ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT cients for the cross flowmeter would have to be determined experimentally. Nomenclature
C
orifice meter coefficient, dimensionless inside diameter, ft. Fanning friction factor, dimensionless local acceleration of gravity, ft./sec.$ meter differential, feet of fluid flowing K = velocity head coefficient, dimensionless AK = section 4 contribution to K due to pipe friction, dimensionless L = length, ft. R = meter differential, inches of carbon tetrachloride minus water Re = Reynolds number, dimensionless t’ = linear velocity of fluid, ft./sec.
= = f = g = AH =
D
literature Cited
(1) (2) (3) (4) (5) (6)
(7)
(8) (9) (10)
Chem. Eng., 60, No. 3, 124 (1953). Gentile, V., Jr., U. S.Patent 2,260,019 (Oct. 21, 1941). Hoopes, J. W., et al., Chem. En#. Progr., 44, 691 (1948). Kroll, A. E., and Fairbanks, H. T’., IND.ENG.CHEM.,37, 588 (1945). Lansford, W. M., Ureiu.Illinois Eng. E x p t . Sta., Bull. 289 (1936). Lapple, C. E., Chem. Eng., 56, No. 5, 96 (1949). Nord, M., Ibid., 57, Ko, 7 , 113 (1950). Perry, J. H., “Chemical Engineers’ Handbook,” p. 377, New York, McGraw-Hill Book Co., 1950. Ibid., p. 415. Steams, R. F., et al., “Flow Measurements with Orifice Meters,” New York, D. Van Kostrand Co., 1951.
RECEIVED for review May 22, 1953.
ACCEPTEDJanuary 22, 1954. Presented before the Regional Conclave of t h e A M E R I C A N CHEMICAL SOCIETY, Kew Orleans, La., Deo. 10-12, 1953.
Kinetics of Coupled GERARD
N. VRIENS
American Cyunumid Co., Bound Brook, N. J
2
M
ANY chemical proccsses of industrial importance behave
kinetically like a system of coupled-i.e., consecutive, or parallel-first-order, reversible reactions. This may be so even if the reactions are truly second order, provided that either one reactant is present in considerable excess or its concentration is maintained a t a constant level throughout the reaction. In still other cases reactions of more complex order may be represented empirically with sufficient accuracy by the first-order equations. Even when a quantitative representation of the reaction kinetics is not given by the simpler first-order equations, considerable insight into the reaction mechanism may often be obtained from them. This is particularly advantageous when, as is the case with coupled reversible reactions, an exact mathematical solution of the kinetics of higher ordered reactions is generally not possible. The system of coupled, first-order, reversible reactions is one of the most complex kinetic systems to be capable of exact mathematical analysis. Solutions for the case of consecutive reactions have been given ( I , 2, 4, 6); unfortunately, in tyro cases they contain errors (2, 6). Furthermore, the solutions referred to are complex and difficult to apply. The case for parallel reactions does not appear to have been solved. The purpose of this paper is to present the solutions to this important kinetic system for both consecutive and parallel reactions in a form which is simplified for ready application by the use of dimensionless ratios and groups. These equations, in the form given, have been conveniently and successfully applied to the study of several commercial reactions in these laboratories. The special conditions under which the kinetics of parallel reactions resemble those of consecutive reactions are discussed in detail. Dimensionless Groups and Ratios Are Employed To Develop Solutions for First-Order Rate Equations
Consider the following system of equations:
ki
k3
A &BaC kz k4 Then
@ dt
April 1954
= kzy -
klx
=
k,x - ksy - kay + k4z
r + y + z = l where 5
y
= mole fraction of mixture present as A a t time t = mole fraction present as B at time 1
= mole fraction present kl = specific rate constant kz = specific rate constant k3 = specific rate constant ka = specific rate constant
z
as C a t time t for conversion for conversion for conversion for conversion
of of of of
A to B B to A B to C C to B
Equations 1 to 4 can be solved simultaneously to give the following second-order differential equations:
(5) d2z
+ (kl + kz + ka + ka) 2 + (kik + kzka + kika) z dz
kika (6)
Solutions to Equations 5 and 6 will be given under the boundary conditions imposed by the t a o most important cases, that of consecutive reactions in which case the starting material consists entirely of A and that of parallel reactions in which case the starting material consists entirely of B. Case I. Consecutive Reversible Reactions. Consecutive reversible reactions are commonly encountered in practice as, for example, in t’he case of alkylations in which 2 moles of alkylating agent react successively with the substrate. If the alkylating agent is present in excess or is maintained a t a constant concentration, or pressure, the first-order equations may be expected to apply. Of course, in such a case the pseudo equilibrium constants for the first-order reactions will be a function of the particular concentration of alkylating agent used. The equations for consecutive, firsborder, reversible reactions were first solved and discussed in detail by Lowry and John (4). Solutions have also been presented, in slightly different forms, by Frost and Pearson ( I ) , Sherwood and Reed (6), and Hougen and Watson ( 2 ) . I n the case of the latter two references, errors in
INDUSTRIAL AND ENGINEERING CHEMISTRY
669
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT sign occur in several of the formulas for the constants of integration. These discrepancies ma>- bc observed by comparing the formulas given in this paper with those in the references cited. I n all of these prior solut,ions, from two to four reaction rate constants appear in the final equations, thue making them estrrmely awkward to apply t o actual reaction rate studies. I n the follo.il-ing solutions these constants have been completely eliminated. Before solving Equations 5 and 6 it is desirable to simplify the constants. Although the four reaction rate constant. will vary wit,h the process variables, it is usually true that ratios of these constants are relatively independent of th(3 temperature, catalyst, etc. For esample, k l / k 2 and k3/k4 represent equilibrium constants which are completely independent oi' the catalyst, and, in most cases, nearly independent' of the temperature over the narrow temperature ranges normally of commercial interest. Similarly, the ratio of t,he forward rat? constants for the two reactions, 1;3/kl: may be expected to h t comparatively constant over varying reaction conditions, provided the consecutive reactions are of the same general t>.pp. Equations 5 and 6 may be rewritten as follows:
u-here
I
