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Evaporation from non-spherical particles: the equivalent-volume and inscribed sphere approximations for nearly spherical particles
Aerosol Science 33 (2002) 307–317
www.elsevier.com/locate/jaerosci
Evaporation from non-spherical particles: the equivalent-volume and inscribed sphere approximations for nearly spherical particles P. Tekasakula; ∗ , S.K. Loyalkab a
Department of Mechanical Engineering, Faculty of Engineering, Thammasat University, Rangsit Campus, Klong Luang, Pathum Thani 12121, Thailand b Particulate Systems Research Center and Nuclear Engineering Program, University of Missouri-Columbia, Columbia, MO 65211, USA Received 9 March 2001; received in revised form 10 July 2001; accepted 11 July 2001
Abstract Although numerical results for evaporation rates from arbitrarily shaped particles have been recently reported in the literature, it remains of interest to explore construction and accuracy of approximate analytical expressions that are simple and of general use. We have analyzed the popular equivalent-volume approximation and also the inscribed sphere approximations in this context, using a technique due to Zhang and Stone (J. Fluid Mech. 367 (1998) 329) who had studied rotatory oscillations of nearly spherical bodies. Comparison of the approximate results with accurate numerical results shows that both these approximations are quite useful. The equivalent-volume sphere approximation is particularly attractive as the error is of the order 2 , where is a measure of departure from the spherical shape. c 2001 Elsevier Science Ltd. All rights reserved. Keywords: Evaporation; Inscribed sphere; Equivalent-volume sphere; Non-spherical particles
1. Introduction The problem of evaporation or condensation of particles has been of major interest in aerosol science (see for example, Davies, 1966; Friedlander, 1977; Ravindran & Davis, 1982; Ray, Lee, & Tilley, 1988; Fuchs, 1989; Ray, Devakottai, Souyri, & Huckaby, 1991; Williams & Loyalka, 1991; Bharat & Ray, 1992; Loyalka & GriAn, 1993; GriAn & Loyalka, 1994, 1996; Li & Davis, 1996; Widmann & Davis, 1997). Generally, one deals here with the transport of ∗
Corresponding author. Present address: Department of Mechanical Engineering, Prince of Songkla University, Hat Yai, Songkla 90112, Thailand. Tel.=fax: +66-74-212893. E-mail addresses: [email protected] (P. Tekasakul), [email protected] (S.K. Loyalka). c 2001 Elsevier Science Ltd. All rights reserved. 0021-8502/01/$ - see front matter PII: S 0 0 2 1 - 8 5 0 2 ( 0 1 ) 0 0 1 7 1 - 9
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a vapor in a background gas in which the vapor does not interact with the gas. There are, however, situations in which the vapor can react with the background gas through a Irst-order reaction under almost isothermal conditions (consider for example, a particle surrounded by other particles whose eJect on the vapor transport can be approximated by a homogenized distribution of reactants). Then the diJusion equation for the problem becomes @ˆ ˆ ˆ · Jˆ − k1 ; (1) =−∇ @t where ˆ (g cm−3 ) is the vapor mass concentration in the gas phase, k1 (s−1 ) is the Irst-order reaction rate constant, and Jˆ = − D∇ˆ (g cm−2 s−1 ) is the mass Mux. Here D (cm2 s−1 ) is the diJusion coeAcient which is a constant for a dilute system. Note that the Stefan Mow associated with moving boundary is not accounted for. At steady state, the diJusion equation becomes (2) ∇ · Jˆ + k1 ˆ = 0: Non-dimensionalizing lengths by the equatorial radius of the spheroidal particle, a, vapor mass concentration by the surface concentration, ˆs , and the mass Mux by Ds =a, we have ∇ · J + 2 = 0:
(3)
Here J is dimensionless mass Mux, the dimensionless vapor concentration in the gas phase, and 2 = k1 a=D is the dimensionless Irst-order reaction rate constant. Boundary conditions for this problem are =1
at surface;
(4a)
=0
as r → ∞;
(4b)
where r is the position from the center of the particle. We note that Eq. (3) is really the Helmholtz equation and it is of fundamental interest in light scattering from aerosols as in ion charging of aerosols. Tekasakul, Tompson, and Loyalka (1999) have previously solved this problem numerically for several geometries, albeit even in a slightly general context, where Eq. (4a) is replaced by the jump condition: @ =1 + at surface (4c) @n and is the dimensionless jump distance (see Williams & Loyalka, 1991 for a discussion). The numerical solutions are, however, not convenient to use; for example, in rate expressions in the general dynamic equation for non-spherical particles, and it is of interest to understand the nature of various approximations that are employed in calculation of the evaporation rates from non-spherical particles. In this paper, we use both the inscribed sphere and equivalent-volume sphere approximations to obtain analytical expressions for total evaporation rates from nearly spherical prolate and oblate spheroidal particles for the no jump condition (4a). Values from both expressions are compared with numerically calculated values ( = 0 case, Tekasakul et al., 1999). An approximate analytical expression is also obtained for cylindrical particles using the equivalent-volume sphere approximation. We have followed here the inscribed sphere and the equivalent-volume
P. Tekasakul, S.K. Loyalka / Aerosol Science 33 (2002) 307–317
309
Fig. 1. Geometry for a unit sphere inscribed in the particle.
sphere approximations employed by Zhang and Stone (1997, 1998) in their study of oscillatory motions of nearly spherical particles. 2. Inscribed sphere approximation In this method, a sphere of unit radius is assumed to be inscribed in the nearly spherical particle in which radius of the sphere is equal to the minimum cross-sectional radius of the nearly spherical particle (as seen in Fig. 1). This method has been suggested by Zhang and Stone (1997) for calculations of forces and torques on oscillating nearly spherical particles. The diJusion equation for the particle (due to the inscribed unit sphere) and its boundary conditions are given in Eqs. (3) and (4) while, the diJusion equation for the inscribed unit sphere (due to the particle) can be written as ∇ · J˜ + 2 ˜ = 0 (5)
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P. Tekasakul, S.K. Loyalka / Aerosol Science 33 (2002) 307–317
and the associated boundary conditions at the surface of the unit sphere and far away are ˜ = 1
at surface;
(6a)
˜ = 0
as r → ∞;
(6b)
where J˜ = − ∇˜ is the dimensionless mass Mux associated with the inscribed unit sphere, and ˜ the dimensionless vapor concentration in the gas phase associated with the inscribed unit sphere. Applying reciprocal theorem to Eqs. (3) and (5), we have ˜ = 0: (∇ · J) ˜ − (∇ · J)
(7)
Let V represent the Muid volume surrounding the arbitrarily shaped particle, V˜ be the Muid volume external to the sphere and Vdif = V˜ − V. Integrating the above equation over the volume ˜ the surface of the inscribed unit sphere, and a distant surface S∞ , and then V˜ enclosed by S, applying the divergence theorem, leads to ˜ (∇ · J) dS + (n · J) dS = 0: (8) S˜
V +Vdif
Here n is the unit vector directing into the Muid volume (Fig. 1). The Muid volume is enclosed ˜ Again, applying by Sp , the surface of the particle and S∞ , and Vdif is enclosed by Sp and S. the divergence theorem, we can write, for the evaporation problem, ˜ ˜ dV = 0: ˜ − (n · J) dS + (n · J) dS + ∇ · (J ) (9) Sp
S˜
Vdif
The surface of the nearly spherical body is described in the spherical coordinates as r = 1 + f(; ’);
(10)
where 1 and f(; ’) is in the order of one and describes the detailed particle shape. Now, let us consider Eq. (9) term by term. The concentration ˜ on Sp in the Irst integral can be written in a Taylor series about r = 1: ˜ @ ˜|r=1+f = ˜|r=1 +f + O(2 ): (11) @r r=1
Since the solution for the sphere gives @˜ = − (1 + ): @r r=1
Therefore, Eq. (11) becomes ˜|r=1+f = 1 − f(1 + ) + O(2 ): The term in the second integral is ˜ @ = (1 + ): n · J˜ |r=1 = n · −n @r r=1
(12)
(13)
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311
The term in the last integral can be written as ˜ = − 2 ˜ − J · J˜ ≈ − 2 ; ˜ = ( ˜ ∇ · J) + J · (∇) ∇ · (J)
(14)
for the volume inside the nearly spherical body in which we assume = 1, ˜ = 1 and J = 0. Therefore, the integral becomes ˜ dV = − ∇ · (J) 2 dV = − 2 f dS + O(2 ): (15) Vdif
S˜
Vdif
Substituting Eqs. (12), (13) and (15) into Eq. (9), we obtain − (n · J)[1 − f(1 + )] dS + (1 + ) dS − 2 f dS = 0 S˜
Sp
or
−
Since
S˜
Sp
Sp
S˜
(n · J) dS + (1 + )
dS = 4, and
f(n · J) dS ≈
S˜
Sp
f(n · J) dS = − (1 + )
˜ dS ≈ (1 + ) f(n · J)
S˜
dS +
2
S˜
f dS:
(16)
S˜
f dS;
the dimensionless total evaporation rate per unit surface area of the prolate spheroidal particle is 2 2 JT =S = (n · J) dS = 4(1 + ) + [ − + (1 + ) ] f dS: (17) Sp
S˜
For a prolate spheroid, we can write = 1 − A, where A is the particle aspect ratio deIned as the ratio of the equatorial radius (a) to the longitudinal radius (b) as shown in Fig. 2. Then we have f = r − 1 ≈ 12 (1 − A2 )cos2 : The dimensionless total evaporation rate per unit surface area of the prolate spheroidal particle becomes 4 (1 − A)[ − 2 + (1 + )2 ]: 3 For an oblate spheroid, we can write = 1 − 1=A, and we have 11 1 1 2 f = r − 1 ≈ 1 − −1 + 1+ sin : A 2A A JT =S = 4(1 + ) +
(18)
The dimensionless total evaporation rate per unit surface area of the oblate spheroidal particle is then 4 1 JT =S = 4(1 + ) − 1− [ − 2 + (1 + )2 ]: (19) 3 A
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P. Tekasakul, S.K. Loyalka / Aerosol Science 33 (2002) 307–317
Fig. 2. The particle geometries that are considered in this work: (a) prolate spheroid, (b) oblate spheroid, and (c) cylinder.
3. Equivalent-volume sphere approximation In this section, we assume the particle to be a sphere of equal volume. Fig. 3 shows the geometry of the particle and its equivalent sphere. In this approach, the sphere center coincides with the center of mass of the nearly spherical particle. S˜ and Sp represent the surfaces of the sphere and the nearly spherical particle, respectively. Portions of the nearly spherical particle that extend beyond the sphere are labeled Vi− ; i = 1; : : : ; N , where N is the number of bumps. These
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313
Fig. 3. Geometry for an equivalent-volume sphere. −
volume elements are enclosed by surfaces S˜ i and Si− . In the same manner, Vi+ represents the volume elements of the sphere that extend beyond the nearly spherical particle and is enclosed + by surfaces S˜ i and Si+ . Using the above assumption of volume equivalent, Eq. (9) of the preceding section reduces to ˜ ˜ dS: (n · J) dS = (n · J) (20) Sp
S˜
The vapor concentration on the particle surface due to the sphere is represented by 1 over Si+ ; ˜|r=1+f = ˜ @ + O(2 ) over Si− ; 1 + f @r r=1
(21)
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since ˜|r=1 = 1. The vapor concentration on the sphere surface because of the sphere can be written as a regular perturbation expansion in : = (0) + (1) + O(2 ):
(22)
A regular perturbation expansion about the boundary condition on the particle surface, as shown by Zhang and Stone (1998), leads to @(0) (1) |r=1 = − f : (23) @r r=1 The concentration Ield evaluated on the surface S˜ of the equivalent-volume sphere (due to the particle) is − over S˜ i ; 1 |S˜ = (24) @(0) 2 ˜+ + O( ) over S : 1 − f i @r r=1 Substituting Eqs. (21) and (24) into Eq. (20), we obtain N N N @˜ (n · J) dS + (n · J) f dS + (n · J) dS @r Si− Si− Si+ i=1
i=1
=
N i=1
˜ dS + (n · J) −
S˜i
i=1
r=1
N i=1
˜ dS − (n · J) +
S˜i
N i=1
˜ (n · J) +
S˜i
i=1
@(0) dS + O(2 ); @r r=1
(0) @ ˜ f (n · J) dS + O(2 ): + @r r=1 S˜i
dS r=1
N i=1
f
(25)
which can be written as N ˜ @ ˜ dS − (n · J) dS = (n · J) (n · J) f @r S˜ Sp Si− +
(26)
− Approximating Si− = S˜ i with the error of O(2 ), the above equation reduces to ˜ @ ˜ dS − (n · J) ˜ f (n · J) dS = (n · J) dS + O(2 ) @r ˜ ˜ S S Sp
or
r=1
˜ @ ˜ f JT = J˜T − (n · J) @r S˜
dS + O(2 ): r=1
Applying Eq. (13) to the above equation, we have 2 ˜ @ f dS + O(2 ): JT = J˜T + @r S˜ r=1
(27)
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315
Fig. 4. Comparison of the dimensionless evaporation rate per unit surface area from spheroidal particles, using numerical, equivalent-volume sphere, and inscribed sphere methods.
Because of volume equivalency, we have JT = J˜T + O(2 )
S˜ f dS
= 0, Eq. (25) then reduces to (28)
which means that the total evaporation rate on the nearly spherical particle can be approximated by the total evaporation rate on a sphere of equal volume with the error of O(2 ). This applies for particle of arbitrary shapes. The total evaporation rate on a sphere can be simply calculated from (Tekasakul et al., 1999) JT = 1 + :
(29)
4. Results and discussions Results of the dimensionless evaporation rate per unit surface area from nearly spherical spheroidal particles are shown in Fig. 4 for the particles with aspect ratios ranging from 0.8 to 1.2, and the dimensionless Irst-order reaction rate constant ( 2 ) ranging from 10−9 to 10−3 . Values obtained from both the equivalent-volume sphere and the inscribed sphere methods are
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P. Tekasakul, S.K. Loyalka / Aerosol Science 33 (2002) 307–317
Fig. 5. Comparison of the dimensionless evaporation rate per unit surface area from cylindrical particles, using numerical, and equivalent-volume sphere methods.
presented in comparison with values obtained numerically by Tekasakul et al. (1999). The values obtained from the equivalent-volume sphere and inscribed sphere agree very well with the numerically calculated values. The results using the equivalent-volume sphere method agree within 0.5% of the numerical results while those using the inscribed sphere method agree within 1.5%. For particles that depart signiIcantly from spherical shape, the agreement becomes poorer and this is not displayed in the Igure. Since the equivalent-volume method works well with spheroids and the calculations are simple, we have calculated the dimensionless evaporation rate per unit surface area for cylindrical particles using this method. The results are shown in Fig. 5 together with corresponding numerical results for A = 0:9–5.0. It can be seen that, for a cylindrical particle, the approximation fails except at A = 2:0, where the agreement is surprisingly good (within 1%). When the aspect ratio is lower than 2.0, the equivalent-volume sphere approximation overpredicts the evaporation rate and the reverse is true for the cylindrical particle with aspect ratio greater than 2.0. At the aspect ratio near 2.0, we have an inMection point.
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317
5. Conclusions Results for nearly spherical spheroidal particles using both the equivalent-volume sphere and inscribed sphere methods have shown excellent agreement with numerical results, though the results for cylindrical particles do not yield the same accuracy. This indicates that the approximations can be used for calculations for evaporation rate from nearly spherical particles without performing lengthy numerical procedure. Acknowledgements This research was carried out with the Inancial support from the Thailand Research Fund (TRF) through Grant PDF=32=2541. Support by the US National Science Foundation of S.K. Loyalka’s eJorts through Grant CTS-99014139 is also acknowledged. References Bharat, V., & Ray, A. K. (1992). Evaporation and growth dynamics of a layered droplet. International Journal of Heat and Mass Transfer, 35, 2389–2402. Davies, C.N.(Ed.) (1966). Aerosol science. London, UK: Academic Press. Friedlander, S. K. (1977). Smoke, dust and haze. New York: Wiley. Fuchs, N. A. (1989). The mechanics of aerosols. New York: Dover. GriAn, J. L., & Loyalka, S. K. (1994). Vapor condensation on multiple spheres and spheroids in the non-continuum regime. Journal of Aerosol Science, 25, 1271–1289. GriAn, J. L., & Loyalka, S. K. (1996). Condensation on aerosol particles: Boundary element formulation. Journal of Aerosol Science, 27, 3–18. Li, W., & Davis, E. J. (1996). Aerosol evaporation in the transition regime. Aerosol Science and Technology, 25, 11–21. Loyalka, S. K., & GriAn, J. L. (1993). Condensation on nonspherical aerosol particles: Numerical solutions in the continuum regime. Nuclear Science and Engineering, 114, 135–140. Ravindran, P., & Davis, E. J. (1982). Multicomponent evaporation of single aerosol droplets. Journal of Colloid and Interface Science, 85, 278–288. Ray, A. K., Devakottai, B., Souyri, A., & Huckaby, J. L. (1991). Evaporation characteristics of droplets coated with immiscible layers of non-volatile liquids. Langmuir, 7, 525–531. Ray, A. K., Lee, J., & Tilley, H. L. (1988). Direct measurements of evaporation rates of single aerosol particles at large Knudsen numbers. Langmuir, 4, 631–637. Tekasakul, P., Tompson, R. V., & Loyalka, S. K. (1999). Evaporation from a nonspherical aerosol particle situated in an absorbing gas. Journal of Aerosol Science, 30, 139–156. Widmann, J. F., & Davis, E. J. (1997). Evaporation of multicomponent droplets. Aerosol Science and Technology, 27, 243–254. Williams, M. M. R., & Loyalka, S. K. (1991). Aerosol science: Theory and practice. New York: Pergamon Press. Zhang, W., & Stone, H. A. (1997). Private communications. Zhang, W., & Stone, H. A. (1998). Oscillatory motions of circular disks and nearly spherical particles in viscous Mows. Journal of Fluid Mechanics, 367, 329–358.
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Evaporation from non-spherical particles: the equivalent-volume and inscribed sphere approximations for nearly spherical particles P. Tekasakula; ∗ , S.K. Loyalkab a
Department of Mechanical Engineering, Faculty of Engineering, Thammasat University, Rangsit Campus, Klong Luang, Pathum Thani 12121, Thailand b Particulate Systems Research Center and Nuclear Engineering Program, University of Missouri-Columbia, Columbia, MO 65211, USA Received 9 March 2001; received in revised form 10 July 2001; accepted 11 July 2001
Abstract Although numerical results for evaporation rates from arbitrarily shaped particles have been recently reported in the literature, it remains of interest to explore construction and accuracy of approximate analytical expressions that are simple and of general use. We have analyzed the popular equivalent-volume approximation and also the inscribed sphere approximations in this context, using a technique due to Zhang and Stone (J. Fluid Mech. 367 (1998) 329) who had studied rotatory oscillations of nearly spherical bodies. Comparison of the approximate results with accurate numerical results shows that both these approximations are quite useful. The equivalent-volume sphere approximation is particularly attractive as the error is of the order 2 , where is a measure of departure from the spherical shape. c 2001 Elsevier Science Ltd. All rights reserved. Keywords: Evaporation; Inscribed sphere; Equivalent-volume sphere; Non-spherical particles
1. Introduction The problem of evaporation or condensation of particles has been of major interest in aerosol science (see for example, Davies, 1966; Friedlander, 1977; Ravindran & Davis, 1982; Ray, Lee, & Tilley, 1988; Fuchs, 1989; Ray, Devakottai, Souyri, & Huckaby, 1991; Williams & Loyalka, 1991; Bharat & Ray, 1992; Loyalka & GriAn, 1993; GriAn & Loyalka, 1994, 1996; Li & Davis, 1996; Widmann & Davis, 1997). Generally, one deals here with the transport of ∗
Corresponding author. Present address: Department of Mechanical Engineering, Prince of Songkla University, Hat Yai, Songkla 90112, Thailand. Tel.=fax: +66-74-212893. E-mail addresses: [email protected] (P. Tekasakul), [email protected] (S.K. Loyalka). c 2001 Elsevier Science Ltd. All rights reserved. 0021-8502/01/$ - see front matter PII: S 0 0 2 1 - 8 5 0 2 ( 0 1 ) 0 0 1 7 1 - 9
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P. Tekasakul, S.K. Loyalka / Aerosol Science 33 (2002) 307–317
a vapor in a background gas in which the vapor does not interact with the gas. There are, however, situations in which the vapor can react with the background gas through a Irst-order reaction under almost isothermal conditions (consider for example, a particle surrounded by other particles whose eJect on the vapor transport can be approximated by a homogenized distribution of reactants). Then the diJusion equation for the problem becomes @ˆ ˆ ˆ · Jˆ − k1 ; (1) =−∇ @t where ˆ (g cm−3 ) is the vapor mass concentration in the gas phase, k1 (s−1 ) is the Irst-order reaction rate constant, and Jˆ = − D∇ˆ (g cm−2 s−1 ) is the mass Mux. Here D (cm2 s−1 ) is the diJusion coeAcient which is a constant for a dilute system. Note that the Stefan Mow associated with moving boundary is not accounted for. At steady state, the diJusion equation becomes (2) ∇ · Jˆ + k1 ˆ = 0: Non-dimensionalizing lengths by the equatorial radius of the spheroidal particle, a, vapor mass concentration by the surface concentration, ˆs , and the mass Mux by Ds =a, we have ∇ · J + 2 = 0:
(3)
Here J is dimensionless mass Mux, the dimensionless vapor concentration in the gas phase, and 2 = k1 a=D is the dimensionless Irst-order reaction rate constant. Boundary conditions for this problem are =1
at surface;
(4a)
=0
as r → ∞;
(4b)
where r is the position from the center of the particle. We note that Eq. (3) is really the Helmholtz equation and it is of fundamental interest in light scattering from aerosols as in ion charging of aerosols. Tekasakul, Tompson, and Loyalka (1999) have previously solved this problem numerically for several geometries, albeit even in a slightly general context, where Eq. (4a) is replaced by the jump condition: @ =1 + at surface (4c) @n and is the dimensionless jump distance (see Williams & Loyalka, 1991 for a discussion). The numerical solutions are, however, not convenient to use; for example, in rate expressions in the general dynamic equation for non-spherical particles, and it is of interest to understand the nature of various approximations that are employed in calculation of the evaporation rates from non-spherical particles. In this paper, we use both the inscribed sphere and equivalent-volume sphere approximations to obtain analytical expressions for total evaporation rates from nearly spherical prolate and oblate spheroidal particles for the no jump condition (4a). Values from both expressions are compared with numerically calculated values ( = 0 case, Tekasakul et al., 1999). An approximate analytical expression is also obtained for cylindrical particles using the equivalent-volume sphere approximation. We have followed here the inscribed sphere and the equivalent-volume
P. Tekasakul, S.K. Loyalka / Aerosol Science 33 (2002) 307–317
309
Fig. 1. Geometry for a unit sphere inscribed in the particle.
sphere approximations employed by Zhang and Stone (1997, 1998) in their study of oscillatory motions of nearly spherical particles. 2. Inscribed sphere approximation In this method, a sphere of unit radius is assumed to be inscribed in the nearly spherical particle in which radius of the sphere is equal to the minimum cross-sectional radius of the nearly spherical particle (as seen in Fig. 1). This method has been suggested by Zhang and Stone (1997) for calculations of forces and torques on oscillating nearly spherical particles. The diJusion equation for the particle (due to the inscribed unit sphere) and its boundary conditions are given in Eqs. (3) and (4) while, the diJusion equation for the inscribed unit sphere (due to the particle) can be written as ∇ · J˜ + 2 ˜ = 0 (5)
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P. Tekasakul, S.K. Loyalka / Aerosol Science 33 (2002) 307–317
and the associated boundary conditions at the surface of the unit sphere and far away are ˜ = 1
at surface;
(6a)
˜ = 0
as r → ∞;
(6b)
where J˜ = − ∇˜ is the dimensionless mass Mux associated with the inscribed unit sphere, and ˜ the dimensionless vapor concentration in the gas phase associated with the inscribed unit sphere. Applying reciprocal theorem to Eqs. (3) and (5), we have ˜ = 0: (∇ · J) ˜ − (∇ · J)
(7)
Let V represent the Muid volume surrounding the arbitrarily shaped particle, V˜ be the Muid volume external to the sphere and Vdif = V˜ − V. Integrating the above equation over the volume ˜ the surface of the inscribed unit sphere, and a distant surface S∞ , and then V˜ enclosed by S, applying the divergence theorem, leads to ˜ (∇ · J) dS + (n · J) dS = 0: (8) S˜
V +Vdif
Here n is the unit vector directing into the Muid volume (Fig. 1). The Muid volume is enclosed ˜ Again, applying by Sp , the surface of the particle and S∞ , and Vdif is enclosed by Sp and S. the divergence theorem, we can write, for the evaporation problem, ˜ ˜ dV = 0: ˜ − (n · J) dS + (n · J) dS + ∇ · (J ) (9) Sp
S˜
Vdif
The surface of the nearly spherical body is described in the spherical coordinates as r = 1 + f(; ’);
(10)
where 1 and f(; ’) is in the order of one and describes the detailed particle shape. Now, let us consider Eq. (9) term by term. The concentration ˜ on Sp in the Irst integral can be written in a Taylor series about r = 1: ˜ @ ˜|r=1+f = ˜|r=1 +f + O(2 ): (11) @r r=1
Since the solution for the sphere gives @˜ = − (1 + ): @r r=1
Therefore, Eq. (11) becomes ˜|r=1+f = 1 − f(1 + ) + O(2 ): The term in the second integral is ˜ @ = (1 + ): n · J˜ |r=1 = n · −n @r r=1
(12)
(13)
P. Tekasakul, S.K. Loyalka / Aerosol Science 33 (2002) 307–317
311
The term in the last integral can be written as ˜ = − 2 ˜ − J · J˜ ≈ − 2 ; ˜ = ( ˜ ∇ · J) + J · (∇) ∇ · (J)
(14)
for the volume inside the nearly spherical body in which we assume = 1, ˜ = 1 and J = 0. Therefore, the integral becomes ˜ dV = − ∇ · (J) 2 dV = − 2 f dS + O(2 ): (15) Vdif
S˜
Vdif
Substituting Eqs. (12), (13) and (15) into Eq. (9), we obtain − (n · J)[1 − f(1 + )] dS + (1 + ) dS − 2 f dS = 0 S˜
Sp
or
−
Since
S˜
Sp
Sp
S˜
(n · J) dS + (1 + )
dS = 4, and
f(n · J) dS ≈
S˜
Sp
f(n · J) dS = − (1 + )
˜ dS ≈ (1 + ) f(n · J)
S˜
dS +
2
S˜
f dS:
(16)
S˜
f dS;
the dimensionless total evaporation rate per unit surface area of the prolate spheroidal particle is 2 2 JT =S = (n · J) dS = 4(1 + ) + [ − + (1 + ) ] f dS: (17) Sp
S˜
For a prolate spheroid, we can write = 1 − A, where A is the particle aspect ratio deIned as the ratio of the equatorial radius (a) to the longitudinal radius (b) as shown in Fig. 2. Then we have f = r − 1 ≈ 12 (1 − A2 )cos2 : The dimensionless total evaporation rate per unit surface area of the prolate spheroidal particle becomes 4 (1 − A)[ − 2 + (1 + )2 ]: 3 For an oblate spheroid, we can write = 1 − 1=A, and we have 11 1 1 2 f = r − 1 ≈ 1 − −1 + 1+ sin : A 2A A JT =S = 4(1 + ) +
(18)
The dimensionless total evaporation rate per unit surface area of the oblate spheroidal particle is then 4 1 JT =S = 4(1 + ) − 1− [ − 2 + (1 + )2 ]: (19) 3 A
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Fig. 2. The particle geometries that are considered in this work: (a) prolate spheroid, (b) oblate spheroid, and (c) cylinder.
3. Equivalent-volume sphere approximation In this section, we assume the particle to be a sphere of equal volume. Fig. 3 shows the geometry of the particle and its equivalent sphere. In this approach, the sphere center coincides with the center of mass of the nearly spherical particle. S˜ and Sp represent the surfaces of the sphere and the nearly spherical particle, respectively. Portions of the nearly spherical particle that extend beyond the sphere are labeled Vi− ; i = 1; : : : ; N , where N is the number of bumps. These
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313
Fig. 3. Geometry for an equivalent-volume sphere. −
volume elements are enclosed by surfaces S˜ i and Si− . In the same manner, Vi+ represents the volume elements of the sphere that extend beyond the nearly spherical particle and is enclosed + by surfaces S˜ i and Si+ . Using the above assumption of volume equivalent, Eq. (9) of the preceding section reduces to ˜ ˜ dS: (n · J) dS = (n · J) (20) Sp
S˜
The vapor concentration on the particle surface due to the sphere is represented by 1 over Si+ ; ˜|r=1+f = ˜ @ + O(2 ) over Si− ; 1 + f @r r=1
(21)
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since ˜|r=1 = 1. The vapor concentration on the sphere surface because of the sphere can be written as a regular perturbation expansion in : = (0) + (1) + O(2 ):
(22)
A regular perturbation expansion about the boundary condition on the particle surface, as shown by Zhang and Stone (1998), leads to @(0) (1) |r=1 = − f : (23) @r r=1 The concentration Ield evaluated on the surface S˜ of the equivalent-volume sphere (due to the particle) is − over S˜ i ; 1 |S˜ = (24) @(0) 2 ˜+ + O( ) over S : 1 − f i @r r=1 Substituting Eqs. (21) and (24) into Eq. (20), we obtain N N N @˜ (n · J) dS + (n · J) f dS + (n · J) dS @r Si− Si− Si+ i=1
i=1
=
N i=1
˜ dS + (n · J) −
S˜i
i=1
r=1
N i=1
˜ dS − (n · J) +
S˜i
N i=1
˜ (n · J) +
S˜i
i=1
@(0) dS + O(2 ); @r r=1
(0) @ ˜ f (n · J) dS + O(2 ): + @r r=1 S˜i
dS r=1
N i=1
f
(25)
which can be written as N ˜ @ ˜ dS − (n · J) dS = (n · J) (n · J) f @r S˜ Sp Si− +
(26)
− Approximating Si− = S˜ i with the error of O(2 ), the above equation reduces to ˜ @ ˜ dS − (n · J) ˜ f (n · J) dS = (n · J) dS + O(2 ) @r ˜ ˜ S S Sp
or
r=1
˜ @ ˜ f JT = J˜T − (n · J) @r S˜
dS + O(2 ): r=1
Applying Eq. (13) to the above equation, we have 2 ˜ @ f dS + O(2 ): JT = J˜T + @r S˜ r=1
(27)
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315
Fig. 4. Comparison of the dimensionless evaporation rate per unit surface area from spheroidal particles, using numerical, equivalent-volume sphere, and inscribed sphere methods.
Because of volume equivalency, we have JT = J˜T + O(2 )
S˜ f dS
= 0, Eq. (25) then reduces to (28)
which means that the total evaporation rate on the nearly spherical particle can be approximated by the total evaporation rate on a sphere of equal volume with the error of O(2 ). This applies for particle of arbitrary shapes. The total evaporation rate on a sphere can be simply calculated from (Tekasakul et al., 1999) JT = 1 + :
(29)
4. Results and discussions Results of the dimensionless evaporation rate per unit surface area from nearly spherical spheroidal particles are shown in Fig. 4 for the particles with aspect ratios ranging from 0.8 to 1.2, and the dimensionless Irst-order reaction rate constant ( 2 ) ranging from 10−9 to 10−3 . Values obtained from both the equivalent-volume sphere and the inscribed sphere methods are
316
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Fig. 5. Comparison of the dimensionless evaporation rate per unit surface area from cylindrical particles, using numerical, and equivalent-volume sphere methods.
presented in comparison with values obtained numerically by Tekasakul et al. (1999). The values obtained from the equivalent-volume sphere and inscribed sphere agree very well with the numerically calculated values. The results using the equivalent-volume sphere method agree within 0.5% of the numerical results while those using the inscribed sphere method agree within 1.5%. For particles that depart signiIcantly from spherical shape, the agreement becomes poorer and this is not displayed in the Igure. Since the equivalent-volume method works well with spheroids and the calculations are simple, we have calculated the dimensionless evaporation rate per unit surface area for cylindrical particles using this method. The results are shown in Fig. 5 together with corresponding numerical results for A = 0:9–5.0. It can be seen that, for a cylindrical particle, the approximation fails except at A = 2:0, where the agreement is surprisingly good (within 1%). When the aspect ratio is lower than 2.0, the equivalent-volume sphere approximation overpredicts the evaporation rate and the reverse is true for the cylindrical particle with aspect ratio greater than 2.0. At the aspect ratio near 2.0, we have an inMection point.
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5. Conclusions Results for nearly spherical spheroidal particles using both the equivalent-volume sphere and inscribed sphere methods have shown excellent agreement with numerical results, though the results for cylindrical particles do not yield the same accuracy. This indicates that the approximations can be used for calculations for evaporation rate from nearly spherical particles without performing lengthy numerical procedure. Acknowledgements This research was carried out with the Inancial support from the Thailand Research Fund (TRF) through Grant PDF=32=2541. Support by the US National Science Foundation of S.K. Loyalka’s eJorts through Grant CTS-99014139 is also acknowledged. References Bharat, V., & Ray, A. K. (1992). Evaporation and growth dynamics of a layered droplet. International Journal of Heat and Mass Transfer, 35, 2389–2402. Davies, C.N.(Ed.) (1966). Aerosol science. London, UK: Academic Press. Friedlander, S. K. (1977). Smoke, dust and haze. New York: Wiley. Fuchs, N. A. (1989). The mechanics of aerosols. New York: Dover. GriAn, J. L., & Loyalka, S. K. (1994). Vapor condensation on multiple spheres and spheroids in the non-continuum regime. Journal of Aerosol Science, 25, 1271–1289. GriAn, J. L., & Loyalka, S. K. (1996). Condensation on aerosol particles: Boundary element formulation. Journal of Aerosol Science, 27, 3–18. Li, W., & Davis, E. J. (1996). Aerosol evaporation in the transition regime. Aerosol Science and Technology, 25, 11–21. Loyalka, S. K., & GriAn, J. L. (1993). Condensation on nonspherical aerosol particles: Numerical solutions in the continuum regime. Nuclear Science and Engineering, 114, 135–140. Ravindran, P., & Davis, E. J. (1982). Multicomponent evaporation of single aerosol droplets. Journal of Colloid and Interface Science, 85, 278–288. Ray, A. K., Devakottai, B., Souyri, A., & Huckaby, J. L. (1991). Evaporation characteristics of droplets coated with immiscible layers of non-volatile liquids. Langmuir, 7, 525–531. Ray, A. K., Lee, J., & Tilley, H. L. (1988). Direct measurements of evaporation rates of single aerosol particles at large Knudsen numbers. Langmuir, 4, 631–637. Tekasakul, P., Tompson, R. V., & Loyalka, S. K. (1999). Evaporation from a nonspherical aerosol particle situated in an absorbing gas. Journal of Aerosol Science, 30, 139–156. Widmann, J. F., & Davis, E. J. (1997). Evaporation of multicomponent droplets. Aerosol Science and Technology, 27, 243–254. Williams, M. M. R., & Loyalka, S. K. (1991). Aerosol science: Theory and practice. New York: Pergamon Press. Zhang, W., & Stone, H. A. (1997). Private communications. Zhang, W., & Stone, H. A. (1998). Oscillatory motions of circular disks and nearly spherical particles in viscous Mows. Journal of Fluid Mechanics, 367, 329–358.