Gol’danskii, V. I., Neimark, I. E., Plachinda, A. S., Suzdaley, I. P., Fourth International Congress on Catalysis, Moscow, 1968; Symposium I11 Novosibirsk, Preprint No. 16, 1968b. Hansford, R. G., Ward, J. W., J . Catal. 13, 316 (1969). Ingalls, R., Phys. Rev. A 133, 787 (1964). Keough, A. H., Sand, L. B., J . Anzer. Chem. SOC.83, 3536 (1961). May, L., Advan. Chem. Ser. No. 68, 52 (1967). hleier, W. M., Z. Kristallogr., Kristallgeometrie, Kristallphys., Kristallchem. 115, 439 (1961). elorice, J. A,, gees, L. V. C., Trans. Faraday SOC.64,1388 (1968). Simanek, E., Sroubek, Z., Phys. REV.163, 275 (1967).
Skalkina, L. V., Suzdalev, L. P., Kolchin, I. K., Margolis, L. Ya., Kinet. Catal. (USSR) 10, 378 (1969). RECEIVED for review June 19, 1972 ACCEPTED March 23, 1973 Presented at the Division of Petroleum Chemistry, 161st National Meeting of the American Chemical Society, Los Angeles, Calif., March 1971. This work has been supported in part by the Office of Army Research under Contract DAHC 0467C0045 and in part by the National Science Foundation under Grant No. GK 17451X.
Stochastic Analysis of the Well-Mixed Tank Reactor with Heat J. E. Berryman and D. M. Himmelblau* Department of Chemical Engineering, The University of Texas, Austin, Texas 78712
The behavior of a continuous flow nonisothermal well-stirred tank reactor when the variables q, (flow rate), concentration), T, (inlet temperature), and k (reaction rate coefficient) were random was studied. Under what circumstances the deterministic solutions for the model differ from the respective sample means is identified as well as the effect of assuming that the random parameters have normal, uniform, or auto-correlated probability distributions. From a comparison with the work of other investigators, it was concluded that using white noise as random coefficients and random inputs, together with the analytical methods developed for communications and control, may not be at all appropriate for the analysis of a real chemical process. How to use measurements of the sample dispersion of the temperature and concentration responses in the design of reactors i s also explained.
CA,(reactant
S t o c h a s t i c mathematical models of chemical reactors, if they represent the process reasonably well, can be used t o study the process in lieu of experimentation. Such models can be simulated on a high-speed large capacity digital or hybrid computer and the results used in various ways by the engineer to help him make decisions relative t o the design and operation of the process. Uncertainty in the variables and parameters in a chemical reactor can result from random fluctuations in the quantity of the input streams or their concentrations and temperatures, from unmeasured ambient conditions affecting heat transfer and the reaction, from inadequate data collection techniques, including instrumentation, or from the least-squares technique used to calculate the interphase heat transfer coefficients and the reaction rate coefficients. Encertainty may also be caused by mixing in the reactor, but we have assumed the reactor is well mixed. If stochastic features are to be introduced into the model t h a t is used to represent the reactor, the input variables, the model coefficients, and perhaps the initial conditions must be assumed to be stochastic variables. The main objective of this study was to determine the effect of random inputs and coefficients on the response of the well-mixed tank reactor with heat transfer. Random coefficients and random inputs with different probability distributions were generated for the parameters in the model of the reactor. The model was solved numerically an adequate number of times (200 or more) to obtain a n estimate of the 310
Ind. Eng. Chem. Fundom., Vol. 12, No. 3, 1973
distribution of the reactor output variables a t a sequence of time periods from which sample means and sample standard deviations could be computed. The sample estimates of the model outputs were related to the corresponding statistics of the model inputs and coefficients. Although the specific values of the temperatures and concentrations obtained from the simulations naturally depended on the deterministic values of the parameters introduced into the reactor model, the stochastic character of the responses depended almost entirely on the stochastic character of the inputs and very little on the ensemble values of the inputs. Answers to several questions relative to the stochastic reactor model were sought of which the following were the most significant. (1) What type of distributions are observed for the reactor outputs (temperatures and concentrations)? (2) Are the sample means of the reactor outputs different from the corresponding deterministic responses, i.e., when the random variables are replaced by their expected values? (3) How does changing the assumptions about the probability distributions of the random parameters and inputs affect the distributions of the outputs of the reactor, and hence any conclusions drawn in the investigation? (4) How can the information generated be specifically applied in the design of well-mixed reactors? I n connection with the last question, in conventional methods of design of reactors, the stochastic nature of the inputs and parameters is ignored. To allow for uncertainty, safety
factors are added to the deterministic volume of the reactor to make sure that the reactor is big enough. This article shows how t o compute safety factors for design in a more quantitative way from information obtained b y Monte Carlo simulation of the response of the stochastic model. It becomes possible to make statements such as: t o obtain a conversion of 9501, in a reactor with a confidence coefficient (probability) of 0.99, the volume required is 1.11 times the volume of the reactor calculated using a deterministic solution of the reactor model. Furthermore, knowledge of the relationship of the statistics of the inputs and parameters t o the statistics of the outputs can be used to obtain confidence limits on the concentration of the reactants for a given reactor volume as, for example, the concentration interval of 0.83 to 0.89 includes the expected value of the output concentration of the product a t a confidence coefficient (probability) of 0.95. Thus the results presented here can be used in design to obtain a more precise knodedge about the design and to permit better decision-making about the magnitude of the safety or over-design factor required for the process design. Furthermore, the analytical tools developed in this study can be applied to many other process models with relatively little cost in engineering or computation time (provided that the basic computer program has been prepared). The Process Model Used
Figure 1 illustrates the model of a continuous flow, nonisothermal, well-mixed tank reactor with heat exchange t h a t served as the basis for analysis. Essentially the model is the same as that of ilris and Xmundson (1958) that has served to illustrate innumerable different types of studies. The equations for this model represent, respectively, a balance on reactant X and a n energy balance on a constant volume system
with CA(t
=
0)
T(t = O )
= CAO
To
=
Figure 1. Well-mixed tank reactor with heat transfer
The model equations assume that the volume of reactor and the density of the reactant fluid remain constant. The constant volume assumption requires that no accumulation of fluid be allowed; hence the instanbaneous flow rate out of the reactor (qf) is equal to the flow rate into the reactor (a;) regardless of fluctuations in q i when p i is used as a random input variable. Studies of linearized, steady-state versions of this reactor have been reported by Xris and Amundson (1958), King (1972), Krambeck (1972), and Pel1 and Aris (1969) as well as experiments by Homan and Tierney (1960). Because the first cited study is so well known, the values of the physical quantities used by h i s and hmundson were adopted for this investigation. The results to be discussed are based on a second-order irreversible reaction, 2A -+ P! taking place in the well-mixed tank. The deterministic values of the physical quantities used are (V) = 100 ft3; (qi) = 0.3 ft3/sec; (7,;) = 1 lb mole/ft3; (Ti)= 650OR; ( A H } = -20,000 Btuilb mole; ((&,,)A) = 60 Btu/(ft3)(OR). In the reaction rate expression the values are ( k o ) = 2.7 X 10"; ( E ) = 44,700 Btu/lb mole; k = 1.987 Btu/(lb mole) (OR).For the cooling coil (A,) = 500 ft2; (V) = 100 Btu/(hr) (it2) (OR);(T'& = 520'R; (qo) = 0.138 ft3/sec. For these conditions, the deterministic steady-state solution gives ( T ) = 825.lOR and (CA) = 0.08399 lb mole/ft3 when the heat removed is calculated using eq 4 and 5. Several initial conditions were examined for a det'erministic numerical solution in order to find a set of initial conditions that would lead to the steady state in the shortest st'art-up time. It was found that using (CAO)= 1.0 and (To) = 650°R with no heat removal (Q = 0) when (T) was less than 690°R gave the shortest start-up time. Shorter start-up times could have been achieved if the reactor contents could have been heated above the feed temperature of 65O"R; however, this was not considered t'o be a practical case and was not used.
where R(CA, T)
=
- C ~ ~ kexp(-E/kT) o
=
C A ~ ~
The rate of heat removal by the cooling coil, Q, may be obtained by determining the amount of heat gained by the coolant as
Q
=
ac(PCp)c(Tcf
- Tci)
(3)
or by calculating the heat transferred through the wall of the cooling coil as
Q
=
L'A,[T
- (Tcf
+ 7',1)/21
(4)
Both eq 3 and 4 are needed because they must be combined to solve for Tcf as a function of T with the resulting equation Tcf
=
2c:AsT
+
Tci(2qc(~Cp)c- GAS) 2qc(PCp)c U:AS
+
(5)
The Tci term of eq 4 mas replaced by the right-hand side of eq 5 to calculate the value of Q in the solution of the model.
Procedure
Stochastic models can be developed by applying probability theory to elements of a fluid and then developin,0' a macroscopic model for the system. Stochastic models can also be generated by taking a deterministic macroscopic model and introducing stochastic coefficients and inputs. Keither trcatment is "better" or theoretically more exact than the other because both models represent the real process only approximately. As long as either approach corresponds as much as possible to the physical phenomena taking place in the process, either can be applied successfully. This study is based on the second approach. Solutions for the moments (mean and variance) of the response of a coupled, or even a single, stochastic differential equation can be obtained by analytical procedures only for a very small class of equations. Two routes for analytical solutions are: (1) solution of the related Kolmogoroff equation to first obtain the probability density function and then Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
31 1
DETERMINISTIC SOLUTION
"AL
--
OENERA'bR
T
CA
0
0
a 0
In Y In
az
700
2
e IL
In
0
i
0
i
DIMENSIONLESS
3 TIME,
8
4 I
Figure 2. Comparison of the sample means of T and CA for the case in which ko, qi*, CAI, and T i were random variables
integration to obtain the moments, and (2) direct solution for the moments of the response. For either method, the requirement that the output variable must be a Markov ensemble places a limitation on the number of real physical processes that may be modeled by stochastic differential equations via an analytical solution. The use of white noise for random variables t o ensure the Markov ensemble property of the output variable causes difficulty in relating the mathematical treatment of the stochastic differential equation to a real process. Since the concepts of white noise and Markov ensembles are mathematical idealizations which can only approximate a real process, the logical conclusion to the question of how to model a real stochastic process is the answer given by Mortensen (1969), who stated t h a t the safest way to obtain the statistics of the output variable of a process is by a Monte Carlo simulation via a representative process model. He further pointed out that any effect less than this is a n attempt t o find a short cut and may yield a n incorrect result. The best criterion by which one can judge the correctness of a model is by how well the predicted output of a model agrees with the observed output of the physical process. T o make the well-stirred tank model as consistent with physical reality as practical and to introduce realistic random variables, i t was decided t o use X o n t e Carlo simulation as a tool for analysis. The equations listed above were solved repetitively on a digital computer using values from random number generators to simulate the random properties of input variables and coefficients. The desired statistics of the model output variables were obtained from a statistical analysis of the repetitive solutions. In this model q l , T , , and CA, were the random variables that were functions of time; V was a constant. Solutions of the model differential equation were obtained by numerical 312
Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
solution of the differential equation using a fourth-order Runge-Kutta method. The preexponential factor ko was a random variable but remained a constant value independent of time for any one trial solution, varying randomly from solution to solution. During one time interval, the random parameters except ko were regarded as having fixed values that might or might not change in the next time increment. The effect of a random activation energy is not discussed here because the variances cited for the activation energies cited in the literature are so high as to obscure effect of the other parameters. The net effect of the above treatment of the well-mixed tank model to make a stochastic model was to discretize the input variables and solve a deterministic model for each time interval At. As long as At is very much less than the time constant of the tank (in the same units), the treatment corresponds t o what presumably is the physical mixing existing in the tank. As might be expected, the computer time required to execute a numerical integration using fixed integration steps was rather long. X relatively large amount of computing time would have been required t o obtain the 200 solutions needed to obtain the sample statistics of the output variables T and CA. Consequently, a variable length integration step procedure was devised based on a maximum allowable change in T or CA during a time step. This procedure greatly reduced the computation time with only a small loss in accuracy because almost all of the solution could be carried out with time steps of t* = t / ~= 0.02, corresponding to 50 possible changes in the random variables per residence time. The critical and lengthy computational part of the solution occurred when the values of T and CA changed very fast a t t* of approximately 1.05. The solid lines in Figure 2 show the rapid change in T and (CA)/(CA~) over a short time period for the unsteady-state deterministic response of the reactor to a step input. Simulation runs were made using At* = 0.02 and At* = 0.10 where At* represents a fraction of the residence time. These two values of At* corresponded t o a frequency of change of 50 and 10 changes per residence time, respectively, in the variables. The following cases (with their corresponding random variables) will be discussed: case 1, ko \vas the only random variable; case 2, ql* was the only random variable; case 3, CA, was the only random variable; case 4, TI was the only random variable; and case 5, ko, ql*, C A ~and , T , were simultaneously random. Each of these cases was run using (1) a normal and a uniform random number generator with a frequency of change of the time-dependent random variables (ql*, C A ~and , TI) of both 10 and 50 changes per residence time and (2) a n auto-correlated generator for the timedependent random variables. The detailed characteristics of the random variables and their generators are described by Berryman and Himmelblau (1971). Each case, except those involving the uniform generator, was carried out to t* = 8. The time period t* = 0 to t* = 4 was considerered t o correspond to the unsteady state and the time period after t* = 4 was considered to be the pseudo-steady state. The coefficient of variation of ko was yro = 0.05, yg,* = 0.10, ycA, = 0.10, and y r , = 0.01 for the results cited here. Results
We will consider the results for the unsteady state first. 1. Unsteady-State Results (t* = 0 to t* = 4). If the distribution of the o u t p u t temperatures and concentrations is not sensitive to changes in the type of distributions of the random inputs and random variables, then t h e engi-
neer does not have t o generate innumerable simulations in order to characterize completely the probability distributions of CA and T . Furthermore, if the observed distribution of the reactor output variable can be demonstrated t o be normal, the designer can use the tables of the standard normal deviate to calculate the confidence limits for the outputs and can avoid executing a large number of runs t o outline the character of the distribution of CA and T with sufficient precision. Of course the designer could apply the Chebyshev inequality (which applies to any distribution) in order to establish the confidence limits on CA and T , but a t the expense of larger over design factors. The relative frequency distributions of CA and T obtained from the Monte Carlo simulations were tested by the x 2 goodness of fit test to ascertain if they could be represented by a normal probability distribution. The hypothesis that the two responses could each be represented b y a normal distribution was accepted a t a confidence level of 0.95 in each instance. The distributions of T(t* = 4) and C A ( ~ *= 4) for both trials of cases 3 and 5 appeared to be slightly skewed but were so near a normal distribution that for the sample size employed (200) the relative frequency distribut'ions passed the x 2 test. h second question of interest was whether the sample means c . k I ( C ~ i }and T,differed significantly from the deterministic solution c A / ' ( C ~ Jand ( T ) ,respectively. If the sample means are not significantly different from the deterministic solutions, t,he designer does not have t o worry about any bias in the reactor volume calculated using the determiiiist'ic values of the parameters and inputs. Figure 2 compares the sample means of T and CA obtained from (1) the trials in which the normal generator was used and (2) tlie trials in which the uniform generat,or was used for case 5 with the deterministic unsteady-state values of T and C A (solid lines). When the hypothesis that the sample means of T and CA for the trials of cases 1, 3, and 5 using the normal generator were the same as the sample means of T and CA for the trials of cases 1, 3 and 5 respectively, using the uniform generator, was tested by the significance test for comparing sample means, the hypot'hesis was accepted for each case except for sample means of T and C A near the time period t* = 1.0. It is believed t h a t if the accuracy of the numerical integration carried out during the time period near t* = 1.0 had been better (a smaller integration time step would have to have been used), the sample means of T and C.4 would have been the same as the deterministic solutions for the entire solution period. When the hypothesis that the sample nieans of T and C.4 from the trials in which the auto-correlated generator was used to obtain yl*, C A , and Ti were different from t'he determinist,ic solutions was tested by the significance test for coniparing a mean t o a known standard, the hypothesis was accepted for the sample means in the t'ime period t* = 0.9 to t* = 1.3 and rejected for the other time periods. When the sample means of T for the time period t* = 0.9 t o t* = 1.3 are compared t o the deterministic solution for the same t'ime period, the sample means of T reach their maximum value a t a later time and have a lower value than tlie deterministic solution. Thus, if the inputs expected in the reactor are not autocorrelated (they were alKays assumed to be independent, that is not cross-correlated), the deterministic solution for T and CA as usually calculated by the designer is unbiased. On the other hand, if the inputs are autocorrelated, the deterministic solution is likely to be a biased solution, and the volume of the reactor should be adjusted for the bias. KOgeneral guides can
Table I. Sample Coefficients of Variation of T and CA at f * 4 for the Normal Random Variables
=
Case
Somple coefficient of variation for frequency of change
Random variable
10
50
CA k0
qi* CAi
Ti All four
0.030 0.072 0.154 0.027 0.190
0.037 0.062 0.013 0.086
0.0006 0.0042 0.0122 0.0021 0.0135
0.0021 0.0048 0.0010 0.00590
T x-0
Pi* CA i
Ti All four
be provided for t'he extent of the bias because the bias is quite sensitive t'o changes in the character of the assumed autocorrelation. The standard deviations of T and C.4 (here expressed as the coefficient of variation, i.e., the standard deviation divided by the mean) provide the information about the dispersion of T and CA needed to compute the overdesign factors. Consequently, it is important to see how sensitive the standard deviations are t o changes in t'he assumptions about the probability distribution of the inputs t o the model. When bhe hypothesis that the sample standard deviations of T and CA from the trials using the normal generator were tlie same as the sample standard deviations of T arid CA from the trials using the uniform generator m-as tested by the F test for comparing variances, the hypothesis was accepted for 7570 of the sets of sample standard deviations. Thus, the normal tables cannot always be used to establish confidence limits for CA and T . Table I was prepared t o show the effect of each random variable and the collective effect on the coefficient of variation T and CA a t t* = 4. (For the case in lvhich X.0 was the only random variable the coefficients of variation of T and C.4 are not associated with a frequency of change although t'he values are listed in the t'able under a frequency of change of 10.) Xote the large effect the frequency of change had on the coefficients of variation of the responses. The frequency of change in the random input has quite a bit t,o do with the extent of the transmission of the random fluctuations inasmuch as high frequencies give substantial daniping n-liereas low frequencies cause less damping. However, one can hardly say that one or two changes in C.4i per residence time corresponds to a realistic random variable. Refer to Berryman and Himmelblau (1971) for further discussion of the effect of the frequency of change. The sample coefficients of variation of T and CA listed in Table I indicate that' the random k0, v i * , C A ~or , Ti had a greater effect on the coefficient of variation of C.4 t,haii on the coefficient of variation of T . A% random C A had ~ the greatest effect on the coefficient of variation of CA.However, if y r i had been (an unrealistic) 0.10 instead of 0.01 (ycAi was 0.10), a random Ti would have had the greatest effect on the coefficient of variation of T and CA. The coefficients of variation of T and CA for the trials usiiig the autocorrelated generator for the case in which X.0, pi*, Cai, and Ti were individually or simultaneously random variables were about the same as those listed in Table I for Ind. Eng. Chem. Fundom., Vol. 12, No. 3, 1973
313
I-1
ni1
0.04
/A\
AUTO-CORRELATED
QENERATOR
1
OJ21
! j 0.04
NORMAL FREQUENCY
GENERATOR
OF
CHANGE
=
10
1
0.0 8820 NORMAL PREQUENCY
OF
I---
NORMAL
GENERATOR
CHANGE
=
0.0 4
FREQUENCY
GENERATOR
Of
CHANGE
=
50
$0 1
5
i
i DIMENSIONLESS
i TIME,
9
1’
Figure 4. Pseudo-steady-state time sequences of T for runs using the autocorrelated generator and the normal generator with frequency of change of 10 and 50
the respective case using the normal generator with a frequency of change of 10 for the time period t* = 0 through t* = 3.0. However, as T and CA approached the pseudo-steady state, the coefficients of variation of T and CA became larger than for the trials using the normal variable. Since real reactor inputs often prove to be autocorrelated, what this observation means is that the overdesign factors must be larger than those for the case of uncorrelated inputs for the same degree of confidence. The engineer can make measurements of the input concentration in typical operating reactors to compute the degree of autocorrelation, or he can make a n assumption, if necessary, and use the resulting information in the Ornstein-Uhlenbeck function (see Berryman and Himmelblau, 1971) to compute the added dispersion over the uncorrelated case. 2. Pseudo-Steady-State Results (t* > 4.0). Figures 3 and 4 illustrate typical model responses for the pseudosteady state (as defined b y t h e deterministic solution). Figure 3 shows t h e time sequences of CA for t h e case in which pi*, CAI,and Ti were random variables for (1) a trial using the autocorrelated generator, ( 2 ) a trial using the normal generator with a frequency of change of pi*, C A ~and Ti of 10 changes per residence time, and (3) a trial using the normal generator with a frequency of change of pi*, C A ~and , Ti of 50 changes per residence time. Figure 4 shows the corresponding sequences of T . Figures 3 and 4 show the effect the frequency of change has on the dispersion of T and CA and indicate that the use of the autocorrelated generator corresponds to using a very low frequency of change. When the hypothesis that the sample means obtained for T and CA were the same as the corresponding deterministic 314 Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
steady-state value of T and CA was tested by the significance test for comparing a mean to a known standard, the hypothesis was accepted for the sample means of each case from the trials involving the normal generator with a frequency of change of 10 and 50 changes per residence time. The hypothesis was also accepted for the trials using the autocorrelated generator for case 2 in which pl* was the only random variable and case 4 in which T , was the only random variable. The hypothesis was rejected for the trials using the autocorrelated generator for case 3 in which C Awas ~ the only random variable and case 5 in which ko, pl*, C A ~and , T I were simultaneously random variables. Sample means and sample standard deviations of T and CA were calculated a t a number of sampling points (different values of t * ) during the pseudo-steady state. For the cases for which the hypothesis that the sample means of T and CA were the same as the corresponding deterministic value was accepted, the differences between the values of T (t* = a)and T(t* = b ) and the values of CA (t* = a) and CA (t* = b ) , where a and b are different values of t* in the pseudo-steady state, were very small and not significant. For cases 3 and 5 for which the hypothesis was rejected that the sample means of T and CA were the same as the corresponding deterministic values, the differences between the values of a t the various sampling points o f t * and of EAa t the various sampling points were relatively large and were significant. For example, when the hypothesis that T(t* = 6) was the same as F(t* = 8) and that c ~ ( t= * 6) was the same as C A ( t * = 8) for case 5 (in which k , q l * , C A ~and , T , were all random variables) was tested by the significance test for comparing sample means, the hypothesis was rejected a t a confidence level of 0.95. It was
Table II. Pseudo-Steady-State Sample Coefficients of Variation of CA and T Normal generator frequency of change 50 10
Random variable
Case
Autocorrelated generator
Table 111. Comparison of the Calculated and Simulated Pseudo-Steady-State Coefficients of Variation of T and CA for Case 5 Coefficient of Variation Trial
Calculated
YCA
ko
1 2 3
0.03577 0.03418 0.06492 0.01391 0.08296
qi* CA i
Ti
4
ko,
5
Qi*, C A I ,
Ti
... 0.07207 0.14460 0,03101 0.18212
... 0.10329 0.76392 0.05445 0.99796
... 0.00423 0.01102 0.00246 0.01317
0.00857 0.03054 0.00420 0.04114
YT
1 2 3 4 5
0.00085 0,00193 0.00489 0.00108 0.00543
ko
qi * CA i
Ti ko, pi*, CAi, Ti
, . .
assumed t h a t a much longer time period would be required (for the trials in which the autocorrelated generator was used) to reach the pseudo-steady state because of the low effective frequency of change associated with the use of the autocorrelated generator. Table I1 lists the sample coefficients of variation of T and CA for the pseudo-steady state. The values of the coefficient of variation of T and CA are relatively large for the cases that used the autocorrelated generator. To see if the propogation of error formula could be employed for design purposes, the pseudo-steady-state values of qCAand qT for cases 1 , 2 , 3, and 4 (in which there was only one random variable) were used to calculate qCA,and qT, respectively, for case 5 in which ko, yl*, CA,, and T , were simultaneously random variables by the following procedure ?T(ko, qI*,
CAl,
TI) =
S T ( ~ O ql*, , Call
Td/T
(6)
where
T T ( ~ oql*, , CA,, T,)=
1 / [ s ~ ( k O ) I 'f
the coefficient of variation of T for the case in which ko, q,*, CA, and T , are all random
[ST(~I*)]'
+
[sT(cAi)12 f [ST(T1)I2
(7)
Since the pseudo-steady state values of T (for the trials using the normal generator) were the same as the deterministic steady-state value of T ( i e . , the ensemble mean ( T ) , eq 7 can be written as ST(kO, y i * C A i ,
TI) =
( T j 1/[9r(ko)l2f [9~(q1*)]' -k [TT(CAI)]*f [ ~ T ( T ~ ) (8) I' and eq 6 becomes
+
1 / [ 9 ~ ( k O'I ) [ ? ~ ( q i * ) 1' f ['fT(CAi) 1' + ['?T(T1)1' (9) Similarly, TcA (ko, yl*, CA,, T , ) can be calculated. The application of the above procedure for the case using the autocorre!' lated generator mas, strictly speaking, not correct because T was equal to (2') for each of the cases in which there n a s only one random variable. Table I11 compares the values of TT and qC4calculated by the above procedure with those from case 5 nhich nere obtained by simulation. When the hypothesis that the calculated coefficients of variation of T and CA were the same as the
Simulated
9CA
Normal-50 Normal-10 Autocorrelated
0 0828 0 168 0 773
0 0829 0 182 0 997
TT
Kormal-50 Normal-10 Autocorrelated
0 00543 0 0120 0 0320
0 00543 0 0131 0 0411
coefficients of variation of T and C-4 obtained by simulation was tested by the E' test for comparing variances, the hypothesis was accepted for the two trials involving the normal generator with frequency of change of 50 and 10 changes per residence time; and the hypothesis was rejected for the trial using the autocorrelated generator. Comparison with the Results of Others
=Iris and Xmundson (1958) linearized the heat exchanger model about the deterministic steady state values of 4 = C A / ( C A ~and ) 7 = T ( p C P ) ~-AH(Cai)). /( They neglected all the squares and products of the noise component of 7 , 6, the noise coniponent of the dimensionless flow rates A, dimensionless inlet conceiitrat,ion F , and dimensionless iiilet temperature, v. They used additive white noise with a spectral density of l / r for A . p , and V , and a n analytical method of solution of the simplified model as out,lined by Laning and Battin (1956) to obtain the autocorrelation and cross-correlation functions of ZEand Z, where Z stands for the noise component. h i s and Amundson reported t'heir result,s as plots of the autocorrelation functions of T and C.4 and the cross-correlation function of T and CA us. T for (I) t'he case in which ZX was the only random variable, (2) the ca.e in which 2, was the only random variable, and (3) the case in which 2, was the only random variable. Since correlat'ion functions of T and CA were not computed in this study, direct comparison of the result's of this study with the work of ilris and hmundsoii is not possible. Xris and hmundson reported that variations in the flow rate cause the great'est variations in the output conceiit'ratioii and temperature. However, this ITork shows that if the coefficient of variation of the flow rate had the same value as the coefficient of variation of Cai, the pseudo-steady-state sample standard deviations of T and CA as a result of a raridoni flow rate would be about one-half the pseudo-steady-state sample standard deviations of T and CA resulting from a random C A ~The . results of this study also imply that if y r , had been as large as ycli or yni$, the effect of a random Ti on the pseudo-steady-state sample standard deviations of T and C.4 would have been larger than the effect of either a random qi* or a random Cai on the pseudo-steady-state standard deviations of T and CA. Xris and hmundson reported t'hat the cross-correlation function vvas negative, meaning that increases of temperature are generally accompanied by decreases in concent,ratioii, and Figure 5 shows typical time sequences of T arid CA that indicate increases in T are accompanied by a decrease in CA (as would be expected from the reaction rate espression) . Airis and -2mundsori also report,ed that the response of concentration lags behind that of temperature by some small Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
3 15
-
I
UPPER C O N F I D E R E L I M I T FOR CONFIDENCE C O E F F I C I E N T = 0.99 PREQUEIPCY OF m N G E
I
10
0.50-
4- 4
v
lu' z
0 2
-
Iu
f U
z
8
VI VI
0.10-
W 4
-
Z
0
0
SAWLE VALUES OF C,/
