AEROSOL CHEMICAL PROCESSES IN THE ENVIRONMENT - CHAPTER 3 pot

47

3

The Estimation of Time-
Dependent (Relaxation)
Processes Related
with Condensation and
Evaporation of Liquid Drop

Mikhail V. Buikov

CONTENTS

Introduction 47
General Equations 48
The Solutions of the Equations 49
Relaxation of Salt Concentration 51
Thermal Relaxation 52
The Rate of Change of Drop Size 55
Intensive Evaporation of the Solution Drop 56
Summary and Conclusions 57
References 58
Nomenclature 58

INTRODUCTION

The theory of condensational growth and evaporation of drops consisting of pure liquid or solution
is a complicated thermodynamical problem that concerns many aspects of kinetic gas theory. A
complete review of the theory of growth and evaporation of drops can be found elsewhere.

1-3

Here,
only one problem will be investigated: the time-dependent processes as a result of which steady-
state temperature, vapor, and salt concentration are reached. Usually, the adopted drop temperature
is constant inside the drop volume and is determined by heat balance between phase transition and
thermal conduction; the salt concentration equal to the average volume value, vapor concentration,
and temperature outside drop obey steady-state distributions in the drop vicinity. This approach is
applicable if the rate of change of droplet size is small enough and the temperature of the environ-
ment varies slowly with time. The drop temperature can be defined as a psychometric one because
it the same as that of aspiration psychrometer in the case of evaporation. The investigation of the
transition to steady-state may be of some interest from a general point of view, and can find
application if the environment temperature is changing fast enough and in the case of intensive
drop growth and evaporation when the deviation of salt concentration near drop surface from mean

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48

Aerosol Chemical Processes in the Environment

value results in substantial influence on the rate of change of drop size. Temperature relaxation has
been investigated

4,5

; the influence of inhomogeneous salt concentration on drop growth has also
been studied.

6-8

Intensive evaporation of solution drop was considered by Buikov and Sigal.

9

Below
are presented the main results of these research efforts. The exact formulation of the problem of
growth (evaporation) of a drop of solution is presented below, the application of the heat potentials
to the solution is described, and an analysis of the time-dependent process resulting in establishing
the steady state is considered. In conclusion, the intensive evaporation of solution drop when the
formation of a salt crust is possible at the drop surface is also considered.

GENERAL EQUATIONS

The equations that describe diffusion of some non-volatile salt in volatile dissolvant (e.g., water),
heat conduction inside and outside the drop, and vapor diffusion in a gas-vapor environment in a
spherical coordinate system are:
(3.1)
(3.2)
(3.3)
(3.4)
The rate of change of drop radius is given by the conventional formula
(3.5)
but the effects under consideration will be displayed through the gradient of the vapor concentration.
If a drop at

t =

0 is placed into a gas-vapor environment, then the initial conditions are:
(3.6)
The boundary conditions are:
(3.7)
c(

r,t

) and T

2

(

r,t

) must be finite at

r =

0.
Temperature, heat flux, and vapor concentration should be continuous at the drop surface (

r =
R

(

t

)); that is,
∂
∂
=
∂
∂
+
∂
∂






c
t
D
c
rr
c
r
1
2
2
2
∂
∂
=
∂

∂
+
∂
∂






T
t
K
T
rr
T
r
1
1
2
1
2
1
2
∂
∂
=
∂
∂
+

∂
∂






T
t
K
T
rr
T
r
2
2
2
2
2
2
2
∂
∂
=
∂
∂
+
∂
∂







q
t
D
q
rr
q
r
2
2
2
2
dR t
dt
D
q
r
rRt
()
()
=
∂
∂
=
2

ρ
cr c T r T T r T qr q R R(, ) , (, ) , (, ) , (, ) , ( ) .00 000
01 102 0
== ===
∞∞
qrt q T rt T r(,) , (,) , .→→→∞
∞∞2

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© 2000 by CRC Press LLC

The Estimation of Time-Dependent (Relaxation) Processes

49

(3.8)
(3.9)
(3.10)
Equation 3.10 takes into account the effect of dissolved salt in the simplest form. The total
amount of salt inside the drop should not change with time:
(3.11)
Taking the derivative of both sides of Equation 3.11 and using Equation 3.1 we have
(3.12)
In a growing (evaporating) drop, the salt concentration decreases (increases) with

r

and does
not change if

R = const.

THE SOLUTIONS OF THE EQUATIONS

There are some specific (characteristic) time intervals in the problem. The first one (

t

1

) is the time
interval during which the steady-state field of the vapor is established in the vicinity of the drop.
The time interval

t

2

describes the relaxation of salt concentration

.

The establishment of the psycho-
metric temperature can be reached after an elapsed time

t

3

. The last time interval is connected with

a substantial increase in drop size (

t

4

); for example,



for a solution drop growing in the saturated
environment, it can be taken as the time when the salt concentration will be smaller than the initial
value. It is well-known

1,3

that

t

1

<<t

4

and

t

1

<< t

2

(

D

2

<< D

1

) and it will be clear later that

t

1

<<t

3

)

.

So, as conventionally adopted, one can use the simplified formula for

q:

(3.13)
The solutions of Equation 3.1 to 3.4 can be presented as thermal potentials

10

(dimensionless
variables):
(3.14)
Trt Trt r Rt
12
(,) (,), (),==at
LD
q
r
k
T
r
k
T
r
rRt
21
1
2
2
0

∂
∂
+
∂
∂
+
∂
∂
==,(); at
qrt q T AcRt r Rt
s
(, ) ( ) ( ()), ()=−
{}
=1 at
4
4
3
23
0
π
π
dr r c r t R c
o
∫
=(,) .
dR
dt
cRt D
c
r

rR
(,)+
∂
∂
=
=
1
0
qrt q q q Rtr
s
(,) () .
–
=+−
()
∞∞
1
xx d K x∑=∂∂∂
∫
(,) () (,, )τστ
τ
0
Kx
xZ xZ
(, , )
()
exp
()
()
exp
()

()
τ
πτ
ττ
∂=
−∂
−
−∂
()
−∂














−−
+∂
()
−∂







1
2
44
22

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50

Aerosol Chemical Processes in the Environment

(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
These solutions satisfy the equations and the initial and boundary conditions at great distances:
(3.20)
(3.21)
The heat balance equation is transformed into:
(3.22)
The temperature continuity and salt conservation equations are, respectively,
(3.23)
(3.24)
The subsidiary functions

σ

(

∂

),

σ

1

(

∂

),

σ

2

(

∂

) can be determined using other boundary conditions.
xY x d K x
11

0
1
(,) () (,, )τστ
τ
=∂∂ ∂
∫
Kx
xZ xZ
(, , )
()
exp
()
()
exp
()
()
τ
πβ τ
βτβτ
∂=
−∂
−
−∂
()
−∂















−−
+∂
()
−∂






1
2
44
1
2
1
2
1
xY x d K x
22
0

2
(,) () (,, )τστ
τ
=∂∂ ∂
∫
Kx
xZ
2
2
2
2
1
2
4
(, , )
()
exp
()
()
τ
πβ τ
βτ
∂=
−∂
−
−∂
()
−∂















Π
ΠΠ
(,)
()
x
Z
x
s
τ
τ
=
−
()
∞
dZ
d
d
dx

xZ
τ
δτ==
Π
()
∑= = = =YY
12
00, τ
Yx
2
0→→∞, .
Γ
Π
Γ
01
12
0
∂
∂
+
∂
∂
+
∂
∂
==
x
Y
x
Y

x
xZ, ( ).τ
ZY d KZ d KZ()() () ( ,, ) () (,, ),ττ σ τ σ τ
ττ
=∂∂ ∂=∂∂ ∂
∫∫
0
11
0
22
dx x x
Z
Z
2
3
0
1
3
∑=−
−
∫
(,) .τ

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The Estimation of Time-Dependent (Relaxation) Processes

51

RELAXATION OF SALT CONCENTRATION

Concentration and thermal relaxations are not connected because the heat of dissolution is not
taken into account. This is in agreement with Equation 3.24, which will be turned into the equation
for

σ

(

θ

):



the dependence on temperature is only through drop size, but not directly. If the dependence
of

Σ



(

x,

τ

)

on

x

in Equation 3.24 is neglected, then we obtain the conventional formula for salt
concentration as:
(3.25)
or,

c

(

r,t

)

≅

R

–3

(

t

).
To get the next approximation taking into account the difference between bulk and surface
concentrations, the kernel

K

(

x,

τ, θ

) is expanded into a series on

x,

keeping the terms

∼

x

3

:
(3.26)
Equation 3.25 is applicable if the following inequality is true:
(3.27)
Because
(3.28)

and
(3.29)
then, instead of Equation 3.27, we have that it is possible to neglect salt concentration relaxation if
(3.30)
This inequality cannot be satisfied for small

τ

;



but because

τ
4
>> τ
2
,

then
(3.31)
Z(τ) can be expanded for small τ and we obtain, from Equation 3.30,
(3.32)
It is a criterion of the applicability of the bulk concentration approximation, when only a small
gradient of salt concentration exists inside a drop, that
∑≅ −(,) ()
–
xZττ
3

1
∑≅∑+∑(,) () ()xxττ τ
1
2
2
∑>> ∑ ∑>> ∑
1
2
21 2
xZ ( ) .or τ
∑≅ −
−
1
3
1() () ,ττZ
∑)≅
∑
2
1
6
(τ
τ
d
d
1
1
3
2
−>>
−

Z
Z
dZ
d
()
()
()
τ
τ
τ
τ
dZ
d
()τ
τ
τ≅+1 const. ,
τ>> >>
1
66
2
1
0
2
or t
D
R
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© 2000 by CRC Press LLC
52 Aerosol Chemical Processes in the Environment
(3.33)

The gradient is negative for growth and positive for evaporation. For a 10-µm drop, t
2
is about 10
–2
s; for a 100-µm drop, t
2
is equal to some seconds; and for a 1-mm drop, t
2
is some minutes.
In the opposite case, when τ << 1/6, it is possible to derive an approximate formula for the salt
concentration, introducing Equation 3.14 into Equation 3.25, after integration, expanding for small
τ, and adopting Z = 1, one can derive:
(3.34)
This means that
(3.35)
So, for the salt concentration, one has
(3.36)
The deviation of the salt concentration from the initial value takes place only in a very thin layer
near the drop surface:
(3.37)
and deeper inside the drop volume c = c
0
.
THERMAL RELAXATION
Leveling of the temperature inside a drop is a more complicated process than salt concentration
relaxation because it involves simultaneous heat exchange inside and outside the drop and is
described by the heat balance equation (Equation 3.24), which can be transformed to the following
form:
(3.38)
where

(3.39)
d
dx
x
Z
dZ
d
∑
≅−
2
()
.
ττ
dZ∂∂=− −
[]
∫
0
3
3
1
τ
σ
π
τ() () .
σπ
τ
τ() ; .∂=− = =ςς
dZ
d
0

xx d
x
∑≅−∂
−
−
∂






∂
∫
(,)
exp
()
τ
π
τ
ς
0
2
1
4
2
x ≅−12τ,
ΓΠ Γ Γ Ψ
01112
10(,) ( )() () () () ,ZYττστσττ++ + + + =

Ψ()
()
()
exp
()
/
τ
σ
πβ τ
β
τ
=
∂∂
−∂
−
−∂






∫
d
1
1
32
0
1
4

1
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The Estimation of Time-Dependent (Relaxation) Processes 53
Two more equations for subsidiary functions σ
1
and σ
2
should be added to Equation 3.38 to find
the steady-state temperature Y(τ):
(3.40)
There are two complications in solving Equations 3.38 through 3.40: (1) non linearity due to
the presence of Π
s
(Y, τ), and (2) time dependence through Σ, which depends on Π
s
(Y(τ), τ). The
first difficulty can be easily overcome because temperature Y is, as a rule, small and Π
s
can be
expanded using the first approximation. Because thermal relaxation is much faster than concentra-
tion relaxation, and because drop size change is very small during thermal relaxation, one obtains:
(3.41)
So, instead of Equation 3.38, one has
(3.42)
Laplace transformation can be used to solve Equations 3.40 to 3.42; and for the Laplace
transforms, one obtains
(3.43)
(3.44)
(3.45)

The complex roots of the equation
(3.46)
can be found for two cases: (1) Γ
0
<< Γ
1
and (2) Γ
0
>> Γ
1
. The parameter λ = Γ
0
/Γ
1
in usual variables is
(3.47)
It is the ratio of two fluxes: condensation heat flux and thermal conductivity flux inside the drop.
So, the inequality λ << 1 means that the real heat flux of phase transition heat is much smaller than
the potential amount of heat that can be transferred by thermal conductivity in the drop. If λ << 1,
there are branch point and two poles with small real parts in Equation 3.46, so the asymptotic
formula for the surface temperature is
Y
dd
()
()
()
()
()
exp
()

τ
σ
πβ τ
σ
πβ τ
βτ
ττ
=
∂∂
−∂
=
∂∂
−∂
−
−∂






∫∫
2
2
0
1
1
0
1
22

1
1
ΠΠΠ
s
ZY(,) ()ττ≅−
∞ 1
10
11 0 111
1
22
1
+−
()
++
()
+
()
+
()
=
∞
−−
ΓΓΠ ΓΠ Γ ΨY στβ στβ τ
σ
β
β
1
1
1
1

2
12
()
()
exp
,s
sY s
s
=
−−
()
−
σβ
22
2() (),ssYs=
Ys
ss
()
()
.=
∞
ΠΓ
Φ
0
Φ()s = 0
λ=
∞
Π LD q T
R
R

Tk
s20
0
0
01
()
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54 Aerosol Chemical Processes in the Environment
(3.48)
Y
p
is steady-state (psychometric) temperature of the drop surface:
(3.49)
(3.50)
It can be shown that a more simple formula (Equation 3.51) is also applicable because it is
possible that τ
3
>> τ >> 1. Temperature relaxation in this case is slow and regular enough. The
primary reason for this is the slow growth or evaporation of the drop, which is determined by the
low value of supersaturation Π
∞
. In the opposite case, when λ >> 1, there are no small poles in
Equation 3.46 and thus,
(3.51)
(3.52)
because the exponential term is absent in this case (the steady-state temperature is reached more
quickly). In both cases, thermal relaxation inside the drop is slower than that outside because β
2
>> β

1
. To derive the conventional formula for the outside temperature from Equation 3.40, one
obtains:
(3.53)
By substituting Equation 3.53 into Equation 3.16 after some transformation, it is easy to derive the
following expression.
(3.54)
If β
2
τ >> (1 – x)
2
, then a formula similar to that for vapor concentration (Equation 3.18)) follows.
(3.55)
YY
p
( ) expτ
τ
τ
π
βτ
=−−






−







1
2
3
0
2
Φ
Y
p
=
+
∞
ΠΓ
Γ
0
1
, and
τ
β
3
1
2
31
=
+
Γ
Γ()

.
YY
p
( ) expτ
τ
τ
=−−












1
3
YY
p
()τ
π
βτ
=−







1
2
0
2
Φ
στ
β
π
τ
τ
2
2
0
2()
()
.=
∂
−∂
∂
∂
∫
ddY
d
xY dz e Y
x
z
x

z
2
1
2
2
2
2
21
4
2
2
()
()
τ
π
τ
β
βτ
=−
−






−
∞
−
∫

Yx
Y
x
2
(,)
()
τ
τ
=
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The Estimation of Time-Dependent (Relaxation) Processes 55
The steady-state temperature Y
p
is obtained from Laplace transform (Equation 3.46) when s →
0; but the same formula can be obtained from Equation 3.42 when t → ∞; the last three terms
vanish and the first two terms represent the conventional heat balance equation of the drop. From
Equation 3.43, it follows that σ
2
(τ) can be connected with dY/dt by an integral equation similar to
Equation 3.53, but with a different kernel. This kernel at large τ will be proportional to τ
1/2
, so
asymptotically, this will be similar to Ψ(τ) and both these terms will be exponentially small (λ <<
1) or as τ
–1
when τ → ∞. More troublesome is the derivation of the formula for the temperature
gradient inside the drop. An approach similar to that used in deriving Equation 3.27 for the salt
concentration can be applied. Expansion of the kernel K
2

(τ, θ, x) in Equation 3.16 leads to:
(3.56)
(3.57)
(3.58)
It is natural that the second term that determines the gradient is proportional to temperature
conductivity. For large τ, from Equation 3.58, one can obtain
(3.59)
The temperature gradient inside the drop is positive (negative) for the growing (evaporating)
drop, smaller near the drop center, and very rapidly decreases with time.
THE RATE OF CHANGE OF DROP SIZE
Thermal relaxation can influence the rate of drop growth only through the saturatation vapor density
in Equation 3.5 and Equation 3.13, which depend on surface temperature. If vapor supersaturation
is small, the deviation of surface temperature from the steady-state value will result in some
retardation of the rate of growth during relaxation (T
10
< T
∞
) and will be greater than at steady-
state. The mirror-reflected situation will take place in the case of evaporation. If the environmental
temperature varies with time for the period (t
∞
) much greater than the relaxation time (λ << 1), then
the drop temperature will follow it. For the period smaller than the relaxation time, the drop will
grow under average environment temperature. In the case λ >> 1, when there is no characteristic
time, the latter case corresponds to t
∞
<< R
0
–1
D

2
and the former to t
∞
>> R
0
–1
D
2
.
Unlike thermal relaxation, deviation in the salt concentration can directly influence drop growth.
For a saturated vapor environment and neglecting thermal relaxation, the following formula is
derived for small time (usual variables).
(3.60)
Yx x
111
2
12
(,) ,τ= +ΠΠ
Π
11
1
4
1
1
2
4
1
2
() ,τ
π

τβτ
βτ
=−
()
()
∞
−
−
∫
ds e Y
r
s
Π
12
1
1
4
2
2
1
1
2
2
3
14
1
2
() .τ
β
π

τβτ
βτ
=−






−
()
()
∞
−
−
∫
ds e s
s
Y
r
s
Yx Y
1
1
23
1
1
96
(,) () .ττ
πβ τ

=+






dR
dt
R
t
R
Rt
D
=+−








0
0
5
2
0
32
1

1
2
3
ω
ω
π
/
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56 Aerosol Chemical Processes in the Environment
The second term in the brackets describes the growth at the initial salt concentration and does not
depend on salt diffusivity. The third term decelerates the growth due to the decrease in salt
concentration near the drop surface. The smaller the salt diffusivity, the greater the salt concentration
gradient. This formula is valid during the first moments of concentration relaxation (t << t
2
); during
the last moments (t >> t
2
), we have the equation to determine Z(τ):
(3.61)
For weak solutions, g << 1, and one obtains for the zero approximation:
(3.62)
The next iteration gives
(3.63)
INTENSIVE EVAPORATION OF THE SOLUTION DROP
An example of the application of the theory of concentration relaxation is intensive evaporation of
the solution drop. Salt concentration enhancement near the drop surface can be large and may result
in the formation of a solid crust. The treatment of this problem was considered by Buikov and
Sigal. Intensive evaporation takes place under high undersaturation in the environment and blowing
of the drop with dry air, so it is possible to assume that the drop radius is a linear function of time;

that is,
(3.64)
Using Equations 3.13 and 3.21, it is possible to obtain the integral equation for the subsidiary
function σ(θ) as:
(3.65)
The salt concentration can be calculated from Equation 3.25. The following formula was derived
for salt concentration at the drop surface:
(3.66)
Crystallization of salt will start when the solution near the surface is saturated (c(R(t
s
), t
s
) =
c
s
) and then the solid crust can grow at the drop surface. For the time for the formation of crust
of thickness δ the following formula can be applied:
Z
g
Zg
53
1
5
3
15() () .τττ−+ −
[]
=
Zgt
0
15

1() ( ) .
/
τ≅ +
ZZ g
Z
Z
() ()
()
()
ττ
τ
τ
≅−
−






0
0
2
0
1
1
Rt R bt() .=−
0
dKZ Z Z a∂∂ ∂ ∂=− −
()

=−
∫
0
4
3
2
3
11
τ
στ τ τ τ() (,) () () ; () .
Σ()
() ()
.τ
τ
πτ π
=+






a
ZDa
1
1
2
L829/frame/ch03 Page 56 Monday, January 31, 2000 2:06 PM
© 2000 by CRC Press LLC
The Estimation of Time-Dependent (Relaxation) Processes 57

(3.67)
Values of t
s
and t
c
calculated for the experimental conditions are given in Tables 3.1 and 3.2.
11,12
SUMMARY AND CONCLUSIONS
The classical formula for the condensation or evaporation rate of a liquid drop derived by
Maxwell and modernized by Fuchs
1
is based on some hypotheses of complete physical lucidity.
This formula is widely used in many branches of aerosol science. Giving up the hypotheses
leads to more complicated mathematical problems and complicates the solution. In the research
work reviewed in this chapter, a new approach is introduced: a heat balance equation on the
drop surface that connects thermal processes inside and outside the drop with phase transition
heat. The application of thermal potentials to solving the system of equations of heat, vapor,
and salt diffusion resulted in the integral equations for subsidiary functions. This system of
equations may be more appropriate for use in numerical methods than the primary system of
differential equations. The approximate analytical analysis of the processes of drop growth
carried out using these integral equations makes it possible to penetrate more deeply into heat
and salt transfer inside the drop, as well as to follow the transition from the initial state of the
drop to steady-state growth and steady-state fields of temperature, vapor, and salt. The asymp-
totical formulas for salt and temperature gradients are also derived. Analytical studies resulted
only in some corrections to the conventional formulas, but nevertheless it has been demonstrated
that the procedure developed can be useful for intensive processes of drop evaporation. It is
hoped that the theory developed can find more wide application, namely for intensive growth
and evaporation — especially if the drop or environmental temperature varies with time rapidly
enough.
TABLE 3.1

The Time to Start Crystallization
Velocity of Blowing
(cm s
–1
)
Experimental Value
(s)
Calculated Value
(s)
40 0.390 0.350
90 0.155 0.145
160 0.100 0.046
TABLE 3.2
The Time to Form Crust
Substance
Experimental
(s)
Calculated
(s)
Na
2
SO
4
210 175
NH
4
NO
3
235 215
tt

R
bR
dc
dr
rR
cs
sc
ss
=+




=
2
2
22
δρ
L829/frame/ch03 Page 57 Monday, January 31, 2000 2:06 PM
© 2000 by CRC Press LLC
58 Aerosol Chemical Processes in the Environment
REFERENCES
1. Fuchs, N.A., Evaporation and Droplet Growth in Gaseous Media, Pergamon Press, New York, 1958.
2. Mason, B.J., The Physics of Clouds, Clarendon Press, London, 1957.
3. Pruppacher, H.R. and Klett, I.D., Microphysics of Clouds and Precipitation, D. Reidel, Dordrecht,
1978.
4. Buikov, M.V. and Dukhin, S.S., Diffusional and heat relaxation of evaporating drop, Eng. Phys. J.,
5(3), 1962 (in Russian).
5. Buikov, M.V., Diffusional and heat relaxation of evaporating drop. Part 2, Eng. Phys. J., 5(4), 1962
(in Russian).

6. Buikov, M.V., Time dependent growth of solution drop. I. Relaxation of concentration, Colloidny J.,
24(6), 1962 (in Russian).
7. Buikov, M.V., Time-dependent growth of solution drop. II. Heat relaxation, Colloidny J., 25(1), 1963
(in Russian).
8. Buikov, M.V., Some Problems of Growth and Evaporation of Drops in Gaseous Media, Thesis of
candidate dissertation. Kiev (in Russian), 1963.
9. Buikov, M.V. and Sigal, V.I., Intensive evaporation of solution drop, Problems of Evaporation, Com-
bustion and Gas Dynamics of Disperse Systems, Editor, V.A. Fedoseev, Naukova Dumka Publ. House,
Kiev (in Russian), 1967.
10. Smirnov, V.I., Course of Higher Mathematics, Vol. 2, Gostechizdat, Moscow (in Russian), 1948.
11. Ranz, W.E. and Marshall, W.R., Chem. Eng Progr., 48(3), 1952.
12. Charlesworth, D.E. and Marshall, W.B., A.I.Ch. E.J., 6(1), p. 1959, 1960.
NOMENCLATURE
AM
v
/M
s
ρ
c(r,t ) Concentration of salt
D
1
Diffusivity of salt
D
2
Diffusivity of vapor
D(a)
v.p. means the main value.
g
κ
1

(K
1
) Heat (temperature) conductivity of liquid
κ
2
(K
2
) Heat (temperature) conductivity of gas-vapor mixture
L Phase transition heat
m Mass of salt in drop
M
s
(M
d
) Molecular weight of salt (dissolvant)
q(r,t ) Vapor concentration
q
s
(T) Density of saturated vapor
r Radial coordinate
R(t) Radius of drop
t Time
t
1
= R
0
2
D
2
–1

t
2
= R
0
2
D
1
–1
T
1
(r, t ) Temperature of liquid inside drop
=− +
()
−
()
∞
−
−
−
∫
144
0
22
1
12
2
v.p. ds e a s a s
s
π
/

=
+
Ac
a
δ
1 Γ
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© 2000 by CRC Press LLC
The Estimation of Time-Dependent (Relaxation) Processes 59
T
2
(r, t ) Temperature of gas-vapor mixture
T
p
Steady-state drop surface temperature
x = rR
0
–1
Y
1
(x, τ)
Y
2
(x, τ)
Y(τ)= (T(R(t)) – T
10
)T
10
–1
Temperature of drop surface

Y
p
= (T
p
– T
10
)T
10
–1
Steady-state surface temperature
Z(τ) = R(t)/R
0
β
1
= κ
1
D
1
–1
β
2
= κ
2
D
1
–1
γ = D
2
D
1

–1
δ
Γ = Γ
0
Π
∞
Γ
1
= κ
1
κ
2
–1
Γ
0
λ
Π(z(τ), τ)
Π
s
(Y)
Π
∞
Π
1
ρ Liquid density
ρ
s
Density of salt
σ(θ),σ
1

(θ),σ
2
(θ) Kernels of thermal potentials
∑(x, τ)
∑
1
(τ)
=+
−
()
()
+
∞∞
T
LD q q T
s
10
2
2
1κ ()Γ
=−
()
−
Trt T T
11010
1
(,)
=−
()
∞∞

−
Trt TT
2
1
(,)
=
Dq T
D
s20
1
()
ρ
=
()
∞∞
−−
LD q T T
s2
1
2
1
κ
=
()







∞∞
−
Π LD q T
R
T
R
s2
0
01
0
1
κ
=−+
[]
+
()
ΠΠΣ
ss
YAC Y x() () (,)
0
11τ
=−
()
()
()
∞
−
∞
qT qT q T
sss

()
1
=
()
−
()
−
∞∞ ∞
∞
q T Ac q
q
1
0
=
−
()
∞
∞
1
0
Ac
q
dq T
dT
s
=−
()
−
Crt C C(,)
00

1
=
−
∫
−
−
dZ
e
Z
θσ θ θ
πτ θ
τ
θ
τθ
2
32
0
4
2
2
()()
()
/
()
()
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© 2000 by CRC Press LLC
60 Aerosol Chemical Processes in the Environment
∑
2

(τ)
τ = t D
1.
R
0
–2
Φ(s)
Φ
0
ω
=
−
−
−






∫
−−
−
dZ
Z
e
Z
θσ θ θ
πτ θ
θ

τθ
τ
θ
τθ
2
32
2
0
4
6
6
1
2
()()
()
()
()
/
()
()
=+ − + +
()
−−−
1
1
12
1
12
1
12

2
12 12
2
12
ΓΓ Γs s cth s
// // //
βββ
=
+
2
1
1
πβ()Γ
=
()
+
∞
qT AmD
R
s 2
0
5
1ρ()Γ
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© 2000 by CRC Press LLC