CHARLES A. SLEICHER, Jr., and STUART W. CHURCHILL Department of Chemical and Metallurgical Engineering, The University of Michigan, Ann Arbor, Mich.
Radiant Heating of Dispersed Particles
W H E N a dispersion of solid or liquid particles in air, such as a spray, smoke, or fog, is subjected to a radiant flux, energy is absorbed by the particles, the temperature of the particles rises, and energy is transferred from the particles to the air. If the particles have an appreciable vapor pressure, mass is also transferred to the gas phase. This article presents the results of a theoretical investigation of the over-all process. Consideration is limited to spherical particles and to applications in which eddy transfer is negligible-i.e., generally to particles of 100 microns or less in diameter. The analysis has several prominent applications. The protection of adjacent buildings against radiation from a fire with a spray of water droplets has been discussed by Thomas ( 7 7 ) . The protection of cities against thermal radiation from a nuclear fireball with a smoke screen was recently mentioned ( 4 ) . The role of radiative transfer in the ignition and combustion of powdered coal and atomized fuels is also of considerable interest (7, 5, 70). Numerous mathematical studies have been reported for heat transfer processes involving a sphere in an infinite medium. Most of these studies have been reviewed by Carslaw and Jaeger (2). However, the complexities arising from simultaneous and transient radiation, conduction, and evaporation in a multiplesphere system have not been resolved. In this study, rate mechanisms are first postulated for each of the individual processes so that the problem can be formulated in terms of differential energy and material balances for a single particle and the associated air, together with the appropriate boundary and initial conditions. The analysis is organized as follows: 1, The temperature distribution within a particle is considered first and its effect is negligible at all times for either internal or surface absorption. 2. An improved solution is obtained for the temperature history of a single nonvolatile particle in an infinite atmosphere. This solution is applicable for the initial period during which the
temperature of the particles rises to a pseudo steady-state value and indicates the significance of the simplifying assumptions made by previous investigators. 3. An approximate equation is developed for the temperature history of a dense dispersion of nonvolatile particles. This solution encompasses the initial period, the pseudo steadystate period, and the final period when the effect of adjacent particles is significant. . 4. The rate of evaporation and the effect of evaporation upon the temperature of volatile particles are finally considered. Illustrative calculations are carried out for coal particles and oil droplets. These results are interpreted in terms of the ignition of the particles and the ultimate disposition of the thermal radiation. Formulations and Assumptions
When a dispersion is exposed to a radiant flux from sources at the boundary or within the dispersion, each individual particle receives radiation from all directions due to scattering by the particles. Within a dense dispersion the reception approaches uniformity. Furthermore, it will be shown that the rate of conduction within a particle is ordinarily sufficient to minimize the temperature gradients which would result from anisotropic radiation. Accordingly, an isotropic flux is assumed for simplicity. The rate of absorption of radiation by an opaque particle in a n isotropic flux can then be written as 47ra20(4, where 01 is the absorptivity. This same expression can be applied for internal absorption by a semitransparent particle, with CY interpreted as an effective absorptivity. The absorptivity is a property of the material and in general will be a function of the wave length of the radiation and the radius of the particle. However, the variation of the absorptivity with radius is omitted in this analysis. Absorption of radiation by air is certainly negligible and the absorption by vapor can be neglected for the concentrations ordinarily encountered.
Heat transfer inside the particle is assumed to take place only by conduction. Internal eddying within a liquid droplet would only serve to decrease already negligible temperature differences. Heat transfer and mass transfer outside the particle are assumed to take place only by diffusional mechanisms and the bulk flow arising from evaporation. The experimental data for forced convection (7) indicate that an air motion relative to the particles corresponding to D V p / y greater than 2 is necessary for appreciable eddy transfer. Generally, D V p / p is less than 2 for particles of 50 to 100 microns in diameter. I n view of the small temperature differences found, reradiation from the particles and free convection also appear negligible. At points of symmetry between particles, the temperature and vapor concentration gradients are zero. This condition can be approximated mathematically by assuming that the radial gradients are zero at all angles at a distance equal to one half of the mean distance between particle centers. All of the physical properties of the particle and surrounding medium, except vapor pressure, are assumed to be constant, and the vapor phase is assumed to follow the ideal gas law. The error resulting from variation in the properties with temperature can be minimized by the use of mean properties. Temperature Distribution Within a Particle
Solutions for the temperature distribution within a particle exposed to a radiant flux can readily be obtained-if the effect of the rate of change of diameter upon the rate of heat and mass transfer is neglected; and if the rate of heat transfer to the surrounding air by conduction and evaporation can be expressed as the product of a constant heat transfer coefficient and the temperature difference between the air at an infinite distance and the surface. The first assumption appears to be very.reasonable. I n the following sections the heat transfer VOL. 48, NO. 10
OCTOBER 1956
181 9
coefficient is shown to have a constant value equal to k / u except for very short times and very high rates of evaporation. For a droplet with uniform internal absorption, an energy balance over a differential spherical shell gives
Table 1.
Properties of Representative Dispersions in Air
a , jft.
\microns for 2000° F. source R , B.t.u./(hr.)(ft.)(" F.) p , Ib./(cu. ft.) C, B.t.u./(lb.)(O F.) D f / R T , 1b.-mole/(hr.)(ft.)(atm.) M , Ib./lb.-mole u, (Ib. force)/ft. A, B.t.u./lb. P d at 68O F . , atm. Normal boiling point, F. n , particles/(cu. ft.) Z = (3/4~ia~n)1'~ 4 = PdCd/JTc 4ra3npdCd/3rrC am
The maximum temperature difference within the particle occurs at steady state. From the steady-state solution of Equation 1, the maximum temperature difference is found to be To - T , = aqa/2kd
(2)
For surface absorution bv an opaque particle, the differential energy balance over a spherical shell reduces to
Coal
Fog Oil
1.64 X 10-6 0.5 0.0005
x
10-5
0.90
3.0 100 0.30
0.072 57 0.44 3.1
x
8.2 25
Dodccane
...
10-4
350 0.0021 94 1.0 x 10-9 770 3 . 5 x 109 250 463 8.9 x 10-6
...
... ... ...
2.2
x
108
25 5 52 0.106
1.64 X 50 0.05 0.086 46.5 0.505 5 . 3 x 10-4 170.3 0,0017 155 1.58 X 10-3 418.1 3 . 5 x 106 25
433 0.083
(3)
and the boundary condition is atr=a
The solution given by Carslaw and Jaeger (3) for the cooling of a sphere by external convection can be reexpressed for surface absorption and external convection as follows:
Assuming uniform internal absorptivity and the same radiant flux, the maximum temperature difference within the 100micron dodecane droplets is estimated to be 3.0' F. and within the 1-micron fog oil droplets is estimated to be only a small fraction of a degree. These temperature differences are too small to affect the energy balance around the particle. Therefore, the
The mean distance between particle centers can be expressed in terms of the concentration of particles as follows 1=
(&)lla
Introducing the dimensionless variables x = r/a (11) C
=
L = l/a X ( T - T,)/T,
(12)
(13)
and
(14)
t = kO/pCaa
reduces Equation 6 to where /3,&are the roots of
Pn cot pn = 1
- ha kd
(5)
According to Equation 4, the temperature difference within a particle will rise to a maximum and then decrease asymptotically to zero at steady state. A simple expression for the maximum temperature difference was not obtained, but numerical calculations indicate that the difference will not exceed aqa/Zkd. Thus the maximum temperature differences for uniform internal and surface absorption are essentially the same, although the gradients are opposite in direction. The maximum difference for nonuniform absorption can be inferred to be of the same order of magnitude. Properties of three typical dispersionspowdered coal, dodecane droplets, and fog oil droplets-are given in Table I. Calculations and assumed ambient conditions are summarized in Table 11. I n a uniform radiant flux of 63,000 B.t.u./(hr.)(sq. ft.) corresponding to radiation from surfaces at 2000' F., the maximum temperature difference within a spherical particle of the coal is then
T,- T, 1 820
=
assumption of a uniform temperature within the particle is justifiable. This assumption is made in all subsequent derivations.
The Laplace transform of Equation 15 with the initial condition is
Transient Behavior
The solution of Equation 16 is
The transient behavior of a nonvolatile particle with a uniform temperature is defined by an energy balance over a differential spherical shell of the surrounding air
and a further change of variables gives v =
d:
jTjT - T o
=
1 x"
{UI =
Transformation and expression of the boundary conditions in terms of the new variable give
the boundary conditions
and and
E = oa t r
where 4
and the initial condition
orqa/2kd = 0.78 I?.
INDUSTRIAL AND ENGINEERING CHEMISTRY
T = To
at 9
=o
(9)
Pdcd/sPc
(21 1
The boundary equations permit elimination of E and F from Equation 18. yielding
For the particle
i i
\1+L
The temperature history of the air at any distance from a particle could be found by inversion of Equation 22. However, principal interest is in the temperature of a particle rather than in the local temperature of the air and attention will be restricted to Equation 23.
Single Particle in an Infinite Medium-Initial Behavior of a Dispersion. Initially, the effect of adjacent particles is negligible and the behavior of a dispersion is typified by a single particle in an infinite medium. I t is advantageous to examine this limiting case before the more general problem. For a single particle in an infinite medium Equation 23 reduces to
a constant h leads only to a small error. The actual variation of the Nusselt number with time can be calculated as follows hD k
Densely Dispersed Particles.
- =
-
-
1)22
++2z4]
I
I
2
2 dU(1,B) dX -
--U
di is
1
I +
A
similar technique may be used to obtain an expression for the complete transient behavior of a particle in a dense dispersion. For L>>1, Equation 23 can be approximated by
(29)
small with
Performing the indicated inversion gives
e -keZl/ p c a 2dz
- (2+
-
f
+
k -
1
T , ) d r
Again assuming respect to 1 4s
hD -?a
7r
-
r
k
'/
dT(a,B) -
2a
(Td
hD 2
The inverse transform of Equation 24 in terms of the original variables is
The Nusselt number, calculated from Equation 28, is plotted in Figure 2 for 6 equal to 500. The Nusselt number is independent of the radiant flux and falls rapidly from an initial value of infinity to within 10% of the steadystate value of 2.0 in about 1 microsecond for the fog oil droplets, 3 milliseconds for the coal particles, and 10 milliseconds for the dodecane droplets (Table I ) . Alternatively, the inverse transform of Equation 24 and of the terms in Equation 27 can be obtained in the form of an infinite series by carrying out the division indicated in the brackets and inverting the resulting series term by term. However, the series obtained for the temperature and Nusselt number do not converge rapidly and prove to be no more convenient for calculating purposes than the integral expressions of Equations 25 and 28.
For the dispersions of powdered coal and dodecane in Table I, Equation 29 differs from Equation 23 by no more than 8% for all values of S from zero to infinity. For the fog oil dispersion, the equations differ by no more than 3%. In all cases the agreement is best for very small and very large S corresponding to long and short times, respectively. The inverse transform of Equation 29 in terms of the original variables is
(25)
0
The integral in Equation 25 must be evaluated graphically or by approximate methods, and Equation 25 is accordingly inconvenient for application. Considerable simplification results from noting that when 6 has a value of 400 or more (Table I), 43.is never more than > 3 . For long times the difference between the two results is probably less than the errors inherent in both analyses. I n particular, the symmetry assumptions may lead to error and the assumption of no relative velocity between the air and particles would lead to error for particles larger than those considered here. The assumption of constant physical properties is increasingly poor a t higher temperatures. On the other hand, this assumption is not really very critical. At higher temperatures only the linear term is important and the rate of increase in temperature is independent of k but dependent on the product, 47ra2naq/pC. If the whole cloud of air and particles expands symmetrically as the temperature increases, then n / p will remain constant. Since a, C, and a are not stronglv dependent upon temperature. the assumption that 4raznaq/pC is constant may not lead to large errors. A different theoretical analysis for a single particle iri a n infinite atmosphere
INDUSTRIAL AND ENGINEERING CHEMISTRY
Negligible evaporation was assumed in all of the above derivations. Evaporation will influence the energy balances and will continuously reduce the diameter of liquid droplets. Coal particles may also lose volatile matter upon radiant heating and change in size and shape. The effect of evaporation upon the behavior of a particle subjected to radiation will now be examined. A nonvolatile particle in a n infinite medium was shown to approach a steady-state temperature. A volatile droplet in an infinite medium approaches a pseudo steady-state temperature which changes as evaporation decreases the radius of the droplet. A specific solution apparently has not been published for this psuedo steady-state temperature although results have been reported for mathematically similar problems (6). Derivations. The energy balance over a differential shell of the gas phase can be written k d
Cw d T
(33)
a mass balance gives
and the boundary conditions are T - t To
as
r
l.),.i-po
ar
r-+
pa
at
r = a
=
@d
+ m
m
(35) (36)
(37)
and
(39)
Integration of Equation 33 and substitution of the boundary conditions yield the temperature profile of the air T -
T
k
(1
("$
Xk
aqaC
)x
- e-mc//i;rrk)eu,~/aink
and the temperature of the droplet is
(40)
I
.05
01 5x 0-2 1O-l
5x16'
5
I
5x0
IO
10'
5x103 IO4
3k e PdCda2
Figure
3. Initial temperature history of a dispersion exposed to thermal radiation Td- To = T [ l k
(euC/annk
(Pd
- po)(l - ewRT/arrPDfM)/ (1 - ewRT/4nuf'D/M) ( 4 2 )
Pd - P o
Equations 41 and 43 must be solved simultaneously together with the vaporpressure data to determine the temperature of the droplet and the rate of evaporation. I n general this solution must be found by trial and error. Owing to surface tension, very small droplets exhibit a higher vapor pressure 'than bulk fluids as follows : =
+ gM/apdRT]
Pa(T)[1
and hD
-
To = aga/k
(51)
the steady-state term in Equations 25, 26, and 30. The temperature and concentration profiles for these special cases can be found by similar reduction of Equations 40 and 42. The rate of evaporation can be expressed as the rate of decrease of radius,
4 4 P d - Po)D/M RT [It
w =
47 reduces to T d
2
and the radius can be found as a function of time by integration-i.e., For negligible absorption of radiation, (q---t zero), Equation 47 reduces to the so-called wet-bulb temperature of a droplet
i
PdT)
+
p a with the
Neglecting the change of radius of a droplet 1
[l
- 1) (41)
If the diffusivity of a partial pressure gradient, D f / R T , rather than the diffusivity of a concentration gradient, D , is assumed to be constant, Equation 34 can be integrated to give the concentration profile
P A - Po =
*]
- 4xa2aq
T d =
T o - *4 ~ a k[1+L 2!
(g) 4rak +
while for a nonvolatile droplet Equation
e
= 4Pd
a2da
J2W
(53)
Interpretation. The dimensionless term wh/4na2aq equals the fraction of the absorbed energy which is converted to latent heat of evaporation. The dimensionless term, w C / 4 n a k , characterizes the effect of evaporation upon the rate of heat transfer to the air, and the
(44)
This correction must be made when appreciable. The Nusselt number based on transfer in the vapor phase only-Le., with the heat transfer coefficient defined as
,
1 U 0
-8 1
Fog Oil 6 0 5 Milliseconds I 5 2 Milliseconds Coal Dodecane 606 Milliseconds
+o
e is thus
ov 0
The effect of evaporation is more apparent if Equations 4 1 , 4 3 , and 46 are expanded in series giving
Figure 4.
0 2
04
06
08
Long-term history of a dispersion exposed to thermal radiation VOL. 48, NO. 10
OCTOBER 1956
1823
Table II. Conditions
Ambient temperature Isotropic radiant flux
Significant terms aqa/2kd, aqa/k,
O
F. F. O
( w a l k ) - (Td - To),O F. Tal O F. PdDjM/aRT lb./(hr.) (sq. it.) Pd
-da/dEJ, microns/sec. = ( P d - p , ) / P , atm.
wX/4~a%q wC/4~ak
uM/aPdRT
D D,
Summary of Illustrative Calculations 63,000
3.4
x x
B.t.u./(hr.)(sq. ft.)
Coal
Fog Oil 3.6
F.
68'
0.78
10-4 10-3
At pseudo equilibrium Negligible 68.0 6.6 X 10-5 9.8 X 10-8 1.0 x 10-9 2 . 0 x 10-4 6.6 X 10-6 9.6 x 10-3
Dodecane 3.0
310
34
0.0 378
3.0 99
... ... ...
.
I
.
... ...
h
= diameter of particle, it. = diffusivity, (sq. ft.)/(hr.) = heat transfer coefficient. B.t.u./
k
= thermal conductivity, B.t.u./(hr.)
I
= one-half mean distance between
(hr.)(sq. ft.)(" F.) (ft.)(" F.)
M n
2.60 4.13
$'
0.0042 0.112 0.0125
r
4.4
x
q
R 10-5
T V dimensionlessterm, ( p d - p,)/P, characterizes the effect of bulk flow on mass transfer within the gas phase. As long as wh/4na2cuq and wC/4sak are small with respect to unity, the transient equations (Equations 25, 26, and 30) derived for nonvolatile particles are applicable for estimation of the temperature of volatile droplets. As wX/4ira2aq approaches unity, the transient equations should still yield a good approximation for the temperature difference if multiplied by a mean value of 1 - wX/4aa2cuq. When wX/4iraaaq exceeds unity the droplet will assume a temperature below the ambient temperature and will fall in temperature as the sensible heat of the air becomes controlling. In all cases, the more volatile is the material, the lower the temperature. For a given droplet temperature, the rate of evaporation, zu, is proportional to the radius of the droplet and w C / 4 ~ a k is independent of size of the droplet. The absorptivity increases with radius, hence wX/47raacuq decreases more rapidly than the reciprocal of the radius. A decrease in radius due to evaporation decreases the rate of absorption but increases the fraction of the absorbed energy converted to latent heat; the temperature rising if wX/4aa2aq is greater than unity and falling if less than unity. In Table I1 the evaporation terms are evaluated a t pseudo-equilibrium for fog oil and dodecane droplets in a n isotropic flux of 63,000 B.t.u./(hr.)(sq. ft.) and with a n ambient temperature of 68' F. The effect of evaporation on the temperature of the 1-micron droplets of fog oil is completely negligible. I n fact, a 500-fold change in vapor pressure is possible before the effect of evaporation on the temperature difference exceeds 10%. Evaporation decreases the temperature of the more volatile, 100-micron droplets of dodecane 3' F. at pseudo-equilibrium and will become increasingly important as the air is heated and the temperature of the droplet again rises. Surface tension does not
1 824
increase the vapor pressure by 10% until the diameter of the droplet is reduced to 0.1 micron for fog oil and 0.05 micron for dodecane. These calculations and analysis indicate that volatile droplets as well as nonvolatile particles below the sizes considered cannot be heated radiantly to the ignition temperature without first heating the associated air, the temperature rise increasing with radius and absorptivity and decreasing with volatility.
w
a
9 6
X p u p
Subscripts
A B
= volatile component = bulk
c
= center of particle = particle = mean = initial = surface
d
Conclusions
m
Reasonably accurate equations are presented for the transient temperature of a dispersion of small particles exposed to a radiant flux. Temperature differences within the particles are generally negligible. The temperature of nonvolatile droplets rises swiftly to a pseudo-equilibrium value and finally rises more slowly as the finite amount of air associated with each particle is heated by conduction from the particle. Evaporation decreases the temperature of volatile droplets throughout the process and for sufficiently volatile materials the temperature may remain below the ambient temperature. I n any event, the radiant energy absorbed by small particles is transferred swiftly to the vapor phase by conduction or evaporation and it is not possible to heat the particles to their ignition point without first raising the temperature of the surrounding air.
o s
Acknowledgment
This work was done in part through the Engineering Research Institute of the University of Michigan under contract with the Army Chemical Corps, Washington 25, D. C. J. H. Chin and C. M. Chu made invaluable suggestions and B. K. Larkin assisted in the calculations. Nomenclature a
= radius of particle, ft.
C
=
INDUSTRIAL AND ENGINEERING CHEMISTRY
heat capacity, B.t.u./(lb.)(" F.)
particles, ft. weight of particle material, lb./lb.-mole = particle concentration, number per cu. ft. = partial pressure, atmospheres = total pressure, atmospheres = radiant flux, B.t.u./(hr.)(sq. ft.) = radial position, ft. = universal gas constant, (atm.) (cu. ft.) (1b.-mole) ( O Rankine) = temperature, O R. or ' F. = velocity of air with respect to particle, ft./hr. = rate of evaporation, Ib./hr. = absorptivity = n&l/3irC = time, hr. = latent heat of vaporization, B.t.u./lb. = density, lb./(cu. ft.) = surface tension, (Ib. force)/(ft.) = viscosity, lb./(ft,)(hr.) = molecular
Operators
d:
= inverse Laplace transform of = Laplace transform of
literature Cited
(1) Berlad, A. L., Hibbard, R. R., Nati. Advisory Comm. Aeronaut. R M E52109, November 13, 1952. (2) Carslaw, H. S., Jaeger, J. C . , "Conduction of Heat in Solids," Oxford Univ. Press, London, 1947. (3) Zbid., p. 203. (4) Chem. Eng. News 33, 833, 948, 2592 (1955 ). (5) GAosh, B., Orning, A. A., IND. ENG. CHEM.47, 117-21 (1955). (6) Godsave, G. A. E., "Combustion," Fourth International Symposium, D. 818. Williams & Wilkins. Baltimore, hd., 1953. (7) McAdams, W. H., 6'Heat Transmission," p. 266, McGraw-Hill, New York, 1954. (8) Nusselt, W., Z.Ver. deut. Ing. 68, 1248 ( 1 924). (9) Omori, T. T., Orning, A. A,, Trans. Am. SOC.Mech. Engrs. 72, 591-7 (1950). (10) Penner, S. S., Weinbaum, S., J . Opt. Sac. Amer. 38. 599-603 (1948). (11) Thomas, P. H.,' Brit. J. A&. Phys. 3, 385-93 (1952). (12) Traustel, S., Feuerungstech. 29, 1-6, 25-31,49-60 (1941). RECEIVED for review November 21, 1955 ACCEPTEDMarch 23, 1956 Division of Industrial and Engineering Chemistry, 129th Meeting, ACS, Dallas, Tex., April 8-13, 1956.
