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(1)

S OME CORRECTIONS TO THE DIFFUSIONAL GROWTH THEORY

1. Kinetic effects 2. Ventilation effects

(2)

Kinetic effects

The mechanisms of heat, mass, and momentum transfer between a drop and its surroundings depend on the Knudsen number .

Knudsen number = l/r: the ration of the molecular mean free path of the gas (l)

to the radius of the drop (r).

The mean free path in air, l, varies approximately inversely with the density and

is about 0.06 µm for normal conditions at sea level.

(3)

Kinetic effects

𝑙

𝑟 ≈ 0.06 𝑟 = 1𝜇𝑚

𝑙 = 0.06𝜇𝑚

• When l/r<<1, as for large drops, the fields of vapor and temperature may be regarded as

continua.

– The equations derived earlier for the transfer of heat and vapor are based on the Maxwell continuum approximation and hence are valid for drops that are much larger than the mean free path.

(4)

Kinetic effects

𝑟 = 0.01𝜇𝑚

𝑙 = 0.06𝜇𝑚

𝑙

𝑟 ≈ 6

• When l/r>>1, as for small drops, the exchange of heat and mass can be calculated from

molecular collision theory.

(5)

Kinetic effects

x10

𝑟 = 0.1𝜇𝑚 𝑙 = 0.06𝜇𝑚

𝑙

𝑟 ≈ 0.6

• Unfortunately, there is no complete theory for heat and mass transfer for l/r ~1.

– Approximations are derived by interpolating between the free molecular theory and the continuum theory, but uncertainities remain in the quantitative results that

follow from the approximations.

(6)

Kinetic correctionsare applied by using an effective diffusivity, D’=D×g(b), and an effective conductivity, K’=K×f(a), in the droplet growth equations.

Functions g(b) i f(a) are semi-empirical corrections that take the form:

𝑙! and 𝑙" define lenght scales

The coefficients a and b are called the accomodation and condensation coefficients respectively and are poorly known.

Standard texts take a » 1 and b » 0.04. The form of g(b) and f(a) guarantees that, D’<D and K’<K , hence the corrections have the effect of slowing the growth of small drops.

As r increases, these factors approach unity and equation of condensational growth reduces to the continuum solution.

𝑔 𝛽 = 𝑟

𝑟 + 𝑙! 𝑓 𝛼 = 𝑟

𝑟 +𝑙"

𝑙! = 𝐷 𝛽

2𝜋 𝑅#

%$

&

𝑙" = 𝐾

𝛼𝑝

2𝜋𝑅𝑇 %$&

𝑐# + ;𝑅 2

(7)

Ventilation effects

We have assumed that the vapor field surrounding each drop is spherically symetrical.

This is appropriate for a droplet at rest.

When the drop is large enough to fall through the surrounding air with a significant speed, ambient air is continually replenished in the vicinity of the drop, and the vapor field becomes distorted.

The rate of heat and mass transfer increase, and are greatest on the upstream side of the drop.

These effects are incorporated into the theory of diffusional growth by

introducing

ventilation coefficients.

(8)

Ventilation coefficient (fv) is described empirically, as a function of the Reynolds

number, Re (the Reynolds number is an estimate of the ratio of inertial to viscous forces in a flow) , defined as:

uµ -the terminal velocity of a drop of size r

n - the dynamic viscosity of the fluid through which the drop is moving

The ventilation correction takes the same form for both the diffusivity and conductivity, so that: ⁄𝐷' 𝐷 = 𝐾'⁄𝐾 = 𝑓# > 1

The ventilation effects are negligible for droplets smaller than 10 µm in radius.

For r=20 µm, fv=1.06

For r=40 µm, fv=1.25. The growth by condensation of large drops is usually negligible compared to growth by collision and coalescence.

Ventilation effects are unimportant in the growth of cloud drops.

Such effects can be very significant in the evaporation of raindrops and other

precipitation particles that have high velocities relative to the air through which they fall.

𝑅𝑒 = 𝑟𝑢( 𝜈

𝑓# = A1.00 + 0.09𝑅𝑒 0 ≤ 𝑅𝑒 ≤ 1.25 0.78 + 0.28 𝑅𝑒 𝑅𝑒 > 1.25

(9)

THE GROWTH OF DROPLETS

POPULATIONS

(10)
(11)

In natural clouds the droplets interact with their environment and with each other that

affect the droplet sizes and concentration

The droplet size

Droplets grow by condensation if the environment has an excess of vapor over the equilibrium value

supersaturation is produced by chilling in ascending

motion

The droplets evaporate if dry environmental air in sufficient amount is mixed with the cloudy air

Concentration

Sedimentation (gravitational settling) is important for the larger droplets and tend to remove them from the cloud.

Coagulation of droplets (coalescence) to form larger drops appears to be

insignificant for droplets smaller than 10µm in radius, but becomes

increasingly important as they grow beyond this size.

(12)

• In the early development of a cloud the droplets are too small for sedimentation or coalescence to be important

Condensation

is the dominant growth process

• The ambient saturation ratio controls the condensation process

• We shall assume that vapor is produced by saturated air in ascent and is lost

by condensation on the growing droplets

(13)

P –production of supersaturation due to the chilling in ascending motion of saturated air; depends on the vertical air velocity

C – condensationon the growing droplets; depends on the condensation rate

the rate of condensation, measured in units of mass of condensate per mass of air per unit time

the vertical air velocity

Q1 and Q2 are thermodynamic variables.

Q1(dz/dt) is the increase of supersaturation due to cooling in adiabatic ascent.

Q2(dql/dt) is the descrease in supersaturation due to the condensation of vapor on droplets.

𝑑𝑆

𝑑𝑡 = 𝑃 − 𝐶

𝑑𝑆

𝑑𝑡 = 𝑄% 𝑑𝑧

𝑑𝑡 − 𝑄& 𝑑𝑞)

𝑑𝑡

(14)

For derivation of Q1 we will assume an ascent with no condensation (ql=const, qv=const);

the change of supersaturation is due only to the vertical motion 𝑑𝑆

𝑑𝑡 = 𝑄% 𝑑𝑧

𝑑𝑡 − 𝑄& 𝑑𝑞) 𝑑𝑡

𝑆 = 𝑒 𝑒*

𝑑𝑆

𝑑𝑡 = 1 𝑒*

𝑑𝑒

𝑑𝑡 − 𝑒 𝑒*&

𝑑𝑒* 𝑑𝑡 𝑞# = 𝜀 𝑒

𝑝 ⟶ 𝑒 = 𝑞#𝑝 𝜀 𝑑𝑒

𝑑𝑡 = 𝑑𝑒 𝑑𝑧

𝑑𝑧

𝑑𝑡 = 𝑑 𝑑𝑧

𝑞#𝑝 𝜀

𝑑𝑧

𝑑𝑡 = 𝑞) 𝜀

𝑑𝑝 𝑑𝑧

𝑑𝑧

𝑑𝑡 = −𝑒

𝑝 R 𝜌𝑔 R 𝑑𝑧

𝑑𝑡 = − 𝑒𝑔 𝑅𝑇

𝑑𝑧 𝑑𝑡

𝑑𝑒*

𝑑𝑡 = 𝑑𝑒* 𝑑𝑇

𝑑𝑇 𝑑𝑧

𝑑𝑧

𝑑𝑡 = 𝐿𝑒*

𝑅#𝑇& − 𝑔 𝑐+

𝑑𝑧 𝑑𝑡

𝑑𝑒*

𝑑𝑇 = 𝐿𝑒* 𝑅#𝑇&

𝑑𝑇

𝑑𝑧 = − 𝑔 𝑐+ 𝑑𝑝

𝑑𝑧 = −𝜌 R 𝑔 qv – specific humidity

(15)

Q1 depends ONLY on temperature 𝑑𝑆

𝑑𝑡 = 𝑄% 𝑑𝑧

𝑑𝑡 − 𝑄& 𝑑𝑞) 𝑑𝑡

𝑑𝑆

𝑑𝑡 = − 1 𝑒*

𝑒𝑔 𝑅𝑇

𝑑𝑧

𝑑𝑡 + 𝑒 𝑒*&

𝐿𝑒* 𝑅#𝑇&

𝑔 𝑐+

𝑑𝑧 𝑑𝑡 𝑑𝑆

𝑑𝑡 = 𝑔 𝑇

𝑒

𝑒* − 1

𝑅 + 𝜀𝐿 𝑅𝑐+𝑇

𝑑𝑧 𝑑𝑡

𝜀 = 𝑅 𝑅# 𝑒

𝑒* ≈ 1 𝑑𝑆

𝑑𝑡 = 𝑔 𝑅𝑇

𝜀𝐿

𝑐+𝑇 − 1 𝑑𝑧 𝑑𝑡 𝑑𝑆

𝑑𝑡 = 1 𝑒*

𝑑𝑒

𝑑𝑡 − 𝑒 𝑒*&

𝑑𝑒* 𝑑𝑡 𝑑𝑒

𝑑𝑡 = − 𝑒𝑔 𝑅𝑇

𝑑𝑧 𝑑𝑡

𝑑𝑒*

𝑑𝑡 = − 𝐿𝑒* 𝑅#𝑇&

𝑔 𝑐+

𝑑𝑧 𝑑𝑡

𝑄% = 𝑔 𝑅𝑇

𝜀𝐿

𝑐+𝑇 − 1

(16)

For derivation of the condensation term we shall consider the saturation change due to condensation assuming no ascent (dz/dt=0, p=const)

Clausius-Clapeyron equation

Q2 depends on

temperature AND pressure 𝑑𝑆

𝑑𝑡 = 𝑄% 𝑑𝑧

𝑑𝑡 − 𝑄& 𝑑𝑞)

𝑑𝑡 𝑆 = 𝑒

𝑒*

𝑑𝑆

𝑑𝑡 = 1 𝑒*

𝑑𝑒

𝑑𝑡 − 𝑒 𝑒*&

𝑑𝑒* 𝑑𝑡 𝑑𝑒

𝑑𝑡 = 𝑝 𝜀

𝑑𝑞#

𝑑𝑡 = −𝑅𝑇𝜌 𝜀

𝑑𝑞)

𝑑𝑡 𝑑𝑞# = −𝑑𝑞)

𝑑𝑒*

𝑑𝑡 = 𝑑𝑒* 𝑑𝑇

𝑑𝑇

𝑑𝑡 = 𝜀𝐿𝑒* 𝑅𝑇&

𝐿 𝑐+

𝑑𝑞)

𝑑𝑡 = 𝜀𝐿&𝑒*𝜌 𝑝𝑇𝑐+

𝑑𝑞) 𝑑𝑡 𝑑𝑒*

𝑑𝑇 = 𝜀𝐿𝑒* 𝑅#𝑇&

𝑒

𝑒* ≈ 1 𝑑𝑆

𝑑𝑡 = −𝑅𝑇𝜌 𝜀

𝑑𝑞)

𝑑𝑡 − 𝜀𝐿&𝜌 𝑝𝑇𝑐+

𝑑𝑞) 𝑑𝑡

𝑐+𝑑𝑇 = 𝐿𝑑𝑞) 𝑑𝑇

𝑑𝑡 = 𝐿 𝑐+

𝑑𝑞) 𝑑𝑡

𝑄& = 𝜌 𝑅𝑇

𝜀𝑒* + 𝜀𝐿&

𝑝𝑇𝑐+

(17)

𝑄% = 𝑔 𝑅𝑇

𝜀𝐿

𝑐+𝑇 − 1 𝑑𝑆

𝑑𝑡 = 𝑄% 𝑑𝑧

𝑑𝑡 − 𝑄& 𝑑𝑞)

𝑑𝑡

𝑄& = 𝜌 𝑅𝑇

𝜀𝑒* + 𝜀𝐿&

𝑝𝑇𝑐+

(18)

Cloud development

in a steady-state updraft

(19)

• the equation of condensational growth AND

• the equation for the change of the saturation ratio

can be solved simultaneousely to get an evolution of droplet spectra in a cloud.

We have to assume an initial CCN distribution and fix a value of ascending motion

In the figure: u= 15 cm/s.

The solid lines indicate the sizes of droplets

growing on nuclei of NaCl having different masses, ranging from 3.7x10-18 to 9.3x10-10g

The dashed envelope bounding the termination of these curves indicates the altitude reached by the different droplets during the simulation. The

smaller ones move essentially with the air at 15 cm/s, but the larger ones fall relative to the air and do not rise to the same altitude.

𝑑𝑆

𝑑𝑡 = 𝑄% 𝑑𝑧

𝑑𝑡 − 𝑄& 𝑑𝑞)

𝑑𝑡 𝑟𝑑𝑟

𝑑𝑡 = 𝑆 − 1 − 𝐴

𝑟 +𝐵* 𝑟, 𝐹- + 𝐹.

(20)

saturation

mean volume radius

x10 µm

concentration x1000 cm-3

𝐿𝑊𝐶 = 𝑐/ R ℎ 𝐿𝑊𝐶 = 4

3𝜋𝜌)𝑁𝑟#, ℎ = 4 3𝜋 𝜌)

𝑐/ 𝑁𝑟#, ⟶ h ∝ 𝑟#,

𝑁 = 𝑐𝑜𝑛𝑠𝑡

(21)

u=0.5 m/s

𝑁 = 𝑐𝑜𝑛𝑠𝑡

ℎ = 4 3𝜋 𝜌)

𝑐/ 𝑁𝑟#,

ℎ ∝ 𝑟#, 𝐿𝑊𝐶 = 𝑐/ R ℎ

(22)

u=1 m/s 𝑁 = 𝑐𝑜𝑛𝑠𝑡

ℎ = 4 3𝜋 𝜌)

𝑐/ 𝑁𝑟#,

ℎ ∝ 𝑟#, 𝐿𝑊𝐶 = 𝑐/ R ℎ

(23)

u=1 m/s u=0.5 m/s

(24)

We can derive an estimate of the limiting, steady value of supersatutration, and the time required to reach this value, by constructing a simple model

assume that all droplets have the same size

the droplets grow only by condensation

the droplets are large enough for the solution and curvature terms in the growth equation can be neglected

let N denote their concentration per unit volume of air and its value is constant

U – constant ascending velocity

(25)

𝑑𝑆

𝑑𝑡 = 𝑄% 𝑑𝑧

𝑑𝑡 − 𝑄& 𝑑𝑞)

𝑑𝑡

𝑑𝑧

𝑑𝑡 = 𝑈

velocity 𝑞) ≈ 𝐿𝑊𝐶

𝜌- = 4

3𝜋𝜌)𝑁𝑟#,/𝜌- 𝑑𝑞)

𝑑𝑡 = 4 3𝜋 𝜌)

𝜌- 𝑁𝑑𝑟,

𝑑𝑡 = 4𝜋 𝜌)

𝜌- 𝑁𝑟 𝑟𝑑𝑟 𝑑𝑡 𝑑𝑞)

𝑑𝑡 = 4𝜋 𝜌)

𝜌- 𝑁𝑟 𝑆 𝐹- + 𝐹.

𝑟# ≈ 𝑟

S - supersaturation

𝜂 = 4𝜋 𝜌)

𝜌- 𝑁𝑟 1

𝐹- + 𝐹. 𝑄&

𝜔 = 𝑄% R 𝑈

𝑑𝑆

𝑑𝑡 = 𝜔 − 𝜂 R 𝑆

(26)

We regard w and h as constant, because they are slowly varying compared to S

The limiting value of the supersaturation (Slim) is when dS/dt=0,i.e. Seq=w/h. time constant or relaxation time of the supersaturation 𝝉=h-1.

For N=300 cm-3, p=800hPa, T=10°C, r=5µm

U (m/s) Slim (%) 𝝉(s)

1 0.11 2.12

5 0.57 2.12

10 1.14 2.12

𝑑𝑆

𝑑𝑡 = 𝜔 − 𝜂 R 𝑆

𝜔 = 𝑄% 𝑇 R 𝑈 𝜂 = 4𝜋 𝜌)

𝜌- 𝑁𝑟 1

𝐹- + 𝐹. 𝑄& 𝑇, 𝑝

𝑆 = 𝑒012 + 𝜔 𝜂

(27)
(28)
(29)

The growth of droplet

population

(30)

• When the droplets are growing by condensation under conditions of steady

supersaturation, it is possible to solve for the droplet distribution function at any time, given its form at an earlier time.

• Let us consider a sample of cloudy air in which the drop-size distribution is

characterized by the function n(r0) where n(r0)dr0 is the number of cloud droplets per unit mass of air with radii in the interval (r0,r0+dr0).

To obtain the expression for the variation of n(r,t) with time, we consider the rate at which the number of droplets in radius interval dr is changing.

We can think of the droplets as flowing through r space as they grow.

• Let I(r,t) denote the droplet curent or number of droplets per unit mass if air passing the point r in radius space.

• The rate of change of the number of droplets in dr is given by:

𝜕

𝜕𝑡 𝑛𝛿𝑟 = − 𝜕𝐼

𝜕𝑟𝛿𝑟

(31)

The droplet current is accounted for entirely by condensation :

𝛿𝑟 𝑖𝑠 independent on time

𝐼 𝑟, 𝑡 = 𝑛𝑑𝑟 𝑑𝑡

𝜕𝐼

𝜕𝑟 = 𝜕

𝜕𝑟 𝑛𝑑𝑟 𝑑𝑡

𝜕

𝜕𝑡 𝑛𝛿𝑟 = − 𝜕𝐼

𝜕𝑟𝛿𝑟

𝜕𝑛

𝜕𝑡 = − 𝜕𝐼

𝜕𝑟

The rate of change of the number of droplets in dr :

𝜕𝑛

𝜕𝑟 = − 𝜕

𝜕𝑟 𝑛𝑑𝑟 𝑑𝑡

𝑑𝑟

𝑑𝑡 = 1

𝑟𝜁 𝜁 = 𝑆 − 1 𝐹- + 𝐹. 𝑟& 𝑡 − 𝑟3& = 2𝜁𝑡

𝑟4 = 𝑟& − 2𝜁𝑡 𝑛 𝑟, 𝑡 = 𝑟

𝑟& − 2𝜁𝑡𝑛3 𝑟& − 2𝜁𝑡

𝑛 𝑟, 𝑡 = 𝑛3 𝑟 𝑟3

(32)

The solution can actually be obtained without recourse to the differential equation that defines the process.

Because growth by condensation is the only process considered, no drops are created by nucleation or lost by precipitation, coagulation, or diffusion out of the cloud. The number of droplets in the interval dr at time t is therefore the same as the number in the interval dr0 at the initial time.

𝑛 𝑟, 𝑡 𝑑𝑟 = 𝑛3 𝑟3 𝑑𝑟3 𝑛 𝑟, 𝑡 = 𝑛3 𝑟3 𝑑𝑟3

𝑑𝑟 𝑟 𝑡 = 𝑟3& + 2𝜁𝑡 ⟶ 𝑟4 𝑡 = 𝑟& − 2𝜁𝑡

𝑛 𝑟, 𝑡 = 𝑟

𝑟& − 2𝜁𝑡𝑛3 𝑟& − 2𝜁𝑡

𝑑𝑟3

𝑑𝑟 = 𝑟

𝑟& − 2𝜁𝑡

(33)

Droplet terminal fall speed

(34)

The rate at which drops fall through air depends both on:

• the size of the drop

• the density of air.

The equilibrium fall speed of a drop can be found by equating its acceleration with the drag it experiences, i.e. when the drag exactly balances the gravitational acceleration.

The gravitational force is a product of gravitational acceleration and the effective mass of the drop, i.e. its mass minus the mass of the fluid it displaces.

The drag is more complicated, but should, on the basis of dimensional arguments, take the form:

µ- the dynamic viscosity Re – the Reynolds number:

the ration of inertial to viscous force 𝐹5 = 𝜌) − 𝜌 𝑉𝑔 = 4

3𝜋𝑟,𝑔 𝜌) − 𝜌

𝐹3 = 𝑟&𝜌𝑢(& 𝑓 𝑅𝑒 𝑅𝑒 = 2𝜌𝑟𝑢 𝜇 4

3𝜌𝑟,𝑔 𝜌) − 𝜌 = 𝑟&𝜌𝑢(& 𝑓 𝑅𝑒 𝑢( = 4𝜌𝑟𝑔

3𝑓 𝑅𝑒 𝜌) − 𝜌

%$

&

≈ 4𝜋𝑟𝑔𝜌) 3𝑓 𝑅𝑒

%$

&

(35)

Re<<1; r<30µm

40µm<r<0.6mm

large Re; 0.6mm<r<2mm

The former equation is misleading, and indeterminate, because the Reynolds number, is itself a function of velocity.

The drag must be obtained empirically.

The limited regime where analytic solutions can be found is appropriate for small droplets moving through a laminar fluid.

In this regime, which is called Stokes flow Re<<1 𝐹3 = 6𝜋𝑟𝜇𝑢(

𝑢( = 𝑘%𝑟&

𝑢( = 𝑘&𝑟

𝑢( = 𝑘,𝑟%$&

𝑘% = 2 9

𝑔𝜌)

𝜇 ≈ 1.19 R 106 𝑐𝑚0%𝑠0%

𝑘& ≈ 8 R 10, 𝑠0%

𝑘, = 2.2 R 10, 𝜌3 𝜌

%$

&

𝑐𝑚%$&𝑠0%

(36)
(37)

Droplet terminal fall speed (Beard; 1977)

1µm£r£20µm 20µm£r£3mm

j m=3 m=7

0 10.5035 6.5639

1 1.08750 -1.0391

2 -0.133245 -1.4001

3 -0.00659969 -0.82736

4 -0.34277

5 -0.083072

6 -0.010583

7 -0.00054208

𝑢( 𝑟 = exp 𝑦 ; 𝑦 = u

783 9

𝑐7 ln 2𝑟 7 𝑟 = 𝑐𝑚

(38)
(39)

p=1000 hPa; T=20 C

(40)

p=1000 hPa; T=20 C

𝑢( = 𝑘%𝑟&

𝑢( = 𝑘&𝑟 𝑢( = 𝑘,𝑟%$&

(41)

p=500 hPa, T=-10 C

(42)

p=500 hPa, T=-10 C

𝑢( = 𝑘%𝑟&

𝑢( = 𝑘&𝑟 𝑢( = 𝑘,𝑟%$&

(43)

p=1000 hPa, T=20 oC p=500 hPa, T=-10 oC

(44)

Evaporation of droplets

The rate of evaporation of droplet is described by the same equation that desribes the condensational growth:

BUT S-1<0 Ž dr/dt<0

By knowing the dependence of droplet fall speed on size, it is possible to solve the equation for the distance a drop falls during the time required for it to evaporate completly.

For drops smaller than about r<50µm in radius the teminal fall speed increases approximately with the square of the radius ( u~r2).

The distance of fall for complete evaporation (l) increases with the fourth power of radius (l ~r 4).

𝑟𝑑𝑟

𝑑𝑡 = 𝑆 − 1 𝐹- + 𝐹.

𝑑𝑟&

𝑑𝑡 = 𝑐𝑜𝑛𝑠𝑡 𝑑𝑟&

𝑑𝑡 = 𝑢𝑑𝑟&

𝑑𝑙 = 𝑐𝑜𝑛𝑠𝑡 𝑢 ∝ 𝑟& ⟶ 𝑟& 𝑑𝑟&

𝑑𝑙 = 𝑐𝑜𝑛𝑠𝑡 ⟶ 𝑙 ∝ 𝑟:

(45)

initial drop radius distance of drop fall

1 µm 2 µm

3 µm 0.17 mm

10 µm 2.1 cm

30 µm 1.69 m

0.1 mm 208 m

0.15 mm 1.05 km

T =280 K,i S=0.8

cloud droplets

drizzle precipitation/rain

Cytaty

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