are identical and are a unique function of pL (Bell, 1970). Curves 3 and 5 demonstrate t h a t at 2000 XHz and no = 1011 cmP3the distribution is sensitive to t h e individual values of both p and L. However, if the central electron density is increased, (curve 4), the distribution becomes coincident with t h a t obtained a t lower frequencies and is again dependent only on the product pL. The distribution of electric field strength is shown in Figure 2. Here again the curves for 20 and 200 M H z coincide. At 2000 V H z there is a very pronounced effect of n+ and p , a s can be seen b y comparing curves 3, 4,and 5. The maximum in the distribution is reduced and shifted towards the center of the discharge for a decrease in noor a n increase in pressure. The behavior demonstrated in Figure 2 can readily be understood in terms of Equation 4,which governs the magnitude of E , as a function of electron density. When r >> a, then
(5) except for a very narrow region near x* = 1 where (T - 1) approaches the magnitude of CY. Where Equation 5 is valid, the electric field distribution is insensitive t o the magnitude of n+, the pressure, or the driving frequency. By contrast, when r 2 a , all three variables have a n effect upon the field distribution. Since t h e local value of the effective field strength determines the local value of the ionization frequency, t h e relative magnitudes of r and a will also influence the behavior of the electron density distribution. The effect of electric field strength distribution on the local rates of a reaction is illustrated in Figure 3. The normalized rate of ionization is plotted as a function of position. As can be seen, the distributions of ionization rates for 20 and 200 RIHz are identical. For the same value of pL the distribution a t 2000 NHz is different and is, furthermore, sensitive to variations in p and L , even though their product is held constant. When no = 10l2ern+ (curve 4),the distribution of rates is almost identical t o t h a t a t the lower frequencies. Based on a simplified model of a high frequency discharge, i t has been shown recently (Bell, 1970) t h a t the value of Eoe/p and the dimensionless distributions of electron density and
electric field strength were uniquely determined by pL. The central electron density, no, was shown to be a function of pL and the ratio of t h e power density to the gas pressure. These results were combined to illustrate that the distribution of rates for electron-molecule reactions occurring in two discharges of different size could be made identical if the power density and the product pL were both held constant. The original results were deduced for a discharge operating a t 20 MHz. The present analysis allows the same criteria of similarity t o be extended t o discharges operating a t other frequencies, provided the inequality ro >> CY is satisfied for both discharges. Nomenclature
D, = ambipolar diffusion coefficient, cm2/sec e E L
charge on electron, coulomb electric field strength, volts/cm plate separation, cm m mass of electron, kg n = electron density, electrons/cm3 n* = nlna p = pressure, torr r = dimensionless electron density 2id = drift velocity, cm/sec x = distance, cm x* = 2x/L a = Townsend ionization coefficient, ionization/cm CY = dimensionless elastic collision frequency e, = permittivity of free space, faradayslm v = elastic collision frequency, sec-1 v i = ionization frequency, sec-l w = driving frequency, sec-l = = = =
literature Cited
Allis, W. P., “Handbuch der Physik,” S. Flugge, ed., Vol. 21, p. 283, Springer-Verlag, Berlin, 1956. Bell, A. T., IND.ENG.CHEM.FUNDAM. 9, 160 (1970). Dinan, F. J., Fridmann, S., Schirmami, P. J., Adaan. Chenz. Ser. No. 80,289 (1969). Eremin, E. N., Vasil’ev, S. S., Kobozev, K.I., Zh. F i z . Khim. 9,48 (1937). Streitweiser, A., Ward, H. R., J . Amer. Chem. SOC.85, 539 (1063). ALEXIS T. BELL University of California Berkeley, Calif. 94720 RECEIVED for review January 14, 1970 ACCEPTED July 20, 1970
Correlation of Kinetic Data from laminar Flow-Tubular Reactors By expressing the residence time of annular elements of the fluid in a laminar flow reactor as a function of reactor length and radial position it i s possible to relate the fractional conversion of reactant to dimensionless groups containing the rate constant, inlet concentrations, reactor volume, and flow rate. The functional dependence will vary with the order of the kinetic expression relating the rate of disappearance of reactant to concentration. For kinetic orders other than zero, the fractional conversion obtained in the laminar flow reactor will be less than that calculated by the plug-flow assumption. This analysis extends the previous treatments to include the general nth-order rate expression and first-order consecutive reaction rate expressions.
In many experimental investigations using liquid reactants, the velocity and concentration profiles are not constant across the reactor and integrated rate expressions based on the plugflow concept, that relate conversion and bulk contact time, are not applicable. I n the laminar flow regime, present in tubular reactors at Reynolds numbers less than 2100, a
parabolic velocity profile can exist over a major portion of the reactor and the residence time of the annular elements of fluid within the reactor will vary from some minimal value a t the center line of the tube where the velocity is a maximum to some high value near the wall where the fluid velocity is zero, This distribution of contact times resulting from laminar Ind. Eng. Chem. Fundom., Vol. 9, No.
4, 1970 681
:iow and the correct relation between conversion and flow ;ate was first considered by Bosworth (1948), who developed Ihe case for zero-order kinetics. Somewhat later this approach ,vas extended to second-order reactions by Denbigh (1951), while the first-order case was developed by Cleland and Kilhelm (1956). As there is a continued interest in this area, it was desirable to pull this information together and to extend the analysis to include the general nth-order reaction and a treatment of first-order consecutive reactions. The reactor is assumed to be long compared to its diameter, ith no change in density due to reaction, and a parabolic .elocity profile established over the entire length.
or expressed in terms of reactor volume and flow rate obtained by multiplying and dividing by rRw2,
This is the same result obtained for plug-flow reactors. For second-order reaction n equals 2 and Equation 7 becomes, after integration,
- 2p2 In ( 1 + l/p) (11) A plot of conversion us. p is given in Figure 1. Also shown,is f = 2p
the curve for plug flow.
f
2P/(1
=
+ 20)
(12)
For reactions whose rates are first-order, the reactant concentration at any time is given by l'hese are the simplifications required for a tractable analysis ,f the problem. The fluid in any annular element behaves like :iconstant volume-isothermal reactor with the molar flov ;ate of reactant in the reactor given by
d,Ta
=
c~*U*2rRdR
CA
=
CAoeekt
(13)
The molar flow rate a t the exit of the reactor becomes, after substitution of Equations 13, 5, and 1 into Equation 2 ,
(2)
,vhere C A = concentration of A and K A = moles of A per mit time. Expressing the rate of disappearance of A by
2
[ - ( 3 1R", 1
-
so1
-
2dR
(14)
Collection of terms into dimensionless groups gives
(3)
=
Cao'-"
+ k ( n - 1)t 1
CA
=
cAo(1
+ k[n - ~ ] C A O " - ' ~ ) ~
(4)
Since the time required for the reactant in each annular element to reach the exit is
L
L
substituting Equations 5, 4, and 1 into Equation 2 gives
[ - (g)2]
2rRdR
1
1
1
2NAo
,ives after integration for n # 1,
C'41-n
- h'a =--
=
TRm'UrnaxCAo
l1+ (1
p/V)l'l-"VdV
s,
Values for the third-order exponential integral, &(a), may be found in the "Handbook of Mathematical Functions" (1964).For most purposes, the plots found in Figures 2 and 3 should be adequate. I n many systems of interest, consecutive, irreversible reactions occur and intermediates are involved, as depicted in the following sequence.
(17)
-d-c a - klCA -
(18)
dt
(7)
Integration of the above and a material balance for component A gives CA = CAoe-k'l (21)
ki C B = CAo(e--Iclt kz - ki CA,
(1 +'p/V)lI1-nVdV
(9) 4, 1970
~-2132~
dCc _ - kzCB
1
Ind. Eng. Chern. Fundam., Vol. 9, No.
VdV
(6)
For a zero-order reaction n is zero and integration of Equation 8 gives
382
a/V
dt
'i'he group of terms in front of the integral sign on the right*land side is twice the molar feed to the reactor, and Equa.ion 7 when written in terms of the fraction reacted becomes
3 = 2
e-
where a = kL/U,,, = kV,/2Qp Substituting 2 for l / V leads to the tabulated function E3 (a).
hich leads to Equation 7 when the above terms are collected
dNA
-
2
to dimensionless groups and integrated.
LYA
-f
= CA
e-kzt)
+ CB + CC
(22) (23)
The fractional conversion of starting material X is given by Equation 16 and similar substitutions for CB and t give the kz related equation for N B with k l
*
10
8.0
6.0
0.4
2 .o
?I
