284
Ind. Eng. Chem. Fundam. 1984, 23, 294-298
Brown. H. C.; Murphey. W. A. J. Am. Chem. SOC. 1951, 73,3308. Crynes, B. L.; Albright, L. F. Znd. Eng. Chem. Process Des. Dev. 1969, 8 , 25. Doue, F.; Guiochon, G. J. Phys. Chem. I968a, 73,2804. Doue, F.; Guiochon, G. J. Chim. fhys. I968b, 65,395. Doue, F.; Guiochon, G. Can. J. Chem. 1969, 47,3477. Fabuss. B. M.; Smith, J. 0.; SatterfieM, C. N. "Advances in Petroleum Chemistry Refining"; Interscience: New York, 1964a. Fabuss, B. M.; Kafesjain, R.; SmRh, J. 0. Znd. Eng. Chem. Process Des. Dev. 1964b, 3 , 248. Fanter, D. L.; Walker, J. Q.; Wolf, C. J. Anal. Chem. 1968, 4 0 , 2189. Fenton, D. M.; Hennlg, H.; Richardson, R. L. "Oil Shale, Tar Sands and Related Materials"; ACS Symposium Series No. 163, Washington, DC, 1981; Chapter 21. Gary, J. H.; Handwerk, G. E. "Petroleum Refining"; Marcell Dekker: New Ynrk . -. .., 1979 .- . - ,. Gillespie, H. M.; Gowenlock, B. G.. Johnson, C. A. F. J. Chem. SOC.,Perkin Trans. 2 1979, 317. Gowenlock, B. G. h g r . React. Kinet. 1966, 3 , 1971. Hazlett, R. N. "Frontiers of Free Radical Chemistry"; Academic Press: New York. - , 1980. ... Hazlett, R. N.; Beai E. "Geochemistry and Chemistry of Oil Shales"; ACS Symposium Series No. 230, Washington, DC, 1983; Chapter 19. Holmes, S. A.; Thompson, L. F. Fuel 1983, 62, 709. Kunzru, D.; Shah, Y. T.; Stuart, E. B. Znd. Eng. Chem. Process Des. Dev. 1972, 11, 605.
Layokun, S. K.; Slater, D. H. Znd. Eng. Chem. Process D e s . D e v . 1979, 18, 232. Marschner, R. F. Znd. Eng. Chem. 1936, 30,554. Miller, D. B. Znd. Eng. Chem. Rod. Res. D e v . 1963, 2, 221. Mushrush, G. W.; Hazlett, R . N. Naval Research Lab. Rept. 8630, Washington, DC, 1982. O'Neal. H. E.: Benson. S. W. "Free Radicals"; Wiley: New York, 1973; Vol. 11. Pease, R. N.; Morton, J. M. J. Am. Chem. SOC. l9S3, 55,3190. Ranzi, E.;Dente, M.; Pierucci, S.; Biardi, G. Znd. Eng. Chem. Fundam. 1983, 22, 132. Rebick, C. "Thermal Hydrocarbon Chemlstry"; ACS Symposium Series No. 183, Washington, DC, 1979; Chapter 1. Rebick, C. "Pyrolysis: Theory and Industrial Practice"; Academic Press: New York, 1983; Chapter 4. Rice, R. 0. J. Am. Chem. SOC. 1933, 55, 3035. Smith, R. D. Combust. Flame 1979, 35, 179. Soiash, J.; Hazlett, R. N.; Hall, J. M.; Nowack, C. J. Fuel 1978. 57, 521. Voge. H. H.; Good,G. M. J. Am. Chem. SOC. 1949, 7 1 , 593.
Received f o r review May 2, 1983 Revised manuscript received November 30, 1983 Accepted January 24, 1984
Heat Transfer between Particles in Packed Beds Sunder M. Rao and Herbert L. Toor' Depatfment of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 152 13
A bed of highly conducting particles which normally behaves like a continuum will not do so if the scale of the boundary conditions is comparable to the size-scale of the particles. Thus the measured steady-state heat flux between a sphere and a surrounding bed of similar sized ceramic or metal spheres is less than half the continuum value and the transient cooling time is approximately 2 to 4 times the continuum value. An elementary discrete model is in satisfactory agreement with measurements.
Introduction Since the work of Maxwell (1904), the complex heat transfer which takes place in a packed bed of particles has been described by an effective conductivity which is usually measured by determining the steady-state heat flux between two parallel planes separated by a large number of particles. This effective conductivity is predicted very well by models which account for the fine scale conduction and (if necessary) radiation in the bed (Schumann and Voss, 1934; Deissler and Boegli, 1958; Hill and Wilhelm, 1959; Schotte, 1960; Kunii and Smith, 1960; Ofuchi and Kunii, 1965; Crane et al., 1977). The concept of effective conductivity assumes that on the large scale (scale of the bed size) the bed behaves as a continuum with an effective conductivity which is a bed property, so for a static bed Fourier's law holds on the large scale
Here t is the average temperature in a volume of bed large enough to give a suitable average but small compared to the bed scale (which implies that the temperature and flux change over a particle must be small). The effective conductivity concept is meaningful and eq 1 is useful so
long as k , is a bed property, which means that ke may only depend upon bed materials, structure, and temperature. Here we discuss a class of practical problems where eq 1 fails. The effective conductivity is not a bed property. These problems may arise in direct contact heat transfer for which one mixes together highly conducting hot and cold particles. The simplest demonstration of the failure occurs when a single hot particle is present in a bed of cold particles. Experimental Section We constructed two different test particles: one was a 6.00 mm diameter cement sphere with a 0.6-ohm nichrome wire coil and a thermocouple imbedded in it. The other was a 6.35 mm diameter aluminum sphere with a 0.95-ohm nichrome wire coil imbedded in a 1.5 mm diameter cement filled hole through the axis and a 36-gauge thermocouple imbedded with cement in a 1.0 mm diameter hole. The resistors were connected to a dc power supply through leads designed to minimize heat losses. In an experiment we suspended a test particle in a 16.5 cm diameter spherical flask immersed in a water bath and filled the flask with bed particles. Thermocouples were placed in the bed, some bare and some imbedded in bed particles. Steady-state data were obtained by measuring temperatures and power consumption of the resistor with a con-
0196-4313/84/1023-0294$01.50/0@ 1984 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 23, No. 3, 1984 295
Table I. Steady-State Data with Fine Powder Beds test particle 6-mm cement 6.35 -mm
I .Ot
sphere bed
aluminum sphere
15-20 GPT, -60 mesh
-200 mesh
combusted shale alumina powder powder
test particle = 4 1 "C temperature boundary temperature = 2 4 " C bed particle size =60 pm
t: . 4 u \
void fraction of bed 'drN
h measured I-
k, measured 5
IO 15 DIMENSIONLESS RADIUS = r / r o
20
(eq 3 and 4)
25
k, from Kunii and Smith (1960)
Figure 1. Steady-statetemperature profiles. Experimental data for powder bed ( 0 )cement test particle with alumina powder bed (e = 0.51); (0)6.35" aluminum test sphere with alumina powder bed (e = 0.33); (A)cement test sphere 6-mm diameter with combusted ahale powder bed (e = 0.60); (m) aluminum test sphere 6.35 mm with alumina powder bed (e = 0.44). Experimental data for large particle beds: ( 0 )6.35-mm aluminum test sphere with 6.35" aluminum sphere bed; (A) 6-mm cement test sphere with 6.35-mm alumina satellites bed,curve a: continuum prediction, eq 2; curve b discrete model prediction, eq 8.
N u measured
stant power input after temperatures became steady, while transient data were obtained by rapidly heating the test particle for 3-5 s and following ita temperature as it cooled. Curve a in Figure 1is the steady-state solution to eq 1 for a constant value of k,/pC,
test particle
7-0 -
-ro
(4)
The values of k, obtained from eq 3 and 4 compare favorably with predicted values (Table I). We now replace the powders by a random packing of 6.35 mm diameter aluminum spheres, using the heated 6.35 mm diameter aluminum sphere as the test particle. The measured temperatures are shown in Figure 1. The internal bed temperatures were obtained by placing three of the 6.35" aluminum spheres which had thermocouplea imbedded in them in a horizontal row adjacent to the test particle and resting on the half-filled bed of spheres, and then filling the rest of the flask with additional spheres. We also repeated the above, using the cement test particle and a surrounding bed of alumina satellites (spheres with slightly projecting bands at the seams) and 6.35-mm cement spheres with imbedded thermocouples. The measured temperatures are also shown in Figure 1. Each point is the average of three separate experiments
satellite
aluminum sphere 6.35." aluminum spheres
spheres - 4 1 "C
-37 "C
3
= 2 1 "C =0.405 9
sphere bed
6.35-mm alumina
temperature boundary temperature = 24 "C void fraction of bed 0.41 number of packing
states h measured
N u measured N u from eq 13
(3)
=21°C = 2 7 fim mean size 0.44 0.0385 58.3 f 2.3 kcal/h mz K 0.177 kcal/h mK 0.194 kcal/h mK 1.91 i 0.08
Table 11. Steady-State Data with Large Particle Beds test particle 6-mm cement 6.35-mm
k, from Kunii and Smith (1960)
The data for the two test spheres and two powders (shale and alumina) fall on the curve. These systems behave as a continuum. The solution to eq 1also gives the steadystate heat flux as
mean size 0.6 0.0364 26 i 0.8 kcal/h mz K 0.077 kcall h mK 0.081 kcal/h mK 1.94 i 0.06
=37"C
f 3.2 kcallh mz K
45.2
0.316 kcal/h mK 0.85 i 0.06 0.95
60.4 k 5.4 kcal/h mz K 0.450 kcal/h mK 0.84 i 0.08
0.95
which differed from each other by less than 3%. The measured temperature profiles do not conform to eq 2, for the temperature falls off more steeply than predicted. The only way eq 1can fit the data is for k, to be made a function of position. But a position-dependent k, is acceptable only if the bed structure or thermal properties are position dependent, and this is not the case. If we heat a different sphere in the same bed, the apparent k, at a point in the bed will change. So an attempt to treat this bed as a continuum leads to the illogical result that k, is a position-dependent function of the boundary conditions. The values of h from eq 3 are given in Table 11and using k, from the model given by Kunii and Smith (1960) (which would presumably be measured in a rectilinear experiment), gives the measured Nusselt numbers in Table 11. These values are less than half of the value of 2.08 predicted by eq 4 for a continuum (and fairly closely realized with powders). Thus, for a given temperature difference the heat flux in beds of particles equal in size to the test particle is less than half the continuum value. An explanation for the lack of continuum behavior is as follows: if the bed did behave as a continuum the temperature profile would be given by eq 2, which says that near the test particle 2/3 of the total temperature drop takes place over a distance of one test particle diameter; the flux change would be ninefold, if the bed particle diameter equals the test particle diameter, as in these experiments. This violates the implicit continuum assumption that the temperature change over any one particle is small. Under these conditions the bed cannot be assumed
298
Ind. Eng. Chem. Fundam., Vol. 23,No. 3, 1984 0t
c;
'
0
\
\
l
,
\
l
h
.
,
o
I
I
l
I
I
1
l
I 2 24 36 48 60 72 84 96 I08 120 I32 144 I56 TIME i S E C i
Figure 2. Transient data. Test particle: 6-mm cement sphere. Experimental data: (0) -60 mesh combusted shale bed ( e = 0.60); (0) 6.35-mm alumina satellites bed ( e = 0.41); curve a: continuum model, shale powder bed (k, = 0.077 kcal/h m K); curve b: discrete model, alumina bed; curve c: continuum model, alumina bed (k,= 0.132 kcal/h m K); curve d: continuum model, alumina bed ( k , = 0.316 kcal/h m K by Kunii model); to@)and toare the temperatures of test particle at time 6 and at time 0, respectively; t~ denotes temperature of surroundings; for these experiments to and t N were 40 "C and 24 "C, respectively.
to be homogeneous. The observed behavior reflects the discreteness of the bed and the proper scaling parameter is the radius of the test particle. For the powders tested rO/rb= 250, rb denoting bed particle radius, and the bed behaves as a continuum, while for the spheres and satellites which showed noncontinuum behavior ro/rb 1. Undoubtedly, with a large enough test particle, a bed made up of large particles would also behave as a continuum, just as it would if the particles were contained between two flat surfaces, while a powder bed surrounding a heated particle of powder would not behave as a continuum. Figures 2 and 3 include transient data for the powder beds (measured dimensionless test particle temperature vs. time). The data cloeely follow the solution to eq 1given by Konopliv and Sparrow (1970) in which we use the values for k, obtained from the steady-state experiments, measured bulk densities and test particle mass, and reported values of the solid heat capacitites. This confirms the steady-state result that the powders behave as a continuum with a large test particle. Transient data for the bed of aluminum spheres are also shown in Figure 3. In the transient experiments there are no thermocouples in the bed. The test particle is suspended in the center of the flask and the aluminum spheres are poured into the flask. The points represent the average of nine separate fillings, Here we cannot obtain a reliable value of k, from the steady-state data since k , is not a bed property, but we ignore this and use eq 3 and 4 to obtain a value of k, which is used in the solution to eq 1 to give curve c. Since that value differs from that given by the model of Kunii and Smith (1960), we also use the value of k , from that model to obtain curve d. Clearly the large aluminum particles surrounding a test particle of the same size do not behave as a continuum, either in the steady or transient state, and eq 1fails. We draw the same conclusions when we use an alumina satellite bed with a cement test particle (Figure
TIME ( S E C I
Figure 3. Transient data. Test particle: 6.35-mm diameter aluminum. Experimental data: (0) alumina powder bed (c = 0.44);( 0 ) 6.35-mm aluminum sphere bed (c = 0.4); curve a: continuum model, alumina powder bed (It, = 0.177 kcal/h m K); curve b: discrete model, 6.35-mm aluminum sphere bed; curve c: continuum model, aluminum sphere bed (k, = 0.185 kcal/h m K; curve d: continuum model, aluminum sphere bed (k, = 0.45 kcal/h m K by Kunii model); nomenclature, same as Figure 2; toand tN values were 37 and 21 "C, respectively.
/ Outer Boundary R a d i u s r~ Temp IN
v
\Bed Particle, Radius r b Test Particle, Radius ro , Temp. 1,
Figure 4. Discrete model.
2). We observe that the time it takes a particle to cool any given amount is about 2 to 4 times as long as that predicted by eq 1. We have done other experiments which agree with the above conclusion that heat transfer from a highly conducting particle follows continuum behavior only if the particle is surrounded by much smaller particles. (We expect a wire in a bed of particles to behave in a similar manner.) When heat transfer is taking place between particles of similar size, the discrete nature of the system must be taken into account. Discrete Model. As the heat flux from the central particle is radial, we model the system as a central sphere surrounded by spherical shells of identical particles, as shown in Figure 4. Since the thermal conductivity of the solid is much greater than that of air, in the absence of radiation, heat transfer is controlled by conduction through the air gaps between contacting particles, primarily adjacent to the points of contact. We assume all particles in layer i to be at the uniform temperature ti. Heat transfer between adjacent layers is through contacts between particles described by a local
Ind. Eng. Chem. Fundam., Vol. 23, No. 3, 1984 297
coefficient, hl, which we base on projected area. In the steady state 4rro2h(t0 - tN) = hlo rrb2no(to - t,) = hlrrb%i(ti - ti+J (i = 1, 2,
...,N - 1) (5)
Although heat transfer between contacting particles depends somewhat upon orientation (Kunii and Smith, 1960), we take hl as independent of orientation and hence layer i. Equation 5 gives
and as rO/rbincreases without limit the Nusselt number reaches the continuum limit of 2/[1- (ro/rN)].When all the particles are the same size and highly conducting, h,, = hl and eq 12 becomes
2mh
1
hl can be independently evaluated by considering the conduction between two contacting spheres in air (Kunii and Smith, 1960) as
and
(7)
-
Equation 6 properly reduces to eq 2 as rb/ro 0 and when all the particles are the same size and highly conducting, hlo = hl and eq 6 reduces to N-1 1 ti -
to
t~
- tN
7 LI
nj (i = 1, 2, El 1 &I=O nj j=i
=-
..e,
N - 1)
(8)
The number of effective contacts for the first shell is the same as the coordination number which is 6 in a randomly packed bed of equal-sized spheres (Haughey and Beveridge, 1969; Powell, 1980). Hence we take no as 6 in the aluminum sphere and alumina satellite experiments. To evaluate the other ni values, we assume that the number of contacts which are effective in transferring heat between two adjacent shells is proportional to the number of particles present. We estimate the number present by assuming the void fraction of any shell to equal the bed void fraction and compute the number of particles in a shell of thickness 2rb which is centered on the outside of the ith shell. The result is
ni = a(1- e)[ 6(:
+2 i ) +21
(9)
We relate hl to ke as measured in a rectilinear system by ke hln'rrb2At= -At
(10)
2rb
Evaluating n' in the same manner as ni gives 3 (1 - 4 n'= -a rrb2
Equations 7, 10, and 11 give the Nusselt number as
-2r0h - -
rb/rO
hlocan be evaluated in a similar manner but this is not done here since all the data presented are for equal sized, highly conducting particles. Equations 10, 11, and 14 give
3(1 - t)(,ln
-1)
When we use an equation given by Kunii and Smith (1960) for k,/k,, we find that for highly conducting materials (k,/k, > 100) and at low temperatures a depends upon void fraction and only slightly upon k,/k,. (As k,/k, approaches one the assumptions of constant values for hl and a become increasingly inaccurate.) We take t = 0.4 for a bed of randomly packed spheres (Haughey and Beveridge, 1969) and obtain a = 1.4 for both the aluminum and alumina beds. With this value of a,eq 8 gives curve b in Figure 1, which has been obtained by drawing a smooth curve through the centers of the calculated discrete steps. This fits the sphere and satellite data reasonably well. (The curve is not very sensitive to the value of a.) When we evaluate eq 13 for both the sphere and satellite experiments, we obtain a Nusselt number of 0.95, which, considering the elementary nature of the model used, does not compare unfavorably with the experimental values of 0.84 and 0.85 given in Table 11. The model is readily extended to include transients by allowing the particle temperatures to vary with time, still retaining the assumption that temperatures within a particle are spatially uniform. The result for the case of equal size test and bed particles is
1,2, ..., N - 1) (16)
When we numerically integrate the above equations for our conditions using the same values of the parameters as before, we find that it successfully predicts the measured temperature-time histories of the test particles in our large particle beds (curve b in Figures 2 and 3). The satisfadory results of the elementary discrete model we have used are undoubtedly due to the localized nature of the thermal resistances for highly conducting particles. When the particle conductivity falls to the fluid conductivity the bed must behave as a continuum for all values of ro/rb, but our discrete model does not account for this situation, in which resistances are distributed throughout the bed.
298
Ind. Eng. Chem. Fundam. 1984, 2 3 , 298-303
We note that the apparatus used in this work offers a rather simple method of measuring the effective conductivity of powders. Nomenclature C, = specific heat of bed h = heat transfer coefficient from test particle to bed, based on surface area of test particle and temperature difference
t = temperature at any radial location r in the bed ti = temperature of ith layer t o = temperature of test particle t N = temperature of outer boundary At = temperature difference over a layer of thickness 2rb Greek Symbols a = number of effective contacts per particle t
(to - tN)
= bed void fraction
bulk density of bed
h, = heat transfer coefficient between two equal size particles
p =
in contact, based on cross sectional area of particles hlo= heat transfer coefficient between test particle and the adjacent bed particle in contact, based on cross-sectional area of bed particle k , = effective thermal conductivity as defined by eq 1 k , = thermal conductivity of gas in bed k , = thermal conductivity of solid particles n, = number of effective contacts between shells i and i + 1 n’ = number of effective contacts per unit superficial area in a rectilinear system no = number of bed particles in contact with test particle N = number of layers corresponding to outer boundary Nu = Nusselt number as defined by eq 4 q = heat flux based on surface area of test particle r = radial distance from center of test particle rb = radius of bed particle ro = radius of test particle rN = radius of outer boundary
Crane, R. A.; Vachon, R. I.; Khader, M. S.Proceedings of the Symposium on Thermophysical Properties, 7th N.B.S., Gaithersburg, MD, ASME Publ., May 1977; pp 109-122. Deissler, R . H.; Boegll, J. S.ASME Trans. Ser. C . 1958, 80, 1417-1425. Haughey, D. P.; Beveridge, G. S. G. Can. J . Chem. Eng. 1989, 4 7 , 130-140. Hill, F. B.; Wilhelm, R. H. AIChEJ. 1959, 5 , 488-496. Konopliv, N.; Sparrow, E. M. Waerme Stoffuebertrag. 1970, 3, 197-210. Kunii, D.; Smith, J. M. AIChE J . 1980, 6 , 71-78. Maxwell, J. C. “A Treatise on Electricity and Magnetism”, Vol. 1. 3rd ed.; Oxford University Press: London, 1904; pp 440-441. Ofuchi, K.; Kunii, D. Int. J . Heat Mass Transfer 1965, 8 , 749-757. Powell, M. J. Powder Technol. 1980, 25, 45-52. Schotte, W. AIChE J . 1860, 6 , 63-67. Schumann, T. E. W.; Voss, V. Fuel Sci. fract. 1934, 13, 249-256.
6 = time
Literature Cited
Receiued for reuiew July 5 , 1983 Accepted January 9, 1984
Inhibition and Catalysis of the Reaction of Steam on the Basal Plane of Graphite by Potassium Carbonate and Potassium Hydroxide Chor Wong and Ralph 1. Yang’ Department of Chemical Engineering, State Universiv of New York at Buffalo, Amherst, New York 14260
The etch decoration/transmission electron microscopy technique has been used to study the kinetics of the C-H,O reaction on the basal plane of slngle crystal graphite. The turnover frequency of the edge carbon surrounding monolayer pits on the basal plane varies with pit density. At 1 pit/pm2, 23 torr of H,O and 600 O C , the values are 0.037 s-‘ for uncoated graphite and 0.01 1 and 0.040 s-l for K2C03and KOH coated samples, respectively. A carbanion mechanism is proposed to explain the phenomenon of inhibition by K2CO3. At 550 OC, the K2C03 forms hexagonal particles with a fixed orientation on the basal plane of graphite, probably due to particle growth in a preferred crystallographic direction of the substrate. These results are discussed in conjunction with the strong catalysis of the overall rates (on all edges and planes of carbon) by both K2C03and KOH, as reported by previous workers.
Introduction The use of microscopy to study the mechanism and kinetics of gasification reactions of graphite single crystals is a powerful technique. It permits direct observation of rates in different crystallographic directions, using specimens of the highest structural perfection. More importantly, it enables one to measure the turnover frequency because the active sites, or the carbon atoms being gasified, are well defined. In fact, the fraction of active sites on the basal plane of graphite is very low, ranging from to lo-@on some of the natural graphite samples (Yang and Wong, 1981a). Optical microscopy was used to obtain some important results concerning the kinetics of both uncatalyzed and catalyzed graphite gasification reactions (Thomas, 1965). However, the invention of the etch decoration-transmission electron microscopy (EDTEM) technique has made it
possible to study the gas-carbon reactions on a truly atomic scale (Hennig, 1966; Evans et al., 1971). The technique requires the use of a thin layer of a single crystal, ca. 800 A thickness, for electron transmission. A one-layer deep (3.35 A) pit is developed on the basal plane by etching or reaction with a gas. The pit originates from a vacancy in the surface, and the active sites are the edge carbon atoms surrounding the pit. The pit is then rendered visible in the TEM by decorating the depression with gold or other electron dense metals. Many important discoveries have been made by use of this technique to measure uncatalyzed carbon gasification kinetics. One recent discovery is the “cooperative effect”, which refers to the phenomenon that the reactivity (or turnover frequency) is orders-of-magnitude higher on a multi-layer edge than on a single-layer edge (Evans et al., 1971). This effect apparently depends on the number of layers involved
0196-4313/84/1023-0298$01.50/0@ 1984 American Chemical Society
