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Evaporation from a nonspherical aerosol particle situated in an absorbing gas
PII: S0021-8502(98)00041-X
J. Aerosol Sci. Vol. 30, No. 2, pp. 139—156, 1999 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0021-8502/98 $19.00#0.00
EVAPORATION FROM A NONSPHERICAL AEROSOL PARTICLE SITUATED IN AN ABSORBING GAS P. Tekasakul,* R. V. Tompson,s and S. K. Loyalkast * Department of Mechanical Engineering, Thammasat University, Klong Luang, Pathum Thani 12121, Thailand s Particulate Systems Research Center and Nuclear, Chemical, and Mechanical & Aerospace Engineering Departments, University of Missouri-Columbia, Columbia, MO 65211, U.S.A. (First received 8 May 1997; and in final form 27 February 1998) Abstract—The problem of an aerosol particle evaporating in an infinite expanse of an absorbing gas is considered. The relevant Helmholtz equation (resulting from the steady-state diffusion equation with an absorption term included) with density jump boundary conditions is converted into a boundary integral equation via the use of the Green’s function. The resulting integral equation is valid for particles of arbitrary shape. Explicit numerical results for the local and average evaporation rates are reported for several axisymmetric particles for a range of values of the dimensionless absorption parameter (j2), where j is the ratio of the radius of the particle (a) to the diffusion length (l). Here, the diffusion length is defined as l"[D/(v& )]1@2, in which v (cm s~1) is the average ! thermal speed of the vapor molecules, & (cm~1) is the cross-section for absorption of the vapor by ! the gas, and D (cm2 s~1) is the diffusion coefficient of the vapor in the gas. Our numerical results for the local and average evaporation rates for a sphere exhibit excellent agreement with the corresponding analytical values (maximum deviation (0.40%). We find that the evaporation rate increases with increasing absorption and that this increase depends on the degree of departure of the particle from a spherical shape. The jump distance has a large impact in that it significantly lowers the evaporation rates as it increases in magnitude. It should be remarked that the results of this paper are also directly applicable to the problem of either neutrons or photons undergoing diffusion from a source situated in an absorbing medium. ( 1998 Elsevier Science Ltd. All rights reserved.
1 . I NT RO D UC T IO N
Evaporation or condensation of particles is a phenomenon of substantial interest in the field of aerosol science. The literature on the subject is extensive and several reviews covering the experimental as well as the theoretical aspects are available (Mason, 1962; Fuchs, 1964; Davies, 1966; Fuchs and Sutugin, 1971; Hidy and Brock, 1972; Friedlander, 1977; Twomey, 1977; Seinfeld, 1986; Williams and Loyalka, 1991). The problem is mainly one in which a particle, situated in an infinite expanse of a background gas and vapor mixture, either grows due to condensation or diminishes in size due to evaporation of the vapor from the particle surface. Generally, regarding the vapor in the mixture, it is assumed first, that it is in trace or small (dilute) amounts and second, that it is not absorbed by the background gas. There can, however, be physically realistic circumstances in which departures from both of the above assumptions occur. In this paper, our purpose is to study the influence of vapor absorption by the background gas on the evaporation rate of a single nonspherical particle. We will assume that the vapor is in dilute amounts, but that the background gas is a mixture of components that can scatter as well as absorb (or react with) the vapor molecules. For example, in a dense aerosol a particle would be surrounded by many other particles, and these particles could act as sinks (or sources) for the vapor. While one could solve the multiple particle problem in the manner we have discussed in previous papers (Griffin and Loyalka, 1996), the effect of the surrounding particles could be approximately represented to the first order by a homogeneously distributed absorbing cloud (Frisch and Collins, 1952). Similarly, in the evaporation of a binary particle (droplet) where one component is in substantially smaller
t Author to whom correspondence should be addressed. 139
140
P. Tekasakul et al.
concentration than the other, the interactions between the two components in the vapor phase could have a measurable effect on the evaporation of the first component. While it would be of interest to study the time-dependent evaporation rate for the entire range of Knudsen number (Kn"j/d, where j is the mean free path and d is the characteristic particle size, i.e. diameter), we confine our attention here to the steady-state continuum (diffusion) equation with jump boundary conditions. For the steady-state, this leads to the Helmholtz equation rather than the Laplace equation obtained in nonabsorbing problems. We note that the Helmholtz equation also arises in studies of the ambipolar diffusion charging of aerosols (Mason and McDaniel, 1988) as well as in the calculation of potential fields around a particle in the context of the Debye—Huckel theory (e.g. Prock and McConkey, 1972). The fundamental nature of the problem is evident in that a very similar circumstance arises when a source of any kind (neutrons, photons, etc.) is situated in an infinite expanse of an absorbing material and one is interested in knowing the rate of loss of the neutrons or photons (Duderstadt and Hamilton, 1976). For our studies here, we convert the Helmholtz equation into a convenient boundary integral equation by the use of a Green’s function technique. This integral equation applies to particles of arbitrary shape without restriction. In particular, particles with concave and occluded surfaces can also be considered. Such particles are frequently encountered in practice. Due to the computational times involved, however, we have obtained accurate numerical solutions of this equation for a selection of axi-symmetric particles only. We have considered spherical, spheroidal, and cylindrical particles. The parameters defining the spheroidal particles can be adjusted to include everything from highly oblate, disk-like particles to highly prolate, needle-like particles. The parameters for cylindrical particles can be adjusted to encompass everything from thin disks to essentially finite fiber-like particles. Fibers, in particular, are of great interest in many branches of aerosol science. We have considered wide ranges of aspect ratios and inverse diffusion lengths, k"(v& /D)1@2 (cm~1), where v (cm s~1) is the vapor molecular average thermal speed, ! & (cm~1) is the cross-section for vapor absorption by the gas, and D (cm2 s~1) is the ! diffusion coefficient of the vapor in the gas. In the next section, we describe the formulation of the problem. In Section 3, the conversion of the Helmholtz equation into an integral equation using the Green’s function is described and the numerical procedure is explained. The results for some selected axisymmetric particles are presented in Section 4. Finally, in Section 5, a discussion of our numerical results is provided along with our conclusions.
2 . ST AT EM EN T O F TH E PRO BL EM AN D AN ALY TI C AL S OL U TI ON S
The dimensional diffusion equation for an absorbing medium is given by (Weinberg and Wigner, 1958; Duderstadt and Hamilton, 1976) L/K "!+ª ) Jª !v & /K , ! LtL
(1)
where /K (d cm~3) is the vapor (molecule) number density, v (cm s~1) is the average thermal speed of the vapor molecules, & (cm~1) is the cross-section for absorption of the vapor by ! the gas, and Jª is the local current defined by Jª "!D +K /K ,
(2)
Here, D (cm2 s~1) is the diffusion coefficient of the vapor in the gas. Note that we use the hat symbol ( ª ) for dimensional quantities which are non-dimensionalized later in this section. At steady state, equation (1) reduces to the Helmholtz equation
A
B
1 +ª 2! /K "0, l2
(3)
Evaporation from a nonspherical aerosol particle
141
where, l"[D/(v & )]1@2 is known as the ‘diffusion length’ or ‘inverse wave number’. The ! density jump boundary condition at the surface of a particle is written as L/K , /K (rˆ )"/K #dK 4 4 % LnL
(4)
where /K (d cm~3) is the saturation density of the vapor, rˆ is a point on the surface of the 4 4 body, nˆ is a unit normal vector on the body surface at rˆ directed into the gas, and dK (cm) is 4 % the jump distance. We assume that both /K and dK are independent of rˆ . Also, far away from 4 % 4 the surface we have the limiting condition lim /K (rˆ )"0. ?=
(5)
rˆ
The dimensional local (d cm~2 s~1) and total (d s~1) evaporation rates at the surface of a particle are defined as L/K JK "!D , (6) L LnL rˆ ! and
K
P
JK " JK drˆ , L 4 T
(7)
respectively. An average evaporation rate can be obtained for a particle by dividing the total evaporation rate by the surface area of the particle, SK . This will produce values of the same order as the local evaporation rates and permits easier comparison of results between particles where the actual dimensions of the particles may be quite different. Expressions for the surface areas of the different bodies considered here are given in the Appendix. In order to simplify the matter, the problem is nondimensionalized. Nondimensionalizing lengths by the body dimension, a (for a sphere, a is the radius, whereas, for other axi-symmetric particles, a is the equatorial radius), vapor number densities by /K , and the 4 local currents by D /K /a, e.g. d "dK /a, r "rˆ /a, S"SK /a2, /"/K //K , and J"Jª /(D/K /a), 4 % % 4 4 4 4 the Helmholtz equation [equation (3)] becomes (+2#k2) /"0,
(8)
where k"$ij and j"a/l. The value of k can be either a positive or a negative imaginary number but the negative value, i.e. k"!ij, is chosen in order to ensure that / vanishes far away from the particle. The jump boundary condition at the surface of the particle and the limiting condition far away from the surface become L/ /(r )"1#d , 4 % dn
(9)
and lim /(r)"0, ?=
(10)
r
respectively. The dimensionless local and total evaporation rates at the surface of a particle then become
K
L/ J " , L dn r and
P
J " J dr , T L 4 respectively.
(11)
4
(12)
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142
An analytical solution for the dimensionless vapor number density around a sphere of radius, a, is easily determined to be
AB
/(r)"
1 [1#d (1#ik)]~1 exp [!k(r!1)], % r
(13)
which leads to the following expressions for the evaporation rates: L/ J "! L dr and 1 J /S" T 4n
K
"(1#ik) [1#d (1#ik)]~1, % r/1
P JL dr4"(1#ik) [1#d% (1#ik)]~1.
(14)
(15)
Typically, at room temperature and atmospheric pressure, one has v+5]104 cm s~1 and D+0.2 cm2 s~1. Since the value of the absorption cross-section is typically in the range, & +0.0—10~2 cm~1, l could typically be expected to range from infinity (no absorption) to 4 about 0.02 cm (very strong absorption). For a typical aerosol particle with a radius of about 0.001 cm, the absorption parameter is then 0.0(j2(2.5]10~3. In this paper, we consider values of j2 in the range from 0.0 to 10~3. For clarity, we note that consideration of the steady state in the present study is not a serious limitation on the results in that the time constants involved in the evaporation and condensation processes of interest are such that the steady-state approximation suffices (Seinfeld, 1986). This remains true even in the presence of the variable curvature of the surface and the variations in the local condensation rate. 3 . N U ME RI CA L T ECH N IQ U E
The Helmholtz equation [equation (8)] together with the specified boundary conditions [equations (9) and (10)] can be solved by the use of the Green’s function technique described previously in conjunction with the problem of particle condensation in a nonabsorbing medium (Loyalka and Griffin, 1993; Griffin and Loyalka, 1996) and in conjunction with the problem of the oscillation of axi-symmetric bodies in a viscous fluid (Tekassakul et al., 1998). The Green’s function is defined by (+2#k2) t (r, r@)"!4nd (r!r@),
(16)
exp (!ikDr!r@D ) t (r, r@)" . Dr!r@D
(17)
and hence,
Using Green’s second identity in conjunction with the condition at infinity, we have the following for values of r exterior to the body:
P Ct(r, r@4 )
!
D
L/(r@ ) Lt (r, r@ ) 4 !/(r@ ) 4 dr@"4n 4 4 Ln@ Ln@
P / (r@4 ) d(r!r@4 ) dr@4 .
For rPr , a point on the body surface, we have 4 L/(r@ ) Lt (r , r@ ) 4 !M/(r@ )!/(r )N 4 4 dr@"4n /(r ). ! t(r , r@ ) 4 4 Ln@ 4 4 4 4 Ln@
PC
D
(18)
(19)
Applying the boundary condition at the surface, equation (19) becomes 4n d %
L/(r ) L/(r@ ) 4 # t(r , r@ ) 4 dr@!d 4 4 Ln@ 4 % Ln
P
P G Ln@4 ! L/(r@ )
H
L/(r ) Lt (r , r@ ) 4 4 dr@"!4n. 4 4 Ln@ Ln (20)
Evaporation from a nonspherical aerosol particle
143
L/(r ) 4 , p8 (r )" 4 Ln
(21)
KI (r , r@ )"t(r , r@ ), 1 4 4 4 4
(22)
Lt (r , r@ ) 4 4 , KI (r , r@ )"! 2 4 4 Ln@
(23)
If we now define
and
equation (20) becomes
P
4n d pJ (r )# KI (r , r@ ) p8 (r@ ) dr@#d % 4 4 1 4 4 % 4
P KI 2 (r4 , r@4 ) [p8(r@4 )!pJ (r4 )] dr@4"!4n .
(24)
Equation (24) is a Fredholm integral equation of the second kind. Since singularities are present, the method of singularity subtraction is employed and the integral equation is rewritten as
P
(gJ (r )#4n d ) p8 (r )# KI (r , r@ ) [p8 (r@ )!p8 (r )] dr@ 4 4 % 4 4 4 1 4 4 #d % or
P KI 2 (r4 , r@4 ) [p8 (r@4 )!p8 (r4 )] dr@4"!4n, P
(gJ (r )#4n d ) p8 (r )# KI (r , r@ ) [p8 (r@ )!p8 (r )] dr@"!4n, 4 % 4 4 4 4 4 4 where
P
gJ (r )" KI (r , r@ ) dr@, 4 1 4 4 and
KI (r , r@ )"KI (r , r@ )#d KI (r , r@ ). 4 4 1 4 4 % 2 4 4 For an axi-symmetric particle
(25)
(26)
(27) (28)
dr "J df du, (29) 4 463& where J is the dimensionless Jacobian of the surface integration (given in Table 1 for 463& a sphere, a spheroid, and a right circular cylinder), u is the azimuthal angle, and f is a coordinate specifying a point on the meridian counter of the body where !1)f)1. Thus, if we let and
p8 (r ) N p(f), 4
(30)
gJ (r )Ng(f), 4
(31)
equation (26) can be rewritten as
where
1
P~1 K(f, f@) [p(f@)!p(f)] df@"!4n,
(g(f)#4n d ) p(f)# %
and
1
P~1 K (f, f@) df@,
g(f)"
K (f, f@)"J@ 463&
2n
P0
KI (r , r@ ) du @. 4 4
(32)
(33)
(34)
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144
Table 1. Dimensionless surface Jacobians for spherical, general spheroidal, and cylindrical coordinates. Here, A"a/b is the aspect ratio of the spheroid in which a is either the minor semiaxis of a prolate spheroid or the maor semiaxis of an oblate spheroid, b is the other semiaxis of the spheroid (i.e. the major semiaxis of a prolate spheroid or the minor semiaxis of an oblate spheroid), - is the dimensionless radial location (from the axis 4 of the cylinder) on the surface of the cylinder, and f is the coordinate specifying a point on the meridian contour of the body Coordinates
Dimensionless surface Jacobian expression
Spherical Spheroidal Cylindrical
J "1 463& J "[f2#A~2 (1!f2)]1@2 463& J "463& 4
Equation (32) can now be expressed as the following system of linear algebraic equations: N (g(f )#4n d ) p(f )# + K(f , f ) [p(f )!p(f )] w "!4n , i % i i j j i j j/1 where w is a weight function. Rewriting equation (35) in matrix form, we have j Ap"!4n,
(35)
(36)
where the coefficient matrix, A, can be written as
G
A " ij
K(f , f ) w , i j j
iOj, (37)
N (g(f )#4n d )! + K(f , f ) w , i"j. i % i k k k/1 kOi By inversion, the solution to equation (36) is then p"A~1 (!4n).
(38)
With p determined in this manner, the local and total evaporation rates can then be calculated from equations (11), (12), and (21). All that remains in this general development of the problem is the evaluation of K (f, f@) in equation (34) which can be written as K (f, f@)"J @ 463&
2n
P0
KK (r , r@ ) du @"K (f, f@)#d K (f, f@) . 4 4 1 % 2
(39)
For an axi-symmetric body (Fig. 1) [see Tekasakul et al., 1997] 2n
P0
K (f, f@)"J @ 463& 1
A
2 t(r , r@ ) du@"J @ Q (c) 463& (- -@ )1@2 ~1@2 4 4 4 4
B
2n exp (!ik [2- -@ (c!cos ( u!u @))]1@2)!1 4 4 du @ , [2- -@ (c!cos(u!u @))]1@2 0 4 4
P
#
(40)
and: K (f, f@)"!J @ 463& 2
2n Lt(r , r@ ) 4 4 du@, Ln@
P0
(41)
in which Q (c) is an associated Legendre function of fractional order with the argument ~1@2 b , (42) c"1# 2- -@ 4 4
Evaporation from a nonspherical aerosol particle
145
Fig. 1. The dimensionless coordinate system used in the present work for a general axi-symmetric particle.
where b"(-@!- )2#(z@!z )2. 4 4 4 4 To determine K (f, f@), we first note that 2 exp(!ikt) t(r , r@ )" , 4 4 t
(43)
(44)
where t"Dr !r@ D. Therefore, 4 4 Lt(r , r@ ) ik Lt L 1 4 4 "! exp (!ikt) #exp (!ikt) Ln@ t Ln@ Ln@ t
C
A BD
ik Lt L 1 "exp (!ikt) ! # t Ln@ Ln@ t
AB
.
(45)
Since
AB
L 1 1 Lt "! , Ln@ t t2 Ln@
(46)
equation (45) can be expressed as
AB
Lt(r , r@ ) L 1 4 4 "(1#ikt) exp (!ikt) . Ln@ Ln@ t
(47)
We note here that (Sobolev, 1964)
where
AB
L 1 (r@!r ) ) n@ 4 0658!3$ , "! 4 Dr !r@ D3 Ln@ t 4 4 (r@!r )"(z@!z ) n #[-@ cos (u @)!- cos (u)] n 4 4 4 ; 4 4 4 x #[-@ sin (u @)!- sin (u)] n , 4 4 y
(48)
(49)
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146
Fig. 2. The particle geometries that are considered in this work. (a) Prolate spheroid. (b) Oblate spheroid. (c) Cylinder.
and (50) Dr !r@ D"[2- -@ (c!cos (u!u @))]1@2 . 4 4 4 4 For any spheroid (sphere, prolate spheroid, or oblate spheroid), the outward pointing unit normal vector can be written as
C
D G
H
~1@2 z@ z@2 4 4 " 1# A2 A n #cos (u @) n #sin (u @) n , n@ 0658!3$ z x y (A~2!z@2 )1@2 (A~2!z@2 ) 4 4 (51) where, A"a/b is the aspect ratio of the spheroid (Fig. 2a and b). Therefore, K can be 2 explicitly written as -@ K (f, f@)" 4 2 A
P0 G(1#ikK Dr4!r@4 D ) exp (!ikK Dr4!r@4 D ) [2-4 -@4 (c!cos (u!u@))]~3@2 2n
C
DH
z@ 4 ] [z@!z ] A#-@!- cos (u!u @) du@. (52) 4 4 [A~2!z@2 ]1@2 4 4 4 For a right circular cylinder (Fig. 2c), the outward pointing unit normal vector can be written as
G
bottom surface, !n z n@ " cos (u @) n #sin(u @) n side surface, 0658!3$ x y n top surface. z
(53)
Evaporation from a nonspherical aerosol particle
147
This results in
G
bottom surface, !(z@!z ) 4 4 " -@!- cos (u!u @ ) side surface, (r@!r ) ) n@ 4 4 4 4 0658!3$ (z@!z ) top surface, 4 4 such that
(54)
G
(z@!z ) 4 4 ! bottom surface, [2- -@ (c!cos (u!u @))]3@2 4 4 -@!- cos (u!u @) (r@!r ) ) n@ 4 4 4 4 0658!3$" side surface, (55) Dr !r@ D3 [ 2- -@ (c!cos (u!u @))]3@2 4 4 4 4 (z@!z ) 4 4 top surface. [2- -@ (c!cos (u!u @))]3@2 4 4 Therefore, K for the case of a cylinder can be calculated from equations (47), (48), and (55). 2 4 . RE SU LT S
Our goal was to calculate the local and average evaporation rates for arbitrary axisymmetric particles in an infinite expanse of an absorbing background gas where the density jump boundary conditions are applicable. The precision of our numerical method was benchmarked against known analytical solutions for a spherical particle for several different values of the dimensionless jump distance (d ) and the dimensionless absorption % parameter (j2). The local and average evaporation rates on spheroidal and cylindrical particles covering a range of aspect ratios (A"a/b) were then calculated as representative cases for our numerical work. The results of benchmarking against a sphere are reported immediately below and are followed by some representative results obtained for spheroids and cylinders. We have performed all of our numerical calculations using Mathematica ' 2.2.2 on a Power Macintosh 7100/66. The results for the local and average evaporation rates are reported in dimensionless forms. The dimensional local ( JK ) and total ( JK ) evaporation L T rates can be readily obtained from JK "(D/K /a) J and JK "(D/K /a) J , respectively. L 4 L T 4 T 4.1. Spheres Since the analytical solution for a spherical particle is known, we first benchmark the precision of the numerical method against the solutions for a sphere for four values of the jump distance (d "0.0, 0.1, 1.0, 10.0). Since the jump condition is analogous to a condition % of surface reaction, and the surface reactions are also of interest, we have included larger values of the jump distance although these are comparable to particle size. No doubt, such a value of the jump distance would generally require consideration of the Boltzmann equation rather than the diffusion equation. The dimensionless local evaporation rates, J , L or the dimensionless average evaporation rates, J /S, are calculated numerically and are T compared to the corresponding analytical values determined from equations (14) or (15). For the case of a sphere, analytical solutions of J and J /S coincide. The computed local L T evaporation rates are virtually constant across the surface of the sphere with only a slight variation. The average value of the local evaporation rate, J , over the entire surface L,!7% agrees exactly with the value of the corresponding average evaporation rate for each value of the absorption parameter j2. This is shown in Fig. 3 together with the analytical values. The values of j2 that were used in these calculations range from 0.0 (implying no absorption present) up to 10~3 (where the absorption is high). The result at j2"0 is not shown because of the logarithmic scale of the abscissa. The agreement of our numerical results (using 20 point Gaussian quadratures) with the corresponding analytical values is excellent in every case with the greatest relative error being less than 0.04%. It is interesting that, with the increase of j2 from 0.0 to 10~3, the local (or total) evaporation rate increases about 3% for
P. Tekasakul et al.
148
Fig. 3. A comparison of the numerically determined local and total evaporation rates for a sphere for four values of the jump distance (d "0.0, 0.1, 1.0 and 10.0) with the corresponding values % determined analytically from equations (14) and (15). The values at j2"0.0 were not shown because of the logarithmic scale of the abscissa but the numerical results agree exactly with the corresponding analytical values. The number of Gaussian quadrature points used in each numerical calculation was 20.
the cases with d "0.0 and 0.1 while, for the case involving the largest jump distance % (d "10.0), the evaporation rate increases only about 0.3%. As indicated by the analytical % solutions and the numerical results, the local evaporation rates are constant across the surface of the sphere. This is clearly not the case for the other bodies discussed later but it does provide a valuable check that the technique is being properly implemented. The excellent agreement between the numerical and analytical results for the sphere shows that the accuracy of this numerical technique is extremely good and that it is appropriate for calculations of this type. We also note that for more complex axi-symmetric bodies, the use of higher-order quadratures to maintain a comparable degree of accuracy is a straightforward matter. 4.2. Spheroids Many particle shapes of interest can be well approximated by spheroids. For example, a disc is well approximated by an oblate spheroid and a needle is well approximated by a prolate spheroid. We have computed the local and average evaporation rates on spheroids with four different aspect ratios, A"a/b"0.1, 0.5, 2.0 and 10.0, (using 20 point Gaussian quadratures) for values of j2"0.0, 10~9, 10~7, 10~5 and 10~3, and values of d "0.0, 0.1, % 1.0, and 10.0. The first two spheroids (A"0.1 and 0.5) are prolate spheroids while the latter two (A"2.0 and 10.0) are oblate spheroids. Numerical results for the average evaporation rates on all four spheroids are shown in Table 2. For the spheroid with the smallest aspect ratio under consideration (A"0.1), the influence of the absorption is the most noticeable. As the value of j2 increases from 0.0 to 10~3 for this spheroid, the average evaporation rates increase by about 10% for d "0.0 % and 0.1, by about 7% for d "1.0, and by about 2% for d "10.0. In contrast to this, for the % % spheroid with the highest aspect ratio considered (A"10.0), the influence of the absorption
Evaporation from a nonspherical aerosol particle
149
Table 2. Numerical results for the dimensionless total evaporation rates per unit surface area from spheroids for four different aspect ratios (A"a/b) and four values of the jump distance (d "0.0, 0.1, 1.0, and 10.0). The absorption parameter considered (j2) ranges from 0.0 to % 10~3. The number of Gaussian quadrature points used in each numerical calculation was 20 J /S T j2
A"0.1
A"0.5
A"2.0
A"10.0
0.0
0.0 10~9 10~7 10~5 10~3
0.4213 0.4213 0.4217 0.4257 0.4654
0.7695 0.7695 0.7698 0.7727 0.8019
1.1984 1.1984 1.1987 1.2015 1.2300
1.3144 1.3144 1.3146 1.3172 1.3426
0.1
0.0 10~9 10~7 10~5 10~3
0.4021 0.4022 0.4025 0.4062 0.4422
0.7132 0.7132 0.7134 0.7159 0.7409
1.0662 1.0662 1.0664 1.0687 1.0911
1.1314 1.1315 1.1317 1.1335 1.1524
1.0
0.0 10~9 10~7 10~5 10~3
0.2916 0.2916 0.2918 0.2937 0.3122
0.4331 0.4331 0.4332 0.4341 0.4432
0.5422 0.5422 0.5423 0.5429 0.5486
0.5531 0.5532 0.5532 0.5536 0.5581
10.0
0.0 10~9 10~7 10~5 10~3
0.0802 0.0802 0.0802 0.0804 0.0817
0.0884 0.0884 0.0884 0.0884 0.0888
0.0922 0.0922 0.0922 0.0922 0.0924
0.0925 0.0925 0.0925 0.0925 0.0926
d %
is significantly less. Here, the average evaporation rates increase by about 2% for the values of d "0.0 and 0.1, by about 1% for d "1.0, and by about 0.1% for d "10.0 as the value % % % of j2 increases from 0.0 to 10~3. It should be noted that a large increase in the evaporation rate occurs when j2'10~5. For values of j2)10~5, the increase of the evaporation rate tends to be only about 1% or less. It should also be noted that the smaller spheroid (higher aspect ratio) has a higher average evaporation rate. This is likely due to the higher curvature associated with the smaller particle. The effect of the jump distance on the evaporation rates is quite pronounced and is most significant for the smaller spheroids as one would expect. For the smallest spheroid (A"10.0), the average evaporation rates decrease by about 90% for all the absorption cases considered (j2"0.0—10~3) as the jump distance increases from 0.0 to 10.0. For the largest spheroid (A"0.1), the average evaporation rates decrease by about 80% for all of the absorption cases as the jump distance increases from 0.0 to 10.0. The results for the other two spheroids (A"0.5 and 2.0) follow an identical trend with values situated appropriately between the results for the more extreme cases described above. Numerical results for the local evaporation rates on the largest and the smallest of the spheroids (A"0.1 and 10.0) are shown in Figs 4 and 5. The influence of the absorption on the evaporation rates of the spheroid with A"0.1 (prolate spheroid) is shown in Fig. 4a for the case without jump (d "0.0). It can be seen that the local evaporation rates are at their % lowest at f"0 and reach their maxima when f"$1 where the curvature of the surface is greatest. Slight increases in the local evaporation rates as the absorption increases can also be detected and are more pronounced at the extremes (f"$1). Figure 4b shows the effect on the local evaporation rates for the same spheroid as the jump distance is increased while the absorption is held constant at the maximum value considered (j2"10~3). It can be seen that there is a pronounced decrease in the evaporation rates as the jump distance increases. Also, the effect of the curvature of the surface at the extremes (f"$1) becomes even more pronounced. Figure 5 presents essentially the same information as Fig. 4 except that it describes the most extreme oblate spheroid considered (A"10.0). For this spheroid the curvature of the surface is greatest at f"0 and the evaporation rates reflect this. The
150
P. Tekasakul et al.
Fig. 4. Numerically determined values of the local evaporation rate for a prolate spheroid with A"(a/b)"0.1. (a) For the jump distance of d "0.0 and three values of absorption (j2"0.0, 10~5 % and 10~3). (b) For the absorption of j2"10~3 and four values of jump distance (d "0.0, 0.1, 1.0 % and 10.0). The number of Gaussian quadrature points used in each numerical calculation was 20. Due to symmetry, only the local evaporation rate values for the upper half of the particle are shown. Here, f is the coordinate that specifies a point on the meridian contour of the particle, and j"a/l.
trends in the computational results remain the same as those observed with the prolate spheroid (A"0.1) with respect to the effects of changing the absorption or jump values.
Evaporation from a nonspherical aerosol particle
151
Fig. 5. Numerically determined values of the local evaporation rate for an oblate spheroid with A"(a/b)"10.0. (a) For the jump distance of d "0.0 and three values of absorption (j2"0.0, 10~5 % and 10~3). (b) For the absorption of j2"10~3 and four values of jump distance (d "0.0, 0.1, 1.0 % and 10.0). The number of Gaussian quadrature points used in each numerical calculation was 20. Due to symmetry, only the local evaporation rate values for the upper half of the particle are shown. Here, f is the coordinate that specifies a point on the meridian contour of the particle, and j"a/l.
4.3. Cylinders Cylindrical particles encompass fibers, which are ubiquitous as aerosols. Fibers undoubtedly are of great interest in all branches of aerosol sciences (e.g. Vincent, 1995). We
P. Tekasakul et al.
152
Table 3. Numerical results for the dimensionless total evaporation rates per unit surface area from cylinders for four different aspect ratios (A"a/b) and four values of the jump distance (d "0.0, 0.1, 1.0, and 10.0). The absorption parameter considered (j2) ranges from 0.0 to % 10~3. The number of Gaussian quadrature points used in each numerical calculation was 48 (N : N : N "12 : 24 : 12) 501 4*$% "0550. J /S T j2
A"0.1
A"0.5
A"2.0
A"10.0
0.0
0.0 10~9 10~7 10~5 10~3
0.3631 0.3631 0.3635 0.3675 0.4067
0.6292 0.6292 0.6295 0.6323 0.6609
0.9639 0.9639 0.9641 0.9668 0.9935
1.2154 1.2154 1.2157 1.2182 1.2436
0.1
0.0 10~9 10~7 10~5 10~3
0.3492 0.3492 0.3496 0.3532 0.3894
0.5886 0.5886 0.5889 0.5913 0.6163
0.8729 0.8729 0.8732 0.8753 0.8972
1.0607 1.0607 1.0609 1.0628 1.0821
1.0
0.0 10~9 10~7 10~5 10~3
0.2637 0.2637 0.2639 0.2660 0.2861
0.3827 0.3827 0.3828 0.3839 0.3943
0.4877 0.4877 0.4878 0.4885 0.4952
0.5372 0.5372 0.5372 0.5377 0.5426
10.0
0.0 10~9 10~7 10~5 10~3
0.0780 0.0780 0.0780 0.0782 0.0799
0.0861 0.0861 0.0861 0.0861 0.0866
0.0905 0.0905 0.0905 0.0905 0.0907
0.0920 0.0920 0.0920 0.0921 0.0922
d %
have computed the local and average evaporation rates for four cylinders having aspect ratios of A"a/b"0.1, 0.5, 2.0 and 10.0 using 48 point Gaussian quadratures (N : N : N "12 : 24 : 12). The absorption and jump distance values considered in 501 4*$% "0550. each case were the same as those considered for the spheroids, i.e. j2"0.0, 10~9, 10~7, 10~5 and 10~3, and d "0.0, 0.1, 1.0 and 10.0. % Our numerical results for the average evaporation rates on the four cylinders are shown in Table 3. For the cylinder with the smallest aspect ratio considered (A"0.1), the influence of absorption is large. The trend as j2 increases from 0.0 to 10~3 remains the same as observed for the spheroids. The average evaporation rates increase by about 12% for d "0.0 and 0.1, by about 8% for d "1.0, and by about 2% for d "10.0. For the cylinder % % % with the largest aspect ratio considered (A"10.0), the influence of the absorption is significantly less. Here, the average evaporation rates increase by about 2% for d "0.0, % and 0.1, by about 1% for d "1.0, and only by about 0.2% for d "10.0 as j2 increases from % % 0.0 to 10~3 in each case. It should be noted that a large increase in the evaporation rate again occurs when j2'10~5 as was observed for the various spheroids. Again, the smaller cylinder has a higher evaporation rate per unit surface area as in the case of the spheroids. The effect of varying the jump distance is again considerable and is most obvious with the smaller cylinders just as it was with the smaller spheroids. For the smallest cylinder considered (A"10.0), the average evaporation rates decrease by about 90% for all of the absorption cases considered (j2"0.0 to 10~3) as the jump distance is increased from 0.0 to 10.0. For the largest cylinder considered (A"0.1), the average evaporation rates decrease by about 80% for all of the absorption cases as the jump distance is increased from 0.0 to 10.0. The values for the other two cylinders considered (A"0.5 and 2.0) lie between these more extreme cases and follow the same trend. Our numerical results for the local evaporation rates of two cylinders (A"0.1 and 10.0) are shown in Figs 6 and 7. The influence of the absorption on the evaporation rates of the cylinder with A"0.1 is demonstrated in Fig. 6a for the case without jump. It is seen that the local evaporation rates are at their maximum at the edges of the cylinder. The increase of
Evaporation from a nonspherical aerosol particle
Fig. 6. Numerically determined values of the local evaporation rate for a cylindrical rod with A"(a/b)"0.1. (a) For the jump distance of d "0.0 and three values of absorption (j2"0.0, 10~5 % and 10~3). (b) For the absorption of j2"10~3 and four values of jump distance (d "0.0, 0.1, 1.0 % and 10.0). The number of Gaussian quadrature points used in each numerical calculation was 48 (N : N : N "12 : 24 : 12). Due to symmetry, only the local evaporation rate values for the 501 4*$% "0550. upper half of the particle are shown. Here, N : N : N represent the number of Gaussian 501 4*$% "0550. quadrature points on the top, side, and bottom surfaces of the cylinder, respectively, f is the coordinate that specifies a point on the meridian contour of the particle, and j"a/l. For the case of a cylinder, we have used f"cos(h) where h is the polar angle to the surface point. Note that several points that were calculated have not been plotted in order to better display the behavior of the evaporation rates near the edges of the body more clearly. The location of the edge is clearly labeled in (a).
153
154
P. Tekasakul et al.
Fig. 7. Numerically determined values of the local evaporation rate for a cylindrical rod with A"(a/b)"10.0. (a) For the jump distance of d "0.0 and three values of absorption (j2"0.0, 10~5 % and 10~3). (b) For the absorption of j2"10~3 and four values of jump distance (d "0.0, 0.1, 1.0 % and 10.0). The number of Gaussian quadrature points used in each numerical calculation was 48 (N : N : N "12 : 24 : 12). Due to symmetry, only the local evaporation rate values for the 501 4*$% "0550. upper half of the particle are shown. Here, N : N : N represent the number of Gaussian 501 4*$% "0550. quadrature points on the top, side, and bottom surfaces of the cylinder, respectively, f is the coordinate that specifies a point on the meridian contour of the particle, and j"a/l. For the case of a cylinder, we have used f"cos(h) where h is the polar angle to the surface point. Note that several points that were calculated have not been plotted in order to better display the behavior of the evaporation rates near the edges of the body more clearly. The location of the edge is clearly labeled in (a).
Evaporation from a nonspherical aerosol particle
155
the evaporation rate with increasing absorption is also most obvious at the edges. Figure 6b shows the effect of the jump distance on the same cylinder for the case of the maximum absorption considered (j2"10~3). It can be seen that the decrease in the evaporation rates as the jump distance increases is also much more prominent along the edges. Figure 7 shows the influences of the absorption and the jump distance on the cylinder with A"10.0. Here again, the trends in our results are the same as those observed for the A"0.1 cylinder. 5 . DI S CU SSI O N AN D CO NC LU SI O N S
The problem of an aerosol particle evaporating in an infinite expanse of an absorbing gas has been considered. The relevant Helmholtz equation, obtained from the steady-state diffusion equation with jump boundary conditions, has been converted into a Fredholm boundary integral equation via the use of the Green’s function. As constructed, this integral equation is valid for particles of arbitrary shape but the calculations in this work have been restricted to axi-symmetric spheroidal and cylindrical particles. The precision of the numerical technique used in this work has been shown, by comparison with the analytical results for a sphere, to be excellent for calculations of both the local and average evaporation rates of a vapor from an axi-symmetric particle situated in an infinite expanse of an absorbing background gas. The numerical results for a sphere subject to jump distances of d "0.0, 0.1, 1.0 and 10.0 are shown in Fig. 3 where the greatest deviation between the % numerical results and the corresponding analytical values is (0.04%. This value should be reasonably indicative of the accuracy of our numerical technique for other axi-symmetric particles as well. It has been shown that absorption can have a significant impact on evaporation from the surface of particles. For the case of a sphere with no jump (d "0.0) or with a very small % jump (d "0.1), the impact of absorption is small and the average evaporation rate % increases only by &3% as the absorption increases from j2"0.0 to 10~3. For spheroids and cylinders, however, the impact of the absorption is much more noticeable with rate increases as high as 12% where the curvature is greatest. With regard to the current results, we can make several observations. First, for all of the particles considered, the same trends were observed in the results. In all cases, higher values of absorption correspond to higher values of both the local and average evaporation rates. Also in all cases, shorter jump distances correspond to higher values for both the local and average evaporation rates. Further, local evaporation rates are obviously strongly dependent on the local curvature of the particle surface as one would expect. Additionally, all of the observed trends in the current quantitative results are consistent with the anticipated behaviors based on previous nonabsorbing studies (Loyalka and Griffin, 1993; Griffin and Loyalka, 1996) and qualitative analyses. However, the results reported here provide new quantitative benchmarks for future research efforts and represent an increased understanding of the evaporation process as it applies to a particle situated in an infinite expanse of gas where absorption is present; a factor commonly not taken into account in previous works. The method used and the results reported here are not confined to the evaporation problem but are applicable to any problem in which the Helmholtz equation needs to be solved. These problems include both neutron and photon diffusion which are of practical interest in nuclear reactors where the diffusion commonly involves axi-symmetric sources of different shapes situated in an absorbing/scattering medium. Ongoing efforts to extend the applicability of the numerical method include its extension beyond the jump regime into the transition and free-molecular regimes. It would be of interest also to explore perturbation solutions of the integral equations [equations (16) and (30)] in the limit of small k. Appropriate approximate solutions could also be obtained by discarding the subtraction terms in equation (30). Acknowledgement—This research was supported by a grant from the Environmental Protection Agency, Office of Exploratory Research (R819722-01-0).
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156
REF ER E NCE S Davies, C. N. (Ed.) (1966) Aerosol Science, Academic Press, London. Duderstadt, J. J. and Hamilton, L. J. (1976) Nuclear Reactor Analysis. Wiley, New York. Friedlander, S. K. (1977) Smoke, Dust and Haze. Wiley, New York. Frisch, H. L. and Collins, F. C. (1952) Diffusional processes in the growth of aerosol particles. J. Chem. Phys. 20, 1797—1803. Fuchs, N. A. (1964) ¹he Mechanics of Aerosols. Pergamon Press, New York. Fuchs, N. A. and Sutugin, A. G. (1971) High-dispersed aerosols. In ¹opics in Current Aerosol Research, Vol. 2 (Edited by Hidy, G. M. and Brock, J. R.). Pergamon Press, Oxford. Griffin, J. L. and Loyalka, S. K. (1996) Vapor condensation on multiple spheres and spheroids in the near continuum regime. J. Aerosol Sci. 27, 3—18. Hidy, G. M. and Brock, J. R. (Eds) (1972) ¹opics in Aerosol Research. Vols. 1—3. Pergamon Press, Oxford, UK. Loyalka, S. K. and Griffin, J. L. (1993) Condensation on nonspherical aerosol particles: Numerical solutions in the continuum regime. Nucl. Sci. Engng 114, 135—140. Mason, B. J. (1962) Clouds, Rain and Rainmaking. Cambridge University Press, Cambridge. Mason, E. A. and McDaniel, E. W. (1988) ¹ransport Properties of Ions in Gases. Wiley, New York. Prock, A. and McConkey, G. (1962) ¹opics in Chemical Physics (Based on the Harvard ¸ectures of P. J. ¼. Debye). Elsevier, New York. Seinfeld, J. H. (1986) Atmospheric Chemistry and Physics of Air Pollution. Wiley, New York. Sobolev, S. L. (1964) Partial Differential Equations of Mathematical Physics. Dover Edition (1989), New York. Tekasakul, P., Tompson, R. V. and Loyalka, S. K. (1998) Rotatory oscillations of arbitrary axi-symmetric bodies in a viscous fluid: numerical solutions. Phys. Fluids A (submitted). Twomey, S. (1977) Atmospheric Aerosols. Elsevier, New York. Vincent, J. H. (1995) Aerosol Science for Industrial Hygienists. Pergamon Press, New York. Weinberg, A. M. and Wigner, E. P. (1958) ¹he Physical ¹heory of Neutron Chain Reactors. University of Chicago Press, Chicago, Illinois. Williams, M. M. R. and Loyalka, S. K. (1991) Aerosol Science: ¹heory and Practice. Pergamon Press, New York.
A PP EN D I X A: SU RFA CE A REA S O F P ART I CL E S The dimensional and dimensionless surface areas of a spherical particle are simply SK "4na2 and S"4n, respectively. For a cylindrical particle, we have SK "2n a2#4nab and S"2n#4n/A, where A"a/b is the aspect ratio. For a spheroidal particle, the surface area is determined by the surface of revolution of the ellipse. The dimensional surface area of a spheroid (Fig. 2a and b) can be obtained from ( Jeffrey, 1995) SK "2n
P
b
~b
xL J1#[xL @(zL )]2 dzL ,
(A1)
using the equation for an ellipse xL 2 zL 2 # "1. a2 b2
(A2)
Nondimensionalizing by the equatorial radius, a, as described in Section 2, the dimensionless surface area of a spheroid is thus determined from S"SK /a2"4n
P
A~1
(1!A2 z2)1@2 J1#A4z2 (1!A2z2)~1 dz,
(A3)
0
which yields
C
D
2n ln 2 ln [2 (A#JA2!1) ] S" A! # . A JA2!1 JA2!1
(A4)
J. Aerosol Sci. Vol. 30, No. 2, pp. 139—156, 1999 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0021-8502/98 $19.00#0.00
EVAPORATION FROM A NONSPHERICAL AEROSOL PARTICLE SITUATED IN AN ABSORBING GAS P. Tekasakul,* R. V. Tompson,s and S. K. Loyalkast * Department of Mechanical Engineering, Thammasat University, Klong Luang, Pathum Thani 12121, Thailand s Particulate Systems Research Center and Nuclear, Chemical, and Mechanical & Aerospace Engineering Departments, University of Missouri-Columbia, Columbia, MO 65211, U.S.A. (First received 8 May 1997; and in final form 27 February 1998) Abstract—The problem of an aerosol particle evaporating in an infinite expanse of an absorbing gas is considered. The relevant Helmholtz equation (resulting from the steady-state diffusion equation with an absorption term included) with density jump boundary conditions is converted into a boundary integral equation via the use of the Green’s function. The resulting integral equation is valid for particles of arbitrary shape. Explicit numerical results for the local and average evaporation rates are reported for several axisymmetric particles for a range of values of the dimensionless absorption parameter (j2), where j is the ratio of the radius of the particle (a) to the diffusion length (l). Here, the diffusion length is defined as l"[D/(v& )]1@2, in which v (cm s~1) is the average ! thermal speed of the vapor molecules, & (cm~1) is the cross-section for absorption of the vapor by ! the gas, and D (cm2 s~1) is the diffusion coefficient of the vapor in the gas. Our numerical results for the local and average evaporation rates for a sphere exhibit excellent agreement with the corresponding analytical values (maximum deviation (0.40%). We find that the evaporation rate increases with increasing absorption and that this increase depends on the degree of departure of the particle from a spherical shape. The jump distance has a large impact in that it significantly lowers the evaporation rates as it increases in magnitude. It should be remarked that the results of this paper are also directly applicable to the problem of either neutrons or photons undergoing diffusion from a source situated in an absorbing medium. ( 1998 Elsevier Science Ltd. All rights reserved.
1 . I NT RO D UC T IO N
Evaporation or condensation of particles is a phenomenon of substantial interest in the field of aerosol science. The literature on the subject is extensive and several reviews covering the experimental as well as the theoretical aspects are available (Mason, 1962; Fuchs, 1964; Davies, 1966; Fuchs and Sutugin, 1971; Hidy and Brock, 1972; Friedlander, 1977; Twomey, 1977; Seinfeld, 1986; Williams and Loyalka, 1991). The problem is mainly one in which a particle, situated in an infinite expanse of a background gas and vapor mixture, either grows due to condensation or diminishes in size due to evaporation of the vapor from the particle surface. Generally, regarding the vapor in the mixture, it is assumed first, that it is in trace or small (dilute) amounts and second, that it is not absorbed by the background gas. There can, however, be physically realistic circumstances in which departures from both of the above assumptions occur. In this paper, our purpose is to study the influence of vapor absorption by the background gas on the evaporation rate of a single nonspherical particle. We will assume that the vapor is in dilute amounts, but that the background gas is a mixture of components that can scatter as well as absorb (or react with) the vapor molecules. For example, in a dense aerosol a particle would be surrounded by many other particles, and these particles could act as sinks (or sources) for the vapor. While one could solve the multiple particle problem in the manner we have discussed in previous papers (Griffin and Loyalka, 1996), the effect of the surrounding particles could be approximately represented to the first order by a homogeneously distributed absorbing cloud (Frisch and Collins, 1952). Similarly, in the evaporation of a binary particle (droplet) where one component is in substantially smaller
t Author to whom correspondence should be addressed. 139
140
P. Tekasakul et al.
concentration than the other, the interactions between the two components in the vapor phase could have a measurable effect on the evaporation of the first component. While it would be of interest to study the time-dependent evaporation rate for the entire range of Knudsen number (Kn"j/d, where j is the mean free path and d is the characteristic particle size, i.e. diameter), we confine our attention here to the steady-state continuum (diffusion) equation with jump boundary conditions. For the steady-state, this leads to the Helmholtz equation rather than the Laplace equation obtained in nonabsorbing problems. We note that the Helmholtz equation also arises in studies of the ambipolar diffusion charging of aerosols (Mason and McDaniel, 1988) as well as in the calculation of potential fields around a particle in the context of the Debye—Huckel theory (e.g. Prock and McConkey, 1972). The fundamental nature of the problem is evident in that a very similar circumstance arises when a source of any kind (neutrons, photons, etc.) is situated in an infinite expanse of an absorbing material and one is interested in knowing the rate of loss of the neutrons or photons (Duderstadt and Hamilton, 1976). For our studies here, we convert the Helmholtz equation into a convenient boundary integral equation by the use of a Green’s function technique. This integral equation applies to particles of arbitrary shape without restriction. In particular, particles with concave and occluded surfaces can also be considered. Such particles are frequently encountered in practice. Due to the computational times involved, however, we have obtained accurate numerical solutions of this equation for a selection of axi-symmetric particles only. We have considered spherical, spheroidal, and cylindrical particles. The parameters defining the spheroidal particles can be adjusted to include everything from highly oblate, disk-like particles to highly prolate, needle-like particles. The parameters for cylindrical particles can be adjusted to encompass everything from thin disks to essentially finite fiber-like particles. Fibers, in particular, are of great interest in many branches of aerosol science. We have considered wide ranges of aspect ratios and inverse diffusion lengths, k"(v& /D)1@2 (cm~1), where v (cm s~1) is the vapor molecular average thermal speed, ! & (cm~1) is the cross-section for vapor absorption by the gas, and D (cm2 s~1) is the ! diffusion coefficient of the vapor in the gas. In the next section, we describe the formulation of the problem. In Section 3, the conversion of the Helmholtz equation into an integral equation using the Green’s function is described and the numerical procedure is explained. The results for some selected axisymmetric particles are presented in Section 4. Finally, in Section 5, a discussion of our numerical results is provided along with our conclusions.
2 . ST AT EM EN T O F TH E PRO BL EM AN D AN ALY TI C AL S OL U TI ON S
The dimensional diffusion equation for an absorbing medium is given by (Weinberg and Wigner, 1958; Duderstadt and Hamilton, 1976) L/K "!+ª ) Jª !v & /K , ! LtL
(1)
where /K (d cm~3) is the vapor (molecule) number density, v (cm s~1) is the average thermal speed of the vapor molecules, & (cm~1) is the cross-section for absorption of the vapor by ! the gas, and Jª is the local current defined by Jª "!D +K /K ,
(2)
Here, D (cm2 s~1) is the diffusion coefficient of the vapor in the gas. Note that we use the hat symbol ( ª ) for dimensional quantities which are non-dimensionalized later in this section. At steady state, equation (1) reduces to the Helmholtz equation
A
B
1 +ª 2! /K "0, l2
(3)
Evaporation from a nonspherical aerosol particle
141
where, l"[D/(v & )]1@2 is known as the ‘diffusion length’ or ‘inverse wave number’. The ! density jump boundary condition at the surface of a particle is written as L/K , /K (rˆ )"/K #dK 4 4 % LnL
(4)
where /K (d cm~3) is the saturation density of the vapor, rˆ is a point on the surface of the 4 4 body, nˆ is a unit normal vector on the body surface at rˆ directed into the gas, and dK (cm) is 4 % the jump distance. We assume that both /K and dK are independent of rˆ . Also, far away from 4 % 4 the surface we have the limiting condition lim /K (rˆ )"0. ?=
(5)
rˆ
The dimensional local (d cm~2 s~1) and total (d s~1) evaporation rates at the surface of a particle are defined as L/K JK "!D , (6) L LnL rˆ ! and
K
P
JK " JK drˆ , L 4 T
(7)
respectively. An average evaporation rate can be obtained for a particle by dividing the total evaporation rate by the surface area of the particle, SK . This will produce values of the same order as the local evaporation rates and permits easier comparison of results between particles where the actual dimensions of the particles may be quite different. Expressions for the surface areas of the different bodies considered here are given in the Appendix. In order to simplify the matter, the problem is nondimensionalized. Nondimensionalizing lengths by the body dimension, a (for a sphere, a is the radius, whereas, for other axi-symmetric particles, a is the equatorial radius), vapor number densities by /K , and the 4 local currents by D /K /a, e.g. d "dK /a, r "rˆ /a, S"SK /a2, /"/K //K , and J"Jª /(D/K /a), 4 % % 4 4 4 4 the Helmholtz equation [equation (3)] becomes (+2#k2) /"0,
(8)
where k"$ij and j"a/l. The value of k can be either a positive or a negative imaginary number but the negative value, i.e. k"!ij, is chosen in order to ensure that / vanishes far away from the particle. The jump boundary condition at the surface of the particle and the limiting condition far away from the surface become L/ /(r )"1#d , 4 % dn
(9)
and lim /(r)"0, ?=
(10)
r
respectively. The dimensionless local and total evaporation rates at the surface of a particle then become
K
L/ J " , L dn r and
P
J " J dr , T L 4 respectively.
(11)
4
(12)
P. Tekasakul et al.
142
An analytical solution for the dimensionless vapor number density around a sphere of radius, a, is easily determined to be
AB
/(r)"
1 [1#d (1#ik)]~1 exp [!k(r!1)], % r
(13)
which leads to the following expressions for the evaporation rates: L/ J "! L dr and 1 J /S" T 4n
K
"(1#ik) [1#d (1#ik)]~1, % r/1
P JL dr4"(1#ik) [1#d% (1#ik)]~1.
(14)
(15)
Typically, at room temperature and atmospheric pressure, one has v+5]104 cm s~1 and D+0.2 cm2 s~1. Since the value of the absorption cross-section is typically in the range, & +0.0—10~2 cm~1, l could typically be expected to range from infinity (no absorption) to 4 about 0.02 cm (very strong absorption). For a typical aerosol particle with a radius of about 0.001 cm, the absorption parameter is then 0.0(j2(2.5]10~3. In this paper, we consider values of j2 in the range from 0.0 to 10~3. For clarity, we note that consideration of the steady state in the present study is not a serious limitation on the results in that the time constants involved in the evaporation and condensation processes of interest are such that the steady-state approximation suffices (Seinfeld, 1986). This remains true even in the presence of the variable curvature of the surface and the variations in the local condensation rate. 3 . N U ME RI CA L T ECH N IQ U E
The Helmholtz equation [equation (8)] together with the specified boundary conditions [equations (9) and (10)] can be solved by the use of the Green’s function technique described previously in conjunction with the problem of particle condensation in a nonabsorbing medium (Loyalka and Griffin, 1993; Griffin and Loyalka, 1996) and in conjunction with the problem of the oscillation of axi-symmetric bodies in a viscous fluid (Tekassakul et al., 1998). The Green’s function is defined by (+2#k2) t (r, r@)"!4nd (r!r@),
(16)
exp (!ikDr!r@D ) t (r, r@)" . Dr!r@D
(17)
and hence,
Using Green’s second identity in conjunction with the condition at infinity, we have the following for values of r exterior to the body:
P Ct(r, r@4 )
!
D
L/(r@ ) Lt (r, r@ ) 4 !/(r@ ) 4 dr@"4n 4 4 Ln@ Ln@
P / (r@4 ) d(r!r@4 ) dr@4 .
For rPr , a point on the body surface, we have 4 L/(r@ ) Lt (r , r@ ) 4 !M/(r@ )!/(r )N 4 4 dr@"4n /(r ). ! t(r , r@ ) 4 4 Ln@ 4 4 4 4 Ln@
PC
D
(18)
(19)
Applying the boundary condition at the surface, equation (19) becomes 4n d %
L/(r ) L/(r@ ) 4 # t(r , r@ ) 4 dr@!d 4 4 Ln@ 4 % Ln
P
P G Ln@4 ! L/(r@ )
H
L/(r ) Lt (r , r@ ) 4 4 dr@"!4n. 4 4 Ln@ Ln (20)
Evaporation from a nonspherical aerosol particle
143
L/(r ) 4 , p8 (r )" 4 Ln
(21)
KI (r , r@ )"t(r , r@ ), 1 4 4 4 4
(22)
Lt (r , r@ ) 4 4 , KI (r , r@ )"! 2 4 4 Ln@
(23)
If we now define
and
equation (20) becomes
P
4n d pJ (r )# KI (r , r@ ) p8 (r@ ) dr@#d % 4 4 1 4 4 % 4
P KI 2 (r4 , r@4 ) [p8(r@4 )!pJ (r4 )] dr@4"!4n .
(24)
Equation (24) is a Fredholm integral equation of the second kind. Since singularities are present, the method of singularity subtraction is employed and the integral equation is rewritten as
P
(gJ (r )#4n d ) p8 (r )# KI (r , r@ ) [p8 (r@ )!p8 (r )] dr@ 4 4 % 4 4 4 1 4 4 #d % or
P KI 2 (r4 , r@4 ) [p8 (r@4 )!p8 (r4 )] dr@4"!4n, P
(gJ (r )#4n d ) p8 (r )# KI (r , r@ ) [p8 (r@ )!p8 (r )] dr@"!4n, 4 % 4 4 4 4 4 4 where
P
gJ (r )" KI (r , r@ ) dr@, 4 1 4 4 and
KI (r , r@ )"KI (r , r@ )#d KI (r , r@ ). 4 4 1 4 4 % 2 4 4 For an axi-symmetric particle
(25)
(26)
(27) (28)
dr "J df du, (29) 4 463& where J is the dimensionless Jacobian of the surface integration (given in Table 1 for 463& a sphere, a spheroid, and a right circular cylinder), u is the azimuthal angle, and f is a coordinate specifying a point on the meridian counter of the body where !1)f)1. Thus, if we let and
p8 (r ) N p(f), 4
(30)
gJ (r )Ng(f), 4
(31)
equation (26) can be rewritten as
where
1
P~1 K(f, f@) [p(f@)!p(f)] df@"!4n,
(g(f)#4n d ) p(f)# %
and
1
P~1 K (f, f@) df@,
g(f)"
K (f, f@)"J@ 463&
2n
P0
KI (r , r@ ) du @. 4 4
(32)
(33)
(34)
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144
Table 1. Dimensionless surface Jacobians for spherical, general spheroidal, and cylindrical coordinates. Here, A"a/b is the aspect ratio of the spheroid in which a is either the minor semiaxis of a prolate spheroid or the maor semiaxis of an oblate spheroid, b is the other semiaxis of the spheroid (i.e. the major semiaxis of a prolate spheroid or the minor semiaxis of an oblate spheroid), - is the dimensionless radial location (from the axis 4 of the cylinder) on the surface of the cylinder, and f is the coordinate specifying a point on the meridian contour of the body Coordinates
Dimensionless surface Jacobian expression
Spherical Spheroidal Cylindrical
J "1 463& J "[f2#A~2 (1!f2)]1@2 463& J "463& 4
Equation (32) can now be expressed as the following system of linear algebraic equations: N (g(f )#4n d ) p(f )# + K(f , f ) [p(f )!p(f )] w "!4n , i % i i j j i j j/1 where w is a weight function. Rewriting equation (35) in matrix form, we have j Ap"!4n,
(35)
(36)
where the coefficient matrix, A, can be written as
G
A " ij
K(f , f ) w , i j j
iOj, (37)
N (g(f )#4n d )! + K(f , f ) w , i"j. i % i k k k/1 kOi By inversion, the solution to equation (36) is then p"A~1 (!4n).
(38)
With p determined in this manner, the local and total evaporation rates can then be calculated from equations (11), (12), and (21). All that remains in this general development of the problem is the evaluation of K (f, f@) in equation (34) which can be written as K (f, f@)"J @ 463&
2n
P0
KK (r , r@ ) du @"K (f, f@)#d K (f, f@) . 4 4 1 % 2
(39)
For an axi-symmetric body (Fig. 1) [see Tekasakul et al., 1997] 2n
P0
K (f, f@)"J @ 463& 1
A
2 t(r , r@ ) du@"J @ Q (c) 463& (- -@ )1@2 ~1@2 4 4 4 4
B
2n exp (!ik [2- -@ (c!cos ( u!u @))]1@2)!1 4 4 du @ , [2- -@ (c!cos(u!u @))]1@2 0 4 4
P
#
(40)
and: K (f, f@)"!J @ 463& 2
2n Lt(r , r@ ) 4 4 du@, Ln@
P0
(41)
in which Q (c) is an associated Legendre function of fractional order with the argument ~1@2 b , (42) c"1# 2- -@ 4 4
Evaporation from a nonspherical aerosol particle
145
Fig. 1. The dimensionless coordinate system used in the present work for a general axi-symmetric particle.
where b"(-@!- )2#(z@!z )2. 4 4 4 4 To determine K (f, f@), we first note that 2 exp(!ikt) t(r , r@ )" , 4 4 t
(43)
(44)
where t"Dr !r@ D. Therefore, 4 4 Lt(r , r@ ) ik Lt L 1 4 4 "! exp (!ikt) #exp (!ikt) Ln@ t Ln@ Ln@ t
C
A BD
ik Lt L 1 "exp (!ikt) ! # t Ln@ Ln@ t
AB
.
(45)
Since
AB
L 1 1 Lt "! , Ln@ t t2 Ln@
(46)
equation (45) can be expressed as
AB
Lt(r , r@ ) L 1 4 4 "(1#ikt) exp (!ikt) . Ln@ Ln@ t
(47)
We note here that (Sobolev, 1964)
where
AB
L 1 (r@!r ) ) n@ 4 0658!3$ , "! 4 Dr !r@ D3 Ln@ t 4 4 (r@!r )"(z@!z ) n #[-@ cos (u @)!- cos (u)] n 4 4 4 ; 4 4 4 x #[-@ sin (u @)!- sin (u)] n , 4 4 y
(48)
(49)
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146
Fig. 2. The particle geometries that are considered in this work. (a) Prolate spheroid. (b) Oblate spheroid. (c) Cylinder.
and (50) Dr !r@ D"[2- -@ (c!cos (u!u @))]1@2 . 4 4 4 4 For any spheroid (sphere, prolate spheroid, or oblate spheroid), the outward pointing unit normal vector can be written as
C
D G
H
~1@2 z@ z@2 4 4 " 1# A2 A n #cos (u @) n #sin (u @) n , n@ 0658!3$ z x y (A~2!z@2 )1@2 (A~2!z@2 ) 4 4 (51) where, A"a/b is the aspect ratio of the spheroid (Fig. 2a and b). Therefore, K can be 2 explicitly written as -@ K (f, f@)" 4 2 A
P0 G(1#ikK Dr4!r@4 D ) exp (!ikK Dr4!r@4 D ) [2-4 -@4 (c!cos (u!u@))]~3@2 2n
C
DH
z@ 4 ] [z@!z ] A#-@!- cos (u!u @) du@. (52) 4 4 [A~2!z@2 ]1@2 4 4 4 For a right circular cylinder (Fig. 2c), the outward pointing unit normal vector can be written as
G
bottom surface, !n z n@ " cos (u @) n #sin(u @) n side surface, 0658!3$ x y n top surface. z
(53)
Evaporation from a nonspherical aerosol particle
147
This results in
G
bottom surface, !(z@!z ) 4 4 " -@!- cos (u!u @ ) side surface, (r@!r ) ) n@ 4 4 4 4 0658!3$ (z@!z ) top surface, 4 4 such that
(54)
G
(z@!z ) 4 4 ! bottom surface, [2- -@ (c!cos (u!u @))]3@2 4 4 -@!- cos (u!u @) (r@!r ) ) n@ 4 4 4 4 0658!3$" side surface, (55) Dr !r@ D3 [ 2- -@ (c!cos (u!u @))]3@2 4 4 4 4 (z@!z ) 4 4 top surface. [2- -@ (c!cos (u!u @))]3@2 4 4 Therefore, K for the case of a cylinder can be calculated from equations (47), (48), and (55). 2 4 . RE SU LT S
Our goal was to calculate the local and average evaporation rates for arbitrary axisymmetric particles in an infinite expanse of an absorbing background gas where the density jump boundary conditions are applicable. The precision of our numerical method was benchmarked against known analytical solutions for a spherical particle for several different values of the dimensionless jump distance (d ) and the dimensionless absorption % parameter (j2). The local and average evaporation rates on spheroidal and cylindrical particles covering a range of aspect ratios (A"a/b) were then calculated as representative cases for our numerical work. The results of benchmarking against a sphere are reported immediately below and are followed by some representative results obtained for spheroids and cylinders. We have performed all of our numerical calculations using Mathematica ' 2.2.2 on a Power Macintosh 7100/66. The results for the local and average evaporation rates are reported in dimensionless forms. The dimensional local ( JK ) and total ( JK ) evaporation L T rates can be readily obtained from JK "(D/K /a) J and JK "(D/K /a) J , respectively. L 4 L T 4 T 4.1. Spheres Since the analytical solution for a spherical particle is known, we first benchmark the precision of the numerical method against the solutions for a sphere for four values of the jump distance (d "0.0, 0.1, 1.0, 10.0). Since the jump condition is analogous to a condition % of surface reaction, and the surface reactions are also of interest, we have included larger values of the jump distance although these are comparable to particle size. No doubt, such a value of the jump distance would generally require consideration of the Boltzmann equation rather than the diffusion equation. The dimensionless local evaporation rates, J , L or the dimensionless average evaporation rates, J /S, are calculated numerically and are T compared to the corresponding analytical values determined from equations (14) or (15). For the case of a sphere, analytical solutions of J and J /S coincide. The computed local L T evaporation rates are virtually constant across the surface of the sphere with only a slight variation. The average value of the local evaporation rate, J , over the entire surface L,!7% agrees exactly with the value of the corresponding average evaporation rate for each value of the absorption parameter j2. This is shown in Fig. 3 together with the analytical values. The values of j2 that were used in these calculations range from 0.0 (implying no absorption present) up to 10~3 (where the absorption is high). The result at j2"0 is not shown because of the logarithmic scale of the abscissa. The agreement of our numerical results (using 20 point Gaussian quadratures) with the corresponding analytical values is excellent in every case with the greatest relative error being less than 0.04%. It is interesting that, with the increase of j2 from 0.0 to 10~3, the local (or total) evaporation rate increases about 3% for
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148
Fig. 3. A comparison of the numerically determined local and total evaporation rates for a sphere for four values of the jump distance (d "0.0, 0.1, 1.0 and 10.0) with the corresponding values % determined analytically from equations (14) and (15). The values at j2"0.0 were not shown because of the logarithmic scale of the abscissa but the numerical results agree exactly with the corresponding analytical values. The number of Gaussian quadrature points used in each numerical calculation was 20.
the cases with d "0.0 and 0.1 while, for the case involving the largest jump distance % (d "10.0), the evaporation rate increases only about 0.3%. As indicated by the analytical % solutions and the numerical results, the local evaporation rates are constant across the surface of the sphere. This is clearly not the case for the other bodies discussed later but it does provide a valuable check that the technique is being properly implemented. The excellent agreement between the numerical and analytical results for the sphere shows that the accuracy of this numerical technique is extremely good and that it is appropriate for calculations of this type. We also note that for more complex axi-symmetric bodies, the use of higher-order quadratures to maintain a comparable degree of accuracy is a straightforward matter. 4.2. Spheroids Many particle shapes of interest can be well approximated by spheroids. For example, a disc is well approximated by an oblate spheroid and a needle is well approximated by a prolate spheroid. We have computed the local and average evaporation rates on spheroids with four different aspect ratios, A"a/b"0.1, 0.5, 2.0 and 10.0, (using 20 point Gaussian quadratures) for values of j2"0.0, 10~9, 10~7, 10~5 and 10~3, and values of d "0.0, 0.1, % 1.0, and 10.0. The first two spheroids (A"0.1 and 0.5) are prolate spheroids while the latter two (A"2.0 and 10.0) are oblate spheroids. Numerical results for the average evaporation rates on all four spheroids are shown in Table 2. For the spheroid with the smallest aspect ratio under consideration (A"0.1), the influence of the absorption is the most noticeable. As the value of j2 increases from 0.0 to 10~3 for this spheroid, the average evaporation rates increase by about 10% for d "0.0 % and 0.1, by about 7% for d "1.0, and by about 2% for d "10.0. In contrast to this, for the % % spheroid with the highest aspect ratio considered (A"10.0), the influence of the absorption
Evaporation from a nonspherical aerosol particle
149
Table 2. Numerical results for the dimensionless total evaporation rates per unit surface area from spheroids for four different aspect ratios (A"a/b) and four values of the jump distance (d "0.0, 0.1, 1.0, and 10.0). The absorption parameter considered (j2) ranges from 0.0 to % 10~3. The number of Gaussian quadrature points used in each numerical calculation was 20 J /S T j2
A"0.1
A"0.5
A"2.0
A"10.0
0.0
0.0 10~9 10~7 10~5 10~3
0.4213 0.4213 0.4217 0.4257 0.4654
0.7695 0.7695 0.7698 0.7727 0.8019
1.1984 1.1984 1.1987 1.2015 1.2300
1.3144 1.3144 1.3146 1.3172 1.3426
0.1
0.0 10~9 10~7 10~5 10~3
0.4021 0.4022 0.4025 0.4062 0.4422
0.7132 0.7132 0.7134 0.7159 0.7409
1.0662 1.0662 1.0664 1.0687 1.0911
1.1314 1.1315 1.1317 1.1335 1.1524
1.0
0.0 10~9 10~7 10~5 10~3
0.2916 0.2916 0.2918 0.2937 0.3122
0.4331 0.4331 0.4332 0.4341 0.4432
0.5422 0.5422 0.5423 0.5429 0.5486
0.5531 0.5532 0.5532 0.5536 0.5581
10.0
0.0 10~9 10~7 10~5 10~3
0.0802 0.0802 0.0802 0.0804 0.0817
0.0884 0.0884 0.0884 0.0884 0.0888
0.0922 0.0922 0.0922 0.0922 0.0924
0.0925 0.0925 0.0925 0.0925 0.0926
d %
is significantly less. Here, the average evaporation rates increase by about 2% for the values of d "0.0 and 0.1, by about 1% for d "1.0, and by about 0.1% for d "10.0 as the value % % % of j2 increases from 0.0 to 10~3. It should be noted that a large increase in the evaporation rate occurs when j2'10~5. For values of j2)10~5, the increase of the evaporation rate tends to be only about 1% or less. It should also be noted that the smaller spheroid (higher aspect ratio) has a higher average evaporation rate. This is likely due to the higher curvature associated with the smaller particle. The effect of the jump distance on the evaporation rates is quite pronounced and is most significant for the smaller spheroids as one would expect. For the smallest spheroid (A"10.0), the average evaporation rates decrease by about 90% for all the absorption cases considered (j2"0.0—10~3) as the jump distance increases from 0.0 to 10.0. For the largest spheroid (A"0.1), the average evaporation rates decrease by about 80% for all of the absorption cases as the jump distance increases from 0.0 to 10.0. The results for the other two spheroids (A"0.5 and 2.0) follow an identical trend with values situated appropriately between the results for the more extreme cases described above. Numerical results for the local evaporation rates on the largest and the smallest of the spheroids (A"0.1 and 10.0) are shown in Figs 4 and 5. The influence of the absorption on the evaporation rates of the spheroid with A"0.1 (prolate spheroid) is shown in Fig. 4a for the case without jump (d "0.0). It can be seen that the local evaporation rates are at their % lowest at f"0 and reach their maxima when f"$1 where the curvature of the surface is greatest. Slight increases in the local evaporation rates as the absorption increases can also be detected and are more pronounced at the extremes (f"$1). Figure 4b shows the effect on the local evaporation rates for the same spheroid as the jump distance is increased while the absorption is held constant at the maximum value considered (j2"10~3). It can be seen that there is a pronounced decrease in the evaporation rates as the jump distance increases. Also, the effect of the curvature of the surface at the extremes (f"$1) becomes even more pronounced. Figure 5 presents essentially the same information as Fig. 4 except that it describes the most extreme oblate spheroid considered (A"10.0). For this spheroid the curvature of the surface is greatest at f"0 and the evaporation rates reflect this. The
150
P. Tekasakul et al.
Fig. 4. Numerically determined values of the local evaporation rate for a prolate spheroid with A"(a/b)"0.1. (a) For the jump distance of d "0.0 and three values of absorption (j2"0.0, 10~5 % and 10~3). (b) For the absorption of j2"10~3 and four values of jump distance (d "0.0, 0.1, 1.0 % and 10.0). The number of Gaussian quadrature points used in each numerical calculation was 20. Due to symmetry, only the local evaporation rate values for the upper half of the particle are shown. Here, f is the coordinate that specifies a point on the meridian contour of the particle, and j"a/l.
trends in the computational results remain the same as those observed with the prolate spheroid (A"0.1) with respect to the effects of changing the absorption or jump values.
Evaporation from a nonspherical aerosol particle
151
Fig. 5. Numerically determined values of the local evaporation rate for an oblate spheroid with A"(a/b)"10.0. (a) For the jump distance of d "0.0 and three values of absorption (j2"0.0, 10~5 % and 10~3). (b) For the absorption of j2"10~3 and four values of jump distance (d "0.0, 0.1, 1.0 % and 10.0). The number of Gaussian quadrature points used in each numerical calculation was 20. Due to symmetry, only the local evaporation rate values for the upper half of the particle are shown. Here, f is the coordinate that specifies a point on the meridian contour of the particle, and j"a/l.
4.3. Cylinders Cylindrical particles encompass fibers, which are ubiquitous as aerosols. Fibers undoubtedly are of great interest in all branches of aerosol sciences (e.g. Vincent, 1995). We
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152
Table 3. Numerical results for the dimensionless total evaporation rates per unit surface area from cylinders for four different aspect ratios (A"a/b) and four values of the jump distance (d "0.0, 0.1, 1.0, and 10.0). The absorption parameter considered (j2) ranges from 0.0 to % 10~3. The number of Gaussian quadrature points used in each numerical calculation was 48 (N : N : N "12 : 24 : 12) 501 4*$% "0550. J /S T j2
A"0.1
A"0.5
A"2.0
A"10.0
0.0
0.0 10~9 10~7 10~5 10~3
0.3631 0.3631 0.3635 0.3675 0.4067
0.6292 0.6292 0.6295 0.6323 0.6609
0.9639 0.9639 0.9641 0.9668 0.9935
1.2154 1.2154 1.2157 1.2182 1.2436
0.1
0.0 10~9 10~7 10~5 10~3
0.3492 0.3492 0.3496 0.3532 0.3894
0.5886 0.5886 0.5889 0.5913 0.6163
0.8729 0.8729 0.8732 0.8753 0.8972
1.0607 1.0607 1.0609 1.0628 1.0821
1.0
0.0 10~9 10~7 10~5 10~3
0.2637 0.2637 0.2639 0.2660 0.2861
0.3827 0.3827 0.3828 0.3839 0.3943
0.4877 0.4877 0.4878 0.4885 0.4952
0.5372 0.5372 0.5372 0.5377 0.5426
10.0
0.0 10~9 10~7 10~5 10~3
0.0780 0.0780 0.0780 0.0782 0.0799
0.0861 0.0861 0.0861 0.0861 0.0866
0.0905 0.0905 0.0905 0.0905 0.0907
0.0920 0.0920 0.0920 0.0921 0.0922
d %
have computed the local and average evaporation rates for four cylinders having aspect ratios of A"a/b"0.1, 0.5, 2.0 and 10.0 using 48 point Gaussian quadratures (N : N : N "12 : 24 : 12). The absorption and jump distance values considered in 501 4*$% "0550. each case were the same as those considered for the spheroids, i.e. j2"0.0, 10~9, 10~7, 10~5 and 10~3, and d "0.0, 0.1, 1.0 and 10.0. % Our numerical results for the average evaporation rates on the four cylinders are shown in Table 3. For the cylinder with the smallest aspect ratio considered (A"0.1), the influence of absorption is large. The trend as j2 increases from 0.0 to 10~3 remains the same as observed for the spheroids. The average evaporation rates increase by about 12% for d "0.0 and 0.1, by about 8% for d "1.0, and by about 2% for d "10.0. For the cylinder % % % with the largest aspect ratio considered (A"10.0), the influence of the absorption is significantly less. Here, the average evaporation rates increase by about 2% for d "0.0, % and 0.1, by about 1% for d "1.0, and only by about 0.2% for d "10.0 as j2 increases from % % 0.0 to 10~3 in each case. It should be noted that a large increase in the evaporation rate again occurs when j2'10~5 as was observed for the various spheroids. Again, the smaller cylinder has a higher evaporation rate per unit surface area as in the case of the spheroids. The effect of varying the jump distance is again considerable and is most obvious with the smaller cylinders just as it was with the smaller spheroids. For the smallest cylinder considered (A"10.0), the average evaporation rates decrease by about 90% for all of the absorption cases considered (j2"0.0 to 10~3) as the jump distance is increased from 0.0 to 10.0. For the largest cylinder considered (A"0.1), the average evaporation rates decrease by about 80% for all of the absorption cases as the jump distance is increased from 0.0 to 10.0. The values for the other two cylinders considered (A"0.5 and 2.0) lie between these more extreme cases and follow the same trend. Our numerical results for the local evaporation rates of two cylinders (A"0.1 and 10.0) are shown in Figs 6 and 7. The influence of the absorption on the evaporation rates of the cylinder with A"0.1 is demonstrated in Fig. 6a for the case without jump. It is seen that the local evaporation rates are at their maximum at the edges of the cylinder. The increase of
Evaporation from a nonspherical aerosol particle
Fig. 6. Numerically determined values of the local evaporation rate for a cylindrical rod with A"(a/b)"0.1. (a) For the jump distance of d "0.0 and three values of absorption (j2"0.0, 10~5 % and 10~3). (b) For the absorption of j2"10~3 and four values of jump distance (d "0.0, 0.1, 1.0 % and 10.0). The number of Gaussian quadrature points used in each numerical calculation was 48 (N : N : N "12 : 24 : 12). Due to symmetry, only the local evaporation rate values for the 501 4*$% "0550. upper half of the particle are shown. Here, N : N : N represent the number of Gaussian 501 4*$% "0550. quadrature points on the top, side, and bottom surfaces of the cylinder, respectively, f is the coordinate that specifies a point on the meridian contour of the particle, and j"a/l. For the case of a cylinder, we have used f"cos(h) where h is the polar angle to the surface point. Note that several points that were calculated have not been plotted in order to better display the behavior of the evaporation rates near the edges of the body more clearly. The location of the edge is clearly labeled in (a).
153
154
P. Tekasakul et al.
Fig. 7. Numerically determined values of the local evaporation rate for a cylindrical rod with A"(a/b)"10.0. (a) For the jump distance of d "0.0 and three values of absorption (j2"0.0, 10~5 % and 10~3). (b) For the absorption of j2"10~3 and four values of jump distance (d "0.0, 0.1, 1.0 % and 10.0). The number of Gaussian quadrature points used in each numerical calculation was 48 (N : N : N "12 : 24 : 12). Due to symmetry, only the local evaporation rate values for the 501 4*$% "0550. upper half of the particle are shown. Here, N : N : N represent the number of Gaussian 501 4*$% "0550. quadrature points on the top, side, and bottom surfaces of the cylinder, respectively, f is the coordinate that specifies a point on the meridian contour of the particle, and j"a/l. For the case of a cylinder, we have used f"cos(h) where h is the polar angle to the surface point. Note that several points that were calculated have not been plotted in order to better display the behavior of the evaporation rates near the edges of the body more clearly. The location of the edge is clearly labeled in (a).
Evaporation from a nonspherical aerosol particle
155
the evaporation rate with increasing absorption is also most obvious at the edges. Figure 6b shows the effect of the jump distance on the same cylinder for the case of the maximum absorption considered (j2"10~3). It can be seen that the decrease in the evaporation rates as the jump distance increases is also much more prominent along the edges. Figure 7 shows the influences of the absorption and the jump distance on the cylinder with A"10.0. Here again, the trends in our results are the same as those observed for the A"0.1 cylinder. 5 . DI S CU SSI O N AN D CO NC LU SI O N S
The problem of an aerosol particle evaporating in an infinite expanse of an absorbing gas has been considered. The relevant Helmholtz equation, obtained from the steady-state diffusion equation with jump boundary conditions, has been converted into a Fredholm boundary integral equation via the use of the Green’s function. As constructed, this integral equation is valid for particles of arbitrary shape but the calculations in this work have been restricted to axi-symmetric spheroidal and cylindrical particles. The precision of the numerical technique used in this work has been shown, by comparison with the analytical results for a sphere, to be excellent for calculations of both the local and average evaporation rates of a vapor from an axi-symmetric particle situated in an infinite expanse of an absorbing background gas. The numerical results for a sphere subject to jump distances of d "0.0, 0.1, 1.0 and 10.0 are shown in Fig. 3 where the greatest deviation between the % numerical results and the corresponding analytical values is (0.04%. This value should be reasonably indicative of the accuracy of our numerical technique for other axi-symmetric particles as well. It has been shown that absorption can have a significant impact on evaporation from the surface of particles. For the case of a sphere with no jump (d "0.0) or with a very small % jump (d "0.1), the impact of absorption is small and the average evaporation rate % increases only by &3% as the absorption increases from j2"0.0 to 10~3. For spheroids and cylinders, however, the impact of the absorption is much more noticeable with rate increases as high as 12% where the curvature is greatest. With regard to the current results, we can make several observations. First, for all of the particles considered, the same trends were observed in the results. In all cases, higher values of absorption correspond to higher values of both the local and average evaporation rates. Also in all cases, shorter jump distances correspond to higher values for both the local and average evaporation rates. Further, local evaporation rates are obviously strongly dependent on the local curvature of the particle surface as one would expect. Additionally, all of the observed trends in the current quantitative results are consistent with the anticipated behaviors based on previous nonabsorbing studies (Loyalka and Griffin, 1993; Griffin and Loyalka, 1996) and qualitative analyses. However, the results reported here provide new quantitative benchmarks for future research efforts and represent an increased understanding of the evaporation process as it applies to a particle situated in an infinite expanse of gas where absorption is present; a factor commonly not taken into account in previous works. The method used and the results reported here are not confined to the evaporation problem but are applicable to any problem in which the Helmholtz equation needs to be solved. These problems include both neutron and photon diffusion which are of practical interest in nuclear reactors where the diffusion commonly involves axi-symmetric sources of different shapes situated in an absorbing/scattering medium. Ongoing efforts to extend the applicability of the numerical method include its extension beyond the jump regime into the transition and free-molecular regimes. It would be of interest also to explore perturbation solutions of the integral equations [equations (16) and (30)] in the limit of small k. Appropriate approximate solutions could also be obtained by discarding the subtraction terms in equation (30). Acknowledgement—This research was supported by a grant from the Environmental Protection Agency, Office of Exploratory Research (R819722-01-0).
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REF ER E NCE S Davies, C. N. (Ed.) (1966) Aerosol Science, Academic Press, London. Duderstadt, J. J. and Hamilton, L. J. (1976) Nuclear Reactor Analysis. Wiley, New York. Friedlander, S. K. (1977) Smoke, Dust and Haze. Wiley, New York. Frisch, H. L. and Collins, F. C. (1952) Diffusional processes in the growth of aerosol particles. J. Chem. Phys. 20, 1797—1803. Fuchs, N. A. (1964) ¹he Mechanics of Aerosols. Pergamon Press, New York. Fuchs, N. A. and Sutugin, A. G. (1971) High-dispersed aerosols. In ¹opics in Current Aerosol Research, Vol. 2 (Edited by Hidy, G. M. and Brock, J. R.). Pergamon Press, Oxford. Griffin, J. L. and Loyalka, S. K. (1996) Vapor condensation on multiple spheres and spheroids in the near continuum regime. J. Aerosol Sci. 27, 3—18. Hidy, G. M. and Brock, J. R. (Eds) (1972) ¹opics in Aerosol Research. Vols. 1—3. Pergamon Press, Oxford, UK. Loyalka, S. K. and Griffin, J. L. (1993) Condensation on nonspherical aerosol particles: Numerical solutions in the continuum regime. Nucl. Sci. Engng 114, 135—140. Mason, B. J. (1962) Clouds, Rain and Rainmaking. Cambridge University Press, Cambridge. Mason, E. A. and McDaniel, E. W. (1988) ¹ransport Properties of Ions in Gases. Wiley, New York. Prock, A. and McConkey, G. (1962) ¹opics in Chemical Physics (Based on the Harvard ¸ectures of P. J. ¼. Debye). Elsevier, New York. Seinfeld, J. H. (1986) Atmospheric Chemistry and Physics of Air Pollution. Wiley, New York. Sobolev, S. L. (1964) Partial Differential Equations of Mathematical Physics. Dover Edition (1989), New York. Tekasakul, P., Tompson, R. V. and Loyalka, S. K. (1998) Rotatory oscillations of arbitrary axi-symmetric bodies in a viscous fluid: numerical solutions. Phys. Fluids A (submitted). Twomey, S. (1977) Atmospheric Aerosols. Elsevier, New York. Vincent, J. H. (1995) Aerosol Science for Industrial Hygienists. Pergamon Press, New York. Weinberg, A. M. and Wigner, E. P. (1958) ¹he Physical ¹heory of Neutron Chain Reactors. University of Chicago Press, Chicago, Illinois. Williams, M. M. R. and Loyalka, S. K. (1991) Aerosol Science: ¹heory and Practice. Pergamon Press, New York.
A PP EN D I X A: SU RFA CE A REA S O F P ART I CL E S The dimensional and dimensionless surface areas of a spherical particle are simply SK "4na2 and S"4n, respectively. For a cylindrical particle, we have SK "2n a2#4nab and S"2n#4n/A, where A"a/b is the aspect ratio. For a spheroidal particle, the surface area is determined by the surface of revolution of the ellipse. The dimensional surface area of a spheroid (Fig. 2a and b) can be obtained from ( Jeffrey, 1995) SK "2n
P
b
~b
xL J1#[xL @(zL )]2 dzL ,
(A1)
using the equation for an ellipse xL 2 zL 2 # "1. a2 b2
(A2)
Nondimensionalizing by the equatorial radius, a, as described in Section 2, the dimensionless surface area of a spheroid is thus determined from S"SK /a2"4n
P
A~1
(1!A2 z2)1@2 J1#A4z2 (1!A2z2)~1 dz,
(A3)
0
which yields
C
D
2n ln 2 ln [2 (A#JA2!1) ] S" A! # . A JA2!1 JA2!1
(A4)