DESIGN AND CONTROL OF FEED-EFFLUENT, EXCHANGER- REACT0 R SYSTEMS . M , D 0 U G L A S Atlantic Refining Co., Philadelphia, Pa. Research Triangle Institute, Durham, V . C. J . C. 0 R CU TT P . W . B E R T H I A U M E , Electronic Associates Inc., Princeton, ,V. J . J
I
I
Simultaneous solutions of the steady-state heat and material balances for a first-order reaction occurring in a feed-effluent, exchanger-reactor system were obtained on an analog computer and used to calculate ihe values of exchanger and reactor lengths which minimized the equipment cost of the system. A dynamic study of the system indicated thai the desired steady-state conditions were metastable. However, a feedback proportional cloniroller could be used to stabilize the process. Computations were made to determine the effect of various controller gains, positions of the sensing element, and time lags in the feedback loop.
factors must be considered during the design of a reactor system for a first-order, exothermal, gas phase reaction. It has been shown that the maximum conversion per unit volume for single, irreversible reactions can be obtained in a n isothermal system (3). However, the high cost of heat transfer equipment (required to preheat the inlet gas stream and to cool the reacting mixture) associated with large scale isothermal operaticn usually makes a process uneconomical. Thus industrial reactors are often designed to operate adiabatically rather than isothermally. Aris ( I ) considered adiabatic systems and showed that a perfectly mixed reactor followed by a tubular (plug flow) reactor gives the highest conversion per unit volume. I n addition, he discussed a method for calculating the optimal design and the optimal reactor inlet temperature for a case where the cost of preheating was proportional to the heat required and where no reaction occurred in the preheater. However, in many situations it is desirable to use the hot effluent stream to preheat the inlet gas mixture. With this type of operation the reaction usually can be made to proceed autothermically (the heat of reaction alone provides the required temperature level) and, thus, the minimum operating cost can be obtained. This paper describes a procedure for determining the optimal design of a concentric tube heat exchanger followed by a tubular reactor. A direct approach for studying stability and control of this type of system is also discussed. Although the methods are described in terms of a specific reactor, the approach is general in nature and can easily be extended to other reactor configurations. ANY
Steady-State Design
There are two limiting cases for the design of a n exchangerreactor system:
1. All of the desired reaction can take place in the heat exchanger (a reactor of zero volume is used). 2. All of the desired ireaction can take place in a reactor (a heat exchanger of zero volume is used).
In the first case, the total volume of the system will often be a minimum; the feedback of heat produces higher average reaction rates than those obtained in a tubular reactor. However, this case does not yield the minimum equipment cost, since the cost per unit volume of a heat exchanger is usually greater than the corresponding cost of a reactor. The second case is also undesirable, because extremely large reactors are required to obtain the desired conversion when the feed stream is cold. Thus, some intermediate case which corresponds to the minimum total cost of the system must be found. Mathematical Model
The system consisted of a concentric tube heat exchanger followed by a tubular reactor (see Figure 1). Assuming plug flow, constant molal flow rate, a constant average heat capacity, and a constant heat of reaction, the steady-state equations which describe the system may be written as:
Heat Exchanger Tube Side Material balance. d x t l d z , = -atktxt Heat balance. d T t / d z , = p( T , - T i ) f
-jtktxl
Shell Side Material balance. dx,/dz. = asksxo Heat balance. dT,/dz, = P ( T s - T t ) - -j8ksx.
Reactor -Material balance. dx,/dz, = -a,k,xI Heat balance. dT,/dz, = y r x r x . Since the equations describing the operation of the heat exchanger are coupled and nonlinear, no attempt was made to obtain an analytical solution. Instead, they were scaled and programmed for a analog computer (a Pace 231R). The reactor equations can be solved analytically (4,6), but, for simplicity they were also programmed for the analog. Using the values listed in Table I, exchanger and reactor profiles were generated on the computer. VOL 1
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Table 1.
Numerical Values
VALUES O F PARAMETERS 1.875
( ~ = i
~t
R
as = 6.548
CY? = 10.91 1 . 3 1 9 x 104 = 2.198 x 104 k = 4.146 X 106 exp ( - 9 8 , 1 0 O / R T )
= 3.776 X 103 =
- 1720
Ya =
0.2609
a
3
BOUNDARY CONDITIONS Exchanger Atr, = 0
~t
Atr. = 0
Tt
Reactor Atz, = 0 At z, = L ,
XI
= 0,1490
xt =
=
t-
a
xs = 0 , 0 3 4 4 T , = 1630" R.
1400 O R.
XI
Tt
Xd
T,
F u-
= =
- 1560
a W n I
T, T.
(see Figure 1 ) 0
I
I
4
8
I1400 12
EXCHANGER LENGTH-FEET
Steady-State Results
Figure 2 shows the profiles for the case where all of the reaction was completed in the heat exchanger. The profiles had the expected shape (8),with the mole fraction of reactant in the tubes decreasing with length (due to reaction) and the temperature in the tubes increasing with length (due to heat transfer from the shell gas). As the gas flowed into the shell (in the minus z direction), the conversion increased rapidly because of the increased cross-sectional area of the shell side. I n this region the rate of heat release due to reaction exceeded the rate of heat loss due to heat transfer and the temperature increased. However, the composition rapidly decreased, so that a t some point in the shell, the rate of heat loss exceeded the rate of heat generation and the temperature decreased. Once the profiles for this case have been obtained, it is possible to calculate the reactor dimensions analytically. By arbitrarily picking a n exchanger length, the correct boundary conditions can be found from Figure 2, and then the tubular reactor size required to obtain these conditions can be either calculated or computed. Three exchanger lengths were picked and their corresponding reactor lengths determined on the analog. The profiles for two of the cases are given in Figures 3 and 4 and the resulting lengths are given in Table 11.
Table II. System Costs .&changer Length, Ft. Reactor Length, Ft. 2 68.7 4 8.5 6 1.8 0 7.8
Total Cost 78.7 28.5 31.8 39 . O
Here, the total cost figures are based on the assumption that the cost per unit length of the exchanger is five times the cost per unit length of the reactor. For the caqe considered, the exchanger length which gives the minimum construction cost for the system is approximately 4 feet Figure 3 shows that this corresponds to a situation where roughly 25% of the conversion occurs in the exchanger. EXCHANGER
F
x . . T.
PRODUCT
Figure 1. 254
REACTOR
Schematic diagram of system
l&EC FUNDAMENTALS
Figure 2.
Profiles in heat exchanger
Dynamic Considerations
The previous results were based on steady-state analysis and gave no consideration to process operability. In situations where there is positive thermal feedback in a system, van Heerden (7) has shown that it is sometimes possible to obtain more than one solution to the steady-state equations. Only one solution will satisfy the prescribed boundary conditions (Table I), but there is a possibility that this solution is metastable-i.e., that the system has a natural tendency always to move away from this state. Hence, it is necessary to study the stability of the system in order to ensure its operability. Several methods can be used to study process control and stability. Considerable thought was given to the problem of picking the most suitable approach for the type of system under consideration, and a simulation approach was selected. This is the most direct method, it gives responses in the time domain so that the designer can determine the effects of any kind of inlet fluctuations rapidly, it will predict the effect of large input perturbations since it can handle the nonlinearities inherent in the problem, it will indicate the final steady-state conditions which a n unstable system will approach, and it provides the simplest method for determining the dynamic characteristics of numerous steady-state base cases. The major disadvantage of the simulation approach is that none of the control synthesis procedures can be applied. A designer must rely solely upon his intuition and judgment when attempting to pick a control system which will stabilize a process or will improve its response characteristics. Perhaps the ideal approach is to undertake a simulation study and also to determine the transfer function, based on a linear analysis, and then use standard synthesis techniques. However, economic considerations often preclude so extensive an investigation. Mathematical Model. The dynamic equations for the process may be written as:
I880
REACTOR
0.24
z 0
I
IV
< 0. I 6 K L
w
J
0
I I-
z a 0.08
c u 4
w K
a 0
8
4
K
1720
*.
w
K 3 I-
< 1560
O
a
a 0.16
0
z
a
0
e
W
a
I-
z 0.08
E
1560
a
I-
IO
a W K
0
11400 12
1400 2
0
4
6
8
LENGTH - F E E T
Figure 4.
Profiles in exchanger-reactor system
Shell Side Material balance. a ( p , x , ) / b B - a( O ~ U , X , ) / ~ Z = , - LaxB Heat balance. d(p,A9c,T,)/a3 - D(p,u,A,c,,T,)/az, = - CLAL(Ts - Ti)$. (-AH)Aaksxa Reactor
( - AH)k+
a a
I-
-
Material balance. d ( p , x , ) / & 9 Heat balance. a(p,c,T,)/BB
w’
1720
LL
LENGTH F E E T
Figure 3.
I
I-
K
z
1 I880
0
W -I
w
EXCHANGER--------~~REXTOW
I
z
+ b(plu7x )/br -k,x, + a(p,uCpTr)/atl = =
These equations were simplified, using the continuity equation and the ideal gas law, and then were replaced by a set of difference-differential equations. T o fit the problem on a single computer, the case where all of the reaction occurred in the heat exchanger was considered. I t was assumed that the molal flow rate was constant, the reciprocal of the temperature could be represented by a linear function of temperature (over the range of Interest), and four difference sections on both the tube side and the shell side would adequately represent the physical situation. A backward difference scheme was used, with the first cell taken as one half of the exchanger length. This was done because a major portion of the curvature in the temperature profiles occurred in the last half of the exchanger for most of the cases considered. In the case reported, however, better results might have been obtained if the initial difference interval had been smaller. System Stability. T o test the stability of the system. the previously computed steady-state profiles were imposed as initial conditions on the analog. If these solutions had been stable, the simulated process would have remained a t steady state, since stable solutions are autoregulatory to small errors. However, when the computer was put into operation, “blowout” occurred-that is, the composition values a t the various points in the exchanger increased until they became equal to the inlet composition. Thus, a new steady state, in which there was no reaction, was obtained. As the inlet temperature was increased to 1450’ R., the same phenomenon was observed. However, a t temperatures above 1450’ R., “blowup” occurred. (The difference between 1450’ and 1400’ R., the actual inlet temperature, was due to the approximations made in the differencing schemes.) At the blowup steady stale, conversion in the tubes was complete and the maximum temperature in the system had in-
Profiles in exchanger-reactor system
creased to 2040’ R. (as compared to 1800’ R.). Since the case under consideration had a maximum temperature limitation of 1860’ R. (imposed by the presence of side reactions and the possibility of metal failure in the exchanger), the blowup steady state was undersirable. Figure 5 compares the composition and temperature profiles for the blowup case with the desired profiles. To check the result that the desired steady-state solution was unstable, a conventional stability investigation was undertaken. The set of 16 difference-differential equations was linearized and the characteristic roots of the coefficient matrix were calculated on a digital computer. Two of the roots had positive real parts, which confirms the analog results. The dynamic model established that two stable steady states existed for this process, one at a very low conversion and the other a t a very high conversion, and that the desired steadystate solution, a t intermediate conversion, was metastable. Similar behavior has been observed in continuous-stirred tank reactors (2, 5 ) and in exchanger-reactor systems where no reaction occurs in the exchanger (7). Proportional control has usually permitted stable operation a t the metastable state in these systems, and therefore, it might prove adequate for this situation. System Control. A proportional controller was simulated on the computer. I t was assumed that the measurement and control dynamics could be represented by a single first-order lag. T h e model was constructed so that the proportional gain could be varied between K , = 0 and K , = 30, the system lag could be varied between T = 0.1 and T = 5 seconds, and the location of the sensing element could be changed. Steady-state operation was obtained when:
1. The temperature-measuring device was located a t the tube outlet.
2. Control was effected a t the tube inlet. 3. The controller gain was K, = 10. 4.
The feedback loop time constant was
T
= 0.1 second.
Figure 6 compares the composition and temperature values with the desired steady-state profiles. The agreement on the tube side was excellent, but fairly large deviations were found near the shell outlet because of the inadequate number of difference sections used in this case. Effect of Controller Gain. At low values of controller gain ( K , = 0 or 1.0) the system was unstable and proceeded VOL. 1
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16
I
TEMPERATURE STEPS A 0
--- DESIRED S T E A D Y S T A T E P R O F I L E S X+
a
z
2200 w*
I-
a
-0
INPUT
VI-
3
a u
k
LL
K J
a a w
0.04
Y O 0-I
SINE PULSES
=ii
1800 b
-
:a5 2
6
4
LENGTH
Figure 5.
ST E A DY, STATE
W K
I400
0
8
- FEET
0.04 0
I
I
I
20
40
60
TIME
Profiles at “blowup”
1
z
0
u 4 a
0.16
/’
-/
L
-
2
1720
C 3
W
I-
-I
a a W a
0
I
I-
z a
A
t
COMPOSITION VALUES
V
a
-
I
1560
W K
0
I
I
4
8
I1400 12
LENGTH-FEET
Figure 6.
z
s
0.08
Heat exchanger with proportional control
TEMPERATURE
1
Kc = 5
I
T‘=-2%
0.06 STATE
0.04
0.02 2 1
STEP INPUT
c COMPOSITION
d m
).06
0.02 1 0
I
I
I
I
20
40
60
80
TIME
Figure 7. 256
I
K, = 5
- SECONDS
Transient response
I&EC FUNDAMENTALS
I
Kc = I O
INPUT
Figure 8.
-I-
Kc: 10
C O M P O S I T I O N VALUES TEMPERATURE VALUES
I 80
- SECONDS
System response
to blowout. Higher gains ( K , = 5.0 or 7.5) gave the desired steady-state profiles and provided adequate control for small temperature fluctuations and for composition fluctuations (The input perturbation was one half of a u p to +IO%. cycle of a sine wave, with a frequency of 0.025 cycle per second.) When large temperature fluctuations were encountered (+IO%), blowout occurred. As the controller gain was increased ( K , = lo), control of large fluctuations was possible. Faster system responses and smaller deviations from the desired values were also obtained. However, the system started to oscillate if the gain was increased too far. Damped oscillations were present when K , = 20, but continuous oscillations occurred at K , = 30. Effect of Feedback Loop Time Constant. When the feedback loop time constant was increased to 5 seconds, a t K, = 10, the process oscillated and went to blowout. -4time constant of 1 second, however, yielded stable operation and provided adequate control of input perturbations. Effect of Sensing Element Location. In the calculations discussed above, the error signal was generated a t the tube outlet. Attempts were made to control the temperatures a t the shell outlet, a t a point two thirds of the way down the tubes, and at a corresponding point in the shell. When the error was measured a t these positions, the system possessed a very sluggish response and in some instances started to oscillate. I t appears most important to control the temperature a t a point where the reaction rate, rather than the heat transfer rate, is predominant. Transient Response. Figures 7 and 8 present the transient response of the system for the case where the feedback loop time constant was 0.1 second and the tube outlet temperature was controlled. The curves indicate that the system response was nonlinear even for small input perturbations and that temperature fluctuations caused greater deviations from the desired steady-state values than composition fluctuations. The actual process has been approximated by a set of differencedifferential equations. This approximation should be acceptable when calculating the response to low frequency perturbations. However, when predicting the response to step inputs, which contain all frequencies, the results would be in error. Conclusion
For the system investigated : To obtain a system with the minimum equipment cost, approximately 257, of the desired conversion should occur in the heat exchanger.
h proportiorial controller with a fast-acting feedback loop could be used to provide adequate control of a metastable condition. Best control is obtained by locating the sensing element a t a position where the reaction rate, rather than the heat transfer rate, is predominant.
GREEK (Y
P Y 0 P 7
Nomenclature
A AL
heat transfer area per unit length, sq. ft./ft.
= heat capacity of mixture, B.t.u./mole O F. GP E = energy of aciivation, Arrhenius law F = molal flow rate, moles/sec. = molal velocity, rnoles/sq. ft. sec. G, (-AH)= heat of reaction, B.t.u./mole k = p exp( - E / R T ) = reaction rate constant, Arrhenius law,
moles/cu. ft. sec.
K, L
PT U
UL X
z
FCP
( - A H ) a / c p = (-AH)/cPG, time, sec. density, moles/cu. ft. = feedback loop time constant, sec. = = =
SUBSCRIPTS
= cross-sectional area, sq. ft. =
= l/Gm, reciprocal molal velocity
- -ULAL
= proportional controller gain
total length, ft. frequency factor, Arrhenius law temperature, O R. velocity, ft./sec. over-all heat transfer coefficient, B.t.u./sq. ft. sec. F. = mole fraction of reactant = length, ft.
= = = = =
e
r S
t
exchanger reactor = shell side = tube side
= =
literature Cited
(1) (2) 3) 4)
Ark, R., Can. J . Chem. Eng. 40, 87 (1962). Aris, R., Amundson, N. R., Chem. Eng. Sci. 7, 121 (1958). Denbigh, K. G., Trans. Faraday Sac. 40, 352 (1944). Douglas, J. M., Eagleton, L. C., IND.ENG.CHEM.,FUNDAMENTALS 1, 116 (1962). (5) Foss, A. S., Chem. Eng. Progr., Symp. Ser. 45, No. 25, 47 (1959). (6) Parts, A. G., Australian J . Chem. 11, 251 (1958). (7) van Heerden, C., Chem. Eng. Sci. 8, 133 (1958). (8) van Heerden, C., Znd. Eng. Chem. 45, 1242 (1953).
I
RECEIVED for review October 26, 1961 ACCEPTEDAugust 10, 1962
BUBBLE DYNAMICS A T THE SURFACE OF A N EXPONENTIALLY HEATED PLATE S. G . BANKOFF Chemical Engineering Department, Northwestern University, Evanston, 111.
The consequences of Zuber’s hypothesis, which states that at every instant the latent heat increase of a bubble growing at a heated surface is equal to the superheat energy of the liquid displaced by the bubble, are evaluated for an exponentially heated plate. For a bubble growing in saturated liquid, the radius increases initially as t3” a result in agreement with Zwick’s relationship for a bubble growing in a liquid with exponential volume heat sources. The effect of plate thickness is also discussed.
N AN EARLIER WORK
( I ) , the heat flux from a solid surface
I to the liquid brought into contact with it as the result of the growth and collapse of a hemispherical bubble was estimated as the integral of a sequence of one-dimensional heat flow problems. Zuber (4) proposed that the superheat content of the liquid layer thus being displaced could be equated, a t every instant, to the gain in latent heat content of the bubble in that instant. Essentially, this is a phenomenological theory, in which it is argued, on physical grounds, that the advancing bubble front cannot displace the heated liquid from its position next to the solid surface, but instead sweeps over it, a thin tongue of evaporating liquid separating a portion of the bubble from the solid surface. The existence of such thin films of liquid at the base of a sessile bubble was first noticed by Derjaguin (2) some years ago, as reported by Frenkel (3). The present work explores the consequences of this hypothesis for bubbles growmg at the surface of an exponentially
heated plate. The problem has bearing on the rapidity of shutdown, and hence the safety, of liquid-cooled nuclear reactors during a power excursion. Zuber’s hypothesis has the novel feature that it focuses attention on the heat flow in the solid, with the result that the bubble growth rates are dependent, at least in rhrory, on the thickness of the solid and its physical properties. Heat Flow in Solid
As a preliminary to the application of Zuber’s hypo thesis, we consider the simple one-dimensional heat flow resulting when a metal slab of thickness 1 is brought in contact with a semi-infinite liquid. Neglecting edge effects, the heat conduction equations may be written
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