50
Ind. Eng. Chem. Fundam. 1980, 19, 50-59
v = kinematic viscosity, m4/s
Argentina Cooperative Research Program of the Consejo Nacional de Investigaciones Cientificas y TBcnicas of Argentina and the National Science Foundation of the United States of America. The support of CONICYT and a Visiting Professorship at the University of Minnesota made possible this collaboration. The suggestions and material assistance from B. G. Higgins are gratefully acknowledged.
= dimensionless coordinate
surface tension coefficient, N/m l? = dimensionless trial functions Literature Cited u =
8,
Brown, D. R., J. Fluid Mech., 10, 297-305 (1961). Cerro, R. L., Scriven, L. E., "Numerical Solution of the Viscous Flow Equations for Free Surface Flows", A.1.Ch.E. Annual Regional Meeting, Bioomington, Minn., 1971. Cerro, R. L., Whtaker, S.,Chem. Eng. Sci., 26, 785-798 (1971). Cerro, R. L., Whitaker, S.,Chem. Eng. Sci., 29, 963-965 (1974). Coyne, J. C., Elroyd, H. G., J. Lubric. Techno/., 92, 451-456 (1970). Esmaii, M. N., Hummei, R. L., AIChE J., 21, 958-965 (1975). Groenveld, P., van Dortmund, R. A., Chem. fng. Sci., 25, 1571-1578 (1970). Higgins, B. G., Slliiman, W. J., Brown, R . A,, Scriven, L. E., Ind. Eng. Chem., 16, 393-401 (1977). Kantorovich, L. V., Kryiov, V. I., "Approximate Methods of Higher Analysis", Interscience, New York, N.Y., 1964. Landau, L. D., Levich, V. G., Acta Physicochim. USSR, 17, 42-54 (1942). Lee, C. Y., Tallmadge, J. A., AIChf J., 18, 1077-1079 (1972). Lin, C. C., Segei, L. A., "Mathematics Applied to Deterministic Problems in the Natural Sciences", Chapter 9, Macmillan, New York, N.Y., 1974. Marques, D., Costanza, V., Cerro, R. L., Chem. Eng. Sci., 33, 87-93 (1978). Morey, F. C., J . Res. NaN. Bur. Stand., 25, 385-393 (1940). Ruschak, K. J., Chem. fng. Sci., 31, 1057-1060 (1976). Stucheii, A., "Ebene isotherme und nichtisotherme Rieseifiimstromung Newton'scher und nicht-Newton'scher Fiussigkeiten", Abhandlung, Eidgenossichen Technischen Hochschuie, Zurich, 1976. Stucheii, A., Ozisik, M. N., Chem. Eng. Sci., 31, 369-372 (1976). Van Rossum, J. J., Appl. Sci. Res., A7, 121-144 (1958). Watson, E. J., J. Fluid. Mech., 13, 481-499 (1964); see p 493 ff. Yimaz, T., Brauer, H., Chem. Ing. Tech., 45, 928-934 (1973).
Nomenclature g = acceleration of gravity, m/s2 h = film thickness, m h* = characteristic film thickness, m H = dimensionless film thickness function
H = curvature of the free surface, m-' L = Landau and Levich's dimensionless film thickness
Nca = capillary number, p u * / u NGV = dimensionless number, gh*2/vu* Nh = Reynolds number, u*h*/u p = pressure, Pa T = Van Rossum's dimensionless film thickness u, u = velocity components, m/s u* = characteristic velocity, m/s U, V = dimensionless velocity x , y = Cartesian coordinates X, Y = dimensionless coordinates Greek Letters p = density, kg/m3 p = viscosity coefficient, Pa s
Received f o r review February 1, 1979 Accepted October 5 , 1979
Detection of the Compensation Effect (6 Rule) Russell R. Krug Chevron Research Company, Richmond, California 94802
Rate and equilibrium constants are sometimes observed not to vary much for systematic series of reactants, catalysts, or solvents even though their respective enthalpies and entropies vary greatly. For these homologous series, the variations in enthalpies approximately cancel the variations in entropies such that the rate or equilibrium constants remain relatively invariant. Such compensations of enthalpies for variations in entropies are most generally referred to a s the compensation effect and are more specifically referred to as the 0 rule for compensations observed in heterogeneous catalysis. The compensation temperature is the temperature at which enthalpy variations precisely cancel entropy variations such that the rate or equilibrium constants are completely invariant. Development of mathematical criteria for detection of the compensation effect and for determination of the numerical value of the compensation temperature have challenged researchers for over a half century. The procedure widely used at this time for calculating the value of the compensation temperature unfortunately gives a number closer to the least probable rather than the most probable value. Given here are both an analysis of variance procedure for estimation of the probability of detection of a compensation effect and a likelihood analysis for determination of the numerical value of the compensation temperature. A statistical procedure is also presented for analysis of troublesome chemical data sets that are typical of those obtained in the study of heterogeneous catalysis. These procedures result from a straightforward application of statistics fundamentals to chemistry fundamentals and give a clear interpretation of thermodynamic and kinetic interrelationships for all homologous chemical series, not just those with linear enthalpy-entropy compensations.
One of the more controversial areas of chemical research is the determination of linear compensations between enthalpies and entropies for homologous series of chemical reactions and equilibria. For these homologous series, which are systematic variations of reactants, catalysts, or solvents, kinetic and thermodynamic data sometimes suggest that variations in enthalpies are opposed to variations in entropies such that they compensate each other and the free energies for the entire series appear to be constant. Such observed compensations may result for chemical reasons or they may result merely as computa-
tional artifacts. Spurious compensations that are computational artifacts occur because measurement errors that randomly influence the values of observed rate and equilibrium constants become structured and highly correlated by the calculation procedures for estimating enthalpies and entropies from the original rate and equilibrium constants. Thus, what may begin as random noise added to rate or equilibrium constants becomes manifested as a highly structured pattern added to enthalpy and entropy values. The challenge addressed here is to properly analyze structured variations between thermodynamic
0019-7874/80/1019-0050$01.00/00 1980 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980 51
variables without confusing structured propagation of measurement errors with true chemical effects. I t is demonstrated in this article that both linear and nonlinear thermodynamic relationships can be uncoupled from structured propagations of measurement errors and can thus be interpreted in a statistically unbiased manner. Over the years many homologous chemical series displayed linear compensations of enthalpies with entropies. Some of the best reviews of the chemical theories that predict such linear compensations were presented by Leffler (1963), Hammett (1970), Lumry and Rajender (1970), and Ritchie and Sager (1964). On the other hand, many criticisms of the standard statistical procedure (plot of enthalpy vs. entropy estimates) used to detect these compensations have also been published. Statistical procedures published by Exner (1964, 1970, 1972, 1973), Wold (1973), Shimulis (1969, 1974), and Krug, Hunter, and Grieger (1976, 1977) are different than the standard procedure and purport to correctly treat the data. The standard procedure of plotting enthalpy estimates vs. entropy estimates presents the data in a highly biased manner. The linear variation of enthalpies with entropies is most commonly referred to as “the compensation effect”, but the various fields of chemistry often refer to it by different names, such as “the 6’ rule” in heterogeneous catalysis. Catalysis textbooks that mention the 6’ rule include texts written by Bond (1962), Boudart (1968), and Thomas and Thomas (1967). The earliest reported example of the 6’ rule dates to Constable (1923), and the controversy of whether these effects are due to statistical artifacts or actual chemical effects has continued because of the difficulty of developing a generally accepted data handling procedure that successfully separates statistical effects from chemical effects. The data analysis problem for heterogeneous catalytic data is especially difficult because it is often either inconvenient or impractical to collect such kinetic or equilibrium data a t identical temperatures for each member of typical homologous series. Hence, data are usually collected, and compensations are reported, for which only the range of thermal variation is noted for each enthalpy-entropy estimate pair. Obtaining kinetic and equilibrium data at identical temperatures does not appear to be as difficult in the other branches of chemistry, however. A recent review of the compensation effect in heterogeneous catalysis (6’ rule) by Galwey (1977) was an update of a previous review by Cremer (1955). Galwey acknowledged the difficulty of analyzing enthalpy-entropy compensatory behavior when data are not sampled at identical temperatures, as is the usual case for heterogeneous catalytic data. For want of an alternate procedure, Galwey proposed a least-squares analysis to detect linear compensations between enthalpy-entropy estimate pairs. A recent development by Krug et al. (19761, however, proved that such a least-squares analysis yields a numerical value of the compensation temperature that is closer to the least probable number rather than the most probable number. Recent developments by Krug et al. (1976,1977) show that the only nonbiased analysis of thermodynamic parameters is between enthalpy-free energy estimates, not enthalpy-entropy estimates. This article shows how an analysis of variance procedure developed by Mandel (1961, 1962) can be used to assess the probability that a linear enthalpy-entropy compensation is detected for chemical kinetic or equilibria data that are sampled at identical temperatures. Of course, this result further emphasizes that whenever practical, if an
experimenter wishes to detect a linear enthalpy-entropy compensation effect, provisions should be made to sample data at identical temperatures for the different reactions or equilibria in the homologous series under study. Presented here is also a “minimum bias” procedure for data analysis if data were obtained at nonidentical temperatures. Theory If a linear compensation due to chemical factors exists between the enthalpies and entropies of a homologous series, Leffler (1955) and Brown and Newsom (1962), showed that the chemical compensation line is characterized by a slope p (the compensation temperature) and an intercept Go (the free energy at T = P ) AH = PAS + AGp where the parameters AH, AS, and AG may be either for chemical reactions or equilibria. Measurement errors associated with the values of rate and equilibrium constants were shown by Krug et al. (1976) also to have a nearly linear distribution in the AH-AS plane AH = ThmAS + AGT, where Thmis the harmonic mean of the experimental temperatures. Thus, data plotted in the enthalpy-entropy plane may appear to display a linear compensation that is due entirely to the propagation of measurement errors with an apparent but false compensation temperature of
@=
Thm.
Data plotted in the AH-AG, plane, however, display structured variations due only to chemical effects because the measurement errors propagate in a random manner cointo this plane. A linear compensation in AH-AG, ordinates is related to that in AH-AS coordinates t y AH = YAG + (1- y)AGp where Y = 1/(1- T h / P )
That is, an AH-AS line with slope has a corresponding AH-AG line with slope y. The values of /3 and y are uniquely related as shown above. Also, if a linear relationship due to chemical effects exists, the Arrhenius or van’t Hoff plots must show a concurrence such as that displayed in Figure 1. Petersen (1964) has sharply criticized those who do not display their data in such a manner so that the raw data can be readily scrutinized for possible concurrence. Such plots are excellent common sense indicators of evaluating whether the raw data display concurrence within measurement error or whether the investigator is deceived by his plot of enthalpies vs. entropies and the apparent correlation that arises from even relatively minor measurement errors. Unfortunately, Petersen’s criticism is leveled against the vast majority of scientists who report linear enthalpy-entropy compensations, for very few subject their raw data to this incisive analysis. One such concurrence which is shown in Figure 2 was presented by Cremer and Kullich (1950) for formic acid decomposition over magnesite. Another such concurrence shown in Figure 3 presented by Germain (1977) was due to catalyst wetting problems in a trickle bed reactor rather than to chemical factors. I. Data Sampled at Identical Temperatures When data are suspected of concurrence as shown in Figure 1 and are sampled at identical temperatures, an assessment of the probability of concurrence may be made by a two-way analysis of variance developed by Mandel (1961, 1962).
52
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
Table I. ANOVA Table for Detection of the Compensation Effecta
ss
DF
sources of variation
P4 - 1 P- 1 4 - 1 (P P- 1
total rows (catalysts) columns (temperatures) interactions slopes (Arrhenius reln) concurrence (compensation effect) nonconcurrence residua 1s
dlS
MST
SST SSR
- 1)
1
MSR
SSC
MSC
ssRC
MSRC
SSS
MSS
SScon
MScon
P - 2 Ssnoncon MSnoncon ( P - 1 ) ( 4 - 2) ss, MS, a D F is degrees of freedom, SS is t h e sum of squares, and M S is t h e mean sum of squares. M S = S S / D F for each source o f variation. HCOOHCATALYST:
t H, M&O, I M ~ O
CO
r
-
-
1 ~
-
Figure 1. If a set of kinetic or equilibrium data are influenced by a linear compensation effect, the Arrhenius or van't Hoff plot must display a single point of concurrence at the compensation temperature p. Such a plot should be used to screen data initially to determine if there is any chance of detecting a linear compensation effect within the measurement accuracy of the original data even before venturing on a more sophisticated analysis.
Consider the case for which p catalysts in a homologous series are evaluated at q temperatures. (The analysis could also be for the action of a single catalyst on p compounds which form a homologous series or of p compounds of a series achieving equilibrium, etc.) The data could then be arranged in a two-way table with the catalysts as row entries and temperatures as column entries such that y,,= In k,, for kinetic rate constants and y,, = In K,, for equilibrium constants, where there are 1 I i I p and 1 I j I q observations. If n replicates of each observation were made, the homoscedasticity (sameness of magnitude of measurement errors) of the data set can also be evaluated. The interested reader is referred to Mandel (1962) for this procedure. Although such replications would be useful, they are rarely reported. From the data y U , an analysis of variance (ANOVA) table such as Table I can be constructed. The entries in Table I are defined as
-3.5 I
I
I
L
L I
Y o l
-0
P Y
SST= C C y I l 2- G 2 / p q SScon = r2SSs 1=1,=1
P
SSR= C ( R I 2 / q -) G 2 / p q
SSnoncon
1=1
= (1- r2)SSs 1 0 3 i ~(
4
SSc = C ( C , ' / p )
-
G 2 / p q S S , = SSRc - SSs
]=I
SSRc = SST - SSR - ssc
sss = m1 1)2E?,2 P
4
-
1=1
J=I
~ - 1 )
Figure 3. Arrhenius plot of the hydrogenation of 2-butanone in a trickle bed reactor for different mass flow rates L of the liquid as plotted by Germain (1977). This compensation is attributed to wetting variations of the catalyst pellets with liquid flow rates, rather than to chemical factors involving the enthalpy or entropy of the chemical reaction. If this concurrence were due to chemical factors, the compensation temperature would be very precisely detected in the middle of the range of experimental temperatures.
Ind. Eng. Chern. Fundarn., Vol. 19, No. 1, 1980
53
of detection of the linear compensation effect and only applies for data sampled a t identical temperatures.
where
11. Data Sampled at Nonidentical Temperatures
Ri = C Y i j j=1 D
C, = C Y i j i=l
cj
G
+ i = P- - - P4
and r is the correlation coefficient between the slopes and intercepts assuming the data obey linear relationships (as is the case when kinetic data fit the Arrhenius relationship or equilibrium data fit the van’t Hoff relationship). These linear relationships are represented by y v. , = pi + Giyj t i j
+
where pi is the average value of yi, for row i, 6i is the slope for row i, yj is the independent variable centered about its mean, and cij is the measurement error associated with the measurement of yip The correlation coefficient, r , is found from 9
CYij
/=1
pi = 4
a
D
r=
i=l D
where the overbars denote an averaged quantity. Note that 6 = 1. A high value of this correlation coefficient indicates that concurrence is detected if the variation of the sums of squares due to nonconcurrence is comparable to that of the measurement errors. From the information supplied by the data in the ANOVA table, probabilities can be assigned for different kinetic effects by evaluation of the F-statistic F(DF1, DF2, 1 - CY)= MS1/MS2 where 1 - CY is the probability that the variation due to effect 1 is greater than that due to effect 2. Thus, the critical tests to perform are: (1) to determine that the probability of concurrence is greater than that of nonconcurrence a t the 100a% level of significance
Mscon > F(1,p - 2,1 - CY) MSnoncon and (2) that the variation due to nonconcurrence is not greater than that due to measurement errors at the 100~1% level of significance
This procedure is merely an adaptation of the test for concurrence using the Tukey degree of freedom as developed by Mandel (1961, 1962) tailored to the problem
When data are not sampled at identical temperatures, the problem of testing for concurrence is difficult. Precise uncertainty statistics may never be strictly applicable to this case. The best experimental design strategy prior to experimentation if the compensation effect is sought is to sample chemical reaction rates or equilibria at identical temperatures for the various members of the homologous series. Exner (1970, 1972, 1973) offers a nonlinear least-squares procedure that does not require the data to be sampled a t identical temperatures. Unfortunately, his procedure uses the reciprocal experimental temperature as the independent variable. Since the range of variation of this independent variable is small and far from its origin, the values of the estimated parameters (including the value of the compensation temperature) are highly correlated with one another and are largely indeterminate. Thus, Exner’s nonlinear regression procedure necessitates choosing the value of the compensation temperature from a very broad and flat minimum on a sum of squares surface from which the value of the compensation temperature and its corresponding confidence interval are largely uncertain. The very short range of variation far from the origin of inverse experimental temperatures is the cause of false observed correlations between enthalpy and entropy estimates in the first place and strongly contributes to the indeterminate quality of compensation temperatures estimated from Exner’s procedure. The dispersion analysis by Shimulis (1969, 1974) is incorrect because that procedure uses biased values of the compensation temperature calculated by the usual but incorrect procedure. Parameter estimates are made less correlated by shifting the origin of the independent variable to the midpoint of the range of variation. When such a transformation is applied to data sampled at identical temperatures, the parameter estimates which we denote here as t are uncorrelated. Since these parameters are measures of (AII,AC;)Thm,the errors associated with the estimates (AH,AG)Tb are also uncorrelated resulting in an unbiased plot of thermodynamic parameters. About these estimates, either the joint confidence region or the standard deviation increments for each estimate serve as adequate delimiters of the certainty of the parameter estimates. When such a transformation is applied to data not sampled at identical temperatures, however, collectively minimum biased estimates are obtained. A discussion of this minimum bias is given in the Appendix. For the case of data sampled at nonidentical temperatures, it is not possible to plot unbiased estimates of any of the thermodynamic potentials vs. each other. The amount of bias is determined by how far apart the experimental temperature spans are from each other. The bias associated with (AH,AG)Thestimates vanishes in the limit that the temperature spans coincide. For data sampled at nonidentical temperatures it is not proper to plot merely the standard deviation increments about the AH-AG estimates, for the bias associated with each estimate must also be displayed and taken into account when fitting these estimates to a model for thermodynamic variations. With such a plot of minimally biased thermodynamic parameter estimates, the experimenter may then visually assess whether or not his data are tending to vindicate or invalidate the assumption of a linear compensation effect. The details of parameter estimation and construction of the concomitant joint probability regions are given below and are virtually the
54
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
same as that presented previously by Krug et al. (1976, 1977) except the data may now be sampled at nonidentical temperatures, Tij. The value of constructing joint confidence regions about parameter values jointly estimated from the same data set has been recognized in chemical investigations by Mandel (1957) and others but has not been widely applied. For the particular case treated here, we must now consider each design matrix separately because each member of the homologous series is sampled at different temperatures. The Arrhenius equation is linearized by taking the logarithm of the rate constants. (The same analysis is also applicable to a van't Hoff analysis of equilibrium data.) yij = In k , = In Ai - Ei/RTij The plot is modified by shifting the origin to the average value of the inverse temperatures yij = (In A - E / R T b ) i - E i / R ( l / T i , - l / T b ) where Thm is the harmonic mean of the experimental temperatures. Allowing for measurement errors ti associated with the observations x, the data may be summarized in matrix notation by yi = Ti + €i = Wifi + ti where {/ = ([ln A - E/RTh,li, -Ei/R) for kinetic data, yij = In kij;f / = (-AGi"/RTb, -AHio/R) for equilibrium data, yij = In Kij,and
B' = ([RTb In (kThe/h) - RTh], -RTb) for kinetic data and B' = (0,O) for equilibrium data. If the data were sampled at identical temperatures, they may be plotted using just their standard deviations as uncertainty delimiters V ( @ J= A(W/Wi)-'As: SD,bi = [V(Afii)I1I2= R T b [ ~ i ~ / q ] l / ~ 4
sD&, =
[v(&i)]1/2 = R [ s i 2 / C ( 1 / T i j- l/Th,)2]1/2 j=l
If the data were not sampled at, identical temperatures, the off-diagonal elements of V(\ki)are nonzero and these biases should be displayed. The joint confidence regions will display these biases and are given by the following elliptic equation which is valid for both the above kinetic and equilibrium expressions (\ki - @ i ) ' ~ - ~ ~ -/ @J ~ i=~2 ~ - :l~(( 2q,-qi2,1 - CY) This equation can be rewritten in quadratic form for either AHi given the relevant calculated values and a guessed value of AGi or vice versa. Choosing the quadratic form where AHi is to be calculated from a guess of AGi, the above equation may be rewritten as ami2 bAHi c = 0
+
' I
-;]
A = [-:Thm
AH,=
+
-b f [b2- 4 a ~ ] ' / ~ 2a 2a
where The values of the regression coefficients are obtained by = (w/wi)-lw/yi =
ti
and
2R2si2F(2,q- 2,l - C Y ) 4
C ( l / T i j- 1 / T b , ) 2
Enthalpy and free energy estimates are then calculated from the slope and intercept estimates. For kinetic data A 6 i * = -RThmtlj + (RTh, In [ k T h m e / h ]- R T b ) AHi* = -Rt2i - EThm
and for equilibrium data
=- R T ~ ~ ~ ~ AHio = -Rt2i
In matrix notion, these relationships may be summarized as \ki = A f i + B where \k/ = (AGi*,mi*) for kinetic data and \k/ = (AGiO, H i ofor ) equilibrium data
j=l
The 1- CY probability region for the location of the true value of the thermodpamic parameters pair ( A H , A G ) T ~ given the estimate (AH,AG)T and the experimental temperatures Ti,may then be cakhlated by assuming a value of AGi and calculating the corresponding values of AHi such that
(%)'- i
10
Such a repetitive calculation is conveniently performed on a programable pocket calculator. Applications to Chemical Examples Reviewed here are two heterogeneous catalysis data sets for which false linear compensation effects (0 rule) were
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980 55 Table 11. Benzene Hydrogenation over Copper-Nickel Alloys Reported b y Caldenhead and Masse (1966)" specific conversion a t temp, "C
sample no.
atom %Ni
112
127
M1 M2
100 92 83 74 56 47 38 29 19 10
6.90 5.81 5.91 6.29 5.33 4.40 3.31 2.18 2.33 0.78
11.2 10.5 11.2 12.2 11.2 9.48 6.72 4.08 3.33 1.10
M3 M4 M6 M7 M8 M9 M10 M11
144
162
18.4 19.5 21.5 24.3 23.8 20.0 13.7 7.84 4.76 1.62
30.0 35.8 40.9 47.7 49.1 42.3 27.8 15.0 6.88 2.35
a T o form a two-way table for the analysis of variance, take the logarithm of t h e entries in this table.
reported by McKee (1965) and Caldenhead and Masse (1966) and the recent review by Galwey (1977) of the compensation effect in heterogeneous catalysis. No linear compensation effects are present for these data sets, however, as substantiated by Arrhenius plots of the original data, lack of linearity on an (AH-AG)Tb plot, and by the low probability of concurrence as calculated using analysis of variance. These authors reporting the linear compensation effects were fooled into believing linear extrathermodynamic effects were present because they plotted their data on AH-AS plots and observed numerically significant but false correlations. Krug et al. (1977) demonstrated that nonlinear variations of thermodynamic potentials are to be generally expected. The analysis presented here confirms that nonlinear variations are observed for the data of McKee (1965) and Caldenhead and Masse (1966), not linear variations as reported by these investigators and Galwey (1977). These examples serve to demonstrate the application of the analytical procedures discussed above. A. Probability Test for Concurrence. A hydrogenation of benzene over alloys of varying relative percentages of copper and nickel was performed at four temperatures by Caldenhead and Masse (1966). The reported data are listed in Table 11. For consistency with the analysis performed by Caldenhead and Masse, common logarithms were taken of the data rather than the natural logarithms used above. The original data are plotted in an Arrhenius plot in Figure 4 and hardly show even a tendency for concurrence within the excellent experimental precision of the data. The traditional but misleading compensation effect or 0 rule plot is shown in Figure 5. There are some minor discrepancies between the reported values of (E,log A ) and those calculated by use of least-squares regression. Perhaps the authors merely estimated these values by graphical techniques. The least-squares values of the compensation ^temperature from the incorrect procedure (regression of E on log A ) are noted on both Figures 4 and 5. These values of the compensation temperatures look deceptively correct in Figure 5, but in Figure 4, the plot of the original data, the improbability of concurrence at this or any temperature is visually evident. The plot in Figure 6 of the (AH*, AG'), estimates along with their concomitant three standard deviation increments (99% confidence intervals) shows that a chemical variation of thermodynamic parameters is indeed caused by changing the composition of the catalyst, but the variation is not linear in the space of thermodynamic potentials. Krug et al. (1976, 1977) showed that a detected variation between any two thermodynamic potentials (e.g., AH-AG) may be reinterpreted as a variation between any other two thermodynamic potentials (e.g., AH-AS) by use of the Maxwell
Thm
101'1 I K - ' I
Figure 4. An Arrhenius plot of benzene hydrogenation over copper-nickel alloys from data reported by Caldenhead and Masse (1966). The data were obtained with excellent precision of fit to the Arrhenius equation but do not display concurrence, even though the original authors and a literature review by Galwey (1977) claim that these data display a linear compensation effect.
8
0
log A
$ Figure 5. A false enthalpy-entropy compensation plot for benzene hydrogenation over copper-nickel alloys from data reported by Caldenhead and Masse (1966). The reported values are denoted by (O),and estimates using least-squares regression on the original data are denoted by (+). The values of the compensation temperatures were computed by the usual but incorrect procedure of regressing E estimates on log A estimates via least squares. The good apparent correlations lead one to conclude incorrectly that the data should display a point of concurrence in the Arrhenius plot of the original {ata in Figure 4. If Al?-AS* values were calculated from the E-log A values, the same analysis would yield the same wrong conclusion regarding the detection of a linear compensation effect.
equations. Because the Maxwell equations are linear relationships between the thermodynamic potentials, only linear relationships between any two thermodynamic potentials will translate to a linear relationship between any other two. For the data displayed in Figure 6, a linear compensation between thermodynamic potentials is not likely. The ANOVA-table for this data set is given in Table 111, and a plot of 6 vs. p is shown in Figure 7. Note the similarity to Figure 6. The mean sum of squares of the residuals in Table 111 (which is an estimate of the square of the standard deviation of the measurement errors, e)
56
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
14
=
-
12-
12-
n
6
-
x
10-
E r
; IC0 5-
I
I
0
0.5
*
1
10
5
ir
Figure 7. A plot of the slope and intercept (at the mean) estimates, 8 and p, for benzene hydrogenation over copper-nickel alloys for the analysis of variance using data reported by Caldenhead and Masse (1966). The similarity of this plot with thaj in Figure 6 results from the fact that the values of AH*% and AG*% are calculated from similar slope and intercept estimates using the chemical identities. A linear compensation effect is not detected by this plot. The arrow indicate the direction of increasing nickel content of the catalysts. Table IV. Exchange of Methane-Deuterium over Palladium-Ruthenium Allow ReDorted by McKee (1965)"
39 8.231 total 9 4.660 rows (catalysts) 3 3.394 column (temperatures) 27 0.1775 interactions 9 0.1773 slopes ( Arrhenius reln) 1 0.09646 concurrence (compensation effect) 8 0.08082 nonconcurrence 18 0.0001756 residuals
0.2111 0.5178 1.131 0.006572 0.01970 0.09646
sample no.
wt %
temp, "C
log k ; ;
1
0
2
5.2
117 127 141 95 105 113 118 128 93 102 121 133 100 110 107 132 97 114 103 125 147 119 127 157 169
10.380 10.988 11.295 10.545 10.826 11.230 11.352 11.491 10.568 10.893 11.395 11.718 10.929 11.238 11.272 11.826 11.220 11.465 10.792 11.161 11.599 10.532 10.875 11.431 11.634
Ru
3
10
4
14.3
5
21
6
40
7
62
8
100
0.01010 0.000009756
Calculations were made using common logarithms instead of natural logarithms for consistency with t h e analysis of t h e original data by t h e authors. Results are reported t o four significant figures, although t h e calculations were made with no less t h a n 10 significant figures. a
is very small relative to the other mean sums of squares. Thus, the data fit well to the Arrhenius equation displaying both catalyst and temperature effects. The probability for concurrence appears to be high relative to that for nonconcurrence
MSCO, -MSnoncon
- 9.55 = F(1,a, 1 - CY = 0.985)
but the probability for nonconcurrence is even higher compared to the precision of the data
MSnoncon -MS,
- 1035 >> F(8, 18, 1 - CY = 0.999)
Thus, to within the precision of the data, no concurrence is detected. If the mean sum of squares due to nonconcurrence were comparable to the mean sum of squares of residuals, the probability of detection of concurrence would be that calculated from the ratio, MScon/MSnoncon* Tables of F at different degrees of freedom are found in various texts such as that by Ostle and Mensing (1975) and are often included in the statistical routines of electronic pocket calculators. B. Evaluating Data Sampled at Nonidentical Temperatures. Data on the exchange of methane hy-
a Additional data reflecting deviations from the linear portion of the Arrhenius plot were deleted from this analysis.
drogen with deuterium catalyzed by a series of catalysts composed of varying amounts of ruthenium and palladium were obtained at different temperatures by McKee (1965). The Arrhenius plots decreased in slope at higher temperatures, so just data that fell in the linear regions of the plots are listed in Table IV. These data are plotted in Figure 8 and display substantially more scatter than those of the first example. The traditional but misleading compensation effect or 8 rule plot for these data shown in Figur? 9 she-ws another slight discrepancy between values of ( E , log A ) reported and those calculated by use of least-squares regression, which may be due to the author's use of a graphical procedure for estimating E and log A . Again, calculation of the compensation temperature using the incorrect proce-
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980 57
2 2
2 4
1 -
2 6
2 8
Thm 10' 1 K -
Figure 8. An Arrhenius plot of methane-deuterium exchange rates over palladium-ruthenium alloys from data reported by McKee (1965). These data do not display concurrence, even though the original author and a literature review by Galwey (1977) claim that these data display a linear compensation effect.
I
'"
24 '09
P
Figure 9. A false enthalpy-entropy compensation plot for methane-deuterium exchange over palladium-ruthenium alloys from data reported by McKee (1965). The reported values are denoted by (01, and estimates using least-squares regression on the original data are denoted by (+). The values of the compensation temperature were computed by the usual but incorrect procedure of regressing E, estimates on log A, estimates via least squares. The good apparent correlations lead one to conclude incorrectly that the data should display a point of concurreQce in the Arrhenius plot of the original data in Figure 8. If H - A S * values were calculated from the E-log A values, the same analysis would yield the same wrong conclusion regarding the detection of a linear compensation effect.
dure, least-squares regression of A", estimates on AS, estimates, gives a deceptively good fit in the A€-AS plot, Figure 9, but concurrence is not substantiated at this or any other temperature by the Arrhenius plot of the original data in Figure 8. The plot of (hH*,AG*)Thestimates in Figure 10 along with the corresponding 95% (1 - LY = 0.95) probability regions for the location of the true values of (LW*,AG*),~ confirm the observation from Figure 8 that a concurrence or linear compensation effect is not detected by these data. The apparent compensation displayed in Figure 9 is spurious and is due to the scatter of the measured data. Notice that when q = 2, there are only two points defining the regression line. The value of F(2,0,1 - cy) is infinitely large, and the uncertainties of the slope and intercept estimates are thus unbounded. For this reason, only five of the (aH*,AG*), pairs in Figure 10 have joint probability regions drawn, and for one of those (0% Ru) the joint
Figure 10. A plot of the enthalpy-free energy estimates for the methane-deuterium exchange over palladium-ruthenium alloys from data reported by McKee (1965). The estimates are delimited by their 95% joint confidence regions. Since the estimates are correlated, the joint confidence regions must be used instead of confidence intervals to display the bias associated with each estimate. I", is the harmonic mean of eTperimental temperatures for all of the data. Since three of these ( A H , * , A G , * ) b pairs were estimated from lines determined by just two points, their joint confidence regions are unbounded and, hence, are not shown. If all such estimate pairs had bounded joint confidence regions, use of 1 / T b as the intercept would result in tilted ellipses such that the overall positive biases would be equal to the overall negative biases. The arrows indicate the direction of increasing ruthenium content of the catalysts.
probability region is very large and exceeds the scale of this figure. Summary and Conclusions Four important points about the analysis for extrathermodynamic relationships were discussed in this article. First, the Arrhenius or van't Hoff plot is a very useful tool for visual inspection of the original data for a point of concurrence (i.e., detection of a linear compensation effect or 8 rule by seeing if the linear Arrhenius or van't Hoff lines intersect at a common point). Second, if detection of any extrathermodynamic relationship (linear or nonlinear) is sought, the data should be sampled a t identical temperatures for the different members of the homologous series under investigation whenever possible. Third, the Tukey degree of freedom allows for an analysis of variance procedure to assess the probability of detection of concurrence on an Arrhenius or van't Hoff plot (which results from a linear compensation effect). Fourth, to check the functionality or structure of the variation of thermodynamic parameters due to chemical effects, with minimal interference from data handling artifacts, the data should be visually inspected in the AH-AG, plane. If the data were sampled at identical temperatures for each member of the homologous series, this is an unbiased plot; if the data were not sampled at identical temperatures, this is a minimum bias plot of the functional relationship between the thermodynamic parameters. Both linear and nonlinear relationships between the thermodynamic parameters are easily identified by a visual inspection of Af-AG, plots. Acknowledgment The author wishes to acknowledge friutful discussions with John Mandel regarding the detection of concurrences. Appendix Demonstration that Choice of 1/ Thmas an Intercept Gives a Minimum Bias Plot of Thermodynamic Potentials. This derivation works equally well for van't Hoff or Arrhenius data. For the sake of brevity, the Ar-
58
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
rhenius data case will only be treated. The linearized Arrhenius equation is yij = In kij = In Ai - Ei/RTij c i j
+
Using 1/Thm rather than 1/T = 0 as the intercept yields yij = (In A - E/RT,,.Ji - E i / R ( l / T i j- 1/Thm) + cij which when placed in matrix notation as in the text is yi = gi + ti = w,ri + ci which is a model vi, linear in parameters ti. The errors ti associated with the observations yi are taken to be normally and independently distributed with zero mean and constant variance ti NID(0,ai2). For the sake of this-example, the case is considered for which the estimates, Ci,result from a constant true set of parameters, 1. The ti differ because the errors ti were different. Thus,this example is treating the case for which observed variations of the thermodynamic potentials are only due to a propagation of measurement errors. Also, for simplicity, let all a? = 2, a constant variance characteristic of sampling from t. Since c NID(0, u2), then yi NID(Wt,u2). The probability distribution of observations for the ith data set is
-
-
-
ti
-
Sinse fi is a linear combination of yi, = (W,lWi)-lW,l?, then ti NID(t,( W[ Wi)-’a2), and the probability distribution of the { estimates is pi(fiIt,u,Wi) =
Finally, the vector of thermodynamic potentials, \k[ = (AGi*,A.Hi*),is a linear combination of ti, \ki = A t + B , so that \ki NID(\k,A(W,lWi)-’Au2)yielding a probability distribution of thermodynamic potentials estimates of
-
For i = p > 1such estimates the joint probability distribution of \E estimates is
Thus, the symmetric variance-covariance matrix is
V(GJ =
v
[
P
= A(ZW,lWi)-1Au2
The numerator of each covariance term is e 9
R 2 ~ 2C(l/Tij C - l/Thm) = 0 i=lj=1
The covariance for any other intercept other than 1/ Thm can likewise be shown to be nonzero. Thus, for the idealized case that thermodynamic parameter estimates are distributed only by experimental measurement errors, choice of 1/Th as the intercept gives an overall nonbiased distribution of estimates. Each es-
timate in general, however, has a covariance that is nonzero when sampled at nonidentical temperatures and, therefore, has an associated bias. Choice of l/Thmas an intercept assures that the sum total of positive biases equals the sum total of negative biases. When data are sampled at identical temperatures, each estimate is unbiased as well. These arguments, for the sake of simplicity, assess the biases introduced by the propagation of measurement errors for the case of invariant true values of the thermodynamic potentials. For the case of the thermodynamic potentials varying due to chemical factors, the consequences regarding the biases introduced by the measurement errors shown above are the same. That is, choice of 1/Thm as the intercept will lead to a bias minimization regardless of the nature of thermodynamic parameter variations due to chemical factors. Choice of 1/Th as the intercept yields balancing positive and negative biases for estimates from data sampled at nonidentical temperatures and zero individual biases for estimates from data sampled at identical temperatures. Nomenclature a = general coefficient in a quadratic equation A = matrix relating thermodynamic potentials with their measures, Arrhenius preexponential b = general coefficient in a quadratic equation B = vector relating thermodynamic potentials with their measures c = general constant in a quadratic expression Cj = sum of column entries in a two-way table for column j DF = degree of freedom e = base of the natural logarithm E = activation energy F(DF,$F2,1 - a ) = F-statistic at degrees of freedom DF,, DF2, and probability 1 - a G = grand sum of all observations in a two-way table h = Planck’s constant i = row (or catalyst) index for a two-way table j = column (or temperature) index for two-way table k i j = rate constant for reaction by catalyst i at temperature j
k = Boltzmann’s constant K = equilibrium constant M S = mean sum of squares n = number of replicate data p = number of rows (or catalysts) in a two-way table p i = probability of an outcome given certain fixed properties q = number of columns (or temperatures) in a two-way table r = correlation coefficient between 6 and p R = gas constant Ri = sum of row entries in a two-way table for row i si = estimate of the standard deviation of observations yi S D = standard deviation SS = sum of squares T = temperature V = variance, variance matrix W = design matrix for linear regression with the inverse experimental temperatures centered about their average value y = observations in the form of the logarithms of the rate or equilibrium constants Greek Letters a = statistical level of significance /3 = compensation temperature y = slope on a AH-AG plot y j = ANOVA independent variable centered about its mean 6 = slope of a line defined by the analysis of variance AG = free energy AH = enthalpy A S = entropy cij = error associated with observation yij 7 = the mathematical model to which data are to be fit p = the mean value of data fitted to a line as defined by the analysis of variance
Ind. Eng. Chem. Fundam. 1980, 19, 59-66
j- = vector of regression parameters that are measures of and AG \k = vector of AH and AG values
AH
59
Caidenhead, D. A., Masse, N. G., J. Phys. Chem., 70, 3558 (1966). Constable, F. H., Proc. R. SOC., London, Ser. A, 106, 355 (1923). Cremer, E., Adv. Catal., 7, 75 (1955). Cremer, E., Kullich, E., Radex Rundschau, 4, 176 (1950). Dzhuntini, E., Shimulis, V. I.,Kinet. Katal., 15, 210 (1974). Exner, O.,Collect. Czech. Chem. Commun., 29, 1094 (1964): Nature(London), 201, 488 (1964). Exner, O., Nature (London), 227, 336 (1970); Collect. Czech. Chem. Commun.. 37. 1425 (1972). -, Exner, 0.; Beranek, V., Collect. Czech. Chem. Commun., 36, 799 (1973). Galwey, A. K;: Adv. Catal., 26, 247 (1977). Germain, A., Extrait de la Collection des Publications de ia Faculte des Sciences Appliquees de I'Unlversite de Liege", No. 65, 1, 1977. Hammett, L. P., "Physical Organic Chemistry", 2nd ed,McGraw-Hill, New York, N.Y., 1970. Krug, R. R., Hunter, W. G., Grieger, R. A., Nature(London), 261, 566 (1976). Krug, R. R., Hunter, W. G., Grieger, R. A., J. Phys. Chem., 60, 2335, 2341 (1976). Krug, R. R., Hunter, W. G., Greiger-Block, R. A,, ACS Symp. Ser., 52, 1972 (1977). Leffler, J. E., J. Org. Chem., 20, 1202 (1955). Leffler, J. E., Grunwald, E., "Rates and Equilibria of Organic Reactions", Wiley, New York, N.Y., 1963. Lumry, R., Rajender. S., Biopolymers. 9, 1125 (1970). Mandel, J., Linning, F. J., Anal. Chem., 29, 743 (1957). Mandel, J., J. Am. Stat. Assoc., 56, 878 (1961). Mandel, J., "The Statistical Analysis of Experimental Data", Interscience, New York, N.Y., 1962. McKee, D. W., Trans. Faraday SOC., 61, 2273 (1965). Ostle, B., Mensing, R. W., "Statistics in Research", 3rd ed, Iowa State University Press, Ames, Iowa, 1975. Petersen, R. C., J. Org. Chem.. 29, 3133 (1964). Ritchie, C. D.. Sager, W., Frog. Pbys. Org. Chem., 2, 323 (1964). Shimulis, V. I., Kinet. Katal.. 10, 1026 (1969). Thomas, J. M., Thomas, W. J., "Introduction to the Principles of Heterogeneous Catalysis", Academic Press, New York, N.Y., 1967. Wold, S.,Exner, O., Chem. Scr., 3, 5 (1973).
Superscripts
- ==estimate arithmetic average
.
-
to equilibrium parameters * == referring referring to kinetic activation parameters
O
Subscripts 1 = first 2 = second C = column con = concurrence hm = harmonic mean i = row (or catalyst) index j = column (or temperature) index noncon = nonconcurrence R = row RC = row-column interaction S = slopes T = total T b = evaluated at T = Thm /3 f evaluated at T = /3 AGi = for the estimate of AGi AHi = for the estimate of AHi t = referring to the measurement errors Literature Cited Bond, G. C., "Catalysis by Metals", Academic Press, New York, N.Y., 1962. Boudart, M., "Kinetics of Chemical Processes", Prentice-Hall, Englewood Cliffs, N.J., 1968. Brown, R. F., Newsom, H. C., J. Org. Chem., 27, 3010 (1962).
~
Received for review March 23, 1979 Accepted October 26, 1979
Fluid Mechanical Description of Fluidized Beds. Experimental Investigation of Convective Instabilities in Bounded Beds G. P. Agarwal, J. L. Hudson, and Roy Jackson' Rice University, Houston, Texas 7700 1
Previous theoretical work on the stability of bounded fluidized beds with uniform distributors has shown that convective instabilities can occur when the bed is wide enough and the distributor pressure drop is small. The present paper describes an experimental investigation of the stability boundary for a two-dimensional bed of glass beads fluidized by water. The observations are compared with the stability boundary predicted by the earlier linearized stability analysis.
Introduction
Over the past two decades a good deal of attention has been paid to the hydrodynamic stability of an unbounded fluidized bed (Jackson, 1963; Pigford and Baron, 1965; Murray, 1965; Molerus, 1967; Anderson and Jackson, 1968; Garg and Pritchett, 1975; Mutsers and Rietema, 1977). It has been shown that such a bed is usually unstable as a result of void fraction waves which rise through the bed and grow exponentially as they propagate. Their rate of growth differs substantially in gas and liquid fluidized beds, and it has been suggested (Jackson, 1964; El Kaissy *Address correspondence to this author at the Department of Chemical Engineering, University of Houston, Houston, Texas 17004. 0019-7874/80/1019-0059$01.00/0
and Homsy, 1976) that this difference may account for the formation of bubbles in gas fluidized beds, in contrast to the comparatively smooth fluidization of most liquid fluidized beds. It has also been shown that interparticle cohesive forces can stabilize the uniformly expanded bed when the particle size is sufficiently small (Mutsers and Rietema, 1977) and this may account for the fact that gas fluidized beds of fine particles can be expanded appreciably beyond the point of minimum fluidization before visible bubbling occurs. Thus, this rather simple stability analysis has enjoyed some success in accounting qualitatively for certain salient features of the behavior of fluidized beds. In the case of liquid fluidized beds of glass beads, where the instability waves can be observed by transmitted light and grow slowly enough for their propagation characteristics to be measured, there is also some 0 1980 American
Chemical Society
