ANALYTICAL SOLUTIONS FOR SOME ADIABATIC REACTOR PROBLEMS .M. L. C .
J
D 0 U G L A S,
T h e Atlantic ReJining Go., Philadelphia, Pa.
E A G L E T 0 N, School of Chemical Engineering, University of Pennsylvania, Philadelphia, Pa.
The analytical solutions which describe adiabatic reactor operation for simple reaction mechanisms have been tabulated in terms of exponential integral functions. Also, the procedure for analytically determining the dynamic response of an adiabatic, tubular reactor has been outlined.
require the simultaneous solution of a t least tkvo?coupled, nonlinear differential equationsLe., the material balance(s) and the total energy balance. In general, this set of equations has no analytical solution. Solutions are available for a large number of special cases, however. The most common simplifying assumption is that of isothermal operation. This leads to solutions for many simultaneous and consecutive reactions, as well as for reversible and irreversible single reactions. Also. analytical solutions have been presented for a type of nonisothermal reactor that has been called a programmed reactor. I n this reactor: the temperature of the reacting medium is a function of time only and not of the degree of reaction. This occurs in practice in situations where the heat of reaction may be neglected-microbiological sterilization: reactions of dilute constituents, and in the preheat zone of reactors. Perhaps the first treatment of the programmed reactor probDeindoerfer and Humphrey lem was by Billingsley et al. ( 7 ) . ( 3 ) have since published a comprehensive list of solutions for many different cases. Both sets of investigators, however, have limited their discussions to first order irreversible reactions. These results may be extended to include all reaction mechanisms for Lvhich the variables are separable in the solution of the analogous isothermal case. This can be done because the exponential expression from the Arrhenius equation appears on the side of the equation Lvith the time derivative, and the other integral, containing the composition terms, is the same as occurs in the isothermal case. The solutions for the programmed reactor are obtained by rearranging the integral involving the exponential into a form identical with a tabulated function-namely, the exponential integral. EACTOR DESIGN P R o B L E m
Adiabatic Reactors
Another type of nonisothermal reactor problem which can be solved, for some simple mechanisms, in terms of exponential integrals is the adiabatic reactor. Apparently, the first work published on this problem was by Parts ( 8 ) . Gratch ( 4 ) and others have obtained some of these solutions, but the authors felt that a more complete list might prove helpful. The method has been applied to two other constant density reactions and to four constant pressure, gas phase cases. I n addition. the 116
l&EC FUNDAMENTALS
method has been extended to give the dynamic response of an adiabatic, gas phase, tubular reactor. The derivations for this case are given in Appendix A. Single Irreversible Reactions of Integral Order
For a single adiabatic reaction the mass and energy balances can be combined and solved to give a linear relationship between composition and temperature. ]%‘hen the Arrhenius expression for the rate constant is substituted into the energy balance equation and the composition term is replaced by the linear relationship in temperature. the resulting differential equation is one in which the variables are separable (see Equations l to 12 in Appendix A). .4 transformation of variables, Xvhere the new variable is set equal to the argument of the exponential, will simplify the composition integral. In fact, for a zero order reaction, the integral is identical to the second order exponential integral. Other reaction mechanisms give compound denominators which, for integral order reactions, can be simplified by the method of partial fractions. This reduces the solution to a summation of integrals, all of which may be expressed in the exponential integral form. In addition to Parts’ results, the solutions for a zero order and a different type of second order reaction in constant density batch or plug flow tubular reactors are listed in Table I. Corresponding solutions for gas phase. constant pressure reactions in plug flow tubular reactcrs are given in Table 11. TWOtables are necessary, since the definitions of the rate constants in the two types of rate equations are different (7). Both sets of solutions are based on the assumptions of constant heat of reaction and average heat capacity. N o second order exponential integrals appear in the tables, since these may always be reduced to an expression containing the first order exponential integral (integration by parts). More Complex Cases
Irreversible Reactions of Fractional Order. Since exponential integrals may be considered as special cases of the incomplete gamma function (which are tabulated for fractional arguments), one might hope to obtain solutions for fractional order rate mechanisms. However, the method of
partial fractions, required to simplify the compound denominator of the integral, will work only for polynomials, and thus the method appears to break down. Reversible Reactions,. We were unable to reduce the integral obtained for a first order reversible reaction to the exponential integral form. However, if either the forward rate constant, the reverse rate constant, or the equilibrium constant can be considered to be independent of temperature, a solution in terms of exponential integrals can be obtained. The procedure is similar to the development in the appendix and therefore the results for these three restricted cases are not given. Simultaneous Reactions. No general solutions have been obtained for cases involving more than one reaction. In the case of two, parallel, first order reactions A B and ,4 C, if one is able to assume that the heat of reaction is negligible for either one of the reactions, it is possible to obtain the ratio of C to B in terms of exponential integrals. Unfortunately. even for this special case, it is not possible to find the variation of -4.13. or C with time. In the case of conB C, the first reaction in secutive reactions. ,4 a sequence might fall under one of the cases given in the tables -i.e., if the heat of reaction for all subsequent reactions could be neglected. However, no other solutions could be found.
-
-
-- -
Table I.
Homogeneous, Batch or Plug Flow, Tubular Reactors at Constant Density
Zero Order vR
.4 7
=
k,
0 = ( p / h A , ) [ T e x p . ( E / R T )- To exp.(E/RT,)] -
(Ep/RXA,)[ E i ( E / R T )- E , ( E / R T , ) ] First Order ( 8 ) A
vR
___+
r = k,Ca
0 = ( l / A , ) [ E , ( E / R T )- Ei(E/RTo)l -
-
[exp.(E1’RT,,i)/.4~]l E i ( Z ) - E i ( z o ) ! Second Ordrr ( 8 ) 2A r
e
vR
=
k,Ca2
XE exp.(E/RT,,~)/pA,RT,d2 X [ e x p . ( Z ) / Z - exp.(Zo)/Zo - E , ( Z ) Smond Ordrr aA bB vR =
+
+ Ei(z$)l
__f
r = kCC.4Cn
+
0 = - [exp.(E/RT,~),/il,](C -~CA,, , b / a ) [ E L ( Z-) E t ( Z O ) l
lexp.(E/RT,dn,)/A,](Cno- C.4, hja)[Ej(TI’)
- El(”,)]
Dynamic Response of a Tubular Reactor
Bilous and Amundsori (1)have published an approximate solution for the response of a tubular reactor, with and ivithout heat transfer through the reactor wall, for the case where the flow is assumed to be incompressible and where the derivations from a knotrn steady state are assumed to be small (linear differential equations \\?re used). In order to use their solution. it is necessarv to have a set of steady state profiles. These can be obtained from an analog (or digital) computer or. if onl) adiabatic reactors with simple reaction mechanisms are considered, the exponential integral solutions may be used. The d>-namic responsia of an adiabatic, plug flow reactor is simply the appearance of the ne\r steady state conditions at some time (the residence time a t the ne\%steady state) later than the corresponding change in the inlet conditions. For simple reaction mechanisms. it is possible to calculate the new effluent conditions by a trial and error application of the exponential integral solutions (outlrt conditions are tried until the calculated length equals the actual reactor length). Also, the residence time can be expressed in terms of exponential integrals (see Equations 13 to 19 in Appendix A). Thus, the dynamic response of an adiabatic tubular reactor can be determined analytically for simple reaction mechanisms.
Table II. Homogeneous, Gas Phase Reaction Rates at Constant Pressure in Plug Flow Tubular Reactors Zpro Order A dvR
r = k V K / F = ( l / X A ) [ T e x p . ( E / R T )- T , esp.(E/RT,\1] ( E / R X A )[E,IE/RT ) - E l ( E / RT o )1 First Order
A-vR;
6 = v - l
r = kaa =
k*P(na/nt)
+
V e / F = - ( 6 / X A * P ) [ T e x p . ( E / R T ) - T oesp.(EIRT,,)] [ ( n o Anao f 6 E / X R ) / A * P ] [ E , ( E / R T) E , ( E / R T + ) I[ ( n o 6na,3)exp.(E/RTad)/-4*P] [ E , i Z )- Ej(Z,)]
+
+
Second Ordrr
Exponential Integral Tables
Perhaps, the two tables of exponential integral functions ivhich are easiest to find are in the ‘*Handbookof Chemistry and Physics” ( 6 ) and in the ‘.Tables of Higher Functions ’ (5). I t is simple to use the tables for mcst design problems, since the arguments of all expcnential intesrals in Tables I and I1 are positive for exothermic reactions. Holvever, for endothermic reactions some arguments ( TI7 and Z ) become negative. Confusion is created b\f the fact that certain tables define the exponential integral as
VOL.
1 NO. 2 M A Y 1 9 6 2
117
To use these tables, the sign of the argument and also the sign
in the last integral gives the results:
in front of each exponential integral in Tables I and I1 should be reversed. Tables are limited to values of the argument beFor absolute values of the argument tween -15 and f 1 5 . greater than 15, the following approximation may be used:
VKA*/Fnt =
Ei(x)
ez
= -
for 1x1
E/RT exp.(y)dy/y S E / R To
- exp.(E/RTad) X exp.(Z)dZ/Z (10)
sz:
> 15
Since
A numerical example, which illustates the proper use of the method, is given in Appendix B.
exp.( t)dt/t
Ei(x) =
(11)
m
VR = ( F n t / A * ) ( E i ( E / R T )- Ei(E/RT,) - exp.(E/RT.d) X [ E t ( Z )- Ei(Z0)lI ( 1 2 )
Conclusions
Analytical solutions for some simple adiabatic reactor problems can be obtained in terms of exponential integral functions. The method of obtaining the solutions is simple and straightforivard, but the use of these solutions sometimes is confusing. If only a single solution is required, it is usually simpler to do a graphical integration, rather than to become familiar with the tables. However, if numerous solutions or greater accuracy is desired? time can be saved by using the exponential integral solutions. Also, when testing parts of reactor design programs, the method can be helpful, since the greater accuracy provides a more conclusive check.
Thus, for any given reactor, the outlet temperature (and from Equation 3, the outlet composition) can be calculated for any set of inlet conditions. Residence Time ( 9 ) d8 =
reactor volume ___ volumetric flow rate
For an ideal gas pnt =
so
8 = P/RF
Acknowledgment
The authors wish to thank Serge Gratch for pointing out that adiabatic reactor problems could be solved analytically in terms of the exponential integral.
pZn dVR/Fnt
P/RT
r
dVR/ntT
Equation 2 may be expressed as: dVz
=
FntdT/Xk*na
sT:
Substituting this into Equation 13 Appendix A
Dynamic Response of an Adiabatic, Gas-Phase, Tubular Reactor, For simplicity, consider a first order reaction where no change in the total number of moles occurs as the reaction proceeds. Material balance dXA/dVR
=
r / F = k a a / F = k*na/Fnt = -dnA/dVR
e
=
P/RA+
exp.(E/RT)dT/T(T.d - T )
Substitute the relationships given in Equation 6
(1) Substitute the relationships given by Equation 9
Energy balance dT/dVR
=
(-AH)r/CpF
=
Xk*na/Fnt
(2)
8 = - P exp.(E/RT,d)/A*RTad
Divide Equation 2 by Equation 1 and integrate T
-
T o = X(nAo
- nA)
(3)
ST:
exp.(E/RT)dT/(XnA,+
0
To - T )
(4)
Let XnA, -k To = Tad y = E / R T , T = E / R y , d T = - E dy/Ry2
Substitute these expressions into Equation 4
Apply the method of partial fractions
(5) (6)
exp.(Z)dZ/Z
Thus,
Equation 2 may be written L 7 ~ = Fnt:A*
sz:
=
- [ P e x p . ( E / R T , d ) / A * R T , d ] [ E i ( Z )- Ei(Z,)]
Appendix B
Problem I d , p. 128 of Smith ( g ) , has been selected to demonstrate the use of the equations in Tables I and 11. I n this problem, the time for 70y0conversion is desired for an adiabatic, first order batch reaction. From problem IC,A = e16.87 min.?, and E / R = 5580' K. Adding the adiabatic temperature rise of 11.4' C. to the inlet temperature, To = 288.0' K., gives T,, = 299.4' K. For 70y0conversion, the outlet temperature is T = 296.0' K. The four arguments for the exponential integrals appearing in the first order case in Table I are computed, and the problem is completed as follows: E/RT
=
5580/296.0 = 18.86
E / R T , = 5580/288.0
=
19.38
Z = E / R ( l / T - l / T a d ) = 5580 (1/296.0 - 1 / 2 9 9 . 4 ) = 0.2165
Z,
=
E/R(l/T,
- l/Tad) =
5580 ( 1 / 2 8 8 . 0
-
1/299.4)
=
0.741
8 = exp.(-l6.87)[Ei(18.86) - E,(19.38)] -
exp.(5580/299.4 - 16.87)[Bt(0.2165) - E t ( 0 . 7 4 1 ) ] 0 = 4.71 X 10-8 (0.868
- 1.320)
X 10'
- 5.87
(-0.7244
1.1819)
118
l&EC FUNDAMENTALS
0 = -0.26
-t 11.19 = 10.93 minutes
Tad
it’hen the recommended approximation (exp. ( x ) / x . 1x1 > 1 5 ) \$as used to obtain the exponential integrals of 18.86 and 19.38, an answer of 10.92 minutes was obtained. Since the approximation is in greatest error for low values of the argument and since the values in question were near the lower limit, the second term in a series (5) (of which the recommended approximation is the first term) was also evaluated. Including the second term, changed the values of the exponential integrals by about 57:, but only changed the final results from 10.92 to 10.93 minutes.
-
Nomenclature
+
rR
aA bB a: h = = a,, b,, = aA! aR A,, A , .4* =
+ .rS
stoichiometric coefficients initial amounts of A and B activities of A and B frequency factors in Arrhenius equation, having the same units as the corresponding values of k . C A ~CR,, ? = initial concentrations of A and B = average heat capacity (B.t.u./lb./” F.) CP E = energy of activation in the Arrhenius equation, k = Ae-E/RT
S:
Ei(x) =
exp.(t)dt, t = first order exponential integral a
exp.(t)dt, t Z = second order exponential integral
E ~ Y=) m
F
feed rate (mass/time) heat of reaction (B.t.u./mol. of A converted) = reaction velocity constants when the rate equation is written in terms of concentrations, activities, and partial pressures, respectively n.4 = n.4” - .YA = mole:; of A unconverted per unit mass of feed n.4”. nB.,, no = initial moles of A and B and total moles of feed per mass, of feed n, = n, 6u.d = total moles of reacting system per unit mass of feed P = pressure in (atm.) r = reaction rate in (moles/volume/time) R = gas constant = initial temperature in O R . T,, T = To ( - J H ) ( C A ~- C A ) / P C = ~ To + , ( - A H ) ( ) i t o - n.4)/Cp - absolute temperature in (” R.) = =
(-AH?; k,. k >A-
+
+
TodBi,
Vb, VI?
Tv, 11’0 XA
Y
2,Z ,
= To
+ +
+
(-AH)Ca,/pCp = T o XnA, in Tables I and 11, respectively = To (-AH)aCB,/pbCp = T o aXnB,/b in Tables I and 11, respectively = volume of system per mass of charge and volume of reactor, respectively = ( E / R )(1/.T- 1 / T a d ~ o )(, E / R )(1/To - l / T o d ~ J , respectively = moles of A converted per mass of reactor feed = =
+
E/RT ( E / R ) ( l / T- 1 / T d and ( E / R ) ( l / T o- l / T o d , respectively
Greek 6
- r+s-a-b -
8
=
x
= (-AH)/Cp = (r s
a
Residence time in flow reactor or reaction time in batch reactor
+ +
) - sum of the stoichiometric coefficients of the reaction products = mass and molar densities in (mass/volume) and p , pm (moles/volume), respectively For some svmbols suggested units are given, but any consistent set of units may be used. V
literature Cited
(1) Billingsley. D. S.,,McLaughlin, LV. S.. Jr., LVelch, N. E., Holland, C. D., Znd. Eng. Chem. 50, 741 (1958). (2) . , Bilous,. O.,. ;\mundson, N. R., A.Z.Ch.E. Journal 2, 117 (1956). (3) Deindorrfer, F. H., Humphrey, A. E.. Appl. Microbiol. 7, 256 (1959). (4) Gratch, S.. Birmingham, Mich., March 1957, personal communication, 32475 Bingham Rd., Birmingham, Mich., March 1957. (5) Jahnke, E., Emde. F.: Losch, F., “Tables ofHigher Functions,” p. 23, McGraw-Hill, New York, 1960. (6) Handbook of Chemistry and Physics, pp. 289-90, C. D. Hodgman, cd., Chemical Rubber Publishing Co., Cleveland, O., 1961. (7) Hougen, 0. -4.; \Vatson, K. M., “Chemical Process Principles,” pp, 816-40, Vol. 111, Kinetics and Catalysis, \Viley, New York, 1947. (8) Parts, A. G., Australian J . Chem. 11, 251 (1958). (9) Smith, J. M., “Chemicd Engineering Kinetics,” pp, 100, 128, McGraw-Hill, Kew York, 1956. RECEIVED for review Sovember 6, 1961 ACCEPTEDMarch 7, 1962
G E N E R A L EQUATIONS FOR TESTING CONSISTENCY OF MULTICOMPONENT VAPOR-LIQUID EQUILIBRIUM DATA LU H C
.
T A 0 , Department of Chemical Engineering, University of Nebraska, Lincoln, Xeb.
General equations are presented for testing the internal consistency along piecewise continuous composition paths in a mullticomponent system b y methods similar to those already used for binary systems. Convenient numerical methods are illustrated b y application to a ternary and a quaternary system. HE DEGREE O F COMPLEXITY in a multicomponent system Tincreases with the total number of components. If the composition of a mixture is represented as a point in a space with coordinates representing concentrations of each component. treatment of data may be simplified to that of a binary system by considering clnly linear paths in a multidimensional space. This geometrical concept is used to obtain general
equations for testing internal consistency of multicomponent, vapor-liquid equilibrium data. Several restricted integration methods have been proposed for testing multicomponent data by using the Gibbs-Duhem equation at constant temperature and pressure; usually they imply a test only for the whole path under consideration. Krishnamurty and Rao (3) suggested a method for linear paths VOL.
1
NO. 2 M A Y 1 9 6 2
119
