. I. NASA Contractor Report 179636 Decoupled Direct Method for Sensitivity Analysis in Combustion Kinetics (BASA-CE- 179636) CECCUPLEC CIGECT METHOD N81-24549 iCB SENSX'IXVIIY AIALPSIS IN CCBBZS'IIOLJ KlIS11;TICS Final Eegort (Sverdrtir Technology) 17 p Avail: PYIS HC A02/11I A01 CSCL 21B Unclas G3/.25 OC8G556 Krishnan Radhakrishnan Sverdrup Technology, Inc. Lewis Research Center Cleveland, Ohio June 1987 Prepared for the Lewis Research Center Under Contract NAS3-24105 National Aeronautics and Space Administration https://ntrs.nasa.gov/search.jsp?R=19870015116 2020-05-31T03:31:24+00:00Z
Transcript
. I. NASA Contractor Report 179636
Decoupled Direct Method for Sensitivity Analysis in Combustion Kinetics
(BASA-CE- 179636) CECCUPLEC CIGECT METHOD N81-24549 i C B SENSX'IXVIIY AIALPSIS IN CCBBZS'IIOLJ KlIS11;TICS F i n a l Eegort (Sverdrtir Technology) 17 p A v a i l : PYIS HC A02/11I A01 CSCL 21B Unclas
G3/.25 OC8G556
Krishnan Radhakrishnan Sverdrup Technology, Inc. Lewis Research Center Cleveland, Ohio
June 1987
Prepared for the Lewis Research Center Under Contract NAS3-24105
DkCOUPLtD I ) l H t C I MtlHOD k O H S E N S I I L V L I Y ANALYSIS I N COMBUSlION K I N € T L C S
t
Krlshnan Radhakrlshnan
Sverdrup Technology, I n c . Lewls Research Center
Cleveland. Ohio 44135 USA
SUMMARY
An e f f i c i e n t , decoupled d i r e c t method f o r c a l c u l a t i n g t h e f i r s t o r d e r s e n s l t l v i t y c o e f f i c i e n t s o f homogeneous, ba tch combust ion k i n e t i c r a t e equa- t i o n s l s presented. I n t h i s method t h e o r d i n a r y d i f f e r e n t i a l equa t ions f o r t h e s e n s l t i v l t y c o e f f i c i e n t s a r e so lved s e p a r a t e l y from, b u t s e q u e n t i a l l y w i t h , those d e s c r i b i n g t h e combustion chemist ry . The o r d i n a r y d i f f e r e n t i a l equa t ions f o r t h e therrnochemlcal v a r i a b l e s are so l ved u s i n g an e f f l c i e n t , I m p l i c i t method (LSODE) t h a t a u t o m a t i c a l l y s e l e c t s t h e s t e p l e n g t h and o r d e r f o r each s o l u t i o n
In s tep . The s o l u t i o n procedure f o r the s e n s l t l v l t y c o e f f l c i e n t s m a i n t a l n s accu- W r a c y and s t a b l l l t y by u s i n g e x a c t l y t h e same s t e p l e n g t h s and numer i ca l a p p r o x l -
combinat ion o f t h e l n l t l a l va lues o f t h e thermochemlcal v a r i a b l e s and t h e t h r e e r a t e cons tan t parameters f o r t h e chemical r e a c t l o n s . The method 1 s i l l u s t r a t e d by a p p l i c a t i o n t o s e v e r a l s lmple problems and, where p o s s i b l e , cornpartsons a r e made w i t h exac t s o l u t i o n s and those ob ta ined by o t h e r techniques.
M
M
w I mat lons . The method computes s e n s l t l v i t y c o e f f i c i e n t s w i t h r e s p e c t t o any
I N 1 RODUCT 10N
l h e model ing o f a homogeneous, b a t c h combustlon system r e q u i r e s t h e s o l u - t l o n o f f l r s t o rde r o r d i n a r y d l f f e r e n t l a l equat lnns ( O D L ' s ) f o r thermochemical v a r i a b l e s such as composi t lon, temperature, and dens7ty. ?he c h e m i s t r y I s rep resen ted by a system o f NR simultaneous r e a c t l o n s among NS d i f f e r e n t spe- c i e s . A l l chemical r e a c t l o n s considered i n t h i s work a r e gas-phase e lementary r e a c t l o n s . The j t h chemlcal r e a c t i o n can be w r i t t e n l n t h e genera l f o r m
NS k , NS
where u i j l and Uij" a r e t h e s t o l c h i o m e t r i c C o e f f i c i e n t s o f spec ies 1 ( w i t h chemlcal symbol X i ) i n r e a c t i o n j as a r e a c t a n t and as a p roduc t , r e s p e c t i v e l y .
The t i m e r a t e o f change o f species 1 can be w r i t t e n as ( r e f s . 1 and 2)
where a i I s t h e mole number o f species i ( i . e . , mole i / g m i x t u r e ) , t t h e t ime, T t h e temperature, and p t h e m i x t u r e mass d e n s i t y . The n e t r a t e o f f o r m a t l o n (fl) of species g i v e n by
i due t o a l l f o rward and r e v e r s e r e a c t i o n s 1 s
where t h e molar forward ( R j ) and r e v e r s e ( R - j ) r a t e s per u n i t volume f o r r e a c t i o n j a r e g iven by
The fo rward r a t e cons tan t ( k j ) i s g i v e n by t h e m o d i f i e d A r rhen ius exp ress ion
k j = A j l n j exp ( - E j / R l ) ( 6 )
where A j , n j , and E j a r e cons tan ts and R i s t h e u n i v e r s a l gas c o n s t a n t . Each r e a c t i o n may be e i t h e r r e v e r s i b l e ( b i d i r e c t i o n a l ) o r i r r e v e r s i b l e ( u n l - d i r e c t i o n a l ) . For r e v e r s i b l e r e a c t i o n s t h e reve rse r a t e cons tan ts ( k - j ) a r e c a l c u l a t e d f r o m k j and t h e c o n c e n t r a t i o n e q u i l i b r i u m cons tan ts ( K j ) u s i n g t h e p r i n c i p l e o f d e t a i l e d b a l a n c i n g ( r e f . 3 )
k - j = k j / K j (7)
where K j i s a f u n c t i o n o f temperature a lone. Two d i f f e r e n t types o f b a t c h r e a c t i o n problems can be I d e n t i f i e d : cons tan t and v a r i a b l e d e n s i t y . For a v a r i a b l e d e n s i t y problem, t h e pressure-versus- t ime p r o f i l e i s g i v e n and t h e O D E ' S f o r temperature and d e n s i t y t a k e t h e fo rm ( r e f . 2)
and
!.le d t = ' ( y p l @ - A + D ) d t
where p i s t h e absolute p ressu re and y, A, B, and D a r e g i v e n by
Y = C / ( C - R/Mw) P P
NS
i = 1 c = & Y c
i P , i
2
( 9 )
i =1
M D=aQ( v-l y ) In these equations hi and cP,i are the molar specific enthalpy and constant-pressure molar specific heat, respectively, of species 1, cp the mixture specific heat, y the mixture specific heat ratio, Mu the mixture mean molar mass, and Q the heat loss rate per unit mass of mixture. The thermodynamic properties hi and cp,i are functions of temperature alone and are computed by using polynomial equations (ref. 2). A can be described as a species production function, and B and D are enthalpy production and loss functions, respectively.
For constant density problems the temperature ODE becomes (ref. 2)
1 ( y - 7 ) A - B - yD dT dt =
and the pressure is obtained from the ideal gas law
The problem I s to determine the therrnochem!cal varfnbles at the end of a prescribed time interval, given the Initlal conditlons and the chemical reac- tion mechanism.
The use of classical methods such as the explicit Runge-Kutta and A d a m methods to solve the O D E ' S arising in combustion chemistry results in prohibi- tive amounts of computer time. This is due to the extremely small steplengths required by these methods due to the "stiffness" exhibited by the O D E ' S (refs. 1 , 4, and 5). The phenomenon of stiffness in chemical kinetic rate equations was first recognized by Curtiss and Hirschfelder (ref. 6) who devel- oped a simple backward differentiation method for handling such equations. Since then, many approaches have been proposed for stiff O D E ' S in general (refs. 7 to 12), and chemical kinetlc rate equations In particular (refs. 1, 2, and 10 to 20). In several recent publications (refs. 1 , 5, 21, and 22) the accuracy and efficiency of many techniques for the solution of stiff O D E ' s arising In combustion chemistry have been examined. These studies showed that the packaged code LSODE (ref. 23) i s at present the most efficient and accurate algorithm for batch combustion chemistry problems. This code has therefore been adopted in the present work.
In addition to solving the O D E ' s for the thermochemical variables, it is often necessary to know how sensitive the solution is to the initial conditions and the chemical reaction mechanism parameters. Such a need arises In the development of reaction mechanisms from experimental data (ref. 24). The rate constants are often not well known and In general, the experimental data are
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not sufficiently detailed to accurately estimate the rate constant parameters. Sensitivity analysis helps to determine the effects of uncertainties in the rate constant parameters and the initial conditions on the solution, and to identify which variables are sensitive to which parameters. The analysis helps identify unimportant elementary reactions which can be discarded, thereby sim- plifying the mechanism and hence the problem. At the same time, reactions that need additional experimental study can be identified. Sensitivity analysis helps also In the understanding of complex reaction mechanisms by showing which parts of the mechanism are important for a given problem.
Another motivation for sensitivity analysis is the increased interest in the numerical simulation of multidimensional chemically reacting flows. The coupling of realistic chemical reaction mechanisms wlth multidimensional flow calculations results in prohibitive amounts of computer time. Sensittvity analysis can help reduce the reaction mechanism by identifying those reactions that are important in computing the variables of interest.
The simplest method for estimating the effect of uncertainties In any parameter is to run the simulation program with two different values of the parameter. Although such a "brute force" method has been used successfully (refs. 25 and 26). it can become very expensive when the number of parameters is large. Many methods (refs. 27 to 39) have been proposed for a more elegant and less expensive sensitivity analysis. The commonly used methods include the direct method (DM) (ref. 27), the Fourier amplitude sensitivity test (FAST) method (ref. 28), the Green's function method (GFM) (refs. 29 and 30) and its variants (refs. 32 and 33). the analytically integrated Magnus modification of the Green's function method (GFM/AIM) (refs. 30 and 34), and the decoupled direct method (DDM) (refs. 37 to 39). For stiff problems of the type examined In the present work, the DDM has shown greater efficiency and stability, with equal or better accuracy than the GFM and the GFM/AIM methods (ref. 37). Advantages of the DDM method over other methods are discussed by Dunker (ref. 37).
Dunker (ref. 40) has recently developed a very efficient computer program CHEMODM which couples the DDM method for sensitivity analysis with the code LSODE for solution of the model equations. However, the code is restricted to constant density and constant temperature problems. This restriction to con- stant temperature problems appears to be a common feature among all sensitivity codes developed to date. However, combustion kinetics i s characterized by a narrow region of rapidly varying temperature. There i s therefore a need to incorporate sensitivity computations into a general kinetics code for noniso- thermal problems. In the present work, the DDM method developed by Dunker (refs. 37 and 40) is extended to calculate the first order sensitivity coeffi- clents ayi/aa, where yi i s the ith thermochemical variable and Q I s a parameter of interest (either a rate constant parameter or an initial value), for combustion (l.e., nonisothermal) kinetic rate equations.
b
DECOUPLtD DIRECl METHOD
The ODE'S for batch combustion chemistry presented in the previous section can be generalized as
4
4
d y k = fi = f ( Y . A , n, E, Q, p
Y ( t = 0) = yo
where y i s t h e s o l u t i o n v e c t o r w i t h NS t 2 components and A,-n, and 5 a r e cons tan t vectors ( c o n t a i n i n g t h e r a t e c o n s t a n t param- e t e r s , eq. 6 ) , each o f which has NR components.
A t p resen t , t h e most e f f i c i e n t method f o r s o l v i n g eq. ( 1 8 ) i s t h e backward d i f f e r e n c e fo rmu la (BDF) i n c l u d e d i n t h e packaged code LSODE ( r e f . 1). T h i s v a r i a b l e - s t e p , v a r i a b l e - o r d e r method computes approx imat ions t o t h e exac t s o l u t i o n Y ( t n ) u s i n g l i n e a r m u l t i s t e p formulas o f t h e t y p e
I n [ = v ( t n ) ]
qn -n y = E QeYn-e + hnBoXn ( 1 9 )
Q =1
where hn ( = t n - t n - 1 ) I s t h e s teps lze, qn i s t h e o r d e r o f t h e numer i ca l app rox ima t ion , Yn[= f(Yn, . . . . ) I i s the approx ima t ion of t h e exac t d e r i v a t i v e y ( t n ) [ = f ( Y ( t n ) , ....)I, and t h e a’s and BO a r e c o e f f i c i e n t s a s s o c i a t e d w i t h t h e o r d e r qn.
A v a r i e t y o f i t e r a t i o n techniques I s i n c l u d e d i n I S O D E t o s o l v e e q u a t i o n ( 1 9 ) . For combustion k i n e t i c s problems t h e m o d l f l e d Newton i t e r a t i o n procedure i s t h e most e f f i c i e n t ( r e f . 1 ) and i s g i v e n by
where I i s t h e i d e n t i t y m a t r i x , J t h e Jacobian m a t r i x w i t h element J
j ’ (m) ,....). For each s o l u t i o n s tep the code a u t o m a t i c a l l y and jhm) = fJ1 s e l e c t s t h e s t e p s l z e and o rde r t o minimize t h e computat ional work w h i l e keep ing t h e es t ima ted l o c a l e r r o r w i t h i n a u s e r - s p e c i f i e d e r r o r t o l e r a n c e .
= aYi/aY t h e s u p e r s c r i p t s (m) and ( m t l ) denote t h e i t e r a t i o n numbers, i j
The d i f f e r e n c e equa t ion f o r t h e f i r s t - o r d e r s e n s i t i v i t y c o e f f i c i e n t where TI i s an i n i t i a l v a l u e o r a r a t e c o n s t a n t
j s (tn) = (av/a, sj ,n[= -j j tn ] parameter, can be ob ta lned by d i f f e r e n t i a t i n g equa t ion ( 1 9 ) w i t h r e s p e c t t o n j . e q u a t i o n (18 ) w i t h r e s p e c t t o nj equa t ion . This g i v e s
The d i f f e r e n c e e q u a t i o n can a l s o be d e r i v e d by f i r s t d i f f e r e n t i a t i n g and then a p p l y i n g t h e BDF t o t h e r e s u l t i n g
where aXn/aqj accounts f o r any e x p l i c i t dependence o f I n on nj. i s an i n i t i a l va lue t h i s t e r m vanishes, and when TI i s a r a t e
When constan “i parameter, i t can be obtained from equa t ion ( 1 8 ) . T i! e i n i t i a l v a l u e
5
o f S Is the jth column o f the ldentlty matrlx I f r( is the jth element o f yo. !f, however, nj is a rate constant parameter, Sjjt = 0) is equal to the null vector.
Equations (20) and (21) show the similarity between the model and sensi- The DDM method exploits this similarity by alternating the
More specifically, the tivity equations. solution of equation (19) with that of equation (21). solution procedure is as follows. is advanced from tn-1 to tn using a standard predictor and the corrector formula equation (20). The solution vn at the new time step is then used in equa- tion (21) to update the sensitivity coefficients. The latter process does not require either a predictor or an iterative procedure. for the sensitivity coefficients the method uses exactly the same stepsize and order as those used for equation (19). This implies that the error control in
tion of equation (19). As discussed in Dunker (ref. 37) the sensitivity coef- ficients calculated from equation (21) are the exact sensitivity of 1 with respect to
For any step the solution ~
~
In solving equation (21)
I the solution of equation (21) is determined by the error control in the solu- I
nj, apart from computer roundoff errors.
At each solution step, equation (21) must be solved as many times as the number o f parameters with respect to which the solution sensitivity is required. However, since the matrix ( I - hnI3oJ) is independent of the sensi- tivity solutions, it has to be LU-decomposed only once, irrespectlve of the number of sensitivity parameters. Hence, although the calculation of the sen- sitivities with respect to the first parameter may require considerable work to form the iteration matrix, perform its LU-decomposltlon, and solve equation (21), the evaluation of the sensitivity coefficients with respect to the second and subsequent parameters is significantly less expensive.
To reduce the computational work associated with the ctlculation and decomposition of the iteration matrix ( I - hnI3oJ) i n equation (20), this matrix is not updated at every iteration in I-SODE. For additional savings it is updated only when the solution to equation (20) does not converge. Hence the iteration matrix i s only accurate enough for the iterations to converge and the same matrix may be used over many steps. However, to maintain accuracy in the computed time step unless J changes slowly (refs. 37 and 39). The updating of this matrix at every time step obviates the need to iterate for separate error tolerances for the sensitivity calculations. otice that because the same Jacobian matrix J is requlred I n both equations (20) and (21) no additional programming Is required by the DDM method for either the calculation of J or the LU-decomposition of the matrix ( I - hnI3oJ) in equation (21 ) .
ZJ, the matrix in equation (21) must be recomputed at every
3 and specify
ILLUSTRATIVE EXAMPLES
The DDM method of sensitivity analysis together with a modified version of the packaged code LSODE (ref. 23) were incorporated into an existing general chemical kinetics computer code GCKP84 (ref. 2). The sensitivity subroutines were adapted from the code CHEMDDM (ref. 40). The new general kinetics code GCKP87 (ref. 41) has been designed to treat a variety o f reaction problems including sensitivity analysis of batch combustion kinetics equations. A t the user's option the code computes the first order sensitivity coefficients ayi/anj where yi is any dependent variable, the species mole numbers,
6
temperature, d e n s i t y , and pressure. The a r e t h e i n p u t parameter va lues . This i n c l u d e s t h e i n i t i a l values species mole numbers, tempera- t u r e and d e n s i t y , as w e l l as t h e values o f t h e r a t e c o n s t a n t parameters A j ,
:!lf:y:iek! a y j / a k j used i n t h e constant temperature computat ions o f o t h e r i n v e s t i g a t o r s , because t h e r a t e constant (eq. 6 ) . The code GCKP87 a l s o computes t h e s e n s i t i v i t y c o e f f i c i e n t s o f t h e temporal d e r i v a t i v e s o f t h e dependent v a r i a b l e s a y i / a n j f o r a l l t h e y i ' s and n j ' s d e f i n e d above.
o f problems t o ensure I t s accuracy and e f f i c i e n c y ( r e f . 4 2 ) . To t e s t t h e sen- s i t i v i t y computat ions many problems have been examined ( r e f . 41), t h r e e o f which a r e presented below f o r i l l u s t r a t i v e purposes. A l l c a l c u l a t i o n s were performed I n double p r e c l s l o n on the NASA Lewis Research C e n t e r ' s I B M 370/3033 c ompu t e r .
f o r t h e j t h r e a c t i o n (eq. 6 ) . The code does n o t compute t h e
k j i s a f u n c t i o n o f t h e temperature
A p r e l i m i n a r y v e r s i o n o f GCKP87 has been t e s t e d e x t e n s i v e l y on a v a r i e t y
Test problem 1, t aken f rom Dunker ( r e f . 37), d e s c r i b e s t h e p y r o l y s i s o f ethane a t a temperature of 923 K . Th is cons tan t temperature, c o n s t a n t d e n s i t y problem c o n s i s t s o f 5 i r r e v e r s i b l e r e a c t i o n s among 7 spec ies . The r e a c t i o n mechanism and r a t e cons tan ts ( a t T = 923 K) a r e g i v e n i n t a b l e I, t o g e t h e r w i t h t h e i n i t l a l c o n d i t i o n s . A l though the mechanism i s q u i t e sma l l t h i s problem i s ve ry s t i f f and o t h e r d i r e c t methods nave produced i n a c c u r a t e r e s u l t s ( r e f . 37). Because t h i s i s a cons tan t temperature problem t h e r a t e cons tan ts k j (eq. 6 ) a r e t i m e i n v a r i a n t . The re fo re i t I s o n l y i;ece;;ary t o compute s e n s i t i v i t y c o e f f i c i e n t s w i t h r e s p e c t t o k j i t s e l f and n o t w i t h r e s p e c t t o t h e i n d i v l d - u a l r a t e cons tan t parameters A j , n j , and E j . Th i s problem was s e l e c t e d as a t e s t f o r t h e m o d i f i c a t i o n s made t o t h e s e n s i t i v i t y r o u t i n e s adapted f r o m CHEMDDM ( r e f . 40) t o ensure t h a t Dunker's r e s u l t s ( r e f . 37) c o u l d be d u p l i c a t e d .
Normal ized s e n s i t i v i t y c o e f f i c i e n t s
c a l c u l a t e d a t two d i f f e r e n t t imes by t h e codes GCKP87 and CHEMDDM a r e g i v e n i n t a b l e 11, t o g e t h e r w i t h t h e r e s u l t s ob ta ined by Dunker ( r e f . 37 ) . A l l sens i - t i v i t y c o e f f i c i e n t s presented i n t h l s t a b l e a r e w i t h r e s p e c t t o t h e r a t e con- s t a n t k l . To generate these c o e f f i c i e n t s w i t h t h e code GCKP87, s e n s i t i v i t y w i t h r e s p e c t t o t h e p reexponen t ia l cons tan t ( A j i n eq. 6 ) i s s p e c i f i e d . For c o n s t a n t temperature problems t h e normal ized s e n s i t i v i t y c o e f f i c i e n t s w i t h r e s p e c t t o A j c i e n t s w i t h r e s p e c t t o k j . b o t h GCKP87 and CHEMDDM were r u n w i t h t h e same values f o r t h e l o c a l e r r o r t o l e r a n c e parameters r e q u i r e d by LSODE. The va lues s p e c i f i e d f o r t hese param- e t e r s were t h e same as those used by Dunker ( r e f . 37 ) : r e l a t i v e e r r o r t o l e r a n c e (RTOL) and 10-8 f o r t h e l o c a l a b s o l u t e e r r o r t o l e r - ance (ATOL) f o r a l l v a r i a b l e s .
a r e i d e n t i c a l l y equal t o t h e no rma l i zed s e n s i t i v j t y c o e f f i - To enable accuracy and e f f i c i e n c y comparisons,
10-6 f o r t h e l o c a l
The agreement between t h e two codes i s e x c e l l e n t a t b o t h 1 and 20 sec, t he reby c o n f i r m i n g t h e r e l i a b i l i t y o f t h e code GCKP87. Comparisons o f t h e com- p u t a t i o n a l work r e q u i r e d by t h e t w o codes showed GCKP87 t o be s i g n i f i c a n t l y
7
more efflclent than CHEMDDM. For 2 output stations ( 1 and 20 sec), GCKP87 required 95 steps with 138 functional and 21 Jacobian matrix evaluations to complete the problem, whereas CHEMDDM required 140 steps with 193 functional and 28 Jacobian matrix evaluations. To solve for the composition and sensitlv- Ity coefficients of all species with respect to all initial values and all rate constants, GCKP87 required approximately 0.65 sec CPU time, whereas CHEMDDM required approximately 1.4 sec. The given execution times do not include the time required for code initialization, preprocessing of the thermochemical data, and input and output. When the number of output stations was increased to 5 (10-3, 10-2, 1 , 10, and 20 sec), the difference in computational work required by the codes was even more marked. GCKP87 required 87 steps, 116 functional and 19 Jacobian matrix evaluations, and 0.61 sec CPU time. CHEMDDM, however, required 153 steps, 207 functional and 27 Jacobian evaluations, and 1.5 sec CPU time. This variation of the computational work required by LSODE with the specified value for the first output station has been observed previ- ously (ref. 5). It i s caused by the procedure used in LSODE to calculate the first stepsize to be attempted for the problem.
The second example, taken from Hwang (ref. 32). i s also a constant temper- ature problem, but it permits a comparison with the exact solution. It i s a simple first-order reversible reaction
k l d A t 6 k -1
kl :: 1000, k-1 = 1
which describes a rapidly changing system. The solution to this problem is
Sensitivity coefficients were generated at the same output statlons as Hwang (ref. 32) and the results are presented in tables I 1 1 and IV, together with the exact solutions. The tolerances used for this problem were RTOL = 10-6 and ATOL = 10-8 to be consistent with Hwang (ref. 32). whose results were generated with RTOL = 10-6. Table 111 gives the sensitivity coefficients of UA with respect to the initial values. In table I V the sensitivity coefficients with respect to the rate constants are presented, along with Hwang's results obtained with a scaled Green's function method (SGFM) (ref. 32). These tables show the excellent agreement between the GCKP87 and exact results. Although the GCKP87 and SGFM results agree well, there are some discrepancies in the SGFM results at early times. In particular, the SGFM sensitivlty coefficients with respect to k-1 are noticeably inaccurate in comparison to the GCKP87 results. GCKP87 was also significantly more efficient than the SGFM method. tlon and the sensitivity coefficients of both species with respect to both ini- tial values and both rate constants GCKP87 required approximately 0.3 sec on the IBM 370/3033 computer. In contrast, Hwang (ref. 31) states that the SGFM
at 1.5~10-4 and 1x10-3 sec
To solve for the composl-
method requlred approximately 8 sec on a CDC-CYBfiR-172 computer. GCKP87 is seen to be significantly faster than the SGFM method even after accounting for the relative slowness o f thls CDC computer. Hwang (ref. 32) also attempted thls problem with a direct method ( D M ) and gives a CPU time of approximately 7 sec, which Is significantly longer than that required by GCKP87. Since no DM results are given in (ref. 32). an accuracy comparison wlth GCKP87 is precluded.
The last example i s also a simple problem for which an analytical solution I s known. It is, however, a nonconstant temperature problem involving a first- order irreverslble reaction
k A + B
k = A Tn exp(-E/R1)
A = 1 , n = 1 , E = 0
To solve the problem analytically the following simplifying assumptions were made: (1) constant pressure, adiabatic reaction, and (2) constant and equal specific heats (cp) for species A and 8. The solution is
aA(t) = CaA(0)e -Act/[ C - XU A ( 0 ) ( 1 - e -Act)]
u (t) = u (0) + UB(0) - UA(t) B A
1 [ A T(t) = T(0) + X ~ ~ ( 0 ) - u (t)
where
C = T ( O ) + huA(0)
where Qc to the reaction.
is the heat of combustion which dictates the temperature rise due
The analytical and computed sensitivity coefficients with respect to A at various times are given in table V . This solution was obtained with values of Qc = 5000 cal/mol and cp = 5 cal/mol K which give a 1000 K temperature rise when reactant A Is completely converted to product B . Also presented in this table is the mixture temperature which gives an indication of the extent of the reaction. The agreement between the analytical and the computed results is excellent at all levels of reactedness. In table V I the sensitivity coeffi- cients of the time derivative of bllity of GCKP87 t o compute these quantities. The analytical solutions were obtained by differentiating the ODE for U A with respect to the rate con- stant parameters. The solution wlth respect to A Is exact, but those with respect to n and E were obtalned by using the computed values for auA/an,
OA are presented to illustrate the capa-
9
aT/an, aaA/aE, and aT/aE, because these q u a n t i t i e s cannot be ob ta ined a n a l y - t i c a l l y . I he n o r m a l i z a t i o n procedure used f o r A cannot be used f o r n and E because these parameters can have zero va lues. The n o r m a l i z a t i o n parameters g i v e n i n t a b l e V I ( l / l n T and -RT f o r n and E, r e s p e c t i v e l y ) produce a 1 pe rcen t change i n k . The normal ized s e n s i t i v i t y c o e f f i c i e n t s w i t h r e s p e c t t o n o r E can t h e r e f o r e be i n t e r p r e t e d as t h e pe rcen t change i n UA due t o t h e change i n t h e parameter n o r E t h a t produces a 1 pe rcen t change i n k . Th is i s analogous t o t h e s e n s i t i v i t y c o e f f i c i e n t alnUA/alnA which rep resen ts t h e percent change i n UA due t o a 1 p e r c e n t change i n A . For a cons tan t temperature problem t h e t h r e e no rma l i zed s e n s i t i v i t y c o e f f i c i e n t s a r e i d e n t i c a l l y equal t o one ano the r . Again, t h e agreement between t h e a n a l y - t i c a l r e s u l t s and those generated by GCKP81 i s e x c e l l e n t , i l l u s t r a t i n g t h e accuracy o f t h i s code f o r s e n s l t i v l t y a n a l y s i s of nonisothermal combust ion k i n e t i c r a t e equat ions. A l though t h i s problem i s s imple, s e l e c t e d because t h e a n a l y t i c a l s o l u t i o n e x i s t s , GCKP87 has been used s u c c e s s f u l l y on a r e a l i s t i c combustion k i n e t i c s problem i n v o l v i n g 110 r e a c t i o n s among 36 species ( r e f . 43).
CONCLUSIONS
An e f f i c i e n t decoupled d i r e c t method f o r c a l c u l a t i n g t h e f i r s t o r d e r sen- s i t i v i t y c o e f f i c i e n t s o f nonlsothermal combustion k i n e t i c r a t e equa t ions has been developed. S e n s i t i v i t y c o e f f i c i e n t s o f a l l thermochemical v a r i a b l e s and t h e i r temporal d e r i v a t i v e s w i t h respec t t o any comb ina t ion c f i n i t i a l va lues o f dependent v a r i a b l e s and t h e r a t e cons tan t parameters o f t h e chemical reac - t i o n s can be computed. The method was i l l u s t r a t e d by a p p l i c a t i o n t o b o t h con- s t a n t and v a r y i n g temperature problems. The computed c o e f f i c i e n t s agreed w e l l w i t h b o t h a n a l y t i c a l s o l u t i o n s and those ob ta ined w i t h o t h e r codes and s o l u t i o n methods. t h a t o f o t h e r s e n s i t i v i t y a n a l y s i s techniques.
The e f f i c i e n c y o f t h e desc r ibed method compared ve ry f a v o r a b l y w i t h
ACKNOWLEDGMENlS
This work was supported by t h e N a t i o n a l Ae ronau t i cs and Space A d m i n i s t r a - t i o n , Lewis Research Center through Con t rac t NAS3-24105. Or. A. Dunker o f t h e General Motors Research L a b o r a t o r i e s p r o v i d e d a copy o f t h e code CHEMDDM. The a u t h o r would l i k e t o thank D r . D.A. B i t t k e r o f NASA Lewis Research Center f o r many h e l p f u l d i scuss ions .
REFERENCES
1. Radhakrishnan, K . : Comparison of Numerical Techniques f o r I n t e g r a t i o n o f S t i f f O rd ina ry D i f f e r e n t i a l Equat ions A r i s i n g i n Combustion Chemistry. NASA TP-2372, 1984.
2. B l t t k e r , D.A. ; and S c u l l i n , V.J.: GCKP84- General Chemical K i n e t i c s Code f o r Gas-Phase F l o w and Batch Processes I n c l u d i n g Heat T r a n s f e r E f f e c t s . NASA TP- 2320, 1984.
3. P r a t t , G.L.: Gas K i n e t i c s . Wi ley, 1969.
4. Gel inas, R.J.: S t i f f Systems of K i n e t i c Equat ions - A P r a c t i t i o n e r ' s V i e w . J. Comput, Phys., v o l . 9, no. 2, Apr. 1972, pp. 222-236.
10
5. Radhakrlshnan, K. : New I n t e g r a t i o n Techniques f o r Chemical K i n e t i c Rate Equat ions. I. E f f i c i e n c y Comparison. Combust. Sc i , Technol., v o l . 46, nos. 1-2, 1986, pp. 59-81.
1
6. C u r t i s s , C . F . ; and H i r s c h f e l d e r , J.O.: I n t e g r a t i o n o f S t i f f Equat ions. Proc. Nat. Acad. S c l . USA, v o l . 38, 1952, pp. 235-243.
7 . S e i n f e l d , J.H.; Lapidus, L.; and Hwang, M.: Revlew o f Numer ica l I n t e g r a t i o n Techniques f o r S t i f f Ord inary D i f f e r e n t i a l Equa t ions . I n d . Eng. Chem. Fundam., v o l . 9, no. 2, 1970, pp. 266-275.
8. Gear, C.W. : Numerical I n i t i a l Value Problems i n Ord ina ry D i f f e r e n t i a l Equat ions. P r e n t i c e H a l l , 1971.
9. Lambert, J.D.: Computat ional Methods I n Ord ina ry D i f f e r e n t i a l Equat ions. Wi ley, 1973.
10. Wi l loughby, R.A., ed.: S t i f f D i f f e r e n t i a l Systems. Plenum Press, 1974.
11. Lapidus, L.; and Schiesser , W.E., eds.: Numerical Methods f o r D i f f e r e n t i a l Systems, Academic Press, 1976.
12. F in layson , B.A.: Non l i nea r Analys ls I n Chemical Eng ineer ing . McGraw H i l l , 1980.
13. Tyson, T.J.: An I m p l l c i t I n t e g r a t l o n Method f o r Chemlcal K i n e t i c s . TRW-9840-6002-RU-000, TRW Space Technology Lab, Sept. 1964.
14 Lomax, H.; and B a l l e y , H.E.: A C r i t i c a l A n a l y s i s o f Var ious Numerical I n t e g r a t i o n Methods f o r Computing t h e Flow o f a Gas i n Chemical Nonequ i l i b r l um. NASA TN-D-4109, 1967.
15, B i t t k e r , D.A. ; and S c u l l i n , V.J.: General Chemical K i n e t i c s Computer Program f o r S t a t i c and Flow Reactions W i t h A p p l i c a t i o n s t o Combustion and Shock-Tube K i n e t i c s . NASA TN-D-6586, 1972.
16. Warner, D.D.: The Numerical S o l u t i o n of t h e Equat ions o f Chemical K i n e t i c s . J. Phys. Chem., v o l . 81, no. 25, Dec. 15, 1977, pp. 2329-2334.
17. Young, T . R . ; and B o r i s , J.P.: A Numerical Technique f o r S o l v i n g Ord-lnary D i f f e r e n t i a l Equat ions Associated Wi th t h e Chemical K i n e t i c s o f React ive- Flow Problems. J. Phys. Chem., vol . 81, no. 25, Dec. 15, 1977, pp. 2424-2427.
18. Eber t , K.H.; Deuf lhard, P.; and Jager, W., eds.: M o d e l l i n g o f Chemlcal React ion Systems. Spr inger-Ver lag, 1981.
19. P r a t t , D.T.; and Radhakrishnan, K. : C R E K I D : A Computer Code f o r T r a n s i e n t , Gas-Phase Cornbustion K i n e t i c s . NASA TM-83806. 1984.
20. P r a t t , D.T.; and Radhakrlshnan, K.: Phys i ca l and Numerical Sources o f Computat ional I n e f f i c i e n c y i n t h e I n t e g r a t i o n of Chemical K i n e t i c Rate Equat ions: E t i o l o g y , Treatment and Prognosis . NASA TP-2590, 1986.
11
4
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
3 3 .
34.
Radhakrishnan, K . : New Integration Techniques for Chemical Kinetic Rate Equations. Part 11-Accuracy Comparison. J. Eng. Gas Turbines Power, vol. 108, no. 2, Apr. 1986, pp. 348-353. (NASA lM--86893).
Radhakrlshnan, K . : A Critical Analysis of the Accuracy of Several Numerical Techniques for Chemical Kinetic Rate Equations. NASA TP , to be published.
Hindmarsh, A.C.: LSODE and LSODI, Two New Initial Value Ordinary Differential Equation Solvers. SIGNUM Newsletter, vol. 15, no. 4, Dec. 1980, pp. 10-11.
Come, G.M.: The Use of Computers in the Analysis and Simulation of Complex Reactions. Comprehensive Chemical Kinetics, C.H. Bamford and C.F.H. Tipper, eds., Elsevier, 1983, pp. 249-332.
Teets, R.E.; and Bechtel, J.H.: Sensitivity Analysis of a Model for the Radical Recombination Region o f Hydrocarbon-Air Flames. Eighteenth Symposium (International) on Combustion, The Combustion Institute, Philadelphia, PA, 1981, pp. 425-432.
Burcat, A.; and Radhakrishnan, K . : High Temperature Oxidation o f Propene. Combust. Flame, vol. 60, no. 2, May 1985, pp. 157-169.
Dickinson, R.P.; and Gellnas, R.J.: Sensitivity Analysis o f Ordinary Differential Equation Systems - A Direct Method. J. Comput. Phys., vol. 21, no. 2, June 1976, pp. 123-143.
Cukier, R.I.; Levine, H . B . ; and Shuler, K.E.: Nonlinear Sensltivlty Analysis of Multiparameter Model Systems. J. Comput. Phys., vol. 26, no. 1 , Jan. 1978, pp. 1-42.
Hwang, J.T., et al.: The Green's Function Method of Sensitivity Analysis in Chemical Kinetlcs. J. Chem. Phys., vol. 69, no. 1 1 , Dec. 1, 1978, pp. 5180--5191.
Dougherty, E.P.; Hwang, J.T.; and Rabitz, H . : Further Developments and Applications of the Green's Function Method o f Sensltivity Analysis in Chemical Kinetics. J . Chem. Phys., vol. 71, no. 4, Aug. 15, 1979, pp. 1794--1808.
Kramer, M.A.; Calo, J . M . ; and Rabitz, H . : An Improved Computational Method for Sensitivity Analysis: The Green's Function Method With 'AIM'. Appl. Math. Modelling, vol. 5, Dec. 1981, pp. 432-441.
Hwang, J.T.: The Scaled Green's Function Method of Sensitivity Analysis and its Application to Chemical Reaction Systems. Proc. Nat. Sci. Council Repub. China Part 8, vol. 6, no. 1 , 1982, pp. 37-44.
Hwang, J.T.; and Chang, Y.S . : The Scaled Green's Function Method of Sensitivity Analysis. 11. Further Developments and App;lcation. Proc. Nat. Scl. Council Repub. China Part 8, vol. 6, no. 3, 1982, pp. 308-318.
Kramer, M.A., et a l . : AIM: The Analytically Integrated Magnus Method for Linear and Second-Order Sensitivity Coefficients. SAND-82-8231, 7982.
12
35. Rabitz, H . ; Kramer, M.A.; and Dacol, D.K.: Sensitivity Analysis in Chemical Kinetics. Ann. Rev. Phys. Chem., vol. 34, 1983, pp. 419-461.
36. Kramer, M.A., et al.: Sensitivity Analysis in Chemical Kinetics: Recent Developments and Computational Comparisons. Int. 3 . Chem. Kinetics, vol. 16, no. 5, 1984, pp. 559-578.
I1 37. Dunker, A.M.: The Decoupled Direct Method for Calculating Sensitivity Coefficients in Chemical Kinetics. J. Chem. Phys., vol. 81, no. 5, Sept. 1 , 1984, pp. 2385-2393.
Differential and Algebraic Equations. Comput. Chem. Eng., vol. 9, no. 1 , 1985, pp. 93-96.
b
38. Leis, J.R.; and Kramer, M.A.: Sensitivity Analysis o f Systems of
39. Caracotsios, M. and Stewart, W . E . : Sensitivity Analysis of Initial Value Problems With Mixed O D € s and Algebraic Equations. Comput. Chem. Eng., vol. 9, no. 4, 1985, pp. 359-365.
40. Dunker, A.M.: A Computer Program for Calculating Sensitivity Coefficients in Chemical Kinetics and Other Stiff Problems by the Decoupled Direct Method. GMR-4831, Env. 192, General Motors Research Laboratories, 1985.
41. Radhakrishnan, K.; and Bittker, D . A . : GCKP87 - An Efficient General Chemical Kinetics and Sensitqvity Analysis Code for Gas Phase Reactjons," NASA TP- , 1987, to be published.
42. Radhakrlshnan, K.; and Bittker, D.A.: GCKP86 - An Efficient Code for General Chemical for General Chemical Kinetics and Sensjtivlty Analysis Computations. Chemical and Physical Processes in Combustlon, Proceedings of the 1986 Fall Technlcal Meeting o f the Eastern Section o f the Combustion Institute, San Juan, Puerto Rlco, Dec. 15-17, 1986, pp. 46-1 - 46-4.
43. Bittker, D.A.: Detailed Mechanism o f Benzene Oxidation, in preparation, 1987.
TAF3L.t I . - RE.ACIION MLCHANlSM F O R
PROBI.tM la
a l n a i / a l n k l a t 1.0 sec
[ 371 GCKPB I CHI-MDDM
-0.044 -0.043 -0.042
1 .ooo 1 .ooo 1 .ooo .976 .97 1 .97 I
.662 .662 .662
.681 -680 .679
.602 .602 .601
. 4 i a .478 . 4 i 9
R e a c t i o n
a l n u i / a l n k l a t
[ 3.1 ] GCKP87
- 0 . 8 2 0 -0 .800
1 .ooo 1.000
.643 .650
-.210 -.191
.323 .329
.221 .%27
.090 . l o o
( 1 ) C2Hb + CH3 t CH3
4 ' C2H5 (2) CH3 + C2H6 -> CH
( 3 ) C2H3 + C H e H
( 4 )
( 5 ) H + H + H2
2 4
2 ' C2H5 H + C2H6 .+ H
Ra te c o n s t a n t ,
k j b
1.14 ( - 2 )
1.19 ( 6 )
1.57 ( 3 )
9.72 (8) 6.99 (13 )
aSee r e f e r e n c e 37. l h e i n i t i a l concen- t r a t i o n o f C2Hg i s 5.951 ( - 6 ) mol cm-3; a l l o t h e r i n i t i a l c o n c e n t r a t i o n s a r e zero.
t u r e i s 923 K . Numbers i n parentheses a r e Dowers o f 10.
b u n i t s a r e mol , cm, s, and t h e tempera
7ABI-E 11 . - S E N S i l I V I l Y COtFFICILNlS F O R EXAMPLE 1
Species
C2H6
CH3
CH4
H2
'2"5
C2H4
H
20.0 sec
C H t M 0 DM
-0 .789
1 .ooo .655
-.1a1
.332
.230
.106
7ABI-E 111. - S L N S I I I V I I Y COEFfICItNlS W I l H RkSPkC1
1 0 I N I l I A L VALUCS FOR EXAMPLE 2
[Numbers i n parentheses a r e powers o f 10 .1
1 . 5 ~ 1 0 -
I
EXACT
0.861
.368
1.04( - 3)
9.99( - 4)
9.99( - 4)
GCKP87
0.861
1 . 0 4 ( - 3 )
9 . 9 9 ( - 4 )
9.99( - 4)
EXACT
1.39( - 4)
6.32( - 4)
9 . 9 9 ( - 4 )
9.99( - 4)
9.99( - 4)
GCKP87
1 .39 ( -4 )
6.32( - 4)
9.99( - 4)
9.99( - 4 )
9.99( - 4 )
4
14
TABLE I V . - S E N S I l I V l l Y COCtFICIENlS WIlH RESPECT TO R A l E
TABLE V . - S k N S I l I V I l Y C O E F F I C I € N l S WIlH RESPEC1 TO RATL PARAMEIER A FOR
LXAMPLE 3
[Numbers I n parentheses a r e powers o f 10.1
a lnag/a lnA I a lna8 /a lnA
EXACT GCKP87
1 .ooo 1 .ooo
.993
1 .ooo 1 .ooo
.992
.550 8.1 50( -8)
9 .990(-1) i 9.990( 4 ) 9 .900(-3) I 9 .899(-3) 9 .003(-2) 8 .978(-2)
4.122(-8) 4 .075(-8) .238 1 .238 .551
8.244( -8)
1ABl.t V I . - SENSI[IVITY C O E F F I C I E N T S OF OA FOR EXAMPLE 3
t, sec
10 -6 10-5 10-4 10-3 10-2 -
E X A C I
1 .ooo 1 .ooo
.980 - .523
-19.00
15
Report Documentation Page 1 Report No
NASA CR-179636 2. Government Accession No
-____- 7. Author($
Kr ishnan Radhakrishnan
19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No of pages
Unc l a s s i f i ed U n c l a s s i f i e d 16
________________
9 Performing organization Name and Address
Sverdrup Technology, Inc. Lewis Research Center Cleveland, Ohio 44135
2 Sponsoring Agency Name and Address
N a t i o n a l Aeronaut lcs and Space A d m i n i s t r a t i o n Lewis Research Center Cleveland, Ohio 44135
22. Price'
A0 2
~
3. Recipient's Catalog No
5. Report Date
6. Performing Organization Code
~
8. Performing Organization Report No.
None (E-3635)
10. Work Unit No.
505-62-21
11. Contract or Grant No.
NAS3-24105 13. Type of Report and Period Covered C o n t r a c t o r Repor t F i n a l
14. Sponsoring Agency Code
5. Supplementary Notes
Project Hanager, Peter M. Sockol, Internal Fluid Mchanics Division, NASA Lewis Research Center. Prepared for the Sixth International Synposiun on Carputer Methods for Partial Differential Equations sponsored by the International Association for Rathematics and Carputers i n Simulation, Bethlehem, Pennsyl vani a, June 23-25, 1987.
6 Abstract
An e f f i c i e n t , decoupled d i r e c t method f o r c a l c u l a t i n g t h e f i r s t o rde r s e n s i t i v i t y c o e f f i c l e n t s o f homogeneous, ba tch combustion k i n e t i c r a t e equat ions i s presented. I n t h i s method t h e o rd ina ry d i f f e r e n t i a l equat ions f o r t h e sensitivity c o e f f i c i e n t s a re so lved s e p a r a t e l y from, b u t s e q u e n t i a l l y with, those d e s c r i b i n g t h e combust ion chemis t ry . The o r d i n a r y d i f f e r e n t i a l equat ions f o r t h e thermochemical v a r i a b l e s a r e so lved u s i n g an e f f i c i e n t , i m p l i c i t method (LSODE) t h a t a u t o m a t i c a l l y s e l e c t s the s t e p l e n g t h and order f o r each s o l u t i o n s tep. The s o l u t i o n procedure f o r t h e s e n s l t i v l t y c o e f f i c i e n t s ma in ta ins accuracy and s t a b i l i t y by u s i n g e x a c t l y t h e same s tep lengths and numer ica l approx imat ions. The method computes s e n s i t i v i t y c o e f f i - c l e n t s w i t h respec t t o any combinat ion o f t h e i n i t i a l va lues o f t h e thermochemical v a r i a b l e s and t h e t h r e e r a t e c o n s t a n t parameters f o r t h e chemical r e a c t i o n s . The method i s i l l u s t r a t e d by a p p l i c a t i o n t o seve ra l s imp le problems and, where p o s s i - b le , comparisons a r e made w l t h exac t s o l u t i o n s and those o b t a i n e d by o t h e r techniques.
17. Key Words (Suggested by Author@))
Combustion k i n e t i c s ; S t i f f ODE'S; Decoupled d i r e c t method
18. Distribution Statement
U n c l a s s i f i e d - u n l i m i t e d STAR Category 25
