(13) Flynn, K . G., h‘enortas, D. R., J . Org. Chem. 28, 3527 (1963). (14) Hall. H. K., J . Phys. Chem. 60, 63 (1956). (15) Houdry Process and Chemical Corp., Data Bull. 4, 16 (1959). (16) Iwakura, Y . ;Okada, H.. Can. J . Chem. 38, 2418 (1960). (17) .Joester, M. D.: Drago, R. S.:J . Am. Chern. Soc. 84, 3817
,.,.,-\
[I YDL).
(18) (19) (20) (21)
Julg, A.: Bonnet. M..Compt. Rend. 250, 1839 (1960). hagakura. S..Gouterman, M., J . Chem. Phys. 26, 881 (1957). Pestemer, M.. Lauerer, D. A n p w . Chem. 72, 612 (1960). Saunders, J. H., Frisch, K. G., ‘.Polyurethanes. Chemistry
and Technology,” Part 1: “Chemistry,” p. 129, Interscience, New York: 1962. (22) Schieler. L.. Jet Propulsion Laboratory, California Institute of Technology, Tech. Rept. 32-129 (July 1, 1961). (23) Smith. J. F.. Friedrich. E. C., J . Am. Chem. SOC.81, 161 (1959). RECEIVED for re\-iew March 4> 1964 ACCEPTED July 29. 1964 Division of Industrial and Engineering Chemistry, 147th Meeting, ACS, Philadelphia, Pa., April 1964.
DIFFERENTIAL THERMAL ANALYSIS AND REACTION KINETICS R O N A L D L. R E E D , ’ LEON WEBER, AND B Y R O N S. G O T T F R I E D
Gulj Research 3 De~elopmentCo., Pzttshurgh, Pa.
Methods a r e presented for the use of differential thermal analysis in the quantitative determination of chemi-
cal reaction kinetic parameters. The theory of Borchardt and Daniels, developed to describe irreversible reactions in stirred systems, is extended, allowing the use of different portions of one or more thermograms in determining activation energies and frequency factors. A widely accepted method devised b y Kissinger is shown to b e incorrect for stirred systems and of questionable value under any circumstances. The Borchardt arid Daniels and the Kissinger methods a r e applied to experimental data, resulting in favorable values for kinetic constants using the Borchardt and Daniels technique; the results of the Kissinger method a r e shown to b e in serious error. The Borchardt and Daniels equations a r e integrated numerically, producing theoretical thermograms which agree well with the corresponding experimental curves. The effects of various pararneters on the thermograms a r e also established b y numerical integration.
differential thermal analysis (DTA) and thermogravimetric analysis (‘PGA) are used for the qualitative characterization of complex chemical reactions, such as the thermal dehydration of clays (79, 20: 25) and the oxidation of crude oil (24). I n the past few years several methods have been suggested for using D‘PA to determine quantitatively the kinetic parameters for certain types of reactions (2. 4. 5, 7 7 , 72). T h e most fundamental approach is that of Borchardt and Daniels (1: 5). where the DT.4 method is modified in such a way that the experiments are accurately described by a simple theory. This paper develops additional theoretical aspects of the Borchardt and Daniels method. Several results obtained with this method are compared with results obtained by follo\ving the method of Kissinger ( 7 7 , 72) and by conventional kinetic studies. OTH
Description of Method
Detailed descriptions of equipment and techniques are not given here, since there is a n abundance of literature on the subject (73: 76-78. 22). Although specific designs of DT.4 equipment vary \videly. depending on what is to be studied, most share the general features which follow. Briefly. the reactive substance to be studied is placed in a sample cell and an inert substance is placed in a reference cell. The inert substance is chosen so that its heat capacity and thermal conductivity approximate those of the reactive material. Both cells are then immersed in a heat bath, and the reaction is initiated by supplying heat to the bath in a prescribed fashion. For simplicity. this is usually done in such Present address. Department of Mechanical Engineering, Drvxel Institute of Technology, Philadelphia, Pa. 38
I&EC
FUNDAMENTALS
a \\.a)- that the temperature of the reference cell, T,? increases linearly \vith time. I n the absence of reaction, the sample temperature, T ! \vi11 also increase linearly and equal T,. \Vhen reaction commences in the sample cell, the heat liberated (or absorbed) causes 7‘ to differ from T,. This difference, A T = 7’ - T,? is recorded by means of a differential thermocouple. T h e actual reference temperature, T,. is simultaneously recorded. l h u s : the outcome of the experiment is a graph of AT LIS. time, t , as shown in Figure 1. T h e rate of reaction is not measured directly, but must be inferred from the D T A curve by suitable theoretical analysis. Theory of Borchardt and Daniels
When the heat bath consists of a metal block and the contents of the cells are a t rest. there is a temperature gradient betlveen the walls of the cells and their centers where the temperatures are measured. \Yhen the appropriate differential equations and auxiliary conditions are specified, this situation is taken into account and the D T X curve A T LIS. t can be computed if the kinetic parameters are known. This procedure was carried through by Sewell (27) for a first-order reaction using a n approximation to the reaction-rate function. T h e result !vas an integro-differential equation ivhich was treated using series approximations. Borchardt and Daniels avoided the mathematical complexity introduced by the finite thermal conductivities of a metal block. stagnant sample. and reference materials by changing the conditions of the experiment (4. 5). They employed a n all-liquid system with a stirred heat bath in which kvere immersed a sample cell containing a stirred reactive liquid and a reference cell containing a stirred inert liquid. LVith such a system the temperatures of reference. sample, and bath are
almost uniform. This permits the system to be described by a first-order differential model. A brief summar)- of the Lvork of Borchardt and Daniels follo\vs. ‘[‘he rate of reaction per unit volume. of order n with respect to a single component. is given by
A,=TOTAL AREA UNDER CURVE
11here .I-is the number of moles of reactant, V the volume of the rample cell. and k the apecific reaction rate “constant.” \\hich is nvumed to conform to the integrated Arrhenius equation
Lvith .1. the frequency factor. and E: the activation energy, assumed constant. .\ heat balance over the system gives 1 d.Y
~~
V dt
-
I’ AH
[ a dLT
dt
+
,,]
(3)
\\here .Yois the number of moles of reactant originally present, E; the heat transfer coefficient of the cell, AH the total heat of reaction. and a the ratio of the cell heat capacity to the heat transfer coefficient. Integrating Equation 3 from zero to infinity gives
-AH
=
KAt
(4)
\vith .gt representing the total area under the D T 4 curve. Thus. the heat of reaction is proportional to the total area under the DTA curve. Equation 4 is the simplest possible result which still retains the basic features of the problem. Relations of this kind have been extensively pursued in the literature and derived under a variety of circumstances more complex than that considered here ( 7 , 3. 8. 27. 23: 26). Combining Equations 3 and 4 to eliminate AH gives
Integrating Equation 5 from t to infinity and combining this result with Equations 1 and 5 to eliminate .V (which is not measured in the course of DT.4) yields the basic equation of Borchardt and Daniels
where 6 represents a portion of the area under the D T A curve, as illustrated in Figure 1. T h e term A t p 1 has been retained to keep the signs correct-i.e.. the signs for A t , AT, and dAT/ dt depend on whether the reaction is exothermic or endothermic. Borchardt and Daniels determine n. E, and A as follo\vs: A value of n is guessed and k is calculated a t a series of temperatures using Equation 6 and graphical determination of ci, A T , and d A T / d t from the DT.4 curve. Then A and E are obtained from Equation 2 by plotting In k ts. 1 / T . If the correct value of n was chosen. this relation is linear. If the volume chanqes significantly. it must be appropriately evaluated a t each temperature for which k is assessed. Freeman and Carroll (9) have pointed out th.at this trial and error procedure is not always essential. since Equations 2 and 6 can be combined in such a way that n and E are obtained from a single graph. Thus [ ( A T ) ’ = d ( A T ) ; d t ] ,
A
f
t
Simple exothermic DTA curve
Figure 1.
At-l[a(AT)’
+ AT]
=
and therefore In { A t - l [ a ( A T ) ’
+ A T ])
;;+
= -
Equation 8 can be evaluated a t any two temperatures using the D‘I‘A curves and the equations so obtained subtracted. ‘Pherefore, if the volume remains constant during the reaction. it is straightforward that for any pair of temperatures T I and
T2> [a(AZ’)’
[a(AT)’
+ A7’I1 + AT]*
[--AT’
+ a],
-
This means that a graph of the quantity to the left us. that in brackets to the right should yield a straight line of slope - E / R and intercept n for any order of reaction. If then, the basic assumptions are strictly applicable to the particular reaction and equipment design selected. the work of Borchardt and Daniels combined with that of Freeman and Carroll provides a scheme for obtaining the kinetic constants from one analysis of a single DTA curve. However, when the experimental data are reduced in the foregoing fashion? d a t a scatter can be so large that it is advisable to use many pairs of T I and 7’2 apd several D T A curves if a good value of n is to be determined. Therefore, it turns out that the method of Freem a n and Carroll offers no practical advantage over that of Borchardt and Daniels. Further Development of Equations
In the foregoing analysis the volume was always explicitly exhibited so that it \ \ o d d be clear what role V plays lvhen there is a volume change during the DTA process. Henceforth: changes in volume are not readily taken into account, so it will be assumed that the reaction takes place a t constant pressure lvith negligible change in volume. .I‘he equations of interest are those for the reaction rate and the heat balance
(5)
VO1. 4
NO. 1
FEBRUARY
1965
39
Since it is clear that the dependence on 4 is manifest a t the low temperature inflection, O 1 . this relation is valid only for temperatures sufficiently below O i . I t can be written
subject to the auxiliary condition
a n d the initial conditions
E
T(0) = To .V(O)
=
1. 4 s &VIvu, = t / a
(Henceforth, the group or variable \\ill be referred to in terms of the basic quantity from which it is derived. For example, t will be called the activation energy and -Or’ dJ do, the reaction rate) and to replace 7 by 0, as the independent variable. Further, there is no significant error introduced by integrating from absolute zero instead of Tu Thus,
L
=
so that sufficiently below Ti. a graph of the quantity to the left us. 1 T should give E and .4 for a reaction of any order n. A practical difficulty is that this portion of the DT.4 curve is sometimes unreliable in view of oscillations induced while achieving the linear heating rate. Reaction Rate a t Peak. .4t the peak of the D T A curve the hich is neither zero nor a rate of reaction has a definite value 1% maximum \$-hen Equations 1 3 and 14 are evaluated a t the peak one obtains
Therefore, the rate of reaction a t the peak of the D T A curve is directly proportional to the temperature deflection there. I t will prove useful to observe further that
LMaximum Reaction Rate. T h e temperature 0, where the reaction rate achieves its maximum value is determined by the
AO(0) = 0
’0)
?‘A,‘
-vo
where To is regarded as a temperature measured just prior to a n observable departure of the D T A curve from the base line I t is convenient to introduce the dimensionless quantities 0 = CY T:I A
.
1
where Or’ is the dimensionless heating rate, and S is f l or -1, depending upon whether the reaction is exothermic or endother mic.
tion 13 it follows that 0, is fixed by
Consequences of Theory
There are many ways in which Equations 13 a n d 14 can be differentiated, integrated. combined, and evaluated a t various characteristic points on D T A a n d T G A curves to provide schemes for determining the kinetic parameters. There is little point in making a n exhaustive tabulation of these, since the riqnificance of each depends on the success of its particular application. T h e following is merely representative of the kinds of results obtainable. Cooling Curve. After substantially all reactant has been conbumed. the reaction rate vanishes. and the shape of the D T A curve is governed solely by
T h e solution is A0
=
( W exp[(O,, f
- 0,)/0,’1
(17)
where [e,,. (AO),] is any point on the D T A curve sufficiently beyond the high temperature inflection point, and 0, 2 07f. This solution can be written 1
Hence a graph of - In 4 T us. T , should have a linear portion \iith slope (aT,’)-I. This provides a measure of any change in cy which may have occurred during the experiment. H e a t i n g C u r v e . LVhen the temperature is still low, the reaction rate is increasing rapidly, in view of exp ( - C / O ) . but not much reactant has been consumed--i.e., $ = 1. With this assumption. it follo\is from Equations 13 and 14 that
I t is instructive to graph AB and its first two derivatives in order to see what is implied for a typical DT,4 curve (see Figure 2). T h e only way Equation 23 can be satisfied is for the first a n d second derivatives of A0 to be of opposite sign. This happens only for values of Or corresponding to the shaded regions. Further, only the left shaded region (exclusive of the end points) is of interest because the right shaded region corresponds to the cooling curve. I t follows that the maximum rate of reaction occurs between the low temperature inflection point and the peak temperature and does not coincide with eitheri.e., B,i < O , , < O r , . Since (AB), < (At?),< (AB),, it also follows that Oi < 0, < 0,. This conclusion has been demonstrated graphically by Vold (26). Recall that the peak of the TGA curve corresponds to 0,; thus it is a further consequence that the temperature a t which the T G A curve peaks is always less than the temperature a t which the D T A curve peaks. Use of H e a t i n g Rate. When multiple peaks occur: there is thermal interference between them and it may not be possible to use the method of Borchardt and Daniels. If the peaks are chemically independent, it would be convenient to have a method whereby several different D T A curves could be generated by changing some parameter, and the collection analyzed ‘only in the immediate vicinity of the peak of interest. T h e heating rate is a parameter that can be used for this purpose. Differentiation of Equation 14 with respect to 0, yields
When the reaction rate is a maximum, this expression must vanish a n d therefore, 40
I&EC
FUNDAMENTALS
This can be \vritten
If both D T A and TGA equipment are available, this expression can be used to determine graphically, n , and t using several pairs of curves, each pa.ir corresponding to a different value of
AB
Oi'.
Llurray and \Vhite ( 7 9 ) assumed n = 1, determined 6, using TG;Z, and put (dAO,'dO,), = 0. I n this event Equation 26 becomes
dol
Kissinger ( 7 7 ) argued (incorrectly) that the maximum reaction rate occurs a t the peak of the DT.4 curve, so that
+
(E),
=
0, 8, + 8, and
$m
+ 4,
(2)7,1
Further it was con-
tended that n~,L,~-l = 1 (actually. it need only be insensitive to 07'). Corresponding to these assumptions one obtains
In
($)
= -
8:. + (+) -
d2A9
de:
In
or, in dimensional form
E
as a n approximate relation for the graphical determination of E using only the peaks of D T A curves obtained by varying the heating rate. It can be shomm that use of Equation 29 leads to large errors in the values obtained for E and A! even though the maximum reacrion rate may occur very close to the peak of the D T A curve. Kissinger appears to have used the method bvith some success in the thermal dehydration of clays in block-type D T A cells. For such equipment it may be that the greater thermal lag introduces a compensating effect not comprehended in the treatment of stirred systems. However, the foregoing considerations suggest that the method be used Lvith caution. T h e strong assumptions required eliminate the heat balance, but no assessment of the validity of the results can be made without it.
t,/&
(1
Om
I .03
1
€=I008
"*'
PI
22
23
24
4
The set of Equations 13 to 15 can be solved bv classical methods, providing thr. nonlinearity in 6 is removed; otherwise. one must resort to numerical techniques Both appr oaches are discussed below. Integrated Source Function Approximation. Since the temperaturr difference. A8, is usually small in comparison to the absolute temperature, A@'Or -, the behavior and sensitivity of the curve to systematic changes in the various dimensionless input parameters \\.ere investigated. Several sets of problems \\-ere solved in Xvhich all of the input parameters for a given set were held constant except one. Results are sho\vn in Figures 4 to 8. T h e numerical values chosen are representative of the decomposition of benzenediazonium chloride, as reported by Borchardt and Daniels (5). I
0.05 I
a
5
IO'*
8: * 0.0594 n=I
Ijl
n
E.840
t-=
I
1008
16
17
18
0.02
19
20
Figure 7.
14
16
20
18
22
24
26
28
30
21
22 8,
23
24
25
26
27
Variction of a
32
8,
Figure 5. 0.10
Variation of
E
0.16
I
0.08 E
IO08
1
n
n:l
0 04
r*10'8
n-0.10
8:=0.0594 E = 1008
0 0594 (I'C./MIN)
0 02
16
17
18
19
20
Figure 6. 42
l&EC
21
22 8,
23
24
Variation of 8,'
FUNDAMENTALS
25
26
27
18
19
20
21
22
23
24
25
26
8,
Figure 8.
Variation of n
27
28
29
Table 1.
Run '$TO
r,
C
CY>
Zlcn
1 2 3 4 5
0 1 1 1
6
1 92 1 94 2 03 3 09 3 92
-,
8
9 10
68 01 03
09 1 85
Wzn 0 0 0 0 1 0 0
1 1 1
890 755 755 -92 058 804 804 125 071 115
Kinetic Parameters Determined Experimentally I'4t1,
C
Ifin
11 503 9 619 9 6'2 9 395 13 763 10 538 11 185 12 30' 7 640 6 440
(2 7 ),>
oT,> K 322 31 326 14 32- 02 328 66 333 64 333 53 334 11 134 08 338 43 342 58
C
0 41' 0 618 0 662 0 652 1 347 1 223 1 31' 1 221 1 208 1 262
.lv,
Standard de\-iation Result from ax,,zss. 1/ T , (Figure 9 ) Result from Kissinqer's method (Figure 10)
r,
and overlapping peaks. T h e differential equations of Borchardt and Daniels can also be solved numerically. resulting in theoretically generated thermograms. Several theoretical thermograms for the decomposition of benzenediazonium chloride were generated and compared to corresponding experimental thermograms. .I he differences between the theoretical and experimental curves are generally \vel1 Lvithin = l o % Lvith regard to peak height and less than 1y0relative to peak location. Numerical values of E and .4 measured for the benzenediazonium chloride decomposition are summarized by E = 30.7 =t 7.6% (kcal. per mole) and log A = 19.' i 8.2% (A in m i n . 3 ) . T h e broad variations are possibly related to catalysis or to the combinations of heating rate. :ample size, and thermal constants employed in the various experiments. By graphing In k , us. l , ' T , (\vhere p refers to the peak of each DT24 curve), a single value for E of 28.7 kcal. per mole is obtained. I t is found that all data conform to the relation E, log A = 1.56 =t 0.9% and thus this ratio is approximately constant.
Table
Per cent error in peak height = 100
Run
1 2 3 4
5 6
'
8 9
10
111.
Comparison of Theoretically Computed and Experimental DTA Curves rc Error Based rc Error Rased on Individual E and 9 on 4 r e r a q r E and d Peak Peak Peak Peakheight location hright location 5 3 0 36 5 3 1 2 -6 0 -0 31 -12 8 1 0 -6 6 -0 35 -16 2 0 5' -2 2 -0 65 -11 8 -0 29 -9 6 0 08 38 n 20 -3 9 -0 16 -9 3 0 24 -7 -0 63 -9 6 0 1-1 8 0 00 10 0 16 -0 61 -1 8 - 0 04 -3 2 0 22 -4 1 -0 42
-
-
-
VOL. 4
NO. 1
FEBRUARY
1965
45
From the above value of E together with this ratio it follows that log A = 18.4 ( A in min.-l), Even the values for E and A derived from application of Kissinger’s method satisfy this relation. although individually they are incorrect. An explanation for the constancy of E ’In A proposed by Moeltqn-Hughes is considered inadequate and it is anticipated that a more detailed consideration of results of this kind \vi11 be the subject of a forthcoming paper. Acknowledgment
(9) Freeman, E. S., Carroll, B.. J . Phys. Chem. 62, 394 (1938). (10) Hildebrand, F. B.. “Introduction to Sumerical Analysis.” McGraw-Hill>New York. 1956. (11) Kissinger, H. E.. .4naI. Chrrn. 29, 1702 (1957). (12) Kissinger. H. E.; J . R ~ s .Vat/. . Bur. Std. 57, 217 (1956). (13) Mackerizie, K. C.. “Differential Thermal Investigation of C h v s . ” Mineralozical Societv. London. 1957. (14) hoel\ryn-Hu$hes, E. .A,. “.Physical Chemistry,” 2nd rev. ed., p. 1245. Pergamon Press. N e w York. 1961. (1 5) hloel\vyn-Hughes. E. A,: .Johnson. P., Trans. Faraday SOC. 36, 948 (1940). (16) Murphy. C. B.. A n a l . Chem. 30, 867 (1958). (17) Ibid., 32, 168K (1960). 1181 Ibid.. 34. 298K (1962). (l9j Murray. P.. \\.h’ite. .J:. Trans. Brit. Guam. Soc. 54, 204 (1955). (20) Se\vell. E. C . , Cloy .\lrnrra/s Bull. 2, 233 (1955). (21) %\\ell. C. C.. Gt. Brit.. Dept. Sci. Ind. Kes.. Bid. Res.. Series of Notrs (1952 ~1056). (22) Smothers, \V. Chiang. Y.. ”Differential Thermal Analysis. Theory a n d Practice.“ Chemical Publishing Co., New York. ,
‘The authors express their appreciation 10 M. R . J. \Vyllie for being the first to suggest that DTA might be useful in the study of crude oil oxidation, A. B. Hartman for designing the electronic components of the DTA unit. and R . L. Gergins for his valuable assistance with the experimental work. Literature Cited
(1) Allison, E. B., Silicates Ind. 19, 363 (1954). (2) Blumberg. A. A . , J . Phys. Chem. 63, 1129 (1959). (3) Boersma, S. L., J . Am. Cernm. Soc. 38, 281 (1955). i4j Borchardt, H. J.. Ph.D. thesis, Uni;,ersitv of IVisconsin, June 1956. (5) Borchardt. H. J., Daniels, F., J . Am. Chem. SOC.79, 41 (1957). (6) Crossley, M . L., Kienle, R. H.: Benbrook, C. H.: Ibzd., 62, 1400 (1940). (7) Denisov, E. T., Zrcest. Akad. n’auk SSSR, Otdel. Khim. .Vauk 1960, 1298. (8) Ellis, B. G., Mortland, M. M., .4m. itilzn~ral.47, 371 (1962). >
I
1958.
(23) Spiel. S., Berkelhamner, L. €I.>Pask. J. A , . Davies. B.? U. S. Bur. Mi,ies, Tech. Paper 664 (1945). (24) ‘Tadema. H. .1.. Erdol Rohie 12, 140 (1959). (25) Vaughn. F.. Cloy .IllineiuIs B d l . 2, 265 (1955). (26j VOIJ, M. .J,, A n a l . Chem. 21, 683 (1949): (27) Vulis, L. A , . Zh. Tvkhri. F i r . 16, 83 (1946). (28) \\‘ads. G.. .Vi$fion Ragaku Zasshi 1960, 1656.
,
RECEIVED for review February 10, 1964 ACCEPTED June 22, 1964
Di\-ision of Petroleum Chemistry, 144th Meeting, XCS, Los Angeles, Calif.; .April 1963.
CONDENSATION OF WATER VAPOR FROM A NONCONDENSING GAS ON VERTICAL TUBES IN A BANK J A M E S T. SCHRODT A N D E A R L R . G E R H A R D Chrmical E n p e e r i n g Ilepartment. C n i i e r s l t y of Louzsriile. Louisr / / l e , Ky
Data are presented on the condensation of water vapor from saturated air streams. under turbulent transverse flow to short cylindrical condensing surfaces. were varied from 2000 to 20,000.
Tests were conducted
Gas-phase Reynolds numbers
Values of JD and JH obtained are expressed graphically as functions
of the appropriate Reynolds numbers, and are shown to agree reasonably well with data from the literature.
s
of cooler-condenser problems are approached in various \cays. I n principle they all follow the accepted theory that the cooling and change of phase occur in three simultaneous steps: removal of sensible heat of the mixture, condensation and removal of latent heat, and removal of the sensible heat of the condensed phase. Colburn and Hougen (2) expressed the heat removal mathematically as OLUTIOSS
+
(1) h , ( t , - ti) k,‘Z.Z, x ( p , - p l ) = h o ’ ( t , - t w ) but because of the lack of diffusion data they relied upon Chilton’s and Colburn‘s J factor correlation ( 7 ) JH
=
Jn
for its solution. T h e resolution of partial condenser problems is therefore greatly dependent upon the reliability of this analogy. 46
I&EC FUNDAMENTALS
Recent studies ( d , 7: 8) on condensing pure vapors from saturated noncondensable gases have substantiated the analogy for streams flo\ving parallel to the condensing surface. but there appeared to be no such data for the more practical situationi.e.. flokc transverse to the cooling surface. This note presents the results of a study of the air-water vapor system \chere condensation occurred from saturated streams at 90’ F. flowing a t right angles to vertical tubes. Five 10.0- X 0.75-inch o.d. brass tubes were aligned in a staggered manner behind 7 ro\vs of false tubes in a 3.25- Y 10.0-inch insulated duct. A vertical rather than a horizontal arrangement \vas chosen after preliminary investigations indicated that condensed liquid falling from horizontal tubes was re-entrained in the air stream. Furthermore. Comings. Clapp. and Taylor ( 3 ) indicated that induced turbulence generated by the false tubes \\.odd providc comparable conditions a t each tube surface for the transfer phenomena. T\centy-fow runs were conducted a t free stream velocities of 0..56 to 47.0 feet per srcond. T h e nearly constant temperatures. concentrations. and mass flow rates in the bulk stream passing through the compact tube bundle justified the
