Combustion Rate of Carbon Combustion of Spheres in Flowing Gas Streams C. M. Tu,' H. DAVIS,~ AND H. C. HOTTEL, Massachusetts I n s t i t u t e of Technology, Cambridge, Mass. HE study of the combustion of carbon under con-
T
Data are presented on the rate of combustion of spheres of carbon in various nitrogen-oxygen mixtures, under conditions in which both the rate of chemical reaction at the surface and the rate of diffusion of oxygen to the surface are facfors. A quantitatice formulation of combustion rate is presented, predicting effects of temperature, gas celocity, und gas composition in agreement with the experimental measurements. The combustion-rate equation is able to establish limits within which the absolute magnitude of the combustion rate must lie, but these limits are rafher broad because of the lack of data o n gas diffusicities at high temperatures and of a n inexact knowledge of the net products of the primary reaction between oxygen and carbon. From similar data f o r the rafe of combustion in carbon dioxide it is concluded that carbon, when burned in air, is consumed by direct oxidation with oxygen rather than by a mechanism in which oxygen is pictured as reaching the surface chiefly in the f o r m of carbon dioxide, Ihere forming carbon monoxide.
ditions closely simulating cases of industrial importance involves simultaneous solutions to interlocked problems in heat and m a t e r i a l t r a n s f e r , fluid dynamics, and reaction kinetics. Mathematical a n a l y s e s can a t best be no better than the underl y i n g a s s u m p t i o n s as to the nature of the mechanism itself; furthermore, it must be borne in mind that the agreement of exp e r i m e n t w i t h a t h e o r y developed mathematically from a certain combination of assumptions does not validate the assumptions since, because of the complexity of the phenomenon, other combinations of assumptions may lead to the same conclusion. On the basis of a thorough review of the then available literature and of their own work, Rhead and Wheeler ( I S ) concluded that neither c a r b o n dioxide nor carbon monoxide could be considered as the sole primary product of the reaction between carbon and oxygen, and that both gases were evolved simultaneously through the formation and subsequent decomposition of the physico-chemical complex C,O,. The data in the literature point strongly to the conclusion that the ratio of carbon dioxide to carbon monoxide in the primary products is a particular characteristic of the carbon investigated. that this ratio increases in general as the degree of graphitization of the carbon increases. The possible variations in this ratio are indicated by the low-pressure experiments of Langmuir (8), Eucken (6), and hileyer (10); these variations appear entirely too large to be attributed solely to the difference in technic of the several investigators. Thus Meyer (10)observed that below 1300" C. carbon was consumed in a reaction kinetically of the first order, corresponding to the stoichiometric equation: 4C
+ 302
= 2COz
+ 2CO
(1)
Above 1500" C. a reaction of zero order sets in, corresponding to 3c
+ 202
=
coz
+ 2co
(2)
Kinetic theory indicates that the number of micromoles of gas impinging on unit surface of the confining vessel per second is given by: (3)
If the gas can react a t all with the surface, a certain fraction of the collisions, given by will be effective, and the
'
Present address, National Central University, Nanking, China. Present address, The Foster-Wheeler Corporation, Carteret, Iri. J.
specific surface r e a c t i o n rate ( g r a m s p e r s q u a r e cm. per second) can be represented by
where c' combines constants and contains a suitable gravimetric conversion factor, and E is the energy of activation. Since the exact significance of "energy of activation" for a heterogeneous reaction is none too obvious, it is best to c o n s i d e r the term e - E I R T as a simple empirical correlation of the v a r i a t i o n with temperature of the effectiveness of collisions of gas molecules with t h e r e a c t i n g s u r f a c e . hfeyer reports heats of activation v a r y i n g f r o m 20,000 to 30,000 calories up to 1520" C. At normal pressures no such simple treatment as that leading to Equation 4 is permissible, since (1) the t r a n s p o r t problem is more c o m p l e x and ( 2 ) there is a Dossibilit!: of reactions in the gas phase. The problem can 6e enorniously simplified by considering the combustion of a geometrically simple shape in a medium flowing by under known conditions. The investigation by Smith and Gudmundsen (14) of the combustion of carbon spheres in air and in air-water vapor mixtures has been the only one of this type hitherto available. I n their experiments, however, the variation of temperature v a s so small as to be completely overshadowed by the variation of sphere diameter and gas velocity. The literature contains several mathematical analyses (1, 2 , 9 , 11, 15) of the problem of the combustion of carbon a t normal pressures, all of which involve the concept of stagnant gas about the burning sample, with diffusion controlling the rate of combustion. Kusselt's analysis (11) has been extended by Wentzel (15)to the powdered-coal engine with good success. The paper of Burke and Schuniann ( 1 ) is in essence and treatment not radically different from that of Xusselt. These investigators, though considering the probability of some carbon monoxide formation, concluded that niathematically it is justifiable t o assume carbon dioxide to be the sole primary product. Likenise, Lewis ( 9 ) ,in his discussion of the mechanism of the combustion of fuel on the grate, assumed carbon dioxide to be the sole primary product. I n their modified double-film theory, Burke and Schumann ( 2 ) assume that practically all oxygen reaching the carbon surface is in the form of carbon dioxide, which is reduced to carbon monoxide. Mathematically the work of the above investigators represents truly ingenious solutions t o a complex problem. At the time of the inception of this work, the concept of a surface, covered by a relatively stagnant film through which oxygen and combustion products must diffuse countercur-
749
INDUSTRIAL AND ENGINEERING CHEMISTRY
750
rently, seemed to offer the most fruitful mental picture. Accordingly, the present work was planned with a view to studying those variables which, in the light of the diffusion picture, seemed most likely to control the rate of combustion. These included primarily temperature, gas velocity, and gas composition, which with size and shape of the specimen constitute the chief factors determining the rate of diffusion to it. Analogous treatments in heat transfer, absorption, humidification, and extraction have met with marked success despite the fact that relatively little is known about the size and shape of the gas films or of the mechanisms of their formation. The characteristics of such a film about a body in a flowing fluid are dependent on such properties as the mass velocity of the fluid, its viscosity, the pattern of flow of the fluid and its temperature distribution, and the size and shape of the body. It is not inferred anywhere that the film about a sphere, for example, is of uniform characteristics, but it is assumed that the diffusion can be approximated by considering a perfectly stagnant film of uniform thickness, the total effect of which is equivalent to that of the actual film.
FILM THEORY The attempt will now be made to evaluate the combustion rate by calculating the diffusion of oxygen through the film and considering the two limiting cases for the formation of either carbon dioxide or carbon monoxide as the sole primary product. The correct answer must be somewhere between these two cases. The assumption of a layer wherein carbon monoxide may burn with counterdiffusing oxygen is temporarily withheld. A sphere is undergoing combustion in a medium flowing with a uniform relative velocity past it. If combustion is slow compared to the total gas flow, the composition of gas everywhere over the outer film boundary may be taken as a constant and equal to that of the main stream. Although the understanding of the actual derivation of the diffusion equation is essential to an evaluation of its merit, the derivation is relegated to the end of this paper in order to minimize the introduction of new nomenclature at this point. The resulting equation representing the diffusion of oxygen through the film surrounding the carbon sphere is (5)
With this must be combined an equation relating the rate of chemical reaction a t the carbon surface to the partial pressure p,, of oxygen at that surface. The earlier discussion has shown that the consumption of oxygen a t the carbon surface a t temperatures below 1500" C. is a first order process and can be represented by:
Vol. 26, No. 7
ture for gases. For convenience d / X may be assumed a power function of the other group-i. e.,
although exponent a may be expected to depend somewhat on the value of dup/,u. The viscosity, k , varies approximately as T1l2so that Equation 8 reduces to
Substitution of the value of X from Equation 9 into 7 and expression of D as a function of temperature gives
where A , B
=
constants
Equation 10 fixes the oxygen consumption as a function of a mean film temperature, the mass velocity of the gas, the gas composition, the diameter of the sphere, and the surface reactivity as measured by e - E J R T . The introduction of a heat balance to permit the evaluation of the surface temperature, T,,as a function of the combustion rate and the gas has been deliberately avoided by the assumptemperature, To, tion of a linear temperature gradient through the film (see section on derivation of equation a t end of paper). If the primary product is all carbon dioxide, the molal rate of combustion, K,, equals N o . If the product is all carbon dioxide and there is no gas phase reaction, K , = 2N.. Since the literature indicates that the ratio of carbon monoxide to carbon dioxide found in the primary products is dependent on the surface temperature, the specific rate of carbon combustion K is given by:
K
=
12K, = 12N&
where 1 S 6 5 2
(11)
Combining Equations 10 and 11, one obtains the final equation for specific combustion rate: 12A'+poO
K =
Pi
aTn-l ( B vd
+ c v T 8eEIRTa
+ i)
(12)
The denominator of Equation 12 contains two additive terms, the first of which corresponds to a diffusional resistance and the second to a chemical resistance. Equation 12 and those preceding indicate that it can assume the following limiting values : (1) When the term E / T , is large, the first term in the denominator of Equation 12 is small relative to the second, the chemical reaction rate at the surface is controlling, and
n
Solving for p,, in Equation 6 and substituting into 5 : Po9
No =
piRT
*D
d/Ta + Ce-E/RT.
($ + i)
(7)
The diffusivity, D , a t the mean film temperature, T, may be replaced by the function D0(T/273)",where Do is the diffusivity a t standard temperature, 273 O K., and TZ is a constant. The film thickness, X , may be evaluated from a consideration of dimensional analysis, which indicates that d / X will be a function of the two dimensionless groups, dup/p and DP/,u. The group D p / p is substantially independent of tempera-
(2) When E / T , is small, the second term in the denominator of ( 1 2 ) is small, and diffusion is the controllingfactor. Two cases now appear: (a) When the curvature of the sphere is small-i. e., when d is relatively large:
K =
+poo x T n - 1 - 0 1 ~ (dUp)'l pid
(b) When d is relatively small:
Wb)
INDUSTRIAL A N D E N GI N E E R I N G CHEMISTRY
July, 1934
FIGURE 1. DIAGRAM OF APPARATUS
It was to test the validity of this type of analysis that the study of the combustion of carbon spheres under controlled conditions of temperature, gas velocity, and gas composition was undertaken b y C. M. Tu.
75 1
with a plate moved by a synchronous motor. A vertical cylindrical lens focuses the beam from the mirror to a point on the photographic plate. The shaft of the motor actuates a rotating switch which flashes a second lamp every 15 seconds to superimpose a time scale on the weight-time record. A record so obtained consists of a single inclined weight line and a series of vertical lines giving the time scale. Horizontal weight calibration lines were superimposed on each record by the addition of known weights to one side of the balance. A typical record is shown in Figure 3. The carbon employed was a brush carbon of high grade having an apparent density of 1.55 and an ash content of 0.17 er cent, the approximate percentage analysis of which was: {me, 34; ferric oxide, 39; silica, 27. A special ball-turning attachment permitted the spheres to be turned out readily in the lathe. The spheres were drilled to receive small platinum hooks fixed in lace by a high-temperature cement and were suspended in the furnace by a fine refractory tube provided a t both ends with platinum hooks. The tube in turn hung from a hooked drill
EXPERIMENTAL PROCEDURE The assembly of the apparatus employed is illustrated in Figure 1: A one-inch (2.5-cm.) sphere was suspended from one arm of an analytical balance of low sensitivity into a horizontal, cylindrical furnace through which metered, preheated gas of known composition was passed. Data were obtained for the rate of combustion in oxygen-nitrogen mixtures, pure carbon dioxide, carbon dioxide-nitrogen mixtures, and nitrogen-water mixtures. This paper is concerned primarily with the oxygen-nitrogen data, the rest being considered as more or less qualitative in nature. Compressed air from the laboratory air line, after being dried by assage through two glass drying towers, A , arranged in series an$ packed with granulated calcium chloride, was metered by orifices a t C. A humidifier, B, was employed in other parts of the work. The nitrogen em loyed was passed through a number of turns of iron pipe, D, pacfed with copper turnings and heated by Meker burners to about 800" C. to remove any oxygen. After several runs the oxidized copper was reduced by passage of hydrogen through the heated unit. The furnace employed was in two sections, a low temperature preheater, E, connected by a short piece of insulated alundum tubing to a globar furnace, F,which was divided by a refractory throat into a high-temperature preheater packed with refractory granules, and an isothermal chamber in which the combustion proceeded. Each section of the latter furnace was a porcelain tube 2.25 inches (5.7 cm.) i. d. and 24 inches (61 cm.) long. When necessary this unit could be forced to 1350" C.
AA.l 1 I I I
I I 1 1 I I I
TIME ( I N T E R V A L S A R E 15 SEC.) FIGURE 3. TYPICAL TIME-WEIGHT RECORD
rod, which terminated in a small ball bearing resting on a drilled agate surface through which the rod passed, the agate bearing being supported by an aluminum disk suspended from the balance. Below the bearing the rod was equipped with long aluminum vanes on which blasts of air could be directed by symmetrically arranged jets. Rotational speeds of 20 r. p. m. were maintained after the effect of rotational velocity was established. The duration of runs was 3 to 12 minutes, long periods being avoided to prevent too great a departure of the carbon balls from sphericity. COMBUSTION RATEIN OXYGEN-NITROGEN MIXTURES T o study the effect of speed of rotation of the spheres, a preliminary series of runs was made in air, over a temperature range in which the rate is sensitive to gas velocity and relatively insensitive t o temperature. The results are given in Table I. Since a n eighteen fold variation of rotational speed was found to have no measurable effect on combustion rate, it was concluded that close control of rotational speed was unnecessary. Speeds of about one-third revolution per second were used in subsequent experiments.
FIGURE 2.
WEIGHT-RECORDINQ MECHANISM
D I A G R h M OF
Platinum, platinum-rhodium thermocouples were placed at the center of the Venturi throat in furnace F where the gas velocity was a maximum, and a t a point 6 inches (15.2 cm.) from the first, in the combustion chamber next t o the sphere. Carbon surface temperatures were obtained by an optical pyrometer sighted through the outlet nozzle opening. The weight-recording unit is indicated in Figure 2: To one arm of a n analytical balance of low sensitivity a spring weighted by a brass disk and dampened in an oil bat,h is attached.
The sphere and its supports are suspended from the other arm. A mirror and horizontal cylindrical lens mounted on one arm reflect and focus a beam of light from a fixed lamp onto a camera
TABLEI. EFFECT OF ROTATION OF SPHERES ON COMBUSTION RATE
(Air velocity, 12.3 cm. per second at S. T. P.; sphere diameter, 25.4 mm.) ,--SPECIFIC COMBUSTION RATEAT:RUN ROTATION 1213' K. 1223' K. 1233' K. 1243' K. 1253' K. Grams per
Rev. /sec. 1 2 3
0.22 2.0 4.0
0.176 0.170 0.174
0.176 0.170 0.174
sq. cm. p e r 8ec. 0.17i 0.177 0.175 0.170 0,174 0.175
0.174 0.171 0.175
The main body of data is presented in Table 11. Because of the desirability, for purposes of analyzing the data, of having results corresponding to fixed values of gas velocity u and sphere diameter d, the actual results were corrected to desired values of u and d by the relation
I pi D U S T R I A L A N D E N G I N E E R 1 N G C H E 321 I S T R Y
752
(13)
in the range in which diffusion controlled the combustion rate. Since the corrections seldom amounted to more than a few per cent, errors due to an error in the form of Equation 13 are negligible. The corrected values of combustion rate are given in the seventh column of Table 11. The data of Table I1 are plotted in Figure 4,on logarithmic scales of specific rate of combustion us. absolute temperature of the furnace, for various values of gas velocity and oxygen content. Inasmuch as the temperature of the carbon surface has weight equal to that of the furnace, part of the data of Figure 4 (those on air) are replotted in Figure 5 with surface temperature as the abscissa. The data of Figure 4 fall into three families of curves, one corresponding to each particular gas mixture used. Each member of any one family breaks away from the common, steep, straight-line portion a t the left of Figure 4 a t a value of temperature that increases with velocity. The steep straight-line portion clearly represents a region in n-hich the chemical resistance is controlling, since the effect of velocity is totally overshadowed by that of temperature.
1-01. 26, No. 7
The slope of the curves, d(lnK)/d(lnT), has a value of 34 in the lowtemperature region, calling for a doubling of the reaction rate for a 15" temperature rise a t 1050" K. This may be compared with Farup's data (6) showing a doubling of rate of combustion of electrode carbon for a 10" rise a t 500" C. and a similar rise observed by Rhead and Wheeler for wood charcoal. It is to be expected that this factor will drop with rise in temperature. As the temperature increases, its effect becomes markedly smaller, and velocity becomes the major factor. The effect of temperature in this region will not be discussed until other factors related to diffusion have been considered. The dotted line of Figure 5 has been calculated from Equation 4 as the maximum reaction rate possible in air on the assumption that the diffusional resistance is negligible and that the pressure of oxygen is therefore the same a t the carbon surface as in the main body of gas, that the energy of activation is 20,000 gram-calories per gram mole of oxygen, and that the product is wholly carbon dioxide. The agreement in order of magnitude between this and the envelope of the air data is surprisingly good. Meyer found E to lie between 20,000 and 30,000 calories for this reaction, and t o increase with temperature. An increase of the energy of activation will lower the curve from its present position.
TABLE11. DATAON RATEOF COMRUSTIOXQ CARBON
AV.
GAB VE- SP. COX-
FUR- SUR- DIAX.LOCITY N A C ~FACE
OF
AT
RUNTEMP.TEMP.SPHERES.T.P. a
K.
a
K.
BUSTION
RATE,
TION
NOTES =
21%
I _
K.
~
947 1038 1179 1245 1340 1338 1251 1272 1348 1135 1209 1120 986 1102 1067 1091 1030 1600 1659 1621 1702
26.1 25.2 24.9 25.5 25.6 23.0 24.1 24.1 24.6 25.7 24.8 25.1 26.1 26.3 25.5 26.3 25.5 19.6 22.4 22.5 24.9
3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.53 3.51 3.51 3.51 3.51 3.67 3.67 3.67 3.67
0.011 0.030 0.103 0.116 0.123 0.130 0.125 0.117 0.128 0.0884 0.112 0.0734 0.0126 0.0822 0.0636 0.0569 0.0256 0.200 0.159 0.156 0.162
0.011 0.030 0.102 0.116 0.124 0.11s 0.122 0.114 0.126 0.0884 0.111 0.0732 0.0126 0.0822 0.0636 0.0569 0.0256 0.172 0.146 0.144 0.156
Rates cor. to 3.51 cm./sec.; sphere diam. 25.4 mm. Corrections applied only on runs i n which furnace temp. > 1050' K.
55 58 59 64 65 70 71 76 82 88 91 100
1004 1026 1044 1074 1096 1207 1166 1111 1550 1648 1603 1712
1027 1124 1137 1193 1213 1341 1353 1253 1607 1686 1649 1720
25.9 25.8 24.5 24.9 24.2 23.4 24.2 23.7 23.5 21.1 21.2 24.6
7.51 7.52 7.52 7.52 7.52 7.52 7.52 7.52 7.81 7.85 7.85 7.85
0.022 0.0975 0.102 0.133 0.144 0.175 0.164 0.150 0.234 0.213 0.290 0.273
0,022 0.0975 0.102 0.132 0.143 0.169 0.160 0.145 0.224 0.195 0.266 0.269
R a t.w ror. to u = - __ .. 7.62 cm./sec.; sphere diam., 25.4 mm. Corrections applied only on runs in Yhich T Q > 1050' K.
103 107 133 156 196
1214 1084 1355 1530
1360 1205 1488 1596 1622
24.9 24.6 24.0 22.3 22.4
27.4 27.4 27.4 27.4 25.0
0,220 0.177 0.331 0.336 0.397
0.218 0.174 0.323 0.316 0.393
Rates cor. to u = 27.4 cm./sec.; sphere diam. 25.4 mm. Correctiohs applied on all runs.
111 134 143 144 154 157 158
1096 1285 1401 1127 1168 1319 1463
1248 1420 1558 1311 1344 1609 1596
24.8 23.5 20.6 22.6 23.2 20.8 21.0
38.9 38.9 38.9 38.9 38.9 38.9 38.9
0.227 0.295 0.389 0.277 0.311 0.378 0.388
0.225 0.285 0.350 0.262 0.297 0.343 0.351
Rates cor. t o 21 = 38.9 cm./sec.; sphere diam., 25.4 m,m. Corrections applied on all runs.
(5 81 87 90 99
1007 1038 1083 109s 1124 1261 1278 1182 1557 1643 1589 1687
26.1 26.0 25.9 24.9 23.3 24.4 24.3 24.6 23.4 22.6 22.6 25.4
7.46 7.46 7.46 7.46 7.46 7.46 7.46 7.46 7.85 7.81 7.81 7.81
23 26 31 34 39 42 45 48 78 84 93 96
1067 1251 1165 1246 1089 1056 952 1019 1584 1627 1603 1705
1093 1290 1209 1289 1151 1086 978 1045 1558 1640 1608 1678
24 6 23.8 25.6 25.5 25.6 25.6 26 0 26 0 22 0 23.2 23.6 25.5
3.49 0.0314 3 . 4 9 0.0600 3.49 0.0522 3.49 0.0616 3 . 4 9 0.0478 3 . 4 9 0.0302 3.49 0 . 0 0 7 8 3 . 4 9 0.026 3 . 6 5 0.0894 3 . 6 5 0.090 3 . 6 5 0.0814 3 . 6 5 0.0832
53 56 61 62 67 73 74 80 86 89 98
963 1014 1028 1055 1090 1175 1212 1093 1525 1630 1592 1726
996 1019 1040 1065 109% 1213 1245
NOTES
RATE
13 14 16 18 20 25 28 30 33 37 38 41 47 77 83 92 95
956 1006 1069 1170 1212 1227 1172 1155 1220 1043 1088 1050 985 1542 1623 1599 1705
0,00874 0.0237 0.0440 0.0441 0,0429 0.0787 0.0817 0.0649 0.116 0.114 0.135 0.121
= 9.69%. _ 02
0.00874 0.0237 0.0446 0.0439 0.0413 0.0774 0.0803 0,0636 0.110 0.0106 0.125 0,119
Rates cor. t o u = 7.52 cm./sec.; sphere diam., 26.4 m m . Corrections applied only on runs i n which To > 10500 K.
0.0310 0.0581 0.0522 0.0616 0,0477 0.0302 0.0078 0.025 0.0835 0.0847 0.0771 0.0816
Rates cor. t o u = 3.51 cm./sec.; sphere diam., 25.4 mm.
RCNS WITH APPROACHING G A S COMPN. = 2.98%
68
0,0196 Rates cor. t o ?I = 54.3 0.0196 1170 2 4 . 9 50.0 cm./sec.; w h e r e 0.418 0.446 1485 2 2 . 5 50 diam 25.4 mm. Cor0.411 0.429 50 1430 2 3 . 2 rectidhs applied only 0.294 0.320 50 1330 2 1 . 4 on runs in whlch T Q > 0.314 0.334 50 1337 22.4 1050° K. 0.406 0.460 50 1443 1604 1 9 . 8 0.451 0.462 50 1548 1596 2 4 . 1 Com5 Furnace temperatures t a k e n aa those of the downstream couple. bustion rate corrected within indlcated limlts a n d t o indicated velocities a n d diameters by:
K
G./sec./
Mm, Cm./sec.cma X 108
956 990 1024 1069 1087 1185 1105 1201 1531 1637 1601 1709
54 57 60 63 66 69 22
__
K.
RUNS WITH A P P R O A C H f N G GA5 C O M P N .
0 2
953 1031 1090 1160 1232 1261 1172 1159 1268 1037 1125 1057 972 1016 995 1015 935 1555 1633 1608 1705
917 1227 1388 1118
REX TESIP.TEIIP.SPHERES.T.P. e
12 15 17 19 21 27 29 32 35 36 40 43 46 49 50 51 52 79 85 94 97
114 135 132 145 155 159 195
.kV. VE- S P . COX- COR. D I A X ? . LOCITY BUGTION CO4IBUSNACE F h C E OF AT R,ATE, TION BON
FUR-S U R -
K RATE G./sec./ Mm. Cm./sec.cm2 X I O a
RUNS V l T H A P P R O A C H I N G O A S CONPX.
GAS
CAR-
COR. COMBUS-
0 2
7.51 7.51 7.52 7.52 7.52 7.52 7.52 7.52 7.85 7.85 7.81 7.81
0.00398 0.00486 0.0133 0.0153 0.0133 0.0260 0.0262 0.0212 0.0355 0.0425 0.0403 0.0359
0.00398 0.00486 0.0133 0.0152 0.0127 0.0258 0.0254 0.0211 0 . 0x34 0.0387 0 0382 0.0358
Rates cor. t o u = 7,52 cm./sec.; sphere diam., 25.4. Corrections ,applied only on runs in which T O > 1050' K.
1516 1615 1571 1622
26.2 26.1 24 3 25.1 22.9 24.9 24.0 25.2 24.2 22.0 23.7 26.1
959 997 1079 1245 1234 1243 1176 1169 1251 1061 1113 1081 1025 1527 1614 1573 1658
25.6 25.3 25.6 25.5 26.3 24.2 23.4 26.1 26.3 25.3 26.1 26.0 26.3 24.6 23.9 23.9 26.6
3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.67 3.67 3.67 3.67
0.0045 0.0061 0.0173 0.0116 0.0188 0,0242 0.0189 0,0182 0,0212 0,0100 0.0165 0.01:36
0.0045
Rates cor. t o u = 3.51 cm./sec: sphere diam. 25.4.' Corrections lapplied only on runs In which T O > 1000° K.
0,00518
0,0234 0.0268 0.0361 0,0277
0.0061
0.0173 0.0116 0.0192 0.0237 0.0181 0.0185 0.0215 0.00991 0.0167 0.0138 0.00518 0.0225 0.0255 0.0344 0.0276
R u n s in which furnace or surface temperatures are not reported have n o t been indicated on corresponding plots, Average diameter of sphere t a k e n as proportional t o d a r i t h m e t i o mean area.
INDUSTRIAL AND ENGINEERING
July, 1931
Inasmuch as diffusion is expected to be important in certain ranges of temperature and gas velocity, i t is desirable to know the combustion rate as a function of the mean temperature of the film rather than the temperature of the furnace T , (Figure 4) or of the carbon surface T , (Figure 5 ) . To permit ready evaluation of film temperature from T , and T,, their difference T , - T,, divided by T,, has been plotted against the specific reaction rate in Figure 6. Data for different gas velocities and oxygen concentrations fall into a band narrow enough to justify the use of a single line to represent all the data. The plot indicates that even a t the highest reaction rates measured, the temperature difference T , T, is a small fraction of the absolute temperature level and has a maximum value of about 0.17. Consequently, the use, for a mean film temperature, of an arithmetic mean of T , and To cannot be in error by more than a few per cent. Figure 7 shows the effect of oxygen concentration in the ambient atmosphere for a constant gas velocity and different temperatures. The data seem to establish fairly well the existence of a proportionality between combustion rate and oxygen concentration. This is not quite the result to be expected from the theory which led to Equation 12, since the presence of the term p , in the denominator of (12) should cause the rate to increase more rapidly than the oxygen concentration, except in the low temperature region in which diffusion is of no importance. However, on account of the fact that pi varies only from 1.0 to 0.9 when the ambient atmosphere varies from pure nitrogen to air, and because the term involving p , is but one of two additive terms in the denominator of Equation 12, the expected upward curvature of the lines of Figure 7 is negligible. The data indicate, if anything, that the rate of combustion increases somewhat 0.60 0.50 0.40
0.30
0.20
CHENISTRY
733
The effect of velocity, for various temperatures and ambient atmosphere of constant composition, is shown in Figure 8, both scales of n-hich are logarithmic. The slope of the isotherms increases with temperature up to about 1 1 0 0 O K., above which i t is substantially independent of temperature. Below 1100" K. the lines become more and more nearly parallel to the velocity axis, showing a lack of dependence of 0.50 0.40
0.30
-
0.20
0
x
W
:
0.10
5
0.06
F
Y
K U
E
0.06 0.05 0.04
U
$
0.03 0.02
0 0 A
- 3.51 CM/SEC. -- 727.4 . 5 2 CM. CM
CARBON SURFACE TEMPERATURE.,*KELVIN
FIGURE 3
rate on gas velocity because of the predominant effect of chemical resistance a t the surface. In the higher temperature range in which the chemical resistance has apparently no effect on the combustion rate, there is an indication that the slope increases with velocity, having values of 0.37 and 0.49 a t velocities of 4 and 40 cm. per second, respectively. The substantial parallelism of the curves above 1100O K. indicates that, if Equation 12 is accepted, the term in the denominator representing chemical resistance has no importance above 1 1 0 0 O K.
n
2
0.10
0.08
2 2
Q $
0.06
0.05 0.04
2
I 0.03
s g-
0.02
2
(0
OF CARBON IN OXYGEN-NITROGEN MIXTURES
0 01
450 -3.51
0.008
b$O-7.51 b
0.006
CM/SEC. CM.bEC.
-27.4 CM./SEC
0.005
0
8
'
0
2
2
-
0.004
0.0031
0
0
0
N
0 0
s
0 0
z
r
0 0
2
0 0
.
0
E
FURNACE TEMP.,'KELVIN
FIGURE4
less rapidly than the oxygen concentration. This may be attributed either to experimental error or to the fact that the mean oxygen concentration a t the outer boundary of the film is somewhat less than the concentration of oxygen in the ambient medium, especially at combustion rates high enough to produce an appreciable concentration of combustion products in the gas downstream from the sample.
The data of Smith and Gudmundsen show more clearly the velocity effect, which totally obscures that of furnace temperature. Equation 12 indicates that K , the combustion rate, may be expected to vary systematically with d/(du)" for constant values of temperature. Furthermore, a t temperatures sufficiently high to make the second term in the denominator negligible, and for sufficiently small ratios of film thickness to diameter, Equation 12b indicates that K will vary linearly with (du)"/d, for constant values of T . The plotting, therefore, of Kd vs. ud on logarithmic paper should produce a straight line having a slope of a, in the region in which d / z is sufficiently large to neglect curvature of the film. Since the data of Gudmundsen and Smith are reported in terms of surface S instead of diameter d of the sphere, K d S is plotted against u d . Figure 9shows their data so plotted. The actual data points of the authors, rather than their smoothed curves, were used in the construction of Figure 9. The plot includes data for furnace temperature varying from 850" to 1000° C. Although all data could have been corrected to a common temperature, the small variation in absolute temperature and its slight effect on diffusion through a gas film are believed to warrant the omission of the temperature correction. For values of u f i > 15, there appears some qualitative indication that the term kfiincreases slightly with temperature as expected from Equation 12b, although the data are badly interwoven. In the region u d < 15, lines with a
INDUSTRIAL AND ENGINEERING CHEMISTRY
754
sloped ( l n K d q / d ( l n u f i = 0.42 follow the data qualitatively. (This is to be compared to a slope of 0.37 to 0.49 found for the authors’ data.) Below u d = 15,the effect of curvature of the film should begin to be appreciable, and Equation 12c r a t h e r t h a n 12b INCREASE OF SURFACE TEMPERATURE should be expected WITH REACTION RATE 0.15 t o h o l d ; i . e., K a l l d , and K d S should become indeTfSURFACE 1 E M P k = r u n u u r TLMR pendent of u a . 0.05 Actually, the d a t a are badly scattered
1
Vol. 26, No. 7
for the present purpose. The remaining exponent of T is n, the power on T in the relation between diffusivity and temperature. Extremes in theory indicate that n = 1.5 or 2.0. One may then conclude that K will vary as a power function of T,with the power lying somewhere between 0.6 and 1.1in the temperature range in which diffusion controls. This is to be compared to the experimental results. The righthand branches of Figure 4 are not strictly parallel, their slopes varying from about 0.4 to 1.0. The right-hand branches of Figure 5 have slopes varying from about 0.7 to 1.3. Since these two figures involve temperatures T , and T., respectively, whereas the mean fXm temperature, T , is the one of present interest, the slopes of the curves of log K us. log T will lie between the above two groups of values, a t about 0.6 to 1.2. Although the slopes of curves corresponding to successively higher lines in Figure 4 may be expected to increase a t a constant temperature, because of the increased relative resistance offered by chemical reaction a t the surface, this effect should be restricted to the vicinity of the point of branching of the curves and should not be apparent at the extreme right of the figure. Whether the variation actually found in the slope of the curves at the right, from 0.6 to 1.2, is due to the necessity for interpreting the rather inadequate data or whether the effect is a real one, cannot at present be decided. Likewise, i t is difficult to conclude positively that the experimentally determined exponent on temperature is in agreement with the value of 0.6 1.1 p r e d i c t e d b y 0.20 theory, because of t h e n e c e s s i t y for 0.1) the questionable asL sumptions concern0.10 ing 4 and the effect of temperature on 0.05 diffusivity Comparisons of theory and experiPERCENT OXYGEN m e n t h a v e been FIGURE 7 limited so far in this discussion to a substantiation of the predicted effects of the various factors, oxygen pressure, gas velocity, and temperature, separately considered. A comparison of the absolute magnitude of the observed combustion rate with that predicted by theory is highly desirable. In the high temperature range of the data of Figure 4 in which diffusion presumably controls the combustion rate, the basic Equation 7, with diffusivity expressed as a function of temperature, reduces to
.
The Reynolds analogy has already been mentioned as a possible basis for evaluating the film thickness, X, from friction data. Although its application to the flow of gases inside pipes yields results in good agreement with heat transfer data, it predicts values of film thickness only 44 to 18 per cent of those demanded by heat transfer data when applied to the case of gases flowing outside of and a t right angles to wires, because of the fact that the analogy involves only that part of the resistance contained in the “skin” adjacent to the wire. Its use on gas flowing past spheres may therefore be expected to give little more than an indication of the order of magnitude of X. The analogy is d
= hd = 8
(--)(:) dup
July, 1934
INDUSTRIAL AND ENGINEERING CHEMISTRY
where \k, the drag coefficient for spheres, is obtainable from Figure 10. The group c p / k is substantially constant for gases, and equal to about 0.8 in the present range of its use. The viscosity, p, is approximated by the Sutherland equation, with constants from International Critical Tables, for air.
The molal rate of oxygen transport, No,can therefore be evaluated. The desired specific rate of combustion of carbon, K , is 12 No@,in which CP lies between 1 and 2. There is, in addition, some question as to the appropriate value to use for n, the exponent on T in the equation relating difhsivity and temperature. Because of these two questionable factors, a zone rather than a line will be predicted by the use of Equation 14. The calculations representing a comparison of experiment with theory are summarized in Figure 11. Four bounding lines are presented for the variation of K with mass velocity a t 1400” K., on assumption of 9 = 1 and 2, and n = 1.6 and 1.9, within which the true n probably lies. The solid line represents the actual data as read from the curves of Figure 4 for the temperature 1400” K. The actual data are seen to lie slightly below the lower boundary of the zone predicted by the Reynolds analogy. This was to be expected in view of the abnormally thin films predicted by application of the Reynolds analogy to data on wires, to which reference has been made. Within the limits of accuracy of interpolating the data, the theoretical limiting curves are parallel with the data curve. A similar comparison of data and theory is desirable for the data of Gudmundsen and Smith, in whose experiments sphere diameters were varied from 1.8to 4.4 mm., air velocity from 1.09 to 10.25 feet per second, and furnace temperature from 850’ to 1000° C. Their data have already been presented in Figure 9. The combination of the Reynolds analogy with the theoretical equation, in a manner similar to that used on Tu’s data, leads to the dotted and dashed lines super-
755
gradient through the film and therefore on the derivation of equations: (1) If, as assumed in the derivation, all the carbon is oxidized by oxygen as such, and if carbon dioxide only is formed, the rate of combustion is given directly by Equation 14 with CP = l. (2) If carbon monoxide only is formed, and the gas-phase reaction between carbon monoxide and counterdiffusing oxygen is negligibly slow, the combustion rate is given by Equation 14 with @ = 2. (3) If carbon monoxide only is formed a t the surface but the gas-phase combustion of carbon monoxide to dioxide is infinitely rapid, the carbon monoxide will burn to carbon dioxide in an infinitesimal zone immediately adjacent to the surface, and the conditions in the main body of the film will differ in no way from case 1 above, CD = 1. (4)If the gas-phase reaction is appreciable in the film, the carbon combustion rate must obviously
n
.l
COMPARISON OF DATA OF S M I T H AND CUDMVNDSEN WITH THEORETKAL PREDICTION OF RATES AT 950’C.BASED O N REYNOLDS ANALOGY
/
_In
2
3
4
5
6
7 8 910
15
20
30
40
50
60
80
IM)
FIGURE 9
lie between cases 2 and 3. One may conclude, therefore, that the equations derived above, with a permitted variation of CP from 1 to 2, cover all cases in which the carbon surface is oxidized by oxygen alone, whether the oxidation is to carbon monoxide, carbon dioxide, or both, and whether the carbon monoxide burns in the film or not. A different picture is possible, however, which is proposed by Burke and Schumann (2) in which no oxygen, as such, reaches the carbon surface, the latter being oxidized only by carbon dioxide. The resulting carbon monoxide formed, diffusing outward, burns with oxygen somewhere midway in the film to form a zone of maximum carbon dioxide concentration, with diffusion of carbon dioxide in both directions from this zone. This zone in effect divides the film into two parts in one of which the oxygen .concentration varies from its value in the main gas stream to substantially zero, in the other of which the carbon monoxide concentration varies from its value a t the carbon surface to substantially zero. The substitution, in experiments of the type described in this paper, of carbon 3 4 5 b 7 8910 I5 20 30 40 M bo70 dioxide for oxygen in the ambient medium should present a basis for testing the validity of this picture of combustion FIGURE8 in a double film. imposed on Figure 9. The predicted zone lies above the Dubinsky (4) has obtained data in this laboratory on the data, as in the case of the data of Tu, further emphasizing rate of oxidation of carbon spheres by carbon dioxide, using the fact that the Reynolds analogy probably predicts unduly the same apparatus and same type of carbon as Tu, described thin films. above. Although failure to remove oxygen completely from the nitrogen used and inability to prevent air leakage into EXPERIMENTS WITH C.4RBON -~ONOXIDE--NITROGEN MIXthe furnace unfortunately prevent a completely quantitative TURES interpretation of the data, the results throw definite light on Although the comparisons made between theory and experi- the mechanism of the reaction. If the picture proposed by ment have indicated substantial agreement both in the effects Burke and Schumann is correct, the substitution of carbon of the various factors studied and in the absolute magnitude dioxide for oxygen in the ambient medium should be the of the results, there may be some question as to the effect of equivalent of the removal of the outer film, through which carbon monoxide formation on the oxygen concentration oxygen is pictured as diffusing, and the extension of the inner
INDUSTRIAL AND ENGINEERING
756
film to occupy the whole film thickness. The velocity of the ambient medium, carbon dioxide-nitrogen, and the temperature of the gas or surface should have the same general effect as the effect with oxygen-nitrogen mixtures, since the same chemical reaction a t the surface is involved, and diffusion should continue to control the rate of combustion in the same 5 4
CHEMISTRY
Vol. 26, No. 7
inert is relatively high and if p i s is replaced by (T - p,,), in which p,, is the partial pressure of oxygen a t the point. This condition is met in nitrogen-oxygen mixtures in which the nitrogen content is high. The diffusivity D, depends on T,, which in turn varies with x. Kinetic theory indicates that a t constant total pressure, D = D,(T/273)", where D is the diffusivity a t 273' K. Making the assumption that the temperature varies linearly from T , a t the outer film surface to T , a t the carbon, and that the temperature T , is therefore equal to T. ( x / X ) (T, - T 8 ) ,one obtains from Equation 17
+
3
2
I 5
-RELATIVE VELOCITY, CM/SEC. -SPHERE DIAMETCR CM.
?IT
9
I D 0 9
which on integration yields
:;
0 8
&
05 04
03 IO
20
30
40
50 10
80
200
100
300 400
BW
800
000
P
FIGURE 10
temperature region. dctually, hoyever, the results of Dubinsky (Figure 12) indicate an enormous effect of temperature on combustion rate in the temperature range 1300' to 1700" K., a range in which temperature has relatively little effect on the rate of combustion of the same kind of carbon in air, as indicated by the dotted lines of Figure 12, taken from Figure 4. This fact points strongly to the conclusion that a change in the ambient atmosphere from air t o carbon dioxide is attended by a change in the nature of the chemical reaction a t the carbon surface, and that the hypothesis that carbon burns by the mechanism of the reduction of carbon dioxide, formed in the film by combustion of carbon monoxide, is untenable. DERIVATION OF
if n # 2.3 A desirable simplification is possible if it is remembered that the ratio of T. to T o is not large. When T,/Tais less than 2 and n lies between 1.1 and 1.9, a negligibly small error is introduced by replacing (2 - n) (T, - T J / (T,*-" - To2-")by T"-l, in which T is the arithmetic
EQUATION O F OXYGEN DIFFUSION THROUGH F I L h l SURROUNDIXG CARBOX
The general conditions covered by the derivation are discussed above. The assumption is temporarily made that the film thickness is small compared to the sphere diameter, and F BRUSH CARBON N CAQBON D i O X DE A N D that the curvature of the film may therefore be neglected, 0 g g g g n . The resistance to diffusion of a FIGURE12 single molecular -x species through mean of T , and To. Furthermore, subsequent interpretation x 0.2 a n o t h e r is pro- of Equation 19 is somewhat simplified by replacing the term portional to the h(ir - p,*)/(?r - peg) by its equivalent (pea - p o s ) / p t ,in 0.1 5 p r o d u c t of the \Thich pa is the logarithmic mean pressure of the "inert," 0.1 0 densities of the equal to : 30 4 0 S O 6 0 00 IO I5 BO 30 YI 8 0 species a n d the UD,CM /SEC -(Up) / O OD129 (n - Pod - (7 - Po64 difference in net FIGURE11. COMPARISON OF Tu's ExIn 6TPERIMENTAL RATESIN AIR AT 1400' K. ~ e l o c It ~ ~ ~ ~ ~ * 7r - Po, WITH THEORETICAL PREDICTION BASED has been shown ON REYNOLDS ANALOGY (7) that the dif- Equation 19, on introduction of those two substitutions, beferential e q u a - comes: tion representing this fact can be reduced to: AIR
0
0
0
0
40
Although Equation 17 applies strictly to diffusion in a twocomponent system in which but one component is being transported, it approximates the more precise but complex equation applicable to a system involving counterdiffusion of two or more components if the pressure of the nondiffusing
Allowance for the fact, that, contrary to the initial assumption, the film thickness is appreciable compared to the diameter of the sphere, is readily made. If d equals the diameter of the sphere, the total rate of diffusion for the case of
-
a When n 2, the bracketed term involving T in Equation 18 will be replaced by the logarithmic mean of T Iand To.
July, 1934
INDUSTRI4L AND ENGINEERING
negligible curvature is obtained by multiplying both sides of Equation 20 by the surface area, 3.14 d2. illlowance for curvature is then made by replacing the surface area 3.14 d2 on the right side of the equation by the geometric mean film area, 4 3 . 1 4 dz x 3.14 (d + 2x)2. Return to a formulation of specific rate of diffusion, based on carbon surface area, is then obtained by dividing both sides of the equation by 3.14 d2, yielding instead of Equation 20:
The first bracketed term will be equal to the dzusivity, D, evaluated at the arithmetic mean film temperature, T. Equation 21 is therefore equivalent to Equation 5 .
COSCLCSIONS 1. A quantitative formulation of the rate of combustion of carbon in air has been presented, which indicates the conditions under which diffusion of oxygen to the surface or chemical resistance a t the surface may be expected to control. 2. In either of the regions in x-hich diffusion or chemical reaction a t the surface controls the combustion rate, the rate has been found t o vary linearly with partial pressure of oxygen in the ambient medium. This is in agreement with the theory presented. 3. The rate of combustion, in the temperature range in which diffusion controls, varies as the 0.4 t o 0.7 power of the mass velocity, in substantial agreement with theory. 4. I n the temperature range in which chemical reaction controls, the rate of combustion doubles for every 15' C. a t 1050" K. 5. I n the temperature range in which diffusion controls, the rate of combustion varies approximately as To.6to T 1 1, lying within the limits established by theory. 6. The combustion rate, in absolute magnitude, lies as near as may be expected to the value predicted by the theory presented, since the predicted values depend on a questionable method of predicting film thickness. 7. Evidence is presented which is interpreted by the authors as rendering untenable the hypothesis that combustion of carbon in air involves the transport of' oxygen to the carbon surface in the form of carbon dioxide alone.
CHEMISTRY
757
NOMESCL.4TCRE
constants a proportiona.lity factor constants d = sphere diam., cm. D = diffusivity of O2through inert, cm.*/sec. = diffusivitv of O2 through inert a t temw existing D, .,a t point i,cm.2jsec. E = energy of activation, cc. atmosphere/mole 0 2 = friction factor in Fanning equation, dimensionless = sp. combustion rate, grams/cm.2 X sec. = molal. sp. combustion rate, gram atoms/cm.2 X sec. Km M = mol. weight n = gas impinging, micromoles/cm.2 X sec. No = 02 diffusing, moles/cm.2 X sec. = abs. pressure, mm. Hg P p,,, p o z ,p,, = pressure O2in gas stream, at point x, and at carbon surface, respectively, atm. p,,, p,,, p,, = pressure of inert, in gas stream, at point x, and at carbon surface respectively, atm. = logarithmic mean pressure of inert in film, atm. fi = gas constant, cc. atmosphere/" K., 82.07. T,, T,, T , = temp. at main stream, a t point x, and at carbon surface, respectively, O K. T = arithmetic mean film temp., O K. ZL = gas velocity, cm./sec. X = distance from carbon surface to point in film, cm. X = film thickness, cm. P = a constant @ = a stoichiometric factor = viscosity, poises (grams/cm. X sec.) P B = total pressure, atm. = density of gas, grams/cc. P = drag coefficient = 4f,dimensionless a,b, B A C, c, c'
= = =
L
*
LITERATURE CITED (1) Burke and Schumann, ISD.ENO.CHEM.., 23, 406 (1931). (2) Burke and Schumann, Proc. Intern. Conf. Bituminous Coal, 2, 485-509 (1932) (3) Chilton and Shepard, private communication from du Pont
Expt. Sta. (4) Dubinsky. M. S., Thesip, Mass. Inst. Tech., 1932. (5) Eucken, 2. angew. Chem., 43, 987 (1930). (6) Farup, Z . anorg. Chem., 50, 276 (1906). (7) Gilliland, D.Sc. Thesis, Mass. Inst. Tech., 1933. ( 8 ) Langmuir, J . Am. Chem. SOC.,37, 1139 (1915). (9) Lewis, W. K., IND.ENG.CHEM.,15, 502 (1923) (10) Meyer, Z . physak. Chem., B17, 385 (1932). (11) Nusselt, Z . V e r . deut. Ing., 68, 124 (1924). (12) Reynolds, Trans. Rous Soc. (London), A190, 67 (1897). (13) Rhead and Wheeler,.J. Chem. Soc., 103, 473 (1913). (14) Smith and Gudmundsen, IND. ENO.CHEX.,23, 277 (1931). (15) Wentzel, Fuel, 11, 177-96, 222-8 (1932).
RECEIVEDOctober 12, 1933. Presented before the Division of Gas and Fuel Chemistry at the 86th Meeting of the American Chemical Society, Chicago, I l l , September 10 to 15, 1933
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REARVIEWOF PITTSBURGH EXPERIMEKT STATION, U. S. BUREAUOF MINES
