May 1950
INDUSTRIAL AND ENGINEERING CHEMISTRY
fusion of sodium naphthalenesulfonates, but also increases the rates of the side reactions. Conversion in fusions with pure potassium hydroxide as agent is greater than in those with sodium hydroxide as agent, but an equimolar mixture of the two gives a higher conversion than either pure alkali.
Literature Cited (1) Czenkusch, E. L., unpublished M.S. thesis in chemical engineering, Purdue University, February 1948. (2) Dziewonski, K., and Chromik, T., Bull. intern. mad. polon. sci ,
classe sei. math., 1938A,541-50. (3) Dziewonski, K., Schoenowna, J., and Waldman, C., Ber., 58B, 1211-18 (1925).
*
817
(4) Groggins, P. H., et al., “Unit Processes in Organic Synthesis,” 3rd ed., New York, McGraw-Hill Book Co., 1947. ( 5 ) Kruder, G. A,, unpublished Ph.D. thesis in chemical engineering, Purdue University, February 1948. (6) Miller, J. R., unpublished M.S. thesis in chemical engineering, Purdue University, February 1948. (7) Royer, R., Hoi, B., and Woi, H. K., Bull. soc. chim., 12, 904-8 (1945).
Shreve, R. N., Color Trade J . , 14, 42-7 (1924). (9) Shreve, R. N., and Lux, J. H., IND.ENG.CHEW,35, 306-12
(8)
(1943). * (10) Stoltenberg, D. R., unpublished M.S. thesis in chemical engi. neering, Purdue University, June 1948. (11) Wilkie, J. M., J. SOC.Chem. Ind., 30, 398-402 (1911). RECEIVEDJanuary 12, 1950.
Continuous flow stirred tank reactor systems Development of Transient Equations DONALD R. MASON’ AND EDGAR L. PIRET UNIVERSITY OF MINNESOTA, MINNEAPOLIS 14, MI”.
Exact mathematical expressions are developed which completely describe the concentration changes that occur in continuous stirred tank reactor systems during transient periods of operation. The reactant and product concentrations are given as functions of time when single, simultaneous, consecutive, or reversible first-order reactions occur in the system, or when a nonreacting system is being purged of its contents. These expressions are applicable when the system is changing from one steady-state operating condition to another, and under certain circumstances when the system is being started
up or shut down. The mathematical solutions are given for systems of vessels all having the same volumes, all having different volumes, and combinations thereof, as well as for multitemperature-level systems. These extremely general equations for the transient states could be readily obtained by using the Laplace transformation method for solving the linear differential rate equation for the system. This method has definite advantages over the more specific mathematical methods which have been used in this field, and it should find applications in many other branches of chemical engineering.
R
equation is obtained which can be applied to systems of vessels all having equal volumes, or unequal volumes, or combinations thereof. The feed component concentration initially in each vessel and in the feed stream may be different from zero, and furthermore be independently different from each other. This equation is applicable when the system is being purged of its contents, or when single or simultaneous first-order irreversible reactions are being carried out in the system. The individual tanks need not be operated at the same temperatures. This most general equation, which is developed below, describes the operation when the system is changing from one steady-state operating condition to another, and under certain conditions when the system is being started up or shut down. Solutions have also bceri worked out for some cases of simultaneous, consecutive, and reversible first-order reactions. The equations that are presented in this paper could also be applied to first-order reactions in fluid catalyst units if no volume change occurs in the reacting system, and if the concentrations are uniform throughout each individual unit. These equations may also be appIied to irreversible secondorder reactions in these types of systems if one reactant is in such large excess that its concentration throughout the system may be assumed to be constant. Continuous stirred tank reactor system have the following characteristics, which must be well understood. The basic assumption is made that the concentration of each individual tank or reaction compartment is uniform throughout its contents, and
ECENTLY considerable interest ( 3 , 4 , 7-9,18, IS) has been shown in the development of design equations for continuOUS stirred tank reactor systems-i.e., for system in which a solution flows continuously through a row of reaction compartments as shown in Figure 1. In a recent paper, Eldridge and Piret ( 4 ) have given a brief review of the field and have presented a useful method for designing such systems. Their method is applicable for the steady-state operating condition only. In addition to the steady-state condition of operation, certain transient conditions arise whenever the system is started up, changed from one set of operating conditions to another, or shut down. At such times the steady-state operating equations do not adequately describe the behavior of the system. To derive the equations that do apply in these circumstances, a rate equation for the system must be solved. To date this rate equation has always been solved by Ham and Coe ( 6 ) , MacMullin and Weber ( I S ) , Kirillov ( 9 ) , Kandiner (8), and Johnson and Edwards (7) using the classical mathematical methods, which become difficult very rapidly as the complexity of the system increases. As a result only a limited number of cases have been solved. These cases and their solutions are included in Tables 111, IV, and V. In this paper the Laplace transform method of solving linear constant coefficient differential difference equations is used to solve in a relatively simple manner the rate equation of the system for a wide variety and number of conditions. A most general 1
Present address, Bell Telephone Laboratories, Inc., Murray Hill, N. J.
INDUSTRIAL AND ENGINEERING CHEMISTRY
818
Vol. 42, No. 5
Table I. Laplace Function Transform Pairs
Transform Function, F ( s )
NO.
1 2
1
1 s+a
1
S6-T-Z
4
(FTa
6
7
Transform Function, F ( s )
KO.
S
3
5
Original Function, j ( t )
te-at
1
~
l!
1 - (I
1 S
+ at)e-Qt
17
a2
m
1
+ a ) 3(s + b )
4s
1
1
+ b ) ( s + u-1 1 s(s + a ) + h)
l8
(s
(8
+ a)2 ( s + b)2 1
(8
s(s
+ a ) z( s + b ) 2
-
+
8 9
1
s(s
+
21
a13
e-bt ( b - a)'
10
1
11
a%
[(b - a) t (b -
-
(b
1 (s
+ h) (s + c )
11 e - a t
F- v a x )
e-*$ (2a b(a - h ) 2 + e --at
12
s(s
te-at
b)e-af
-
e-bl
- a ) (c - a )
+
(a
e-C1
- b) (c - b)
is equal to the outlet concentration. Denbigh (3)and Kirillov (9) have briefly investigated this problem, and recently MacDonald and Piret ( I 0 , I i ) found this assumption to be valid under most practical conditions of operation for their homogeneous liquid system. Ss a result of the rapid mixing, the molecules of
(a
- C) ( b - C)
the feed stream to each vessel are uniformly dispersed almost instantaneously, so that some of them appear immediately in the effluent stream and are lost from the vessel. This loss has been called short-circuiting (13) or "mathematical by-passing" (3). Other molecules of the feed stream entering the vessel at the same instant may be retained in the vessel for extremely long periods of time, so that the history of the individual molecules in the system is not uniform. This variation in the retention times of the molecules gives rise to a time-distribution effect (IS'). This timedistribution effect is usually not present in tubular-type continuous reaction systems because the individual feed molecules are retained for essentially the same periods of time.
Derivations of Transient Equations for Continuous Stirred Tank Reactor Systems
I I
\
II
'
I
AGITATOR DRIVES IN c
FEE
PFODUCT
The rate equation for continuous stirred tank reactor systems is given below, and its solution by the Laplace transformation method is illustrated. In addition to equations that describe the reactant concentrations in the effluent streams, equations that are applicable for the product material are also given. The results are summarized in Tables 111, IV, and V. Rate Equation for Continuous Stirred Tank Reactor Systems. The rate equation for the reactant concentration in any one vessel of a continuous stirred tank reactor system is obtained from material balance considerations for the case where an irreversible first-order reaction occurs.
--ErTREAM Change in mass of reactant in vessel n
I Figure 1.
2 3 n Systems of Reaction Compartments
1
=
[,,,E added] in feed
] - [by reaction
- [.mass lost in effluent
INDUSTRIAL AND ENGINEERING CHEMISTRY
May 1950
819 On the other hand, if no reaction occurs in the system, k is equal to zero, and Equation 1 describes the purging operation of the vessel.
Table 1. Laplace Function Transform Pairs (Contintced)
Original Function, f(t) tse-at
__
3!
Use of Laplace Transformation Method. In using the Laplace transformation method for solving differential equations, the equation is transformed into an 1 e-at e-bt algebraic equation; this transc(a c ) ( b c ) & - [ a ( b - a)( c - a ) b(a - b ) ( c - b ) e-ct formed equation is solved algebraically, and then subjected to an inverse transformation operation to give the solution of the original differential equation. 1 e-hl The result is that a differential equation is solved by using simple algebra only. In this respect the Laplace transformation method is similar to the use of logarithms, in that a transformation is made to reduce one mathematical operation to a simpler operation. (2a - b - c ) e - a t te-at e-bt e-ct The use of the Laplace trans(a - b)2 ( a - c ) 2 + ( a - b ) ( a - c ) + ( a - b)2 ( c - b ) + ( a - c ) 2 ( b - c ) formation method for the solution of transient, nonsteady-state e-bt a problems is relatively new, and + at1 + b(a - b ) 2 ( c - b ) is rapidly displacing the use of e-et other methods, especially in the I field of electrical engineering, c(a - c y ( b - c ) / Although it has appeared only e-"t e-bt e-Ct a limited number of times in the chemical engineering literature (a - c ) ( b - c ) ( d - c ) ( h - a ) ( c - a ) ( d - a ) + (a - b ) ( e - b ) ( d - b ) ( 1 , Z), its advantages are such e-dt that it will appear with increas(a - d ) ( b - d ) ( C - d ) ing frequency. For this reason a brief discussion of its properties e-ot e-bt . . 1 may be in order. a m - i(b - a)( c - a ) ( d - a ) b(a - b ) ( c - b ) ( d - b ) Two slightly different definie-C1 e-dt tibns of the LaDlace transformation have grown up, and care d(a - d ) ( b - d ) ( C - d ) C(U - C ) ( b - C ) ( d - C ) must be exercised to follow onlv one system when solving pro& lems by this method. The definition used in this paper is the same as that used by Gardner and Barnes ( 6 ) ,rather than that This is written mathematically as follows: used by Marshall and Pigford (14). Gardner and Barnes (6) define and represent the Laplace transform as follows: dEn = FC,,-, - FC,, - kW,, (1) dt Pm
1
a+@T$
+
+
where
W.,
C,, V, F k
=
mass of reactant A in vessel n
= reactant concentration in vessel n - 1 = reactant concentration in vessel n = volume of vessel n = =
volumetric flow rate of solution through system first-order irreversible reaction velocity constant
By dividing Equation 1 by V n to convert t o concentration units, letting Vn/F equal On which is the nominal holding time for vessel n, and rearranging, one obtains:
This is a linear differential difference equation, and the coefficients xi11 be constant if neither k nor e, is a function of time; they may, however, be functions of n. Can_,,which is the feed concentration to vessel n, must be a known explicit function of time. In Equation 1, the volumetric flow rates into and out of the vessel are assumed equal; in other words, the tanks are initially full and no appreciable change in density occurs in the vessel. Changes in density would normally be functions of the concentration, and the substitution of these functions into Equation 1 would give rise to a nonlinear equation which could not be solved in the general case by known mathematical processes. When second- or higher order reactions occur in the system the equation again becomes nonlinear, so that these cases are also excluded.
+
~ [ f ( t ) =l
j o- f(t)e-atdt
= ~ ( s )
(3)
in which s is a complex variable. This integral transformation provides the means for determining the transform function, F ( s ) , corresponding to the original function, f(t). The functional correspondences which are useful in this work are tabulated as Laplace function-transform pairs such as in Table I. In using the table, the inverse Laplace transform may be taken by starting with a given transform function, F ( s ) ,and looking up its corresponding function,f(t). The inverse Laplace transform operation is analogous to the taking of an antilogarithm, and is indicated by: Besides the property of transforming mathematical functions, the Laplace transformation has the important property of simplifying certain mathematical operations. Consider the transformation of a derivative:
which, after integration by parts, gives:
L[f'(t)l = slY.3)
- f(0)
The result is that a differentiation has been transformed into a 'multiplication, and in a like manner an integration may be transformed into a division operation. Some useful relationships of this general type are summarized in Table 11, as Laplace operation-transform pairs.
INDUSTRIAL AND ENGINEERING CHEMISTRY
a20
Table 11. Laplace Operation-Transform Pairs (5) Same of Original Transform Equivalent, F ( s ) Operation Operation, f(t)
Linearity
+ Sdt)
aF(s ) F d s ) Fz(s)
=
sJY5)
af(t)
fdt)
+
dm f'(t)
Real differentiation
dt
--f(O+)
f -YO+)
Real integration S
Scale change Real translation
S
aF(as)
f(t/a) f(t - a) if f(t - a) = 0
e - a a F ( s)
We shall now take the Laplace transformation of the differential difference Equation 2, wherein C, is considered to be f&), By defining An as the value of C,, when the time is taken as zero, and using the Laplace transform pair tables, we have:
- An
sFn(s)
in which the group
Qn
+ 8, Fn(s) Qfl
is defined as (ken
F'n-l(~)/@n
r%+
+ l),or
Vol. 42, No. 5
With Equation 11 and an adequate table of Laplace function transform pairs (Table I), it is possible t o solve an exceedingly wide variety of transient problems. In this work four types of boundary conditions are i o be considered: (1 ) one-vessel systems, (2) n-vessel systems, all vessels having cqual volumes, ( 3 ) n-veswl systems, all vessels having unequal volumes, and (4) n-vessel system, the vessels having both equal and unequal volumes. Type 4 is also applicable t o systems wherein all the tanks do not have the same reaction velocity constant, as IS true of nonisothermal systems. These systems of vessels are listed under the respectively numbered columns in Tables 111, IV, and V. By taking the inverse Laplace transform after the appropriate boundary conditions for a physical system have been placed in Equation 11, one obtains the expression for the outlet concentration of the feed component from the system as a function of time. For example, when all tank volumes are equal to V and all reaction velocity constants are equal to k-e.g., an isothermal system--it is apparent that:
(7) Equation 11 then reduces to:
1).
By solving Equation 7 algebraically for F n ( s ) we obtain:
Equation 8 is recognizable as an algebraic difference equation which can be solved by an iteration scheme-that is, by establishing the relationship between the equations for successive vessels in the system. For the first vessel, it will be assumed that the concentration of the reactant in the feed stream is constant. This is not an essential assumption from the mathematical viewpoint but is made in the interest of simplicity. Equation 8 then gives :
By using the expression for F , ( s ) from Equation 9, the ident,ity which is obtained for P,(s) is:
F,(s)
= A A ___A,-,/@
(s
+ &/e)
+
(s
An - 2 / f l '
+ Q / @ ) 2 + &/e)3 +
(8
' '
'
the inverse Laplace transform of which gives:
c,
=
Ca &"
+e-wo
i
A,
+
An-1
(i) + 3j-(5)' + An - 2
(14) Equation 14 expresses the outlet reactant concentration from vessel n of an equal volume isothermal continuous flow stirred tank reactor system as a function of the time, 1, when the concentration, C, in the feed stream is constant, and wherein the concentrations initially present in the individual vessels are given by the Ai terms. All the equations represented in columns I and 2 of Tables 111, IV, and V may be derived from Equation 14 by the substitution of the initial conditions which are indicated on the left side of the tables-for example, Equation 3i in Table IV is obtained when all the A, terms are zero. For systems where all the vessels have different volumes from each other, Equation 11 is transformed directly to give:
By proceeding in a similar manner, the expression obtained for vessel n is:
AZ/@nOn- 1'
n T(S
2
+
. .e3
+
AI/@,'
n Qi/%t)
T(S
I
+
. .e&
CG,/@?J
+
n Qiiei)
ST(S
1
+ Q~/oi)
Equation 11 is, for first-order irreversible reactions, the most general Laplace transform equation in which all boundary and initial conditions have been accounted for. These boundary conditions are the time-invariant properties such as the feed rate, number of vessels in the system, tank volumes, and the reaction velocity constants-Le., temperatures-in the different tanks. The initial conditions depend on the condition of the feed component in the system when time is chosen as zero. This is expressed by the concentration in the feed stream and the various initial concentrations which might be present in the different vessels.
All the equations represented in column 3 of Tables 111, IV, and V may be derived in a similar manner from Equation 15 by the substitution of the initial conditions.
May 1950
821
INDUSTRIAL AND ENGINEERING CHEMISTRY Table 111. Transient Equations for Purging
(Summary of trrtnsisnt equations giving effluent concentrations as functions of time for the purging of conhnuous stirred tank reactor systems)
Section A
Case So. 1
2 3 4 5
Eq. No
Conditions of Operation of System Feed Concentrations initially present concenin vessels tration C,, = 0 Ai A2 = . An = A , Ca, 0 A1 Z A2 # . . A n Z 0 Ca, # 0 A1 AS = . . A n = 0 Ca, f O AI = A2 An=Ao Ca, # 0 AI f Az f An # 0
Column 1, one vessel only 2 2 3a 3b 3b
System of Vesselsa Column 2, n vessels Column 3, n vessels all having unequal all having equal volumes volumes 1 5a 3c 5b 3d 4 3e -5c 3f .5 (i
Section B (Equations established by system of vessels and conditions of operation, Reference Concentration of Effluent as a Function of Time
itx
Column 4,n vessels having equal and unequal volumes 5e 5f 5g 5h
5i
~ I M I UI I i n SwtIor1 A )
a Numbers refer to equations given in Section B. Equations 5e, 5f, 5g, 5h, and 5i listed in Section A must be derived from the general Laplace transform Equation 11, for any particular system.
For combination volume systems and nonisothermal systems, which are represented by column 4 of Tables 111, IV, and V, the correct inverse transformations to be made from Equation 11 will depend on the particular sequence of the vessels of different volumes or of reaction velocity constants in the system, or both. The list of Laplace function-transform pairs (Table I) has been compiled to cover all possible combinations of four-vessel systems. Terms for additional tanks may be derived with the Cauchy-Heaviside expansion theorem (6). Five types of initial conditions for the system have been considered in these tables. These are: concentration of component A in the feed stream is zero, concentration in the feed stream is constant, concentrations in all vessels are initially zero, concentrations in all vessels are initially equal, and concentrations in all vessels are initially unequal. When the material under consideration in the system is inert or nonreacting the reaction velocity constant, k , becomes zero, and the dimensionless group Q reduces to unity in a11 the equations. This is the operation of the purging of the system, all possible cases of which have been considered in Table 111, and the complete mathematical solutions are indicated. Purging is understood here to mean the replacement of the original contents of a vessel with new material that is introduced in the feed stream. It will not be considered to be accompanied by a chemical reaction unless specifically stated. When a firsborder reaction is taking place in the system, the value of group Q is not unity and 1). The particular can be evaluated, because it equals (ken equations which describe the reactant concentrations as functions of time for this type of operation are summarized in Table IV for the most important combinations of initial conditions.
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
822 Table IV.
Vol. 42, No. 5
Transient Equations for Reactant Material Undergoing First-Order Irreversible Reaction
(Summary of transient equations giving effluent concentrations as functions of time for reactant material undergoing first-order irreversible reactions in continuous stirred tank reactor systems)
Section A System of Vessels"
Case NO. 1 2 3
Conditions of Operation of System Feed Concentrations conceninitially present tration in vessels Ai # An # . . A , # 0 Ca, = 0 Ai = A B = . . A n = 0 Ca, f 0 Ai # A2 f . . A , # 0 Ca, # 0
Column 1, one vessel only 5j 3g 3h
Section B Concentration of Reactant in Effluent as Function of Time
Eq. S O .
3g 3h
3i
Q + (A,
-
($!)
Qn
1 - e-Qt/Q
5j
C,, = A,e-Qt/Q
5k
Can = e - Q W
2
A , ( t / e p -1
1=1
51,
C,,,
a, = C -
Q"
+ e-Vf/d
5111
expressions for the product material, component B , of the first-order irreversible reaction, A --+B.
We shall first consider the situation where the initial concentrations of component B in the vessels and in the feed stream are zero. In order to do !Qt/e)n-i] ( (9) this we must return for a moment to consider ( n - l)! \ component A . If component A were inert, its concentration would be described by the purging equation, but because 1 mole of A reacts to form 1 mole of B , the sum of the concentrations of A and B must obey the purging equation. The true concentration of component A is known from the transient equation wherein account has been taken of the reaction, and it follows directly then that the concentration of the material B produced by the reaction is described by the difference between purging and reaction equations written for component A. If Cbn is defined as the concentration of component B in and from vessel n, it can be represented as: (9)
e-Qt/O
\ Can = L6, - 1
2
5r
Ref. (9)
C,,
Column 3, n vessels all having unequal volumes 5m 5n 50
Column 2, n vessels all having equal volumes 5k 3i 5L
Column 4, n vessels having equal and unequal volumes, nonisothermal systems, or both
+
Cbll
= (Can)purging conditions
- (cas) rezcting conditions
(16)
when there is no component B initially present in the system. 5n
50
Now if the initial concentrations of the inert product B in the system are not zero there will be an additional factor contributing to the total product concentration, which is determined from a purging equation written for component B based on these initial conditions. For the case where A ----f mB, the right-hand side of Equation I6 must be multiplied by the factor rn. By adding these two refinements, the expression for Ch, is:
+
Ch,, = (Cb,,) purging conditions m[(C,,) purging conditions - (Can) reacting conditions] (17)
" Numbers refer to equations given in Section B.
Equations 5p, 5q, and 5r, listed in Table IV. Section A, must be derived from the general Laplace transform Equation 11, for any particular system.
. An inspection of the most complicated transient expressions (Equations 14 and 15) shows that each is composed of a constant term and a series of t e r m that contain a negative exponential of the time factor. As time approaches infinity the exponential terms approach zero, leaving only the constant term, which is identical with the previously reported design equations ( 4 ) for the steady-state condition of operation. The exponential terms then are the transient portions of the total solution. Transient Expressions for Concentration of Product Material in EfEuent from Continuous Stirred Tank Reactor Systems. In all the preceding discussion, the focus of attention has been on the reactant material, component A , exclusively. I t is, however, a relatively simple matter to obtain the corresponding transient
To define Ca, completely in Equation 17, four independent initial conditions must be established: the concentrations in the feed stream, C,, and Ca,, and the concentrations initially in the vessels, the At's and the Bz's. There are 36 possible combinations of these initial conditions and the transient equations could be written for all of them. Houever, because a large number of cases are felt to be of little practical interest, a dozen of the ones n-hich should be the most valuable have been given in Table V. Cases which are not considered may also be derived in t h e manner described above. Simultaneous First-Order Reactions. The previous discussion has been limited to single irreversible first-order reactions. If simultaneous first-order reactions occur, the situation becomcs: h.1
A
--t
yB and A
k:
--+ p C
If the reaction velocity constant, k , that was used in the preceding analysis for single irreversible reactions is now replaced by the sum, kl kz, the concentration of reactant in t,he system is
+
I N D U S T R I A L A N D E N G I N E E R I N G ,C!H E M I S T R Y
May 1950
a23
Table V. Transient Equations for Product Material Obtained from a First-Order IrreversibIe Reaction (Summary of transient equations giving effluent concentrations as functions of time for produot material obtained from a first-order irreversible reaction in a continuous stirred tank reactor system)
Section A Systems of Vessels”
Case No. 1
Conditions of Operation of System Feed Concentrations conceninitially present tration in vessels
C,,
=
Cbo
2
0
@Cao# 0
0
Cb,
.I
0
3
Ca, # 0 cb,, = 0
4
Ca, # 0 Cb, = 0
5
Ca, # 0 cbo =
0
6
Cc,,#O Cb, = 0
7
Ca, = 0 Cbo Z O
8
Ca,
9
C, # 0 Cb, # 0
10
Ca, # 0 Ca, # 0
11
C #O C;: # 0
12
Cao # 0 Cb, Z 0
# 0 Cb, # 0
Column 4, n vessels having equal and Column 2, TZ vessels Column 3, n vessels unequal volumes, all having unequal nonisothermal all having equal v o1u m es volumes systems, or both
Column 1, one vessel only
+ m(3c-5k)
+ m(5b-5m)
+ m(5f-5p)
AI # Az # , .A, # 0 Bi # Ba # .Bn # O ..An = 0 Ai = An B1 = B2 a . , B n = 0
2’ 4- m(2-5j)b
m(3k) = m(3a-3g) ’m(3L) = m(3d-3i)
m(4-5n)
m(5g-5q)
A I # Az # . . A n # 0 B1 = Bz = . . B n = 0 ..An = 0 A I = Az B1 = Bz = . . B n = Bo
m(3j) = m(3b3h) m(3f-5L)
m(5d-50)
m( 5i-5r)
=i
=i
Ai = Az ..A, = 0 BI # Bz # . .Bn # 0 Ai#Az#..A,#O BI # Bz # . . B , # 0
3c’
+ m(3L) 3c’ + m(3L) 3 ~+ ‘ m(3f-5L) 3f‘ + m(3c-5k) 3d’ + m(3L) 3d’ + m(3f-5L)
+ m(3k) 2’ + m(3k)
1’
2‘
2’
+ m(3j)
3b’ A I # Az # . . A , # 0 Bi # Bz # ..Bn # 0 3a’ Ai = Az ..A, = 0 B1 = Bz . .Bn = 0 3a’ Ai # Az # . . A n # 0 BI = Bz = . . B , = 0 3b‘ A i = Az = , . A , = 0 Bi = Bz = . .B, = 8, 3b‘ Ai=Az=..A,=O B1 # BZ # . . B , # 0 3b‘ Ai Z Az # . . A n # 0 Bi # BZ # . .Bn # 0
+ m(2-5j)
+ m(3k) + m(3j) + m(3k) . + m(3k) + m(3j)
+ m(3L) 3f’ + m(3L) 3e’
3f’
+ m(3f-5L)
5b’
+ m(4-5n) .5b’ + m(4-5n) 5b’ + m(5d-50) 5a’
+ m(5h-5m) 4’ + 44-511) 4’ + m(5d-50) 5c’ + m(4-5n) 5d’ + m(4-5n) 5d’ + m(5d-50) 5d‘
5f’
+ m(5g-5q) 5f’ + m(5g-5q)
tie‘
5f’
+ m(5i-5r)
5i’
+ m(5f-5p)
5g‘
+ m(5g-5q)
5g‘
+ m(5i-5r)
+ m(5g-5q) 5i‘ + m(5g-5q) hi’ + m(5i-5r) 5h‘
Section B
Concentration of Product in Effluent as Function of Time
Ref.
3k
(9)
3L Q
b
Numbers refer to equations given in Section B of Tables 111, IV, and V. Primed equations are written for component B, unprimed for component A .
The other equations shown in Section A of this table are combinations of equations which are listed in Tables I11 and IV, and need not be listed separately. The ones listed above serve as examples.
again described by the equations which have been derived and tabulated in Table IV. The product concentrations for this case may also be derived by the same process as before, with the exception that the amount of component B produced in the reaction is in the ratio of
(
(A), -
-,
is in the ratio of kA)The i+kz * final expression for the concentration of component B in and from vessel n is:
and the amount of component
Cbn = (Cb,,)purging conditions
conditions
+x [(c,,)purging ki + kz gkl
- (*can) reacting conditions]
(19)
Equation 19 is the same as Equation 17 where the term m in
Equation 17 is replaced by the group
+
qki in Equation 19. By hi kz using this substitution, the equations in Table V may be used for simultaneous reactions as well as for single Consecutive First-Order Reactions. A conaecutive first-order reaction is one wherein the original reactant A forms a product B in a first-order reaction, and B in turn produces a second product C in another first-order reaction-that is:
ka
kb
A-+-B-+C
(20)
For the reaction of A forming B , the transient concentration of component A is described by the equations in Table Iv for a single irreversible reaction. The rate equation is the same as Equation l.
INDUSTRIAL AND ENGINEERING CHEMISTRY
824
Table VI.
Vol. 42, No. 5
Transient Equations for Consecutive Reactions in One-Vessel Systems
(The equationr in this table are applied only to the intermediate reactant. The primary reactant concentrations may be calculated using the equations ful single irreversible firat-order reactions)
Eq. No.
Feed Concentrations
Initial Concentrations in Tank
Concentration of Component B as Function of Time
1
When component B is considered, however, a complication arises from the fact that it is being fornied by reaction from component A. The rate equation for component B in the first vessel
&
(8
f (S
dt
V
V
I
(21)
But Fl(s) is given in Equation 9 and may be substitut,ed into Equation 22, giving:
+
S(S
f
+
&,/e) kdAo/e
kaAl
is:
_ dCb1 - Ca,F - Cb,F - kbCbl + k,C,
Cbo/e
&de) f
&b/e)(s
f &./e) 4-s(s f
&b/O)(s
4- &*/e)
(23)
The inverse transform of Equation 23 can be reduced to:
This is a relatively complicated equation for only one vessel,
INDUSTRIAL AND ENGINEERING CHEMISTRY
May 1950
but all possible finite initial conditions are accounted for. Naturally, simplifications will result if some of the initial conditions are zero, and some of these special cases are outlined in Table VI. The equations for more vessels can be derived by an iterative process, but the equations become cumbersome. Reversible First-Order Reactions. The condition of a reversible first-order reaction is described by the kinetic equation: k.
A*R
(25)
kb
This is somewhat similar to the case of consecutive reactions considered in the preceding section, with the exception that component B reacts to give component A back again instead of a new component, C. The rate equation for component B will be the same as Equation 21, but account must now be taken of the formation of A from B in the writing of the rate equation for A . The equation becomes:
and it is noted that Equation 26 is of the same form as 21, the only difference being the exchange of b’s for a’s. The Laplace transform of 26 is:
which is likewise similar to Equation 22. In order to determine Fl(s)or G,(s) Equations 22 and 27 must be solved simultaneously. The simultaneous solutions of these equations will be similar in mathematical form, but the reference conditions must be evaluated for either component A or component B. The explicit solution for F1(8) is:
and the inverse Laplace transform gives the concentration of component A in the first vessel as a function of time, which can be simplified to:
c.,
=
825
ferent methods for using these equations. Some new concepts will also be developed in order to describe adequately several methods for starting up and shutting down equal volume systems in which a first-order irreversibl: reaction is being carried out
Acknowledgment This work was supported in part by a fellowship granted by E. I. du Pont de Nemours & Company, Inc., and by fundc granted by the Graduate School of the University of Minnesota The final manuscript was prepared with the aid of the Engineering Experiment Station at the University of Minnesota.
Nomenclature A = reactant component in system A , = initial concentration of A in vessel n of a CSTR system B, C = product materials derived from A in firsborder reactions B, = initial concentration of B in vessel Ca,, Cbo = concentrations of A and B in the feed stream Ca,, Cb, = concentrations of A and B i n and from vessel n f(t) = original function in Laplace transform equation F,(s) = Laplace transform function for component A in vessel n G,(s) = Laplace transform function for component B in vessel n k = firsborder irreversible reaction velocity constant IC,, kl = reaction velocity constants for simultaneous irreversible firsborder reactions ka, kb = reaction velocity constants for steps in consecutive or reversible firsborder reactions L = indicates performance of Laplace transform operation rn, p = product coefficients in kinetic equations kV Q = dimensionless group, k8 1 = 1 F s = complex variable in Laplace transform equation t = time V., = effective volume of reaction vessel, tank, or compare ment number n in a continuous stirred tank reactor system Wan = mass of reactant A in vessel n 0% = (V,/F), nominal holding time for vessel n
+
+
Subscripts refers to component A refers to component B variables for indicating limits of summation of series refers to vessel n in a CSTR system o = refers to feed stream, or a constant initial condition throughout the system
a = b = i, j , k = n =
Literature Cited As in the case of the consecutive reactions, all the initial conditions have been finitely satisfied, and the complexity of the equation will be reduced if some of the initial conditions are zero. Some special cases are presented in Table VII. The iterative solution for succeeding vessels becomes even more complex than for the case of consecutive reactions.
Summary By using the Laplace transformation it has been possible to develop general equations which can be used to describe a wide variety of transient conditions in continuous stirred tank reactor systems. These equations are applicable to equal, unequal, and combination volume types of systems, and multitemperaturelevel systems, which are being purged of their contents or in which any of several types of first-order reactions are being carried out. In a subsequent article, examples will be given of several dif-
(1) Berg, C., and James, I. J., Jr., Chem. Eng. Progress, 44, 307 (1948). ( 2 ) Brin, M. S., Friedman, S. J., Gluckert, F.A., and Pigford, R. L., IND.ENG.CHEM.,40, 1050 (1948). (3) Denbigh, K. G., Trana. Faraday Soc., 40, 352 (1944). (4) Eldridge, J. W., and Piret, E. L., Chem. Eng. Progress, in
press.
( 5 ) Gardner, M. F., and Barnes, J. L., ‘‘Transient# in Linear Systems,” New York, John miley &Sons, 1942. (6) Ham, A., and Coe, H. S., Chem. & Met. Eng., 19,663 (1918). (7) Johnson, J. D., and Edwards, L. J., Trans. Faraday SOC.,45, 286 (1949).
Kandiner. E€. J., Chem. Eng. Progress, 44,383 (1948). Kirillov, N. I.,J . Apptied Chem. (U.S.S.R.), 13,978 (1940). MacDonald, R. W., thesis, University of Minnesota, 1949. MacDonald, R. W., and Piret, E. L., to be published. (12) MacMullin, R. B., Chem. Eng. Progress, 44, 183 (1948). (13) MacMullin, R. B., and Weber, M., Jr., Trans. Am. Inst. Chem.
(8) (9) (10) (11)
’ Engrs., 31, 409 (1935). (14)Marshall, W. R., and Pigford, R. L., “Aaulications of Differential Equations to Chemioal Engineering,” University of Delaware Press, 1946.
RECEIVED January 12, 1950.
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