CORRESPONDENCE Bubbling Bed Model-Model
for the Flow of Gas through a Fluidized Bed
SIR:liuiiii and Levenspiel (1968) are to be coiigratiilated on prcwitiiig a bubbling bed model which unifies much of the reccnt adv:ince in the subject hi an extremely useful forin. One rather rninor aspect of the model for slow bubbles (ub/urnl< 1) inay require further coninient. In their derivation of Equation 9 it \vas assumed that a tyl)ical cross sectioii of a fluidized bed cuts through bubbles at their niasiniuin cross sections a t which the bubble throughflow velocity given by Equation 6 is 3 umf.In taking any cross scctioii of thc bed, it is more realistic to assume that it cuts through bubbles at random positions. I n this case the mean throiigliflo~velocity- i.e., nieaii velocity of gas in a bubble, across a plane normal to the vertical axis of the bubbleshould be used in Equation 9 in place of the throughflow velocity a t the ntaxiinuni cross seetion of the bubble. In Davidson’s theory used by Kunii and Levenspiel, both these velocitieq m e equal t o 3 urnland their Equation 9 is therefore nuinet-ically correct. However, the two velocities are different
if the more comprehensive theory of Jackson is used (Leung and Sandford, 1969). The point is made here, as Kuiiii and Levenspiel’s excellent model may be modified by using the theory of Jackson in place of that of Davidson. Nomenclature ‘ u b = rise velocity of a bubble in a bubbling bed u m j = superficial gas velocity at niinimuni fluidization conditions
literature Cited
Kunii, I)., Levenspiel, O., IND.ENG.CHEWFUNDAMENTALS 7, 446 (1968).
Leung, L. S., Sandford, I. C . , Chem. Eng. Sci. 24, 1391 (1969). L. S. L e w g Department of Chemical Engineering University of Queensland Brisbane, Australia
Strategy for Estimation of Rate Constants from Isothermal Reaction Data S I R : 111a recent article (Eakman, 1969) two criteria for estiiiiat ing 1)arameters froni multiresponse data-niininiizatioii of the total sum of the squared errors and niinimization of the determinant of the irintrix of sums of squares and cross products (130s and Draper, 1965)-are used in an example for which they are not applicable. Although the author correctly states a number of assuniptions iniplicit in these two criteria, one iniportant assumption that he neglects to state, and appear3 t o be unaware of since it is violated, is that the error variances of each of the res1)orises must rernaiii coiist:mt from run to run. If the variances :ire not coilstant, t’he situation should be rectified by traiisforrnatioii of the respoiises or the use of weights. The coininon ewe of coilstant per cent error, as in the example which Eakman considers, can easily be handled by fitting the logarithnis of t’heobservations. .llso, if one of the purposes of this research is to compare the two estiiriation criteria, a very poor example was chosen (even disregarding the violation of t,he constant variance assumption). The clcterniiiiant Criterion is superior iri most cases to the total sun1 of squares criterion, because it allows the responses to have unequal (but constant) variances, and takes into account correlation antong the responses. However, in
Case I of the example, the variances of the reqponses are a1 equal, and in both Caie I and Case I1 there are no correlations present among the responses. Therefore, in Case I we would expect absolutely no advantage from using the determinant criterion, and in Case I1 me would expect some advantage but not as much as would usually be experienced. Thus, thiq example seems to offer very little bask for comparing the two criteria. 111 general, the determinant criterion is the one to be recommended, since in almost all practical situations correlations anioiig the responses are present arid often the variances of the responses are not equal. Hunter (1967) discuwes these two estimation criteria aiid others along 1%ith R detailing of the assumptions implicit in each. literature Cited
Box, G. E. P., Draper, N. R., Biometrika 52, 355-63 (1965). Eakman, J. RI.,IND. ENG.CHEM.FUNDAMENTALS 8, 53-8 (1969). Ilunter, W. G., IND. ENC.CHEM.FUNDAMENTALS 6, 461-3 (1967).
John Erjaver University of Wisconsin Madison, W i s . 65706
Strategy for Estimation of Rate Constants from Isothermal Reaction Data SIR:Erjavec is rorrect in his statement that constaiit error variance is one of the assumptions upon which the niinimuiu deterniinant and sum of squares criteria are derived. I n the article (Eakman, 1969) to which he refers, the logarithms of the observations a.oultl provide a theoretically better expression upoii wliich to base the determiiiaiit criterion used it1 step 2 of the algorithni. There is little justification, however, for ~~roposing that this traiisforiiiation also be used with the total suin of squares criterion iii step 1. As illustrated in
the article, the purpose of using the sum of squares criterion is that it can be readily clifferetitiatecl to provide ~pproxitiinte expressions for the elernents of the gradient of the siiiu of squares surface. This allows the diiect application of a inole efficient gradient iriiniinisation algorithni t o this “approximate” criterion. Hence “good” paranieter estimates are rapidly obtained. These “good” estimates are then refined by minimization of the determinant criterion. The logarithmic transforrnation suggested for the deVOL. 9 NO. 1 FEBRUARY 1970
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187
terminant criterion may be readily used in the algorithm presented. It requires the modification of one statement in the corresponding FORTRAN program. There is a computational danger here, however, in that this transformation may cause loss of accuracy due to formation of the differences of the logarithms of the concentration as opposed to the differences of the concentrations. T o test the consequences of the logarithmic transformation partially, the second step in the example reported in Table I11 (Case 11) of the paper was again run with the following results after 49 integrations: Parameter
Correct value
Original estimate
Transformed estimate
kl kz
0.3000 0.1000
0.3040 0.1011
0.3065 0.1044
The second step of the example reported in Table I1 (Case I) was run with the following results after 67 integrations: Parameter
Correct value
Original estimate
Transformed estimate
k1 kZ
0 3000 0.1000
0.2987 0.0995
0,3000 0.0997
A somewhat poorer estimate was obtained in Case 11, where the more highly scattered data were considered; an improved estimate was obtained with the more exact data. This seems to
indicate no clear advantage of one approach over the other in the problem considered. To emphasize, the purpose of this paper was not a comparison of the generalized least squares and multiresponse determinant criteria, but rather the reporting of a computational procedure which uses the two criteria in turn more efficiently to arrive a t final parameter estimates. The only place a comparison of the two methods might be implied is a t the point where the estimation algorithm switches from the least squares to the determinant criterion. I n line with its stated purpose, the paper explicitly sets forth an approximate expression for the gradient of the sum of squares surface, which is shown to be effective in the miiiimization niethod used. As pointed out in the article on page 54, this approxirnation can also be used to improve other gradient procedurese.g., Ball and Groenweghe (1967). literature Cited
w.
~ ~ 1 1 , E., ~ ~ L, c. D,, ~ ~ F ~~ MENTALS 5, 181-4 (1966). Eakman, J. M., IND. ENG.CHEM.FUNDAMENTALS 8, 53-8 (1969).
James M.Eakinan University of Nebraska Lincoln, Neb. 68508
Momentum, Heat, and Mass Transfer Analogy for Turbulent Flow in Circular Pipes SIR: In a recent article, Hughmark (1969) attempted to adapt models by Marchello and Toor (1963) and Harriott (1962), both of which stem from the elementary renewal penetration model (Danckwerts, 1951, 1955), to mass and heat transfer for turbulent pipe flow. Hughmark’s -models 1 and 2 (Hughmark, 1969) are based on the classical analogical approach where expressions for the momentum and mass diffusivities were taken from Xarchello and Toor (1963)-i.e.,
and
where 1 / represents ~ the mean frequency of renewal or mixing and 21 represents the depth of a mixing layer. Significantly, Equations 1 and 2 were developed on the basis that mixing is exclusively of an internal nature-that is, transfer through a mixing layer was assumed to occur by instantaneous lateral mixing of the layer a t random intervals and molecular diffusion during the time between mixing. Consequently, at the instant of mixing, the driving (initial) potential, was assumed to be of the form
low momentum and high momentum fluid have been observed to occur as close to the wall as y + = 2.0. Accordingly, expressions for the mean Sherwood number developed by hIarchello and Toor and Hughmark (models 1 and 2) on the basis of Equations 1 and 2 have been found to underpredict the experimental data as illustrated by Figure 1. The failure to consider the external mixing feature of turbulence becomes very important for high Prandtl number fluids. I n an attempt to improve the correlation of the data for high Prandtl number fluids, Hughniark introduced the fairly well known surface rejuvenation model by Harriott (1962). It is important to understand the significance of the approach distance distribution used in this model. The results of Harriott’s model are shown in Figure 2 for a uniform approach distance and a random distribution. The form of the approach distance distribution selected is seen to be significant. Hughmark based his model 2 h on Harriott’s results for an assumed uniform approach distance distribution. The resulting expression may be approximated as
0
e,,
(3)
188
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VOL. 9 NO. 1 FEBRUARY 1970
1965)
MODEL
.?A
10
c
2 10
Contrary to the important premise of internal mixing upon which Equations 1 and 2 are based, recent experimental data by Popovich and Hummel (1967) and Kline et al. (Runstadles et al., 1963; Schraub and Kline, 1965) strongly suggest the presence of external mixing. Indeed, exchanges of
H A R R l O T B HAMILTON
L,
I
MODEL I
I
IO
I
3
2 0
I
10
4 10
I
SC
Figure 1.
Heat and mass transfer data a t Re = 10,000
~ ~ ~
