a5
Langmuir 1987,3, 85-93
bons. The problems that plague the direct IR measurement of the carbon (samples are highly scattering, strongly IR absorbing, and not homogeneous) were expected to obscure the IR spectrum of an adsorbate. To ascertain whether IR can be used to measure uptake, we have attempted to quantify a nonfavorable case: the comparison of different kinds of carbon adsorbents. Our results show that the relative uptake by different carbons for toluene can be estimated with IR spectroscopy to approximately 15% accuracy. When studying the relative uptake of a single type of carbon, we estimate that 5% accuracy can be achieved with IR spectroscopy of adsorbed toluene or
other aromatic hydrocarbon. Finally, when a single sample is studied in situe even greater measurement accuracy can be achieved. The spectra of toluene adsorbed to the different carbons also highlight the relative contributions of the adsorbate in the first monolayer and the multilayer, revealing significant differences among the carbons tested.
Acknowledgment. I am indebted to Leo Volpe for his technical insights and to Haik Marcar for his suggestions for improvement of this manuscript. Registry No. C, 7440-44-0;toluene, 108-88-3.
Internal Heating of Multilayered Aerosol Particles by Electromagnetic Radiation Marek Sitarski Clarkson University, Potsdam, New York
13676
Received July 18, 1986. I n Final Form: October 17, 1986 Expressions for the heat source function inside an irradiated multilayered spherical particle have been derived. The proposed new algorithm circumvents the typical numerical difficulties with generation of the Riccati-Bessel functions of complex arguments by reformulation of the series expansions of the electromagnetic fields and the corresponding systems of complex algebraic equations for the coefficients of the expansions. The developed computer code appears to be stable in a wide range of size parameters. The results of the test computations for homogeneous coal spheres, two-layered coal-water and water-coal spheres,and three-layered coal-vapor-water concentric spheres prove the correctnessof the derived equations as well as the computer code. The computations for the two- and three-layered particles simulate optical properties of an explosively boiling colloidal coal-water droplet exposed to high-intensityinfrared radiation.
Introduction Layered particles have been found in many man-made and natural systems. Ultrafine coal particles wetted by water or liquid fuels in combustors, soot and fly ash particles coated by water in the polluted atmosphere, metallic fine particles covered by oxides, and multilayered combustion aerosols resulting from fast multicomponent nucleation and subsequent condensation are typical examples. None of them are perfect concentric spheres; however, an understanding of their basic radiative properties can be obtained by studying the simple geometry of multilayered concentric spheres. Internal radiant heating of layered particles causes a number of interesting phenomena such as linear photophoresis, nonlinear photophoresis, and explosive evaporation of two-phase drop l e t ~ . ~ -Numerical ~ difficulties in computations of the internal fields have restricted the studied systems to relatively small size parameters. The expressions for electromagnetic fields evaluated by Aden and Kerkef for two concentric spheres involve expansion series whose terms contain Riccati-Bessel functions and their derivatives of complex arguments and first and second derivatives of the Legendre polynomials. The most serious obstacle in writing a computer program for an arbitrary layered sphere is that the expressions for the fields have to be put in a form that avoids the computation of excessively small or large numbers beyond the limits of the computer employed.
* Present address: Morgantown Energy Technology Center of the U.S. Department of Energy, Morgantown, WV 26505. 0743-7463/87/2403-0085$01.50/0
Accordingly, Bohren and Huffmans call such a program “explosion-proof’. On the other hand, the values of internal fields inside large and absorbing particles can become excessively small on physical grounds. To cope with these and other numerical problems, we first reformulate the derived expressions to avoid generation of the Riccati-Bessel functions of complex arguments. This approach makes the program explosion-proof within certain limits, and then we apply the published experience in Mie scattering computations of Toon and Ackerman? Wiscombe,’ and Bohren and H ~ f f m a n . ~ The purpose of this paper is threefold: (1)to present a very brief derivation of the working expressions for the heat source function for irradiated multilayered spheres; (2) to construct an algorithm for computations of the heat source function applicable to large (in terms of size parameter) and absorbing multilayered spheres; and (3) to perform test calculations for one-, two-, and three-layered spheres checking correctness of the derived equations, algorithm, and computer code. The particular systems chosen for test computations are coal-water particles irradiated by infrared unpolarized light. Results of the computations can be related to the absorption properties of the colloidal coal-water dtrafiie slurry droplets exposed (1)Sitarski, M.; Kerker, M. J. Atmos. Sci. 1984, 41, 2250. (2)Sitarski, M. J. Colloid Interface Sci. 1985, 105, 502. (3)Sitarski, M. Proceedinas of the 16th Annual Meeting_ of. the Fine Particle Society, Miami Beach, 1986; p 32.10. (4) Aden, A. L.; Kerker, M. J. Appl. Phys. 1951,22, 1242. (5)Bohren, C.F.;Huffman, D. R. Absorption and Scattering of Light by Small Particles; Wiley: New York, 1983. (6) Toon, 0.B.; Ackerman, T. P. Appl. Opt. 1981,20, 3657. (7) Wiscombe, W. J. Appl. Opt. 1980, 19, 1505.
0 1987 American Chemical Society
86 Langmuir, Vol. 3, No. 1, 1987
Sitarski
componenb Eb, ED, and E, of the complex electric vectors El, we have to solve the corresponding boundary values problem for the Maxwell equations. The geometry of the multilayered sphere is depicted on Figure 1. The incident parallel electromagnetic beam propagating along the z axis is a monochromatic (2-2) plane polarized wave. The medium surrounding the particle has optical properties of a vacuum: m = 1.0 + 0.0i. The magnetic permeabilities of all the layers are assumed to be equal to that of the medium. By the notation of Bohren and Huffman,6 the following series expansions describe the normalized electromagnetic fields: incident electric field, m
E = CEn(Mon(l) - iNen(I))
(5)
n=l
electric field in the core, m
El = CEn(alnMon(l) - iblnNen(l))
(6)
n=l
electric field in the lth layer, 1 = 2, m
El =
t t t t t t t t t t t
..., L,
En(alnMon(l) - ibInN en(l) + clnMon(2) - idlnNen(2))
n=l
(7)
Incident Radiation
Figure 1. Geometry and parameters used in calculations of absorption and scattering of electromagnetic radiation by mul-
scattered electric field, m
E, = CEn(ianNeL3) - bnMon(3))
tilayered particles.
(8)
n=l
where
to high-intensity thermal radiation.
Expressions for Electromagnetic Fields and Heat Sources inside Multilayered Concentric Spheres Expressions for the distribution of heat sources inside the irradiated particle are derived from the electromagnetic energy balance e q ~ a t i o n : ~ w(r) div S(r) = -u(r) Re E(r) Re E(r) (1)
+
where Sfr) = Re E(r) Re H(r) is the Poynting vector and W(r) is a sum of electric and magnetic energy density. The distribution of sinks of the electromagnetic energy inside the absorbing particle is expressed by the right-hand side of eq 1 and is equivalent to the Joule heat or heat source function Q(r) with an opposite sign. Taking a n average over a time interval long compared with 1 / w for time-harmonic fields we obtain
Q(r) = (a/2)IE(r)I2
Re (ml) Im (ml)
x 107
2n
+1
n(n + 1) The complex vector harmonics Mon(i),
and N on(i) in the above series are expressed in spherical coordinates as follows:
Mon(i)= cos $rnzn(’)(p)ee - sin $~~z,(’)(p)e, n(n + 1) z,(’)(p)e,+
= sin &rn sin 8
Nen(’) = cos $T,, sin f3
P
n(n + 1) P
zn(i)(p)e,+
(2)
The local electric conductivity of the particle a t the given frequency of the electric field is expressed in SI units as follows: a(r) =
E, = in
where the superscripts i = 1, 2 , 3 denote spherical Bessel functions of the first, second, and third kind, respectively. z n ( l ) ( p )= j ,
(3)
XOC
where al-l < r < al. Substituting eq 3 into eq 2 and taking the definition of irradiance of the incident plane wave as the magnitude of the corresponding Poynting vector we get the following expression for the heat source function in spherical coordinates:
(4)
Q(r) has a dimension ( l l m ) which means the power density source ( W/m3) a t location r inside the particle is due to absorption of the incident plane electromagnetic wave of unit irradiance ( W/m2). To derive expressions for the
~ , ( ~ ) ( p=) yn
~ , ( ~ ) ( p=) j ,
+ iy,
(11)
The complex argument is expressed by the relation 2aml P=r A0
where the radial coordinate is inside the lth layer: al-l C r C al. The angular eigenfunctions rnand 7, are defined by the associated Legendre functions P,’(cos 0) as follows: 7, = dPnl(cos @ / d e (12) A, = Pnl(cos@/sin 6 The analogous expressions to eq 5-8 are derived for magnetic vector fields H by applying the Maxwell relation: 1 H = -(V X E) (13) Substituting eq 11 into eq 10 and these into eq 6 and 7,
Langmuir, Vol. 3, No. 1, 1987 87
Internal Heating of Multilayered Aerosol Particles we get the following expressions for the components of the electric vector El inside the Ith layer of the particle irradiated by unpolarized light: sin 8 " El,0(r,8) = -C in-'(2n + l)rn[bln+n + d l n ~ n I p 2 n=l
E&r,O) = 1"
- CEn[rn(aln+n+ ClnXn) Pn=l
- iTn(blnJ/ln+ dlnx'n)l
El,0(r,O) = 1"
- CEn[irn(blnJ/ln+ dlnx'n) - Tn(aln+n + c l n ~ n ) l(14) Pn=1
where 1 = 1, ...,L and cln = d l , = 0. For unpolarized light the fields as well as the heat source function have higher symmetry and depend only on coordinates r and 0:
IEloo(r,0)12+ IEl,O(r,0)I2)(15) The Riccati-Bessel functions are defined, according to Abramovitz and Stegun? as follows: 3;, = p l j n b ) + i ~ n b ) l Xn = pYn(p) +n = P~,(P) (16)
Boundary Conditions At the interfaces between the layers of the particle and a t the outer interface, the tangential components of the electric and magnetic fields must be continuous, which results in the following systems of equations: E,, + Eio = EL0 E8, + Ei, = EL, H,g Hie = HLe H,, Hi, HL$
+
+
EI+1,0= El9
El+l,, = El,
Hi+i,e= Hie
Hi+i,+= Hi,
(17)
Substituting into eq 17 the corresponding 0- and +cornponents of the electric fields by use of eq 5-10 and their magnetic counterparts by applying eq 13, we get the following two sets of complex linear algebraic equations with the unknowns being the expansion coefficients, a,, b,, al,, bin, Cln, and din, A , X , = C, (18) B, Y, = K , (19) n = 1, 2, ..., NmB, where N,, results from the required accuracy of the expansion series and is usually determined by computer experimentation with the a l g ~ r i t h m . ~ ~ ' C, and K, are the 2Lth order column vectors of the constant terms and X, and Y, are the 2Lth order column vectors of the unknowns:
The complex coefficients of the unknowns in the two systems of algebraic equations (eq 18 and 19), expressed by the Riccati-Bessel functions and their derivatives, form 2Lth order square matrixes A , and B, shown in Tables I and 11. The expansion coefficients al,, bl,, cl,, dl,, a,, and b, obtained by solution of eq 18 and 19 together with eq 6-10 and 14 provide the complete set of expressions for the scattered and internal fields. For L = 2, i.e., for coated spheres, eq 18 and 19 reduce exactly to those derived by PluchinolO and tested in his computations of internal energy densities. Similar to eq 18 and 19, a set of linear algebraic equations for internal field coefficients and scattering amplitudes has been given by Kerker.lla The differences might be a result of different notations used in both the derivations. Very recently, the system of linear equations of Kerker has been solved analytically by Bhandari,12 applying recursive method. This analytical solution has become a basis for his algorithm for determination of scattering amplitudes for the multilayered spheres. In our algorithm, described below, instead of an analytical solution to eq 18 and 19, we rearrange them linearly into a form favorable for computations and then in the corresponding computer code the subroutine LEQTlC from IMSL library13 is called in order to solve them numerically.
New Algorithm for Computations of Internal Fields The above derived expressions for absorption and scattering of radiation by multilayered spheres can be considered as an algorithm in numerical computations. The necessary generation of series of Riccati-Bessel functions of complex arguments will, unfortunately, produce excessively large numbers which restrict the computations to relatively small size parameters for strongly absorbing particles. Toon and Ackerman6have found that ratios of the Riccati-Bessel functions of different complex arguments can be generated by the stable recursive relation and the round-off errors do not propagate during the upward recursion. According to this observation, we rearrange the equations for internal electric field (eq 14) in the following, favorable for computation, form: sin 0 " El,O(r,O)= -C in-'(2n + l)a,[b*l,R,(l)(p,mlcl) + p2
n=l
d*lnRn(2)(~,m~aJI
1" El~O(r,o)= - C E n (an[a*lnRn(l)(p,mlel)+ Pn=1
C*J?,(2)(p,ml~l)] - i~,[ b*lnDn(l)Rn(l)(p,mlc~) + d*lnDn(2)Rn(2)(p,mlcul)l 1 1" E12(r,0)= - C E , (i7rn[b*lnDn(1)Rn(1)(p,mlcl) + Pn=l
d*lnDn(2)Rn(2)(P,mlcul)l - Tn[a*lnRn(')(P,mlel) + c*lnRn(2)(P,mleJ I ] (21)
where Dn(l)are logarithmic derivatives of the Riccati-Bessel functions introduced to light scattering by Aden14 Dn(l) = V n b ) / J / n b ) D n (') = X ' n b ) / X n b ) (22) and R,(i) are the ratios of Riccati-Bessel functions
(8) Abramovitz, M.; Stegun, I. A. Handbook of Mathematical Functions; National Bureau of Standards: Washington, DC, 1964. (9)Sommerfeld, A. Electrodynamics; Academic Press: New York, 1952.
(10)Pluchino, A. B. Appl. Opt. 1981,20, 2986. (11)Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969;(a) pp 221-223,(b) 121,122. (12)Bhandari, R.Appl. Opt. 1985,24, 1960. (13)IMSL Library; International Mathematical and Statistical Literature: Houston, TX, 1980. (14)Aden, A. L. J.Appl. Phys. 1951,22, 601.
88 Langmuir, Vol. 3, No. 1, 1987
Sitarski Rn("(z1,zz) = + n ( Z 1 ) / + n ( Z z ) Rn(2)(z1,zz) =
Xn(Zl)/Xn(ZZ)
(23) The starred coefficients of the series expansions are defined as follows: a*ln
=
aln+n(mlal)
h
h
b*ln = b i n + n ( m l d
d*ln dlnxntmlal) (24) Let us notice that the coefficients with asterisks can be evaluated by solution of the following systems of complex linear algebraic equations obtained by simple transformations of eq 18 and 19. A*,X*, = C, (25) B*,Y*, = K , (26) where X*, and Y*, are the 2Lth order column vectors of the unknowns: C*ln
= CloXn(mlal)
y* =
-
h
A $A
s1 = mpl, the coefficients a t the unknowns in eq 25 and
75
26 form the square 2Lth order matrices A*, and B*, presented in Tables I11 and IV, respectively.
2.c
i
- 5
AE *
h
- u
A
ss
;+
2%
iPE
.. .. .. .. .. .. .. .. .. e
.
.
.
.
.
sb
h -
v c .c
5 5
f f i
If the complex arguments are denoted by zf = mpl-l, and
2 %
.
.
.
Brief Description of t h e Computer Program The computer code basically is devoted to evaluation of eq 14 and 15. Solution of the systems of linear complex algebraic equations (eq 25 and 26) is performed by application of the standard IMSL subroutine,13 called LEQTlC. In evaluation of terms in the series expansions of eq 14 and the elements of the series of matrices A*, and B*, and column vectors C, and K,, several recursion procedures have been applied. Solutions of the systems of eq 25 and 26 give the vectors of unknowns X*, and Y*,, the elements of which are the internal field and scattering amplitudes. The numerical values of the electric field for a given set of internal grid points are subsequently substituted into eq 15 to obtain the required heat source function. We numerically integrate the heat source function inside every shell of the particle obtaining the total amount of radiant energy absorbed by each individual shell. This value divided by the geometrical cross section of the particle gives us the contribution of the lth layer, Qab,l, to the total absorption efficiency, Qab, of the particle:
For the purpose of testing the obtained formulas, the proposed algorithm, and the corresponding computer code we also calculate absorption, extinction, scattering, and backscattering efficiencies of the multilayered particle using the following well-known expressions with the expansion coefficients a, and b, of the scattered fields only: Qabs = Qext - Qscat (29)
Langmuir, Vol. 3, No. 1, 1987 89
Internal Heating of Multilayered Aerosol Particles
The number of terms in the above series has been adjusted by the following rule: N,, = Max (N,=(l), ..., (33) where
h
A
I m
Y
i
I
i
E
22 h
Y
A
B
3
il .. .. .. .. .... ... ... ... ... 4=c
:E 9
( I
. . . . .,. ... ... ... .. a a
( I
2
OeI
. .
Nmax(*) = Nint (Imp~l+ 4.05)mp111/3) + 2 (34) Max (...). denotes the maximum value from among the arguments list and Nint (...) denotes the nearest integer for the given real argument, which is an application of the convergence analysis of Wiscombe7 to multilayered spheres. This formula worked very well for all of the layered particles of our interest. However, when repeating results of Fenn and Oser15 we encountered problems with convergence in some cases when the outer radius of the shell was much larger than the radius of the core. The criterion for the number of terms applied in summations of the expansion series of eq 21 is given by eq 34 for each individual layer. For every given internal point of the particle, the relative contribution of the successive terms of the expansion series to the value of the heat source function of eq 15 is calculated. When this contribution is less than lo4 the expansion series of eq 21 are terminated even in the situation when n I NmC Numerical studies by Green et a1.16 of light absorption in homogeneous spheres show that the heat source function is very sensitive to numerical errors. Therefore, in our computer code, double precision together with stable recursion procedures have been applied to assure accuracy at the computed source function. The calculations were carried out on a VAX computer a t the U.S.Department of Energy, Morgantown Energy Technology Center, by using a VAX-11 FORTRAN/G-FLOATING compiler. A. Subroutines. The subroutines used in the computer program are the coded recursion formulas applied to generate the following series of functions. Logarithmic derivatives Dn(l)are evaluated by the downward recurrence relation Dn-l(*)(z)= n / z - l / ( D i l ) ( z ) + n / z ) (35) which starts with An alternative technique of continued fractions17in computations of the logarithmic derivatives Dn(l)(z)can be applied as well.7 Logarithmic derivatives DL2)are evaluated by the upward recurrence relation Dn(2)(z) = - n / z + l / ( n / z - Dn-l(2)(z)) (36) starting with Do(2)(z)= -sin (z)/cos (2). Ratios of the Riccati-Bessel functions Rn(l)of different arguments are evaluated by the upward recurrence
starting with Ro(')(z1,z2) = sin (zl)/sin (z2). Ratios of the Riccati-Bessel functions, R i 2 ) ,of different arguments are evaluated by the upward recurrence h
/,f starting with Ro(2)(~l,~2) = cos (zl)/cos ~~~
(22).
~
(15) Fenn, R. W.; Oser, H. Appl. Opt. 1965, 4, 1504. (16)Green, W. M.; Spjut, R.E.;Bar-Ziv, E.;Sarofim, A. F.; Longwell, J. P.J. Opt. SOC.Am. 1985, B2,998. (17) Lentz, W. J. Appl. Opt. 1976, 15, 668.
90 Langmuir, Vol. 3, No. 1, 1987
Sitarski
Riccati-Bessel functions of the first kind, +, of the real arguments are evaluated by the downward recurrence scheme (39) this stable procedure starts with J / ~ , , ( x )=
o
$N8t-l(~)=
10-15
(40)
where NE,= 2(Nm, + 1)+ Nint (1x1). After generation of N,, functions $, we multiply every one of them by the constant factor f given by
In the computations of the first derivatives of the Riccati-Bessel functions, we use the relation Vn(x)= k l ( x ) - (n/x)+,(x) (42)
vn(x),
Riccati-Bessel functions of the second kind, x,, of the real arguments are evaluated by upward recurrence relation 2n + 1 (43) X,+l(X) = X , b ) - X,-l(X) X
with the starting functions xl(x) = -sin ( x ) - cos ( x ) / x xo(x) = -cos ( x )
(44)
Riccati-Bessel functions of the third kind, [, of the real argument are calculated according to their definition (45) lnb)= $,(x) + ixn(x) Their derivatives, r,(x), are evaluated by the relation ?‘n(x) = ln-1(x) - (n/x)Cn(x) The angular eigenfunctions, T, and T,, are evaluated by the upward recurrence of the following equations:’ t = p - T,-] p = cos (O)T, n + l n with the starting functions r0 = 0, r1= 1. B. T h e Crucial Tests. Test 1. A major check of correctness and stability of the algorithm is the comparison of the absorption efficiency of a given particle calculated by eq 29-31 with the efficiency obtained by summation of the volume integrals of the heat source function inside the individual layers of the particle. T,+1
= P + -t
1=1
(47)
The two numbers, Qabsand Qabs,hteg, should agree within the numerical error of integrations and series summations. Test 2. Another valuable test is the examination of the limiting behavior of the backscatter gain, Q b , calculated by eq 32. For large particles with a thick and absorbing outer shell and for homogeneous large absorbing particles, the Qbscashould approach the ray optics limit, Qbsca,lim, equal to the reflectivity of the material of the outer sphere a t perpendicular incidence. - (Re (mL) - 1)2 + Im (m# (48) QbscaJim (Re (mL)+ 1)2+ Im (mL)2 Test 3. When the complex indices of refraction for every layer of the particle are the same, then the result should be independent of the number of assumed layers and of their relative dimensions and equal to the absorption and
scattering properties of the homogeneous sphere.
Test Computations for One-, Two-, and Three-Layered Spheres Published results of computations of the heat source function or equivalent internal fields for two-layered absorbing spheres1p2J0concern particles with size parameters not exceeding 9. However, in many situations one must estimate internal radiant heating rates for particles with much larger size parameters. Combustion of a colloidal suspension of ultrafine coal particles in water is an actual example. The colloid is atomized and injected into the combustor where the dispersed fine droplets are exposed to high-intensity thermal radiation. To predict the physicochemical response of the coal-water microdroplets to the high-intensity electromagnetic radiation, one must know the distribution of heat sources inside the droplets. Experimental registration of dynamical structures of fine coal-water slurry droplets exposed to high-intensity electromagnetic radiation is difficult because of the time scale of the processes s) and size of the physical system m). However, recent experiments of Sarofim et al.ls on secondary atomization and Maloney et al.19on explosive evaporation of coal-water sluny droplets suggest formation of layered particles during combustion. To provide some preliminary information about the internal heating rates in the layered particles and to verify numerically the derived analytical expressions, the algorithm, and the computer code, test computations were performed of scattering and absorption of unpolarized infrared light by (1) homogeneous coal particles of m = 1.72 + 0.3i; (2) two-layered particles (coal particles coated by water shell of ml = 1.72 + 0.3i and m2 = 1.325 (1.07 X lO+)i and water drop coated by a coal shell of ml = 1.325 + (1.07 X lO+)i and m2 = 1.72 + 0.3i); and (3) three-layered particles, i.e., coal spheres separated from the outer shell of water by a concentric layer of water vapor of ml = 1.72 + 0.3i, m2 = 1.0 + O.Oi, and m3 = 1.325 + (1.07 X 104)i. The results of these test calculations help to interpret some of the experimental observations.20 A. Homogeneous Coal Particles. Computations for homogeneous spheres by use of the multilayered spheres code can be performed by taking the complex refractive indices the same in every one of the assumed layers. Changing the number of layers L = 2, 3, ..., we should obtain the same numerical results (test 3). In all but one case, we assumed the lowest permitted by the code number of layers L = 2. In this one case, for coal particles of the radius a = 1.5 pm, we divided the particle into three layers and got exactly the same numerical values of scattering and absorption as for two-layered spheres of the same size. Therefore the results are independent of the way the homogeneous sphere is subdivided and test 3 has been completed positively. Similar calculations repeated the sample results of WiscombeZ1for absorbing and nonabsorbing particles up to seven significant digits. Computed scattering and absorption properties of homogeneous coal spheres of radii ranging from 1.0 to 8.0 pm,
+
(18) Sarofim, A.; Beer, J. M.; Teare, J. D.; Timothy, L. D.; Kang,S. W.; Srinivasachar, S. Proceedings of AR&TD Direct Utilization Contractors Reoiew Meeting; National Technical Information Service, US. Department of Energy, 1985; p 7. (19) Maloney, D. J.; Spann, J. F.; Lawson, L. 0.; Fasching, G. E.; Sitarski, M. Proceedings of AR&TD Direct Utiliation Contractors Review Meeting; NTIS, US.Department of Energy, 1985; p 26. (20) Sitarski, M. Proceedings of AR&TD Direct Utilization Contractors Reoiew Meeting; NTIS, US. Department of Energy, 1985; p 15. (21) Wiscombe, W. J. NCAR Technical Note, PB301388, Boulder, CO, 1979.
Langmuir, Vol. 3, No. 1, 1987 91
Internal Heating of Multilayered Aerosol Particles
Table V. Efficiencies for Extinction, Scattering, and Absorption for Homogeneous Coal Spheres (m= 1.72 by Infrared Unpolarized Light (X, = 1.06 pm) a, p m
a
Qea
Qmt
1.0 1.5 2.0 3.0 3.5 4.0 5.04 8.0
5.93 8.89 11.86 17.78 20.75 23.71 29.87 47.42
2.533 2.425 2.354 2.274 2.248 2.228 2.197 2.146
1.196 1.215 1.112 1.207 1.205 1.202 1.199 1.190
Qk
8.h.
Qatm,intdg
1.337 1.211 1.142 1.067 1.044 1.026 0.9982 0.9562
1.338 1.212 1.143 1.069 1.045 1.027 0.9993 0.9574
Q k , ~ i m
0.08651 0.08388 0.08180 0.081 12 0.08143 0.081 25 0.08117 0.08127
0.08125
+ 0.3i) Irradiated
+
Table VI. Efficiencies for Extinction, Scattering, and Absorption for Water Droplet (ml = 1.325 (1.07 X 10b)i) Coated with Concentric Shells of Coal (m,= 1.72 0.3i) Irradiated by Infrared Unpolarized Light (X, = 1.06 pm)
+
al, wn
Pm 2.0 2.5 3.0 3.5
a2
1.0 1.0 1.0 1.0
a2
8e.t
11.86 14.82 17.78 20.75
Qk
QSCat
2.334 2.308 2.274 2.248
1.191 1.210 1.207 1.205
0.08287 0.08167 0.081 18 0.081 43
Qab0
Qatm,integ
Qabs,i
Qabs,2
1.142 1.098 1.067 1.044
1.142 1.098 1.067 1.044
1.599 X 10” 1.657 X 1.888 X lo4 2.298 X
1.142 1.098 1.067 1.044
Table VII. Extinction, Scattering, and Absorption Efficiencies for a Coated Sphere with an Absorbing Coal Core (m = 1.72 0.3i) and Concentric Water Shell ( m z= 1.325 (1.07 X Irradiated by Infrared (A, = 1.06 rrm) UnDolarized Light
+
+
al, wm
4.0 10.0 20.0
wm 5.04 12.6 25.2
02,
a2
Qsrt
Qmt
QbW
Qab
29.87 74.69 149.36
2.128 2.115 2.046
1.087 1.150 1.111
2.968 X lo-’ 6.003 X 1.776 X lo-’
1.041 0.9641 0.9337
irradiated by an unpolarized infrared light beam (X,= 1.06 pm), are summarized in Table V. Comparing the absorption efficiencies, Qab, calculated from the scattered fields with the absorption efficiencies, Qaba,inte, calculated by numerical integration of the heat source functions obtained from internal fields, we observe excellent agreement between those two sets of numbers (test 1). Qualitatively, the changes of the efficiency factors Qert, Qat, Qh, and Qabwith size parameter are the same as depicted by Kerker.llb Comparison of the backscatter gain, Qbca, with the values of Qbaca,lim obtained by the geometrical optics limit (eq 48) shows that beginning from the size parameter 11.86, the backscatter gain oscillates around the reflectivity of the coal within 1% tolerance (test 2). For smaller particles, the backscatter gain is higher than that predicted by the geometrical optics limit because the rays refracted by the sphere are not completely attenuated and are partially returned backward due to internal reflections. The oscillations for size parameters 25.93 are strongly damped because of high absorption index of coal. B. Two-Layered Water-Coal Particles. In this test, spherical coal particles are assumed to have a 1.0-pm radius concentric hole filled with water. Results of the test computations of scattering and absorption properties of the particles with increasing thickness of the coal shell are listed in Table VI. In all the cases, the absorption efficiencies, Qab, calculated from scattered fields are in excellent agreement with those, Qabs,mteg, obtained by numerical volume integrations of the heat source functions. When we compare the scattering and absorption efficiencies for the water-coal particles with the corresponding values calculated for the homogeneous coal particles of the same size (Table V) we denote that the numerical values become the same starting from the particles of radius 3.0 pm. The central water drop shows the strong influence on the backscatter efficiency, Qbsca,because of the additional interface. However, for larger particles of radius 22.5 pm, the backscatter gain is approaching the rays optics limit, QbscaJim, for coal within 1% tolerance. The computer code gives the opportunity to calculate the contributions to the total absorption efficiency due to
’
Qab,intdg
Qatm.eore
1.048 0.9648 0.9374
1.664 1.532 1.488
Qate,i
1.048 0.9648 0.9373
Qab,2
2.705 X 10” 5.473 X loT5 8.997 X
the absorption in the individual shells, Qabs,l, computed numerically by eq 28. From the tabulated values of Qab,l, we observe very fast decay of the radiation absorption in the central water drop with increasing thickness of the strongly attenuating coal shell. C. Two-Layered Coal-Water Particles. The water-coated coal particles have an opposite structure to the ones discussed in section B. The volume of the absorbing coal core is assumed to be equal to the volume of the water shell which corresponds to the 50150 composition of the ultrafine coal-water slurry. The results of our test computations of efficiencies for extinction, scattering, and absorption for these particles are summarized in Table VII. The absorption efficiencies, Qabs, calculated from scattered fields agree very well with those, Qabs,integ, obtained by numerical integration of the heat source functions. The differences of about 0.5%are a result of numerical error of integration depending on the integrated function, number of interpolation points, and the applied method of integration (we use the two-dimensional Simpson’s rule). I t is worthwhile to mention that the magnitudes of the computed values of the heat source function for particles of 12.6 pm in radius differ as much as 17 orders depending on the location inside the particle. The relative error of computations of the heat source function in all the points is less than lo+. The backscatter gain, Qba, for all the particles is higher than the reflectivity of water (Qbca,Km = 0.019 54) because some of the refracted rays are reflected from the coal-water interface. Absorption efficiencies for two-layered soot-water particles studied by Fenn and Oser15have been recalculated for testing purposes and were found to be in agreement up to three significant digits. In a number of cases, however, when the radius of the outer sphere is much larger than the radius of the core, problems arise with convergence of the series. In those cases, special care must be taken as discussed in Appendix B of the book of Bohren and H ~ f f m a n . ~ An interesting phenomenon is observed when comparing the absorption efficiency of the core, Qab,core, which is the integrated amount of energy absorbed in the core divided by its geometrical cross section, with the absorption effi-
92 Langmuir, Vol. 3, No. 1, 1987
Sitarski
Table VIII. Efficiencies for Extinction, Scattering, and Absorption for Three-Layered Spheres with an Absorbing Coal Core ( m ,= 1.72 0.3i) Separated from a Concentric Outer Shell of Water (m,= 1.325 (1.07 X 10b)i) by a Concentric Layer of Water Vauor ( m ,= 1.0 0.0i) Irradiated by Infrared ( i n= 1.06 um)UnDolarized Light
+
al, pm 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0
+
+
~ 2 pm ,
~ 3 Ct ,m
a3
Qelt
Qscat
Qbsca
Qabs
Qabs,intag
Qsbs,care
4.01 4.1 4.3 4.5 5.0 6.0 7.0 8.0
5.05 5.10 5.22 5.34 5.68 6.43 7.26 8.12
29.93 30.23 30.94 31.65 33.67 38.11 43.03 48.13
2.137 2.166 1.984 1.738 1.737 2.789 2.360 1.437
1.107 1.247 1.330 1.166 1.257 2.393 2.068 1.198
0.2706 0.4700 0.0486 1.752 2.279 4.145 0.1475 0.1005
1.030 0.9187 0.6539 0.5720 0.4798 0.3952 0.2920 0.2395
1.040 0.9459 0.6608 0.5753 0.4798 0.3952 0.2919 0.2391
1.658 1.538 1.125 1.025 0.967 1.021 0.961 0.985
ciency, Qabs, of the same size homogeneous coal particle (Table V). The corresponding numerical values of the absorption efficiencies have been increased very markedly by the presence of the water shell. This enhancement of absorption can be attributed to focusing of the incident radiation onto the core by the relatively transparent water shell. The focusing effect has been noted before in computations of the total absorption cross-sections of sootwater sphere^.'^^^^ When a coal-water particle is exposed to high-intensity radiation the focusing effect can facilitate the onset of explosive boiling at the coal-water interfa~e.~,'~ Assuming the incident radiation to be isotropic, as approximately is the thermal radiation in combustors or other high-temperature chemical reactors, the optical properties of the explosively boiling coal-water particle can be simulated by a three-layered model: coal core separated from the water shell by a concentric layer of water vapor. Evidence for this type of structure has been recently registered in the high-energy short-pulse experiments of Maloney et a1.l' The response of the coated particle to high-intensity radiation by the development of the vapor layer changes dramatically thermal properties of the particle insulating the hot core from the relatively cold water layer by its vapor. The question is raised, how are the optical properties of the particle changed by the above mentioned change of the structure? Is the refracted radiation focused onto the core in the resulting three-layered particle? D. Three-Layered Coal-Vapor-Water Particles. According to the above proposed three-layered model for explosive boiling of the coal-water particle, the starting structure is the coal sphere coated by a concentric water shell of equal volume. Then, due to the intense radiant heating of the particle, part of the water is evaporated by boiling a t the coal-water interface. This occurs a t the temperature of maximum superheat for water which according to SkripovZ3is equal to 575 K. The quantity of evaporated water is related to the volume of the vapor shell by the equation of state. Evaporation at the external interface, water-gas, has been neglected. By this procedure we calculate radii u2 and u3of the concentric shells of vapor and water listed in Table VIII. Efficiencies for absorption, Qabs,calculated from scattered fields and those, Qab,integ,obtained by volume integrations of the heat source functions match very well, which proves correctness of the procedures used. The backscatter gain, Qbsca,for the three-layered particles is much higher than the ray optics limits for both condensed media, Le., for coal (Qbsca,lim= 0.08125) and for water (Qb8ca,i!m = 0.019 541, because of the additional internal reflections at the interfaces. Interestingly, the significant enhancement of the backscatter efficiency was observed for the particle of dimensions a, = 4.0, u2 = 6.0, and u3 = 6.43 pm (Qbsca = 4.145). ~~~~~
(22) Ackerman, T. P.; Toon,0.B.Appl. Opt. 1981, 20, 3661. (23) Skripov, V. P. Metastable Liquids; Wiley: New York, 1974.
&s,i
1.040 0.9459 0.6607 0.5752 0.4798 0.3951 0.2918 0.2391
Qabs,i
Qabs,i
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.676 X 3.623 X 3.668 X 6.533 X 4.528 X 7.566 X 1.455 X 7.251 X lo*
1
t
r) 1 . 4 n! 5
-:1 . 3s 1.2 *
1 .o
Figure 2. Efficiency for absorption, Qabs,core, of the 4.0-fim-radius coal sphere inside an expanding three-layered,coal-vapor-water droplets irradiated by infrared unpolarized light (A, = 1.06 l m ) .
Computations of the absorption efficiency of the core, Qab,core,indicate clearly that the focusing effect observed for the two-layered coal-water particles has been quenched quickly by the formation of the blanketing shell of water vapor. This phenomenon is depicted in Figure 2. The developing layer of vapor a t the coal-water interface, as thin as 0.5 pm, brings the amount of radiant energy absorbed by the coal core to the level of a pure coal particle. The observation resembles strikingly the conclusion of Le Chatelier, who studied thermodynamic systems taken out of equilibrium by some imposed perturbation. The perturbation directly induces a process that, by the Le Chatelier principle, reduces the perturbation. In our case, the water-coated coal particle is taken out of equilibrium by absorption of high-intensity infrared radiation. The focusing of the water shell facilitates induction of the process of boiling which, by changing the structure of the particle, subsequently reduces absorption of the radiation. Conclusions The complete set of expressions for heat source function inside the irradiated multilayered particles has been obtained. The derived equations have been rearranged into a form favorable for computations. The transformed series expansions for the internal fields and the systems of complex linear algebraic equations for determination of the scattering and internal field coefficients do not involve Riccati-Bessel functions of complex arguments. The algorithm applies numerically stable procedures for generation of logarithmic derivatives of the Riccati-Bessel functions and their ratios of different arguments and Riccati-Bessel functions of real arguments. In this way, the constructed computer code avoids computing excessively large numbers and does not set any upper bound on the size parameters. However, for very large particles, composed of strongly absorbing shells, location of interior points for which the electromagnetic fields and the heat
Internal Heating of Multilayered Aerosol Particles source function are calculated needs some caution. For these particles, the fields and the source function decay very fast from the surface. Therefore, if the interior point is located much deeper than the penetration thickness of the radiation, the magnitudes of the fields can become smaller than the numbers handled by the computer. The excellent agreement between the absorption efficiencies, Qaba,calculated from scattered fields and the absorption efficiencies, calculated by volume integration of the heat source function obtained from the internal fields prove correctness of the derived equations, the proposed algorithm, and the constructed computer code. When the size parameters for the concentric shells differ very much from that of the core, however, the computer code might become unstable and special care must be taken. Results of computations of the backscatter gain, Q b , for one- and two-layered absorbing particles provide additional support of the reliability of the presented method. The numerical values of backscatter listed in Tables V and VI rapidly approach the corresponding ray optics limit within a tolerance of 1%. The developed computer code can be applied for other particles possessing spherical symmetry with radial change of the complex refractive index by proper discretization. The test computations of the energy dissipated inside coal spheres show the focusing effect of the water layer on the distribution of the energy. During intense radiant heating the developing shell of water vapor at the coalwater interface rapidly quenches the focusing effect. This phenomenon can be an example of the Le Chatelier principle in optics.
Acknowledgment. The financial support by US.Department of Energy, Morgantown Energy Technology Center, Contracts 40552501 and DE-AC21-85MC22073,is gratefully acknowledged. Nomenclature radius of the lth concentric sphere coefficients of the series expansions 2L X 2L square matrixes of the coefficients, at the unknowns coefficients in the series expansions 2L X 2L square matrices of the coefficients, at the unknowns velocity of light in vacuo coefficients in the series expansions column vectors of constant terms coefficients in the series expansion logarithmic derivatives of Riccati-Bessel functions complex electric field vector constant factor complex magnetic field vector
Langmuir, Vol. 3, No. 1, 1987 93 imaginary unit column vectors of constant terms number of layers in the particle complex refractive index of the lth layer odd spherical vector harmonics numerator of the term in series expansion even spherical vector harmonics odd spherical vector harmonics number of terms in the series expansions of the fields number of terms in calculations of the heat source function in the lth layer starting term number in downward recurrence auxiliary function associated Legendre functions heat source function efficiency for extinction of the particle efficiency for absorption of the particle efficiency for absorption obtained by volume integration of the heat source function contribution to the absorption efficiency of the lth layer absorption efficiency of the core of the particle efficiency for scattering of the particle efficiency for backscattering limit value of backscatter gain radial spherical coordinate ratios of the Riccati-Bessel functions complex argument Poynting vector auxiliary function electromagnetic energy density real argument complex arguments spherical Bessel functions of the ith kind angular spherical coordinates Riccati-Bessel functions angular eigenfunctions size parameter of the Zth layer electric conductivity of the lth layer at given frequency of the field magnetic permeability of the vacuo wavelength in vacuo complex argument Subscripts pertaining to components of the vectors in spherical r, 0, 4 coordinates 1 pertaining to the layer number n pertaining to the term number in series expansions Superscripts 1 order of the spherical Bessel function 0 pertaining to unpolarized light * pertaining to transformed matrices and coefficients
