Combustion Rate of Carbon Combustion of Spheres in Flowing Gas Streams C. .
Tu,1 H. Davis,2
and
study of the combustion of carbon under conditions closely simulating of industrial importance
THE cases
involves simultaneous solutions to interlocked problems in heat
H. C. Data
Hottel,
Massachusetts Institute of Technology, Cambridge, Mass.
presented on the rate of combustion of in various nitrogen-oxygen mixcarbon spheres of in which both the rate of under conditions tures, chemical reaction at the surface and the rate of are
oxygen to the surface are
factors. diffusion of A quantitative formulation of combustion rate is presented, predicting effects of temperature, gas velocity, and gas composition in agreement with
specific surface reaction
(grams per square second)
K
can =
cm.
be represented c pe
Vt
rate per
by (4)
and material transfer, fluid where c' combines constants and contains a suitable gravimetric dynamics, and reaction kinetics. conversion factor, and E is the Mathematical analyses can at best be no better than the underenergy of activation. Since the the experimental measurements. The combusexact significance of “energy of lying assumptions as to the tion-rate equation is able to establish limits within activation” for a heterogeneous nature of the mechanism itself; reaction is none too obvious, it which the absolute magnitude of the combustion furthermore, it must be borne in mind that the agreement of exis best to consider the term rather broad berate must lie, but these limits are e~E /rt ag a simple empirical corperiment with a theory deat cause of the lack of data on gas diffusivities from a relation of the variation with veloped mathematically high temperatures and of an inexact knowledge certain combination of assumptemperature of the effectiveness of collisions of gas molecules of the net products of the primary reaction betions does not validate the asFrom similar data with the reacting surface. tween oxygen and carbon. sumptions since, because of the Meyer reports heats of activacomplexity of the phenomenon, for the rate of combustion in carbon dioxide it is tion varying from 20,000 to other combinations of assumpconcluded that carbon, when burned in air, is tions may lead to the same con30,000 calories up to 1520° C. consumed by direct oxidation with oxygen rather clusion. At normal pressures no such than by a mechanism in which oxygen is pictured On the basis of a thorough resimple treatment as that leadas reaching the surface chiefly in the form of carview of the then available literaing to Equation 4 is permissible, since (1) the transport probture and of their own work, bon dioxide, there forming carbon monoxide. lem is more complex and (2) Rhead and Wheeler (18) conthere is a possibility of reactions cluded that neither carbon dioxide nor carbon monoxide could be considered as the sole pri- in the gas phase. The problem can be enormously simplimary product of the reaction between carbon and oxygen, and fied by considering the combustion of a geometrically simple that both gases were evolved simultaneously through the for- shape in a medium flowing by under known conditions. The mation and subsequent decomposition of the ph)rsico-chemical investigation by Smith and Gudmundsen (14) of the combuscomplex CkO». The data in the literature point strongly to the tion of carbon spheres in air and in air-water vapor mixtures conclusion that the ratio of carbon dioxide to carbon monoxide has been the only one of this type hitherto available. In in the primary products is a particular characteristic of the their experiments, however, the variation of temperature was carbon investigated, that this ratio increases in general as the so small as to be completely overshadowed by the variation of sphere diameter and gas velocity. degree of graphitization of the carbon increases. The possible The literature contains several mathematical analyses variations in this ratio are indicated by the low-pressure experiments of Langmuir (8), Eucken (s), and Meyer (10); these (1,2,9, 11, 15) of the problem of the combustion of carbon at variations appear entirely too large to be attributed solely normal pressures, all of which involve the concept of stagnant to the difference in technic of the several investigators. Thus gas about the burning sample, with diffusion controlling the rate of combustion. Nusselt’s analysis (11) has been exMeyer (10) observed that below 1300° C. carbon was consumed in a reaction kinetically of the first order, corresponding tended by Wentzel (15) to the powdered-coal engine with good to the stoichiometric equation: The paper of Burke and Schumann (1) is in essence success. and treatment not radically different from that of Nusselt. 4C + 302 2C02 + 2CO (1) These investigators, though considering the probability of Above 1500° C. a reaction of zero order sets in, corresponding some carbon monoxide formation, concluded that matheto matically it is justifiable to assume carbon dioxide to be the sole primary product. Likewise, Lewis (9), in his discussion 3C + 202 2CO C02 + (2) of the mechanism of the combustion of fuel on the grate, Kinetic theory indicates that the number of micromoles of assumed carbon dioxide to be the sole primary product. In their modified double-film theory, Burke and Schumann (2) gas impinging on unit surface of the confining vessel per second is given by: assume that practically all oxygen reaching the carbon surface is in the form of carbon dioxide, which is reduced to carbon 9720 p monoxide. Mathematically the work of the above investi(3) \/MT gators represents truly ingenious solutions to a complex probIf the gas can react at all with the surface, a certain fraction lem. At the time of the inception of this W'ork, the concept of a of the collisions, given by e~E /RT, will be effective, and the surface, covered by a relatively stagnant film through which Present address, National Central University, Nanking, China. Present address, The Foster-Wheeler Corporation, Carteret, N. J. oxygen and combustion products must diffuse countercur=
=
1
2
749
INDUSTRIAL
750
AND
ENGINEERING
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 Theoey 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 tempo-
rarily 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
CHEMISTRY
ture for gases. For convenience d/X may be assumed power function of the other group—i. e., d
No
=
ß ,
=
Q
e-EIM,Vos
Solving for p,> in Equation jy
=
6
PiRT
The diffusivity, D, at the
mean
^
y/T,
C7i
film temperature, T, may
be replaced by the function Do(77/273)", where Do is the diffusivity at standard temperature, 2730 K., and 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/µ and Dp/µ. The group Dp/µ is substantially independent of temperá-
a
although exponent a may be expected to depend somewhat the value of dup/µ. The viscosity, µ, varies approximately as Tin so that Equation 8 reduces to on
dT°'2
v A
(9)
(duPy
Substitution of the value of X from Equation 9 into 7 and expression of D as a function of temperature gives
_Ap0g
No
where A, B
cVT,
Pi_
(B
(dupY
e~~E /RTg
(10)
dToTz
constants
=
Equation 10 fixes the oxygen consumption as a function of 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/RT. 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 temperature, T¡, has been deliberately avoided by the assumption of a linear temperature gradient through the film (see section on derivation of equation at end of paper). If the primary product is all carbon dioxide, the molal rate of combustion, Km, equals No. If the product is all carbon dioxide a mean
and there is
gas phase reaction, Km
no
=
2No.
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 where
á
1
g
12 Km
=
=
12
(11)
2
Combining Equations
10
and 11,
one
obtains the final equa-
tion for specific combustion rate: K
12 '
=
Pi (
+ cv'T,
B (dupY +
d/
eE
/rt‘
(12)
1)
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
and substituting into 5:
_Vil_
*°(i+i)
(6)
7
(8)
X
T"-1
With this must be combined an equation relating the rate of chemical reaction at the carbon surface to the partial pressure Pot of oxygen at that surface. The earlier discussion has shown that the consumption of oxygen at the carbon surface at temperatures below 1500° C. is a first order process and can be represented by:
Yol. 26, No.
e
is
j?/RT‘
Vt.
(12a)
(2) When E/T, is small, the second term in the denominator of (12) is small, and diffusion is the controlling factor. Two cases now appear: (a) When the curvature of the sphere is small—i. e., when d is relatively large: cc
Wo*
"-!--/2
(dupY_
Vid
(b)
(12b)
When d is relatively small:
K
oc
WisE^l Vid
(12c)
INDUSTRIAL
July, 1934
AND
CHEMISTRY
ENGINEERING
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 by C. M. Tu.
751
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 per cent, the approximate percentage analysis of which was: lime, 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 place by a high-temperature cement and were suspended in the furnace by a fine refractory tube provided at both ends with platinum hooks. The tube in turn hung from a hooked drill -i
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 passage through two glass drying towers, A, arranged in series and packed with granulated calcium chloride, was metered by orifices at C. A humidifier, B, was employed in other parts of the work. The nitrogen employed was passed through a number of turns of iron pipe, D, packed 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.
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 Rate
in
Oxygen-Nitrogen Mixtures
To 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 to temperature. The results are given in Table I. Since an 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.
Diagram of Weight-Recording Mechanism
Platinum, platinum-rhodium thermocouples were placed at the center of the Venturi throat in furnace F where the gas velocity was a maximum, and at a point 6 inches (15.2 cm.) from the first, in the combustion chamber next to 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 an analytical balance of low sensitivity a spring weighted by a brass disk and dampened in an oil bath 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
Table I.
Effect
(Air velocity, Run 1
2 3
12.3
Rotation Rev./sec. 0.22 2.0 4.0
cm.
of
Rotation of Spheres Rate
on
Combustion
per second at S. T. P.; sphere diameter, 25.4 mm.) ,--Specific Combustion Rate at:-·. 1213° K. 1223° K. 1233° K. 1243° K. 1253° K. Grams per sq. cm. per sec. 0.174 0.176 0.177 0.177 0.176 0.171 0.175 0.170 0.170 0.170 0.175 0.174 0.174 0.175 0.174
The main body of data is presented in Table II. 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
INDUSTRIAL
752
ENGINEERING
AND
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 II. The data of Table II are plotted in Figure 4, on logarithmic scales of specific rate of combustion vs. 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 at the left of Figure 4 at a value of temperature that increases with velocity. The steep straight-line portion clearly represents a region in which the chemical resistance is controlling, since the effect of velocity is totally overshadowed by that of temperature. Table II. CarFur-
bon Av. Sur- Diam.
NACE FACE
OF
Gas Ye- Sp. Comlocity AT
K.
0
K. runs
with
947 1038 1179 1245 1340 1338 1251 1272 1348 1135 1209 1120 986 1102 1067 1091 1030 1600 1659 1621 1702
26.1
46 49 50 51 52 79 85 94 97
953 1031 1090 1160 1232 1261 1172 1159 1268 1037 1125 1057 972 1016 995 1015 935 1555 1633 1608 1705
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
103 1214 107 1084 133 1355 156 196 1530
1360 1205 1488 1596 1622
24.9 24.6 24.0 22,3 22.4
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
12
15 17 19 21 27 29 32
35 36 40 43
111 134 143 144 154 157 158
RATE,
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
25.8 24.5 24.9 24.2 23.4 24.2
23.7 23.5 21.1 21.2
24.6
21.0
APPROACHING
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
GAS. COMPN.
0.200 0.159 0.156 0.162
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
27.4
0.220 0.177 0.331 0.336 0.397
27.4
27.4
27.4
25.0 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
0.011 0.030
917 1227 1388 1118
1170 1485 1430 1330 1337 1604 1596
24.9 22.5 23.2 21.4
54.3
of
Notes
K.
21% O2
0
RUNS
to 3.51 Rates cor. cm./sec.; sphere diam. 25.4 mm. Corrections applied only on runs in which furnace temp. > 1050° K.
WITH
26.3 26.0 25.9 24.9 23.3
0.0884 0.111 0.0732
87 90 99
1007 1038 1083 1098 1124 1261 1278 1182 1557 1643 1589 1687
0.0822 0.0636 0.0569 0.0256 0.172 0.146 0.144
23 1067 26 1251 31 1165 34 1246 39 1089 42 1056 45 952 48 1019 78 1534 84 1627 93 1603 96 1705
1093 1290 1209 1289 1151 1086 978 1045 1558 1640 1608 1678
24.6
953 1014 1028 1055 1090 1175 1212 1093 1525 1630 1592 1726
996 1019 1040 1065 1092 1213 1245
26.2
1515 1615 1571 1622
24.2
956 1006 1069 1170 1212 1227 1172 1155 1220 1043 1088 1050 985 1542 1623 1599 1705
959 997 1079 1245 1234 1243 1176 1169 1251 1061 1113 1081 1025 1527 1614 1573 1658
0.116
0.124 0.118 0.122 0.114 0.126
81
0.156
0.169
0.160
Rates
cor.
to
u
—
7.52 cm./sec.; sphere Cordiam., 25.4 mm.
rections applied only on runs in which Tg > 1050° K.
0.224 0.195 0.266 0.269
0.316
Rates cor. to u 27.4 cm./sec.; sphere Cordiam., 25.4 mm. rections applied on all =
0.393
runs.
0,225
Rates
0.285 0.350 0.262
0.297 0.343
to u cor. 38.9 cm./sec.; sphere Cordiam., 25.4 mm.
rections applied
=
on
all
0.351
0.0196
=
50 50 50 50 50 50
)-
24.4
24.3 24.6 23.4 22.6 22.6 25.4
23.8
25.6
25.5
25.6 25.6 26.0 26.0 22.0
23.2 23.6 25.5
RUNS WITH
53 56 61 62
0.145
0.218 0.174 0.323
54 57 60 63 66 69 72
75
0.0126
0.022 0.0975 0.102 0.132 0.143
at
Mm. Cm./sec.
K.
956 990 1024 1069 1087 1188 1105 1201 1531 1637 1601 1709
0.102
Gas VeLOCITY
Run Temp.Temp. Sphere S.T.P. 6
=
7
Combustion0
of
0.0196 Rates cor. to u 50.0 cm./sec.; sphere 0.445 0.418 Cor0.411 0.429 diam., 25.4 mm. rections applied only 0.320 0.294 in which Tg > on runs 0.334 0.314 22.4 1050° K. 0.406 0.460 19.8 1443 0.451 0.462 24.1 1548 ° Fuinace temperatures taken as those of the downstream, couple. Combustion rate corrected within indicated limits and to indicated velocities and diameters by: U standard K X ^obsvd. U obsvd. ^standard (^actual 114 135 132 145 155 159 195
Rate
Av. Fur- Sue- Diam.
TION
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
on
Vol. 26, No.
The slope of the curves, d(lnK)/d(lnT), has a value of 34 in the low-temperature region, calling for a doubling of the reaction rate for a 15° temperature rise at 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 at 500° G. 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 at 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 to increase with temperature. An increase of the energy of activation will lower the curve from its present position.
Cor. Combus-
K Rate G./sec./ 103 Mm. Cm./sec. cm2 X
Run Temp.Temp. Sphere S.T.P. 0
bustion
Data
CHEMISTRY
67 68 73 74 80 86 89 98 13 14 16 18
20 25 28 30 33 37
38 41 47 77 83 92
95
26.1 24.3 25.1 22.9
24.9 24.0
25.2 22.0 23.7
26.1 25.6 25.3 25.6 25.5
26.3
7.46
7.46
7.46 7.46
7.46 7.46
7.46 7.46
7.85 7.81 7.81 7.81
3.49 3.49 3.49 3.49
3.49 3.49
3.49 3.49
3.65 3.65 3.65 3.65
7.51 7.51 7.52
7.52 7.52 7.52 7.52 7.52 7.85 7.85 7.81 7.81
3.51 3.51 3.51
3.51
3.51
26.1
3.51 3.51
24.6 23.9
23.9
26.6
Notes
Rate
3.51 3.51 3.51
3.67 3.67 3.67 3.67
=
9.69% O2
0.00874 0.0237 0.0440 0.0441 0.0429 0.0787 0.0817 0.0649 0.116 0.114 0.135 0.121
0.00874 Rates cor. to u 0.0237 7.52 cm./sec.; sphere Cor0.0446 diam., 25.4 mm. 0.0439 rections applied only 0.0413 on runs in which Tg > 0.0774 1050° K. 0.0803 0.0636
0.0314 0.0600 0.0522 0.0616 0.0478 0.0302 0.0078 0.025 0.0894 0.090 0.0814 0.0832
0.0310 0.0581 0.0522 0.0616 0.0477 0.0302 0.0078 0.025 0.0835 0.0847 0.0771 0.0816
APPROACHING
3.51 3.51 3.51
26.0 26.3
K G./sec./ cm2 X 103
APPROACHING GAS COMPN.
24.2 23.4 26.1 26.3 25.3
Sp. Com- Cor. BUSTION COMBUS-
1
gas
=
0.110
0.0106 0.125 0.119
COMPN.
= Rates cor. to u 3.51 cm./sec.; sphere diam., 25.4 mm.
= =
2.98% O2
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 Rates cor. to u 0.00486 7.52 cm./sec.; sphere 0.0133 diam., 25.4. Correc0.0152 tions applied only on 0.0127 runs in which Tg > 0.0258 1050° K. 0.0254 0.0211 0.0339 0.0387 0.0382 0.0358
0.0045 0.0061 0.0173 0.0116 0.0188 0.0242 0.0189 0.0182 0.0212 0.0100 0.0165 0.0136 0.00518 0.0234 0.0268 0.0361 0.0277
0.0045
=
0.0061 0.0173
0.0116 0.0192 0.0237 0.0181 0.0185
Rates
cor.
to
3.51 cm./sec.;
u
=
sphere
diam., 25.4. Corrections applied only on in which Tg > 1000° K.
runs
0.0215
0.00991 0.0167 0.0138 0.00518
0.0225 0.0255 0.0344 0.0276
not Runs in which furnace or surface temperatures are not reported have Average diameter of sphere taken on corresponding plots.
been indicated as
proportional to Varithmetic
mean
area.
INDUSTRIAL
July, 1934
AND
ENGINEERING
Inasmuch as diffusion is expected to be important in certain ranges of temperature and gas velocity, it is desirable to know the combustion rate as a function of the mean temperature of the film rather than the temperature of the furnace Tg (Figure 4) or of the carbon surface T, (Figure ). To permit ready evaluation of film temperature from Ts and Te, their difference T, Tg, divided by Ts, 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, at the highest reaction rates measured, the temperature difference T, Ta 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 Tg 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 —
CHEMISTRY
733
The effect of velocity, for various temperatures and ambient atmosphere of constant composition, is shown in Figure 8, both scales of which are logarithmic. The slope of the isotherms increases with temperature up to about 1100° K., above which it 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
—
Figure
5
rate on gas velocity because of the predominant effect of chemical resistance at 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 at velocities of 4 and 40 cm. per second, respectively. The substantial parallelism of the curves above 1100° K. indicates that, if Equation 12 is accepted, the term in the denominator representing chemical resistance has no importance above 1100° K.
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)a for constant values of temperature. Furthermore, at 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)a/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/x 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\/S plotted against u^/S. Figure 9 shows 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\/~S > 15, there appears some qualitative indication that the term ky/s increases slightly with temperature as expected from Equation 12b, although the data are badly interwoven. In the region u\/~S < 15, lines with a is
rapidly than the oxygen concentration. This may be attributed either to experimental error or to the fact that the mean oxygen concentration at 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 less
produce an appreciable concentration of combustion products in the gas downstream from the sample.
754
slope d
tively.
INDUSTRIAL
AND
ENGINEERING
0.42 follow the data qualita(This is to be compared to a slope of 0.37 to 0.49
(InKy/S)/d (lnu\/S)
=
found for the authors’ data.) Below u\/s 15, the effect of curvature of the film should begin to be appreciable, and Equation 12c =
rather
than
12b
should be expected to hold; i. e., Kccl/d, and Ky/S should become independent of uy/S. Actually, the data are badly scattered
but indicate qualitatively the conclusion drawn from Equation 12c, that k'\/s'varies less with variation in uy/S as the latter decreases in value. The effect of velocity on combustion under conditions causing diffusion to control the rate may be further studied by the application of certain analogies among heat transfer, diffusion, and fluid friction, such as that proposed by Reynolds (12). These analogies permit the prediction of the effect of a variable, such as velocity, on one of these phenomena from its effect on another, because of the similarity in the basic processes involved—namely, transfer of energy, matter, or momentum through a fluid film. The Reynolds analogy indicates that the effective thickness of a film through which heat or material transfer takes place from a body to the ambient atmosphere should be inversely proportional to the product of the mass velocity of the ambient medium past the body by the drag coefficient, , which latter term is obtainable from data on friction. A comprehensive correlation of data on drag coefficients for spheres is available in the form of a plot, Figure 10, by Chilton and Shepard (3) of drag coefficient vs. Reynolds number dup/µ, on logarithmic scales. The work of the present authors is in the range of Reynolds numbers of 20 to 400, in dimensionless units, at which values the curve of the drag coefficient vs. dup/µ has the slopes of —0.6 and —0.34. Therefore, the film thickness is proportional to 1/m1-0·6 or 1/m1-0·34. Since at constant temperature the combustion rate is inversely proportional to the film thickness, K is expected to vary as w0·4 tow0·66, provided d/X is large. The values 0.4 to 0.66 which are obtained from friction data but which correspond to exponent a of the combustion Equation 12, are to be compared to the values 0.37 to _
0.49, at the corresponding Reynolds numbers, obtained from the present data on combustion, and to the value of 0.42 obtained from Smith and Gudmundsen’s combustion data at slightly higher Reynolds numbers. Restrictions on the
applicability of the Reynolds analogy to this problem will be considered later. The remaining effect to be considered in the range in which diffusion apparently controls the reaction rate is that of temperature. Equation 12b indicates that, when diffusion is controlling and d is large relative to film thickness, the specific combustion rate varies as j”>-i-“/2 . Although little can other than that it lies between be said positively about 1 and 2, Meyer’s work indicates that it changes gradually from about 1.25 at 1100° K. to 1.5 at 2000°. Roughly, therefore, may be assumed to vary as T0·3 over the temperature range covered by the present work. The numerical value of a has been discussed and has values of 0.37 to 0.49 based on the present data, 0.4 to 0.66 based on the Reynolds analogy and data on drag coefficients for spheres, or 0.42 based on the present analysis of Smith and Gudmundsen’s combustion data. A mean value of 0.4 is probably adequate
CHEMISTRY
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 tem1.5 or perature. Extremes in theory indicate that 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 1.1 in 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 Tg and T,, respectively, whereas the mean film temperature, T, is the one of present interest, the slopes of the curves of log K vs. log T will lie between the above two groups of values, at about 0.6 to 1.2. Although the slopes of curves corresponding to successively higher lines in Figure 4 may be expected to increase at a constant temperature, because of the increased relative resistance offered by chemical reaction at 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, it is difficult to conclude positively that the experimentally determined exponent on temperature is in agreement with the value of 0.6 =
—
1.1
predicted by
theory, because of the necessity for the questionable assumptions concerning and the effect of temperature on
diffusivity. Comparisons
of
theory and experiment have 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 IT
No
Pou
+ d/X'
°
RT
Pi
1
T"\
(14)
d/2
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 at 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
x
~
hd k
'S' =
8
(dup\ f µ\
It; U;
,1K.
(15)
where
,
AND
INDUSTRIAL
July, 1934
ENGINEERING
the drag coefficient for spheres, is obtainable from
Figure 10. The group ßµ/k is substantially constant for gases, and equal to about 0.8 in the present range of its use. The viscosity, µ, is approximated by the Sutherland equation,
with constants from International Critical Tables, for air. "
* 1°·
-
170·9 (*
2- 5 ) (mT
™
CHEMISTRY
735
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 Equa1. tion 14 with (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 at 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, 1. (4) If the gas-phase reaction is appreciable in the film, the carbon combustion rate must obviously =
=
The molal rate of oxygen transport, No, can therefore be evaluated. The desired specific rate of combustion of carbon, K, is 12 , in which 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 diffusivity 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 1 and 2, and n velocity at 1400° K., on assumption of 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 1.8 to 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-
AND GUDMUNDSEN WITH THEORETICAL PREDICTION RATES AT 950’C.BASED ON REYNOLDS ANALOGY /
COMPARISON OF DATA OF SMITH OF
rpp I
=
=
imposed on Figure 9. The predicted zone lies above the data, as in the case of the data of Tu, further emphasizing the fact that the Reynolds analogy probably predicts unduly thin films.
Experiments
with
Carbon Monoxide-Nitrogen
Mix-
tures
Although the comparisons made between theory and experiment have indicated substantial agreement both in the effects of the various factors studied and in the absolute magnitude of the results, there may be some question as to the effect of carbon monoxide formation on the oxygen concentration
K, REACTION RATE, MGSVSQ.MM·,SEC. U AIR VELOCITY FT/SEC. AT 20*C. AND l ATM. S, SURFACE AREA, SQ.MM. .
,
|
/
I
I
L /
' I
*
-rHl J-
2
i
"
/
|
i
oc o
!
L"?
L-cfi56 A*
_La
""
i
o'?
44
V
A
i
3. u,
° £>
,
0
i
2
3
J_L 4
5
6
7
u'/T
8 9
10
Figure
_l_
j
20
15
30
40
50
_I_I__ 60
80
IOO
9
lie between cases 2 and 3. One may conclude, therefore, that the equations derived above, with a permitted variation of 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 at the carbon surface to substantially zero. The substitution, in experiments of the type described in this paper, of carbon dioxide for oxygen in the ambient medium should present a basis for testing the validity of this picture of combustion in a double film. Dubinsky (4) has obtained data in this laboratory on the rate of oxidation of carbon spheres by carbon dioxide, using the same apparatus and same type of carbon as Tu, described above. Although failure to remove oxygen completely from the nitrogen used and inability to prevent air leakage into the furnace unfortunately prevent a completely quantitative interpretation of the data, the results throw definite light on the mechanism of the reaction. If the picture proposed by Burke and Schumann is correct, the substitution of carbon dioxide for oxygen in the ambient medium should be the equivalent of the removal of the outer film, through which oxygen is pictured as diffusing, and the extension of the inner
INDUSTRIAL
756
ENGINEERING
AND
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 at the surface is involved, and diffusion should continue to control the rate of combustion in the same
CHEMISTRY
Vol. 26, No.
7
inert is relatively high and if p,-z is replaced by (x p„z), in which pox is the partial pressure of oxygen at the point. This condition is met in nitrogen-oxygen mixtures in which the nitrogen content is high. The diffusivity Dx depends on Tx, which in turn varies with x. Kinetic theory indicates that at constant total pressure, D 0( /273) , where D is the diffusivity at 273° K. Making the assumption that the temperature varies linearly from TQ at the outer film surface to T, at the carbon, and that the temperature Tx is therefore equal to T. + (x/X) {To Ta), one obtains from Equation 17 —
=
—
din(x
N0R 273" Ts
p„x)
—
which
on
No
=
(18)
_
dx
D0
integration yields In
x
—
—
Pot
X-Do
pog
RX 273"
l_(2V-“
27-")
-
(19) Uy;
J
if Figure
10
n ¿é 2.3 A desirable simplification is possible if it is remembered that the ratio of T, to Ta is not large. When Ts/Tg is less than 2 and n lies between 1.1 and 1.9, a negligibly small error is introduced by replacing (2 ) {T, Tg)/{T,2~n 7V~n) by Tn~1, in which T is the arithmetic —
—
temperature region. Actually, however, 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 to carbon dioxide is attended by a change in the nature of the chemical reaction at 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.
—
1
U(
j I CM./SEC, AT y?» n
f?--
AlR DATA OF 1U
—
xV // /' // i
l?_-
j
-r
_
^
-?L·
3,
Rvrx
t a5
2
/
'>·"
/
OM C02 DATA OF DUBINS 1
Derivation
of
Equation of Oxygen Diffusion Film Surrounding Carbon
through
/
through is pro-
portional to the
Comparison of Tu’s ExRates in Air at 1400° K. perimental Prediction Based with Theoretical on Reynolds Analogy Figure
11.
tion representing this fact —
din (pix) dx
can
product of the densities of the species and the difference in net velocities.
ferential equabe reduced to:
N„RTx Dx
U '
applies strictly to diffusion in a twoAlthough Equation component 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 17
-2.1; CNVSEC. s.1 s
;
i
t
y=
^
1
¡
¡
i
I
i 8
\
i
VRISON OF RATES
OF COMBUST,ON
CARBON DIOXIDE AND
OF BRUSH CARBON IN
t 1
1
'
1
i
i
|
i
1
TEMPERATURE *K.
cjrnacE
Figure
;
12
of T, and Ta. Furthermore, subsequent interpretation of Equation 19 is somewhat simplified by replacing the term p«,)/p¿, in pog) by its equivalent {pog p„s)/(t ln(x which p¡ is the logarithmic mean pressure of the “inert,” equal to: mean
—
—
—
(x
—
Pot)
(x
—
—
In
It
has been shown (7) that the dif-
U
1
neglected. The resistance to diffusion of a single molecular
another
0
°
/
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 that the curvature of the film may therefore be
species
/
Equation comes
19, on
7
—
Pog)
Pot pog
introduction of those two substitutions, be-
:
No
=
Tt{Pog
RT {
Pot)
^
L"° U73/ JX
(20)
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 T9 and Tg.
INDUSTRIAL
July, 1934
AND
CHEMISTRY
ENGINEERING
negligible curvature is obtained by multiplying both sides of Equation 20 by the surface area, 3.14 d2. Allowance 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, V3.14 d2 X 3.14 id -1- 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:
N OMENCLATURE a, b, B
constants
No
MSI:
1+^ r
A
d
(21)
a
C, c, c'
constants sphere diam., cm. diffusivity of 02 through inert, cm.2/sec. diffusivity of O2 through inert at temp, existing at point x, cm.2/sec. energy of activation, cc. atmosphere/mole Oj friction factor in Fanning equation, dimensionless sp. combustion rate, grams/cm.2 X sec. molal. sp. combustion rate, gram atoms/cm.2 X sec. mol. weight gas impinging, micromoles/cm.2 X sec. O2 diffusing, moles/cm.2 X sec. abs. pressure, mm. Hg pressure O2 in gas stream, at point x, and at carbon surface, respectively, atm. pressure of inert, in gas stream, at point x, and at carbon surface respectively, atm. logarithmic mean pressure of inert in film, atm. gas constant, cc. atmosphere/0 K., 82.07. temp, at main stream, 0at point x, and at carbon surface, respectively, K. 0
d
D Dx
E
f
Km
M
=
The first bracketed term will be equal to the diffusivity, D, evaluated at the arithmetic mean film temperature, T. Equation 21 is therefore equivalent to Equation .
Conclusions
No P Pogy PoXy Pos
Pigy Pixy Pis
Tg, To, Ts
3. The rate of combustion, in the temperature range in which diffusion controls, varies as the 0.4 to 0.7 power of the mass velocity, in substantial agreement with theory. 4. In the temperature range in which chemical reaction controls, the rate of combustion doubles for every 15° C. at 1050° K. 5. In the temperature range in which diffusion controls, the rate of combustion varies approximately as T0·6 to T 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.
A ß
a
gas
cm.
constant
stoichiometric factor viscosity, poises (grams/cm. X sec.) total pressure, atm. density of gas, grams/cc. drag coefficient 4/, dimensionless a
µ
p
=
Literature
Cited
(1) Burke and Schumann, Ind. Enq. Chem., 23, 406 (1931). (2) Burke and Schumann, Proc. Intern. Conf. Bituminous Coal, 2,
485-509 (1932).
Chilton and Shepard, private communication from du Pont Expt. Sta. (4) Dubinsky, M. S., Thesis, Mass. Inst. Tech., 1932. (3)
(5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
Eucken, Z. angew. Chem., 43, 987 (1930). Farup, Z. anorg. Chem., 50, 276 (1906). Gilliland, D.Sc. Thesis, Mass. Inst. Tech., 1933. Langmuir, J. Am. Chem. Soc., 37, 1139 (1915). Lewis, W. K., Ind. Eng. Chem., 15, 502 (1923). Meyer, Z. physik. Chem., B17, 385 (1932). Nusselt, Z. Ver. deut. Ing., 68, 124 (1924). Reynolds, Trans. Roy. Soc. (London), A190, 67 (1897). Rhead and Wheeler,_ J. Chem. Soc., 103, 473 (1913). Smith and Gudmundsen, Ind. Eng. Chem., 23, 277 (1931). Wentzel, Fuel, 11, 177-96, 222-8 (1932).
Received October 12, 1933. Presented before the Division of Gas and Fuel Chemistry at the 86th Meeting of the American Chemical Society, Chicago, 111.,
Pittsburgh
=
arithmetic mean film temp., K. velocity, cm./sec. distance from carbon surface to point in film, film thickness, cm.
T u x
presented.
of
—
R
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 at the surface may be expected to control. 2. In either of the regions in which diffusion or chemical reaction at the surface controls the combustion rate, the rate has been found to vary linearly with partial pressure of oxygen in the ambient medium. This is in agreement with the theory
Rear View
~~
Pi
1.
proportionality factor
A
K
(, ß Vo.) RT pi ~
=
757
September 10 to 15, 1933
Experiment Station, U.
S.
Bureau
of
Mines
