PERIODIC REACTOR OPERATION J .
M. DOUGLAS
Department of Chemical Engineering, University of Rochester, Rochester, N . Y .
An analytical procedure for determining the frequency response of a simple nonlinear reactor is described in detail. The time average value of the reactor effluent is different from the steady-state value, and therefore the optimum steady-state reactor design does not always give the most profitable system. Sometimes it is better to design control systems which amplify the effects of disturbances, than to force the system to operate close to the optimum steady-state conditions.
HE conventional approach used in the development of a Tchemical reactor is first to determine the optimum steadystate design. Some of the reactor inputs vary with time, but steady-state analysis considers only the average values of these variables. Then a cointrol system is designed to compensate for the effects of the varying inputs. I n this way the controlled reactor is forced to have a relatively constant output close to the desired opti.mum steady-state conditions. Although the steady-state design procedures normally take into account the nonlinearities in the system equations, the control system design is usually based on a linearized model. This paper describes a method for evaluating dynamic effects in nonlinear systems. 'These results are then used to determine the effects of time-varying inputs on the reactor design. T h e system chosen for investigation was a second-order reaction in an isothermal continuous stirred tank reactor. Previous work has shown that the time average behavior of this system is different from the steady-state value when the input variables fluctuate sinusoidally (2). Also, this system has the advantage of simplicity, although the approach is valid for more complex processes.
Steady-State Design
Suppose we are asked to design a reactor for producing G lb. moles per hour of a product P by the reaction 2 A -+ P, where the reaction rate is given by the expression r = kA2
(11
and k = (cu. ft)/(lb. mole)(hr.), A = (lb. mole)/(cu. ft.). T h e reaction will be carried out in a backmix reactor having a cost Cv($)/(cu. ft.) (hr.), which includes all the operating costs and the capital costs on a depreciated basis. T h e feed mixture has an average composition A,,, and the cost of the feed stream may be taken as Cl($)/(lb. mole A). Ifwe assume that the cost required to separate A and P in the product stream is negligible and that A cannot be recycled, it is a simple matter to calculate the optimum reactor design (5). The total cost, CT, is given by the equation
s = A/Af,,
Q = qAf,/G, and
V
=
kVRAfs/q
(5)
Q
(6)
our equations become
(Cv/CfkAf,2) QV f
CT/CfG (1 1
- s) = vs2 = Q(l - X )
(7) (8)
Using Equations 7 and 8 to eliminate Q and V from Equation 6 we obtain
CT/CfG = (Cv/CfkAfs2)(1/x2)f 1 / ( 1
-
S)
(9)
Then, the optimum conversion is obtained when
b(c,/C,G)/b~= 0 or
s3 = ( 2 C,/C,kAf,')(l
-s)~
(10)
This cubic equation can be used to determine snpt..Then, Equations 7 and 8 can be used to find V,,,. and QOpt. Thus the optimum steady-state design procedure is straightforward. Feed Composition Disturbances
In some processes, such as a catalytic cracking unit, the feed composition actually varies with time. Then attempts are made to design control systems which will compensate for the effects of this disturbance. However, if we assume that the feed composition in our simple problem fluctuates according to the equation A,
=
Afs (1
+ a sin w,t)
(11)
and attempt to determine the optimum design, we get a different result. Providing that the feed rate remains unchanged, the average cost of reactants is not affected. T h e average production rate over one period, T , will be
or where V , is the reactor volume (cubic feet) and q is the feed rate (cubic feet per hour). A steady-state material balance on the reactor shows that
q(Ay,, - A )
=
kVRA2
(3)
and the production rate may be written as
G = q(Afs - A )
(4)
These relationships provide sufficient information to determine the values of q, V,, and A which minimize the total cost. If we let
1 = Q
-:LTrdt
Thus we must develop an expression for the time dependence of the fraction of unconverted material in the reactor effluent. This required expression is actually the frequency response of the nonlinear reactor. However, for a case where the disturbances have a very low frequency, the accumulation terms in the material balance can be neglected and it is possible to find a simple approximate equation. The material balance is VOL. 6
NO. 1
JANUARY 1967
43
or, letting x f = Af/Afs= 1
+ a sin w,t
(13)
0
(14)
we obtain
+x -
vx2
XI =
v = kVRAf.s -, w
Solving explicitly for x gives x =
-(I/*
V ) [l
- (1
+ 4 VXf)1’2]
( 1 5)
+ a(1 + 4 V)-1’2 sin oat a2V (1 + 4 Y)-3’2sin2 w,t + . . .
(16)
where
-
xs =
(1/2
V )[ l
-
(1
- 4 V )+1’2]
(17)
Thus
dx
dr
+ x + Vx2 = x f
x =
= q ( A f s - A,)
+ 21 qAfsa2V(1+ 4 V ) ) - - 3 / 2 -
[
1
]
- x s + -21 a2V(1 + 4 V)--3’2
(23)
= 1
+ a sin wr
(24)
Xa
+ y,
a = 2
vx,
(25)
Equation 23 can be put into the form
(19)
This result shows that the production rate is higher than that predicted by the optimum steady-state design. If the additional production is not desirable, it is possible to maintain the initial production rate using a smaller amount of feed, and thereby decrease the raw material cost. Equation 2 can be written as 1 = Q
q
This nonlinear equation for the frequency response is a nonhomogeneous Riccati equation. I t can be transformed into a second-order linear equation with periodic coefficients, similar to a Mathieu equation (7). However, the frequency response of nonlinear systems can often be approximated in a satisfactory manner by merely considering the linearized system equations. Therefore, it should be possible to develop a simple approximate solution using perturbation techniques (6). If we let
and G,
V R wn
and substituting Equation 1 1 for A , into Equation 21 gives
Now, if we expand this result in a Taylor’s series about x j = 1 , we obtain x = xs
=
Q
(20)
2 + (I + a ) y = a sin wr - p ~ y 2 dr
(26)
Parameter I.( has been arbitrarily introduced in front of the nonlinear term in this equation. Obviously, if p was equal to zero it would be a simple matter to solve this equation, and the frequency response of the linearized system would be obtained. I t is apparent that p must be equal to unity, but if we consider that p is a small parameter in order to keep track of the order of magnitude of various correction terms, we can assume a solution having the form
and this expression can be used to eliminate Q from the cost equation. Thus Substitution of this expression into the system equation gives
(1
+ a)bo + py1 + a
and the optimum conditions are obtained when b ( C T / C , G ) / ~ V = 0. This result is similar to that obtained from a slightly different formulation of the optimum steady-state design problem except for the last term in the denominator. Since this term depends upon the amplitude of the input disturbance, conventional design procedure does not always correspond to the reactor having the minimum cost. Also, it is more profitable to let the disturbances enter the process than to design a control system which will damp out the input fluctuations ( 2 ) . Although the approach described above neglects accumulation in the system and is based on the curvature of the steadystate response surface, it should indicate the proper type of behavior unless it is possible to have resonance effects in the system. D y n a m i c s of Nonlinear Reactor
To determine the effect of high frequency fluctuations on the process it is necessary to include the accumulation term in the material balance equation. 44
I & E C PROCESS D E S I G N A N D DEVELOPMENT
sin w r
P2.n
- pV
(YO
+ ...) + + + . . .) =
pyx
p2yz
and if we equate terms with like coefficients of p-i.e., terms having the same order of magnitude-we obtain the linear frequency response and the equations for the correction terms
9 + (1 + a) dr
y1
= -vvoz
(29)
The steady-state solution of Equation 28 is yo =
+ + 4[(l + a) sin a
[(l
wr
- w cos
UT]
(31)
This result can be substituted into Equation 29, which can then be solved for y l .
y1 =
2(1
-a2V + a)[(I + 4'+ + W'I
+ a)Z -
[2(1
a'V
so that
W']
sin 2
WT
+ CfG
1
+ -1 [I - (1 + 4 V)l/ZI +
a' V
Now, Equations 31 and 32 can be substituted into Equation 30, and y2 can be evaluated. I t is not difficult to show that the expression for y2 contains the first three harmonics of W T , but does not contain a constant term. Hence the solution for y can be approximately written as: 3' = yo
+ + + P'Y2
jly1
(27)
, , ,
or, after substituting Equations 31 and 32,
Y=-
a
+ a)' + w'l
[(I
[(I
+ a ) sin
WT
+ + +
2(?1 a)' - W 2 (1 a)' 4 W'
(*/*I (1 + a)[(I
-
[(I
-w
1+
-
cos U T ]
sin 2 W T
+
+ a) sin
UT
+ a)' + 4
-- w cos U T ]
2(1 + a)Z - W' {(li a)' -+ 4 W'
'/'(I
+ a) [(I + [(l + a)' + 4
c1)'
- 5
I'@
-5
W']
COS
I ' W
+ [(I +aa' )'V + sin 2
x
2 WT
t
-2(1
WT
cos 2
UT
= 0
b(C,/CfG)/dV
For very low frequencies, as w + 0, Equation 34 becomes identical to Equation 18, derived from the steady-state equation, and Equation 36 reduces to Equation 21. Also, for very high frequency inputs, as w ---* m , the average concentration is equal to the unperturbed steady-state value, xs, and the optimum design problem degenerates to its original form. This type of behavior is to be expected, since high frequency disturbances are normally completely damped in chemical processes, whereas low frequency fluctuations pass through the system undamped.
Another problem of some interest is to examine the response of the system to simultaneous disturbances in feed composition and flow rate. If we consider a case where the feed composition is given by Equation 11 and the flow rate is described by the equation
q
X W ' I
=
qs [I
+ b sin(Wbt + e ) ]
X
=
A/Afs,
Q
+ . . .}
V
= qsA/s/G,
a1 = vRWa/qs,
V
+ 4 V)l"(l + 4 v +
(37)
W2
=
=
kvRAfs/qs,
vRWb/qs,
7
= qst/vR
(38)
+ @dT]
(39)
the cost equation becomes (33)
(34)
[ 4- f 1
T
ab sin
WIT
sin(m7
Since
ilTab
sin
O ~ T sin(w27
0 if
w1 #
WZ,
+ e)&
=
for large values of T
'/zab cos 0 if W I
=
WZ,
for one period
(40)
then
where the term in parenthesis is present only if w1 = w 2 . If initially we consider only very low frequency disturbances, so that the accumulation term in the material balance can be neglected, then
W')
x=-lL[l-(l+4V-qs 2 v
and +4V)-'/'(l
Again, the optimum reactor is obtained when
+
wZ1
a'
(36)
where 0 is the phase angle between the composition and flow fluctuations, again it is possible to obtain a higher average conversion. If we let
This result shows that the frequency response of this simple nonlinear system is a periodic function with a nonzero mean value and contains higher order harmonic terms in addition to the fundamental component. T h e procedure for evaluating additional correction terms is straightforward, but the algebraic manipulations become unwieldy. However, for mild nonlinearities (functions which deviate only slightly from a linear approximation in the region of interest) only one or two correction terms should provide a sufficiently accurate approximate solution. Obviously, we must verify the solution by comparing it with a more exact numeric I integration before using the results in a design calculation. This type of numerical study is described below. T h e quantity of gre,atest interest in the approximate solution for the purpose of determining a better reactor design is the average value ofy. Using Equations 12, 17, 25, and 33, we find that x,\. = xs
+ 4 V)-l/'(l + 4 v +
Feed Composition and Flow Rate Disturbances a)'
Now we let p be equal .to unity, but require that the amplitudes of the terms involving be smaller than the amplitudes of the linear frequency response terms, so that our assumed solution, Equation 27, will be a convergent series. Thus we find that
[(I
- (1 2
2v
+4V+d)-']
(35)
AfS
qs)"2] 4
(42)
Expanding this equation in a Taylor's series about the steadystate condition gives: VOL. 6
NO. 1 J A N U A R Y 1 9 6 7
45
sin + + a(1 + 4 + 2 V - ( 1 + 4 V ) l @ ] [2 V(1 + 4 V)'i2]-1sin(~27+ 0) 2 V ( 1 + 4 V)--3l2sin' - b2V(1 + 4 V)-3i2sin2(w2T + 0) + ab 2 V ( l + 4 sin w1r sin(w2T + 0) + . . . (43) x = xs
WIT
b[l
WIT
The average fraction of material unconverted is
xsv = x,
- ' / z V(l
+ 4 V)-3i2(a2+ bZ - 2 ab cos 0)
(44)
where the term ab cos 6 is present only if w i = wz. I t is apparent that the maximum improvement in conversion is obtained when the amplitudes, a and b, are large, when w1 = W Z , and when6 = 180'. However, the production rate, which is given by the expression
yo = m l sin wr
+ a sin w l r ] [ l + b sin(w27 + e)] q s A f s [ l+ b sin(w27 + O ) ] x
-
G = qsAfs[l
-Q
a2V
-
- x s + 2(1 + 4
V)3i2
(45)
ml =
+
+
+ 4 V)3iz- ( 1 + 6 V ) ]
,
+ a)' + 4 w' .
+
cos 2 w r
where
m2 = ab cos 0 [(I 2(1 4 V)3'2
.
\
(1
has an average value
1
UT
+ +
or
Gsv qsAfs
+ mz cos
mr(l a) 2 m5 w m3 sin 2 wr yl = 1+a+ (l+a)'+4wZ
- A)
G = q (AI
Equation 48 can be solved without difficulty. The solution of this equation is then substituted into the right-hand side of Equation 49, and Equation 49 is solved for y l . The procedure is repeated until a solution of the required accuracy is obtained. For the case where w 1 = w2,we find that
m3 =
[a
+ b(l - x,) cosO](l + a ) + b ( l - x,)sinO (1
b(l
- xs)sinO(l
+ + a)'
0'
+ a) - [a + b(l - x,)cosO]
(1
+ +
w2
- bm1 cos 0 - bmz sin 0
' / z [ a b cos 0
- mI2V - m2'V]
(46)
where the last term is present only if w1 = w2. The conditions for the optimum reactor can be obtained by substituting the above equation into the total cost expression, Equation 41, and then finding the value of V which makes b(CT,,/CfG)/bV = 0. This approach can also be used to determine the optimum phase angle, 0, which affects both the cost of reactants and the production rate.
(51) m 4 = l l z [ a bcos 0
- bml sin 0
m5 = ' / z [ - a b cos 0
- bmz cos 0 - m l m z V ]
+ bml cos 6 - bmz sin 6 -I- m?V
- rnz'v]
Then m3 + m l sin wr + mz cos wr + + 1+a + a ) - 2 mpw 4 1 + 4 + 2 m w sin 2 + ms(l ( 1 + a)' + 4 (1 + +4
x = xs
COS
2
WT
UT
Nonlinear Dynamics
a)2
T o determine the effect of frequency on the optimum design it is again necessary to include the accumulation term in the material balance equation. Thus
dA VE - = q ( A f - A ) dt
- ~VRA'
(22)
Substituting the definitions given by Equations 1 1 , 25, 37, and 38 we obtain
dr + (1 + a)y = a sin w1r + b(l
dr
pub sin W I T sin(w27
- x,)
sin
(WZT
+ 0) +
+ 6 ) - p b sin(w2r + 0)y - pVy2
w2
0'
(52) Thus we find that the fraction of unconverted material in the output has a mean value which is different from the steadystate value and has higher harmonic terms in addition to the fundamental component. I t is possible to show that for very low values of w , the average value of x becomes identical to Equation 44. Substitution of Equation 52 into 45 allows an evaluation of the average production rate.
(47)
where parameter p has been introduced in front of the terms which make an analytical solution difficult. Proceeding as before we assume a solution having the form
y = yo
+ + + py1
pzyz
,
.,
substitute this solution into the nonlinear equation, and equate terms having like coefficients of 1.1. This leads to the set of equations
dvo + (1 + a)y0 = a sin dr 46
WIT
+ b(l - x,)
sin
ab
(27)
(WZT
+ 0)
l & E C PROCESS D E S I G N A N D DEVELOPMENT
(48)
2(1 (b(1
-
xl)(l
+ a ) [ ( l + a)' +
X w21
+ a) + a [ ( l + a ) cos 0 - w sin 61)
(53)
Now, Q can be eliminated from Equation 41, and the optimum reactor volume determined using the relationship b (CT/CfG)/ bV = 0.
Numerical Evaluation of Approximate Solutions
I n order to assess the accuracy of the approximate solutions, particularly since we have used only a first-order correction term, it is necessary to compare the predicted values with a numerical integration of the equation. A simulation of the nonlinear reactor problem under investigation was undertaken by Douglas and Rippin (2) on an analog computer. T h e values used in that study were k = 1.2, q = 10, V = 100, and A,8 = 1.0. Similar results can be obtained by choosing C,/Cv = 60, k = 1.2, and A I , = 1.0. T h e computer results are given in Figures 1 through 5, and are compared with the analytical estimates for the very low frequency case and the more accurate solution. I t is apparent that the analytical techniques described above provide sufficient accuracy for engineering purposes for the simple system under investigation.
1
. 0
- --
ANALYTICAL SOLUTION, w I
0
A N A L Y T I C A L SOLUTION, wI
:2
-*
2i-
0
1'
- A N A L Y T I C A L SOLUTION o
COMPUTER SOLUTION
Y
1
I
I
I
/
COMPUTER SOLUTION,/ w , I, =, 2
/
x
/
/
I
I
0
90
I
180 PHASE ANGLE
I
I
2 70
360
(-8)
Figure 5. Simultaneous feed composition and flow rate disturbances F E E D COMPOSITION A M P L I T U D E ,
a
Figure 1. Effect of feed composition perturbation amplitude I
I
I
I
- A N A L Y T I C A L SOLUTION I 0
0
I
COMPUTER SOLUTION 2 6 8 F E E D COMPOSITION FREOUENCY, w ,
IO
Figure 2. Effect of feed composition perturbation frequency
ANALYTICAL
SOLUTION, w e I O
x
4
0
I I 0.05 0.10 0.15 FLOW R A T E A M P L I T U D E ,
I 0.20
0 !5
b
Figure 3. Effect of flow rate perturbation amplitude
Conclusions
Approximate analytical solutions for the frequency response of nonlinear systems can be obtained using the techniques of nonlinear mechanics (6). The average value of the output is different from the steady-state value, although this difference is small for mild nonlinearities; for highly nonlinear systems or those which exhibit resonance, the deviations might be very significant. Whenever disturbances cause a shift in the mean value of the output (7, 2), the optimum steady-state design does not necessarily correspond to the system yielding the maximum profit. Instead, the design must be based on the time average values of the production rate. Similarly, rather than designing control systems to damp out the effects of disturbances and maintain the reactor outputs constant a t their optimum steady-state values, it is sometimes better to design a control system which will amplify the effects of the disturbances-Le., simultaneous disturbances in feed composition and flow rate sometimes lead to higher deviations than are obtained with composition fluctuations alone. Of course, the final product must satisfy certain specifications, but damping and blending can be provided near the end of the process stream, rather than a t each individual unit. Hence, from the investigation of this very simple problem it is possible to conclude that neither the conventional approach of optimum design nor the normal definition of the control problem always leads to the most profitable system. However, for systems which do not exhibit resonance, it is possible to estimate the difference between the steady-state behavior and the time average behavior using only the steadystate equations or correlations describing the system. This VOL. 6
NO. 1
JANUARY 1967
47
estimate will give the maximum discrepancy between the two cases for inputs of very low frequency. A proper formulation of the design problem should include a description of the disturbances and the control system should be designed simultaneously with the process equipment. This leads to a problem in variational calculus, which can be attacked using the methods described by Horn (3, 4) for periodic processes. Nomenclature
A! A ,
reactor composition and feed composition, respectively, lb. mole/cu. ft. a = dimensionless amplitude of feed composition disturbance b = dimensionless amplitude of flow rate disturbance Cr, Cv,C, = total cost, cost of reactor volume, and raw material cost, respectively G = production rate, lb. mole/hr. k = reaction rate constant, (cu. ft.)/(lb. mole)(hr.) ml,m2,m3,m4,r n j = constants defined by Equation 51 = see Equation 5 Q = flow rate, cu. ft./hr. q r = reaction rate, lb. mole/(cu. ft.)(hr.) T = period of operation t = time, hr. V = see Equation 5 = reactor volume, cu. ft. VR x> XI = dimensionless reactor composition and feed Composition, respectively Y = deviation from dimensionless steady-state reactor composition =
.YO, YI,
y~
= components of y, see Equation 27
GREEKLETTERS see Equation 25 phase angle between feed composition and flow rate disturbances small parameter, see Equations 26 and 27 dimensionless time, see Equation 23 frequencies of feed composition and flow rate disturbances, respectively, radians/hr. w, Wlr w2 = dimensionless frequencies of disturbances, see Equations 23 and 38 SUBSCRIPTS av - average - steady state S literature Cited
(1) Blum, E. W., il.Z.Ch.E. J . 11, 532 (1965). (2) Douglas, J. M., Rippin, D. W. T., Chem. Eng. Sci. 21, 305 (1966). ( 3 ) Horn, F., 56th National Meeting, A.I.Ch.E., San Francisco, Calif., Mav 1965. ( 4 ) Horn, F., Division of Industrial and Engineering Chemistry, 151st Meeting, ACS, Pittsburgh, Pa., March 1966. ( 5 ) Levenspiel, O., “Chemical Reaction Engineering,” Chap. 6, Wiley, New York, 1962. (6) Minorsky, N., “Nonlinear Oscillations,” Chap. 9, Van Nostrand, New York, 1962.
RECEIVED for review May 2, 1966 ACCEPTED November 9, 1966 Presented in part at Division of Industrial and Engineering Chemistry, 151st Meeting, ACS, Pittsburgh, Pa., March 1966.
END OF SYMPOSIUM
CLOGGING OF FILTER MEDIA EPHRAIM KEHAT, ARIEH L I N , AND ABRAHAM KAPLAN
Department of Chemical Engineering, Technion-Zsrael Institute of Technology, Haifa, Israel
A simple relationship was derived for the effective resistance of a partly clogged filter medium and the resistance of the filter medium a t the beginning of a filtration cycle. This relationship holds for both complete blocking and standard blocking mechanisms and was verified b y experimental work. Five woven wool filter cloths were tested a t constant low pressures, 1 1.2 to 61.2 cm. of water, and feed concentrations of 2 to 10 grams per liter of ground polystyrene in water. The effective resistance of open filter cloths increased to a relatively low constant value. The effective resistance of tight filter cloths was independent o f pressure and feed concentration, for the range of variables in this work, and increased from cycle to cycle. A simple correction to the classical filtration theory that includes the effect of clogging of filter media i s suggested. HE object of this work was to develop a correction to the Tbasic filtration equations far the increase of the resistance of the filter medium because of clogging (also called blinding or plugging) of the filter medium. Clogging of filter media is a particularly severe problem in the paper industry, where the expense of time and material in changing a clogged “felt” in a paper machine is considerable. Hermans and Bredte (4)were the first to study this problem. They suggested two possible mechanisms for clogging of filter media : complete blocking, in which single particles, some\chat larger than the holes in the filter medium, plug u p individual holes; and standard blocking, also called semiblocking ( 7 , 6) or depth filtration (5),in which particles, smaller than the holes, are attached to the fibers along or within holes, or to other particles previously retained. They showed that for
48
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
constant pressure filtration, the inverse of the flow rate of the filtrate was proportional to the volume of the filtrate raised by an exponent. The exponent for complete blocking was 2, for standard blocking was 1.5, and for cake filtration was 1. They used the values of these exponents to determine the mechanism of clogging of filter media and suggested that standard blocking is more likely to dominate than complete blocking for most media. Heertjes and Haas (3) derived a rate equation for complete blocking and experimentally found that the exponent in the Hermans and Bredte equations can have any value between 2 and l-i.e., between complete blocking and cake filtrationand is a function of the concentration of the suspension. Grace (2) studied the increase of the resistance of a variety of types of filter media for very low concentrations of solids
