I
LIANG-TSENG FAN Department of Chemical Engineering, Kansas State University, Manhattan, Kan.
Calculating Conversion in
...
Continuous Stirred Tank Reactors A numerical method applicable to 0
a single reactor
0
reactors in series
0
reactions of any order-integral
CONTINUOUS
stirred tank reactors are widely employed in the process industries. This type of reactor is operated either singly or in series, as shown. Usually an attempt is made to maintain uniform temperature and composition in each reactor, and under these conditions perfect mixing is assumed to exist. This article presents a general numerical method for solving the material balance equations for stirred tank reactors. The method employs NewtonRaphson's iterative technique, which converges rapidly. The material balance equation is transformed into a dimensionless form to facilitate the compu tation. The technique is applicable to reactions of any order, including both integral and fractional orders. Reactions with fractional order are not uncommon ( 7 ) . The widespread use of automatic computing machines makes the execution of such numerical solutions easy and rapid. The method developed here v
or fractional
is generally applicable not only to reactions for which analytical solutions are available but also to those for which no analytical solutions have been obtained. Analytical solutions are also extended to third order and to some fractional order reactions.
Theoretical
CO- C = k
(g)
Cn
(2)
Equation 2 can be transformed into &mensionless form as:
or
When the effective volume of the reacting mixture, V, and the volumetric flow rate, F, are constant, a material balance on a single reactor may be written as (2) :
co
Then Equation 1 can be written as:
-
c = r (V/F)
$ 2+ y - - l = O
4)
where
(1)
For many types of reactions the rate can be suitably expressed by an equation of the form: r = kC"
where k is the rate constant and n is an exponent having a value of 0 5 n 5 3 .
y = -
C
CO
Analytical and graphical solutions of y from either Equation 2 or Equation 4 are available for zero, first, and second order reactions ( 2 ) .
C or
Typical system of reactors in series shows values used to determine degree of conversion of a reactant VOL. 52, NO. 1 1
NOVEMBER 1960
921
Single - Reactor
When the system has only one reactor,
the solution of Equation 4 yields the fraction of a reactant unreacted and, consequently, the degree of conversion of the reactant. Because the value of function f(7) defined as
1.0
-
I
I
I
I I
J
increases monotonically as y changes from its minimum limit of zero to h e maximum Of (Figure NewtonRaphson's iterative method (4 'Onverges quite to the Of Equation 4, especially when the iteration is started from = Because
-/;
I
= Ya -
I / I
I
I
I
/
I
I I
I
Ya
nR
~
--
'.
I
I
;+ -/a_L 1
I
I I
the iteration is represented as :
2
y&-1
- 1 (9)
+1
where a = the number of consecutivc
'4nalytical solutions, obtained by other investigators and by this author, for systems with a single reactor are presented below for single (not simultaneous or consecutive) irreversible reactions involving only one reactant or equivalent reactant concentration. Zero O r d e r Reactions.
0 5 t h Order Reactions. Substitution of 70.5 = X in Equation 4 yields a quadratic equation. It is known ( 3 ) that 0 5 y 2 1 : Therefore,
0.4
First Order Reactions.
0.2
1
Y =
P R
1
+Z
1.5th Order Reactions. Transformation, 70.5 = X , yields a cubic equation from Equation 4. As 0 5 7 5 1 ( 3 ): =
{[(k
8
-
m3)+
(& -
whenR2 2 Y =
li3
&)"2]
+
16 27
-
27R2
4
(s
{% C O LS ~ ( ~ / ~ ) ~ C O S - ~ - 1)
-
]:4
&I2
+
(14)
when R2 < 16 27 Second Order Reactions. Equation 4 yields a quadratic equation ; as 0 5 y 5 1 ( 3 ): Figure 1. Because f(y) increases monotonically as y changes from 0 to 1, solution of Equation 4 by the Newton-Raphson method is rapid
922
INDUSTRIAL AND ENGINEERING CHEMISTRY
y=---4 m R
R
-
1
(15)
CONTINUOUS STIRRED T A N K REACTORS
A Figure 2. unreacted reaction
Determination of fraction for a given order of
In combination with Figure 3, this plot can b e used for graphical solution o f Equation 4 for any other order of reaction
For the third order reactions, Equation 4 becomes a cubic equation, and as 0 2 y 5 1 (3):
+ [k
-
( 41 + &-4’’’ (16)
1 -
The results of analytical solutions and some of the numerical solutions are plotted in Figures 2 and 3, using the order of reactions n and R, respectively, as parameters. T h e combination of these two figures allows graphical interpolation for solutions of Equation 4 for reactions with any order when great accuracy is not demanded.
Application
Example 1. A reaction, the rate of which is expressed as r = k c 0 7 6 = 1.6Co.76 (gram-moles)/(minute) (liter), is carried out in a 200-liter continuous stirred tank reactor. Calculate the percentage conversion of a reactant when the flow rate of the reacting mixture is 400 liters per minute and the initial concentration is 1.2 grammoles per liter. SOLUTION :
Numerical solution by means of Equation 9 gives: y = 0.53 Therefore, %conversion = IOO(1 - y ) = 47% From Figure 3, y = 0.53 or 1 - y = 0.47 Example 2. The same reaction is carried out in two reactors connected in series. Calculate the percentage conversion of the reactant. The temperature and volume of each reactor and the flow rate of the reacting mixture are identical to those in the previous example. SOLUTION : From the previous example, for the first reactor:
Reactors in Series When a number of continuous stirred tank reactors are operated in series under steady-state condition, Equations 4 and 5 can be written for each successive reactor repeatedly. For each reactor !3$+yd-1 2
=o,
Rl
= 1.53
71
= 0.53
For the second reactor:
Numerical solution by means of Equation 9 gives: yz =
0.485
Therefore, from Equation 20 y = y l y z = (0.53)(0.485) = 0.257
Equation 17 can be solved for each reactor numerically by the Newton-Raphson method, as mentioned for a single reactor, starting from the first reactor. The value of Rifor each successive reactor is calculated by Eqnation 18.
or
% conversion
=
IOO(1 - y) = 74.3%
Figure 2 also yields approximately 0.485 as the value of y2.
WOL. 52, NO. 1 1
NOVEMBER 1960
923
Nomenclature
RLO
I .o
I
1
I
l
l
I
I
I
I
I
I
C = exit concentration of reactant, also
0.5
0
1.5
2.5
2.0
n
30
x
Determination of fraction unreacted when R i s constant
Figure 3. Figures 2 and is not essential
IO
concentration of reactant in reactor, gram-molesiliter CO= inlet concentration of reactant, gram-moles/liter C, = exit concentration of reactor from ith reactor, also concentration of reactant in ith reactor, grammoles/liter J ( 7 )= function of y F = volumetric rate of flow of reacting mixture, liters/minute k = specific reaction rate constant m = number of reactors operated in series n = order of reaction Y = reaction rate, gram-moles/(liter) (minute) R = dimensionless auantitv defined bv Equation 5 I/ = effective volume o i reacting mixture, liters
3 can be used for graphical interpolation o f solutions to Equation 4 when great accuracy
When the volume of each reactor, Vi, and the temperature in each reactor are
been obtained for some reactions with integral orders (2). First Order Reactions.
Sub script
i = ith reactor i - 1 = (i - 1)th reactor 0 = feed to first reactor 1, 2, 3 = first, second, or third reactor, respectively
I
identical, the stepwise numerical solutions are further simplified because :
= 9 . 5 = 0th iteration in numerical solution y = C/C, Yz = Cd:C,-l CY
For 0.5th, 1.5th, and second order reactions, Equations 11, 13, 14, and 16 can be repeatedly applied to each reactor in the system consecutively to obtain y i ; y is then calculated by Equation 20.
Acknowledgment
The suggestions of James M. Church and George R. Marr, Jr., Columbia University, and R . E. Greenhalgh, Dow Corning Corp., are appreciated.
Identical Reactors
IVhen the volume and temperature in every reactor in the system are identical, further simplification of the solution can be obtained (2).
or
Rt
R(qiiqi2
...
yp-
2
+iz-i)”-’
in which R is the dimensionless parameter expressed by Equation 5. The final concentration of the reactant or fraction of reactant unreacted, y,in the exit stream, which is C,/Co, can be calculated as :
cm c 1c z c3 - -.-cti Ca c1 cz
-
,
,
Zero Order Reactions. ? = I - - mR 2
f23)
First Order Reactions.
(1) Chang, W. H., 819 (1957’1. (2) Eldridgk, J. W., Eng. Progr. 46, 290 (3) HodgFan, C.
J. Phys. Chem. 61,
Piret, E. L., Chem. (1950). D., “Mathematical Tables, 9th ed., p. 272, Chemical Rubber Publishing Co., Cleveland, 1952. (4) Kunz, K. S., “Numerical Analysis,” pp. 10-15, McGraw-Hill, New York, 1957. RECEIVED for review February 12, 1960
cm- c m 1
cm-2
literature Cited
ACCEPTED July 3, 1960
em-1
Second Order Reactions.
or y = y1
y2 y3
. . . ym-1 y m
(20)
The numerical solution is completely general, regardless of the order of reaction and the number of reactors. The procedure for numerical solutions can also be carried out graphically by repeatedly using Figure 2 or 3. The exact analytical solutions for y have also
Correction Repeating this operation from i = m to i = 1:
Elastomeric-Binder and Mechan ical-Property Requirements In this article by Thor L. Smith [IsD. ENG.CHEM.52, S o . 9, 776 (September
Zero Order Reactions.
in which a
T=(1-%)(1-$)..+!5)
(21 )
924
INDUSTRIAL AND ENGINEERING
=
4 1 f 2R
The square root operation in Equation 25 is repeated m times.
CHEMISTRY
1960)], an error occurred in the box material on page 777. Line 4 should read as follows “grain must nor creep appreciably during storage.”
