February 1948
INDUSTRIAL A N D ENGINEERING CHEMISTRY
o
HOURS
Figure 2
heated to 55-60' C. to the red phosphorus then add thc stdchiometric quantity of aluminum sulfate in the form of a 10% solution heated to 55-60' c.,agitate the mixture for about an hour, Blter, and dry. Since copper and iron are mainly found on the surface of rcd phosphorus, and since the rate of oxidation increases with specific surface, it is advantageous to remove as much of the fines as the use of the particular red phosphorus will permit. In the irwesti-
303
gation described, the particles less than 10 microns in diameter were kept below a maximum of about 1% by weight. This can be accomplished by sedimentation, contrifuging, elutriation, or any other similar procedure. In conclusion, by the processes of size classification, removal of iron and coppcr, and the addition of hydrated alumina, a stable red phosphorus can be produced. Such a material has been produced on a pilot plant scale in cooperation with the Oldbury Electrochemical Company and the Chemical Engineering Department of the Tennessee Valley Authority. This new product remains free-flowing for at least five years when exposed to normal atmospheric conditions. Furthermore, the disadvantages of the dsual commercial red phoaphorus-namely, the generation of acidic, hygroscopic, and poisonous oxidation products and the hazards of spontaneous combustion-have been minimized to the point where they are, for practical purposes, nonexistent. ACKNOWLEDGMENT
The authors wish to express their appreciation to C. C. Fawcett, E. R. Reehel, and J. W. Mitchell of the Frankford Arsenal Ordnance Laboratory for their helpful suggestions and cooperation, and to the Ordnance Department for permission to publish this paper. REOEIVED January 8,1947.
Contact Times of Continuous-Flow Reacting Systems with Volume Change STUART R. BRINKLEY, JR. Central Experiment Station,
U. S. Bureau of Mines, Pittsburgh, Pa.
A
general relation between contact time and space velocity is obtained by comparing the solutions of the Euler and Lagrange forms of the hydrodynamical steady-state equations for the composition of a flowing system which undergoes chemical reaction with accompanying volume change. It is assumed that the reactor is isothermal and isobaric and that the flow is one-dimensional. This relation is applied to the calculation of contact times for particular rate laws. An approximate formula is developed for the estimation of contact times when the rate law is unknown, and the resulting error is obtained for several cases by a direct numerical comparison with the correct expressions. The application of these methods to reactions on a granular catalyst is discussed.
S
TUDIES of reactions taking place in continuous-flow sys-
tems, which are conducted with a v:ew to the determination of the governing rate law, are most appropriately correlated in terms of thc contact time, since the rate law in terms of this quantity has the same form as in the static case. If the reaction proceeds without volume change, the contact time for one-dimensional flow with negligible diffusion is simply the reciprocal of the space velocity based on the feed and on the free volume of the
reactor. However, if a volume change accompanies the reap tion, the relation between contact time and space velocity is somewhat more complicated, and involves the degrce of reaction in a manner determined by the rate law. The hydrodynamical equations for continuous-flow system in which chemical reaction occurs have been obtained by Eckart (1). These relations have been employed by Hulburt ( X ) in s discussion of the kinetics of continuous flow reacting systems. Instcad of the contact time, Hulburt employs as independent variable the distance from the inlet of the reactor, and he d i e cusscs in detail thc application of the resulting relations to typical systems. The use of contact time as independent variable leade to an alternative description of continuous-flow systems which is simpler in some particulars than that provided by the use of distance (or space velocity), In the present communication a relation is developed between the space velocity and the contact time for systcms in which a volume change accompanies the reaction. Examples are given of the application of this general relation to three simple types of reaations, two of which have been previously considered in detail by Hulbkrt. An approximate formula is developed for the estimation of contact times where the rate law is unknown, and the usefulness of the approximate relation is illustrated by numerical comparison with the exact results for the three simple types of reactions.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
304
Alternative statements of the fundamental hydrodynamical equations are provided by the formulations of Euler and Lagrange. The independent variable in the steady-state form of the Erst is distance, which is readily convertible to space velocity. The steady-state form of the second involves the time a9 indepndent variable. Therefore, a comparison of the solutions of the two formulations of the equations expressing the degree of reaction will result in a parametric relation betmeen space veloritg snd contact time, with the degree of reaction as parameter rWE I X h D A H E > T A L KELATTO\
The fundamental hydrodynamical Equations 1 describt: the transport of energy, momentum, total mass, and the mass of each constituent by the flowing mixture. We shall consider a reactor in the form of IL cylindrical tube within which reaction occurs in a homogeneous flaid flowing isothermally with negliqible piessure drop. We shall aswme that the composition, density, and velocity of the fluid are uniform over any cross ser?ionof the reactor and that the effect of diffusion can be neglected These assumptions take the place of the flow pattern and temperature distribution. which would be obtained by a solution of ?he equations for energy and momentum transport, and n-e need *orisider only the solution of the two remaining fundamental zquations subject to these assumptions. The Euler formulation for the system under consideration of the equations for mass transport and the transport of the individual constituents may be written in the form (Z),
mined by the quantity $. Therefore, these quantities map considered t o be fuiictions of q only, 0 =
PW, ra
=
*b
raw
and the integration of Equation 5 along a contour of cortstant time over the length of the reactor leads to the expremion.
where O* = 1/S is the apparent contact time (based on the velocity of the feed gas) for the reactor of length L, and q L is the fraction of the reference constituent remaining a t the exib of the reactor. The Lagrangian formulation (3) provides an alternative statement of the fundamental hydrodynamical equations. The Lagrange coordinate xo of a fluid particle is equal to the value of the Euler coordinate of that particle a t a reference time lo at which the description of tho syst,em i s to begin, In view of the definition
the Lagrange partial time derivative is related to the Euler partis1 derivatives (for the one-dimensional case) by
which reduces t o
mere p u
=
density of mixture, mass/unit volume
= linear velocity of mixture, distance/unit time
= concentration of kth constituent of mixture (moles/ unit weight of mixture) r k = rate of formation of kth constituent of mixture, moles/unit weight of mixture/unit time 1 = time z = distance coordinate, measured parallel to the axis of the reactor ck
if it be further assumed that the reaction and flow have reached the steady state, then the density, velocity, and composition a t a €ked point are independent of the time, and Equations 1 become
cormtant = C
Integrating Equation 8 along the path zo = 0 from fo to time for which x(20
t ~ , ,the
= 0, t t ) = L
one obtains
(3)
where 6
(4)
Equations 6 and 8 relate the true and apparent contact timee to functions of the fraction of the reference constituent remaining a t the exit of the converter.
Equation 2 can be integrated a t once, with the result pu =
for the steady stale. The Lagrangian form of the steady-state equation for the reference constituent is then obtained a t once from Equation 3.
wtiere G = mass current, mass/uuit cross section area/uIut t h e
Hulburt ( 2 ) has shoFn that the composition of the mixture is completcly determined by the specification of a reference constituent which can be designated by the subscript a. Employing the abbreviations 7 = ca/call, f = x / L , S = G/paL, where caa and po are the values of the concentration of the reference constituent and the density, respectively, at the inlet to the reactor, 6 is the length of the reactor, and S is the space velocity bascd on the feed gas, Equation 3 for the reference constituent can be mitten in the form.
(5) The density of the mixture and the rate of formation of the refer*ace constituent are functions of the composition which is deter-
=
contact time for the reactor of length L
0 = F(TL),
o*
= G(T1,)
(in)
where
and Equations 10 represent in parametric form a relation between the contact time and the apparent contact time bwed on the space velocity of the feed. However, a more useful relation is obtained by division ill)
305
INDUSTRIAL AND ENGINEERING CHEMISTRY
February 1948
ZEROth-ORDER REACTION. For this Case,
pra=
-k0
(14)
Substituting in Equations 10, one obtains
The ratio of the real and apparent contact times is obtained by substitution in Equation 11.
The factor e/6* is plotted as the full curves of Figure 1, as a function of T L for various values of the parameter P. FIRST-ORDERREACTION.We consider the simple case for which pror =
- kt(pca) =
- 4ca0 (~17)
(17)
From Equations 10 one obtains
0.03
0.05 0.07 ai
0.2
7L
0,3
Q5
0.7
6 = F ( ~ L= )
I
Figure 1. Ratio of Real and Apparent Contact Times of Zeroth- and First-Order Reactions
expressing the correction factor to the apparent contact time as a function of the fraction of reference constituent remaining. I t will develop that the reaction-rate constant does not appear in this expression, and it will be shown that this circumstance makes possible the application of Equations 10 to catalytic reactions in reactors containing granular catalysts.
- 1 1%
(18)
qL,
The expression for 8* is identical with Hulburt's (2) solution for the first-order reaction. The ratio of the real and apparent contact times is a function of q L , given by
The results of Equation 19 are shown by the broken lines of Figure 1.
CALCULATION OF CONTACT TIMES FOR TYPICAL RATE LAWS
For the case of a single over-all reaction, Hulburt (2) has shown that the density and concentration of each constituent may be simply expressed in terms of the concentration of a reference constituent, which we take to be a reactant. We omit the details of his analysis and reproduce here only his results. The chemicalreaction may be represented by ZkPkAk
=
0
where Ak represents the symbol of the kth constituent of the mixture and where Vk, the coefficient to this symbol in the chemical equation, is taken to be positive for products of the reaction, negative for reactants, and zero for inert constituents. The concentration of the kth constituent is related to that of the reference constituent by ck = ChO
Vk -Cora VU
(1
- 7)
(12)
where C ~ and O core are the values of c i ~and cor, respectively, at the inlet to the reactor. The density of the mixture is expressed by
and MO= mean molecular weight of the feed
If the rate law for the chemical reaction is known, the rate of formation of the reference constituent Forcan now be expressed as a function of q and the integrations represented by Equations 10 performed. We consider, as examples, three simple cases.
0.03 Figure 2.
0.05 Q07 QI
?t
0.2
Q3
05 0.7
I
Ratio of Real and Apparent Contact Times of Second-Order Reactions
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
306
Vol. 40, No. 2
IO
5
5
g o
0
c 2
W
0
a a
0
a
IW
z
a W
pW -5
-5
a
[L
0
a
-10 0
0.4
0.2
as
0.6 0.7 Q0 CONVERSION, I -?,.
8=2
I
Figure 3. Error in Estimation by Approximation Formula of Contact Times for Zeroth-Order Reactions
f SECOND-ORDER R E A C T I O l . Hulbui 1 ( 2 ) has considered the case for which the rate law is
P r a= - X 2 ( P L n ) ( ~ l ( ~ )
(20) -5
From Equations 10 one obtains'
Q
0.2
0.4
Q.6
0.7
CONVERSION, I
0 = F(9L) =
- 7 (1 + 8 ) log,
?lL
PSI
(21)
=
G ( ~ L=) + ~ ) z l O y , 9L
- (,
+
log, 11 - (1 - 9 L 1 ]
-I
_l_____l______l____l_
li?cgopo? (1
- 1_-__--__ d2 (1 - ? \ ( I- ,)
if r # 1, where r = caovp/cp3va. \\'e note that it is always possible to assign the subscripts N and a so'that T < 1. If T = 1, the results are
e
F ( , , ~)
=
___ (1
+ 3)(1 - +
O L )-.--'''ga i j 9 L
=
KL-+.p)"i_d%dJ
s(t)
- 9L) t 2 8 9 L ( 1 kxp,PO17 L
e-
i-2
+ $1 loge 9 L
+ p ) log, 9~ - + 8 ) log, [ I - r(1 - VL)I + ~ ) 2 1 o g , - + 8)*log,__-[I - ((1 -
+(I (I
I(T
TI,
(T
r p 2 (1 - r ) ( l
-
- w,)
(24)
where
The ratio of the real and apparent m i t a c t 1,imrsis given by -0 --
N e--k(t--ia)
Equation 24 is identical with the expression for an isothermal, isobaric, first-order reaction if k is interpreted as the rate constant rather than the initial slope of q. Equation 4 may be solved for the linear velocity and combined with Equation 13. Then,
(22) = (&&)
function. Since the initial value of q is unity, there results the approximation
91
k? ('BnP07lL
o*
0.9
Figure 4. Error in Estim;ition hy Approximation Formula of Contact Times for Second-Order Reactions
- 0 $. 3 ) loa, i _1_ _- -/ (-1___ - 9L)l k2cg, (1 - r )
_____I_______
e"
0.0
- (IL
VI,)~
u (238)
ifr # l a n d b y
if r = 1. The results of Equations 23 are presented graphically in Figure 2. These examples suffice to illustrate the methods by which the contact time corresponding to a particular space velocity may be calculated if the rate law for the reaction is known. I.:STI~144?'IOI*iOF CONTACT TIMES FOR REACTIOYS OF UNKNOWN RATE LAW
The form of the curves of q L tis. e suggests that the peak approximation may frequently represent with sufficient accuracy the & m e curve. This approximation is given by an expansion in a Taylor series of the logarithm of q about its initial value at t o with retention of the first two terms, and it is a natural assumption for the representation of a monotone, decreasing
=
2PO 11 + so - ?)I
(25)
Substituting Equations 7 and 24 and introducing the abbreviations = x / L and the apparent contact t h e 8* = l/S = poL/G, one obtains ~~
r
Integrating, as for Equation 9, along the path 20 = 0 between the time limits t o and t~,,corresponding to the limits = 0 and = 1, respectively, one obtains
r
r
The constant 12 may be expressed in terms of the quantity q L by means of Equation 24. The resulting expression, after rearrangement is 1 Hulburt's discussion ( d ) of the second-order reaction contsins two algebraic errors. The integrand of his Equation 21 is the reciprocal of the corroot expression. The solution for r # 1, Equation 22, is correct, but tha solution for T = 1, Equation 23, is incorrect. The correot form may be obtained by rearrangement of thesecond of our Equations 2 2 .
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
February 1948
307
ACKNOWLEDGMENT
The approximate expression, Equation 28, is of course identical with Equation 19 for the first-wder reaction. The result could have been obtained by the assumption that the conversion follows a first-order course to a sufficiently good approximation, a n assumption that is essentially physical in nature. We have deduced Equation 28 from an assumption of mathematical nature involving a different interpretation of the quantity k. The errors that would be incurred through the use of the approximate expression for zeroth-order and second-order reactions are shown by Figures 3 and 4. For considerable ranges of the amount of conversion, 1 q L , the approximation provides an expression sufficiently precise for practical purposes. It seems to be a reasonable assumption that Equation 28 may be used 6 t h equal confidence for reactions of orders ranging from zero to two. .
-
REACTIONS O b GRANULAR CATALYST
These considerations apply t o catalytic reactions in reactors packed with granular catalyst if apparent contact time is calculated from a space velocity defined as the volume of gas at the temperature and pressure of the feed gas per free volume of the catalyst bed per unit of time. If the space velocity is based on the total volume of the catalyst bed, the values of the contact time resulting from the application of Equation 28 must be multiplied by the ratio of the free volume of the catalyst bed to the total volume. If the uncorrected values are used in a kinetic analysis of operating data, the resulting rate constant will be the product of the true rate constant and the ratio of the free t o the total volume.
The author is grateful to C. R. Siple for assistance in the preparation of the figures.
.
NO.MENCLATURE
concentration of kth constituent of mixture, moles/unit weight of mixture initial concentration of kth constituent of mixture G = mass current, mass/unit cross section area/unit time L = reactor length S = space velocity based on feed, volume feed/volume reactor/ unit time t = time u = linear velocity, distance/unit time x = Euler distance coordinate, measured parallel to the axis of the reactor 20 = Lagrange distance coordinate p = relative residual volume l?k = rate of formation of kth constituent of mixture, moles/ unit weight mixture/unit time p = density of mixture, mass/unit volume PO = initial density of mixture 6 = contact time e* apparent contact time = 1/S 7 = fraction of reference constituent unreacted = reduced Euler distance coordinate = x / L ck =
C ~ = O
=i
r
LITERATURE CITED
(1) Eckart, C., Phus, Rev.,58,269 (1940). (2) Hulburt, H. M., IND. ENG.CHEM., 36,1012 (1944). (3) Lamb, H . , “Hydrodynamics,” 5th ed., p. 12, Cambridge, Cambridge Univ. Press, 1924. RECEIVED November 22, 1946. Published by permission of the Director, Bureau of Mines, U. 9. Department of the Interior.
GR-S 65, a Low Water Absorption Cosolvmer I J
J. C. MADIGAN, E. L. BORG, R. L. PROVOST, W. J. MUELLER, AND G. U. GLASGOW U.S. Rubber Company, Naugatuck, Conn. D u r i n g the war, military and other essential requirements for insulation placed emphasis on the need for a large volume of a new GR-S type of synthetic with a lower water-absorbing tendency t h a n the standard material. I t was assumed t h a t creaming salt retained by the finished polymer was responsible for this tendency. Attempts to coagulate with sulfuric acid alone in conventional equipment resulted in a floc t h a t was extremely tacky and could not be satisfactorily handled in subsequent operations a t the copolymer plant. Experiments were r u n using equipment employed in the alum coagulation of GR-S, consisting essentially of two concentric pipes so arranged as t o introduce compressed air and latex beneath the acid solution surface. While some improvement was realized, the subsequent operations were still difficult because of floc tackiness. I n combination with this equipment, various protective agents were added to the coagulant to reduce stickiness. Glue was found the most satisfactory. By properly adjusting latex and air flow rates and glue concentration, a suitable coaguIum was obtained. As predicted, the polymer thus obtained had
a water absorption lower t h a n other GR-S types available. Successful plant operation was realized a t a Cost comparable t o GR-S. The product, GR-S 65, was used in various applications by the wire and cable industry and has been manufactured i n large quantities since December 1944. I t represents a large step in t h e development of a “pyramid” polymer for wire and cable insulation.
D
URING the recent war, military and other essential requirements placed great emphasis upon the need for a large volume of a GR-S type of synthetic elastomer with good electrical properties for wire and cable insulation. To attain this end, it was decided to attempt production of a copolymer with a lower water absorption than that characteristic of available types of GR-S. One type of GR-S was reasonably acceptable from this standpoint but had been diverted to uses of even higher priority. This was made at the Naugatuck plant opcratt I by United States Rubber Company, which is specially equipped with machinery that strains the wet and dry stock. Regular GR-S, made in a “standard” plant and specially leached at great effort
