1634
MORTON H. FRIEDMAN AND HILBERT J. UNGER
From Figure 1 it can be seen that the range of the image forces can be large in dilute solutions and decreases rapidly for more concentrated solutions. This is due to the more effective screening of the ionic charge by its ionic atmosphere in concentrated solutions. The concentration of electrolyte rises sharply in the immediate neighborhood of the metallic surface, and for an ion with a distance of closest approach of 2.0-2.5 A, its concentration at the OHP is almost twice its value in the bulk of the solution up to solution concentrations of M . I n certain cases, this effect might prove to be of importance in the study of electrode kinetics in the absence of a supporting electrolyte. The effect of this rapid increase in concentration near the electrode on the surface excess and hence on the interfacial tension can be seen in Figure 2. The concentration a t which the value of the surface excess changes sign, and hence the interfacial tension curves show a minimum, is shifted to the more dilute concentrations. For example, the position of the minimum in the interfacial tension changes from 0.05 to 0.001 M when the position of the OHP is
shifted from 2 to 4 A. Also, the decrease in the interfacial tension values due to image forces is much less. (The maximum decrease is 0.02 dyn/cm for 2 and 0.0002 dyn/cm for 4 A,) Two concurring effects produce this large change: (1) an increase of over-all adsorption due to imaging in the metal and (2) an increase of the electrolyte desorption due to the presence of an ion-free layer at the metal surface given by the term xOCsalt in eq 6 . The largest expected decrease of the interfacial tension at the ECM for a 1:1 electrolyte in the absence of specific adsorption, when the concentration is varied from to 5 X M , would be of the order of 0.02 dyn/cm. If this were the case, it should be possible to measure this change with a sufficiently accurate differential electrometer. Acknowledgments. The author wishes to thank Dr. J. T. D'Alessio and the CNEA for providing working facilities and the Consejo Nacional de Investigaciones, Buenos i 'res, Argentina, for financial support during part of this work.
Light-Induced Boiling of Blackened Liquidsla by Morton H. Friedmanlb and Hilbert J. UngerlO Applied Physics Laboratory, The Johns Hopkina University, Silver Spring, Maryland
2091 0
(Received June 27, 1968)
The differential equations which describe the flash heating process are formulated and solved analytically for the case of a spherical absorber surrounded by a transparent medium and subjected omnidirectionally to an exponentially decaying heat flux. Heat is transferred by conduction M ithin the absorber and its surroundings. The solution is used to study the variation in time and space of the temperature of the surroundings. The dependence of the peak temperature at the absorber surface on the 1 roperties of the absorber, the absorber particle size, and the time constant of the flash are elucidated and tke existence of an optimum particle size, which maximizes this temperature, is predicted. The above soluti, n is confirmed experimentally by using the radiant energy from a xenon flashtube to boil liquids in which are suspended carbon particles of a known size. The conditions required just to cause the onset cf boiling are found and are presented as a plot of Rashtube discharge voltage us. absorber particle size. The data confirm the theoretical prediction that there is an optimum absorber particle size. The intensity and duration of the radiant output of the flashtube are measured and used with the analytical solution to construct theoretical curves of the threshold discharge voltage us. particle size. The agreement between the theoretical and experimental curves is good. The radiant output of capacitor discharge lamps has been used for many years in physical chemistry research. The effects of this output can be photochemical or thermal. The processes corresponding to these two effects of flashlamp radiation are flash photolysis and flash heating. This paper deals with the latter process. I n flash heating, two classes of sample response can be identified. The primary response is that of the flashed absorber. If the absorber is surrounded by a transThe Journal of Physical Chemistry
parent medium of low thermal conductivity, or by a vacuum, the absorber can reach temperatures so high that reradiation becomes important. In this mode, flash heating has been used to melt and evaporate metals and carbon,2'9 pyrolyze c0aI,~-6 and induce reac(1) (a) This work was supported by the Department of the Navy under Contract NOW 62-0604-c. (b) Principal Engineer, Research Center. (c) Senior Physicist, Aeronautics Division.
LIGHT-INDUCED BOILINGOF BLACKENED LIQUIDS tions between the absorber and its surrounding^.^ The secondary response of the system is that caused by conduction heating of the material adjacent to the absorber. Conduction from an irradiated absorber has been used to heat6 and pyrolyze2&gases, initiate gaseous mixtures, 2a pyrolyze liquids,2a and d e c ~ m p o s e ~ ~and > ’ -va~ porizel0solids. If the full potential of flash heating in physical chemical research is to be realized, an analytical description of the temperature history in the absorber and its immediate neighborhood must be developed. I n the experiments referenced above, the absorber was in the form of either powder, filaments or wool, or foils; that is, all three one-dimensional geometries are of interest. This paper treats only spherical geometry; the corresponding cylindrical and plane-parallel problems can be solved similarly. Assuming that the thermophysical properties of the absorber and its surroundings are temperature-independent, a satisfactory analysis of the flash heating process must include the following features. (1) Temperature gradients within the particle should be included, if the analysis is to be correct when the “thermal relaxation time” is comparable to or exceeds the flash duration. The duration of a typical flashtube light pulse is 300 psec; this is the thermal relaxation time of a carbon particle 40 pm in diameter. Even when the flash duration is long or small particles are used, the inclusion of conduction within the particle always provides a more accurate solution than is obtained by assuming the particle temperature to be uniform. Temperature gradients in the absorber must be included in any event if the thermal history during the early portion of the discharge is to be followed. (2) The shape of the incident flux waveform should correspond at least grossly with the waveform of the experimental device which is to be modeled. (3) Heat transfer from the absorber to its immediate surroundings must be included if the solution is to be highly versatile. (4) Similarly, reradiation from the absorber should be included. Approximate solutions to the flash heating problem have been reported by Nelson and Lundberg2&I1land by Burkig.O Nelson and Lundberg characterized the absorber temperature by a single value and computed its magnitude from an integrated heat balance, which included reradiation and a simplified representation of the heat transferred t o the surroundings. Burkig assumed that the particle temperature was uniform and included reradiation and an arbitrary incident flux in her analysis. However, her treatment 6f the flux from the particle to the surroundings was based on the unsupported assumption that the particle is in effect surrounded by an adiabatic cell in which the temperature profile is quasi-steady and is determined by the particle temperature. Neither of the above analyses is entirely satisfactory. The principal difficulty encountered in trying to de-
1635 velop a more accurate solution is that when reradiation is included in the model, a complicating nonlinearity is consequently introduced into the differential equations. However, in many applications of flash heating, the thermal properties of the surroundings are such that the absorber never reaches a temperature at which reradiation becomes important. This is generally so when the surroundings are condensed. The purpose of this paper is to derive and verify experimentally a general expression for the variation in space and time of the temperature of the medium surrounding an irradiated spherical absorber, under conditions such that reradiation is negligible. The first part of this paper describes the heat transfer problem, its solution, and the properties of the solution. Heat is transferred from the absorber only by conduction. The absorber is spherical, is surrounded by an inert medium, and is subjected omnidirectionally to an exponentially decaying heat flux. The second part of the paper describes an experimental program in which liquid suspensions of carbon particles were flash irradiated and the variation with absorber particle size of the threshold energies required to cause the liquids to boil was found. Last, the experimental results are compared with the present and earlier models of the flash heating process.
Theory Heat Transfer Solution. Consider a gray absorber (subscript a) and its surroundings (subscript s) in intimate contact asd initially a t a relative temperature us, = v, = 0. The surface of the absorber is subjected to an omnidirectional flux Q‘ = Q’o exp( - Xt),12 where &’o is the initial irradiance of the source, X - l is the time constant of the flux, and t is time. Heat is conducted (2) (a) L. 6. Nelson and J. L. Lundberg, J . Phys. Chem., 63, 433 (1969); (b) L. S. Nelson and N. A. Kuebler, Rev. Sci. Instr., 34, 806 (1963). (3) L. 8. Nelson and N. L. Richardson, J . Phys. Chem., 68, 1268 (1964). (4) C. 0. Hawk, et al., “Flash Irradiation of Coal,” Report of Investigations, No. 6264, U. 9. Bureau of Mines, Pittsburgh, Pa., 1963. (6) E. Rau and L. Seglin, Fuel, 43, 147 (1964). (6) V. C. Burkig, “Thermal Absorption in Seeded Gases,’’ NASA CR-811, June 1967. (7) L. S. Nelson and N. A. Kuebler, J . Chem. Phys., 37, 47 (1962). (8) R. V. Petrella and T. L. Spink, ibid., 47, 1488 (1967). (9) G. L. Pellett and A. R. Saunders, “Heterogeneous Decomposition of Ammonium Perchlorate-Catalyst Mixtures using Pulsed Laser Mass Spectrometry,” presented at the AIAA 6th Aerospace Sciences Meeting, New York, N. Y . , Jan 1968. (10) K. A. Lincoln, Anal. Chem., 37, 641 (1965). (11) J. L. Lundberg, et al., Proc. Conf. Carbon, 3rd, Buffalo, 1967, 411 (1959). (12) The flux waveform used in the theoretical analysis clearly deviates from the output of a real flashtube at short times. When low inductance discharge circuits are used, the rise time of the light pulse is only a small fraction of the decay time and the light output decays more or less exponentially during more than SO% of the flash duration. The waveform Q = Qat exp( - A t ) was found to give no better fit to the experimental traces. Volume 79, Number 6 June 1969
MORTON H. FRIEDMAN AND HILBERT J. UNGER
1636 from the surface of the absorber to its interior and to the surroundings. The temperatures v(r, t ) , where r is radial distance from the center of the absorber, are given by
Table I : Thermal Properties of Absorbers and Surroundings Thermal conductivity, cal/sec cm deg
Absorbers 0.012 0.36
Carbon Nominal metal Nominal liquid Monochlorobenzene Glycerin
with the initial and boundary conditions
v&, 0) = 0, 0 _< r 5 a
Thermal diffusivity, cmE/sec
0.015
0.24
Surroundings 0.0006
0.0006 0.001008 0.000932
0,000343 0.000677
vs(r, 0 ) = 0, r 2 a
v,(a, t )
=
vs(a, t )
lim vs(r, t ) = 0, v,(O, t ) e
i < 0.8
3Ji Y
YI
V 3
0.6
st
3w 0.4 I-
0.2 I
2.!i
3.0
I
I
3.5 CONDENSER VOLTAGE, (kV)'
4.0
Figure 6. Transducer response for various values of absorber particle size and discharg? voltage for carbon in monochlorobenzene. Particle diameters: *, 3-5 pm; A, 5-10 pm (two sets, A and A ); U, 10-20 pm (averaged data); m, 20-30 pm (averaged data); X, 30-63 pm.
(16) N. A. Kuebler and L. S. Nelson, J . Opt. SOC.Am., 51, 1411 (1961). (17) J. H.Goncz and E'. B.Newell, ibid., 56,87 (1966).
Volume 79, Number 6 June 1969
1640 applied because even in the absence of boiling, signals greater than the blank were observed as a consequence of the increased opacity of the sample when carbon was added. Two sets of data are presented for suspensions of 5-10 pm carbon in monochlorobenzene. Though the slopes of the regression lines differ, reflecting differing carbon loadings, the lines converge to give very similar values of Vb.
Results and Discussion Theoretical curves of V b us. 2a for carbon in monochlorobenzene and glycerine were constructed from eq 2 and 5, with vim = 2)b, and using experimental E ( V ) and te(V) data obtained as described above. These curves are plotted in Figure 7 and are labeled “THEORY.” The abscissa is particle diameter. Reradiation from the absorber is negligible under the conditions of these experiments. The experimental values of v b computed from the transducer data are presented in Figure 7 as a series of the horizontal lines, each extending from a’min to dmPx; ordinates are the threshold voltages corresponding to each size range, obtained in most cases as an rms average of the Vb’s of several samples from the same cut. Error bars, of length equal to twice the standard error of a single measurement, are indicated on the figure. Where no error bars are given, the value of lib was found by lumping all of the data obtained with a single sample subjected to replicated voltages in random sequence. The value of Vb obtained using the 10 X 10 pm cylinders is also given in Figure 7 . The abscissa, 10 pm, is the diameter of a sphere having the same surface-to-volume ratio as the cylinders. The experimental data confirm the theoretical prediction that there is an “optimum” absorber particle size which minimizes the voltage requirement. The value of this optimum size is rather well predicted by the theory and indeed the threshold voltages are in satisfactory agreement. The shape of the theoretical curves derived from of the flash heating process can be earlier analyses2*~6~11 deduced from Figure 2. When the absorber particles are small, these curves will be close to those in Figure 7. As the particle size increases, they will fall below the present curves, only to rise again and cross them when the particle size is much greater than a*. Based on the carbon-in-liquid calculations, the values of VI,(a*) predicted by the Kelson-Lundberg and Burkig models are probably between 50 and 70% of those predicted here. It should be remarked that the experimental results cannot be interpreted in terms of the theory unless the experimental suspensions are dilute. In the present work, v b was found to be independent of particle concentration in the range studied, and the suspensions were visibly transparent. If t,he suspension is too highly concentrated, particles will interact by shadowing one another and the temperature fields associated with each The Journal of Physical Chemistry
MORTON H. FRIEDMAN AND HILBERT J. UNGER particle will no longer satisfy in approximation the first boundary condition, eq If. The experimental values of Vb are consistently higher than the theoretical ones. This discrepancy can be attributed to differencesbetween the experimental setup and the mathematical model (for instance, the absorber particles are not spherical) and/or to an imperfect interpretation of the S ( V ) data. The value of Vb obtained using the carbon cylinders is less than the other experimental values, but it too exceeds the corresponding theoretical value. The first of these observations is probably a consequence of the size uniformity of the cylindrical particles, whose surface-to-volume ratio is close to that of a sphere of radius a*. The 5-10 and 10-20 pm cuts also contain particles of near-optimum size, but these are diluted with other particles whose individual Vb’s exceed Vt, (a*).
t
2s5
ABSORBER PARTICLE DIAUETER, pn
Figure 7. Theoretical and experimental threshold discharge voltages for boiling: 0 , 10 X 10 pm cylinders.
Even when samples containing particles of uniform size are used, the experimental threshold voltage exceeds the theoretical value. This difference between the theoretical and experimental results is explained if the absorber surface temperature must exceed the boiling point of the liquid for some short time, if detectable boiling is to result. As the duration of the heat pulse is very short, it is reasonable to expect that higher superheats are required for boiling to occur in the available time than would be the case if the heating were steady. The discrepancy is also explained if some small critical mass of the liquid must be heated to temperatures above the normal boiling point. If the sense of this interpretation is correct, the discrepancies in Figure 7 reflect not so much inadequacies in the model but rather a failing in the experimental thermometry. I n any event, the agreement between theory and experiment shown by Figure 7 indicates that the theoretical analysis can be used to design and interpret further flash irradiation experiments.
1641
LIGHT-INDUCED BOILINGOF BLACKENED LIQUIDS
Consider the function F ( r , 1) = d/dt (extv,). Then F - A), since us(?, 0) = 0. Equation A1 can be written as G(r, p) = f(p)/(p A), so F = f(p - A). Now construct the inversion integral for F
p = X+iY
f
/
= p%(r,p
X
+
where ( q ’ J 2 = (p - A)/K,. F meets all the requirements for the validity of eq A2 except at r = a, because F‘ is not well behaved at (a, 0). However, for t > 0 v,(a, t ) = lim vs(r, t ) . The integrand in eq A2 has a r-a
branch point at p = A and no poles. Thus the contour shown in Figure 8 may be used. The integral around p = A is zero and the integrals along the real axis give rn Figure 8. Integration contour for F
Appendix. Derivation of the Heat Transfer Solution I n this section, eq l a and l b are used with boundary and initial conditions lc-lg to find us(r, t ) . The substitution ui= rui is made to simplify eq l a and lb. The Laplace transform is applied to eq l a and lb, giving two simultaneous ordinary differential equations in the transformed variables a, and zis. These equations are solved with the transforms of eq lc-lg t o give, for the surroundings
&(r, p)
21,
= - =
+
- a ) ] ) du
(A3)
where R = K , - k,(l - uKa COS (uKa)),I = k,au, and K = (Ks/Ka)l’a, From the definition of F and eq Id t
v,(r, t ) = e - ” J F
dt
0
From eq A3 and A4, us(r, t) can be written as a double integral, over u and then over 2. The order of integration can be interchanged; the integration over t gives
r
Q0az
(P
+
exp[t(A - KSu2)]u{Rsin [u(r - a ) ] I cos [u(r R2 I 2
exp(q,a
- qsr)
+ A) [ks(aqs + 1) - k ( 1 - qaa coth (naa)) IT
(-41)
where qt2 = p/Ki and p is the Laplace variable. Bc c tuse of the form of the flux function, the solution will be valid everywhere but at (a, 0). The inversion integral derived from eq A1 has a pole at p = - A and a branch point at the origin and is not amenable t o the usual evaluation procedures. Hence, the following approach is used.
[exp(-K,u2t)
- exp(-At)]u{Rsin
[u(r - a ) ]
+
Using the nondimensional variables w = ua, 4 = r/a, A = a2A/Ks, T = K,t/a2, IC = k,/k,, and 0 = rC,v,/ (Qoa), eq A5 becomes eq 2.
Volume. 78, Number 6 June 1960
