Equation AS the relationship simplifres to Equation A6 3 P2 M1 Dd A6 P1 Mz D3 611 l i The saturation ratio over the solution droplet S is the product of water activity and the Kelvin effect For the latter we assume that the interfacial energy 039 is not altered by the presence of dissolved salt With these assumptions the Kohler Equation becomes 3 M D 4M S 139L d 710 A7 3 3 6X P1 39M2 lDw Dd P1RTDw where R is the universal gas constant and T is the temperature For most applications the Kelvin term can be linearized in the following way 3 M D 4M P2 1 d 10 J A8 S 1717 3 3 17 P1 39Mz lDw Ddl P1RTDw After multiplying the two terms on the right side of Equation A8 it is concluded that one of the four terms is small in comparison to the other three Neglecting that small term the right side of Equation A8 becomes 3 M D 4M S 1 P2 1 d 1039 i A9 p1 M2 lDE Dj PIRTDw The nal approximation is to neglect the D31 relative to the D3 3 D 314M710P2 4417 P1RTDw P1 M2 Dw A10 Often we prefer to use radius instead of diameter and in that case Equation A10 becomes 24110 P2 M1 5 77 S1 3 P1RTVx P1 M 2 rx A11 Finally we define two constants 51 2M10p1RT and b i p2 M1 4quot p1 M2 The latter definition is deceptive because 5 scales with the volume or mass of the salt With these definitions the Kohler Equation becomes 3 33 A12 r x rx In summary the Kohler Equation has two terms which alter the saturation ratio over a solution droplet The first is the curvature term this enhances the saturation ratio and the second is the solute term or Raoult term this reduces the saturation ratio see Young section 34 McDonald JE Erroneous cloudphysics applications of Raoult s Law JMeteor 10 6870 1953 Prob06 ATSC5006 In this problem session we examine Kohler theory Kohler theory relates dry particle size wet particle size and the saturation ratio fractional relative humidity over the wet particle surface The theory is essential for describing heterogeneous condensation nucleation on the cloud condensation nuclei CCN Mathematically speaking the Kohler curve describes the vapor boundary condition at over wet particle surface The boundary condition described by Kohler theory is different from the boundary condition we assumed in our solution to the diffusion equation see notes from ATSC5005 and Young Equation 52 Recall that we took the boundary condition at the droplet surface as that of saturation over a at surface of water containing no dissolved material Since cloud droplets are curved and since they form on hygroscopic nuclei the CCN this boundary condition is inconsistent with Kohler theory Assignment Starting with the procedure called testikohleristudentspro reproduce the following three graphs 101 3 1 w 100 004 pun 021 pm N 099 Saturation Ratio 098 D D C Growth Factor 097 1 1 001 010 100 08 09 10 Wet Diameter D F pm Saturation Ratio S 10000 1 01000 7 7 00100 7 7 00010 7 7 Curvature and Solute Terms 00001 001 010 100 Wet Diameter D F pm
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