COM M UN I CAT1ON
APPROXIMATE ANALYTICAL SOLUTION TO REACTOR DESIGN PROBLEM FOR CERTAIN VARIABLE T E M P E R A T U R E REACTORS Elementary chemical kinetics gives analytical solutions to reactor design problems for isothermal reactors. This paper describes an approximate analytical soluiion to some simple reactor design problems in which reactor temperature varies.
By combining a simple exponential expression for the temperature dependence
of reaction rate constant with use of the exponential integral function it is possible to obtain analytical solutions to problems involving first- and second-order irreversible reactions. The method is restricted to reactors to which heat is supplied at a constant rate. The approximate solutions should be useful for preliminary design work or as a starting point for more exact computer solutions.
E
chemical kinetics gives exact analytical solutions to reactor design problems for isothermal reactions. This paper describes a n approximate solution for some simple cases in which reactor temperature varies. Specifically we are concerned icith first- and second-order irreversible reactions conducted in reactors to which heat is added at a constant rate. Heat is supplied a t approximately constant rate to reactors, such as cracking furnaces, in which the heat source is a t a considerably higher temperature than the reaction mixture. Hougen and Watson (2) have described a trial and error numerical method for this calculation. Rase and Perkins (4) have developed a simplified numerical solution. T h e present analytical method is based on approximations, but the author feels there is a real need for quick approximate solutions to complex problems.
irreversible and first-order. are made:
Three equations are required for a mathematical statement of this problem
dxa - _ - k c ( n ~o dt xaAH
=
Cp(T- To)
=
k,ebs ( n a o -
for mathematical work. Barkelew ( 7 ) has suggested the use of a simple exponential function over short temperature kOeb:''- To)
(2)
ranges. T h e author has found this approximation satisfactory for design work over temperature differences of 100' to 200' F. depending on the size of the activation energy. Application to Batch Reactions
1. T h e reaction A y B . Using the simple exponential equation far variation of rate constant \ve \vi11 now solve a reactor design problem. Consider the simple batch reaction, ---f
98
l&EC
FUNDAMENTALS
+ Qt
13)
Integrated energy balance
(4)
(5)
\'ariation of k Lzith temperature
Substituting Equations 6 and 5 into 3 and letting ( T - T o )= 0 gives
dXA
=
Rate equation
XA)
If Me differentiate Equation 4 and solve for dt lie get
T h e key to this method is the use of an approximate equation for expressing the variation of reaction rate constant with temperature. I n most practical design work a relation of the Arrhenius type is used for predicting rate constant as a function of temperature. This is a troublesome function
k
T h e folloit ing approximations
No volume change associated n i t h the reaction S o change in heat capacity per unit mass of reaction mixture Negligible variation of enthalpy change for the reaction n i t h temperature Constant pressure. Constant rate of heating of the reactor
k c -- k,eb(T-To)
Approximate Equation for Variation of Rate Constant with Temperature
+ -yB
A
LEMENTARY
AH XA)
-
Q
dXa
+ k,ebs-
CP
Q
(nAo - XA)
do
(7)
LVe can linearize the equation by substituting z = eba, dz = bebedo.
dxa
=
k,
AH
-z ( ~ A ~xa)dxa
Q
f
k,C p P
(nAo -
bQ
After some rearrangement the equation becomes
xa)dz
(8)
Nomenclature Table I.
Comparison of Solutions
0.00 0.10 0.20 0,30 0.40 0 50 0 60 0.70
0.00 49.38 62 56 73.42 87.11 98.85 110.8 123.3
0.00 51.89 64.43 76.05 88.28 100.2 112.5 125.3
Since this equation is first-order and linear, it can be integrated readily. T h e result is +
= e-.ifL4
A\.e.wn'4"o--*A'
- E1(,ZlnAg,)]
(10)
'l'he function E l can be obtained in terms of the exponential integral function. For endothermic reactions M is positive and El(,\)
=
r= :e-'ds
El(,~= )
Lvhere s
> 0.
For exo-
..X
thermic reactions hl is negative a n d E i ( x ) t'dS ~~~~
S
\\here
Y
is negative.
=
= = =
AE AH
= =
k, k,
= =
31
=
&V
=
nAo
=
Q
=
constant in .irrhenius equation constant in empirical equation for reaction rate constant heat capacity a t constant pressure, B.t.u./lb. m of reacting system energy of activation in Arrhenius equation entha1p)- change associated \iith conversion of 1 lb. mole of component A reaction rate constant: based on concentration value of k, a t To bAH ~~~. constant CP bQ--. constant k"C, initial moles of component A per unit mass feed constant heating rate: B . t . u per lb. initial charge to batch reactor universal gas constant variable of integration absolute temperature initial temperature time conversion of component A . moles converted per unit mass initial charge of feed to reactor ~
R
=
s
.xA
= = = = =
2
= eh0
7 T(,
1El [ n f ( n ~ -o XA)]
A b C,
t
GREEK e = ( 7 - - T(,) = stoichiometric coefficient
-,
El(x) =
T h e functions E l ( x ) and
a(u)are
available in tables of the higher transcendental functions ( 3 ) . Equation 10 gives T for a given x.4. T h e time is then found directly from Equation 4. the integrated energy balance. 2 . To check the validity of the above method an illustrative problem \\.as kvorked. Consider the reaction A + B . Since there is no change in number of moles in this reaction. it is convenient to use a basis of 1 Ib. mole of initial reactor charge. \Vith this basis the parameters used \\-ere: k = 6.4 X 10'j exp (-91.000 R T ) approximated by k = 0.000134 exp 0.0398 ( 7 ' - T ( > )C, ; = 20 B.t.u., lb. mole ' F . ; Q = 50 B.t.u. sec. mole reactant charged; AH = 5030 B.t.u. lb. mole; and 7'" = 1010" R . 'l'able I compares the analytical solution Equation 10 \vith a numerical solution by the Runge-Kutta method. T h e agreement is satisfactory. ?he above approach can readily be extended to higher order irreversible reactions and to continuous reactors.
literature Cited
(1) Barkelew, C. H., Chem. Eug. Progr. Symp. Ser. .Yo. 25, 5 5 , 37 (1959). (2) Hougen. 0. A: ]Vatson. K. M., "Chemical Process Principles." Vol. 111, 1st ed., pp. 8'4-83 LViley, S e w York, 1947. (3) Jahnke, E.; Emde. F.. "Tables of Functions," 4th ed., pp. 1-9. Dover: New- York, 1945. (4) Perkins, T. K., Rase? H . F., Chem. En?. Progr. 52, 105-M (1956).
FR.iSK R . GROVES? J R . Louisiana State Vniz'ersitj Baton R o u f e , La. REC:EIVED for review .April 20. 1964 ACCEPTED September 25, 1964 Southwest Regional Meeting, ACS. Houston. Tex.. December 1963.
VOL. 4
NO. 1
FEBRUARY
1965
99
